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gas in giant molecular clouds ( gmcs ) is distributed non - uniformly and appears to aggregate itself into isolated dense clumps or more contiguous and elongated , filament - like structures . detailed observations of potential star - forming clouds have demonstrated the ubiquitous nature of filamentary clouds . for example , giant molecular filaments on scales of a few parsecs have been reported in the inter - arm regions of the milky - way ( ragan _ et al . _ 2014 ; higuchi _ et al . _ 2014 and other references therein ) . on the other hand , relatively small ( by about an order of magnitude compared to the former ) , dense filaments have also been reported within star - forming clouds in the local neighbourhood ( e.g. schneider & elmegreen 1979 ; nutter _ et al . _ 2008 ; myers 2009 ; andr ' e _ et . _ 2010 ; jackson _ et . _ 2010 ; arzoumanian _ et . _ 2011 ; kainulainen _ et al . _ 2011 , 2013 and kirk _ et . _ 2013 are only a few authors among an exhaustive number of them ) . star - forming sites within gmcs are often found located within dense filamentary clouds or at the junctions of such clouds and further , these filamentary clouds usually show multiplicity , in other words , striations roughly orthogonal to the main filament , and form hubs ( e.g. palmeirim _ et al . _ 2013 ; hacar _ et al . _ 2013 ; schneider _ et al . _ 2012 & 2010 ; myers 2009 ) . in fact , inferences drawn from detailed observations of star - forming clouds have led some authors to suggest that turbulence - driven filaments could possibly represent the first phase in the episode of stellar - birth , followed by gravitational fragmentation of the densest filaments to form prestellar cores . filamentary clouds therefore form a crucial part of the star - formation cycle . consequently , a significant observational , theoretical and/or numerical effort has been directed towards understanding these somewhat peculiar clouds . in the last few years we have significantly improved our understanding about these clouds as a number of them have been studied in different wavebands of the infrared regime of the electromagnetic spectrum using sub - millimeter arrays on the jcmt and the herschel ( e.g. nutter & ward - thompson 2007 ; andr ' e _ et al . _ 2010 ; menschikov _ et al . _ 2010 ; also see review by andr ' e _ et al . _ 2014 ) . the stability and possible evolution of filamentary clouds has also been studied analytically in the past and in more recent times . however , these models were usually developed under simplifying assumptions . for example , ostriker ( 1964 ) , developed one of the earliest models by approximating a filamentary cloud as an infinite self - gravitating cylinder described by a polytropic equation of state and derived its density distribution . in a later contribution , bastien ( 1983 ) , bastien _ et al . _ ( 1991 ) and inutsuka & miyama ( 1992 ) studied the stability criteria of filamentary clouds under the assumption of isothermality and suggested that such clouds were more likely to form via the radial collapse of an initial cylindrical distribution of molecular gas . these models also demonstrated formation of prestellar cores along the dense axial filament via jeans - fragmentation . formation of dense filaments via interaction between turbulent fluid flows has been demonstrated numerically by a number of authors ( e.g. klessen _ et al . _ 2000 ; bate _ et al . _ 2003 ; price & bate 2008 , 2009 ; federrath _ et al . _ 2010a ; padoan & nordlund 2011 and federrath & klessen 2012 ) . similarly in other recent contributions ( e.g. heitsch _ et al . _ ( 2009 ) ; peters _ et al . _ 2012 and heitsch 2013 ) , respective authors specifically investigated the process that is likely to assemble a dense filament . the conclusion of these latter authors supports the idea of filament formation via radial collapse of gas followed by an accretional phase during which the filament acquires mass even as it continues to self - gravitate . in fact , peters _ ( 2012 ) demonstrated the formation of filamentary clouds on the cosmic scale and argued that a filament was more likely to collapse radially and form stars along its length when confined by pressure of relatively small magnitude . in another recent contribution , smith _ et al . _ ( 2014 ) , have demonstrated the formation of dense filaments in turbulent gas , however , they have not addressed the other crucial issue about the temperature profile of these filaments ; observations have revealed that gas in the interiors of dense filaments is cold at a temperature on the order of 10 k ( e.g. arzoumanian _ et al . ( 2013 ) also address the issue of tidal - forces acting on the sides of a filamentary cloud having finite length and suggest that gas accreted at the ends can possibly give rise to fan - like features often found at the ends of infrared dark clouds ( irdcs ) . in a semi - analytic calculation , jog(2013 ) demonstrated that tidal - forces also tend to raise the canonical jeans mass , though compressional force - fields are more likely to raise the local density and therefore lower the jeans mass . ( 2014 ) derived a dispersion relation for a rotating non - magnetised filamentary cloud idealised as a polytropic cylinder with localised density perturbations . under these simplifying assumptions , the authors demonstrated that the filament indeed developed jeans - type instability with propensity to fragment on the scale of the local jeans length . these conclusions are in fact , consistent with those drawn in an earlier work by pon _ ( 2011 ) , or even those by bastien _ ( 1991 ) who arrived at similar conclusions from their analytical treatment of the problem . on the other hand , inutsuka & miyama ( 1997 ) , showed that a cylindrical distribution of gas is unlikely to become self - gravitating as long as its mass per unit length was less than a certain critical value . similar conclusions were also drawn by fischera & martin ( 2012 ) when they performed a stability analysis of clouds idealised as isothermal cylinders . in the present work we study the dynamical stability of an initially non - self gravitating cylindrical distribution of molecular gas maintained initially in pressure equilibrium . in particular , we are interested to see how this distribution of gas behaves as it is allowed to cool in the presence of self - gravity . we would like to examine if a radial collapse does indeed ensue and the nature of the new state of equilibrium attained by the gas . thus , we would like to investigate the physical conditions under which radial collapse is likely to be the favoured mode of evolution . although the objective we have set ourselves is similar to that of peters _ et al . _ ( 2012 ) , our work differs on three counts : ( i ) unlike peters _ et al . _ who assumed a polytropic equation of state to calculate gas temperature , we calculate gas temperature by solving the energy equation coupled with a cooling function , ( ii ) we do not use a filament that already is dynamically evolving instead , we begin by setting up a cylindrical cloud in pressure equilibrium and then examine if it does indeed collapse to form a thin dense filamentary cloud . this relatively simple set - up represents the intermediate stage of filament - formation , in other words , the stage after supersonically moving turbulent flows in a molecular cloud ( mc ) collide to form a filamentary distribution of warm gas ( see e.g. klessen _ et al . _ 2000 ; bate _ et al . _ 2003 ; price & bate 2008,2009 ; federrath _ et al . _ 2010a ; federrath & klessen 2012 ; smith _ et al . _ 2014 ) , and more importantly , ( iii ) we investigate if gravitational supercriticality is a necessary pre - condition for filament - formation and follow the evolution of the dense filament further to study the formation of prestellar cores in it . we observe that when the magnitude of the confining pressure is relatively small such that the initial volume of gas is at least critically stable , it does indeed collapse radially to assemble a thin dense filament which then fragments via a jeans - like instability on the scale of the fastest growing unstable mode to form prestellar cores . we also investigate physical properties such as the distribution of gas density and temperature within the post - collapse filament . we demonstrate that the density profile of the post - collapse isothermal filamentary cloud is indeed similar to the classic profile suggested by ostriker ( 1964 ) ; the profile is plummer - like for a filament that is allowed to cool dynamically . the article is divided as follows : in section 2 we discuss the stability of a pressure - confined isothermal cylinder . then in section 3 we present the initial conditions used for this work and briefly describe the numerical algorithm used for the simulations . the results are presented and discussed in respectively , sections 4 and 5 before concluding in section 6 . we consider a cylindrical distribution of isothermal molecular gas composed of the usual cosmic mixture ( approximately 90% hydrogen and 10% helium ) . the cylinder has initial radius , @xmath0 , length , @xmath1 , and maintained at temperature , @xmath2 . in a purely non - magnetic case gas within this cylinder is supported by thermal pressure , @xmath3 , against self - gravity and is confined externally by finite pressure exerted by the inter - cloud medium ( icm ) , @xmath4 . we assume an initial state of equilibrium so that the external pressure , @xmath4 , balances the internal pressure , @xmath3 . the internal and external pressure are both thermal by nature since particles representing the two media are not imparted any initial momentum . the stability of a spherical gas body against self - gravity is defined in terms of the thermal jeans mass . now , for a cylindrical distribution of gas its mass line - density , @xmath5 , defined as @xmath6 this quantity is analogous to the mass of a spherical cloud and as will be seen below , is useful in describing the stability of a cylindrical cloud against self - gravity . here , @xmath7 , is the initial density of the cylindrical distribution of gas , assumed to be uniform . a gas cylinder is stable against gravitational collapse as long as its mass line - density is smaller than the maximum value @xmath8 where @xmath9 is the isothermal sound - speed for the cylinder . we define the stability factor , @xmath10 , as the ratio of the mass line - density for a cylinder against its maximum value . thus , @xmath11 when @xmath12 exceeds unity , @xmath13 , a radial collapse ensues and the cylinder may be described as _ super - critical _ ; cylinders with @xmath12 less than unity are described as _ sub - critical _ ( fischera & martin 2012 ) . we describe cylinders with @xmath14 1 as _ critically stable_. as in our earlier work investigating the stability of starless cores ( anathpindika & di francesco 2013 ) , in the following sections we will calculate the stability factor , @xmath10 , at different radii within the model cylindrical cloud over the course of its evolution to describe its stability against self - gravity . we now calculate the radius of the initial cylindrical distribution of gas . keeping in mind that the pressure exerted by the confining icm balances the thermal pressure within the cylinder , we can write @xmath15 so that its radius , @xmath16 finally , following a simple dimensional analysis we define the free - fall time , @xmath17 @xmath18 @xmath19 , which is slightly faster than the free - fall time in jeans analysis ; @xmath20 , is the density of the post - collapse filament having radius , @xmath21 , and @xmath9 , the sound - speed of gas within it . a free - fall ensues if @xmath22 , where @xmath23 is the sound - crossing time , @xmath24 . using the above inequality we obtain the condition for jeans instability as , @xmath25 the corresponding mass of the fragment is @xmath26 , which , using eqn . ( 3 ) above , can be re - cast as @xmath27 @xmath28 , being the radius of the filament over which gas density remains approximately uniform before turning over in its wings . [ cols= " > , < , < , > , < , < , < , < , < , < , < , < , > " , ] in this demonstrative work we began by placing a non - self gravitating uniform - density cylindrical distribution of molecular gas in pressure equilibrium with its surrounding medium . to this initial distribution of gas we assigned a feducial temperature ( see table 1 ) . in this work the gas pressure and pressure exerted by the external medium are purely of thermal nature . while ambient physical conditions such as the turbulent mach number in a mc could possibly influence filament properties , we leave this aspect of the problem for future investigation . other free parameters in the set - up were , the length of the cylinder , @xmath1= 5 pc , contrast , @xmath29=10 , between the initial gas density within the cylinder and its surrounding medium , the average initial gas density within the cylinder , @xmath30 = 600 @xmath31 and the initial mass line - density , @xmath32 . this choice of gas density is also consistent with densities found in a typical star - forming cloud ( e.g. andr ' e _ et al . the temperature of the confining medium is calculated by using the condition of pressure - balance at the surface of the cylinder . the initial radius of the cylinder was calculated using eqn . ( 4 ) above . though simple , this set of initial conditions is suitable for the extant purposes of studying the origin of the density profile and the temperature profile of a typical filament and the formation of prestellar cores along the length of this filament . parameters used in the realisations performed in this work have been listed in table 1 . sinusoidal density perturbations along the length of the initial distribution of gas were imposed in five simulations ; see table 1 . these perturbations were imposed by revising particle positions , @xmath33 , to @xmath34 , such that the following expression was satisfied , @xmath35 ( hubber _ et al . _ 2006 ) , where the wave - number , @xmath36 , the integer , @xmath37 = 1 , 2 for respectively , the fundamental mode and the second harmonic ; @xmath38 = 0.1 , is the amplitude of perturbations . the number of gas particles in a simulation are calculated according to - @xmath39 where , @xmath40 , is the volume of the initial distribution of gas , while the initial average smoothing length , @xmath41 , was set equal to @xmath42 ; @xmath43 being the resolution parameter listed in column 11 of table 1 and defined by eqn . ( 10 ) below ; the jeans length , @xmath44 pc , at the density threshold(@xmath45 g @xmath31 ) , adopted for representing prestellar cores in this work at a temperature of 10 k. particles representing the initial distribution of gas were assembled by extracting a cylinder of unit size and length from a pre - settled glass like distribution . this unit - cylinder was then stretched to the desired dimensions and then placed in a confining medium of external - pressure , modelled using the icm particles . the envelope of icm particles had a thickness , 30@xmath41 , along each spatial dimension of the cylinder . the icm particles are special sph particles that exert only hydrodynamic force on the normal gas particles ; gas particles on the other hand , exert both , gravitational and hydrodynamic forces . the cylinder and its confining medium were then placed in a box lined with boundary particles that prevent sph particles from escaping . unlike the gas and the icm particles , the boundary particles are dead and do not contribute to sph forces . the layer of dead boundary particles was there to simply hold the gas+icm particles in the box and prevent them from diffusing away . particles within the set - up were initially stationary as no external velocity field was introduced . simulations were performed using our well tested sph algorithm , seren , in its conservative energy mode ( hubber _ et al . the sph calculations in this work also include contributions due to the artificial conductivity prescribed by price ( 2008 ) , to avoid any spurious numerical artefacts possible due to the creation of a gap between multi - phase fluids . an sph particle , in this work , had a fixed number of neighbours , @xmath46 = 50 . gas cooling was implemented by employing the cooling function given by the equation below , @xmath47 @xmath48 ( koyama & inutsuka 2002 ) . constant background heating due to cosmic rays is provided by the heating function , @xmath49 erg s@xmath50 . the temperature of an sph particle is calculated by revising its internal energy . the equilibrium temperature , @xmath51 , attained by * a * small sph particle corresponds to equilibrium energy , @xmath52 , and the timescale , @xmath53 , over which energy is either radiated or acquired by a particle is , @xmath54 then , after a time - step @xmath55 , the energy @xmath56 of a particle is revised according to , @xmath57 ( v ' azquez - semadeni _ et al . _ 2007 ) . however , a more accurate calculation of the gas temperature demands solving the radiative transfer equation(e.g . _ 2007 ; price & bate 2009 ; bate 2012 ) , and accounting for the molecular chemistry in the cloud ( glover _ et al . we leave this for a future work . + _ the sph sink particle _ + we represent a prestellar core by an sph sink particle in this work , for it is not our intention to follow the internal dynamics of a core . an sph particle is replaced with a sink only when the particle satisfies the default criteria for sink - formation which include , negative divergence of velocity and acceleration of the seed particle ( see bate , bonnell & price 1995 ; bate & burkert 1997 and federrath _ et al . _ ( 2010b ) ) . we set the density threshold for sink - formation at @xmath58 g @xmath31 ( @xmath59 @xmath31 ) , which despite being on the lower side , is consistent with the average density of a typical prestellar core . the interested reader is referred to hubber _ ( 2011 ) for other technical details related to implementation of this prescription in our algorithm seren . the radius of a sink particle was set at , @xmath60 , where @xmath61 is the smoothing length of the sph particle that seeds a sink . sink particles in the simulation therefore have different radii . a sink interacts with normal gas particles only via gravitational interaction and accretes gas particles that come within the pre - defined sink - accretion radius , @xmath62 . + + the average initial smoothing length , @xmath41 , for the ensemble of calculations discussed in this article is calculated as - @xmath63 where the number of neighbours , @xmath64 = 50 . the diameter of an sph particle , @xmath65 , so that the resolution criterion becomes , @xmath66 @xmath67 , and @xmath68 , the typical length - scale of fragmentation has been defined by eqn . ( 5 ) above . evidently , higher the value of @xmath69 , better the resolution . value of the resolution factor , @xmath69 , for each simulation has been listed in column 10 of table 1 . the resolution criterion defined by eqn . ( 10 ) above is the sph equivalent of the truelove criterion ( truelove _ et al . _ 1997 ) , defined originally for the grid - based adaptive mesh refinement algorithm . it has been shown that the gravitational instability is reasonably well - resolved for @xmath43 = 1 , and need be @xmath182 , for a good resolution of this instability . note that some authors prefer to define the resolution factor , @xmath69 , as the inverse of the fraction on the right - hand side of eqn . ( 10 ) , in which case the criterion for good resolution becomes , @xmath70 ( e.g. hubber _ et al . all our simulations discussed in this work satisfy this resolution criterion . however , this choice of resolution is reportedly insufficient to achieve convergence in calculations of crucial physical parameters such as the energy and momentum , but sufficient to prevent artificial fragmentation of collapsing gas ( federrath _ et al . _ 2014 ) . in order to alleviate this latter problem these authors suggest a more stringent resolution criterion , @xmath71 15 . however , our extant purpose here is limited to studying the stability of a filamentary cloud and its fragmentation so that the adopted choice of numerical resolution(@xmath72 2 ) , should be sufficient . in fact , in 4.1 below we demonstrate numerical convergence over the observed fragmentation of the post - collapse filament . finally , the sink radius , @xmath62 , defined in 3.2 above is @xmath73 pc in simulations developed with the least number of particles and @xmath74 pc in cases 6 and 7 , those developed with the highest resolution in this work . although prestellar cores of smaller size are known to exist in star - forming clouds ( see for e.g. jijina _ et al . _ 1999 ) , the resolution adopted here is sufficient for a demonstrative calculation of the kind presented in this work . the typical width of the post - collapse filament , @xmath180.1 pc , is also consistently resolved by a minimum of 80 particles along its length . since this volume of gas is allowed to cool , what follows , is the fragmentation of cooling gas . the length - scale of this fragmentation is comparable to the thermal jeans length ( see for e.g. v ' azquez - semadeni _ et al . _ 2007 ) , which is much larger than the initial width of the cylinder so that there is no fragmentation in the radial direction , as will be demonstrated below . furthermore , we have set our resolution to the jeans length , @xmath68 , evaluated at 10 k , which is lower than the average gas temperature within interiors of the filament as will be evident in the following section . our simulations therefore have the desired spatial resolution to represent core - formation in the post - collapse filament . our interest lies particularly in examining the possibility of forming a dense filamentary cloud out of a cylindrical volume of gas confined by thermal pressure . this possibility is explored for different choices of initial conditions ; first , when the volume of gas is initially critically stable , then marginally super - critical and finally , sub - critical . interestingly though , the initial cylindrical distribution of gas evolves radically differently if it is at / above critical stability as compared to when it is sub - critical . we also vary the spatial resolution and compare results from simulations developed by using a cooling function with that from an isothermal calculation . the initial cylindrical distribution of gas , in all these test cases , has comparable mass and an identical magnitude of confining pressure , @xmath4 . one of these , test realisation 1 , was developed under the assumption of isothermality while the cooling function defined by eqn . ( 7 ) above , was used in the remaining test cases . the mass line - density , @xmath77 , defined by eqn . ( 2 ) is slightly less than that for the initial distribution of gas , @xmath32 , for all the test realisations , except for the first where the two are of comparable magnitudes . the initial distribution of gas is therefore marginally super - critical in all realisations ; gas in the first realisation is approximately in equilibrium in the sense , @xmath78 ( see table 1 ) , @xmath0 being the radius of the cylinder . in all these realisations the radius of the initial cylindrical distribution of gas begins to shrink due to the onset of an inwardly propagating compressional wave . this radial contraction of the initial cylindrical distribution in outside - in fashion is visible in the rendered density images of the cross - section through the mid - plane of the collapsing cylinder shown in fig . 1 . overlaid on these plots are the velocity vectors that denote the direction of local gas - motion . this contraction soon assembles a thin dense filament along the axis of the cylinder and prestellar cores , represented by sph sink particles , begin to form along the length of the filament . at this point a word of caution would be appropriate . by a compressional wave we do not imply that the external pressure compresses the initial cylindrical distribution of gas in the radial direction . that hydrostatic cylinders can not be compressed in this manner has been demonstrated by fiege & pudritz ( 2000 ) . in this work density enhancement along the cylinder - axis is the result of gas being squeezed during the radial collapse of the initial configuration . we reserve the attribute of a compressional wave to describe this process . shown on the upper - panel of fig . 2 is a rendered density plot of the post - collapse filament in realisation 1 . the black blobs on this image represent the location of prestellar cores that begin to form along the length of this filament . evidently , the cores that had formed in this filament at the time of terminating the calculation are separated by approximately a jeans length(@xmath18 0.2 pc at @xmath2 = 25 k ; avg . density @xmath45 g @xmath31 ) , defined by eqn . the image on the lower - panel of fig . 2 is a plot of the radial distribution of gas density within the collapsing cylindrical cloud in this realisation . this plot and all other similar plots in this paper were generated by taking a transverse section through the mid - plane of the filament . the inwardly propagating compressional disturbance is evident from the density plots at early epochs of the collapse . as the gas is steadily accumulated in the central plane of the cylinder , the density of gas there rises and eventually develops a profile that matches very well with the one suggested by ostriker ( 1964 ) , @xmath79^{p/2}};\ ] ] with @xmath80 = 4 . here @xmath81 ; @xmath28 being the radius at which the density profile develops a knee . the density of this filament falls - off relatively steeply in the outer regions which is consistent with that reported by malinen _ ( 2012 ) for typical filamentary clouds in the taurus mc . furthermore , this figure also shows , @xmath82 0.1 pc , which is comparable to the radius of typical filaments found in eight nearby star - forming clouds in the gould - belt ( e.g. arzoumanian _ et al_. 2011 ; andre _ et al . _ 2014 ) . another interesting conclusion that can be drawn from this realisation is that about the stability of a filamentary cloud . it can be seen from fig . 2 that the centrally assembled , post - collapse dense filament at the epoch when calculations for this realisation were terminated , has a radius on the order of 0.1 pc , the point at which the density - profile develops a knee . an average density of @xmath45 g @xmath31 , the threshold for core - formation in these calculations , would suggest a mass line - density , @xmath83 50 m@xmath84 pc@xmath50 , for this post - collapse filament which is significantly higher than , @xmath85 41 m@xmath84 pc@xmath50 . the filament would therefore be expected to be in free - fall and should collapse rapidly to form a singular line . however , such is not to be the case and fig . 3 shows that the collapsing cylindrical cloud , or indeed the post - collapse filament , in fact , remains sub - critical at all radii , @xmath86 . we will discuss later in 5.1 the implications of this finding for the classification of observed filamentary clouds in typical star - forming regions . in fig . 3 we have shown the radial variation of the stability factor , @xmath10 , defined by eqn . ( 3 ) , and integrated over the radius of the collapsing cylindrical cloud . for the purpose , using the sturges criterion for optimal bin - size ( sturges 1926 ) , we divided the cylindrical distribution of gas into a number of concentric cylinders having incremental radii in the radially outward direction . number of these concentric cylinders is given by , @xmath87 ; @xmath88 being the number of gas particles in a realisation . the innermost cylinder would therefore have the smallest radius and radii of successive cylinders in the outward direction would be incremented by a small increment , @xmath89 . thus , if @xmath90 is the radius of the @xmath91 cylinder , then the radius of the @xmath92 cylinder is simply , @xmath93 ; @xmath94 . the initial distribution of gas in this case was characterised by @xmath78 . evident from the plot shown in fig . 3 is the fact that the gravitational state of the collapsing cylinder does not vary during the process . within the cylinder gas always remains gravitationally sub - critical so that , @xmath95 , although it does rise steadily close to the centre where the density is the highest in the post - collapse filament . in other words , shells of gas within the collapsing cylinder though self - gravitating , are not essentially in free - fall . that this must be the case is also evident from the density plots shown in fig . 1 . from these plots it can be seen that gas inside the collapsing cylinder has a relatively small velocity in comparison with that in the outer regions that is pushed in by the compressional wave . also , the observed outside - in type of collapse lends greater credence to the conclusion that gas within the collapsing cylinder must be gravitationally sub - critical , for had it been super - critical , the collapse would have been of the inside - out type , i.e. , the innermost regions of the cylinder would have collapsed rapidly followed by the outer regions . we had presented a similar argument in an earlier work ( anathpindika & di francesco 2013 ) , where hydrodynamic models developed to explain why some prestellar cores failed to form stars despite having masses well in excess of their respective jeans mass , were discussed . in that work it was demonstrated that cores could indeed contract in the radial direction without becoming singular ( i.e. , a free - fall collapse was not seen in these cores ) , if the mass of infalling gas at radii within the core was smaller than the local jeans mass , or equivalently , we had suggested that the jeans condition had to be satisfied at all radii within a collapsing core for it to become singular ( protostellar ) . in the present work we have used the mass line - density instead of the thermal jeans mass to discuss the stability of a cylindrical volume of gas . we observe , the collapsing cylinder appears to evolve through quasi - equilibrium configurations where thermal pressure provides support against self - gravity and a radial free - fall does not ensue . interestingly , we observe that prestellar cores can indeed form in a filamentary cloud that is not in radial free - fall . these cores acquire mass by accreting gas and shown in fig . 4 is the accretion history of these cores . , calculated for different radii at various epochs of the collapsing filament in realisation 2.,scaledwidth=50.0% ] realisation 2 is a repetition of the previous simulation but the gas temperature was now calculated by solving the energy equation and further , the energy of a gas particle was corrected according to eqn . ( 9 ) to account for gas cooling / heating . as with the isothermal cylindrical cloud in the previous realisation , the marginally super - critical initial distribution of gas in this case also begins to collapse in the radial direction . and , as in the previous case the inwardly propagating compressional wave assembles a dense filament of gas along the axis of the initial cylindrical cloud . shown in fig . 5 is the radial distribution of gas density within the collapsing cylindrical cloud which , though remarkably similar to that shown in fig . 2 for the isothermal filament , is relatively shallow ( plummer - like ; @xmath80 = 2 in eqn . the radial variation of the stability factor , @xmath10 , within the collapsing cylinder for this realisation has been shown in fig . this plot is qualitatively similar to the one produced for simulation 1 and shown earlier in fig . 3 . as with this former plot , gas within the collapsing cylinder in this case also always remains less than unity , @xmath96 , suggesting that gas is not in free - fall . also , the filament in this case is assembled on a timescale comparable to that in the isothermal realisation . this is probably because , despite the gas - cooling , the collapsing gas is still gravitationally sub - critical . a direct comparison of the mass line - density , @xmath97 , against the threshold for stability , @xmath77 , however , suggests otherwise . this is because , implicit in a comparison of this kind is the assumption of isothermality and uniform distribution of gas density within the post - collapse filament . on the contrary , and as is evident from fig . 7 , the gas temperature within the post - collapse filament is hardly uniform . the cold central region of the filament is cocooned by the relatively warm jacket of gas . the innermost regions of this filament acquire an average temperature of 10 k which is consistent with that reported from observations of filamentary clouds in typical star - forming regions ( see for e.g. palmeirim _ et al . _ 2013 ; arzoumanian _ et al . _ 2011 ; andr ' e _ et al . _ 2010 ) . finally , shown in fig . 8 is the history of core - formation in this filament . it is qualitatively similar to that for the isothermal filament plotted in fig . to this end , following the formation of the first core , the next set of cores form relatively quickly . also , the filament in the second realisation , where gas was allowed to cool , begins to form cores earlier than the isothermal filament and in general , cores in the latter are somewhat less massive than those in the former . this is the result of a higher mass line - density of the cooling filament relative to the isothermal filament in the earlier realisation . = 1 . ) in realisation 6 ( _ only a small volume of the medium confining the filament has been shown on this plot _ ) . fine black blobs on top of the image represent the positions of cores in the post - collapse filament at the time of termination of calculations . perturbations , in this realisation , were imposed on the length - scale @xmath68 , defined by eqn . not all density perturbations have condensed at this epoch , but those that are relatively closely spaced are separated on a scale on the order of @xmath44 pc._lower - panel : _ same as the picture shown in the upper- panel , but now for the realisation 4 that was developed with a somewhat lower resolution , though sufficient to avoid artificial fragmentation along the length of the post - collapse filament.,title="fig:",scaledwidth=50.0% ] = 1 . ) in realisation 6 ( _ only a small volume of the medium confining the filament has been shown on this plot _ ) . fine black blobs on top of the image represent the positions of cores in the post - collapse filament at the time of termination of calculations . perturbations , in this realisation , were imposed on the length - scale @xmath68 , defined by eqn . not all density perturbations have condensed at this epoch , but those that are relatively closely spaced are separated on a scale on the order of @xmath44 pc._lower - panel : _ same as the picture shown in the upper- panel , but now for the realisation 4 that was developed with a somewhat lower resolution , though sufficient to avoid artificial fragmentation along the length of the post - collapse filament.,title="fig:",scaledwidth=50.0% ] ( the second harmonic ) , in realisation 7 . again , not all density perturbations have condensed at this epoch , but those that are relatively closely spaced are separated on a scale on the order of @xmath98 pc . , scaledwidth=50.0% ] in realisation 1 , for instance , where no external perturbations were imposed , we observed that the separation between cores in the post - collapse filament was on the order of the corresponding jeans length . next , we use the same set of initial conditions as those for realisation 2 in this next set of four realisations namely , 4 , 5 , 6 and 7 , but now by imposing perturbations on the initial distribution of gas as described in 3.1 . perturbations for realisation 4 were imposed on the scale of the fragmentation length defined by eqn . ( 5 ) , which at 10 k and average density , @xmath45 g @xmath31 , the threshold for core - formation , is @xmath99 0.08 pc . we define this as the fundamental mode of perturbation . in the next realisation , numbered 5 in table 1 , we imposed the second harmonic of this perturbation . we repeat this set of two calculations with a higher number of particles , labelled 6 and 7 respectively , in table 1 . as with the gas in realisation 2 that was subject to cooling , in this case also , we observe that the initial distribution of gas collapsed radially to form a thin filament aligned with the central axis of the cylinder . shown in figs . 9 and 10 are the rendered images of the post - collapse filaments that form in realisations 6 and 7 , those that have the best resolution in this set of calculations . for comparison purposes we have also shown a plot of the post - collapse filament from realisation 4 on the lower - panel of fig . the physical parameters for this realisation are the same as those for realisation 6 , but was developed with a slightly lower , albeit sufficient resolution to prevent artificial fragmentation ( truelove criterion satisfied ) . a comparison of the plots on the two panels of fig . 9 suggests little qualitative difference between the outcomes from the respective realisations . we must , however , point out that while the satisfaction of the resolution criterion defined by eqn . ( 10 ) above is sufficient to ensure there is no artificial fragmentation , as is indeed seen in this work , it is unlikely to be sufficient to obtain convergence in calculations of energy and momentum of the collapsing gas ( federrath _ et al . _ 2014 ) . in fact , below we compare the sink - formation timescales and the magnitude of gas - velocity in the radial direction within the post - collapse filament . evidently , cores in the post - collapse filament indeed , appear like beads on a string . however , we note that not all density perturbations had collapsed to form cores when calculations were terminated . a few more would be expected to form in this filament , but the extant purpose of demonstrating a jeans - type fragmentation of this post - collapse filament is served with the set of cores that can be seen on either rendered image . finally , shown in figs . 11 and 12 are the accretion histories for cores that form in realisations 6 and 7 , respectively . there is not much difference between the timescale of fragmentation , as reflected by the epoch at which the first core appears in either realisation . although , the formation of cores after the first one , in realisation 7 is somewhat delayed and form over a timescale of 10@xmath100 yrs , as against those in case 6 which form relatively quickly after the first core appears . however , it is likely that a more significant difference between timescales would be visible for a higher harmonic . also evident from the comparative plots shown in fig . 11 is the absence of convergence in the timescale of sink - formation and the mass of sink - particles in the two realisations likely due to inadequate numerical resolution as was discussed in 3.3 . crucially , the number of sink particles remains the same irrespective of the choice of resolution which , it was argued previously is sufficient to prevent artificial fragmentation even for the realisation with the lowest realisation in this work . also , the accretion histories for sink - particles in either case are qualitatively similar . in this second case we raised the gas temperature such that the initial cylindrical distribution of gas was rendered sub - critical . we performed one realisation for this choice of @xmath4 and repeated it with a higher resolution , numbered 8 in table 1 . here we discuss this latter realisation . in contrast to the simulations discussed under case 1 , the radial collapse of the cylinder could not be sustained as it became to be squashed by the confining pressure into a spheroidal globule that was in approximate equilibrium with the external medium . the other notable feature being , this spheroidal globule was assembled on a relatively longer timescale , @xmath102 2.5 myrs ; in comparison to that observed in the simulations tested under case 1 . as with the realisations discussed earlier under case 1 , shown in fig.13 is the stability factor , @xmath10 , for this realisation . this is plot is similar to the ones shown earlier in figs . 3 and 6 in the sense that gas within the cylindrical cloud always remains sub - critical as @xmath95 . however , it differs from the former plots in one respect ; the magnitude of @xmath10 , in the outer regions , away from the central axis of the globule , shows a significant fall relative to that for the initial configuration . this , as we will demonstrate later , is because gas in the outer regions is significantly warmer than that observed in realisations grouped under case 1 . , calculated at different epochs for the cylindrical distribution of gas in realisation 8 . as was seen in figs . 2 and 5 for realisations grouped under case 1 , in this realisation as well the magnitude of @xmath10 remains unchanged and gas always remains sub - critical even as the initial cylindrical distribution is squashed into a spheroidal globule.,scaledwidth=50.0% ] myrs).,scaledwidth=50.0% ] = 2.5 myrs ) . , scaledwidth=50.0% ] shown in fig . 14 is the density averaged gas temperature within this globule and was made at the time of terminating the calculations in this case ( @xmath102 2.5 myrs ) . from this plot it is clear that though the interiors of this globule are indeed cold , comparable to the central regions of the dense filament in case 1 , gas away from the central axis is significantly warmer . the relatively large gas temperature renders it gravitationally sub - critical and so , it is unable to collapse radially to assemble a thin dense filament along the axis of the initial distribution . instead , it begins to be squashed from either side and proceeds to form an elongated spheroidal gas - body as can be seen in the rendered density image shown in fig . 15 . overlaid on this image are density contours that readily reveal evidence of sub - fragmentation to form smaller cores along the cold central region of the globule ; velocity vectors on this image indicate the direction of local gas - flow within the elongated globule . the magnitude of this velocity field has been plotted in fig . 15 . note that the magnitude of this velocity field is consistent with that derived for typical cores ( e.g. motte _ et al . _ 1998 ; jijina _ et al . _ 1999 ) . however , the local direction of the velocity field is unlikely to reveal much about the boundedness of a core which reinforces our suggestion made in an earlier work related to modelling starless cores . there it was shown , a core could remain starless despite exhibiting inwardly pointed velocity field which usually , is indicative of a gravitational collapse ( anathpindika & difrancesco 2013 ) . formation of sub - structure within this globule can be identified with the aid of density contours . this indicates the onset of sub - fragmentation . the fragments , at this epoch though , have not reached the density threshold for cores adopted in this work . prestellar cores usually appear embedded within dense filamentary clouds . understanding the morphology of these clouds is therefore crucial towards unravelling the details of the star - forming process . in the present work we explore the possibility of formation of prestellar cores along the length of a filamentary cloud . to this end we developed hydrodynamic simulations starting with a cylindrical distribution of gas and separately studied its evolution when it was gravitationally sub - critical and super - critical . we identify two possible modes of evolution : one , when the initial cylindrical distribution of gas is either marginally super - critical(@xmath103 1 ) , or even critically stable(@xmath14 1 ) , a thin dense filament forms as a result of radial collapse of the initial distribution of gas . the post - collapse filament continues to accrete gas during in - fall and prestellar cores form along the length of this filament via a jeans - like instability . in literature this mode has been identified as a tendency to form a spindle ( e.g. inutsuka & miyama 1998 ) ; and two , when the initial distribution of gas is gravitationally sub - critical(@xmath1041 ) , it tends to get squashed and forms a spheroidal globule that is in approximate pressure equilibrium with its confining medium . the relatively cooler regions of this globule show evidence for further fragmentation into smaller cores . similar examples of fragmentation within larger cores to form smaller ones has been identified in for instance , the serpens north ( duarte - cabral et al . 2010 ) , or within a number of infrared dark clouds ( e.g. wilcock et al . 2011 , 2012 ) . previous numerical work by a number of authors on the formation of filamentary clouds , and their stability against self - gravity , has culminated in at least three possible scenarios : ( i ) filamentary clouds are often seen in magneto-/hydrodynamic simulations of turbulent gas . these clouds are believed to be generated due to interaction between turbulent gas , followed by enhancement of self - gravity once the gas becomes sufficiently dense ( see for e.g. klessen & burkert 2000 ; klessen et . al . 2000 ; federrath & klessen 2013 ) . this model favours filament formation via interaction between turbulent flows so that there is no global collapse ; dense filamentary clouds instead , appear locally ( e.g. federrath et al . 2010 ; peters et al . 2012 ) , ( ii ) heitsch et al . ( 2010 , 2013 ) , have demonstrated that filamentary clouds could also form within a self - gravitating mc , and ( iii ) fragmentation of gas - sheets is also a possible mode of formation for elongated clouds . this has been demonstrated in both , the simple case of an isothermal sheet ( e.g. schmid - burgk 1967 ; myers 2009 ) , as well as in the one confined by shocks ( e.g. v ' azquez - semadeni et al . 2007 ; anathpindika 2009 ; heitsch 2010 ) . although there is little doubt about the fact that dense filaments are a result of fragmentation of larger clouds and that prestellar cores usually form within such filaments like beads strung on a wire , the process(es ) leading to the formation of a thin filament are somewhat unclear and is still a matter of debate . heitsch ( 2013 ) , for instance has suggested that filaments , during their formation , do exhibit density enhancement in their central regions , but this density enhancement is not associated with a corresponding reduction in the filament - width . smith et al . ( 2014 ) , partially agree with this conclusion but have suggested that dense filaments are likely to have a static density - profile , one that does not vary with time . in other words filament - formation is probably a one step process and once created , dense filaments remain as they are . in simulations grouped under case 1 of this work we have demonstrated the formation of a thin dense filamentary cloud via radial collapse of the initial cylindrical distribution of molecular gas . et al._(2012 ) argue that the magnitude of the externally confining pressure probably determines if whether a dense filament could possibly form . these authors have suggested that a relatively small magnitude of the external pressure is likely to support the formation of a dense filament . on the contrary , a filamentary distribution of gas is more likely to end up as a spheroidal globule when the magnitude of confining pressure is relatively large . however , the possible cause(s ) for such occurrence is(are ) not clear from their work . conclusions drawn from the simulations presented in this work are broadly consistent with those of peters _ et al._(2012 ) , but with an added qualification about the gravitational state of the distribution of gas that precedes a dense filament . this leads us to the next question , that about the propensity of a cylindrical cloud to collapse radially and assemble a dense filament . we observe that when the initial distribution of gas is gravitationally sub - critical , apart from a higher magnitude of confining pressure , the gas does not collapse in the radial direction , instead , it forms a spheroidal globule . we therefore believe , the gravitational state of the initial distribution of gas is likely to hold the key to its future evolution . we argue , a radial collapse leading to the formation of a dense filament is possible when the gas is at least critically stable(@xmath14 1 ) , initially . interestingly , the mass line - density for the post - collapse filament in this case exceeds its maximum value , @xmath77 , for stability under the assumption of a uniform temperature of 10 k , as is indeed seen in the interior of the filament ; though gas in the wings of the filament is warmer . under this assumption of isothermality and uniform density the filament could be described as gravitationally _ super - critical _ ( fischera & martin 2012 ) . although the two sets of simulations discussed in this work demonstrate that an initially sub - critical cylindrical volume of gas can not be induced to collapse in the radial direction , plots shown in figs . 2 and 5 suggest that gas in a radially collapsing cylindrical cloud does not become gravitationally super - critical ( @xmath105 1 ) . in other words , the mass line - density , @xmath106 , within the cylinder never exceeds its maximum value , @xmath77 , at any radius within the cylinder . simulations grouped under this case(case 1 ) also demonstrate that a filament need not be in radial free - fall in order to form prestellar cores along its length . interestingly , evidence for global in - fall has recently been reported in the filamentary cloud dr21 ( schneider et al . 2010 ) , while palmeirim et al . ( 2013 ) and andre et al . ( 2014 ) also report observations of some filamentary clouds that exhibit radial collapse . inutsuka & miyama ( 1997 ) , by assuming super - critical initial conditions and an isothermal gas had demonstrated such a collapse . however , this latter set of initial conditions would always be predisposed to radial collapse leading to the formation of a thin dense filament . despite their propensity to collapse radially , globally super - critical filaments are unlikely to be in free - fall . the difficulty with the idea of a radially free - falling filament is that that filament would collapse rapidly to form a thin line and it would be a challenge to explain the formation of cores in it . it is therefore difficult to reconcile the suggestion of free - falling filaments . we will revisit this point in the context of observations of filaments in the following subsection . that filamentary clouds are unlikely to experience a free - fall collapse can simply be shown by deriving an expression for the radial component of gravitational acceleration , @xmath107 , within a typical filamentary cloud . for the sake of argument , it would be safe to adopt a plummer - density distribution for gas within a filamentary cloud so that the radial distribution of thermal pressure within this cloud may be written as , @xmath108 from the plummer - density profile we have , @xmath109 combining eqns . ( 12 ) and ( 13 ) , we wind up with an expression for the gravitational acceleration as , @xmath110 then in the limit of @xmath111 , eqn . ( 14 ) reduces to @xmath112 which suggests that filamentary clouds are unlikely to have a true super - critical state , i.e. , filaments are unlikely to be in free - fall even if they do collapse radially . we have shown that prestellar cores are likely to form via a jeans - type fragmentation of the filamentary cloud and have demonstrated this by developing simulations with and without initial perturbations to the density field . this scenario is consistent with the argument presented by freundlich & jog ( 2014 ) . we also note that a plummer - like profile ( @xmath80 = 2 ) , fits the radial density distribution of the post - collapse filament very well when gas temperature is calculated by accounting for gas - cooling . on the other hand , the steeper ostriker - profile ( @xmath80 = 4 ) , fits the radial density profile for the isothermal filament . furthermore , the typical radius of the post - collapse filament in our simulations at the epoch when calculations were terminated , is on the order of @xmath180.1 pc . both these findings are consistent with corresponding values derived for filamentary clouds in the gould - belt ( e.g. arzoumanian _ _ 2011 ; malinen _ et al . _ 2012 ; palmeirim _ et al . _ 2013 ; andr ' e _ et al . although a few handful number of exceptions have been reported with relatively steep density profiles in outer regions and have slopes in excess of 2 , e.g. the b211/3(@xmath80 = 2.27 ) filament in the taurus mc that also has a relatively small radius of @xmath180.04 pc ( malinen _ et al . _ 2012 ; palmeirim _ et al . however , it remains to be seen if these post - collapse dense filaments would continue to self - gravitate and acquire even smaller radii ; investigation into this question is best deferred for a future work . we also note that purely hydrodynamic simulations such as the ones discussed in this work succeed in reproducing the typical plummer - like radial density profile for filamentary clouds ( also see smith _ et al . _ 2014 ) . this is contrary to some of the earlier suggestions about the propensity of formation of such filaments in magneto - hydrodynamic simulations ( e.g. fiege & pudritz 2000 ; tilley & pudritz 2003 ; hennebelle 2003 ) . it appears that the slope of the radial profiles of filamentary clouds is unlikely to be influenced by the presence / absence of the magnetic field . also , unlike much of the earlier work we have also derived the radial distribution of gas density and temperature for the post - collapse filament and showed that is indeed consistent with that derived observationally for typical filamentary clouds observed in nearby star - forming regions . an interesting point to which attention must be drawn is about the temporal evolution of the density profile of filamentary clouds . radial density profiles shown in figs . 2 and 5 demonstrate that the collapse of the initial cylindrical distribution of gas is accompanied with a rising central density . this is consistent with the finding of heitsch ( 2013 ) . however , we also observe that the width of the filament shrinks during the process which is inconsistent with the other finding by the same author and smith _ et al . _ as can be seen from these plots , the problem could be circumvented by adopting a value of the @xmath28 , the point where the density - profile turns - over , smaller than where it is actually seen to lie . ( 2014 ) have argued that rotation tends to stabilise a filamentary cloud against self - gravity and leads to a density - profile shallower than the ostriker - profile . on the contrary , we have shown that even a simple prescription of gas - cooling can indeed reproduce a plummer - like density distribution for the post - collapse filamentary cloud . on the other hand , upon raising the magnitude of external pressure , @xmath113 , such that the initial distribution of gas was rendered sub - critical ( because the condition of pressure balance at the gas - icm interface demands that gas temperature be raised ) , we observed , such a volume of gas was unable to sustain a collapse in the radial direction . instead , after initially shrinking along the radial direction , gas was squashed laterally and therefore tended to rebound . confined by external pressure , this gas then assembled a pressure - supported spheroidal globule on a time - scale larger than that for simulations discussed above . interestingly , arzoumanian et al . ( 2013 ) have recently identified filamentary clouds with a greater internal pressure in the ic5146 , aquila and the polaris molecular clouds . these filaments have a relatively large velocity dispersion and therefore , are gravitationally unbound . some of these in fact , exhibit signs of lateral expansion with little evidence for core - formation along the filament - axis as is usually envisaged in the _ beads - on - string _ analogy . on the basis of results obtained from our simulations we suggest , factors controlling the efficiency with which molecular gas cools would determine the propensity to form dense filamentary clouds which is simply a restatement of the stability argument presented in 2 . a larger thermal pressure could have an impact on the molecular chemistry of a cloud and therefore , on the efficiency with which it could possibly cool . however , in the absence of a complete treatment of the molecular chemistry we can not address this issue in the present work . irrespective of whether the initial cylindrical distribution of gas ends up as a dense filament , we observe that gas in the collapsing cylinder remains gravitationally sub - critical , characterised by the stability factor , @xmath114 , at all epochs ( see figs . 3 and 6 ) . this suggests , the gravitational state of the gas remains unaltered over the course of its evolution . of particular interest is the implication of this finding for dense filaments . we have seen that the formation of a dense filament is the result of a radial contraction of the initial cylindrical distribution of gas . this contraction is associated with a rise in the central density along the axis of the cylinder whence a filamentary cloud is assembled . although this is our observation in the simulations discussed here , we note that the radially collapsing gas is not in a free - fall and is therefore characterised by @xmath96 , at all radii within the volume of gas . this sequence supports the hypothesis that filamentary clouds are probably assembled via gravitational contraction during which mass is steadily accreted by the centrally located filamentary cloud . this scenario is further reinforced by observational findings of accreting filaments found in taurus and cygnus x ( goldsmith _ et al . _ 2008 ; nakamura & li 2008 ; schneider _ et al . _ 2010 ) . a comparison of the radial velocity field in the two cases discussed in this work would be instructive . for case 1 we take the sixth realisation , one of the two in this set with the highest resolution , as a representative calculation . the radial velocity field in the post - collapse filamentary cloud has been shown in fig . we note that the magnitude of velocity is relatively large in the central region and peters - off in the wings of the filament . in fact , the radial distribution of gas velocity can be approximated by a power - law of the type , @xmath115 , which is very close to that for a bound object ( @xmath116 ) . a similar plot for simulation 8 under case 2 was shown previously in fig . the difference between these plots is evident ; in this latter case , the gas close to the centre is moving radially outward where as that in the outer regions is transonic and moving inwards with approximately constant velocity ( sound speed @xmath180.55 km / s in this region ) , as a consequence of being squashed . the findings reported here from simulations in case 1 are consistent with those for typical filamentary clouds ( see for e.g. arzoumanian _ _ 2013 ) . filaments of the type seen in case 1 are conventionally described as gravitationally bound . this brings us to the next question - as to whether gravitationally bound filaments are also likely to experience a free - fall collapse . for the purpose of illustration we consider the examples of four filamentary clouds . ( i ) ic5146 in the cygnus region . this filament with a mass line - density , @xmath83 152 m@xmath84 pc@xmath50 , which is an order of magnitude in excess of the maximum mass line - density , @xmath77 , evaluated at 10 k , and therefore should be expected to be in a free - fall collapse in the radial direction . however , it seems , this is not really the case as the filament does not appear to be in radial free - fall ( arzoumanian _ et al . _ 2011 ; fischera & martin 2012 ) . ( ii ) the filamentary cloud dr21 is the next example . this is a massive filament that has two sub - filaments that are gravitationally super - critical and show signs of in - fall , but no apparent signs of radial free - fall ( schneider _ et al . this filament appears more akin with the outcome of simulations in case 2 of this work . ( iii ) next , we consider the example of the perseus mc . it has a mass line - density , @xmath83 50 m@xmath84 pc@xmath50 to 100 m@xmath84 pc@xmath50 , a good factor of 3 - 5 larger than the maximum mass line - density , @xmath77 , for the deduced isothermal temperature of @xmath1812 k ( hatchell _ et al . _ 2005 ) . again , a direct comparison of @xmath106 against @xmath77 for the perseus mc would suggest that the filament is gravitationally bound and must collapse radially . however , there is no corroborative evidence to suggest that the filament is in free - fall though several cores have been detected along its length . in fact , estimates of line - width of the optically thick c@xmath117o emission suggests that that filament is probably thermally supported . ( iv ) finally , we consider the example of the serpens mc , recent observations of which , have revealed evidence of radially infalling gas onto the dense filaments in this cloud . yet , there is no evidence to suggest that these filaments are in free - fall , however , prestellar cores have been detected along their length and appear to have formed on the scale of the local jeans length ( friesen _ et al . results from our simulations grouped under case 1 corroborate these observations . it therefore appears to us that the condition , @xmath118 , i.e. , @xmath119 , is probably necessary to induce radial collapse in a cylindrical volume of gas . such a volume of gas could become super - critical by accreting mass from its surroundings ( heitsch & hartmann 2014 ) . however , this aspect of the problem is beyond the scope of present investigation and therefore best left for a future work . significantly though , this collapsing gas is unlikely to be in free - fall and @xmath120 which is consistent with the suggestion of mckrea(1957 ) ; see also toci & galli ( 2014 ) . in fact , if indeed this gas is in radial free - fall , filamentary clouds would rapidly end up in a thin line and formation of cores in such a collapsing cloud would be difficult to reconcile . on the contrary , a steady in - fall , as gas within the collapsing filament is allowed to cool appears to be a more promising mode of evolution of filamentary clouds where gas is initially at least critically stable . as we have seen in the simulations grouped under case 1 , such filaments can indeed form cores along their length via a jeans - like fragmentation . this evolutionary scenario also circumvents the uneasy question as to why filaments with extremely high column densities , on the order of a few times 10@xmath121 @xmath122 , are not found . the likely solution perhaps is that filaments are not in radial free - fall . those that are gravitationally bound , shrink in the radial direction and eventually acquire a thermally supported configuration . if indeed filaments were to become pressure - supported during their evolutionary sequence , it would also explain why filaments across star - forming regions tend to have comparable widths , on the order of @xmath180.1 pc ( e.g. arzoumanian _ et al . _ 2011 ) . at 10 k , the temperature typically found in filamentary clouds and an average density @xmath1810@xmath123 g @xmath31 , a representative density for prestellar cores , the thermal jeans length , @xmath99 0.075 pc , which is comparable to the typical radius of filamentary clouds . we make a similar observation in simulations grouped under case 1 . for instance , figs . 2 and 5 , where the radial density profile for the post - collapse filament in simulations 1 and 2 have been plotted , demonstrate that the filament radius , @xmath124 , at the epoch when calculations were terminated in respective simulations . although , we note that magnetic field could also have a key role to play in the evolution of filamentary clouds . the point though , is beyond the scope of this article . in this work we have demonstrated that an initial cylindrical distribution of molecular gas can indeed collapse radially to assemble a thin dense filament along its axis when the magnitude of confining pressure is relatively small such that gas is initially at least critically stable . in this case a radial collapse ensues and the gas is able to cool on a relatively short timescale . equivalently , it may also be argued that a collapse of this kind is likely when the line mass exceeds its critical value required to maintain stability . however , there is a caveat to this argument . this argument is valid only if gas is assumed to be isothermal and has uniform density as was the case with our initial cylindrical distribution of gas . for a typical dense filament this argument can not be applied since its density and temperature distribution is far from uniform . we argue , a filamentary cloud even if gravitationally bound , is unlikely to be in radial free - fall , instead it is likely to contract in the radial direction and consequently , the density along its central axis steadily rises as gas is accreted on to it . we argue that this evolutionary sequence of a typical filamentary cloud could possibly explain - ( i ) the dearth of filamentary clouds that have extremely high extinction , and ( ii ) why the width of filamentary clouds shows little variation across star - forming regions . we suggest that filamentary clouds are likely to be thermally - supported and their radii(width ) are likely to be on the order of the local thermal jeans length ( twice the local thermal jeans length ) . a plummer - like density profile appears to fit very well the radial distribution of gas density in the post - collapse filament so that magnetic field may not be necessary to generate filaments with a relatively shallow slope , as has been suggested in the past ( e.g. fiege & pudritz 2000 ) . we also demonstrate that prestellar cores are likely to form via a jeans - like instability while it is accreting gas . it also appears , cores are likely to form along the length of a dense filament if it is at least ( gravitationally)critically stable . processes via which filaments acquire mass from their parent cloud are therefore likely to hold the key to determining the efficiency of star - formation ( see also tafalla & hacar 2014 ) on the other hand , when the magnitude of the confining pressure was increased such that the initial distribution of gas was rendered gravitationally sub - critical , we observed that a radial collapse was unsustainable . instead , the gas distribution was squashed from either side and demonstrated a propensity to expand laterally . the result was an elongated spheroidal globule which then showed signs of sub - fragmentation . such objects are more likely to form out of dynamic interactions between gas flows within molecular clouds . the upshot therefore is that potential star - forming dense filaments are likely to be found in molecular clouds where ambient conditions contrive to render the volume of gas immediately preceding the dense filament critically stable , at least ( i.e. mass line - density approximately equal to its maximum stable value ) . the possible effect of the ambient conditions on the chemistry within a molecular cloud must directly impact the gas thermodynamics and therefore , the dynamical stability of that cloud . we will examine this issue in a future contribution . the authors thank an anonymous referee for a helpful and prompt report . j.f . acknowledges support from the indo - french centre for the promotion of advanced research ( ifcpar / cefipra ) through a raman - charpak fellowship over the course of which this project was conceived . anathpindika , s , 2009 , a&a , 504 , 437 anathpindika , s & di francesco , j. , 2013 , mnras , 430 , 1854 andr ' e , ph . , 2010 , a&a , 518 , l102 andr ' e , ph . , di francesco , j. , ward - thompson , d. , inutsuka , s. , pudritz , r & pineda , j. , astroph 1312.6232 , to appear in _ protostars and planets vi , university of arizona press ( 2014 ) _ , eds . h. beuther , r. klessen , c. dullemond , th . henning arzoumanian , d. , andr ' e , ph . , peretto , n & k@xmath125nyves , v. , 2013 , a&a , 553 , 119a arzoumanian , d. , andr ' e , ph . , didelon , p. , k@xmath125nyves , v. , schneider , n. , menshchikov , a _ et al . _ , 2011 , a&a , 529 , l6 bastien , p. , 1983 a&a , 119 , 109 bastien , p. , arcoragi , j. , benz , w. , bonnell , i & martel , h. , 1991 , apj , 378 , 255 bate , m. r. , bonnell , i. a & price , n. m. , 1995 , mnras , 277 , 362 bate , m. r. & burkert , a. , 1997 , mnras , 288 , 1060 bate , m. r. , bonnell , i. a. , bromm , v. , 2003 , mnras , 339 , 577 bate , m. r. , 2012 , mnras , 419 , 3115 duarte - 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it is now widely accepted that dense filaments of molecular gas are integral to the process of stellar birth and potential star - forming cores often appear embedded within these filaments . although numerical simulations have largely succeeded in reproducing filamentary structure in dynamic environments such as in turbulent gas and while analytic calculations predict the formation of dense gas filaments via radial collapse , the exact process(es ) that generate / s such filaments which then form prestellar cores within them , is unclear . in this work we therefore study numerically the formation of a dense filament using a relatively simple set - up of a uniform - density cylinder in pressure equilibrium with its confining medium . in particular , we examine if its propensity to form a dense filament and further , to the formation of prestellar cores within this filament bears on the gravitational state of the initial volume of gas . we report a radial collapse leading to the formation of a dense filamentary cloud is likely when the initial volume of gas is at least critically stable ( characterised by the approximate equality between the mass line - density for this volume and its maximum value ) . though self - gravitating , this volume of gas , however , is not seen to be in free - fall . this post - collapse filament then fragments along its length due to the growth of a jeans - like instability to form prestellar cores like _ beads on a string_. we suggest , dense filaments in typical star - forming clouds classified as gravitationally super - critical under the assumption of : ( i ) isothermality when in fact , they are not , and ( ii ) extended radial profiles as against one that is pressure - truncated , thereby causing significant over - estimation of their mass line - density , are unlikely to experience gravitational free - fall . the radial density and temperature profile derived for this post - collapse filament is consistent with that deduced for typical filamentary clouds mapped in recent surveys of nearby star - forming regions . this profile is also in agreement with a plummer - like density profile . for an isothermal filament though , the density profile is much steeper , consistent with the classic density profile suggested by ostriker ( 1964 ) . on the other hand , increasing the magnitude of the confining pressure such that the initial volume of gas is rendered gravitationally sub - critical is unable to collapse radially and tends to expand laterally which could possibly explain similar gas filaments found in recent surveys of some molecular clouds . simulations were performed using smoothed particle hydrodynamics ( sph ) and convergence of results is demonstrated by repeating them at higher resolution . unlike some of the earlier work reported in literature , here we calculate gas temperature by solving the sph energy equation and allow it to cool according to a cooling function . gravitation hydrodynamics ism : structure ism : clouds prestellar cores

1501.03248

shortly after the discovery of the cosmic microwave background radiation ( cbr ) , it was shown that cosmic ray protons above @xmath160 eev ( @xmath2ev ) should be attenuated by photomeson interactions with cbr photons ( @xcite ) . it was later calculated that heavier cosmic ray nuclei with similar _ total _ energies would also be attenuated , but by a different process , _ viz . _ , photodisintegration interactions with ibr photons ( @xcite ) , hereafter designated psb ) . we will refer to such cosmic rays of total energies above 10 eev as ultrahigh energy cosmic rays ( uhcr ) . in the more conventional scenario , uhcrs are charged particles which must be accelerated to ultrahigh energies by electromagnetic processes at extragalactic sites , both because there are no known sites in our galaxy which can accelerate and magnetically contain them and also because most of the observed uhcr air shower events arrive from directions outside of the galactic plane . although such acceleration of charged particles to energies above 100 eev in cosmic sources pushes our present theoretical ideas to their extreme , it has been suggested that it may occur in hot spots in the lobes of radio galaxies ( @xcite ) . the detection of the two highest energy air shower events yet observed , with energies of @xmath3 ( between 170 and 260 ) eev ( @xcite ) and @xmath4 eev ( @xcite ) has aggravated both the acceleration and propagation problems for cosmic - ray physicists . ( very recently , the agasa group has presented a total of 6 events of energies between @xmath5 and @xmath3 eev , including the one cited above , observed since 1990 ( @xcite ) . ) how does nature accelerate particles to these extreme energies and how do they get here from extragalactic sources ( @xcite ) ? to answer these questions , new physics has been invoked , physics involving the formation and annihilation of topological defects ( tds ) which may have been produced in the very earliest stages of the big bang , perhaps as a result of grand unification . a td annihilation or decay scenario has unique observational consequences , such as the copious production of uhcr neutrinos and @xmath0-rays ( @xcite and refs . therein ; @xcite ) . a new ground - based detector array experiment named after pierre auger ( @xcite ) and an interesting satellite experiment called _ owl _ ( @xcite ) have been proposed to test look for such consequences . a uhcr _ proton _ of energy @xmath1 200 eev has a lifetime against photomeson losses of @xmath6s ; one of energy 300 eev has a lifetime of about half that stecker ( 1968 ) . these values correspond to linear propagation distances of @xmath1 30 and 15 mpc respectively . even shorter lifetimes were calculated for fe nuclei , based on photodisintegration off the ibr ( psb ) . recent estimates of the lifetimes of uhcr _ @xmath0-rays _ against electron - positron pair production interactions with background radio photons give values below @xmath7s ( @xcite ) . within such distances , it is difficult to find candidate sources for uhcrs of such energies . in this paper , we reexamine a part of the propagation problem by presenting the results of a new calculation of the photodisintegration of uhcr _ nuclei _ through the cbr and ibr in intergalactic space . in order to do this , we have made use of a new determination of the ibr based on empirical data , primarily from iras galaxies , recently calculated by malkan & stecker ( 1998 ) . they calculated the intensity and spectral energy distribution ( sed ) of the ibr based on empirical data , some of which was obtained for almost 3000 iras galaxies . it is these sources which produce the ibr . the data used for the new ibr calculation included ( 1 ) the luminosity dependent seds of these galaxies , ( 2 ) the 60 @xmath8 m luminosity function for these galaxies , and ( 3 ) the redshift distribution of these galaxies . the magnitude of the ibr flux derived by malkan & stecker ( 1998 ) is is considerably lower than that used in psb in their extensive examination of the photodisintegration of uhcr nuclei . a search for absorption in the high energy @xmath0-ray spectra of extragalactic sources can also be used to help determine the value of the ibr or to place constraints on the magnitude of its flux ( @xcite ) . the observed lack of strong absorption in the @xmath0-ray spectra of the active galaxies mrk 421 ( @xcite ) and mrk 501 ( @xcite ) up to an energy greater than @xmath1 5 - 10 tev is consistent with the new , lower value for the ibr used here ( @xcite ) . the sed calculated by malkan & stecker ( 1998 ) agrees with direct estimates of the far infrared background obtained from the _ cobe / firas _ observations ( @xcite ) . recent fluxes reported from _ cobe / dirbe _ obervations at 140 and 240 @xmath8 m ( @xcite ) are roughly a factor of 2 higher than the malkan & stecker ( 1998 ) predictions , but are consistent with them if one considers the systematic uncertainties in the observational results ( @xcite ) . in justifying our reexamination of the photodisintegration problem using the new ibr sed , we point out that it may reasonable to expect that the highest energy cosmic rays may be nuclei . this is because the maximum energy to which a particle can be accelerated in a source of a given size and magnetic field strength is proportional to its charge , @xmath9 . that charge is 26 times larger for fe than it is for protons . although some composition measurements in the energy range 0.1 - 10 eev appear to indicate a transition from heavier to lighter nuclei with increased energy ( @xcite ) , this and other data appear to be consistent with a `` mixed '' composition of both protons and heavier nuclei ( @xcite ) . in any case , at the `` lower '' energies for which composition measurements have been attempted , most of the cosmic rays may be galactic in origin . we have now done a full monte carlo calculation similar to that presented in psb , but using the new intergalactic infrared spectrum given by malkan & stecker ( 1998 ) . in this new calculation , we have also specifically included the threshold energies of the various nuclear species to photodisintegration . the reason that this is now important is that with the cbr playing a relatively more important role , interactions of the steeply falling cosmic - ray spectrum near threshold with the wien tail of the cbr become important relative to the much flatter ibr photon spectrum . in fact , as we will show , taking account of the measured higher threshold energies ( as opposed to the artificial value of 2 mev taken by psb ) increases the value of the cutoff energy for heavy uhcr nuclei . our intent is to both update and improve the psb results and also to determine if the highest energy cr events seen by the fly s eye and akeno groups are consistent with their being heavy nuclei that have propagated to us from candidate active galactic nuclei ( @xcite ) . the energy loss from photodisintegration has a much stronger dependence on lorentz factor than on atomic weight , and increases strongly with lorentz factor @xmath0 ( psb ) . therefore , to maximize the propagation distance for a given total particle energy , @xmath10 ( where @xmath11 is the nucleon mass ) , one takes the largest possible mass number . given the abundances of the elements , this nucleus is fe . therefore ( as psb have done ) we chose to examine the propagation history of nuclei originating as @xmath12fe , as this nuclide offers the best chance for providing a conventional explanation for the uhcr events . the nuclear photodisintegration process is dominated by the giant dipole resonance ( gdr ) , which peaks in the @xmath0-ray energy range of 10 to 30 mev ( nuclear rest frame ) . experimental data are generally consistent with a two - step process : photoabsorption by the nucleus to form a compound state , followed by a statistical decay process involving the emission of one or more nucleons from the nucleus ( @xcite ) . the photoabsorption cross section roughly obeys a thomas - reiche - kuhn ( trk ) sum rule , _ viz_. , @xmath13 where @xmath14 is the mass number , @xmath15 is the nuclear charge , @xmath16 , @xmath11 is the nucleon mass and @xmath17 is the photon energy in the rest system of the nucleus . the trk sum rule is not exact , however , owing to the presence of nuclear exchange forces ( @xcite ) , as can be seen in table 1 of psb ( which is based on the data tabulations of fuller , gerstenberg , vander molen , and dunn ( 1973 ) and related material supplied by e. fuller . ) psb approximated the gdr cross section for a given nuclide ( @xmath18 ) with the parameterization @xmath19 where the @xmath15 and @xmath14 dependence of the width @xmath20 , the peak energy @xmath21 , and the dimensionless integrated cross sections @xmath22 and @xmath23 are understood , and are listed in table 1 of psb . @xmath24 and @xmath25 are the heaviside step functions , and the normalization constants @xmath26 are given by @xmath27.\ ] ] the index @xmath28 takes the values 1 or 2 , corresponding to single or double nucleon emission during the photodisintegration reaction . for energies @xmath29 mev , the measured cross sections are dominated by single [ @xmath30 or @xmath31 or double [ @xmath32 , @xmath33 , @xmath34 nucleon loss , since the threshold energies for the emission of larger numbers of nucleons are close to or exceed @xmath35 ( @xcite ) . from @xmath35 to @xmath36 mev , the cross section is approximated as flat , normalized so that the integrated cross section matches experimental values . the probability of emission of @xmath37 nucleons in this region is given by a distribution function which is independent of the @xmath0-ray energy ( see table 2 of psb ) . above @xmath38 , detailed cross section data are more scarce ; we follow psb by approximating this smaller residual cross section by zero . interactions with photons of energy greater than @xmath39 will have a negligible contribution to the photodisintegration process for uhcr energies below 1000 eev owing to the fact that the density of the background photons seen by the uhcr near the peak of the gdr cross section falls rapidly with energy ( exponentially along the wien tail of the cbr , and roughly as @xmath40 in ir - optical region ) , along with the fact that the photodissociation cross section is about two orders of magnitude lower in the @xmath0-ray energy region from @xmath38 to @xmath11 gev than at the gdr peak ( @xcite ) . we have verified by numerical tests that interactions with photons of energy greater than @xmath39 indeed have a negligible effect on our calculation . in the psb calculation , the gdr threshold energies were taken to be @xmath41 mev for _ all _ reaction channels . this value is far smaller than the true thresholds ; single - nucleon emission has a typical threshold of @xmath42 mev , while the double - nucleon emission energy threshold is typically @xmath43 mev . table 1 lists the energy thresholds for the @xmath30 , @xmath44 , @xmath32 , @xmath33 , @xmath45 , and @xmath46 channels for each nuclide in the @xmath12fe decay chain ( taken from forkman & petersson , 1987 ) . owing to the increased importance of the cbr relative to the ibr that follows from the new , lower ibr estimates , and the presence of the wien tail that exponentially increases the target photon density at the gdr threshold ( see figure 1 ) , increasing this threshold energy may significantly lengthen the propagation distance of a highly relativistic nucleus . therefore we have used the measured threshold energies for a subset of the reaction channels that are given in table 1 . unfortunately , photodisintegration cross section data are incomplete . for many reaction channels , @xmath47 data do not exist . also , integrated cross section strengths are not available for all of the exclusive channels . the most complete compilation of the world s gdr cross section data exists in the 15 volumes of fuller & gerstenberg ( 1983 ) . in these volumes gdr cross section data for @xmath12fe , for example , are given only for the @xmath48 channel and the inverse channels @xmath49 and @xmath50 . one can not perform the monte carlo calculations in such a way as to distinguish a @xmath30 interaction from a @xmath44 reaction when the relative branching ratios are not known . instead , we separately consider only single - nucleon emission ( as a single channel ) and double - nucleon emission ( as a single channel ) in the gdr region up to @xmath35 . we take the conservative approach of choosing the energy threshold for single - nucleon emission to be the _ lowest _ of the two for @xmath30 and @xmath44 for each nuclide in the decay chain , with a similar choice for the two - nucleon emission channel . along most of the decay chain from @xmath12fe to @xmath51h there is only one stable isotope for a given mass @xmath14 . since the radioactive decay time to the line of stability is less than the one - nucleon photodisintegration loss time for all but three unstable nuclei , @xmath52mn , @xmath53al , and @xmath54be , we assume decay has brought the daughter nucleus to the line of stability before the next photon collision , so that for any given mass @xmath14 there is a unique charge @xmath15 . this clearly is not the case for mass values @xmath14 = 54 , 50 , 48 , 46 , 40 , and 36 , where more than one stable isotope exists , but the absence of cross section data for each of these isotopes makes this a moot point . we also note that although the thresholds are lowest for @xmath55 emission ( due to the @xmath55 s large binding energy ) , the integrated cross section for @xmath55 emission from @xmath12fe , for example , is over two orders of magnitude lower than the @xmath56 value for that nuclide ( skopic , asai , & murphy , 1980 ; see also fuller & gerstenberg , 1983 ) . we therefore neglect the @xmath55 emission channels entirely in our calculation . for a uhcr nucleus with lorentz factor @xmath57 propagating through an isotropic soft photon background with differential number density @xmath58 , the photodisintegration rate @xmath59 ( lab frame ) is given by ( stecker 1969 ) @xmath60 where @xmath61 is the total cross section , summed over the number of emitted nucleons . in our calculations we construct the soft photon background by summing three components : ( 1 ) the @xmath62k cosmic background radiation ( cbr ) from lab frame energies of @xmath63 ev to @xmath64 ev , ( 2 ) the infrared background radiation ( ibr ) estimated by malkan & stecker ( 1998 ) from @xmath65 ev to 0.33 ev ( using the two `` best estimates '' shown by heavy lines in their figure 2 , which we denote as the `` high ibr '' and `` low ibr '' cases ) , and ( 3 ) the optical to uv diffuse , extragalactic photon background estimated by salamon & stecker ( 1998 ) ( taking their no - metallicity - correction case ) from @xmath66 ev to 13 ev . figure 1 shows the sed of the entire low energy background radiation . a salient feature of this figure is the roughly constant energy flux down to the dramatic rise of the wien tail of the cbr . for uhcr nuclei with lorentz factors large enough so that the cbr photons are above photodisintegration threshold , most of the reaction rate @xmath59 is dominated by collisions with cbr photons this is particularly true with the new , smaller ir photon background levels , compared to those used by psb . [ fig1 ] the energy loss time @xmath67 is defined as @xmath68 where @xmath69 , @xmath70 , and @xmath71 are the reaction rates for one- and two - nucleon emission , and @xmath71 is the reaction rate for @xmath72 nucleon loss . the reduction in @xmath0 comes from two effects : nuclear energy loss due to electron - positron pair production off the cbr background , and the @xmath0-ray momentum absorbed by the nucleus during the formation of the excited compound nuclear state that preceeds nucleon emission . this latter effect is much smaller ( of order @xmath73 ) than the energy loss from nucleon emission and will therefore be neglected . for the former mechanism , we use the results given in figure 3 of blumenthal ( 1970 ) , which gives the loss rate for relativistic nuclei off the cbr calculated in the first born approximation . ( we note that the coulomb corrections to the born approximation ( @xcite ) have a negligible effect on the pair production loss rate for ultrarelativistic fe nuclei . ) figure 2 shows the energy loss rates due to single - nucleon , double - nucleon , and pair production processes for @xmath12fe as a function of energy , along with the total energy loss rate . also shown is the total energy loss rate when the photodisintegration thresholds @xmath74 are all set to 2 mev ; this indicates the effect of incorporating more realistic threshold energies in the monte carlo calculation , compared to those of psb . [ fig2 ] figure 3 shows the spectra of uhcr nuclei which started out as @xmath12fe after propagating linear distances from 1 to 1000 mpc ( corresponding to lifetimes between @xmath75 and @xmath76 s ) , assuming a source differential energy spectrum @xmath77 over the energy interval 10 to 1000 eev . the effect of the cbr `` wall '' at the wien side of the 2.73k blackbody spectrum is to produce a sharp cutoff at a rather well defined energy , @xmath78 . [ fig3 ] figure 4 shows the cutoff energy as a function of propagation time , where @xmath79 is defined as the cr energy at which the propagated differential flux is @xmath80 that of the unpropagated flux . for comparison , the cutoff energies calculated by psb are also shown . it can be seen that ( except for energies above @xmath3 eev ) for a given energy , the propagation time increases by a substantial factor over that calculated by psb , who assumed larger ibr fluxes and a 2 mev threshold for all photodisintegration interactions . we also note that our new results do not differ significantly for the two seds adopted from malkan & stecker ( 1998 ) and , except for the longest propagation times , they do not differ significantly from the no - ibr case shown in the figure . this is because the new values obtained for the ibr are so low . as can be seen from figure 4 , our use of the new , lower values for the intergalactic infrared photon flux , together with the explicit inclusion of the measured threshold energies for photodisintegration of the individual nuclides involved in the calculation , has the effect of increasing the cutoff energy ( for a given propagation time ) of heavy uhcr nuclei over that originally calculated by psb . this increase may have significant consequences for understanding the origin of the highest energy cosmic ray air shower events . stanev ( 1995 ) and biermann ( 1998 ) have examined the arrival directions of the highest energy events . they point out that the @xmath1200 eev event is within 10@xmath81 of the direction of the strong radio galaxy ngc315 . ngc315 lies at a distance of only @xmath1 60 mpc from us . for that distance , our results indicate that heavy nuclei would have a cutoff energy of @xmath1 130 eev , which may be within the uncertainty in the energy determination for this event . the @xmath1300 eev event is within 12@xmath81 of the strong radio galaxy 3c134 . the distance to 3c134 is unknown because its location behind a dense molecular cloud in our galaxy obscures the spectral lines required for a redshift measurement . it may therefore be possible that _ either _ cosmic ray protons ( stecker 1968 ) _ or _ heavy nuclei originated in these sources and produced these highest energy air shower events . biermann , p.l . 1998 , in _ workshop on observing giant cosmic ray air showers from @xmath82 ev particles from space _ a.i.p . krizmanic , j.f . ormes and r.e . streitmatter ( new york : amer . inst . phys . ) p. 22 rrrrrrrr 26 & 56 & 11.2 & 10.2 & 20.5 & 20.4 & 18.3 & 7.6 26 & 54 & 13.4 & 8.9 & 24.1 & 20.9 & 15.4 & 8.4 25 & 55 & 10.2 & 8.1 & 19.2 & 17.8 & 20.4 & 7.9 24 & 54 & 9.7 & 12.4 & 17.7 & 20.9 & 22.0 & 7.9 24 & 53 & 7.9 & 11.1 & 20.0 & 18.4 & 20.1 & 9.1 24 & 52 & 12.0 & 10.5 & 21.3 & 21.6 & 18.6 & 9.4 24 & 50 & 13.0 & 9.6 & 23.6 & 21.1 & 16.3 & 8.6 23 & 51 & 11.1 & 8.1 & 20.4 & 19.0 & 20.2 & 10.3 23 & 50 & 9.3 & 7.9 & 20.9 & 16.1 & 19.3 & 9.9 22 & 50 & 10.9 & 12.2 & 19.1 & 22.3 & 21.8 & 10.7 22 & 49 & 8.1 & 11.4 & 19.8 & 19.6 & 20.8 & 10.2 22 & 48 & 11.6 & 11.4 & 20.5 & 22.1 & 19.9 & 9.4 22 & 47 & 8.9 & 10.5 & 22.1 & 19.2 & 18.7 & 9.0 22 & 46 & 13.2 & 10.3 & 22.7 & 21.7 & 17.2 & 8.0 21 & 45 & 11.3 & 6.9 & 21.0 & 18.0 & 19.1 & 7.9 20 & 48 & 9.9 & 15.8 & 17.2 & 24.2 & 29.1 & 14.4 20 & 46 & 10.4 & 13.8 & 17.8 & 22.7 & 22.7 & 11.1 20 & 44 & 11.1 & 12.2 & 19.1 & 21.8 & 21.6 & 8.8 20 & 43 & 7.9 & 10.7 & 19.4 & 18.2 & 19.9 & 7.6 20 & 42 & 11.5 & 10.3 & 19.8 & 20.4 & 18.1 & 6.2 20 & 40 & 15.6 & 8.3 & 29.0 & 21.4 & 14.7 & 7.0 19 & 41 & 10.1 & 7.8 & 17.9 & 17.7 & 20.3 & 6.2 19 & 40 & 7.8 & 7.6 & 20.9 & 14.2 & 18.3 & 6.4 19 & 39 & 13.1 & 6.4 & 25.2 & 18.2 & 16.6 & 7.2 18 & 40 & 9.9 & 12.5 & 16.5 & 20.6 & 22.8 & 6.8 18 & 38 & 11.8 & 10.2 & 20.6 & 20.6 & 18.6 & 7.2 18 & 36 & 15.3 & 8.5 & 28.0 & 21.2 & 14.9 & 6.6 17 & 37 & 10.3 & 8.4 & 18.9 & 18.3 & 21.4 & 7.8 17 & 35 & 12.6 & 6.4 & 24.2 & 17.8 & 17.3 & 7.0 16 & 36 & 9.9 & 13.0 & 16.9 & 21.5 & 25.0 & 9.0 16 & 34 & 11.4 & 10.9 & 20.1 & 21.0 & 20.4 & 7.9 16 & 33 & 8.6 & 9.6 & 23.7 & 17.5 & 18.2 & 7.1 16 & 32 & 15.0 & 8.9 & 28.1 & 21.2 & 16.2 & 6.9 15 & 31 & 12.3 & 7.3 & 23.6 & 17.9 & 20.8 & 9.7 14 & 30 & 10.6 & 13.5 & 19.1 & 22.9 & 24.0 & 10.6 14 & 29 & 8.5 & 12.3 & 25.7 & 20.1 & 21.9 & 11.1 14 & 28 & 17.2 & 11.6 & 30.5 & 24.6 & 19.9 & 10.0 13 & 27 & 13.1 & 8.3 & 24.4 & 19.4 & 22.4 & 10.1 12 & 26 & 11.1 & 14.1 & 18.4 & 23.2 & 24.8 & 10.6 12 & 25 & 7.3 & 12.1 & 23.9 & 19.0 & 22.6 & 9.9 12 & 24 & 16.5 & 11.7 & 29.7 & 24.1 & 20.5 & 9.2 11 & 23 & 12.4 & 8.8 & 23.5 & 19.2 & 24.1 & 10.5 10 & 22 & 10.4 & 15.3 & 17.1 & 23.4 & 26.4 & 9.7 10 & 21 & 6.8 & 13.0 & 23.6 & 19.6 & 23.6 & 7.3 10 & 20 & 16.9 & 12.8 & 28.5 & 23.3 & 20.8 & 4.7 9 & 19 & 10.4 & 8.0 & 19.6 & 16.0 & 23.9 & 4.0 8 & 18 & 8.0 & 15.9 & 12.2 & 21.8 & 29.1 & 6.2 8 & 17 & 4.1 & 13.8 & 19.8 & 16.3 & 25.3 & 6.4 8 & 16 & 15.7 & 12.1 & 28.9 & 23.0 & 22.3 & 7.2 7 & 15 & 10.8 & 10.2 & 21.4 & 18.4 & 31.0 & 11.0 7 & 14 & 10.6 & 7.6 & 30.6 & 12.5 & 25.1 & 11.6 6 & 13 & 4.9 & 17.5 & 23.7 & 20.9 & 31.6 & 10.6 6 & 12 & 18.7 & 16.0 & 31.8 & 27.4 & 27.2 & 7.4 5 & 11 & 11.5 & 11.2 & 19.9 & 18.0 & 30.9 & 8.7 5 & 10 & 8.4 & 6.6 & 27.0 & 8.3 & 23.5 & 4.5 4 & 9 & 1.7 & 16.9 & 20.6 & 18.9 & 29.3 & 2.5 3 & 7 & 7.3 & 10.0 & 12.9 & 11.8 & 33.5 & 2.5 3 & 6 & 5.7 & 4.6 & 27.2 & 3.7 & 26.4 & 1.5 2 & 4 & 20.6 & 19.8 & 28.3 & 26.1 & & 2 & 3 & 7.7 & 5.5 & & & & 1 & 2 & 2.2 & & & & &

we present the results of a new calculation of the photodisintegration of ultrahigh energy cosmic - ray ( uhcr ) nuclei in intergalactic space . the critical interactions for energy loss and photodisintegration of uhcr nuclei occur with photons of the 2.73 k cosmic background radiation ( cbr ) and with photons of the infrared background radiation ( ibr ) . we have reexamined this problem making use of a new determination of the ibr based on empirical data , primarily from iras galaxies , consistent with direct measurements and upper limits from tev @xmath0-ray observations . we have also improved the calculation by including the specific threshold energies for the various photodisintegration interactions in our monte carlo calculation . with the new smaller ibr flux , the steepness of the wien side of the now relatively more important cbr makes their inclusion essential for more accurate results . our results indicate a significant increase in the propagation time of uhcr nuclei of a given energy over previous results . we discuss the possible significance of this for uhcr origin theory . # 1#23.6pt

astro-ph9808110

any _ in vivo _ quantum system is in a contact with its environment . although typically weak , this interaction becomes relevant when studying the evolution of a system over long time scales . in particular , the asymptotic state of such an _ open _ system depends both on the unitary action induced by the system hamiltonian , and the action of the environment , conventionally called ` dissipation ' . a recent idea of `` engineering by dissipation '' @xcite , the creation of designated pure and highly entangled states of many - body quantum systems by using specially designed dissipative operators , has promoted dissipation to the same level of importance as the underlying unitary dynamics . the use of time - periodic modulations constitutes another channel to impact states of a quantum system . in the coherent limit , when the system is isolated from the environment , modulations imply an explicit time - periodicity of the system hamiltonian , @xmath3 . the dynamics of the system are determined by the basis of time - periodic _ floquet eigenstates _ the properties of the floquet states depend on various modulation parameters . modulations being resonant with intrinsic system frequencies can create a set of non - equilibrium eigenstates with properties drastically different from those with time - independent hamiltonians . modulations enrich the physics occurring in fields such as quantum optics , optomechanics , solid state and ultra - cold atom physics @xcite and disclose a spectrum of new phenomena @xcite . as an object of mathematical physics , ( [ lind ] ) has a specific structure and possesses a variety of important properties @xcite . in the case of a time - independent , stationary hamiltonian @xmath15 , the generator @xmath5 induces a continuous set of completely positive quantum maps @xmath16 @xcite . under some conditions ( ` quantum ergodicity ' ) , the system evolves from an initial state @xmath17 to a unique and time - independent asymptotic state @xmath18 , @xmath19 @xcite . when time - periodic modulations are present , eq . ( [ lind ] ) preserves the complete positivity of the evolution if all coupling rates are non - negative at any instance of time , @xmath20 @xcite . under some suitable , experimentally relevant assumptions , a set - up `` time - dependent hamiltonian and time - independent dissipation '' provides a valid approximation @xcite . here , we address the particular case of quench - like , time - periodic dependence of the hamiltonian , @xmath21 , corresponding to periodic switches between several constant hamiltonians @xcite . a popular choice is the set - up composed of two hamiltonians , @xmath22 where @xmath23 , @xmath24 $ ] . this minimal form has recently been used to investigate the connection between integrability and thermalization @xcite or , alike , for disorder - induced localization @xcite in _ isolated _ periodically modulated many - body systems . from a mathematical point of view , ( [ lind],[pc ] ) define a linear equation with a time - periodic generator @xmath25 . therefore , floquet theory applies and asymptotic solutions of the equation are all time - periodic with temporal period @xmath26 @xcite . @xmath25 is a dissipative operator and , in the absence of relevant symmetries @xcite , the system evolution in the asymptotic limit @xmath27 is determined by a unique ` quantum attractor ' , i.e. , by an asymptotic , time - periodic density operator , @xmath28 , @xmath24 $ ] and @xmath29 . the challenge here consists in explicit numerical evaluation of the matrix form of this operator . to use spectral methods ( complete / partial diagonalization and different kinds of iterative algorithms @xcite ) to calculate @xmath30 as eigen - element of a superoperator would mean to deal with @xmath31 scaling of computationally expensive operations . in the case of periodically modulated systems it restricts the use of spectral methods to @xmath32 , @xmath33 and the corresponding lindblad operators @xmath34,@xmath35 are sparse , the floquet map @xmath36 is a dense matrix . therefore , the numerical evaluation can not benefit from sparse - matrix methods . ] . a direct propagation of eq . ( [ lind ] ) for a time span long enough for @xmath37 to approach the attractor is not feasible for @xmath1 for two reasons : first , direct propagation requires to numerically propagate @xmath38 complex differential equations with time - dependent coefficients , and second , accuracy might become a problem for large times . although the accuracy may be improved by implementing high - order integration schemes @xcite or faber and newton polynomial integrators @xcite , this approach is hardly parallelizable ) by using the so - called time evolving block decimation ( tebd ) technique @xcite . the numerical effort scales as @xmath39 . however , this algorithm can only be used for lattice systems , i.e. , systems that can be partitioned into @xmath40 ` pieces ' coupled by next - neighbor interactions , both unitary and dissipative . in this case it gives a correct answer when the asymptotic state is characterized by a low entanglement . ] . systems with @xmath41 states may still be too small , for example , to explore mbl effects in open periodically - modulated systems . is it possible to go beyond this limit ? and if so how far ? we attempt to answer these questions by unraveling of the quantum master equation ( [ lind ] ) into a set of stochastic realizations , called `` quantum trajectories '' @xcite . this method allows us to transform the problem of the numerical solution of eqs . ( [ lind],[pc ] ) into a task of statistical sampling over quantum trajectories which form vectors of the size @xmath4 . the price to pay for the reduction from @xmath31 to @xmath4 is that we now have to sample over many realizations . this problem is very well suited for parallelization @xcite and we can definitely benefit from the use of a computation cluster . if the number of realizations @xmath42 becomes large , the sampling of the density operator @xmath37 with initial condition @xmath43 converges to the solution of eq . ( [ lind ] ) @xcite . this method is popular in the field of quantum optics where it adequately describes the physics of experiments @xcite . [ cols="^,^ " , ] finally , we estimated the computation time to propagate a single trajectory on a single - core as a function of system size @xmath4 , see table [ table2 ] . for the model specified by eqs . ( [ eq : hamiltonian ] , [ eq : jump ] ) this time scales as @xmath44 ; this is due to the multiplication of the quadratic scaling of a dense matrix - vector multiplication and a linear scaling of the jump frequency . the latter scaling is , however , model specific and may differ for other physical systems so that the overall computation time may vary substantially with the type of hamiltonian and/or dissipators under study . on top , the numbers we present in table [ table1 ] depend on the values of the coupling constant @xmath45 and the period of modulations @xmath26 . this is so because these parameters control the rate of the jumps . therefore , the obtained estimates are specific and can not be used as universal quantifiers . we now report the results of our simulations obtained for the physical setup with eqs . ( [ eq : hamiltonian ] , [ eq : jump ] ) . we start with the performance of the algorithm , table [ table1 ] . the idea of the algorithm mimics a float : the algorithm constantly attempts to ` float to the surface ' , i.e. , to increase the time step of integration towards its maximal value @xmath46 while every next jump pulls it downwards to @xmath47 , see fig . [ fig : husimi ] . the average time between two consequent jumps is the mean of the local maxima in the depicted saw - like time sequence of @xmath48 . there is no problem in overestimating @xmath46 , simply because the time step will rarely reach its maximum . the shortest time step , @xmath47 , or , equivalently , the depth @xmath49 , is tuned to the values needed to reach the desired accuracy . next we turn to the averages @xmath50 over the realizations and the corresponding statistical variances @xmath51 $ ] of the matrix elements , as discussed in detail in section [ sampling ] . both quantities converge to `` limit cycles '' if the propagation time @xmath52 , @xmath53 , is much larger than all relaxation times . this means that for @xmath54 the density matrix converges to a time - periodic quantum attractor , i.e. , @xmath55 , and the variances also become time - periodic functions , @xmath56 \backsimeq \operatorname{var}[\varrho^\mathrm{att}_{kl}(\tau)]$ ] , see fig . [ fig:3]b . the crumpled caustic - like shapes of the limit cycles is a result of the projection on a plane of a global limit - cycle living in @xmath57-dimensional space . this limit - cycle is not a topological product of @xmath31 two - dimensional limit cycles ; elements of the asymptotic density matrix do not evolve independently , they do interact so that their means and variances are coupled . in the asymptotic regime sampling can be performed stroboscopically , i.e. , after every period @xmath26 . in our simulations we used as the transient time @xmath58 and then performed the stroboscopic sampling of @xmath59 . the attractor density matrix at any other instant of time @xmath60 can be sampled by shifting the starting time of the sampling , @xmath61 , or simply by performing an extra - sampling at all needed intermediate points . with @xmath62 samples per trajectory ( that amounts to an additional propagation for the time @xmath63 ) it became possible to collect @xmath64 samples for the model of dimension @xmath65 ( i.e. , with @xmath66 indistinguishable bosons ) by running the program on @xmath67 cores during three days . for some choice of parameters the sample density matrix displays a standard diagonal - dominated structure , see fig . [ fig:3]a . the husimi distribution of the sample density matrix is depicted in fig . [ fig:4 ] . there is an intriguing similarity between the distribution of the quantum attractor and the phase - space structure of the classical attractor ( its stroboscopic section , to be more precise ) produced by the mean - field equations , eqs . ( [ eq : thetadot ] ) . this allows us to conjecture that the attractor density matrix is resolved with good accuracy . the @xmath68 cores allowed us to sample the same number of realizations for the model of the dimension @xmath69 during approximately one week , [ eq : jump ] ) involve summation over series of binomial coefficients of the order @xmath4 . it was not possible , however , to go beyond @xmath70 when evaluating the corresponding husimi distributions . ] . the primary aim of this study was to estimate the numerical horizon of a high - accuracy sampling of non - equilibrium dissipative quantum attractors , i.e. , the asymptotic solutions of periodically driven open systems , by using the quantum trajectory method . our main result is that it is possible to resolve asymptotic density matrices of systems of several thousand of states by using even a small cluster ( with @xmath71 cores ) on a time scale of a few days . the benefit of having access to the whole density matrix is the possibility to extract more detailed information about non - equilibrium regimes encoded in the matrix structure such as purity and many - body entanglement @xcite . naturally , the next step must be a systematic analysis of the errors , statistical and numerical ones . we reserve this objective for further studies . here , we would like to surmise on further optimization of the sampling procedure . an immediate idea is to use a reduced adaptive basis , constructed by using coherent states @xcite . this however demands a priori knowledge of the attractor s structure . this insight can be obtained by using models of small sizes , @xmath72 . another possibility is the choice of more optimal initial conditions which in turn will substantially reduce the transient time @xmath73 ( for example , one can use an extrapolation of the largest - eigenvalue eigenstate of the density matrix for a small system ) . altogether , we expect that such improvements could increase @xmath4 further by a factor of @xmath74 . research areas where quantum attractors are of potential interest were already mentioned in section [ introduction ] . here , we briefly recall them . first , this is many - body localization @xcite where the action of temporal modulations @xcite and dissipation @xcite so far have been considered separately . a combined action of both factors presents an intriguing challenge . a complementary task is to extend the idea of `` dissipative engineering '' @xcite to periodically - modulated quantum systems . finally , a survival of floquet topological insulators @xcite in the presence of dissipation is a timely question . the authors acknowledge support by the russian science foundation ( grant no . 15 - 12 - 20029 ) . we thank i. vakulchik for the help with preparing the figures . f. pastawski , l. clemente , and j. i. cirac , phys . a * 83 * ( 2011 ) 012304 ; m. j. kastoryano , m. m. wolf , and j. eisert , phys . rev * 110 * ( 2013 ) 110501 ; e. kapit , phys * 116 * ( 2016)150501 . k. j. strm and b. bernhardsson , _ comparison of periodic and event based sampling for first - order stochastic systems_. in preprints 14th world congress of ifac , beijing , p.r . china , july 1999 . ; k. j. strm and b. bernhardsson , lecture notes in control and information sciences * 286 * ( 2002 ) 1 . d. poletti , j .- s . bernier , a. georges , c. kollath , phys . * 109 * ( 2012 ) 045302 . c. gross , t. zibold , e. nicklas , j. esteve , m.k . oberthaler , nature * 464 * ( 2010 ) 1165 . j. tomkovic , w. muessel , h. strobel , s. lock , p. schlagheck , r. ketzmerick , m. k. oberthaler , _ observing the emergence of chaos in a many - particle quantum system _ ,

quantum systems out of equilibrium are now a subject of intensive research both in theoretical and experimental physics . in this paper we study periodically modulated quantum systems which are in contact with a stationary environment . within the framework of lindblad quantum master equation , the asymptotic states of such systems are described by time - periodic density operators . resolution of these operators is a non - trivial computational task . approaches based on spectral and iterative methods are restricted to systems with the dimension of the hosting hilbert space @xmath0 , while the direct long - time integration of the master equation becomes problematic for @xmath1 . to overcome these limitations we use the quantum trajectory method which unravels the deterministic master equation for the density operator into a set of stochastic processes for wave functions . this method avoids calculations of the kernel of the floquet superoperator ; instead the asymptotic density matrix is calculated by performing a statistical sampling preceded by a long transient propagation . we present a high - accuracy realization of this idea based on exponential propagators combined with a time - stepping technique . employing a scalable model of interacting bosons hoping over a dimer , we test the performance of the algorithm on a supercomputer . we demonstrate that the algorithm allows to resolve non - equilibrium asymptotic states of model systems with @xmath2 on a small computer cluster thus reaching the scale on which numerical studies of isolated periodically - modulated systems are currently performed . open quantum systems , lindblad equation , periodic modulations , quantum trajectories

1612.03848

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

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

0712.4255

our bubble chambers are insensitive to minimum ionizing particles , allowing us to exploit a new calibration technique using charged pions as wimp surrogates to produce nuclear recoils by strong elastic scattering . we measure the pion scattering angle using silicon pixel detectors . the nuclear recoil kinetic energy can be calculated by @xmath26 on an event by event basis , where @xmath27 is the beam momentum , @xmath28 the scattering angle , and @xmath29 the nuclear mass of the target . for a cf@xmath0i target , a measured scattering angle corresponds to a different recoil energy depending on which nucleus is involved in the interaction ; in this paper , we will refer to iodine equivalent recoil energy , @xmath30 , as the energy given to an iodine nucleus for a given pion scattering angle . for a @xmath1 pion beam , approximately @xmath31 of the rate of pions scattering into angles corresponding to @xmath30 between 5 and 20 kev is due to elastic scattering on iodine , with smaller contributions from carbon , fluorine , and inelastic scattering @xcite . therefore , the bubble nucleation efficiency for iodine recoils in a bubble chamber with seitz threshold between 5 and 20 kev can be inferred from a measurement of the fraction of pion - scattering events that nucleate bubbles in the chamber as a function of @xmath30 . the measurement was performed in the fermilab test beam facility @xcite using a @xmath1 mainly @xmath32 beam with @xmath33 and an angular spread of @xmath34 mrad . the absolute momentum of the beam is known to @xmath21 . the pions were tracked with a silicon pixel telescope @xcite consisting of 4 upstream and 4 downstream silicon pixel plaquettes , with a spatial coverage of 14 mm x 14 mm . the total length of the telescope was 90 cm . the angular resolution was 0.6 mrad ( @xmath6 ) in the horizontal ( @xmath35 ) direction and 0.7 mrad in the vertical ( @xmath36 ) direction , with roughly equal contributions from multiple coulomb scattering ( mcs ) in the target and the spatial resolution of the telescope . plastic scintillators triggered the pixel telescope on each beam particle . a small bubble chamber was designed for this measurement consisting of a quartz test tube with inner diameter 10 mm and 1-mm - thick wall , filled with 7 @xmath37 of cf@xmath0i . the small size is required to minimize mcs in the short radiation length of cf@xmath0i ( @xmath38 mm ) . the bubble chamber was operated at a pressure of @xmath39 psia and a temperature of @xmath40 c with a nominal seitz threshold of @xmath2 . the iodine equivalent threshold scattering angle is 4.7 mrad . an acoustic transducer was attached to the top of the test tube to record the acoustic emission produced during bubble formation , providing the time of bubble nucleation with @xmath2010 @xmath41s resolution . temperature control was provided by a water bath around the bubble chamber . bubble chamber data were taken between march 14 and march 28 , 2012 , with a beam flux of @xmath201000 particles per 4-second beam spill with one spill per minute . the size of the beam spot was wider than both the bubble chamber and the pixel telescope . the chamber was expanded to the superheated state 22 seconds before the arrival of the beam , allowing time for pressure and temperature transients to dissipate after expansion . the observation of bubbles by a 100-hz video camera system created a bubble chamber trigger , causing the video images and associated data to be recorded and the chamber to be recompressed . after recompression , the chamber was dead for the remainder of the beam spill , allowing us to collect at most one bubble event per minute . we collected about four good single - bubble events per hour , with the primary losses due to premature bubble chamber triggers , bubbles forming outside of the region covered by the telescope planes , multiple bubble events and large - angle scatters outside the acceptance of the downstream plaquettes . the last two categories are predominantly the result of inelastic interactions . figure [ fig : signals ] shows an example scattering event . at the end of the run the cf@xmath0i was removed and a target empty data set was taken . in addition , data were taken in a test run in december 2011 with no target , as well as solid targets of quartz , graphite , teflon or ( c@xmath42f@xmath24)@xmath43 , and crystalline iodine . ( color online ) an example event ( @xmath44 mrad ) , including the relative timing of the telescope trigger and acoustic signal , one camera image of the bubble , and the @xmath36 and @xmath45 positions of the telescope hits . the pion beam is in the @xmath46 direction . the camera image is not to scale but the test tube has inner diameter of 10 mm . beam tubes in the water bath to minimize the material traversed by the pion beam are visible to either side of the bubble chamber . ] the primary analysis output is the bubble nucleation fraction as a function of @xmath30 , given by the ratio @xmath47 where @xmath48 is the observed number of pion tracks creating single bubbles , @xmath49 is the total number of pion tracks , @xmath50 is the number of tracks creating multiple bubbles , and @xmath51 is the fraction of scatters that occur in the active cf@xmath52i volume , determined by a comparison of the number of scatters in the target - full data set to the number in the target - empty data set normalized to the number of pion tracks ( @xmath53 ) : @xmath54 an angular smearing correction is made to @xmath53 to include the mcs from the absent cf@xmath0i by convolution with the standard gaussian approximation for mcs @xcite . each pion track is fitted for an upstream and downstream component , with an associated scattering angle and 3-d point of closest approach of the two components . the upstream and downstream track segments are required to have exactly one hit cluster in at least three of the four pixel planes , good fits to straight lines ( @xmath55 ) , and to meet in space to within @xmath56 mm . to exclude pions that passed through little or no cf@xmath0i , the upstream track is required to pass within @xmath57 mm of the center of the @xmath58-mm - diameter bubble chamber in the @xmath35 direction . the @xmath36 location of the track is limited by the vertical extent of the pixel planes . because the uncertainty on the location of the point of closest approach in the beam direction ( @xmath45 ) depends strongly on the scattering angle , we require the @xmath45 location to be within 3@xmath59 of the bubble chamber , where @xmath59 is the uncertainty on @xmath45 for each individual event . events with more than one track are rejected . as these track cuts are applied without regard to the presence of nucleations in the bubble chamber , their efficiency applies equally to @xmath48 , @xmath49 , @xmath50 , and @xmath53 , and therefore cancels in the final ratio , @xmath60 . the next step is to associate a bubble with a unique track using both time and space correlations . the timing requirement for correlating a track with a bubble is chosen to be @xmath61 @xmath41s . the bubble locations are reconstructed using standard coupp techniques @xcite , and the difference between reconstructed bubble position and point of closest approach of the track components is required to be less than 2.1 mm in the @xmath35 direction and less than 0.9 mm in the @xmath36 direction . the combined event acceptance of these timing and spatial cuts is @xmath62 . after these data selection and quality cuts , @xmath63 good single bubble events remain . the final bubble nucleation fraction , @xmath64 , is shown as the points in fig . [ results ] . ( color online ) the fraction of pion scattering events that produced bubbles as a function of iodine equivalent recoil energy . the solid curves show the simulated contribution from individual recoil species ( from high to low at 20 kev , red for iodine , green for fluorine , and pink for carbon and inelastics ) , with the blue dashed curve showing the sum . the iodine curve shown takes a step function efficiency model for iodine recoils using the best fit threshold of @xmath65 . ] to disentangle the iodine component from carbon , fluorine and inelastic scattering , we perform a full simulation using geant4.9.5 @xcite . the simulation was validated by comparing the simulated scattering angular distributions to data for no target , target empty , target full , and the solid targets . in all cases , in the mcs - dominated small scattering angle region there is good ( few percent ) agreement with no adjustable parameters , suggesting that the telescope geometry is accurately modeled in the simulation . in the larger scattering angle region dominated by strong elastic scattering , the simulation systematically overestimates the observed scattering rate by @xmath66 . as this ratio is measured to be the same for teflon ( @xmath67 ) and iodine ( @xmath68 ) , we assume that the relative contributions of iodine , fluorine and carbon are being accurately described by the mc . a significant systematic uncertainty is introduced by our developing understanding of the carbon and fluorine recoil nucleation efficiency in this low energy regime . ongoing studies with ad hoc neutron sources @xcite will reduce this uncertainty in the future , but here we apply the exponential carbon and fluorine efficiency model from @xcite : we test the hypothesis that the iodine recoil nucleation efficiency follows the nominal seitz model of a step function ( @xmath72 ) with 100% efficiency above the seitz threshold by fitting a step function to the data in the region @xmath73 kev , allowing @xmath18 to float . the fit returns @xmath74 , where the error bars are statistical . this value is 2.1@xmath6 higher than the seitz model threshold @xmath75 , where the systematic error includes absolute energy scale uncertainties of 3% in the beam momentum and 1% in the scattering angle stemming from uncertainty in the @xmath45 positions of the plaquettes . the fit is shown as the dashed blue line in fig . [ results ] . figure [ contour ] shows the inferred iodine nucleation efficiency as a function of @xmath30 with the iodine component isolated by subtracting the simulated contributions from carbon , fluorine and inelastic scatters . the dashed blue curve is the best fit step function with @xmath74 . for comparison , the red region represents a step function at the predicted seitz threshold , where the range represents the 1@xmath6 band including the thermodynamic uncertainty and the scale uncertainties in the absolute energy scale of the experiment . given the energy resolution smearing induced by mcs in this experiment , the preference for a value of @xmath18 higher than the prediction can not be easily distinguished from an exponential model like eq . ( [ eq : expo ] ) for iodine nucleation efficiency with a lower threshold energy and a finite value of @xmath76 . previous studies have shown that the seitz model accurately predicts the threshold at which bubble nucleation begins for heavy radon daughter nuclei in cf@xmath0i @xcite . we therefore perform a second fit applying the exponential model to iodine recoils , taking the seitz threshold calculation as an external input to the analysis to explore the allowed range of @xmath76 . the best fit is shown as the black curve in fig . [ contour ] . the inset shows 2@xmath6 contours for fits to the exponential model with the threshold constrained by our prediction ( shaded region ) and free ( unshaded region ) , along with the best fit points . ( color online ) the data points represent the measured iodine nucleation efficiency as a function of iodine equivalent recoil energy , where the contributions from carbon , fluorine and inelastic scatters have been subtracted . the gradual turn on is predominantly due to the angular resolution of the experiment , as illustrated by both the red region , representing the step function model with the threshold varied within the uncertainty on the seitz theory prediction , @xmath77 , and the dashed blue curve , representing the best fit step function with @xmath78 kev . the black curve shows the best fit exponential model with the threshold constrained by the theory as described in the text . the inset shows 2@xmath6 contours for a fit to the exponential model with the threshold allowed to float ( pink ) or constrained by the theory ( solid cyan ) . the colored dots represent the corresponding curves in the main plot . ] to assess the systematic errors associated with carbon and fluorine recoils , we refit the data with two alternative models for carbon and fluorine efficiency : the flat model from @xcite with energy - independent nucleation efficiency , @xmath79 , above threshold , and a step function with @xmath80 . the latter case represents the worst possible scenario for the response of the bubble chamber to iodine recoils , as the response to carbon and fluorine is maximized . we use the exponential model for iodine recoils , allowing @xmath76 to float and treating @xmath18 as a nuisance parameter constrained by the prediction . the results of these fits are summarized in table [ tab : summary_alpha ] . extended fits over the energy interval @xmath81 kev have a negligible effect on the iodine fit parameters but disfavor the flat c / f efficiency model with @xmath79 . .summary of fits to @xmath76 , including @xmath82 lower limits on @xmath76 . the three different c / f efficiency models described in the text are tested , and in all cases the predicted seitz threshold is treated as a nuisance parameter . by maximizing the subtracted c / f contribution , the step function with @xmath80 represents the worst case for iodine efficiency . [ cols="<,^,^,^",options="header " , ] in conclusion , we have directly measured the efficiency for iodine bubble nucleation in a cf@xmath0i bubble chamber operated with a nominal threshold of @xmath2 . for some models of carbon and fluorine efficiency , the response to iodine recoils is consistent with a step function at the seitz threshold , but in all cases there is a preference for either a softer turn on or a slightly higher threshold . even in the worst case scenario for iodine , however , the response of the chamber to iodine recoils is much closer to the nominal seitz model than it is for carbon and fluorine recoils . this was expected from the considerably larger stopping power of iodine , which facilitates the concentration of energy that leads to critical bubble formation . systematic uncertainties from both the absolute beam momentum calibration and the carbon and fluorine response limit the present measurement . this measurement provides confirmation of the sensitivity of coupp bubble chambers to spin - independent wimp interactions with iodine nuclei , a confirmation that was not attainable using standard neutron source calibrations . the technique of employing hadron elastic scattering as a tool to measure bubble nucleation thresholds is now established , enabling the measurement of bubble nucleation energies on an event by event basis . we have begun studies of the feasibility to repeat this technique with different fluids . the coupp collaboration would like to thank fermi national accelerator laboratory , the department of energy and the national science foundation for their support including grants phy-0856273 , phy-1205987 , phy-0937500 and phy-0919526 . we acknowledge technical assistance from fermilab s accelerator , computing , and particle physics divisions , and from a. behnke at iusb .

we have directly measured the energy threshold and efficiency for bubble nucleation from iodine recoils in a cf@xmath0i bubble chamber in the energy range of interest for a dark matter search . these interactions can not be probed by standard neutron calibration methods , so we develop a new technique by observing the elastic scattering of @xmath1 negative pions . the pions are tracked with a silicon pixel telescope and the reconstructed scattering angle provides a measure of the nuclear recoil kinetic energy . the bubble chamber was operated with a nominal threshold of @xmath2 . interpretation of the results depends on the response to fluorine and carbon recoils , but in general we find agreement with the predictions of the classical bubble nucleation theory . this measurement confirms the applicability of cf@xmath0i as a target for spin - independent dark matter interactions and represents a novel technique for calibration of superheated fluid detectors . recent years have seen a resurgence in the use of superheated liquids and bubble chambers as continuously sensitive nuclear recoil detectors searching for dark matter in the form of weakly interacting massive particles ( wimps)@xcite . at a low degree of superheat , bubble chambers are insensitive to minimum ionizing backgrounds that normally plague wimp searches but retain sensitivity to the nuclear recoils that would be characteristic of wimp scattering . in a superheated liquid the process of radiation - induced bubble nucleation is described by the classical `` hot spike '' model @xcite . for the phase transition to occur , the energy deposited by the particle must create a critically sized bubble , requiring a minimum energy deposition in a volume smaller than the critical bubble . under mildly superheated conditions , the latter requirement renders the bubble chamber insensitive to minimum ionizing particles . the radius of the critical bubble is given by the condition that the bubble be in ( unstable ) equilibrium with the surrounding superheated fluid @xcite . this demands the pressure balance @xmath3 where @xmath4 is the pressure inside the bubble , @xmath5 is the pressure in the liquid , @xmath6 is the bubble surface tension , and @xmath7 is the critical bubble radius . the pressure @xmath4 is fixed by the condition that the chemical potential inside and outside the bubble be equal , giving @xmath8 where @xmath9 is the pressure in a saturated system at the given temperature , and @xmath10 and @xmath11 are the liquid and vapor densities in the saturated system @xcite . in seitz s `` hot spike '' model for bubble nucleation , the entire energy necessary to create the critical bubble must come from the particle interaction that nucleates the bubble . this is in contrast to earlier models that required only the work ( free energy ) to come from the particle interaction , with the remaining bubble - formation energy supplied by heat flowing in from the surrounding superheated fluid @xcite . as the name `` hot spike '' implies , the nucleation site in seitz s model begins as a high - temperature seed , so it can not draw heat from the surrounding fluid . once the decision is made to consider the total bubble creation energy rather than just the free energy , the threshold energy calculation is completely described by gibbs @xcite . this energy is given by @xmath12 here , @xmath13 is the temperature of the system , @xmath14 is the bubble vapor density , and @xmath15 and @xmath16 are the specific enthalpies of the bubble vapor and superheated liquid . the surface tension @xmath6 and temperature derivative are taken along the usual saturation curve . the three terms give , from left to right , the heat necessary to create the bubble surface , the heat needed to vaporize the fluid to make the bubble interior , and a reversible work term done in expanding the bubble to the critical size that must be subtracted to avoid double - counting work present in both of the first two terms . to good approximation @xmath17 may be replaced by the normal heat of vaporization at temperature @xmath13 . the greatest uncertainty in determining the thermodynamic @xmath18 is the relation between the surface tension at a flat liquid - vapor interface and the surface tension for a very small bubble . this relation is described by the `` tolman length '' @xmath19 , which is unknown but is expected to be some fraction of the intermolecular distance @xcite . this translates to an uncertainty on @xmath18 of @xmath20@xmath21 . for the rest of this paper , we refer to the calculated threshold in eq . ( [ eq : seitz ] ) as the seitz threshold . the seitz model assumes the efficiency for bubble nucleation is @xmath22 for all interactions that deposit @xmath23 over a volume small compared to the critical bubble . the length scales for nuclear recoil cascades in the energy region between 5 and 20 kev relevant for a wimp search are similar to the critical radius , so the seitz model may or may not give a good description of bubble nucleation , and direct calibrations of bubble nucleation efficiency are necessary . the working fluid discussed in this paper is iodotrifluoromethane or cf@xmath0i , which contains two highly sensitive wimp target nuclei : fluorine , for spin - dependent interactions , and iodine , for spin - independent interactions . neutrons are typically used to mimic wimps in calibrating the nuclear recoil response of a wimp detector , and neutron sources have been used to measure the nucleation threshold for carbon and fluorine recoils in cf@xmath0i , cf@xmath0br @xcite and c@xmath24f@xmath25 @xcite under various superheat conditions . however , iodine recoils contribute only a small fraction to the total neutron - nucleated bubble rate in cf@xmath0i . therefore , neutron sources are an ineffective calibration tool for iodine recoils in coupp . we have used heavy daughter nuclei produced in alpha decays as a proxy @xcite , but these are high energy recoils of @xmath20100 kev . this paper describes a measurement of bubble nucleation efficiency for iodine recoils near our dark matter search thresholds .

1304.6001

with the advent of high temperature superconductivity in the cuprates and the possibility of exotic gap symmetry including nodal behavior , a renewed effort to find novel experimental probes of order parameter symmetry has ensued . one result of this effort was the proposal by sauls and co - workers@xcite to examine the nonlinear current response of d - wave superconductors . they showed that a nonanalyticity in the current - velocity relation at temperature @xmath1 is introduced by the presence of nodes in the order parameter . one prediction was that an anisotropy should exist in the nonlinear current as a function of the direction of the superfluid velocity relative to the position of the node . this would be reflected in an anisotropy of a term in the inverse penetration depth which is linear in the magnetic field @xmath2 . early experimental work did not verify these predictions@xcite and it was suggested that impurity scattering@xcite or nonlocal effects@xcite may be responsible . however , a more recent reanalysis of experiment has claimed to confirm the predictions@xcite . an alternative proposal was given by dahm and scalapino@xcite who examined the quadratic term in the magnetic response of the penetration depth , which shows a @xmath3 dependence at low @xmath4 as first discussed by xu et al.@xcite . dahm and scalapino demonstrated that this upturn would provide a clear and unique signature of the nodes in the d - wave gap and that this feature could be measured directly via microwave intermodulation effects . indeed , experimental verification of this has been obtained@xcite confirming that nonlinear microwave current response can be used as a sensitive probe of issues associated with the order parameter symmetry . thus , we are led to consider further cases of gap anisotropy and turn our attention to the two - band superconductor mgb@xmath0 which is already under scrutiny for possible applications , including passive microwave filter technology@xcite . mgb@xmath0 was discovered in 2001@xcite and since this time an enormous scientific effort has focused on this material . on the basis of the evidence that is available , it is now thought that this material may be our best candidate for a classic two - band electron - phonon superconductor , with s - wave pairing in each channel@xcite . a heightened interest in two - band superconductivity has led to claims of possible two - band effects in many other materials , both old@xcite and new@xcite . our goal is to compare in detail the differences between one - band and two - band s - wave superconductors in terms of their nonlinear response , that would be measured in the coefficients defined by xu et al.@xcite and dahm and scalapino@xcite . this leads us to reconsider the one - band s - wave case , where we study issues of dimensionality , impurities , and strong electron - phonon coupling . we find new effects due to strong - coupling at both high and low @xmath4 . we then examine the situation for two - band superconductors , starting from a case of highly decoupled bands . here , we are looking for signatures of the low energy scale due to the smaller gap , the effect of integration of the bands , and the response to inter- and intraband impurities . unusual behavior exists distinctly different from the one - band case and not necessarily understood as a superposition of two separate superconductors . finally , we return to the case of mgb@xmath0 which was studied previously via a more approximate approach@xcite . in the current work , we are able to use the complete microscopic theory with the parameters and the electron - phonon spectral functions taken from band structure@xcite . in this way , we provide more detailed predictions for the nonlinear coefficient of mgb@xmath0 . in section ii , we briefly summarize the necessary theory for calculating the gap and renormalization function in two - band superconductors , from which the current as a function of the superfluid velocity @xmath5 is then derived . in section iii , we explain our procedure for extracting the temperature - dependent nonlinear term from the current and we examine the characteristic features for one - band superconductors in light of issues of dimensionality , impurity scattering and strong coupling . section iv presents the results of two - band superconductors and simple formulas are given for limiting cases which aid in illuminating the effects of anisotropy . the case of mgb@xmath0 is also discussed . we form our conclusions in section v. the superfluid current has been considered theoretically in the past by many authors for s - wave@xcite and for other order parameters , such as d - wave and f - wave@xcite . most recently , the case of two - band superconductivity has been examined@xcite with good agreement obtained between theory and experiment for the temperature dependence of the critical current@xcite . in this work , we wish to calculate the superfluid current as a function of superfluid velocity @xmath5 or momentum @xmath6 and extract from this the nonlinear term . to do this , we choose to evaluate the expression for the superfluid current density @xmath7 that is written on the imaginary axis in terms of matsubara quantities.@xcite this naturally allows for the inclusion of impurity scattering and strong electron - phonon coupling in a numerically efficient manner . written in general for two - bands having a current @xmath8 and @xmath9 , for the first and second band , respectively , we have : @xmath10 where @xmath11 is the electric charge , @xmath12 is the electron mass , @xmath4 is the temperature , @xmath13 , @xmath14 is the electron density and @xmath15 is the fermi velocity of the @xmath16th band ( @xmath17,2 ) . the @xmath18 represents an integration for the @xmath16th band which is given as @xmath19 for a 3d band and @xmath20 for a 2d band , with @xmath21 in the 2d case . also , in the expression for the current , the 3 should be changed to a 2 for 2d . this is done within a mean - field treatment and ignoring critical fluctuations near @xmath22 . here , we have taken the approximation of a spherical fermi surface in 3d and a cylindrical one in 2d as we will see further on that the differences between 2d and 3d are not significant to more than a overall numerical factor and so providing more precise fermi surface averages will not changes the results in a meaningful way . to evaluate this expression , we require the solution of the standard s - wave eliashberg equations for the renormalized gaps and frequencies @xmath23 and @xmath24 , respectively . these have been generalized to two bands and must also include the effect of the current through @xmath6 . with further details given in refs . @xcite , we merely state them here : @xmath25\nonumber\\ & \times&\biggl\langle \frac{\tilde\delta_j(m)}{\sqrt{(\tilde\omega_j(m)-is_jz)^2+\tilde\delta_j^2(m)}}\biggr\rangle_j\nonumber\\ & + & \pi\sum_jt^+_{lj}\biggl\langle \frac{\tilde\delta_j(n ) } { \sqrt{(\tilde\omega_j(n)-is_jz)^2+\tilde\delta_j^2(n)}}\biggr\rangle_j\label{eq : del}\end{aligned}\ ] ] and @xmath26 where @xmath27 sums over the number of bands and the sum over @xmath12 is from @xmath28 to @xmath29 . here , @xmath30 is the ordinary impurity scattering rate and @xmath31 indexes the @xmath31th matsubara frequency @xmath32 , with @xmath33 , where @xmath34 . the @xmath35 are coulomb repulsions , which require a high energy cutoff @xmath36 , taken to be about six to ten times the maximum phonon frequency , and the electron - phonon interaction enters through @xmath37 with @xmath38 the electron - phonon spectral functions and @xmath39 the phonon energy . note that the dimensionality does not change the gap equations when there is no current . for finite @xmath6 , it does and we will see later the result of this effect . likewise , an essential ingredient is that the current enters the eliashberg equations and provides the bulk of the nonlinear effect for temperatures above @xmath40 . indeed , at @xmath22 all of the nonlinearity arises from the gap . we now proceed to the case of one - band superconductors , to illustrate the generic features of the superfluid current and demonstrate how we extract the nonlinear term . in the section following , we will return to the two - band case . ( 250,200 ) in fig . [ fig1 ] , we illustrate that these equations reproduce the standard results for @xmath7 versus @xmath6 for a one - band superconductor in the weak coupling bcs limit . equations ( [ eq : js])-([eq : z ] ) were solved for both the 2d and 3d cases at @xmath41 and 0.95 . the @xmath1 result of past literature@xcite is recovered in the case of 3d . one sees for @xmath42 at low @xmath6 , the curve is essentially linear , reflecting the relationship of @xmath43 , with @xmath44 the superfluid density . for strong coupling , the slope would be reduced by approximately @xmath45 as the superfluid condensate is also reduced by this factor . likewise the reduction in the slope with temperature would reflect the temperature dependence of the superfluid density . indeed , to provide these curves using the eliashberg equations , we used the @xmath46 spectrum of al and made the corrections for the @xmath45 factor . al is a classic bcs weak coupling superconductor , that agrees with bcs in every way and is generally used for bcs tests of the eliashberg equations . the @xmath47 for al is 0.43 . while at low @xmath4 the curves show little deviation from linearity at low @xmath6 , and thus the nonlinear correction will be essentially zero ( exponentially so with temperature in bcs theory ) , at @xmath4 near @xmath22 , one sees that there is more curvature for @xmath48 and hence a larger nonlinear term is expected . however , while the 2d and 3d curves differ in behavior near the maximum in @xmath7 , one finds that the behavior at low @xmath6 is very similar . indeed , the nonlinearity is a very small effect on these plots and hard to discern , however , it will be borne out in our paper that the nonlinear current does not show significant differences in the @xmath4-dependence between 2d and 3d . nevertheless , we will still include both the 2d and 3d calculation in our two - band calculations as mgb@xmath0 has a 2d @xmath49-band and a 3d @xmath50-band , and there is a overall factor of 2/3 between the two in the nonlinear term due to dimensionality . to obtain the nonlinear current as @xmath48 , the general expression for the current can be expanded to second lowest order in powers of @xmath6 leading to the general formula @xmath51 , \label{eq : jexpand}\ ] ] where only first and third order terms arise . here by choice @xmath52 and the variable for the expansion was taken as @xmath53 , where @xmath54 . @xmath55 and @xmath56 are temperature - dependent coefficients which follow when solutions of the eliashberg equations ( [ eq : del ] ) and ( [ eq : z ] ) are substituted in the expression ( [ eq : js ] ) for the current . in practice , it is complicated to expand eqs . ( [ eq : js])-([eq : z ] ) to obtain an explicit form @xmath56 and so we chose to extract @xmath55 and @xmath56 numerically by solving our full set of equations with no approximations for @xmath7 versus @xmath6 . from this numerical data , we find the intercept and slope of @xmath57 versus @xmath58 for @xmath48 from which we obtain the @xmath55 and @xmath56 , respectively . ( 250,200 ) the results for @xmath59 and @xmath60 as a function of temperature are shown in fig . one sees , in the inset , @xmath59 which , in the one - band case , is just the superfluid density @xmath44 normalized to the clean bcs value at @xmath1 . there is no difference between 2d and 3d bcs . also , shown is the @xmath59 extracted for the strong electron - phonon coupling superconductor pb with no impurities and with impurity scattering of @xmath61 . one sees that strong coupling pushes the temperature dependence of the curve higher , even slightly so at @xmath1 , and this is a well - documented effect@xcite . with impurities , the superfluid density is reduced in accordance with standard theory . these curves were obtained from our @xmath7 calculations and agree exactly with bcs and eliashberg calculations done with the standard penetration depth formulas@xcite , confirming that our numerical procedure is accurate . the second term in eq . ( [ eq : jexpand ] ) gives the nonlinear current and the coefficient @xmath60 , which is a measure of this , is also shown for the four cases . [ note that @xmath60 is the same as the @xmath62 of ref . @xcite to within a constant of proportionality . ] here one does find a difference between the 2d and 3d bcs curves showing that dimensionality can affect the nonlinear current . in the case of strong coupling one finds an increase in the nonlinear piece near @xmath22 and also a finite contribution at low @xmath4 which is unexpected in the usual bcs scenario . impurities have the effect of further increasing the low @xmath4 contribution and reducing the curve near @xmath22 . near @xmath4 equal to @xmath22 ( @xmath63 ) in bcs , it can be shown analytically that @xmath64 and for 3d @xmath65 as obtained in our previous paper , ref . the value of @xmath56 for 2d is increased by a factor of @xmath66 . these numbers agree with the numerical calculations in fig . [ fig2 ] , where the 3d bcs curve goes to 0.21 for 2d and 0.14 in 3d . there are two definitions in the literature for the nonlinear coefficient : one is denoted as @xmath67 due to xu et al.@xcite and the other , @xmath68 , used by dahm and scalapino@xcite , is the one that is related to the intermodulation power in microstrip resonators . rewriting eq . ( [ eq : jexpand ] ) in the form @xmath69 , \label{eq : jexpand2}\ ] ] dahm and scalapino define@xcite @xmath70 xu , yip and sauls@xcite keep the form of eq . ( [ eq : jexpand ] ) but define a variable @xmath71 , where @xmath72 is the temperature dependent gap equal to @xmath73 . with this they identify the coefficient @xmath74 in this work , we always take @xmath75 to be the usual bcs temperature dependence of the gap function . ( 250,200 ) in fig . [ fig3 ] , we show the calculations for the @xmath67 coefficient of xu et al .. here , we have made a number of points . first , the 2d bcs curve derived from our procedure agrees with that shown by xu et al.@xcite , once again validating our numerical work for extracting the very tiny nonlinear coefficient . second , for bcs one sees a difference between 2d and 3d in the nonlinear coefficient . the 2d curve goes to 1 at @xmath22 and to 2/3 for the 3d case . the question arises as to whether the difference between 2d and 3d is simply a numerical factor and so with the dotted curve , we show the 3d case scaled up by 3/2 . we do note that there is a small difference in the temperature variations at an intermediate range of @xmath4 , but the major difference between 2d and 3d is the overall numerical factor of 2/3 . third , one might question the necessity of including the effect of the current on the gap itself and to answer this , we show the long - dashed curve where the @xmath6 dependence was omitted in the eliashberg equations ( [ eq : del ] ) and ( [ eq : z ] ) . one finds that the nonlinear coefficient is reduced substantially at temperatures above @xmath76 and disappears at @xmath22 . thus , without the @xmath6 dependence in the gap , the true nonlinear effects will not be obtained for high temperatures as the gap provides the major contribution to the nonlinearity . in the lower frame of fig . [ fig3 ] , we examine the case of pb to illustrate strong electron - phonon coupling and impurity effects . it is seen that the strong coupling increases the value at @xmath22 and also gives a finite value at low @xmath4 . the behavior at low @xmath4 is surprising in light of the bcs result@xcite , but is related to the inelastic electron - phonon scattering which appears to increase the nonlinear coefficient at small @xmath4 in a similar way to what is already known about the effect of impurities in bcs@xcite . the strong coupling behavior near @xmath22 is similar to that seen for other quantities such as the specific heat@xcite , where the downward bcs curvature is now turned concave upward to higher values at @xmath22 . impurities have the effect of reducing the nonlinearity near @xmath22 and increasing it at low @xmath4 . once again , in bcs we can provide some analytic results near and at @xmath22 for @xmath77 which provide a useful check on our numerical work . for three dimensions near @xmath22 : @xmath78 and , upon substituting for @xmath75 , @xmath79 doing the same algebra for the two - dimensional case corrects these expressions by a factor of 3/2 and gives 1 instead of 2/3 for @xmath80 . to characterize the strong - coupling effects seen in the figure for pb , we can develop a strong - coupling correction formula for @xmath81 and @xmath80 . these formulas have been provided in the past for many quantities and form a useful tool for experimentalists and others to estimate the strong coupling corrections.@xcite this was done by evaluating this quantity for ten superconductors using their known @xmath46 spectra and their @xmath22 values . we used al , v , sn , in , nb , v@xmath82ga , nb@xmath82ge , pb , pb@xmath83bi@xmath84 , and pb@xmath85bi@xmath86 . these materials were chosen to span the range of typical s - wave superconductors with strong coupling parameter @xmath87 ranging from 0.004 to 0.2 . the details of these materials and references for the spectra may be found in the review by carbotte@xcite . the parameter @xmath88 is defined as : @xmath89.\ ] ] by fitting to these materials , we arrived at the following strong coupling correction formulas for three dimensions : @xmath90\ ] ] and @xmath91 note that , even though @xmath92 is the usual form of the strong coupling correction , in this last equation , we have found no advantage in fitting with the additional parameter offered by the log factor . these formula should be seen as approximate tools to give the trend for @xmath87 for values restricted to the range of 0 to 0.2 . pb has @xmath87 value of 0.128 and is intermediate to this range , and al is a weak coupling superconductor with a value of 0.004 . ( 250,200 ) in fig . [ fig4 ] , we show the coefficient used by dahm and scalapino@xcite for the same cases as previously considered . with this coefficient one finds qualitatively similar curves . the 2d and 3d bcs curves go to zero rapidly at low temperature , but once again the strong coupling effects in pb give a finite value for @xmath68 at low @xmath4 . with impurities the tail at low temperature is raised significantly . due to the divergence in @xmath68 near @xmath22 because of the division by three powers of the superfluid density which is going to zero at @xmath22 , we prefer to work with a new quantity @xmath93 , which removes this divergence . thus , we define @xmath94 and this is shown in the inset in fig . it has the advantage of illustrating the detailed differences between the curves more clearly and providing finite values at @xmath22 which can be evaluated analytically in bcs theory . in this instance , we obtain @xmath95 for three dimensions in agreement with what we obtain from our numerical work , shown in the fig . [ fig4 ] . once again we can develop strong coupling formulas for this quantity and they are given as : @xmath96\ ] ] and @xmath97 for three dimensions . once again , there was no extra advantage to fitting @xmath98 with the usual form that includes the log factor . this last quantity @xmath99 is related to the intermodulation power in microstrip resonators and hence can be measured directly . having identified the features of one - band superconductors , we now turn to the two - band case where signatures of the two - band nature may occur in these nonlinear coefficients . the generalization of eq . ( [ eq : jexpand2 ] ) to the two - band case proceeds as follows . the total current @xmath7 is the sum of the two partial currents @xmath100 , @xmath101 with @xmath102 for the two - dimensional @xmath49- and three - dimensional @xmath50-band , respectively . for our numerical work , we do take into account the different dimensionality of the bands but , for simplicity in our analytic work below , we take them both to be three dimensional . a decision needs to be taken about the normalization of the current @xmath7 in the second term . dahm and scalapino have used @xmath103 . here instead , we prefer to use the more symmetric form @xmath104 which reduces properly to the one - band case when our two bands are taken to be identical , with @xmath105 , where @xmath31 is the total electron density per unit volume . for the combined system , eqs . ( [ eq : dougb ] ) and ( [ eq : saulsa ] ) still hold with @xmath55 and @xmath56 modified as follows : @xmath106 and @xmath107 with these definitions eq . ( [ eq : jexpand ] ) also holds with @xmath108 replacing @xmath109 and the xu , yip , and sauls variable , @xmath71 , of the one - band case is replaced by @xmath110 , with @xmath75 the usual temperature profile of the bcs gap . other choices could be made . the superfluid density @xmath44 is proportional to @xmath55 for the combined system , specifically @xmath111 is given by eq . ( [ eq : atb ] ) with the first two factors omitted . ( 250,200 ) in fig . [ fig5 ] , we show both the @xmath60 and the @xmath68 for a model which uses truncated lorentzians for the @xmath112 spectra . this same model was used in our previous work@xcite and so we refer the reader to that paper for details . also , in ref . @xcite may be found the curves for the @xmath113 , the penetration depth , and other quantities for the same parameters used here . the essential parameters of this model are @xmath114 , @xmath115 and the interband electron - phonon coupling is varied from @xmath116 ( nearly decoupled case ) to 0.1 ( more integrated case ) . in addition , the @xmath117 , @xmath118 and @xmath119 . in the nearly decoupled case of @xmath116 , it can be seen that the solid curve looks like a superposition of two separate superconductors , one with a @xmath22 which is about 0.33 of the bulk @xmath22 . the lower temperature part of this curve is primarily due to the @xmath50-band ( or band 2 ) which is three dimensional , and indeed , when examined in detail , it has the characteristic behavior of the 3d example studied in the one - band case . the part of the curve at higher temperatures above about @xmath120 is due to the @xmath49-band ( or band 1 ) which is taken to be 2d and indeed , in the case of @xmath60 it shows a dependence approaching @xmath22 that expected for 2d strong - coupling with some interband anisotropy effects . the relative scale of the two sections of the curve is set by the value of the gap anisotropy @xmath121 and the ratio @xmath122 . the overall scale on the y - axis for @xmath68 differs from that of fig . [ fig4 ] due to our choice of @xmath108 for the normalization in the nonlinear term . indeed , for nearly decoupled bands ( solid curve ) , the value of the nonlinear coefficient @xmath68 is small at reduced temperature @xmath123 just above the sharp peak due to band 2 . specifically , it is of order 0.5 . if it had been referred to @xmath124 instead of @xmath108 , it would be smaller still by a factor of 1.7 and comparable to the single band 2d bcs result at the same reduced temperature ( fig . [ fig4 ] bottom frame , solid curve ) . however , as the non - diagonal electron - phonon couplings @xmath125 and @xmath126 are increased and a better integration of two bands proceeds , @xmath68 at @xmath123 can increase by an order of magnitude as , for example , in the dashed curve . the actual scale in this region is set by the details of the electron - phonon coupling ( see later the specific case of mgb@xmath0 ) . with more integration between the bands , one finds that the sharp peak at lower @xmath4 is reduced and rounded with a tail reaching to @xmath22 . when @xmath127 , the feature characteristic of the @xmath50-band @xmath22 is almost gone in @xmath60 and absent entirely in @xmath68 , even for modest interband coupling . the same conclusion holds for the effects of interband scattering ( shown in fig . [ fig5 ] for a value of @xmath128 for the nearly decoupled case ) which also integrates the bands and eliminates the lower energy scale . however , while the structure at the lower @xmath22 is now reduced to the point of giving a monotonic curve for @xmath68 , there still remains a large nonlinear contribution well above that for the one - band s - wave case , which marks the presence of the second band . we can have further insight into these results and check our work by developing some simple analytic results in renormalized bcs theory ( rbcs ) . for a summary of the approximations of rbcs and a comparison with full numerical solution for various properties including @xmath7 , we refer the reader to our previous work@xcite . for simplicity , we take both bands to be three dimensional in the following . near @xmath129 , eqs . ( [ eq : rbcsa ] ) and ( [ eq : rbcsb ] ) are modified for each band to : @xmath130 and @xmath131 where the functions @xmath132 and @xmath133 have been derived in ref . @xcite and @xmath134 is independent of @xmath135 , where @xmath136 . the @xmath137 s depend on the microscopic parameters of the theory . in rbcs , they are @xmath138 , @xmath139 , @xmath140 , and @xmath141 , from which @xmath22 and @xmath142 follow . while the expressions obtained for the @xmath132 and @xmath133 are lengthy , and hence we do not repeat them here , they are explicit algebraic forms . it is useful in this work to consider several simplifying limits . for decoupled bands @xmath143 and @xmath144 . as the band 2 does not contribute near @xmath22 , @xmath145 and @xmath146 take on the form of the single band case ( eqs . ( [ eq : rbcsa ] ) and ( [ eq : rbcsb ] ) ) . another limiting case is the separable anisotropy model.@xcite in this model , there are only two gap values with a ratio of @xmath147 , with @xmath148 an anisotropy parameter often assumed small . in this case , @xmath149 , @xmath150 and @xmath151 , where @xmath152 and @xmath153 . as a result @xmath154 and @xmath155 in this model , taking in addition that @xmath118 and @xmath156 leads to the one - band case and this can be used as a check of our algebra . to see the consequences of this algebra for our nonlinear coefficient @xmath68 , we begin with the decoupled band case near @xmath129 for which @xmath144 and @xmath145 and @xmath146 reduce to their single band value . in this limit of @xmath157 , @xmath158 ^ 2 , \label{eq : dougdc}\ ] ] where @xmath159 is the gap anisotropy parameter @xmath121 , and @xmath142 is the gap at @xmath1 . this expression shows explicitly the corrections introduced by the two - band nature of the system over the pure one - band case . note that @xmath68 is always increased by the presence of the correction term . in ( [ eq : dougdc ] ) , @xmath159 can never be taken to be one since we have assumed band 2 is weaker than band 1 . before leaving the decoupled case , it is worth noting that @xmath68 will show a change at the band 2 critical temperature @xmath160 . for @xmath4 below @xmath160 , @xmath161 and @xmath162 will be finite while above this temperature they are both zero . when the coupling @xmath125 and @xmath126 is switched on but still small , we expect that these quantities will acquire small tails and that they vanish only at @xmath22 . this is the hallmark of nearly decoupled bands . for the anisotropic @xmath163 model near @xmath22 @xmath164 ^ 3},\ ] ] where the average gap @xmath165 is related to @xmath22 by @xmath166 $ ] . for @xmath167 this expression reduces properly to the one - band limit . therefore , it is seen that anisotropy increases @xmath68 for @xmath4 near @xmath22 . another interesting limiting case is to assume both bands are the same , i.e. isotropic gap case , but that the fermi velocities differ in the two bands . near @xmath22 , we obtain @xmath168 ^ 3 } \frac{1}{8}\frac{(v_{f1}+v_{f2})^2}{(v_{f1}v_{f2})^2}(v_{f1}^2+v_{f2}^2).\ ] ] in this case , the fermi velocity anisotropy changes the nonlinear coefficient , but when @xmath118 the expression reduces properly to the one - band result . we find that the fermi velocity anisotropy increases @xmath68 near @xmath22 , a result that is seen in one of our calculations for mgb@xmath0 shown in fig . [ fig6 ] . ( 250,200 ) with fig . [ fig6 ] , we now turn to the specific case of mgb@xmath0 , where we have used the parameters and @xmath112 given by band structure calculations , and as a result , there are , in principle , no free parameters other than varying the impurity scattering rate . the basic parameters are @xmath169 , @xmath170 , @xmath171 , @xmath172 , @xmath173 , @xmath174 , @xmath175 , @xmath176 , with a @xmath177 k and a gap anisotropy of @xmath178 . the ratio of the two density of states is @xmath179 and of the fermi velocities is @xmath180 . we have found excellent agreement between theory and experiment for these parameters , as have other authors@xcite . as we have found in our previous work , mgb@xmath0 is quite integrated between the bands . it is also an intermediate strong coupler with @xmath181 and thus there is competition between the strong coupling effects and the anisotropy@xcite . in fig . 6 , the solid curve gives the prediction for mgb@xmath0 for @xmath60 and @xmath68 . a strong nonmonotonic feature around the lower band energy scale is observed in @xmath60 , but the @xmath68 is monotonically increasing with temperature . to see the second band effects in @xmath68 , it is better to plot @xmath182 ( the inset ) which accentuates the subtle variations found at the lower energy scale associated with the @xmath50 band . also shown in the inset for the upper frame is @xmath59 , which gives the temperature dependence of the superfluid density . the solid curve agrees with our previous calculation by other means@xcite . the variation in @xmath59 appears to be sufficient to remove the bump in @xmath60 when divided by three factors of @xmath59 to obtain the definition of @xmath68 . also , shown are the effects of intraband scattering with @xmath183 for the dashed curve and @xmath184 for the dot - dashed . scattering in the @xmath50-band reduces its contribution and provides an impurity tail at low @xmath4 , as found for the one - band case . however , scattering in the @xmath49-band , while lowering @xmath60 near @xmath22 as expected , does not appear to add weight at low @xmath4 . this is because the parameters for mgb@xmath0 heavily weight the @xmath50-band and the @xmath49-band is a small component . thus , upon comparison between @xmath49- and @xmath50-band scattering , @xmath68 could be lowered at @xmath185 , for example , by putting impurities in the @xmath50-band , but it would be raised if the impurity scattering is in the @xmath49-band . the dotted curve in the figure is for pure mgb@xmath0 but where we have taken @xmath186 to mimic a case where transport may happen along the c - axis . in this instance , the bump in @xmath60 remains , but is gone in @xmath68 . we see that @xmath68 is large due to the higher power of the fermi velocity ratio that enters the calculation , and , as a result , the nonlinearity is greatly increased . as @xmath68 is a relevant quantity for microwave filter design , this study provides some insight into which factors may be used to optimize the material and reduce the nonlinear effects . study of nonlinear current response is important for device applications and for providing fundamental signatures of order parameter symmetry , such as have been examined in the case of d - wave superconductivity . in this paper , we have considered the case of two - band superconductors . in so doing , we also reexamined the one - band case and discovered that there can exist extra nonlinearity at both low and high temperatures due to strong electron - phonon coupling , for which we have provided strong - coupling correction formulas , whereas the excess nonlinearity induced by impurity scattering occurs primarily at low temperatures . at @xmath22 , impurities will give an enhanced or decreased contribution depending on the particular nonlinear coefficient discussed . in this paper , we have examined two nonlinear coefficients defined in the literature , one due to xu et al.@xcite and one defined by dahm and scalapino , with an emphasis on the latter as it is related to the intermodulation power in microstrip resonators@xcite . we have also studied issues associated with dimensionality motivated by the two - band superconductor mgb@xmath0 , which has a two - dimensional @xmath49-band and a three - dimensional @xmath50-band . within our one - band calculation , aside from an overall factor of 2/3 , we find little difference in the temperature variation of the nonlinear coefficient in mean - field between 2d and 3d . this is further reduced by strong coupling effects . for two - band superconductors , we show that for nearly decoupled bands a strong signature of the small gap @xmath50-band will appear in the nonlinear coefficients , but with increased interband coupling or interband scattering , such a signature will rapidly disappear . likewise , intraband impurities in the @xmath50-band will wash out the temperature variation of the @xmath50-band , whereas the intraband impurities in the @xmath49-band largely effect the nonlinearity at higher temperatures above the energy scale of the @xmath50-band , for the parameters typical to mgb@xmath0 . we provide a prediction for the nonlinear coefficient in mgb@xmath0 using the parameters set by band structure calculations . as the bands in mgb@xmath0 are quite integrated , we find that the nonlinear coefficient @xmath68 is monotonically increasing in contrast to a previous prediction , which was based on a number of approximations,@xcite and we find that the increased nonlinearity due to the @xmath50-band is best reduced at @xmath185 by adding impurities to the @xmath50-band . should the supercurrent sample the c - axis direction , a larger anisotropy in the fermi velocity ratio between the bands would result and this effect is found to increase the nonlinearity . finally , several simple formulas have been provided for near @xmath22 which aid in the understanding of the range of behavior observed in the numerical calculations . we await experimental verification of our predictions . we thank dr . ove jepsen for supplying us with the mgb@xmath0 electron - phonon spectral functions . djs would also like to acknowledge useful discussions with thomas dahm . ejn acknowledges funding from nserc , the government of ontario ( prea ) , and the university of guelph . jpc acknowledges support from nserc and the ciar . djs acknowledges nsf support under grant no . dmr02 - 11166 . this research was supported in part by the national science foundation under grant no . phy99 - 07949 and we thank the hospitality of the kitp , where this work was initiated . a. maeda , y. iino , t. hanaguri , n. motohira , k. kishio , and t. fukase , phys . lett . * 74 * , 1202 ( 1995 ) ; a. maeda , t. hanaguri , y. iino , s. matsuoka , y. kokata , j. shimoyama , k. kishio , h. asaoka , y. matsushita , m. hasegawa , and h. takei , j. phys . jpn . * 65 * , 3638 ( 1996 ) ; a. bhattacharya , i. zuti , o.t . valls , a.m. goldman , u. welp , and b. veal , phys . lett . * 82 * , 3132 ( 1999 ) ; a. bhattacharya , i. zuti , o.t . valls , a.m. goldman , phys . * 83 * , 887 ( 1999 ) ; c.p . bidinosti , w.n . hardy , d.a . bonn , and r. liang , phys . rev . lett . * 83 * , 3277 ( 1999 ) ; a. carrington , r.w . giannetta , j.t . kim , and j. giapintzakis , phys . b * 59 * , r14173 ( 1999 ) . james c. booth , k.t . leung , sang young lee , j.h . lee , b. oh , h.n . lee , and s.h . moon , supercond . * 16 * , 1518 ( 2003 ) ; g. lamura , a.j . purnell , l.f . cohen , a. andreone , f. chiarella , e. di gennaro , and r. vaglio , appl . phys . lett . * 82 * , 4525 ( 2003 ) . for example , yu.a . nefyodev , a.m. shuvaev , and m.r . trunin , cond - mat/0509244 ; v. guritanu , w. goldacker , f. bouquet , y. wang , r. lortz , g. goll , and a. junod , phys . b * 70 * , 184526 ( 2004 ) ; etienne boaknin , m.a . tanatar , johnpierre paglione , d. hawthorn , f. ronning , r.w . hill , m. sutherland , louis taillefer , jeff sonier , s.m . hayden , and j.w . brill , phys . rev . lett . * 90 * , 117003 ( 2003 ) . for example , m.a . tanatar , johnpierre paglione , s. nakatsuji , d.g . hawthorn , e. boaknin , r.w . hill , f. ronning , m. sutherland , louis taillefer , c. petrovic , p.c . canfield , and z. fisk , phys . lett . * 95 * , 067002 ( 2005 ) ; p.m.c . rourke , m.a . tanatar , c.s . turel , j. berdeklis , c. petrovic , and j.y.t . wei , phys . lett . * 94 * , 107005 ( 2005 ) .

we have calculated the nonlinear current of a number of single band s - wave electron - phonon superconductors . among issues considered were those of dimensionality , strong electron - phonon coupling , impurities , and comparison with bcs . for the case of two bands , particular attention is paid to the role of anisotropy , the integration effects of the off - diagonal electron - phonon interaction , as well as inter- and intraband impurities . for the specific case of mgb@xmath0 , we present results based on the known microscopic parameters of band theory .

cond-mat0510250

for proper , locally flat embeddings in @xmath1 , it is well - known that : 1 . [ rlt ] a ray (= copy of @xmath2 ) knots if and only if @xmath3 . [ hlt ] a hyperplane (= copy of @xmath4 ) knots if and only if @xmath3 . both facts hold in the smooth , piecewise linear , and topological categories @xcite . fox and artin discovered the first knotted ray @xcite . the boundary of a closed regular neighborhood of any knotted ray is a knotted hyperplane . for @xmath5 , fact [ hlt ] is the cantrell - stallings hyperplane unknotting theorem , an enhancement of the famous schoenflies theorem of mazur and brown @xcite , . embeddings in @xmath1 , @xmath6 , of at most countably many rays or hyperplanes were recently classified by king , siebenmann , and the first author @xcite . in @xmath0 , no classification is known or even conjectured . + a ray or multiray @xmath7 is _ unknotted _ if and only if an automorphism of @xmath0 carries @xmath8 to a union of radial rays . unknotted multirays with the same number of components are ambient isotopic ( * ? ? ? * lemma 4.1 ) . + a rich collection of knotted rays may be obtained from wild arcs . let @xmath9 be an arc with one wild (= non - locally flat ) point @xmath10 . consider @xmath11 in @xmath12 . if @xmath10 is an endpoint of @xmath13 , then @xmath8 is a knotted ray . if @xmath10 is an interior point of @xmath13 , then @xmath8 is a knotted , two component multiray . hence , in @xmath0 : 1 . there exist infinitely many knot types of a ray @xcite . 2 . there exist uncountably many knot types of a ray @xcite . 3 . there exist uncountably many knot types of two component multirays with unknotted components @xcite . a three component multiray @xmath7 will be called _ borromean rays _ provided @xmath8 is knotted , but any two components of @xmath8 form an unknotted multiray . debrunner and fox constructed an example equivalent to borromean rays @xcite . earlier , doyle attempted a construction @xcite , but his argument contained a gap @xcite . we prove that there exist uncountably many knot types of borromean rays . the following is an overview . + consider the four blocks in figure [ four_blocks ] . , @xmath14 , @xmath15 , and @xmath16 . each block is a three component tangle in a thickened @xmath17-sphere . the set of these four blocks is denoted by @xmath18 . ] the block @xmath19 consists of a three component tangle @xmath20 in a thickened @xmath17-sphere @xmath21}$ ] . any two components of @xmath20 can be straightened by an ambient isotopy of @xmath21}$ ] relative to boundary . however , no diffeomorphism of @xmath21}$ ] sends @xmath20 to a radial tangle ( corollary [ anottrivial ] ) . the blocks @xmath14 , @xmath15 , and @xmath16 are reflections of @xmath19 . let @xmath22 be the set of these four blocks . let @xmath23 , @xmath24 , be a sequence of blocks in @xmath18 . the _ infinite concatenation _ @xmath25 is obtained by gluing the inner boundary sphere of @xmath26 to the boundary of a @xmath27-disk , and gluing the inner boundary sphere of @xmath28 to the outer boundary sphere of @xmath23 for each @xmath24 . this yields the pair : @xmath29 where @xmath30 is a three component multiray . each such @xmath31 forms borromean rays ( corollary [ borr_blocks_yield_borr_rays ] ) . let @xmath32 be the borromean rays determined by another such sequence @xmath33 , @xmath24 . we prove that if @xmath34 is a diffeomorphism of pairs , then there is an isotopy of @xmath35 to a diffeomorphism @xmath36 and an integer @xmath37 such that : @xmath38 hence , the existence of @xmath35 boils down to : ( i ) the tails of the sequences @xmath23 and @xmath33 , and ( ii ) possible diffeomorphisms between individual blocks in @xmath18 . the latter are studied in section [ s : diffeo_blocks ] . our main result , theorem [ borr_rays_thm ] , gives necessary and sufficient conditions for two such sequences to yield equivalent borromean rays . care is taken to account for orientation . as an application , we give necessary and sufficient conditions for our borromean rays to be achiral ( corollary [ chiral_cor ] ) . while most turn out to be chiral , we give a countably infinite family of pairwise inequivalent , achiral borromean rays . + the notion of an _ irreducible block _ plays a central role . a block @xmath39 is _ irreducible _ provided : if @xmath39 is diffeomorphic to a concatenation @xmath40 , then @xmath26 or @xmath41 is diffeomorphic to a _ trivial block _ (= block with a radial tangle ) . trivial blocks are irreducible ( proposition [ sigma_en ] ) . we use this fact to prove that each @xmath31 in forms borromean rays . the block @xmath19 is also irreducible ( theorem [ a_irred ] ) , although the proof is more technical . thus , blocks in @xmath18 are irreducible . this fact is used to improve diffeomorphisms as in . + we are unaware of a general method for detecting irreducibility . for instance , let @xmath26 and @xmath41 be blocks containing @xmath37 component tangles @xmath42 and @xmath43 respectively . let @xmath31 be the tangle in the concatenation @xmath40 . let @xmath44 , @xmath45 , and @xmath46 be the fundamental groups of @xmath47 , @xmath48 , and @xmath49 respectively . let @xmath50 be the @xmath17-sphere where @xmath26 and @xmath41 meet in @xmath40 . then , @xmath51 is an @xmath37-punctured sphere and @xmath52 is free of rank @xmath53 . using dehn s lemma and the loop theorem @xcite , one may show that the inclusions @xmath54 induce injective homomorphisms on fundamental groups . by van kampen s theorem , @xmath55 is the free product of @xmath44 and @xmath45 amalgamated over @xmath56 ( see @xcite ) . by grushko s theorem @xcite , the rank of the free product @xmath57 equals @xmath58 . thus , one might hope that @xmath59 . however , no such relation holds in general for free products with amalgamation @xcite . still , rank behaves better when the amalgamating subgroup is malnormal in each factor @xcite , @xcite . for knot groups , malnormality of the peripheral subgroups was studied recently by weidmann @xcite and de la harpe and weber @xcite . it is unclear to us whether @xmath60 is malnormal in @xmath44 for an arbitrary block @xmath26 . it would be interesting to find block invariants sensitive to irreducibility . + we discovered the block @xmath19 as follows . consider a three component multiray @xmath30 with the property : 1 . [ two_comps_std ] any two components of @xmath31 form an unknotted multiray . let @xmath61 , @xmath62 , denote the components of @xmath31 . property [ two_comps_std ] implies that for each pair @xmath61 and @xmath63 of components of @xmath31 there is a _ (= properly embedded copy of @xmath65\times[0,\infty)$ ] ) whose _ stringers _ (= @xmath66 and @xmath67 ) equal @xmath61 and @xmath63 . the interior of @xmath68 probably intersects the third component of @xmath31 ( if not , then @xmath31 is unknotted ) . using a small regular neighborhood of @xmath31 , one may twist these strips about their stringers so that they patch together to form a general position immersion @xmath69 . let @xmath70 , @xmath62 , be equally spaced points in @xmath71 . the immersion @xmath35 sends the radial ray @xmath72 to @xmath61 for each @xmath62 and is an embedding on each of the three closed sectors between such radial rays . in an attempt to unknot @xmath31 , one may try to eliminate multiple points of @xmath35 . certain types of multiple points can be eliminated . however , difficulties arise from essential circles of double points . figure [ immersion ] displays the relevant , compact part of a simple configuration where two essential circles of double points are identified under @xmath35 . from the compact , @xmath17-dimensional annulus @xmath73 into the thickened sphere @xmath21}$ ] . the image of @xmath35 is a torus sitting atop an annulus and containing the tangle @xmath20 . ] here , the domain of @xmath35 is the compact , @xmath17-dimensional annulus @xmath73 containing three radial arcs . the image of these three radial arcs is the tangle @xmath20 in the block @xmath19 . having found @xmath19 , a fundamental group calculation shows that @xmath19 is not a trivial block ( see section [ equiv_rel_blocks ] ) . our proof that @xmath19 is irreducible ( see section [ ab_irred ] ) makes essential use of the immersion @xmath35 in figure [ immersion ] . + after we discovered the block @xmath19 , we found debrunner and fox s _ mildly wild @xmath27-frame _ @xcite , @xcite and doyle s attempted example @xcite . we pause to make some observations on these two examples . 1 . debrunner and fox s mildly wild @xmath27-frame @xmath74 is compact , periodic , and contains one wild point @xmath75 . in our notation , their building block is the concatenation @xmath76 . put @xmath77 in @xmath78 . we define the _ debrunner - fox borromean rays _ to be @xmath79 in @xmath80 . in our notation : @xmath81 by corollary [ chiral_cor ] below , @xmath82 is achiral . debrunner and fox s proof that @xmath77 is wild hinges on showing a certain group is not finitely generated . their approach yields a mildly wild @xmath37-frame for each @xmath83 . on the other hand , it is not clear how one can use it to distinguish between two wild @xmath27-frames . in section [ irred_blocks ] , we use irreducibility of trivial blocks to prove our multirays ( including @xmath82 ) are knotted . then , we use irreducibility of blocks in @xmath18 to distinguish between multirays . bing showed that doyle s @xmath27-frame is standard @xcite , though bing s argument is not indicated . lemma [ multiray_straightening ] below is useful for recognizing unknotted multirays and applies to doyle s @xmath27-frame . this paper is organized as follows . section [ definitions ] presents conventions and notation , introduces blocks ( including several examples ) , and proves some basic properties concerning blocks . section [ s : diffeo_blocks ] studies diffeomorphisms between individual blocks in @xmath18 . section [ irred_blocks ] introduces irreducible blocks , proves trivial blocks are irreducible , deduces some corollaries , and constructs infinitely many irreducible blocks containing two component tangles . section [ ball_arc_pairs ] identifies some unknotted ball - arc pairs in blocks . section [ ab_irred ] proves that blocks in @xmath18 are irreducible . section [ improve_spheres ] simplifies certain spheres in concatenations of borromean blocks and deduces two useful corollaries . section [ brbh ] classifies borromean rays arising from sequences of blocks in @xmath18 and then uses regular neighborhoods of multirays to obtain results on knotted multiple hyperplane embeddings . in particular , we prove that there exist uncountably many pairwise inequivalent so - called _ borromean hyperplanes _ in @xmath0 . we work in the smooth (= @xmath84 ) category . throughout , @xmath85 denotes diffeomorphism of manifolds or manifold pairs . a map is * proper * provided the inverse image of each compact set is compact . all isotopies will be smooth and proper . a submanifold @xmath86 is * neat * provided @xmath87 and this intersection is transverse @xcite , @xcite . + a * ray * is a proper embedding of @xmath2 . a * multiray * is a proper embedding of @xmath88 where @xmath89 is a finite or countably infinite discrete space . indeed , each embedded submanifold of @xmath1 contains at most countably many components since @xmath1 is a separable metric space . a ray in @xmath1 is * radial * provided either it is straight and emanates from the origin , or it is contained in such a ray . in particular , a radial ray can meet the origin only at its endpoint . a collection of intervals embedded in @xmath1 is * radial * provided each component lies in a radial ray . + the standard euclidean norm on @xmath1 is @xmath90 . on @xmath0 , the euclidean norm function will be denoted : @xmath91 ^ -{\eta } & \mathbb{r}\\ x \ar@{|-{>}}[r ] & \left\|x\right\| } \end{split}\ ] ] all lengths come from the standard euclidean metric . the unit @xmath37-disk @xmath92 consists of all points @xmath93 such that @xmath94 . the unit @xmath95-sphere is @xmath96 . the sphere of radius @xmath97 in @xmath1 about @xmath98 is denoted @xmath99}$ ] and is called a * level sphere*. in particular , @xmath100}$ ] . let @xmath101}$ ] , where @xmath102 , denote the thickened sphere of points @xmath93 such that @xmath103 . in particular , @xmath104}$ ] equals the disjoint union of the spheres @xmath105}$ ] and @xmath106}$ ] . let @xmath107 denote the half - infinite annulus of points @xmath93 such that @xmath108 . let @xmath109}$ ] , where @xmath102 , be a thickened sphere in @xmath0 . a * tangle * @xmath31 is an embedding of the disjoint union of @xmath110 copies of @xmath65 $ ] as a neat submanifold of @xmath109}$ ] . if @xmath111 is a component of @xmath31 , then the initial point @xmath112 of @xmath111 must lie in @xmath113}$ ] and the terminal point must equal @xmath114}$ ] . so , @xmath111 stretches between the two boundary @xmath17-spheres of @xmath115}$ ] , and its initial and terminal points lie on a radial ray . + a * block * is a pair @xmath116},\tau)$ ] where @xmath31 is a tangle . each block is oriented : @xmath115}$ ] inherits its orientation from the standard one on @xmath0 , and each component of @xmath31 is oriented to point out from the inner boundary @xmath17-sphere . a * diffeomorphism * of blocks is any diffeomorphism of the corresponding pairs of spaces , not necessarily orientation or boundary preserving in any sense . + a * trivial block * is any pair @xmath117 consisting of an @xmath37 component , radial tangle in a thickened sphere ( see figure [ trivial_block ] ) . + containing a three component , radial tangle . ] by our convention , in every displayed block , the positive @xmath118-axis points horizontally to the right , the positive @xmath119-axis points vertically up , and the positive @xmath120-axis points out of the page towards the reader . + given any block @xmath121},\tau)$ ] , define two blocks : 1 . @xmath122},\overline{\tau})$ ] is the reflection of @xmath39 across the @xmath123-plane . @xmath124},\tau^{\ast})$ ] is the inversion of @xmath39 across @xmath125}$ ] . if @xmath126},\tau)$ ] , then inversion is @xmath127 . components of @xmath128 are still oriented out from the inner boundary @xmath17-sphere . evidently , the bar and star operations commute and are involutions : @xmath129 figure [ four_blocks ] above introduced four blocks important for our purposes . let : @xmath130 be the set of these four blocks . note that @xmath18 is closed under the bar and star operations . by construction , blocks in @xmath18 are pairwise diffeomorphic . they are pairwise distinct , though , up to finer equivalence relations , as explained below . the coarsest equivalence relation on blocks we consider is that of _ diffeomorphism _ ( defined above ) . finer diffeomorphism relations , involving orientation and/or boundary preservation , arise in section [ s : diffeo_blocks ] . on blocks with the same underlying thickened spheres , the finest relation we consider ( short of equality ) is that of * ambient isotopy relative to boundary * , meaning ambient isotopy of tangles fixing both boundary @xmath17-spheres pointwise at all times . [ one_comp ] any block @xmath131},\tau\right)}$ ] , where @xmath31 has one component , is ambient isotopic relative to boundary to @xmath132 . this fact is well - known ( e.g. , it s an exercise in rolfsen @xcite ) . we are not aware of a published proof , so we sketch one . all isotopies are ambient and relative to boundary . by a preliminary isotopy ( left to the reader ) , we assume @xmath39 appears as in figure [ knot_block_e1 ] where @xmath133 is a crossing diagram in general position . with a one component tangle @xmath31 . ] it suffices to prove that crossings of @xmath133 may be switched by isotopy , since then we may arrange that @xmath133 has monotonic @xmath120-coordinate and the result follows . so , consider a crossing @xmath134 of @xmath133 with over arc @xmath135 and under arc @xmath136 . push @xmath135 and @xmath136 sufficiently close together in the @xmath120-direction . let @xmath10 and @xmath137 be the midpoints of @xmath135 and @xmath136 respectively , where @xmath10 lies directly above @xmath137 . let @xmath138 $ ] denote the subarc of @xmath31 where @xmath139 and @xmath140 . let @xmath141 $ ] if @xmath142 $ ] , and let @xmath143 $ ] if @xmath144 $ ] . in other words , @xmath145 is the unique subarc of @xmath31 originating on @xmath146 and terminating at the first point , @xmath10 or @xmath137 , encountered by @xmath145 . assume @xmath141 $ ] ( otherwise , flip the picture over ) . isotop @xmath137 close to @xmath146 by following just underneath @xmath145 and stretching @xmath136 . then , loop @xmath136 under @xmath146 and isotop @xmath137 back ( again using @xmath145 as a guide ) to lie above @xmath10 . the crossing @xmath134 has been switched , completing the proof . [ knots_tied ] example [ one_comp ] has the following possibly surprising corollary , which appears to be due to wilder @xcite . if the ray @xmath7 is obtained by tying successive knots in a radial ray ( see figure [ knots_ray ] ) , obtained by tying successive knots in a radial ray . ] then @xmath8 is ambient isotopic to a radial ray . proof : let @xmath147}$ ] for each @xmath148 . simultaneously apply the straightening process from example [ one_comp ] to each @xmath149}$ ] . @xmath150 [ knot_blocks ] let @xmath151 be a knot . let @xmath152 be a @xmath27-disk such that : ( i ) @xmath153 is a neatly embedded arc in @xmath77 , and ( ii ) @xmath154 is an unknotted ball - arc pair . let @xmath155 be a small , round @xmath27-disk meeting one tangle component in an arc . the * knot block * @xmath156 is obtained from @xmath157 by replacing @xmath158 with @xmath159 as in figure [ knot_block ] . where @xmath133 is a diagram yielding @xmath160 . ] in general , @xmath156 is well - defined up to diffeomorphism . if @xmath160 itself is oriented , then one could define @xmath156 more carefully . let @xmath31 be the tangle in a knot block @xmath156 . evidently , deleting the boundary from @xmath161 yields @xmath162 . in particular , @xmath163 implies @xmath164 . so , knots with nonisomorphic groups ( e.g. , torus knots @xcite ) yield nondiffeomorphic knot blocks . finally , let @xmath131},\tau\right)}$ ] be a block where @xmath31 has two components , @xmath42 and @xmath43 . then , @xmath39 is ambient isotopic relative to boundary to some knot block . to see this , straighten @xmath43 using the process in example [ one_comp ] . then push @xmath42 away from @xmath43 by integrating a suitable vector field tangent to level spheres . [ block_d ] consider the block @xmath77 in figure [ one_twist ] . . ] the thickened sphere underlying @xmath77 is @xmath21}$ ] , and @xmath77 is obtained from @xmath165 by fixing the inner boundary @xmath17-sphere pointwise and rigidly rotating the outer boundary sphere one revolution about the @xmath118-axis . in particular , @xmath166 , in fact by an orientation preserving diffeomorphism that is pointwise the identity on both boundary @xmath17-spheres . on the other hand , @xmath165 and @xmath77 are not ambient isotopic relative to boundary , as proved by newman @xcite and fadell @xcite ( see also @xcite and @xcite ) . if @xmath77 had been obtained from @xmath165 by two complete twists , rather than just one , then @xmath77 would have been ambient isotopic relative to boundary to @xmath165 by dirac s belt trick . a diffeomorphism between thickened spheres is : ( i ) * radial * provided it sends radial arcs to radial arcs , and ( ii ) * level * provided it sends level @xmath17-spheres to level @xmath17-spheres . the next lemma says that there is essentially just one trivial block @xmath117 for each @xmath167 . [ en_unique ] let @xmath168},r)$ ] and @xmath169},r')$ ] be trivial blocks . then , there is a radial , level , orientation preserving diffeomorphism @xmath170 sending @xmath171}$ ] to @xmath172}$ ] for @xmath173 . if @xmath174}={\left[t'_1,t'_2\right]}$ ] , then there is an ambient isotopy @xmath175 , @xmath176 , of @xmath115}$ ] such that : 1 . [ eh0id ] @xmath177 and @xmath178 . [ h_tradial ] @xmath175 is a radial , level diffeomorphism for all @xmath176 . let @xmath179}\to{\left[t'_1,t'_2\right]}$ ] be the unique affine , orientation preserving diffeomorphism . then , @xmath180 is a radial , level diffeomorphism @xmath115}\to s^2{\left[t'_1,t'_2\right]}$ ] . so , it suffices to consider the case @xmath174}={\left[t'_1,t'_2\right]}=[1,2]$ ] . let @xmath181 , @xmath182 , denote the initial points of the components @xmath183 of @xmath8 . define @xmath184 similarly for @xmath185 . let @xmath186 be a smooth , simple path from @xmath187 to @xmath188 in @xmath146 . let @xmath189 be a smooth regular neighborhood of @xmath186 in @xmath146 . there is an ambient isotopy of @xmath146 , with support in @xmath189 , carrying @xmath187 to @xmath188 . for instance , begin with a suitable nonzero tangent vector field to @xmath186 , extend to a vector field @xmath190 on @xmath146 that vanishes outside of @xmath189 , and then integrate @xmath190 ( cf . extend this isotopy radially to get an ambient isotopy of @xmath21}$ ] carrying @xmath191 to @xmath192 . any component @xmath183 , @xmath193 , that moved during this isotopy is still radial and is still denoted @xmath183 . repeat this procedure , while choosing @xmath194 disjoint from @xmath195 . to distinguish @xmath19 from @xmath165 up to diffeomorphism , it suffices to distinguish the fundamental groups of their tangle complements up to isomorphism . presentations of such groups are obtained using wirtinger s algorithm . consider the diagram of @xmath19 in figure [ wirt_a ] . with oriented and labeled arcs . ] as usual , labels of arcs correspond to generators of @xmath196 . the basepoint is above the page . the based loop representing a generator @xmath197 first penetrates the plane of the page at a point just to the right of the oriented arc labeled @xmath197 , and has linking number @xmath198 with this oriented arc . a presentation of @xmath199 is : @xmath200 each crossing of @xmath20 contributes a relation . the last relation is evident topologically . vertex relation _ for the fundamental group of the complement of the graph obtained by crushing the inner boundary @xmath17-sphere to a point , , . the outer vertex relation , @xmath201 , is redundant . + for the trivial tangle , @xmath202 is free of rank @xmath17 . to distinguish @xmath196 from @xmath203 , we count their _ classes _ of homomorphisms into small symmetric groups @xmath204 using the computer algebra system magma ( a finite problem ) . two homomorphisms @xmath205 are considered _ equivalent _ provided there exists @xmath206 such that @xmath207 for all @xmath208 . table [ magma_data_1 ] collects this data . .numbers of classes of homomorphisms into @xmath204 . [ cols="^,^,^",options="header " , ] [ non_exist_types ] let @xmath209 be blocks in @xmath18 . then , the type @xmath210 is not in the image of @xmath211 . suppose otherwise . then , there is a diffeomorphism @xmath212 of type @xmath210 . + let @xmath213 denote the @xmath17-sphere @xmath146 with three marked points @xmath214 . the mapping class group @xmath215 is the group of orientation preserving diffeomorphisms of @xmath146 that send @xmath216 , modulo isotopies of @xmath146 fixing @xmath187 , @xmath217 , and @xmath218 at all times @xcite . recall that the natural map @xmath219^{\cong } & { \textnormal{sym}}(3)}\ ] ] which sends an element of @xmath215 to its action on the three marked points , is an isomorphism @xcite . + by hypothesis , @xmath220 preserves boundary @xmath17-spheres componentwise and preserves orientation of @xmath21}$ ] . so , @xmath221 is an orientation preserving diffeomorphism . as the tangle permutation of @xmath220 is the identity , permits us to assume @xmath220 is the identity on inner boundary @xmath17-spheres . pasting together the diffeomorphisms @xmath222 and @xmath212 yields a diffeomorphism @xmath223 . this contradicts the last row of table [ magma_data_2 ] since @xmath209 . consider two types @xmath224 such that @xmath225 is defined . as the image of @xmath211 is a subgroupoid of @xmath226 , if any two of @xmath135 , @xmath136 , or @xmath225 lie in the image of @xmath211 , then the third does as well . therefore , if @xmath135 is any type forbidden by lemma [ non_exist_types ] , @xmath136 is any of the already realized @xmath227 types in the image of @xmath211 , and @xmath225 is defined in @xmath226 , then @xmath225 is not in the image of @xmath211 . a tedious , but completely straightforward calculation ( facilitated by magma ) , shows that this yields @xmath228 types not in the image of @xmath211 . this completes our proof of table [ hom_block ] , and yields the following . [ blocks_inequiv_isotopy ] the blocks @xmath19 , @xmath14 , @xmath15 , and @xmath16 are pairwise distinct up to ambient isotopy relative to boundary . this section introduces the notion of an _ irreducible block_. such blocks play a central role in our construction of borromean rays . first , recall the following standard definitions and accompanying lemma . + a @xmath17-sphere @xmath50 in a thickened sphere is * essential * provided it does not bound a @xmath27-disk in the thickened sphere . otherwise , @xmath50 is * inessential*. let @xmath13 be an arc transverse to a @xmath17-manifold @xmath50 in a @xmath27-manifold . then , @xmath229 denotes the number of points in @xmath230 ( ignoring any orientations ) . the * mod @xmath17 intersection number * of @xmath13 and @xmath50 , denoted @xmath231 , is @xmath232 . [ ess_sphere ] let @xmath13 be a neatly embedded arc in @xmath115}$ ] . assume @xmath13 has one boundary point in @xmath113}$ ] and the other in @xmath233}$ ] . let @xmath50 be a @xmath17-sphere embedded in the interior of @xmath115}$ ] and transverse to @xmath13 . the following are equivalent : ( i ) @xmath50 is essential in @xmath115}$ ] , ( ii ) @xmath234 , and ( iii ) there is a neighborhood @xmath235 of @xmath236}$ ] in @xmath115}$ ] and an ambient isotopy @xmath175 , @xmath237 , of @xmath115}$ ] such that : 1 . @xmath177 . @xmath238 for all @xmath176 . 3 . @xmath239 is a level @xmath17-sphere in @xmath240}}$ ] . assume , without loss of generality , that @xmath174}=[1,2]$ ] . let @xmath77 denote , the @xmath27-disk of radius @xmath17 . by the @xmath27-dimensional schoenflies theorem ( * ? iii ) , @xcite , ( * ? ? ? 1.1 ) , @xmath50 bounds a unique @xmath27-disk @xmath241 . let @xmath242 . so , @xmath243 and @xmath244.case 1 . then , @xmath246 lies in the interior of @xmath247 $ ] . so , @xmath50 is inessential and clearly @xmath248.case 2 . @xmath249 then , @xmath50 can not bound a @xmath27-disk in @xmath247 $ ] and , hence , is essential in @xmath247 $ ] . the arc @xmath13 has one boundary point in @xmath250 and one outside @xmath246 , so @xmath251 . let @xmath252 be a level @xmath17-sphere between @xmath50 and @xmath253}$ ] . let @xmath158 be the @xmath27-disk in @xmath254 with boundary @xmath252 . let @xmath255 be the compact region in @xmath254 with boundary @xmath256 . by uniqueness of disk embeddings @xcite , there is an ambient isotopy of @xmath158 carrying @xmath246 to a round @xmath27-disk . hence , there is a diffeomorphism @xmath257 $ ] sending @xmath50 to @xmath258 . construct a vector field @xmath190 on @xmath247 $ ] as follows . on @xmath259 , @xmath190 is the pushforward by @xmath260 of the constant vector field @xmath261 on @xmath262 $ ] . extend @xmath190 to the rest of @xmath21}$ ] , making it @xmath98 outside a small neighborhood of @xmath259 . the isotopy generated by @xmath190 is the desired @xmath175 . next , we give two definitions of * irreducible block * and then we prove they are equivalent . let @xmath121},\tau)$ ] be a block where @xmath31 has @xmath110 components . [ irred_def_1 ] the block @xmath39 is * irreducible * provided : if @xmath39 is diffeomorphic to a concatenation of blocks @xmath40 , then @xmath26 or @xmath41 ( or both ) is diffeomorphic to a trivial block @xmath117 . [ irred_def_2 ] let @xmath50 be a @xmath17-sphere embedded in the interior of @xmath115}$ ] and transverse to @xmath31 . assume @xmath50 meets each component of @xmath31 at exactly one point . the block @xmath39 is * irreducible * provided : there is a neighborhood @xmath235 of @xmath236}$ ] in @xmath115}$ ] and an ambient isotopy @xmath175 , @xmath237 , of @xmath115}$ ] such that : 1 . [ h0id ] @xmath177 . [ hrelu ] @xmath238 for all @xmath176 . [ r_setwise ] @xmath175 fixes @xmath31 setwise for all @xmath237 . 4 . @xmath239 is a level @xmath17-sphere in @xmath115}$ ] . @xmath263 has no critical points between @xmath239 and @xmath113}$ ] , or between @xmath239 and @xmath233}$ ] . [ defs_equiv ] the two definitions of irreducible block are equivalent . assume @xmath39 is irreducible according to the second definition . suppose @xmath264 is a diffeomorphism . assume @xmath220 sends the inner boundary @xmath17-sphere to the inner boundary @xmath17-sphere , the other case being similar . let @xmath252 be the level @xmath17-sphere in @xmath40 along which the concatenation takes place . let @xmath265 . then , @xmath50 satisfies the hypotheses of the second definition . let @xmath175 be the isotopy provided by the second definition . so , @xmath239 is a level @xmath17-sphere in @xmath115}$ ] and , say , @xmath263 has no critical points in the compact region @xmath259 with boundary @xmath266}$ ] . by corollary [ trivial_block_cor ] , the block @xmath267 is diffeomorphic to a trivial block @xmath117 . hence , @xmath268 , as desired . + next , assume @xmath39 is irreducible according to the first definition . let @xmath50 satisfy the hypotheses of the second definition . by lemma [ ess_sphere ] , @xmath50 is essential in @xmath115}$ ] , and there is an isotopy of @xmath115}$ ] ( probably disturbing @xmath31 ) which carries @xmath50 to a level @xmath17-sphere , @xmath252 . let @xmath269 be the image of @xmath31 under this isotopy . this @xmath252 divides @xmath116},\tau')$ ] into two obvious blocks @xmath26 and @xmath41 . evidently , @xmath39 is diffeomorphic to the concatenation @xmath40 . the first definition of irreducible block implies that , say , @xmath26 is diffeomorphic to @xmath117 . by lemma [ en_unique ] ( trivial block uniqueness ) , we can and do assume @xmath117 has underlying thickened sphere @xmath247 $ ] . let @xmath259 denote the compact region in @xmath115}$ ] between @xmath50 and @xmath113}$ ] . hence , there is a diffeomorphism of pairs @xmath270 , and @xmath197 sends @xmath113}$ ] to @xmath146 . let @xmath271}$ ] for some small @xmath272 to be specified . first , choose @xmath135 small enough so @xmath273 lies between @xmath50 and @xmath113}$ ] . next , reduce @xmath135 if necessary so that : 1 . [ nocp ] @xmath263 has no critical points on or between @xmath273 and @xmath113}$ ] . this reduction is possible since @xmath31 is neatly embedded . note that futher reducing @xmath135 maintains condition [ nocp ] . finally , reduce @xmath135 if necessary so that : 1 . [ nottangent ] @xmath274 is nowhere tangent to any radial arc in @xmath21}$ ] . this last reduction is possible since : ( i ) @xmath197 is a diffeomorphism , ( ii ) @xmath275})=s^2 $ ] , and ( iii ) @xmath113}$ ] is compact . as @xmath274 is essential in @xmath21}$ ] , condition [ nottangent ] implies that : 1 . [ onepointtransverse ] each radial arc in @xmath21}$ ] of length @xmath276 intersects @xmath274 in exactly one point and transversely . condition [ onepointtransverse ] permits construction of an ambient isotopy of @xmath21}$ ] that carries @xmath274 to @xmath277}$ ] and merely slides points along radial arcs . so , by an abuse of notation , we further assume the diffeomorphism of pairs @xmath197 itself sends @xmath273 to @xmath277}$ ] . let @xmath278 be the compact region in @xmath115}$ ] between @xmath50 and @xmath273 . construct a vector field @xmath190 on @xmath115}$ ] as follows . on @xmath278 , @xmath190 is the pushforward by @xmath260 of the vector field @xmath279 on @xmath280}$ ] . note that @xmath190 is tangent to @xmath31 on @xmath278 . extend @xmath190 to the rest of @xmath115}$ ] , making it @xmath98 outside a small neighborhood of @xmath278 and ensuring tangency to @xmath31 . the isotopy generated by @xmath190 is the desired @xmath175 . in general , it appears to be a difficult problem to decide whether a given block is irreducible . we prove next that trivial blocks are irreducible and then observe some corollaries . [ sigma_en ] each trivial block @xmath168},\tau)$ ] , @xmath110 , is irreducible . let @xmath281 denote the annulus where @xmath21}$ ] meets the @xmath123-plane . lemma [ en_unique ] reduces us to the case where @xmath117 has underlying thickened sphere @xmath21}$ ] and @xmath31 consists of @xmath37 equally spaced radial arcs in @xmath281 . we prove @xmath117 is irreducible according to the first definition . by the first paragraph of the proof of proposition [ defs_equiv ] , it suffices to consider a @xmath17-sphere , @xmath50 , embedded in the interior of @xmath21}$ ] , transverse to @xmath31 , and intersecting each component @xmath61 of @xmath31 in one point @xmath70 . let @xmath282}$ ] be the compact set between @xmath50 and @xmath146 . it suffices to produce a diffeomorphism @xmath283 . + we will improve @xmath50 ( and , hence , @xmath284 ) by ambient isotopies of @xmath21}$ ] . improved spaces will be denoted by their original names , except @xmath281 always denotes @xmath21 } \cap { \textnormal{($xy$-plane)}}$ ] . so , assume @xmath50 intersects @xmath281 transversely . thus , @xmath285 is a closed @xmath276-manifold and one component , @xmath133 , of @xmath285 must contain all of the points @xmath70 . + if @xmath286 , then consider a component , @xmath134 , of @xmath287 that is innermost in its component of @xmath288 . let @xmath289 be the @xmath17-disk in @xmath288 with boundary @xmath134 . let @xmath290 be the @xmath17-disk in @xmath281 with boundary @xmath134 . then , @xmath291 is an embedded @xmath17-sphere in the interior of @xmath21}$ ] disjoint from @xmath31 . by lemma [ ess_sphere ] , @xmath291 is inessential in @xmath21}$ ] . let @xmath77 be the @xmath27-disk in @xmath21}$ ] with boundary @xmath291 . this @xmath77 permits construction of an isotopy of @xmath21}$ ] , with support near @xmath77 , that carries @xmath289 past @xmath290 to a parallel copy of @xmath290 . thus , @xmath134 ( at least ) has been eliminated from @xmath292 . repeating this operation finitely many times , we get @xmath293 . + now , we give a bootstrapping definition of the required diffeomorphism @xmath197 . first , @xmath197 sends @xmath294 by the identity . second , @xmath197 sends @xmath295 to @xmath61 by an affine diffeomorphism for each @xmath296 . third , @xmath197 sends @xmath133 to @xmath253}\cap\pi$ ] . fourth , @xmath197 sends a smooth , regular neighborhood of @xmath297 in @xmath298 to a smooth , regular neighborhood of @xmath299}\cap\pi\right)}$ ] in @xmath281 . this step may be accomplished , quite concretely , by judiciously choosing ( closed ) collars @xcite and ambiently rounding corners . fifth , @xmath197 sends @xmath298 to @xmath281 . by the smooth @xmath17-dimensional schoenflies theorem ( * remark 9.19 ) , this step evidently requires extension of @xmath197 over @xmath37 smooth @xmath17-disks @xmath300 , @xmath301 . let @xmath302 denote the @xmath17-disk with boundary @xmath303 . as every diffeomorphism of @xmath71 extends to one of @xmath304 , each diffeomorphism @xmath305 extends to a diffeomorphism @xmath306 . extending @xmath197 over @xmath307 by @xmath308 yields a well - defined homeomorphism ( smooth except possibly at @xmath309 ) . by an isotopy of @xmath308 , relative to @xmath309 and with support in a collar of @xmath309 @xcite , this extension is a diffeomorphism . sixth , @xmath197 sends @xmath310}$ ] . seventh , @xmath197 sends a smooth , regular neighborhood of @xmath311 in @xmath284 to a smooth , regular neighborhood of @xmath312}$ ] in @xmath21}$ ] . this is done as in step four ( product a nice corner rounding with @xmath71 ) . finally , @xmath197 sends @xmath284 to @xmath21}$ ] . by the smooth @xmath27-dimensional schoenflies theorem , this step requires extension of @xmath197 over two smooth @xmath27-disks . this is done as in step five , except using the fact that every diffeomorphism of @xmath146 extends to one of @xmath313 @xcite , @xcite , @xcite . by construction , the diffeomorphism @xmath314}$ ] sends @xmath315 to @xmath31 . [ cor1 ] let @xmath23 , @xmath316 , be any blocks . if @xmath317 , then @xmath318 for each @xmath316 . by induction , it suffices to consider the case @xmath319 . the case @xmath319 follows from the proof of proposition [ sigma_en ] , since @xmath284 can be the compact region between @xmath50 and @xmath146 , or between @xmath50 and @xmath253}$ ] . alternatively , it is instructive to see that the case @xmath319 follows from the statement of proposition [ sigma_en ] as follows . proposition [ sigma_en ] implies that @xmath268 or @xmath320 . assume @xmath268 ( the other case is similar ) . we have diffeomorphisms : @xmath321 where the first exists by hypothesis , the second follows from lemma [ diff_impl_diff ] since @xmath268 and @xmath322 , and the last is given by lemma [ e_nidentity ] . [ cor2 ] let @xmath23 , @xmath148 , be any blocks . if @xmath323 , then @xmath324 for each @xmath325 . let @xmath326 be a diffeomorphism ( of pairs ) . let @xmath31 and @xmath8 be the @xmath37-component multirays determined by these concatenations respectively . clearly , @xmath8 is radial . isotopies of @xmath35 will send @xmath31 to @xmath8 at all times . note that @xmath327 since @xmath35 must restrict to a diffeomorphism on the boundaries of the total spaces . let @xmath328}\right)}$ ] . by a radial isotopy , relative to @xmath146 , we may assume @xmath329}}$ ] . this @xmath50 satisfies the hypotheses in the second definition of irreducible block . propositions [ defs_equiv ] and [ sigma_en ] permit us to isotop @xmath35 , relative to @xmath146 , so that @xmath330}$ ] . having isotoped @xmath35 so that @xmath331}\right)}=s^2{\left[i\right]}$ ] for @xmath316 and some @xmath332 , the same argument permits us to further isotop @xmath35 , relative to @xmath333}$ ] , so that @xmath334}\right)}=s^2{\left[k+1\right]}$ ] . evidently , the composition of all of these ( infinitely many ) isotopies is a well - defined , smooth , proper isotopy . so , we can and do assume @xmath335}\right)}=s^2{\left[k\right]}$ ] for all @xmath336 . the result is now immediate . [ cor3 ] let @xmath23 , @xmath148 , be any blocks . if @xmath337 , then @xmath324 for all sufficiently large @xmath24 . let @xmath338 be a diffeomorphism ( of pairs ) . let @xmath31 and @xmath8 be the @xmath37-component multirays determined by these concatenations respectively . isotopies of @xmath35 will send @xmath31 to @xmath8 at all times . by compactness , there exists @xmath332 such that @xmath335}\right)}$ ] is disjoint from @xmath313 . as in the previous proof , we may isotop @xmath35 , relative to @xmath313 , so that @xmath335}\right)}=s^2{\left[k\right]}$ ] . restricting @xmath35 to @xmath339 implies @xmath340 . now , apply the previous corollary . borromean blocks were defined in section [ borr_blocks ] . a multiray @xmath30 forms * borromean rays * provided : ( i ) no diffeomorphism of @xmath0 carries @xmath31 to a radial multiray , and ( ii ) each multiray obtained from @xmath31 by forgetting one component is ambiently isotopic to a two component , radial multiray . [ borr_blocks_yield_borr_rays ] let @xmath23 , @xmath148 , be borromean blocks and let : @xmath341 then , @xmath31 forms borromean rays . in particular , the conclusion holds if each @xmath342 . by definition of borromean block , corollary [ cor3 ] implies that @xmath344 . hence , no diffeomorphism of @xmath0 carries @xmath31 to a radial multiray . next , let @xmath32 be obtained from @xmath31 by forgetting any one component . let @xmath33 be obtained from @xmath23 by forgetting the corresponding tangle component . by definition of borromean block , each @xmath33 is ambient isotopic ( relative to boundary ) to @xmath157 . performing these isotopies , for @xmath24 , simultaneously yields an ambient isotopy of @xmath0 carrying @xmath32 to a radial multiray . thus , @xmath31 forms borromean rays . lastly , blocks in @xmath18 are borromean by corollary [ four_blocks_borromean ] . 1 . corollary [ cor3 ] reduces the _ infinite _ problem of constructing knotted multirays to the _ finite _ problem of constructing nontrivial blocks . for instance , the infinitely generated group theory in @xcite may be replaced by finitely generated group theory ( as used in section [ equiv_rel_blocks ] above ) . 2 . the converse of corollary [ cor3 ] holds by remarks [ concat_remarks ] item [ tail_det_type ] . we close this section by constructing infinitely many irreducible blocks containing two component tangles . recall the notion of a knot block from example [ knot_blocks ] . [ pk_ib ] if @xmath151 is a prime knot , then @xmath156 is an irreducible block . let @xmath345},\tau\right)}$ ] where @xmath43 is radial and @xmath42 contains the diagram @xmath133 for @xmath160 as in figure [ knot_block ] . let @xmath50 be a @xmath17-sphere embedded in the interior of @xmath21}$ ] , transverse to @xmath31 , and intersecting each component @xmath61 of @xmath31 in one point @xmath70 . perturb @xmath50 so it coincides with the level sphere through @xmath217 near @xmath43 . all isotopies will be ambient and relative to a neighborhood of both @xmath346 $ ] and @xmath43 . subsets that move will be called by their original names . as in the proof of proposition [ sigma_en ] , we isotop @xmath50 to the level sphere containing @xmath217 . push @xmath42 away from @xmath43 by integrating a vector field tangent to level spheres . the result is shown in figure [ knot_block_comp ] . after ambient isotopy carrying @xmath50 to a level sphere . ] as @xmath160 is prime , one of the diagrams @xmath347 or @xmath348 must be trivial . [ inf_ib ] there exists a countably infinite collection of irreducible knot blocks , pairwise distinct up to diffeomorphism . let @xmath349 denote the set of torus knots @xmath350 where @xmath351 and @xmath352 . as torus knots are prime @xcite , lemma [ pk_ib ] implies that each @xmath156 , @xmath353 , is irreducible . the fundamental groups of these torus knots are pairwise nonisomorphic @xcite . by example [ knot_blocks ] , these knot blocks are pairwise distinct up to diffeomorphism . under concatenation , knot blocks commute , unlike distinct blocks in @xmath18 ( recall table [ magma_data_2 ] ) . concatenating infinitely many knot blocks yields a multiray in @xmath0 known as _ wilder rays_. they were classified by fox and harrold @xcite . this section identifies some unknotted ball - arc pairs in blocks . these tools will be used in the next two sections . recall that a * ball - arc pair * is a pair @xmath354 such that @xmath355 is neatly embedded in @xmath356 . such a pair is * unknotted * provided it is diffeomorphic to the standard pair @xmath357 , and otherwise it is * knotted*. if @xmath151 is a smooth knot ( not the unknot ) and @xmath358 is an unknotted ball - arc pair such that @xmath359 , then @xmath360 is a knotted ball - arc pair . every knotted ball - arc pair arises this way up to diffeomorphism . + [ ball - arc ] let @xmath361},\tau\right)}$ ] . let @xmath50 be a @xmath17-sphere embedded in the interior of @xmath21}$ ] and transverse to @xmath31 . assume @xmath50 meets @xmath31 at exactly two points @xmath10 and @xmath137 , both of which lie on one component of @xmath31 , say @xmath42 . then , @xmath50 bounds a @xmath27-disk , @xmath246 , in @xmath21}$ ] and @xmath362 is an unknotted ball - arc pair . by lemma [ ess_sphere ] , @xmath50 is inessential in @xmath21}$ ] . let @xmath281 denote the annulus where @xmath21}$ ] meets the @xmath123-plane . without loss of generality , @xmath363 and @xmath50 is transverse to @xmath281 . thus , @xmath285 is a closed @xmath276-manifold and one component , @xmath133 , of @xmath285 must contain @xmath10 and @xmath137 ; this is where the hypothesis @xmath364 is used . as in the proof of proposition [ sigma_en ] ( paragraph three ) , we may arrange that @xmath365 . now , it is straightforward to construct the required diffeomorphism ( cf . paragraph four of the proof of proposition [ sigma_en ] ) . 1 . lemma [ ball - arc ] becomes false without the hypothesis @xmath364 ( i.e. , with @xmath157 replaced by @xmath132 ) . to see this , consider the block @xmath131},\tau'\right)}$ ] in figure [ sphere_two_points ] ( left ) . + meeting the tangle @xmath269 at two points in the block @xmath366 . at right is the result of an ambient isotopy that fixes @xmath269 setwise . ] + the indicated sphere @xmath252 meets @xmath269 in two points and bounds the @xmath27-ball @xmath367 . let @xmath133 be any crossing diagram such that @xmath368 is a knotted ball - arc pair . straighten @xmath269 using the argument in example [ one_comp ] . let @xmath31 , @xmath50 , and @xmath246 denote the respective images of @xmath269 , @xmath252 , and @xmath367 under this ambient isotopy . then , @xmath50 is a @xmath17-sphere in @xmath132 meeting @xmath31 in exactly two points and transversely . however , @xmath369 is a knotted ball - arc pair . 2 . lemma [ ball - arc ] and the previous remark may be recast in @xmath0 as follows . consider a @xmath17-sphere @xmath370 . let @xmath371 be the @xmath27-disk with @xmath372 . suppose @xmath13 is a straight arc in @xmath0 that is neatly embedded in @xmath246 . let @xmath373 be the straight line containing @xmath13 . if @xmath50 is disjoint from @xmath374 , then @xmath354 is an unknotted ball - arc pair . if @xmath50 meets @xmath374 , then @xmath354 may be a knotted ball - arc pair . in fact , every knotted ball - arc pair @xmath375 in @xmath0 is ambient isotopic to some such @xmath354 . proof : ( i ) straighten @xmath376 near an endpoint @xmath377 , ( ii ) let @xmath378 be a point in the straightened end of @xmath376 , and ( iii ) ambiently isotop the other endpoint of @xmath376 along @xmath376 until it concides with @xmath379 ( cf . figure [ sphere_two_points ] ( right ) ) . @xmath150 [ disk_ess_sphere ] let @xmath13 be a neatly embedded arc in @xmath115}$ ] . assume @xmath13 has one boundary point in @xmath113}$ ] and the other in @xmath233}$ ] . let @xmath289 be a @xmath17-disk embedded in the interior of @xmath115}$ ] . assume that @xmath380 lies in some @xmath381}$ ] , @xmath134 is disjoint from @xmath13 , @xmath382 is disjoint from @xmath381}$ ] , and @xmath289 is transverse to @xmath381}$ ] . let @xmath290 and @xmath383 be the two @xmath17-disks in @xmath381}$ ] bounded by @xmath134 . then : 1 . [ indneq ] the intersection numbers @xmath384 and @xmath385 are unequal . 2 . the intersection numbers @xmath386 and @xmath384 are equal ( after possibly interchanging the names of @xmath290 and @xmath383 ) . the sphere @xmath291 is inessential in @xmath115}$ ] . the sphere @xmath387 is essential in @xmath115}$ ] . immediate by lemma [ ess_sphere ] . let @xmath388 denote closed upper half space . the closed upper half disk is @xmath389 . [ ball - arc_diffeo ] let @xmath361},\tau\right)}$ ] . let @xmath42 be a component of @xmath31 and let @xmath390}$ ] . suppose @xmath289 is a @xmath17-disk neatly embedded in @xmath21}$ ] such that : ( i ) @xmath289 is transverse to @xmath31 , ( ii ) @xmath289 meets @xmath31 at one point @xmath391 , and ( iii ) @xmath392 lies in @xmath253}$ ] . let @xmath290 be the @xmath17-disk in @xmath253}$ ] with boundary @xmath134 and containing @xmath137 . then , @xmath393 bounds a piecewise smooth @xmath27-disk @xmath394}$ ] and @xmath395 . further , there is a diffeomorphism of pairs @xmath396 that sends @xmath289 to the upper hemisphere and @xmath290 to @xmath397 . by lemma [ disk_ess_sphere ] with @xmath398 , @xmath393 is inessential in @xmath21}$ ] . by hypothesis , so , @xmath400 , and @xmath395 as well . the required diffeomorphism @xmath197 is constructed in bootstrapping fashion ( cf . paragraph four of the proof of proposition [ sigma_en ] ) : define @xmath197 on @xmath401 , extend to a smooth , regular neigborhood of @xmath401 in @xmath77 , and extend to the rest of @xmath77 utilizing lemma [ ball - arc ] . [ a_irred ] each block in @xmath18 is irreducible . the remainder of this section is devoted to proving @xmath19 is irreducible , which suffices to prove theorem [ a_irred ] . recall the block @xmath402},t\right)}$ ] from figure [ four_blocks ] . the ( general position ) immersion @xmath35 yielding @xmath19 ( see figure [ immersion ] ) plays a central role in our proof . we pause to explain @xmath35 and fix some notation . we assume the reader has figure [ immersion ] at hand . + the domain of @xmath35 is @xmath403}\subset\r^2 $ ] . the compact annulus @xmath73 contains three equally spaced radial arcs , @xmath42 , @xmath43 , and @xmath404 , as in figure [ immersion_domain ] . of the immersion @xmath35 . ] let @xmath405 . note that @xmath406 , the tangle in @xmath19 . we let @xmath407}$ ] , the subannulus of @xmath73 that is unshaded in figure [ immersion_domain ] . the boundary of @xmath408 is @xmath409 , the disjoint union of @xmath410}$ ] and @xmath411}$ ] . + given a subset @xmath412 , it will be convenient to let @xmath413 denote @xmath414 . ( a notable exception is @xmath415 . ) in particular , @xmath416 , @xmath417 , and @xmath418 . + for each @xmath419}$ ] , @xmath420}$ ] is an embedding , namely the composition of : a rigid rotation , a homothety , and a translation in the @xmath120-direction . we will see that : 1 . the multiple points of @xmath35 are double points where @xmath421}\right)}=f{\left(s^1{\left[\sfrac{5}{3}\right]}\right)}=\omega'$ ] . 2 . @xmath422}\times{\left\{\varepsilon\right\}}$ ] where @xmath423 . 3 . @xmath424 is a torus in @xmath21}$ ] , smooth except for corners along @xmath425 . on @xmath426 , @xmath35 is inclusion @xmath427 . with plane containing @xmath120-axis . ] figure [ slice ] shows the intersection of @xmath428 with any plane in @xmath0 containing the @xmath120-axis . on each of the three subannuli of @xmath73 in figure [ immersion_domain ] , @xmath35 is defined as follows . the annulus @xmath429}$ ] is stretched radially to @xmath430}$ ] , then the outer boundary component is twisted by @xmath431 radians ccw while fixing the inner boundary component , then level circles near the outer boundary component are lifted up a bit in the @xmath120-direction ( to yield general position ) . the annulus @xmath432}$ ] maps into @xmath21}$ ] by @xmath433 , then level circles near the inner boundary component are lifted up a bit in the @xmath120-direction ( again , to yield general position ) . finally , @xmath35 is defined on @xmath408 , interpolating @xmath434}$ ] and @xmath435}$ ] , so as to yield a torus @xmath436 as in figures [ immersion ] and [ slice ] . the two components of @xmath437 are identified under @xmath35 after half a rotation of @xmath410}$ ] . this completes our description of @xmath35 . + for distinct @xmath438 , let @xmath68 denote the closed sector in @xmath73 between @xmath61 and @xmath63 of angular measure @xmath439 . note that @xmath440 is an embedding . fix distinct @xmath441 . observe that @xmath442 meets @xmath443 ( transversely ) at exactly two points . for example , using the labelings in figure [ immersion_domain ] , @xmath444 meets @xmath445 at the two points : @xmath446 similarly , @xmath447 and @xmath448 for @xmath449 and @xmath17 . the points @xmath450 , @xmath451 , and @xmath452 , where @xmath449 and @xmath17 , will be referred to as * special points*. + we prove @xmath19 is irreducible according to definition [ irred_def_1 ] . it suffices to consider a @xmath17-sphere , @xmath50 , embedded in @xmath453}$ ] , transverse to @xmath20 , and meeting each component of @xmath20 at exactly one point . we improve @xmath50 by ambient isotopies of @xmath21}$ ] that fix @xmath20 setwise at all times . by an abuse , we refer to each improved @xmath50 as @xmath50 . we view @xmath416 as an auxiliary object , unaffected by these isotopies . perturb @xmath50 so that @xmath454 is disjoint from @xmath425 . perturb @xmath50 again so that further @xmath50 meets @xmath428 in general position . in particular , @xmath455 is an immersed , closed @xmath276-manifold in @xmath50 in general position . define : @xmath456 which is an embedded , closed @xmath276-manifold in @xmath457 , transverse to @xmath437 and @xmath31 . each component , @xmath61 , of @xmath31 meets @xmath32 at exactly one point ( not in @xmath437 ) . so , there exists one component , @xmath133 , of @xmath32 that meets each @xmath61 at one point ( transversely ) and @xmath133 is essential in @xmath73 . [ suff_claim ] it suffices to arrange that @xmath458 and @xmath459 . similar to the argument in paragraph four of the proof of proposition [ sigma_en ] . we give three operations for improving @xmath50 . define the * complexity * of @xmath50 to be : @xmath460 . ] [ taco0 ] suppose that @xmath461 and @xmath462 are arcs , and @xmath463 is a simple closed curve bounding a disk @xmath464 . assume that @xmath77 contains no special points , @xmath465 , and @xmath466 ( see figure [ taco_hypo ] ) . then , the points of @xmath467 can be eliminated from @xmath468 , and @xmath469 decreases by at least @xmath27 . as @xmath77 intersects only one component of @xmath437 , @xmath470 is an embedding . the disk @xmath471 permits construction of an isotopy of @xmath50 , with support near @xmath158 , that carries @xmath472 past @xmath473 to a parallel copy of @xmath473 . if @xmath158 intersects @xmath20 , then @xmath474 is a disk - arc pair ( all of which are unknotted ) . so , @xmath475 is an unknotted disk - arc pair , and the isotopy fixes @xmath20 setwise . the reduction in @xmath469 follows from figure [ taco_complexity ] . of @xmath476}$ ] correspond to two points of @xmath477}$ ] . ] four points of @xmath468 are eliminated , and , at worst , the number of components of @xmath32 increases by one . [ inessential ] among the components of @xmath32 that are inessential in @xmath73 and disjoint from @xmath437 , let @xmath134 be one that is innermost in @xmath73 . then , @xmath134 can be eliminated from @xmath32 , and @xmath469 decreases by at least @xmath276 . let @xmath464 be the @xmath17-disk with @xmath478 . note that @xmath77 is disjoint from @xmath437 and @xmath31 . so , @xmath470 is an embedding and @xmath471 is a @xmath17-disk disjoint from @xmath20 and bounding @xmath479 . the circle @xmath480 bounds two @xmath17-disks , @xmath289 and @xmath290 , in @xmath50 . the arc @xmath444 meets @xmath50 at one point . so , without loss of generality , @xmath444 meets @xmath289 at one point ( transversely , and in @xmath481 ) , and @xmath482 . by hypothesis , so , @xmath484 and @xmath485 are embedded @xmath17-spheres . by lemma [ ess_sphere ] , @xmath486 is essential in @xmath21}$ ] and @xmath485 is inessential . so , @xmath485 bounds an embedded @xmath27-disk @xmath487}$ ] , and @xmath488 . the @xmath27-disk @xmath246 permits construction of an isotopy of @xmath50 , with support near @xmath246 , that carries @xmath290 past @xmath158 to a parallel copy of @xmath158 . [ one_special_point ] let @xmath134 be a component of @xmath32 that bounds a @xmath17-disk @xmath464 . assume that @xmath489 is a neatly embedded arc in @xmath77 , @xmath77 contains exactly one special point @xmath118 , @xmath490 , and @xmath491 . then , @xmath134 can be eliminated from @xmath32 , and @xmath469 decreases by at least @xmath492 . without loss of generality , @xmath493 as in figure [ ball - arc - push ] . correspond to two points of @xmath477}$ ] . ] the embedded @xmath17-disk @xmath471 meets @xmath20 at exactly @xmath494 . the circle @xmath479 bounds two @xmath17-disks , @xmath289 and @xmath290 , in @xmath50 . without loss of generality , @xmath495 and @xmath444 meets @xmath290 at one point ( transversely ) . by lemma [ ess_sphere ] , @xmath485 bounds an embedded @xmath27-disk @xmath496}$ ] and @xmath497 is a neatly embedded arc in @xmath246 . by lemma [ ball - arc ] , @xmath498 is an unknotted ball - arc pair . the pair @xmath498 permits construction of an isotopy of @xmath50 , with support near @xmath246 , that carries @xmath290 past @xmath158 to a parallel copy of @xmath158 . this isotopy fixes @xmath444 setwise and fixes @xmath499 and @xmath500 pointwise . improve @xmath50 by applying lemmas [ taco0 ] , [ inessential ] , and [ one_special_point ] _ in any order and as long as possible_. this is a finite process since the initial complexity of @xmath50 is a positive integer and each operation strictly reduces the complexity . the complexity of the resulting improved @xmath50 is @xmath501 . the rest of this section shows that @xmath502 and @xmath459 , which suffices to prove theorem [ a_irred ] by claim [ suff_claim ] . [ taco ] there do not exist arcs @xmath461 and @xmath462 such that @xmath463 is a simple closed curve bounding a disk @xmath464 where @xmath77 is disjoint from the special points . we have @xmath503 where @xmath39 is a finite disjoint union of neatly embedded arcs in @xmath77 . we claim that @xmath254 contains no closed component of @xmath32 . otherwise , let @xmath504 be the @xmath17-disk bounded by an innermost such component @xmath134 . if @xmath505 , then lemma [ inessential ] applies to @xmath134 , a contradiction . thus , @xmath506 is a nonempty finite disjoint union of neatly embedded arcs in @xmath246 . let @xmath507 be an arc of @xmath506 that is outermost in @xmath246 in the sense that @xmath508 subtends an arc @xmath509 such that : @xmath510 lemma [ taco0 ] applies to @xmath511 and @xmath507 , a contradiction . the proof of the claim is complete . + if @xmath512 , then lemma [ taco0 ] applies to @xmath13 and @xmath513 , a contradiction . otherwise , there exists a component @xmath507 of @xmath39 that is outermost in @xmath77 in the sense that @xmath508 subtends an arc @xmath514 such that : @xmath515 lemma [ taco0 ] applies to @xmath511 and @xmath507 , a contradiction . next , we show that @xmath32 contains no component inessential in @xmath73 . suppose , by way of contradiction , that @xmath32 contains component(s ) inessential in @xmath73 . among these components , there must be one , call it @xmath134 , that is innermost in @xmath73 . by lemma [ inessential ] , @xmath134 meets @xmath437 . so , @xmath516 is positive and even . let @xmath464 be the @xmath17-disk with @xmath478 . note that @xmath517 . also , @xmath518 is a nonempty , finite disjoint union of neatly embedded arcs in @xmath77 . recall that the only component of @xmath32 that meets @xmath31 is @xmath133 , and @xmath133 is essential in @xmath73 . so , @xmath77 is contained in the interior of a sector @xmath68 and @xmath470 is an embedding . without loss of generality , assume @xmath519 . let @xmath471 , an embedded @xmath17-disk with @xmath520 . the circle @xmath480 also bounds two @xmath17-disks , @xmath289 and @xmath290 , in @xmath50 . as @xmath521 , @xmath486 and @xmath485 are embedded @xmath17-spheres in @xmath21}$ ] . by lemma [ ess_sphere ] ( using the arc @xmath444 , say ) , one of these spheres is essential in @xmath21}$ ] and the other is inessential . without loss of generality , assume @xmath486 is essential in @xmath21}$ ] and @xmath485 is inessential . let @xmath246 be the @xmath27-disk in @xmath21}$ ] with @xmath522 . [ no_inessential ] the disk @xmath77 can not be disjoint from the special points @xmath523 and @xmath524 . otherwise , let @xmath513 be a component of @xmath39 . let @xmath525 be an arc with @xmath526 . the arcs @xmath13 and @xmath513 contradict lemma [ taco ] . [ 2special ] the disk @xmath77 can not contain both special points @xmath523 and @xmath524 . suppose otherwise . note that @xmath444 meets @xmath158 twice ( transversely in @xmath527 ) , and @xmath499 and @xmath500 are disjoint from @xmath158 . by lemma [ ess_sphere ] , @xmath20 is disjoint from @xmath290 . the disk @xmath246 permits construction of an an ambient isotopy @xmath528 , @xmath176 , of @xmath21}$ ] that carries @xmath290 past @xmath158 to a parallel copy of @xmath158 . this isotopy has support near @xmath246 , is relative to a neighborhood of @xmath529}$ ] , but does _ not _ fix @xmath444 setwise . note that @xmath530 where @xmath531 still denotes @xmath532 . as @xmath533 is a borromean block , there is an ambient isotopy @xmath534 , @xmath535 , of @xmath21}$ ] , relative to a neighborhood of @xmath536}$ ] , that straightens both @xmath537 and @xmath538 . the strip @xmath539 permits construction of an ambient isotopy @xmath175 , @xmath176 , of @xmath21}$ ] that carries @xmath540 to an arc close to and winding around @xmath541 . this isotopy has support near @xmath539 , is relative to both @xmath542 and @xmath541 , but otherwise is not relative to @xmath536}$ ] . the resulting block @xmath543 is diffeomorphic to @xmath165 by untwisting tangle components @xmath17 and @xmath27 . so , @xmath544 , which contradicts corollary [ anottrivial ] . [ one ] the disk @xmath77 can not contain exactly one of the special points @xmath523 or @xmath524 . assume , without loss of generality , that @xmath254 contains @xmath523 but not @xmath524 . recall that @xmath517 . if @xmath39 is connected , then lemma [ one_special_point ] applies to @xmath77 , a contradiction . if @xmath39 is disconnected , then @xmath39 contains a component , @xmath513 , that does not contain @xmath523 . let @xmath525 be the arc with @xmath526 and such that @xmath523 does not lie inside the simple closed curve @xmath463 . the arcs @xmath13 and @xmath513 contradict lemma [ taco ] . taking stock , @xmath32 contains no component inessential in @xmath73 . so , @xmath458 is a single , essential circle in @xmath73 , and @xmath133 meets each @xmath61 at one point ( transversely ) . it remains to prove that @xmath459 . a * segment * will mean an arc @xmath545 for which @xmath546 . each inline figure , such as , represents a segment of @xmath133 in a sector ; vertical lines represent adjacent components of @xmath31 , upper and lower horizontal lines represent arcs of @xmath437 , dots represent special points , and arrows indicate reflected cases of the entire figure . for example , represents six cases ( three choices of sector and a possible vertical reflection ) , and represents twelve cases . [ scin ] none of the following appear : , , or . observe that points of @xmath73 inside ( outside ) @xmath133 map under @xmath35 to lie inside ( outside ) @xmath50 respectively . suppose there is . without loss of generality , assume the indicated special points are @xmath523 and @xmath524 . then , @xmath547 is outside @xmath50 and @xmath548 is inside @xmath50 . there are three possibilities for the location of @xmath549 ( see figure [ annulus_omo ] at right ) . the inner option implies @xmath550 and @xmath551 both lie outside @xmath50 . the middle option implies @xmath550 lies inside @xmath50 and @xmath551 lies outside @xmath50 . the outer option implies @xmath550 and @xmath551 both lie inside @xmath50 . all three are contradictions since @xmath552 and @xmath553 . so , no appears . we claim that : 1 . [ not_both ] there do not exist and in the same sector and with the same reflection . to see this , suppose , by way of contradiction , that and both appear in the same sector , say @xmath554 . let @xmath555 and @xmath556 denote these segments respectively . assume , without loss of generality , that the special point just below @xmath555 is @xmath523 . let @xmath557 be the unique arc such that @xmath558 and @xmath559 . lemma [ taco ] implies that @xmath560 bounds segments @xmath561 parallel to @xmath555 and @xmath562 parallel to @xmath556 ( @xmath563 and @xmath564 are possible ) . then , @xmath565 is an inessential component of @xmath32 , a contradiction . this completes our proof of [ not_both ] . + suppose there is . assume , without loss of generality , that this segment @xmath566 of @xmath133 lies in @xmath554 . by lemma [ taco ] , the only possible segment of @xmath133 that meets both @xmath408 and @xmath567 , and is disjoint from @xmath31 , is . by [ not_both ] , the boundary points of @xmath566 lie in distinct segments , @xmath568 and @xmath569 , of @xmath133 , where @xmath568 meets @xmath43 at one endpoint and @xmath569 meets @xmath404 at one endpoint ( see figure [ annulus_omo ] ) . . ] so , @xmath547 is outside @xmath50 and @xmath548 is inside @xmath50 . this yields the same contradiction as for above . hence , no appears . + suppose there is . call this segment @xmath566 and assume , without loss of generality , that @xmath523 is the special point pictured above @xmath566 . let @xmath557 denote the short arc with @xmath570 ( see figure [ annulus_omm ] ) . we claim that no segment of @xmath133 contained in @xmath408 may meet @xmath120 . this follows from : ( i ) lemma [ taco ] , ( ii ) the nonexistence of , ( iii ) since @xmath566 meets @xmath43 , and ( iv ) since @xmath133 meets @xmath43 exactly once . therefore , @xmath571 appears as in figure [ annulus_omm ] . can not exit the annulus @xmath408 between @xmath572 and @xmath523 . ] so , @xmath547 is outside @xmath50 and @xmath548 is inside @xmath50 , a familiar contradiction . hence , no appears . recall that @xmath573}$ ] is an embedded torus , smooth except for corners along @xmath425 . so , @xmath574 is a closed @xmath276-manifold , smooth except for corners , and embedded in @xmath50 and in @xmath424 . we introduce a based longitude @xmath575 and a based meridian @xmath576 on @xmath424 . both originate and terminate at @xmath577 ( recall figures [ immersion ] , [ immersion_domain ] , and [ slice ] ) . the longitude @xmath575 runs once along @xmath425 in the cw direction about the @xmath120-axis when viewed from the point @xmath578 . the meridian @xmath576 is the right loop in figure [ slice ] , oriented ccw , where the plane of intersection is the @xmath579-plane . in particular , a parallel pushoff @xmath580 of @xmath575 into the inside of @xmath424 has linking number @xmath198 with @xmath576 in @xmath0 . an oriented loop in @xmath424 has * type * @xmath581 provided it is freely homotopic in @xmath424 to @xmath582 . [ essential ] let @xmath134 be a component of @xmath574 , equipped with an orientation and of type @xmath581 . then , @xmath583 . assume @xmath134 is essential in @xmath424 ( otherwise , the result is clear ) . note that @xmath584 . focus attention on the submanifolds @xmath50 and @xmath424 of @xmath21}\subset\r^3 $ ] . we view @xmath50 and @xmath424 as submanifolds of @xmath585 , where @xmath586 and @xmath587 . as @xmath134 is essential in @xmath424 , @xmath134 is not null - homologous ( denoted @xmath588 ) in both @xmath589 and @xmath590 . let @xmath284 denote @xmath589 or @xmath590 where @xmath588 in @xmath284 . exactly one component of @xmath591 contains @xmath134 in its frontier and lies in @xmath284 ; let @xmath592 denote the closure of this component in @xmath50 . note that @xmath592 is a compact @xmath17-disk with holes , @xmath593 , @xmath594 , and @xmath595 . as @xmath588 in @xmath284 , there must be another component @xmath596 of @xmath597 such that @xmath598 in @xmath284 . in particular , @xmath596 is essential in @xmath424 . choose an orientation of @xmath596 . then , @xmath596 has type @xmath599 where @xmath600 . the algebraic intersection number of @xmath134 and @xmath596 in @xmath424 equals @xmath601 , which must vanish since @xmath574 is embedded in @xmath424 . it follows that @xmath596 has type @xmath602 . switching the orientation of @xmath596 if necessary , @xmath596 has type @xmath581 . thus , @xmath134 and @xmath596 are parallel in @xmath424 . an exercise ( left to the reader ) shows that the linking number @xmath603 in @xmath0 equals @xmath604 . as @xmath134 and @xmath596 are disjointly embedded in the sphere @xmath370 , @xmath605 . hence , @xmath583 as desired . we define an * extended segment * to be an arc component of @xmath606 of the form : @xmath607{bar_arrow.eps } , \quad \includegraphics[scale=0.6]{scurve_arrow.eps } , \quad { \textnormal{or}}\quad \includegraphics[scale=0.6]{scurve_straddle_arrow.eps}\ ] ] in the latter type , the vertical line is a component of @xmath31 and the horizontal lines are arcs of @xmath437 not meeting special points . [ int_ext_seg ] any component of @xmath606 is an extended segment . let @xmath566 be a component of @xmath606 . by lemma [ scin ] , no appears . therefore , @xmath608 . hence , @xmath566 is an arc neatly embedded in @xmath408 . note that @xmath609 , and @xmath610 is disjoint from @xmath31 and from special points . by lemma [ taco ] , no appears . the result now follows from lemma [ scin ] by considering the possible locations of points in @xmath610 . [ disjoint ] @xmath459 . suppose , by way of contradiction , that @xmath133 meets @xmath408 . then , there is a circle @xmath611 . by corollary [ int_ext_seg ] , there exists a finite , disjoint collection @xmath612 , @xmath613 , of extended segments such that : @xmath614 orient all segments @xmath615 , @xmath616 , to point out from @xmath617}$ ] . note that this yields a coherent orientation of @xmath134 . a moment of reflection on the immersion @xmath35 ( cf . figure [ immersion ] ) reveals that each @xmath618 winds around @xmath424 by : ( i ) @xmath198 revolutions in the @xmath576 direction , and ( ii ) @xmath619 revolutions in the @xmath575 direction where @xmath620 . hence , @xmath134 has type @xmath581 where @xmath621 and @xmath622 , which contradicts lemma [ essential ] . this completes our proof that @xmath19 is irreducible and our proof of theorem [ a_irred ] . this section proves that certain spheres in concatenations of borromean blocks may be ambiently isotoped into a single block summand , while fixing the tangle setwise . [ borr_prime ] let @xmath23 , @xmath623 , be borromean blocks where @xmath332 . consider the concatenation : @xmath624},\tau\right)}\ ] ] let @xmath50 be a @xmath17-sphere embedded in the interior of @xmath625}$ ] , transverse to @xmath31 , and meeting each component @xmath61 of @xmath31 at exactly one point @xmath626 for @xmath449 , @xmath17 , and @xmath27 . then , there is a neighborhood @xmath235 of @xmath627}$ ] in @xmath625}$ ] and an ambient isotopy @xmath175 , @xmath237 , of @xmath625}$ ] such that : 1 . @xmath177 . @xmath238 for all @xmath176 . 3 . @xmath175 fixes @xmath31 setwise for all @xmath237 . 4 . @xmath628}}$ ] for some @xmath629 . for the definition of _ borromean block _ , see section [ borr_blocks ] . without loss of generality , no point @xmath626 lies in a sphere @xmath630}$ ] where @xmath631 , and @xmath50 is transverse to these spheres . so , @xmath632}\right)}$ ] is a closed @xmath276-manifold . it suffices to improve @xmath50 , by appropriate ambient isotopies of @xmath625}$ ] , so that @xmath89 becomes empty . we will employ the following two operations . + * operation 1 . * suppose @xmath134 is a component of @xmath89 bounding a disk @xmath633 such that : ( i ) @xmath289 is disjoint from @xmath634 and ( ii ) @xmath289 is disjoint from @xmath31 . the circle @xmath134 lies in @xmath630}$ ] for some @xmath631 and bounds two @xmath17-disks , @xmath290 and @xmath383 , in @xmath630}$ ] . by lemma [ disk_ess_sphere ] ( with @xmath635 ) , we have @xmath636 and @xmath291 is inessential in @xmath625}$ ] . so , @xmath637 ( since each @xmath61 meets @xmath290 at most once ) and there is a @xmath27-disk @xmath77 in @xmath625}$ ] bounded by @xmath291 . hence , @xmath31 is disjoint from @xmath77 and @xmath77 permits construction of an ambient isotopy , with support near @xmath77 , that carries @xmath289 past @xmath290 to a parallel copy of @xmath290 . thus , @xmath134 ( at least ) has been eliminated from @xmath89 . + * operation 2 . * suppose @xmath134 is a component of @xmath89 bounding a disk @xmath633 such that : ( i ) @xmath289 is disjoint from @xmath634 and ( ii ) @xmath289 intersects @xmath31 at one point . for notational convenience , we assume @xmath638 . note that @xmath639 . the circle @xmath134 lies in @xmath630}$ ] for some @xmath631 . let @xmath640}$ ] be the @xmath17-disk whose boundary is @xmath134 and whose interior meets @xmath42 ( necessarily at one point , call it @xmath137 ) . now , @xmath289 lies in @xmath641}$ ] or in @xmath642}$ ] . without loss of generality , assume the latter . consider the block : @xmath643},r\right)}\ ] ] where @xmath8 has components : @xmath644 } { \textnormal { for $ i=1 $ , $ 2 $ , and $ 3$.}}\ ] ] let @xmath413 be the block obtained from @xmath284 by forgetting @xmath645 . then , @xmath646 since @xmath647 is a borromean block . hence , lemma [ ball - arc_diffeo ] implies that @xmath291 bounds the @xmath27-disk @xmath648}$ ] , that @xmath649 , and there exists a diffeomorphism @xmath650 . the first two of these consequences imply that @xmath651 . forgetting @xmath652 instead of @xmath645 , we get @xmath653 . therefore , we have a diffeomorphism : @xmath654 this diffeomorphism permits construction of an ambient isotopy of @xmath625}$ ] that : ( i ) has support near @xmath77 , ( ii ) fixes @xmath43 and @xmath404 pointwise , ( iii ) fixes @xmath42 setwise , and ( iv ) carries @xmath289 past @xmath290 to a parallel copy of @xmath290 . thus , @xmath134 ( at least ) has been eliminated from @xmath89 . + observe that if @xmath655 , then operation 1 or 2 is applicable . indeed , let @xmath596 be a component of @xmath89 . one component , @xmath656 , of @xmath657 contains at most one of the points @xmath658 , @xmath659 , or @xmath660 . if @xmath656 contains components of @xmath89 , then operation 1 or 2 applies to any innermost component of @xmath89 in @xmath656 . if @xmath656 contains no component of @xmath89 , then operation 1 or 2 applies to @xmath596 itself . so , by finitely many applications of operations 1 and 2 , we get @xmath661 and the proof is complete . [ fin_cor ] let @xmath23 , @xmath662 , and @xmath663 , @xmath664 , be irreducible borromean blocks . consider the concatenations : @xmath665},\tau\right)}\\ c:=&c_1 c_2 \cdots c_n = { \left(s^2{\left[1,n+1\right]},\sigma\right)}\end{aligned}\ ] ] suppose there is a diffeomorphism @xmath666 such that @xmath667 . then , @xmath668 and there is an isotopy of @xmath35 , sending @xmath31 to @xmath32 setwise at all times , to a diffeomorphism @xmath669 such that @xmath670 for each @xmath671 . assume @xmath672 ( otherwise , consider @xmath673 ) . if @xmath674 , then @xmath675 contradicts irreducibility of @xmath26 ( recall that each @xmath676 since each @xmath663 is a borromean block ) . so , @xmath677 and we are done . next , let @xmath678 . all isotopies of @xmath35 send @xmath31 to @xmath32 setwise at all times . all isotopies of @xmath134 are ambient and fix @xmath32 setwise at all times . by an abuse of notation , the corresponding improved @xmath35 will still be denoted by @xmath35 . by proposition [ borr_prime ] , we can isotop @xmath35 such that @xmath679\right)}\subset{{\textnormal{int } } \ , c}_j$ ] for some @xmath680 . as @xmath663 is irreducible , we can isotop @xmath35 so that @xmath679\right)}$ ] is a level @xmath17-sphere in @xmath663 very close to one boundary sphere of @xmath663 , namely : ( i ) @xmath681 $ ] , or ( ii ) @xmath682$].case 1 . @xmath683 and ( i ) then , @xmath684 , a contradiction since @xmath26 is a borromean block.case 2 . @xmath683 and ( ii ) occurs , or @xmath685 and ( i ) occurs . then , we may further isotop @xmath35 so that @xmath679\right)}=s^2[2]$].case 3 . @xmath685 and ( ii ) occurs , or @xmath686 . these cases contradict irreducibility of @xmath26.in any case , we have arranged that @xmath679\right)}=s^2[2]$ ] . repeat this process with @xmath687\right)}$ ] and so forth , and it must terminate with @xmath668 . [ workhorse ] let @xmath23 and @xmath33 , @xmath148 , be sequences of irreducible borromean blocks . consider the two borromean rays : @xmath688 suppose there is a diffeomorphism @xmath34 . then , there exist @xmath689 and @xmath690 , and there is an isotopy of @xmath35 , sending @xmath31 to @xmath32 setwise at all times , to a diffeomorphism @xmath36 such that @xmath691 for each @xmath692 . all isotopies send @xmath31 to @xmath32 setwise at all times , and are relative to @xmath313 . by compactness , there exists an integer @xmath693 such that @xmath694})$ ] is disjoint from @xmath695 . by proposition [ borr_prime ] , we may isotop @xmath35 so that @xmath696\right)}\subset { { \textnormal{int } } \ , c}_i$ ] for some @xmath697 . as @xmath33 is irreducible , we may further isotop @xmath35 so that @xmath696\right)}=s^2[m]$ ] for some integer @xmath698 . define @xmath699 . use proposition [ borr_prime ] and irreducibility repeatedly to get @xmath700\right)}=s^2[i+n]$ ] for each integer @xmath692 . this section proves theorem [ borr_rays_thm ] , the first of our main results . if @xmath32 is a multiray given by a concatenation of blocks : @xmath701 then the * mirror * of @xmath32 , denoted @xmath702 , is defined by : @xmath703 [ borr_rays_thm ] let @xmath23 and @xmath33 , @xmath24 , be sequences of blocks in @xmath18 . consider the two borromean rays : @xmath688 there exists a diffeomorphism @xmath34 preserving orientation of @xmath0 if and only if there exists @xmath689 such that one of the following holds for all sufficiently large @xmath24 : if [ beqc ] holds , then @xmath35 exists by remarks [ concat_remarks ] item [ tail_det_type ] . assume [ beqcbarstar ] holds . then : @xmath711 the last diffeomorphism is @xmath712 (= rotation of @xmath0 about @xmath118-axis ) , followed by a simple ambient isotopy in each block relative to boundary @xmath17-spheres ( cf . section [ s : diffeo_blocks ] ) . the other two diffeomorphisms come from remarks [ concat_remarks ] item [ tail_det_type ] . the composition is the required @xmath35 . + assume [ beqcbar ] holds . the first orientation preserving case above yields a diffeomorphism @xmath713 that preserves orientation of @xmath0 . composing @xmath197 with @xmath714 yields the required @xmath35 . assume [ beqcstar ] holds . then : @xmath715 for sufficiently large @xmath296 . the second orientation preserving case yields a diffeomorphism @xmath713 that preserves orientation of @xmath0 . again , compose @xmath197 with @xmath714 to obtain the required @xmath35 . + for the forward implications , note that blocks in @xmath18 are irreducible by theorem [ a_irred ] . by corollary [ workhorse ] , there exist @xmath689 , @xmath690 , and a diffeomorphism @xmath36 , isotopic to @xmath35 , such that @xmath691 for each @xmath692 . assume @xmath35 preserves orientation of @xmath0 . then , @xmath197 preserves orientation of @xmath0 . so , each of the diffeomorphisms : @xmath716 preserves orientation , preserves boundary @xmath17-spheres setwise , and has the same tangle permutation @xmath717 ( see section [ s : diffeo_blocks ] ) . if @xmath718 , then [ beqc ] holds by table [ hom_block ] in section [ s : diffeo_blocks ] . if @xmath719 , then [ beqcbarstar ] holds by table [ hom_block ] . + finally , assume the given @xmath35 reverses orientation of @xmath0 . the composition of @xmath35 with @xmath714 is a diffeomorphism @xmath720 preserving orientation of @xmath0 . now , apply the orientation preserving case . let @xmath721 denote the set of all sequences @xmath23 , @xmath24 , of blocks in @xmath18 . declare two sequences to be equivalent , written @xmath722 , if and only if their corresponding borromean rays are equivalent by some diffeomorphism of @xmath0 ( not necessarily orientation preserving ) . a multiray @xmath724 is * achiral * provided there exists a diffeomorphism @xmath725 preserving orientation of @xmath0 . otherwise , @xmath31 is * chiral*. equivalently , @xmath31 is * achiral * provided there exists a diffeomorphism @xmath726 reversing orientation of @xmath0 . [ chiral_cor ] let @xmath727 for a sequence @xmath23 , @xmath24 , of blocks in @xmath18 . then , @xmath31 is achiral if and only if ( i ) there exists a block @xmath728 where @xmath729 and each @xmath730 , and ( ii ) a tail of the sequence @xmath23 equals one of the following : @xmath731 in particular , if @xmath31 is achiral , then @xmath23 is eventually periodic . so , at most countably many achiral @xmath31 arise this way . by theorem [ borr_rays ] , @xmath31 is achiral if and only if ( i ) @xmath23 and @xmath732 have identical tails , or ( ii ) @xmath23 and @xmath733 have identical tails . so , if a tail of @xmath23 has the form , then @xmath31 is achiral . conversely , suppose @xmath23 and @xmath732 have identical tails ( the other case is similar ) . then , there exists @xmath689 and @xmath734 such that @xmath735 as @xmath736 , we get @xmath737 . without loss of generality , assume @xmath738 ( otherwise , apply bar to ) . note that : @xmath739 where the second equality used . repeating this argument , we get that if @xmath740 and @xmath741 , then : @xmath742 therefore , @xmath743 . [ achiral_br ] for each @xmath744 , define : @xmath745 and define : @xmath746 by corollary [ chiral_cor ] , @xmath747 is achiral . by theorem [ borr_rays_thm ] , @xmath748 is a diffeomorphism invariant of @xmath747 . so , @xmath747 , @xmath744 , is a countably infinite family of achiral borromean rays , pairwise distinct up to diffeomorphism . a * hyperplane * is a smooth , proper embedding of @xmath4 in @xmath749 . a * multiple hyperplane * @xmath750 is a smooth , proper embedding of a disjoint union of ( at most countably many ) copies of @xmath4 . the basic invariant of @xmath751 is an associated tree @xmath752 . the vertices of @xmath752 are the components of @xmath753 . two vertices are adjacent provided their closures in @xmath1 share a component of @xmath751 . figure [ hyp_graphs ] depicts some multiple hyperplanes and their trees . multiple hyperplanes in @xmath1 , @xmath6 , are classified by their associated trees @xcite . more precisely , if @xmath751 and @xmath754 are multiple hyperplanes in @xmath1 , @xmath755 , and the trees @xmath752 and @xmath756 are isomorphic , then there is a diffeomorphism @xmath757 that preserves orientation of @xmath1 . for @xmath758 , these trees are naturally planar ( i.e. , the edges incident with a given vertex are cyclically ordered ) . the result then holds provided @xmath752 and @xmath756 are isomorphic as planar trees . + let @xmath759 denote klein s model of hyperbolic @xmath27-space . namely , @xmath759 is the open unit @xmath27-disk in @xmath0 , and a * hyperbolic hyperplane * is the nonempty intersection of @xmath759 with an affine plane in @xmath0 . a * hyperbolic multiple hyperplane * is a properly embedded submanifold of @xmath759 , each component of which is a hyperbolic hyperplane . the multiple hyperplanes in figure [ hyp_graphs ] are hyperbolic . a multiple hyperplane @xmath760 is * unknotted * provided there exists a diffeomorphism @xmath761 where @xmath762 is a hyperbolic multiple hyperplane . let @xmath751 and @xmath754 be unknotted multiple hyperplanes in @xmath0 . there exists a diffeomorphism @xmath763 preserving orientation of @xmath0 if and only if @xmath752 and @xmath756 are isomorphic trees . given a tree @xmath20 that is at most countable and is not necessarily locally finite , it is not difficult to construct a hyperbolic multiple hyperplane @xmath764 such that @xmath765 is isomorphic to @xmath20 . hence , unknotted multiple hyperplanes in @xmath0 are classified , up to diffeomorphism , by isomorphism classes of such trees . up to isomorphism , there is a unique tree with @xmath37 vertices for @xmath766 . for @xmath767 , there are two : a linear tree and the @xmath27-prong ( see figure [ hyp_graphs ] ) . + let @xmath768 denote closed upper half space . throughout this section , @xmath769 denotes the ray @xmath770 in the positive @xmath120-axis . + if @xmath771 is a multiray with @xmath772 components , then @xmath773 denotes a smooth , closed regular neighborhood of @xmath8 in @xmath0 @xcite ( see also hirsch @xcite ) . if @xmath183 is a component of @xmath8 , then @xmath774 denotes the component of @xmath773 containing @xmath183 . basic properties of @xmath773 include : 1 . [ mnr1 ] for each @xmath183 , there is a diffeomorphism of pairs @xmath775 . [ mnr2 ] @xmath773 is unique up to ambient isotopy of @xmath0 relative to @xmath8 . [ mnr3 ] the boundary of @xmath773 , denoted @xmath776 , is an @xmath37 component multiple hyperplane . [ mnr4 ] the tree @xmath777 is an @xmath37-prong . let @xmath30 be the ray @xmath783 in the positive @xmath118-axis . let @xmath784 denote the points of norm @xmath785 in the first quadrant of the @xmath579-plane . we identify @xmath784 with @xmath65\times[1,\infty)$ ] so that @xmath786 and @xmath787 . all isotopies of @xmath781 will be ambient and relative to a neighborhood of @xmath780 in @xmath781 . isotoped subsets will be called by their original names . it suffices to construct an isotopy of @xmath781 that carries @xmath788 to @xmath789 . let @xmath790 be orthogonal projection . the submanifold @xmath791 permits construction of an isotopy of @xmath781 that carries @xmath788 close enough to @xmath792 so that @xmath793 is an embedding . this is possible since @xmath792 is properly embedded . next , by integrating a suitable vector field of the form @xmath794 , we get an isotopy carrying @xmath788 into @xmath795 . rays do not knot in @xmath796 ( * ? ? ? . 9.13 ) . so , there is an ambient isotopy @xmath211 of @xmath795 that carries @xmath788 to a straight ray . use a small tube about @xmath795 in @xmath797 and a suitable bump function to extend @xmath211 to an isotopy of @xmath781 . the rest is elementary . 1 . lemma [ ztoz ] is very different from ` uniqueness of regular neighborhoods ' . let @xmath133 be a smooth subcomplex of a smooth manifold @xmath798 . it is not true , in general , that each orientation preserving diffeomorphism @xmath799 is isotopic to a diffeomorphism @xmath800 . for a simple counterexample , let @xmath133 be a bouquet of three circles embedded in @xmath801 in such a way that no circle of @xmath133 is inside another . then , @xmath802 is a smooth , compact @xmath17-disk with three holes . let @xmath803 be a simple closed curve such that two boundary components of @xmath804 are inside @xmath134 . let @xmath805 be a dehn twist about @xmath134 . then , @xmath220 is not isotopic to a diffeomorphism @xmath800 . otherwise , @xmath220 would be isotopic to the identity , which is false @xcite . counterexamples exist even when @xmath133 is a smooth submanifold . we are indebted to bob gompf for these examples . let @xmath284 be a simply - connected , closed , symplectic @xmath806-manifold with positive signature , denoted @xmath807 , and @xmath808 . ( many such manifolds are known even khler examples have been around for several decades . ) as @xmath284 is symplectic , @xmath284 is smooth and oriented , and @xmath809 is odd . let @xmath133 be @xmath284 blown up @xmath810 times . then , @xmath811 and @xmath133 is homeomorphic to @xmath812 for some @xmath678 ( and very large in practice ) . since @xmath133 is symplectic , @xmath133 has nonvanishing seiberg - witten invariants , denoted @xmath813 . however , @xmath814 (= @xmath133 with reversed orientation ) splits off a @xmath815 summand ( from the blowup of @xmath284 ) and has @xmath808 . therefore , @xmath816 and @xmath133 admits no orientation reversing self diffeomorphism . fix any @xmath817 . then , @xmath818 is a smooth , closed , oriented manifold containing @xmath819 , and @xmath820 is identified with @xmath821 . now , there is a smooth @xmath220-cobordism @xmath656 between @xmath133 and @xmath89 . so , @xmath822 is smoothly a product , and , working relative to boundary , @xmath823 is smoothly a product . hence , there is a diffeomorphism @xmath824 . let @xmath135 be an orientation reversing self diffeomorphism of @xmath89 ( e.g. , @xmath135 permutes the summands and is otherwise the identity ) . let @xmath136 be an orientation reversing self diffeomorphism of @xmath92 . then , @xmath825 is an orientation preserving self diffeomorphism of @xmath826 , and @xmath827 is multiplication by @xmath828 on @xmath829 . let @xmath220 be the corresponding orientation preserving self diffeomorphism of @xmath804 . on @xmath830 , @xmath831 is multiplication by @xmath828 . so , @xmath220 is not isotopic to a diffeomorphism @xmath800 . otherwise , we get a forbidden orientation reversing self diffeomorphism of @xmath133 . an alternative approach to proving lemma [ ztoz ] uses the following lemma together with some collaring arguments . only two implications require proof . + [ ediffeo ] @xmath842 [ eisotopy ] : we may assume @xmath35 preserves orientation of @xmath1 ( otherwise , compose @xmath35 with reflection through @xmath843 ) . by milnor @xcite , @xmath35 is isotopic to the identity . + [ ehalves ] @xmath842 [ ediffeo ] : let @xmath844 denote closed lower half space . replacing @xmath845 and @xmath846 with their compositions with appropriate reflections , we can and do assume @xmath840 and @xmath847 are orientation preserving diffeomorphisms . let @xmath848 , an orientation preserving automorphism of @xmath843 . let @xmath849 be the orientation preserving automorphism of @xmath4 given by the following composition where @xmath850 is the obvious projection : @xmath851 let @xmath852 be the identity map on @xmath2 . then , @xmath853 is an orientation preserving automorphism of @xmath854 . replacing @xmath845 with @xmath855 , we can and do further assume @xmath856 . define @xmath857 by @xmath858 if @xmath859 and @xmath860 if @xmath861 . then , @xmath220 is an orientation preserving autohomeomorphism of @xmath1 . by construction , @xmath220 is smooth on @xmath284 and @xmath220 is smooth on @xmath836 . using collaring uniqueness ( * ? ? ? 8.1.9 ) , we may adjust @xmath220 ( by isotoping @xmath845 and @xmath846 near @xmath751 and relative to @xmath751 ) to obtain the desired diffeomorphism @xmath35 . [ rtor ] let @xmath8 and @xmath185 be multirays in @xmath0 . let @xmath862 be a diffeomorphism . let @xmath778 and @xmath863 . then , @xmath220 is isotopic , relative to a neighborhood of @xmath751 in @xmath773 , to a diffeomorphism of pairs @xmath864 . it suffices to consider @xmath865 where @xmath183 is a component of @xmath8 . reindex the components of @xmath185 so that @xmath866 . let @xmath775 and @xmath867 be diffeomorphisms . lemma [ ztoz ] yields an isotopy , relative to a neighborhood of @xmath780 , such that @xmath868 and @xmath869 . then , @xmath870 is the desired isotopy of @xmath865 . suppose @xmath751 is unknotted . then , there is a diffeomorphism @xmath872 where @xmath762 is a hyperbolic multiple hyperplane . as @xmath752 is an @xmath37-prong , so is @xmath765 . without loss of generality , the origin of @xmath759 does not lie in @xmath873 . for each component @xmath183 of @xmath8 , let @xmath874 and let @xmath875 . let @xmath70 be the point of @xmath876 closest to the origin in @xmath759 ( for the euclidean metric ) . let @xmath877 be the radial ray in @xmath759 with initial point @xmath70 . let @xmath878 be the radial ray in @xmath879 that is half as long as @xmath879 ( for the euclidean metric ) . notice that @xmath33 is a smooth , closed regular neighborhood of @xmath61 in @xmath759 . let @xmath880 be the radial multiray with components @xmath61 . notice that @xmath873 is a smooth , closed regular neighborhood of @xmath31 in @xmath759 . as in corollary [ rtor ] , we may isotop @xmath220 to a diffeomorphism @xmath881 . but , this implies @xmath8 is unknotted , a contradiction . [ rays_planes ] let @xmath31 and @xmath269 be multirays in @xmath0 , each containing @xmath37 components where @xmath882 . let @xmath883 and let @xmath884 . if @xmath885 is a diffeomorphism , then @xmath35 is isotopic relative to @xmath31 to a diffeomorphism @xmath886 . conversely , suppose @xmath886 is a diffeomorphism . if any of the following conditions are met , then @xmath197 is isotopic relative to @xmath887 to a diffeomorphism @xmath885 . the case @xmath677 where @xmath31 and @xmath269 are unknotted is exceptional for trivial reasons . for example , let @xmath31 and @xmath269 both equal @xmath789 . let @xmath889 and @xmath890 both equal @xmath781 , so @xmath891 . then , @xmath892 is an orientation preserving diffeomorphism of @xmath0 sending @xmath751 to @xmath754 . however , @xmath197 is not isotopic relative to @xmath751 to a diffeomorphism @xmath893 . the forward implication is immediate by ambient uniqueness of closed regular neighborhoods @xcite . next , let @xmath886 be a diffeomorphism . first , assume condition [ cond1 ] . then , the result is immediate by corollary [ rtor ] . second , assume condition [ cond2 ] . as @xmath752 and @xmath756 are both @xmath37-prongs , condition [ cond1 ] is satisfied and the result follows . third , assume condition [ cond3 ] . without loss of generality , assume @xmath31 is knotted ( otherwise , consider @xmath260 ) . let @xmath894 and let @xmath895 . by corollary [ krikh ] , @xmath751 is knotted . by lemma [ unknot_equiv ] , @xmath896 . as @xmath897 , we see that condition [ cond1 ] is satisfied and the result follows . proposition [ rays_planes ] permits us to translate results on knotted multirays in @xmath0 to results on knotted multiple hyperplanes in @xmath0 . by mcpherson @xcite , there exist uncountably many knot types of a ray : so there exist uncountably many knot types of a hyperplane . by fox and harrold @xcite , there exist uncountably many knot types of two component multirays with unknotted components ( see fox and artin @xcite for a nice example ) : so there exist uncountably many knot types of two component multiple hyperplanes with unknotted components . + proposition [ rays_planes ] is proved via ambient isotopies , so it also yields results on chirality . let @xmath898 be a multiple hyperplane . we say @xmath751 is * achiral * provided there is a diffeomorphism @xmath899 that reverses orientation of @xmath0 . otherwise , @xmath751 is * chiral*. for example , it is an exercise to show that each unknotted multiple hyperplane @xmath760 is achiral . + a multiple hyperplane @xmath751 in @xmath0 forms * borromean hyperplanes * provided @xmath751 is knotted , but any two components of @xmath751 form an unknotted multiple hyperplane . proposition [ rays_planes ] implies that if @xmath30 forms borromean rays , then @xmath776 forms borromean hyperplanes . thus , we obtain our second main result . there exist uncountably many borromean hyperplanes , pairwise distinct up to diffeomorphism of @xmath0 . there exists a countably infinite family of achiral borromean hyperplanes , pairwise distinct up to diffeomorphism of @xmath0 . calcut , h.c . king , and l.c . siebenmann , _ connected sum at infinity and cantrell - stallings hyperplane unknotting _ , accepted at rocky mountain j. math . ( 2011 ) , available at http://www.oberlin.edu/faculty/jcalcut/csi.pdf[http://www.oberlin.edu/faculty/jcalcut/csi.pdf ] , 42 pp .

three disjoint rays in @xmath0 form _ borromean rays _ provided their union is knotted , but the union of any two components is unknotted . we construct infinitely many borromean rays , uncountably many of which are pairwise inequivalent . we obtain uncountably many borromean hyperplanes .

1211.6465

theoretical descriptions of photonic crystal fibers ( pcfs ) have traditionally been restricted to numerical evaluation of maxwell s equations . in the most general case , a plane wave expansion method with periodic boundary conditions is employed @xcite while other methods , such as the multipole method @xcite , take advantage of the localized nature of the guided modes and to some extend the circular shape of the air - holes . the reason for the application of these methods is the relatively complex dielectric cross section of a pcf for which rotational symmetry is absent . the aim of this work is to provide a set of numerically based empirical expressions describing the basic properties such as cutoff and mode - field radius of a pcf based on the fundamental geometrical parameters only . we consider the fiber structure first studied by knight _ _ @xcite and restrict our study to fibers that consist of pure silica with a refractive index of 1.444 . the air holes of diameter @xmath1 are arranged on a triangular grid with a pitch , @xmath2 . in the center an air hole is omitted creating a central high index defect serving as the fiber core . a schematic drawing of such a structure is shown in the inset of the right panel in fig . [ fig1 ] . depending on the dimensions , the structure comprises both single- and multi - mode fibers with large mode area as well as nonlinear fibers . the results presented here cover relative air hole sizes , @xmath3 , from 0.2 to 0.9 and normalized wavelengths , @xmath4 , from around 0.05 to 2 . the modeling is based on the plane - wave expansion method with periodic boundary conditions @xcite . for the calculations of guided modes presented the size of the super cell was @xmath5 resolved by @xmath6 plane waves while for calculations on the cladding structure only , the super cell was reduced to a simple cell resolved by @xmath7 planes waves . when attempting to establish a simple formalism for the pcf it is natural to strive for a result similar to the @xmath0parameter known from standard fibers @xcite . however , a simple translation is not straight forward since no wavelength - independent core- or cladding index can be defined . recently , we instead proposed a formulation of the @xmath0parameter for a pcf given by @xcite @xmath8 although this expression has the same overall mathematical form as known from standard fibers , the unique nature of the pcf is taken into account . in eq . ( [ vpcf ] ) , @xmath9 is the wavelength dependent effective index of the fundamental mode ( fm ) and @xmath10 is the corresponding effective index of the first cladding mode in the infinite periodic cladding structure often denoted the fundamental space filling mode ( fsm ) . for a more detailed discussion of this expression and its relation to previous work we refer to ref . @xcite and references therein . we have recently argued that the higher - order mode cut - off can be associated with a value of @xmath11 @xcite and showed that this criterion is indeed identical to the single - mode boundary calculated from the multipole method @xcite . recently the cut off results have also been confirmed experimentally @xcite . further supporting the definition of @xmath12 is the recent observation @xcite that the relative equivalent mode field radius of the fundamental mode , @xmath13 as function of @xmath12 fold over a single curve independent of @xmath3 . the mode field radius @xmath14 is defined as @xmath15 and corresponds to the @xmath16 width of a gaussian intensity distribution with the same effective area , @xmath17 , as the fundamental mode itself @xcite . in the left panel of fig . [ fig1 ] , calculated curves of @xmath12 as function of @xmath18 are shown for @xmath3 ranging from 0.20 to 0.70 in steps of 0.05 . in general , all curves are seen to approach constant levels dependent on @xmath3 . the horizontal dashed line indicates the single - mode boundary @xmath11 . in the right panel , @xmath13 is plotted as function of @xmath12 for each of the 9 curves in the left panel and as seen all curves fold over a single curve . an empirical expression for @xmath13 can be found in ref . the mode is seen to expand rapidly for small values of @xmath12 and the mode - field radius saturates toward a constant value when @xmath12 becomes large . in fact , it turns out that @xmath19 for @xmath20 and @xmath21 for @xmath11 . in the left panel of fig . [ fig2 ] , curves corresponding to constant values of @xmath12 are shown in a @xmath4 versus @xmath3 plot . in the right panel , curves of constant @xmath13 is shown , also in a @xmath4 versus @xmath3 plot . since there is a unique relation between @xmath13 and @xmath12 @xcite the curves naturally have the same shape . when designing a pcf any combination of @xmath1 and @xmath2 is in principle possible . however , in some cases the guiding will be weak causing the mode to expand beyond the core and into the cladding region @xcite corresponding to a low value of @xmath12 . in the other extreme , the confinement will be too strong allowing for the guiding of higher - order modes @xcite . since both situations are governed by @xmath12 the design relevant region in a @xmath4 versus @xmath3 plot can be defined . this is done in fig . [ fig3 ] where the low limit is chosen to be @xmath22 where @xmath23 . how large a mode that can be tolerated is of course not unambiguous . however , for @xmath24 leakage - loss typically becomes a potential problem in pcfs with a finite cladding structure . in non - linear pcfs it is for dispersion reasons often advantageous operating the pcf at @xmath25 and then a high number of air - hole rings is needed to achieve an acceptable level of leakage loss @xcite . finally , we note that the practical operational regime is also limited from the low wavelength side . in ref . @xcite a low - loss criterion was formulated in terms of the coupling length @xmath26 $ ] between the fm and the fsm . in general scattering - loss due to longitudinal non - uniformities increases when @xmath27 increases and a pcf with a low @xmath27 will in general be more stable compared to one with a larger @xmath27 . using @xmath28 we can rewrite eq . ( [ vpcf ] ) as @xmath29 from which it is seen that a high value of the @xmath0parameter is preferred over a smaller value . in fig . ( [ fig3 ] ) it is thus preferable to stay close to the single - mode boundary ( @xmath30 ) but in general there is a practical lower limit to the value of @xmath4 which can be realized because when @xmath31 one generally has that @xmath32 @xcite . although the @xmath0parameter offers a simple way to design a pcf , a limiting factor for using eq . ( [ vpcf ] ) is that a numerical method is still required for obtaining the effective indices . in analogy with expressions for standard fibers @xcite it would therefore be convenient to have an alternative expression only dependent on the wavelength , @xmath33 , and the structural parameters @xmath1 and @xmath2 . in fig . [ fig4 ] , we show @xmath12 as function of @xmath4 ( data are shown by open circles ) for @xmath3 ranging from 0.20 to 0.80 in steps of 0.05 . each data set in fig . [ fig4 ] is fitted to a function of the form [ vpcf_fit ] @xmath34 + 1}\ ] ] and the result is indicated by the full red lines . ( [ vpcf_fit_v ] ) is not based on considerations of the physics of the v - parameter but merely obtained by trial and error in order to obtain the best representation of calculated data with the lowest possible number of free parameters . prior to the fit , the data sets are truncated at @xmath35 since @xmath36 in this region ( see left panel in fig . [ fig1 ] ) and the data is thus not practically relevant . in eq . ( [ vpcf_fit_v ] ) the fitting parameters @xmath37 , @xmath38 , and @xmath39 depend on @xmath3 only . in order to extract this dependency , suitable functions ( again obtained by trial and error ) are fitted to the data sets for @xmath37 , @xmath38 , and @xmath39 . we find that the data are well described by the following expressions @xmath40 @xmath41 @xmath42 the above set of expressions , eqs . ( [ vpcf_fit ] ) , constitute our empirical expression for the @xmath0parameter in a pcf with @xmath4 and @xmath3 being the only input parameters . for @xmath43 and @xmath44 the expression gives values of @xmath12 which deviates less than @xmath45 from the correct values obtained from eq . ( [ vpcf ] ) . the term endlessly single - mode ( esm ) refers to pcfs which regardless of wavelength only support the two degenerate polarization states of the fundamental mode @xcite . in the framework of the @xmath0parameter this corresponds to structures for which @xmath46 for any @xmath4 @xcite . as seen in the left panel of fig . [ fig1 ] this corresponds to sufficiently small air holes . however , from the plot in fig . [ fig1 ] it is quite difficult to determine the exact @xmath3 value for which @xmath11 for @xmath33 approaching 0 . from eq . ( [ vpcf_fit ] ) it is easily seen that the value may be obtained from @xmath47 fig . [ fig5 ] illustrates this equation graphically where we have extrapolated the data in fig . [ fig4 ] to @xmath48 . from the intersection of the full line with the dashed line we find that @xmath49 bounds the esm regime . solving eq . ( [ d_esm ] ) we get @xmath50 and the deviation from the numerically obtained value is within the accuracy of the empirical expression . there are several issues to consider when designing a pcf . in this work we have addressed the single / multi - mode issue as well as those related to mode - field radius / field - confinement , and mode - spacing . we have shown how these properties can be quantified via the @xmath0parameter . based on extensive numerics we have established an empirical expression which facilitate an easy evaluation of the @xmath0-parameter with the normalized wavelength and hole - size as the only input parameters . we believe that this expression provides a major step away from the need of heavy numerical computations in design of solid core pcfs with triangular air - hole cladding . we thank j. r. folkenberg for stimulating discussion and m. d. nielsen acknowledges financial support by the danish academy of technical sciences .

based on a recent formulation of the @xmath0parameter of a photonic crystal fiber we provide numerically based empirical expressions for this quantity only dependent on the two structural parameters the air hole diameter and the hole - to - hole center spacing . based on the unique relation between the @xmath0parameter and the equivalent mode field radius we identify how the parameter space for these fibers is restricted in order for the fibers to remain single mode while still having a guided mode confined to the core region . 10 s. g. johnson and j. d. joannopoulos , `` block - iterative frequency - domain methods for axwell s equations in a planewave basis , '' opt . express * 8 * , 173 ( 2001 ) , + http://www.opticsexpress.org/abstract.cfm?uri=opex-8-3-173 . t. p. white , b. t. kuhlmey , r. c. mcphedran , d. maystre , g. renversez , c. m. de sterke , and l. c. botton , `` multipole method for microstructured optical fibers . i. formulation , '' j. opt . soc . am . b * 19 * , 2322 ( 2002 ) . j. c. knight , t. a. birks , p. s. j. russell , and d. m. atkin , `` all - silica single - mode optical fiber with photonic crystal cladding , '' opt . lett . * 21 * , 1547 ( 1996 ) . a. w. snyder and j. d. love , _ optical waveguide theory _ ( chapman & hall , new york , 1983 ) . d. marcuse , `` gaussian approximation of the fundamental modes of graded - index fibers , '' j. opt . . am . * 68 * , 103 ( 1978 ) . n. a. mortensen , j. r. folkenberg , m. d. nielsen , and k. p. hansen , `` modal cut - off and the @xmath0parameter in photonic crystal fibers , '' opt . lett . * 28 * , 1879 ( 2003 ) . b. t. kuhlmey , r. c. mcphedran , and c. m. de sterke , `` modal cutoff in microstructured optical fibers , '' opt . lett . * 27 * , 1684 ( 2002 ) . j. r. folkenberg , n. a. mortensen , k. p. hansen , t. p. hansen , h. r. simonsen , and c. jakobsen , `` experimental investigation of cut - off phenomena in non - linear photonic crystal fibers , '' opt . lett . * 28 * , 1882 ( 2003 ) . m. d. nielsen , n. a. mortensen , j. r. folkenberg , and a. bjarklev , `` mode - field radius of photonic crystal fibers expressed by the @xmath0parameter , '' opt . lett . * 28 * , in press ( 2003 ) , + http://arxiv.org/abs/physics/0309030 . t. p. white , r. c. mcphedran , c. m. de sterke , l. c. botton , and m. j. steel , `` confinement losses in microstructured optical fibers , '' opt . lett . * 26 * , 1660 ( 2001 ) . b. t. kuhlmey , r. c. mcphedran , c. m. de sterke , p. a. robinson , g. renversez , and d. maystre , `` microstructured optical fibers : where s the edge ? , '' opt . express * 10 * , 1285 ( 2002 ) , + http://www.opticsexpress.org/abstract.cfm?uri=opex-10-22-1285 . w. h. reeves , j. c. knight , p. s. j. russell , and p. j. roberts , `` demonstration of ultra - flattened dispersion in photonic crystal fibers , '' opt . express * 10 * , 609 ( 2002 ) , + http://www.opticsexpress.org/abstract.cfm?uri=opex-10-14-609 . n. a. mortensen and j. r. folkenberg , `` low - loss criterion and effective area considerations for photonic crystal fibers , '' j. opt . a : pure appl . opt . * 5 * , 163 ( 2003 ) . t. a. birks , j. c. knight , and p. s. j. russell , `` endlessly single mode photonic crystal fibre , '' opt . lett . * 22 * , 961 ( 1997 ) .

physics0310065

classification problem is one of the most important tasks in time series data mining . a well - known 1-nearest neighbor ( 1-nn ) with dynamic time warping ( dtw ) distance is one of the best classifier to classify time series data , among other approaches , such as support vector machine ( svm ) @xcite , artificial neural network ( ann ) @xcite , and decision tree @xcite . for the 1-nn classification , selecting an appropriate distance measure is very crucial ; however , the selection criteria still depends largely on the nature of data itself , especially in time series data . though the euclidean distance is commonly used to measure the dissimilarity between two time series , it has been shown that dtw distance is more appropriate and produces more accurate results . sakoe - chiba band ( s - c band ) @xcite originally speeds up the dtw calculation and later has been introduced to be used as a dtw global constraint . in addition , the s - c band was first implemented for the speech community , and the width of the global constraint was fixed to be 10% of time series length . however , recent work @xcite reveals that the classification accuracy depends solely on this global constraint ; the size of the constraint depends on the properties of the data at hands . to determine a suitable size , all possible widths of the global constraint are tested , and the band with the maximum training accuracy is selected . ratanamahatana - keogh band ( r - k band ) @xcite has been introduced to generalize the global constraint model represented by a one - dimensional array . the size of the array and the maximum constraint value is limited to the length of the time series . and the main feature of the r - k band is the multi bands , where each band is representing each class of data . unlike the single s - c band , this multi r - k bands can be adjusted as needed according to its own class warping path . although the r - k band allows great flexibility to adjust the global constraint , a learning algorithm is needed to discover the best multi r - k bands . in the original work of r - k band , a hill climbing search algorithm with two heuristic functions ( accuracy and distance metrics ) is proposed . the search algorithm climbs though a space by trying to increase / decrease specific parts of the bands until terminal conditions are met . however , this learning algorithm still suffers from an overfitting phenomenon since an accuracy metric is used as a heuristic function to guide the search . to solve this problem , we propose two new learning algorithms , i.e. , band boundary extraction and iterative learning . the band boundary extraction method first obtains a maximum , mean , and mode of the paths positions on the dtw distance matrix , and the iterative learning , band s structures are adjusted in each round of the iteration to a silhouette index @xcite . we run both algorithms and the band that gives better results . in prediction step , the 1-nn using dynamic time warping distance with this discovered band is used to classify unlabeled data . note that a lower bound , lb_keogh @xcite , is also used to speed up our 1-nn classification . the rest of this paper is organized as follows . section 2 gives some important background for our proposed work . in section 3 , we introduce our approach , the two novel learning algorithms . section 4 contains an experimental evaluation including some examples of each dataset . finally , we conclude this paper in section 5 . our novel learning algorithms are based on four major fundamental concepts , i.e. , dynamic time warping ( dtw ) distance , sakoe - chiba band ( s - c band ) , ratanamahatana - keogh band ( r - k band ) , and silhouette index , which are briefly described in the following sections . dynamic time warping ( dtw ) @xcite distance is a well - known similarity measure based on shape . it uses a dynamic programming technique to find all possible warping paths , and selects the one with the minimum distance between two time series . to calculate the distance , it first creates a distance matrix , where each element in the matrix is a cumulative distance of the minimum of three surrounding neighbors . suppose we have two time series , a sequence @xmath0 of length @xmath1 ( @xmath2 ) and a sequence @xmath3 of length @xmath4 ( @xmath5 ) . first , we create an @xmath1-by-@xmath4 matrix , where every ( @xmath6 ) element of the matrix is the cumulative distance of the distance at ( @xmath6 ) and the minimum of three neighboring elements , where @xmath7 and @xmath8 . we can define the ( @xmath6 ) element , @xmath9 , of the matrix as : @xmath10 where @xmath11 is the squared distance of @xmath12 and @xmath13 , and @xmath9 is the summation of @xmath14 and the the minimum cumulative distance of three elements surrounding the ( @xmath6 ) element . then , to find an optimal path , we choose the path that yields a minimum cumulative distance at ( @xmath15 ) , which is defined as : @xmath16 where @xmath17 is a set of all possible warping paths , @xmath18 is ( @xmath6 ) at @xmath19 element of a warping path , and @xmath20 is the length of the warping path . in reality , dtw may not give the best mapping according to our need because it will try its best to find the minimum distance . it may generate the unwanted path . for example , in figure [ flo : dtw1 ] @xcite , without global constraint , dtw will find its optimal mapping between the two time series . however , in many cases , this is probably not what we intend , when the two time series are expected to be of different classes . we can resolve this problem by limiting the permissible warping paths using a global constraint . two well - known global constraints , sakoe - chiba band and itakura parallelogram @xcite , and a recent representation , ratanamahatana - keogh band ( r - k band ) , have been proposed , figure [ flo : dtw2 ] @xcite shows an example for each type of the constraints . [ cols="^,^ " , ] [ flo : result ] in this work , we propose a new efficient time series classification algorithm based on 1-nearest neighbor classification using the dynamic time warping distance with multi r - k bands as a global constraint . to select the best r - k band , we use our two proposed learning algorithms , i.e. , band boundary extraction algorithm and iterative learning . silhouette index is used as a heuristic function for selecting the band that yields the best prediction accuracy . the lb_keogh lower bound is also used in data prediction step to speed up the computation . we would like to thank the scientific parallel computer engineering ( space ) laboratory , chulalongkorn university for providing a cluster we have used in this contest . 1 fumitada itakura . minimum prediction residual principle applied to speech recognition . , 23(1):6772 , 1975 . eamonn j. keogh and chotirat ann ratanamahatana . exact indexing of dynamic time warping . , 7(3):358386 , 2005 . alex nanopoulos , rob alcock , and yannis manolopoulos . feature - based classification of time - series data . , pages 4961 , 2001 . chotirat ann ratanamahatana and eamonn j. keogh . making time - series classification more accurate using learned constraints . in _ proceedings of the fourth siam international conference on data mining ( sdm 2004 ) _ , pages 1122 , lake buena vista , fl , usa , april 22 - 24 2004 . chotirat ann ratanamahatana and eamonn j. keogh . three myths about dynamic time warping data mining . in _ proceedings of 2005 siam international data mining conference ( sdm 2005 ) _ , pages 506510 , newport beach , cl , usa , april 21 - 23 2005 . juan jos rodrguez and carlos j. alonso . interval and dynamic time warping - based decision trees . in _ proceedings of the 2004 acm symposium on applied computing ( sac 2004 ) _ , pages 548552 , nicosia , cyprus , march 14 - 17 2004 . peter rousseeuw . silhouettes : a graphical aid to the interpretation and validation of cluster analysis . , 20(1):5365 , 1987 . hiroaki sakoe and seibi chiba . dynamic programming algorithm optimization for spoken word recognition . , 26(1):4349 , 1978 . yi wu and edward y. chang . distance - function design and fusion for sequence data . in _ proceedings of the 2004 acm cikm international conference on information and knowledge management ( cikm 2004 ) _ , pages 324333 , washington , dc , usa , november 8 - 13 2004 .

1-nearest neighbor with the dynamic time warping ( dtw ) distance is one of the most effective classifiers on time series domain . since the global constraint has been introduced in speech community , many global constraint models have been proposed including sakoe - chiba ( s - c ) band , itakura parallelogram , and ratanamahatana - keogh ( r - k ) band . the r - k band is a general global constraint model that can represent any global constraints with arbitrary shape and size effectively . however , we need a good learning algorithm to discover the most suitable set of r - k bands , and the current r - k band learning algorithm still suffers from an overfitting phenomenon . in this paper , we propose two new learning algorithms , i.e. , band boundary extraction algorithm and iterative learning algorithm . the band boundary extraction is calculated from the bound of all possible warping paths in each class , and the iterative learning is adjusted from the original r - k band learning . we also use a silhouette index , a well - known clustering validation technique , as a heuristic function , and the lower bound function , lb_keogh , to enhance the prediction speed . twenty datasets , from the workshop and challenge on time series classification , held in conjunction of the sigkdd 2007 , are used to evaluate our approach .

0903.0041

matrices with random ( or pseudo - random ) elements appear naturally in many different problems and are well investigated in physical and mathematical literature ( see e.g. @xcite ) . in this note we consider a special class of sparse random matrices , namely , matrices associated with tree ( or tree - like ) structures . a fundamental property of such matrices is that the number of non - zero elements in each row and column either remains finite or grows much slowly than the matrix dimension when the latter increases . as usual , a ( connected ) tree is a graph where any pair of vertices is connected by only one path without repeating vertices . vertices are labeled by integers . if a symmetric ( or hermitian ) matrix @xmath0 is such that its entries @xmath1 , @xmath2 , are non - zero if and only if vertices @xmath3 and @xmath4 are connected on a given tree , then the matrix @xmath0 is said to be associated with the tree . an example of such a matrix is the adjacency matrix of the tree . in general , diagonal entries of @xmath0 are nonzero . let @xmath5 be the green function for matrix @xmath0 , namely @xmath6 where @xmath7 is the identity matrix . it is well established that for matrices associated with a tree there exist recursive relations which connect the diagonal elements @xmath8 of the green function with similar quantities but for smaller matrices . probably the simplest way to derive such relations on a tree is to use an easily verified identity ( sometimes called the schur complement formula ) , which states that for any matrix @xmath0 one has @xmath9 where @xmath10 is the green function as in but for the matrix @xmath11 obtained from @xmath0 by removing row @xmath12 and column @xmath12 . for a tree , removing one vertex splits the remaining graph into a disjoint union of smaller trees . thus , @xmath11 is block - diagonal and the green function @xmath10 has no matrix elements between different neighbors of a fixed site @xmath12 . it means that for trees eq . takes the form @xmath13 where the sum is taken over all neighbors of site @xmath12 and we assume that matrix @xmath0 is hermitian . applying the same arguments to @xmath10 leads to a similar equation for each neighbor @xmath14 of @xmath12 , namely @xmath15 where the sum is over all neighbors of @xmath14 except @xmath12 ( which has already been removed ) . in principle , @xmath16 in the right - hand side of this equation is the green function element for a matrix obtained from @xmath0 by removing two connected sites @xmath12 and @xmath14 . but because we are on a tree , @xmath16 is the same whether only @xmath14 or all its ancestors are removed ; thus it is sufficient to indicate only the last removed site . it is this property which permits to write the recursive relation where on both sides similar quantities are present . for finite trees the above relations allow to calculate the green function recursively , but their the most important application corresponds to infinite ( or very large ) trees where matrix elements of @xmath17 are assumed to be independent random variables ( or constants ) . in this case , because of the disjoint nature of different sub - trees , eq . for all @xmath14 only involves variables @xmath18 from the sub - tree to which @xmath14 belongs . thus for large uniform trees , one can assume that all @xmath19 are independent random variables having the same distribution . let @xmath20 be the coordination number of vertex @xmath14 , and @xmath21 , @xmath22 , be random variables distributed according to that distribution . for simplicity we assume below that the diagonal matrix element @xmath23 is a random variable @xmath24 , off - diagonal elements @xmath18 are real i.i.d . variables @xmath25 , and all coordination numbers are equal to @xmath20 . then eq . means that the random variable @xmath26 has the same distribution as the @xmath21 . this equation is the main tool for the investigation of random uniform trees . it has been obtained initially by abou - chacra , thouless , and anderson in their study of self - consistent theory of localization @xcite and later it has been re - derived by many different methods : replica formalism @xcite , @xmath27-model @xcite , ricatti equation @xcite , rank - one perturbation @xcite , cavity method @xcite etc . we shall refer below to as the tree equation . strictly speaking , the tree equation is valid only for infinite uniform trees ( the bethe lattice ) but in many cases it is applied to models which have only tree - like structure , i.e. which can locally be approximated by trees but may have loops of large length . a typical example is that of random regular graphs , where each vertex has the same number of neighbors , as in a tree with constant coordination number , but where the boundary shell present in finite trees is absent ( see e.g. @xcite ) . for certain tree - like models the validity of the tree equation can be proved rigorously @xcite . there exist three main types of tree - like problems , corresponding to three possible sources of randomness in the tree equation . the first corresponds to a regular tree with a fixed coordination number and with only diagonal disorder ( i.e. @xmath24 is a random variable and @xmath28 ) . this model has been proposed in a seminal paper @xcite and was recently investigated in @xcite and @xcite . the second class ( see e.g. @xcite ) corresponds to models which are also defined on a fixed regular tree but have only off - diagonal disorder ( i.e. all @xmath25 are i.i.d . random variables and @xmath29 ) . finally , the third type of models includes trees without disorder but with fluctuating coordination number . characteristic examples of such models are erds - rnyi graphs @xcite or sparse random matrices with a finite connectivity @xcite . in this note , as an example , we will restrict ourselves to random regular graphs . a general method for numerically solving the tree equation has been proposed in @xcite and it is commonly used now under the name of belief propagation method @xcite . the main steps of this method are as follows . first , fix arbitrarily an initial sample of a large number , say @xmath30 , of elements @xmath31 , @xmath32 . second , choose randomly @xmath33 integers from @xmath34 to @xmath30 and variables @xmath24 and @xmath25 from their known distributions . third , calculate @xmath35 from the tree equation . fourth , choose randomly an element from the initial sample and replace it by the calculated @xmath35 . repeat these steps till the convergence of the resulting distribution to the distribution of @xmath35 . there exist two types of solutions of . the first one corresponds to real values of the energy @xmath36 : at each step of the iteration , the variables @xmath31 are real and after iteration one obtains the distribution of the real variable @xmath35 . the second type of solution corresponds to adding a small positive imaginary part to the energy , that is , put @xmath37 in . in this case , upon iteration the @xmath31 become complex variables and the result of the iteration will yield the distribution of both the real and the imaginary part of @xmath35 , which may and will depend on @xmath38 . if the energy @xmath36 corresponds to a region of localized states , then when @xmath39 the imaginary part of @xmath35 will tend to zero almost everywhere , while if the energy corresponds to a region of non - localized states the imaginary part of @xmath35 goes to a finite distribution . here we restrict ourselves to real values of the energy , with @xmath40 . the purpose of this note is to discuss the calculation of the mean spectral density of typical tree - like models by construction of approximate solution of the tree equation . strictly speaking , the method is valid for trees with a large coordination number but often gives good results even at small coordination numbers . the method itself is not new and has been used widely for random erds - rnyi graphs ( see e.g. @xcite@xcite and references therein ) . we first clarify certain important points which seem not to be discussed in the literature and then investigate in detail the application of the method to random regular graphs and compare different types of approximations . the plan of the paper is the following . after setting some definitions in section [ rhodee ] , section [ mean_field ] is devoted to the general discussion of the method . it is demonstrated that the commonly used mean - field solution of the tree equation corresponds to the approximation of the exact solution by a symmetric cauchy distribution whose parameters are calculated self - consistently . different useful formulas for the mean spectral density are briefly discussed in this section . in section [ diagonal ] the case of regular graphs with diagonal disorder is considered . by comparing results of direct numerical calculations with various kinds of approximations we check their precision and found that at zeroth order the best results for regular trees with diagonal disorder is given by the so - called single defect approximation proposed in @xcite . section [ off_diagonal ] treats the case of regular graphs with off - diagonal disorder . for these models the best results are obtained by using the modified effective medium approximation introduced in section [ mean_field ] . in all considered cases , the next order approximation for the tree equation solution , though it agrees much better with the numerical solution obtained by belief propagation , improves noticeably the spectral density only at lowest coordination numbers . the conclusion of this note ( stated in section [ conclusion ] ) is that the approximate solution of the tree equation is useful , flexible , and general method of calculation of the mean spectral density in various uniform tree - like models . the mean spectral density is defined as usual by @xmath41 where @xmath42 are eigenvalues of @xmath0 and the average is performed over random realizations of the matrix entries . as mentioned in the introduction , here we only consider real solutions of eq . . let us denote by @xmath43 and @xmath44 the ( known ) probability densities of @xmath24 and @xmath45 . the probability density of variable @xmath46 is given by some function @xmath47 , and that of variable @xmath48 by a function @xmath49 . by definition , @xmath50 is @xmath51 the tree equation implies that for real @xmath52 the probability density @xmath47 satisfies the equation @xmath53 or equivalently @xmath54 eigenvalues of @xmath0 correspond to values of @xmath36 where @xmath55 gets singular . assuming that @xmath56 and @xmath57 in can be replaced by random variables , which we denote by @xmath24 and @xmath58 respectively , we see that these singularities occur at @xmath59 . under these assumptions , the definition gives @xmath60 where the summation is performed over all @xmath20 neighbors of a given site . another useful expression for the mean spectral density is obtained by transforming eq . into the form @xmath61 performing the integral over @xmath62 in eq . yields @xmath63 of course , the above formulas can be rewritten in many equivalent forms . therefore the knowledge of the spectral density can directly be deduced from that of the distribution @xmath47 which is a solution of the tree equation , or from that of @xmath49 . in the next section we consider simple solutions of these equations . the tree equation may have a constant complex solution , that is , a @xmath64 ( depending on @xmath36 ) such that @xmath65 if such a @xmath66 exists , the mean spectral density associated with this solution is obtained from as @xmath67 using contour integration , this expression can be equivalently rewritten as @xmath68 when the imaginary part of @xmath69 is positive . equation is known as the single defect approximation ( sda ) @xcite . to further simplify this expression one can simply argue that for large @xmath33 ( which , as we see below , is the parameter which control this type of approximation ) the sum over @xmath20 terms in eq . can be approximated by a sum over @xmath33 terms , so that from one concludes that the mean spectral density takes the form @xmath70 this type of approximation is widely applied for sparse random matrices , where eq . is called the effective medium approximation ( ema ) @xcite . physically , this solution is a kind of a mean field and often these equations are obtained by arguing that the random variable @xmath52 distributed according to @xmath47 fluctuates slowly around its mean value @xmath64 ( see e.g. @xcite ) . such an approach is simple , physically transparent , does not require heavy numerical calculations , and gives , as a rule , quite good results . the trouble is that its main assumption that variable @xmath52 fluctuates only slowly around its mean value @xmath64 can not be , in general , correct for tree - like models . indeed , it follows from eq . that if the function @xmath71 is smooth , then @xmath47 has to decrease as @xmath72 in particular , this means that @xmath47 belongs to the class of heavy - tail distributions , for which the mean value @xmath73 does not exist . therefore , the meaning of a ( complex ) mean field solution and , especially , its relation to a direct numerical solution of the tree equation for real @xmath52 remains obscure . the above mean field approach is equivalent to the assumption that the function @xmath47 can be approximated by the symmetric cauchy distribution @xmath74 with @xmath75 and @xmath76 real parameters , @xmath77 , and @xmath78 . indeed , for a test function @xmath79 without singularities in the upper - half plane one has the identity @xmath80 in other words , the cauchy distribution is indistinguishable from the @xmath81-function when acting on a large class of functions . in particular , when @xmath82 and @xmath83 are real and @xmath84 , contour integration yields the following formula @xmath85 other useful elementary properties of cauchy distributions that we will make use of are the following identities @xmath86 and the convolution property @xmath87 the characteristic function of is given by @xmath88 let us calculate the characteristic function of @xmath47 , @xmath89 using one obtains the exact functional relation @xmath90 if in the right - hand side of this equation @xmath47 is replaced by the cauchy distribution @xmath91 given by eq . , the consecutive use of gives a first - order approximation of the characteristic function of @xmath92 as @xmath93 in general @xmath94 . requiring that parameters @xmath75 and @xmath76 be such that at small argument @xmath95 is exactly equivalent to requiring that @xmath96 satisfies the mean field equation . these arguments can be reformulated as follows . let @xmath64 be a ( complex ) constant solution of and @xmath97 the corresponding cauchy distribution . then an approximate solution to such that @xmath98 for small arguments @xmath99 is obtained by replacing @xmath92 by @xmath100 in the right - hand side of eq . . it means that @xmath64 is not a mean value of a random variable as in the usual mean field approach but just a pole of the cauchy distribution which reproduces the behavior of the characteristic function @xmath101 at small @xmath102 . ( it is interesting to notice that for the usual goe ensemble of random matrices the local green function also has a cauchy distribution , see e.g. @xcite . ) the knowledge of @xmath64 permits to calculate easily the mean spectral density from eqs . or . in the cases considered in the next sections we found that approximation does not give good results at small values of @xmath33 . instead , we propose to use another simple approximation obtained by substituting the cauchy approximation into eq . . simple transformations based on eqs . lead to the following approximate formula for the mean spectral density @xmath103 this expression is almost as simple as ema but often gives better results . we refer to it as to the modified effective medium approximation ( mema ) . a few formulas are useful to mention . when @xmath97 is substituted into eq . one gets , after integration over @xmath104 , a first order approximation of @xmath49 , as @xmath105 the first iteration of the initial cauchy distribution , @xmath106 , is then straightforwardly obtained either from this expression and eq . , or directly from eq . : if we define @xmath107 then eq . can be rewritten @xmath108 where average is taken over @xmath24 and the @xmath109 . this is exactly the characteristic function of the cauchy distribution with parameters @xmath110 ( see eq . ) . thus one directly gets @xmath111 the next iterations can be calculated in the same manner . below we refer to the cauchy function @xmath91 with parameters from eq . as the zeroth order approximation of @xmath47 and to the first iteration as the first order approximation . for each order of approximation one can calculate the mean spectral density by using either eq . or eq . . both formulas are exact when @xmath47 obeys the tree equation , but for approximate expressions they may and will give different results . though , in principle , one can rely on the results only when these two expressions are close to each other , for a given problem usually one of these formulas works better and we shall indicate it . let us now consider in detail the example of a regular tree ( which will be approximated by a regular graph to avoid boundary effects ) where each vertex has @xmath20 neighbors . the matrix we consider is the adjacency matrix of the graph , to which we add a diagonal part given by i.i.d . random variables with e.g. the uniform distribution @xmath112 it is this model which has been considered in @xcite and later has been investigated in many places ( see e.g. @xcite and references therein ) . such a model corresponds to the situation discussed in section [ mean_field ] with fixed @xmath113 . therefore the mean field equation reads @xmath114 equation implies that @xmath115 is the the probability distribution of a sum of @xmath33 i.i.d . variables distributed according to the law @xmath47 . from it follows that @xmath47 is determined by the @xmath33-fold convolution of @xmath47 ( cf . ) . when @xmath33 is large , the generalized central limit theorem ( see e.g. @xcite ) can be applied . it states that a properly normalized sum of i.i.d . random variables tends to the one of known stable distributions . if the second moment is finite , the limiting distribution is the gaussian . for heavy - tail distributions , when the second moment does not exist , the limiting distribution is one of the levy distributions . the asymptotics means that @xmath47 belongs to the domain of attraction of the symmetric cauchy distribution , so at large @xmath33 the probability density of @xmath33-copies of @xmath52 in the bulk is close to @xmath116 characterized by the two parameters @xmath117 and @xmath118 , with @xmath119 . important is that these parameters are determined only from the behavior of the characteristic function of @xmath47 at small @xmath102 , @xmath120 as in . the mean spectral density is then given by which , using the approximation , yields @xmath121 which corresponds exactly to sda . note that the function @xmath122 has the meaning of a strength function @xcite where one uses as initial wave function the one localized at a site with energy @xmath123 . the simplest case of the above formalism corresponds to the absence of disorder , i.e. @xmath124 . in this case . gives , for @xmath125 , a solution @xmath119 with @xmath126 while for @xmath127 , @xmath128 . from eq . , the mean spectral density in this case is given at order 0 by @xmath129 which agrees with kesten - mckay law for random regular graphs @xcite . for a general distribution @xmath43 of diagonal elements , such as , parameters @xmath76 and @xmath75 are easily calculated numerically from eq . . qualitatively , they have a shape ( as a function of @xmath36 ) similar to . a simple way to find @xmath64 is the direct iteration of eq . starting from a complex initial guess @xmath130 , i.e. @xmath131 where @xmath132 denotes the rhs of eq . . to ensure stability it is convenient to use a slightly different iteration scheme , @xmath133 with a certain @xmath134 . once parameters @xmath76 and @xmath75 are tabulated as functions of energy , the spectral density can be calculated either from eq . or from eq . we found that eq . gives better results at small @xmath33 . the sda is plotted for a few values of @xmath33 and @xmath135 in figs . [ fig_diag_0.3 ] and [ fig_graph ] for the uniform disorder distribution . the agreement between the above simple formulas and the results of direct numerical diagonalization of the corresponding matrices is quite good in the bulk of the spectra even for the smallest value of the coordination number . deviations are clearly present only near the spectral ends , where in all cases the discussed approach can not be applied . and @xmath136 ( black circles ) , @xmath137 ( red squares ) , and @xmath138 ( blue triangles ) , obtained from computations done on @xmath139-vertex regular graphs , with @xmath140 different realizations of graphs and disorder . dashed lines of corresponding color indicate zeroth order sda formulas . solid lines corresponds to the next iteration of the density . for @xmath141 these two curves are practically the same and only sda approximation is shown . ] but for @xmath142 ( left panel ) and @xmath143 ( right panel ) . solid lines of corresponding color indicate zeroth order sda formulas . the next order approximations at the scale of the figures are hardly distinguishable from the sda and are not presented . ] but for @xmath142 ( left panel ) and @xmath143 ( right panel ) . solid lines of corresponding color indicate zeroth order sda formulas . the next order approximations at the scale of the figures are hardly distinguishable from the sda and are not presented . ] to investigate more precisely the accuracy of these different approximations , we calculate numerically the distribution @xmath47 directly from the tree equation by the belief propagation method explained in section [ introduction ] . the results for various disorder strengths are presented in figs . [ w_0.3 ] and [ w_12 ] . for each @xmath144 and each energy @xmath36 , we calculate the zeroth order and the first order approximations to @xmath47 . for small values of @xmath144 ( fig . [ w_0.3 ] ) the two approximations are practically the same but for larger @xmath144 they are clearly different . in all considered cases the first order approximation @xmath106 given by eq . is in a good qualitative agreement with direct numerical solution of the tree equation . the first order approximation for the mean spectral density can be calculated e.g. from eq . which , for our model , reads @xmath145 where @xmath106 is calculated from . calculated numerically from the tree equation for regular random graphs with @xmath146 and @xmath141 at different energies , from left to right @xmath147 , @xmath148 and @xmath149 . black circles indicate results of direct numerical solution of the tree equation by belief propagation method , using a sample of @xmath150 initial values and performing @xmath151 iterations . red dashed lines show the zeroth order ( cauchy ) approximation . the next approximation is practically indistinguishable from the zeroth one and is not shown on the plots.,title="fig : " ] + but with @xmath152 . from left to right : @xmath147 , @xmath153 , and @xmath154 . black circles indicate results of direct numerical solution of the tree equation by belief propagation . red dashed lines show the zeroth - order ( cauchy ) approximation . black solid lines are the first order approximation . , title="fig : " ] + from fig . [ fig_diag_0.3 ] it is clear that this approximation agrees better with direct numerical calculations at small @xmath33 . for larger @xmath33 the zeroth order and the first order formulas are practically indistinguishable in the scale of the figures . this robustness can be explained as follows . as has been mentioned in section [ mean_field ] , the choice of the cauchy distribution as the zeroth order approximation is to a large degree arbitrary ( what matters is only the asymptotics ) . on the other hand , according to the generalized central limit theorem the @xmath33-fold convolution , @xmath71 , of the chosen zeroth order function is in the bulk universal and is not sensitive to details of the initial function . therefore , quantities which can be expressed through @xmath71 and @xmath155 at a low order are more precisely described by low - order approximations than those related directly with the initial function . in this section we consider a different type of tree - like models , namely matrices associated with regular graphs with fixed coordination number @xmath20 , with diagonal elements set to zero and off - diagonal matrix elements defined as i.i.d . random variables . the densities @xmath47 and @xmath156 are given by eqs . and with @xmath124 ( since diagonal elements are zero ) and a certain function @xmath44 which determines the probability density of off - diagonal elements . we choose as @xmath44 the gaussian distribution with variance @xmath27 and zero mean @xmath157 this choice is not essential but simplifies analytical calculations . note that changing the variance corresponds to a rescaling of the energy ; therefore in what follows we set @xmath158 . it is convenient to define the integral @xmath159 which for integer @xmath33 reduces to well - known standard functions . many quantities defined in section [ mean_field ] can be expressed through this integral . the transformations are straightforward and results are as follows . the mean - field equation becomes @xmath160 the first order of the @xmath33-fold convolution , eq . , reads @xmath161\ .\ ] ] the two approximations and proposed in section [ mean_field ] for the spectral density are now given by @xmath162 \label{rho_mema}\ ] ] and @xmath163\ . \label{sda}\ ] ] the first order approximation of the distribution @xmath47 , obtained from the first iteration of the cauchy function , is readily expressed from eq . as @xmath164\ . \label{g_1_integral}\ ] ] when @xmath165 or @xmath166 , the main contribution in integral comes from regions where @xmath167 , and one can approximate @xmath168 as @xmath169 the solution of eq . in this regime reads @xmath170 one can then check , using again , that in this limit @xmath171 and @xmath172 given by and tend to the kesten - mckay law for the regular graph @xmath173 where for convenience we reintroduce the variance @xmath27 of off - diagonal elements . the limit @xmath174 gives the semi - circle density @xmath175 as should be from general considerations @xcite . for small values of @xmath33 the situation is different . it is known ( see e.g. @xcite ) that in graphs with off - diagonal disorder , where the probability of small values of @xmath45 is non - zero , the mean density of states has a singularity at small energies of the form @xmath176 where @xmath177 is a certain positive constant . this peak is related with a possibility of approximate localization at small sub - graphs which are isolated from the bulk due to small values of corresponding off - diagonal elements . numerical calculations for different @xmath33 and off - diagonal disorder given by are presented in fig . [ off_fig ] . in the same figure mema expressions are plotted . we checked that sda formulas give worse results at small @xmath33 . the overall agreement of approximate formulas and numerics is reasonably good but the presence of the peak is clearly visible especially at small @xmath33 . , red triangles to @xmath178 , and blue squares to @xmath143 . solid lines of the same color indicate the zeroth order mema approximations . dashed black line shows the next order approximation for @xmath146 . for other values of @xmath33 the difference between the zeroth and the first order approximations is small and only the zeroth order approximation is indicated in the figure . ] as in the previous section we also calculate the function @xmath47 by direct belief propagation method . the results are indicated in fig . [ fig_functions ] for a few values of energy . the same figure also contains the zeroth - order ( cauchy ) approximation and the first iteration of it given by . in all cases considered , the first order approximation is in a better agreement with numerical simulations . this better agreement of the first - order approximation for @xmath47 motivated us to calculate the next approximation to the mean spectral density . we obtain it from as @xmath179 with @xmath106 given by . introducing probability density @xmath180 of @xmath33 i.i.d . variables @xmath181 with distribution @xmath44 , @xmath182 one can rewrite as @xmath183 with @xmath184 ( note that since @xmath185 and im@xmath186 one has im@xmath187 ) . using the elementary identities for the cauchy distribution , one directly gets @xmath188 for the gaussian disorder with @xmath158 , @xmath189 for @xmath190 , zero otherwise . the triple integral in are simplified by the transformation @xmath191p_1(t ) p_k(t_1)p_k(t_2 ) \mathrm{d}t\ , \mathrm{d}t_1\ , \mathrm{d}t_2\ ] ] together with the integration over @xmath192 and @xmath193 using . it leads ( after changing @xmath194 ) to @xmath195 this expression is plotted in fig . [ off_fig ] for @xmath146 . it clearly agrees better with the direct numerical calculation of the mean spectral density . for larger values of @xmath33 the zeroth and the first order formulas are close to each other . it confirms the statement observed in the previous section that though the first order approximation much better agrees with probability density @xmath47 obtained by the belief propagation , the difference between the two approximations in the mean density is noticeable only for the smallest values of coordination number . for regular graphs with @xmath146 and off - diagonal disorder , at different energies : ( a ) @xmath147 , ( b ) @xmath196 , ( c ) @xmath197 , ( d ) @xmath154 . solid black lines show results of direct numerical solution of the tree equation . dashed - dotted blue lines indicate the zeroth order ( cauchy ) approximation with parameters obtained from the mean - field equation . dashed red lines are the first iteration of the above cauchy distribution given by eq . . , title="fig : " ] + ( a ) for regular graphs with @xmath146 and off - diagonal disorder , at different energies : ( a ) @xmath147 , ( b ) @xmath196 , ( c ) @xmath197 , ( d ) @xmath154 . solid black lines show results of direct numerical solution of the tree equation . dashed - dotted blue lines indicate the zeroth order ( cauchy ) approximation with parameters obtained from the mean - field equation . dashed red lines are the first iteration of the above cauchy distribution given by eq . . , title="fig : " ] + ( b ) for regular graphs with @xmath146 and off - diagonal disorder , at different energies : ( a ) @xmath147 , ( b ) @xmath196 , ( c ) @xmath197 , ( d ) @xmath154 . solid black lines show results of direct numerical solution of the tree equation . dashed - dotted blue lines indicate the zeroth order ( cauchy ) approximation with parameters obtained from the mean - field equation . dashed red lines are the first iteration of the above cauchy distribution given by eq . . , title="fig : " ] + ( c ) for regular graphs with @xmath146 and off - diagonal disorder , at different energies : ( a ) @xmath147 , ( b ) @xmath196 , ( c ) @xmath197 , ( d ) @xmath154 . solid black lines show results of direct numerical solution of the tree equation . dashed - dotted blue lines indicate the zeroth order ( cauchy ) approximation with parameters obtained from the mean - field equation . dashed red lines are the first iteration of the above cauchy distribution given by eq . . , title="fig : " ] + ( d ) in the same way it is possible to investigate the general case of regular graphs with both diagonal and off - diagonal disorders . we mention only a special case of matrices of the following form @xmath198 where @xmath199 are i.i.d . random variables with probability density @xmath43 and @xmath200 are i.i.d . real symmetric variables with zero mean and finite variance @xmath201 pastur proved in @xcite that in the limit @xmath202 and under certain mild conditions the mean green function @xmath203 obeys the equation @xmath204 in the formalism discussed here this case corresponds to a regular graph with coordination number @xmath205 . as in the limit @xmath202 @xmath206 in tends to @xmath69 ( cf . and since from the definition @xmath207 it is easy to check that eq . coincides with the mean - field equation , which gives another confirmation of the tree - equation universality . we investigated the mean spectral density for random regular graphs by finding an approximate real solution of the tree equation associated with these graphs . this equation is general and is valid for any uniform tree - like models . the mean spectral density for these models is determined by the @xmath208-fold convolution of real tree equation solution , where @xmath20 is the coordination number of the tree . for large @xmath33 the generalized central limit theorem states that in the bulk such convolution depends only on a few parameters which can be calculated directly from the initial function . from the structure of the tree equation it follows that the required solution belongs to the domain of attraction of the symmetric cauchy distribution . therefore , it is natural to use , as the zeroth order approximation of the tree equation solution , the cauchy distribution itself , whose parameters are calculated self - consistently . iterations of that initial cauchy distribution give next order approximations . when a good approximation for tree equation solution is found , the mean spectral density can be calculated using one of the exact formulas relating it to the tree equation solution ( see section [ introduction ] ) . we applied this scheme for the calculation of mean spectral density for regular graphs with diagonal or off - diagonal disorder , and compared the zeroth and first order approximations with results of direct numerical calculations . as expected , for large coordination number the zeroth order approximation for the density gives quite good results , which are rather accurate even at the smallest @xmath33 . for regular graphs with diagonal disorder the sda gives a slightly better results but for graphs with off - diagonal disorder the mema is closer to the numerics than other zeroth order approximations . as for the tree equation solution itself , the first order approximation is always much closer to the numerical solution obtained by the belief propagation than the zeroth order cauchy approximation . nevertheless , the corresponding discrepancies for the mean spectral density are usually noticeable only at small coordination numbers . the statement that the mean spectral density of random graphs is related to the solution of an equation which , strictly speaking , is valid only for corresponding trees is physically quite natural and could be proved rigorously in certain cases without disorder and with diagonal disorder . further investigation of this and related questions in the spirit of trace formulas on graphs ( see e.g. @xcite ) is of interest . m. l. mehta , _ random matrix theory _ , ( springer , new york , 1990 ) . g. akemann , j. baik , and p. di francesco , _ the oxford handbook of random matrix theory _ ( oxford university press , 2011 ) . r. abou - chacra , d. j. thouless , and p. w. anderson , journal of physics c : solid state physics * 6 * , 1734 - 1752 ( 1973 ) g. j. rodgers and a. j. bray , b * 37 * , 3557 - 3562 ( 1988 ) . mirlin , and y.v . fyodorov , nucl . b * 366 * , 507 - 532 ( 1991 ) . j. d. miller and b. derrida , j. stat . phys . * 75 * , 357 - 388 ( 1994 ) . v. v. flambaum and f. m. izrailev , phys . e * 61 * , 2539 ( 2000 ) . m. mzard and a. montanari , _ information , physics , and computation _ , ( oxford university press , 2009 ) . h. kesten , trans . am . soc . * 92 * , 336 - 354 ( 1959 ) . b. d. mckay , j. lin . . appl . * 40 * , 203 - 216 ( 1981 ) . m. mzard and g. parisi , eur . j. b * 20 * , 217 - 233 ( 2001 ) . c. bordenave and m. lelarge , random structures and algorithms , * 37 * , 332 - 352 , ( 2010 ) . l. geisinger , arxiv:1305.1039 ( 2013 ) . g. biroli , g. semerjian , and m. tarzia , prog . * 184 * , 187 - 199 ( 2010 ) . g. biroli , a. c. r. teixeira , and m. tarzia , arxiv:1211.7334 ( 2012 ) . b. v.gnedenko and a. n. kolmogorov , _ limit distributions for sums of independent random variables _ , ( cambridge , addison - wesley , 1954 ) . g. samorodnitsky and m. taqqu , _ stable non - gaussian random processes _ , ( new york : chapman and hall , 1994 ) . e. p. wigner , annals of mathematics * 62 * , 548 - 564 ( 1955 ) . v. bapst and g. semerjian , j. stat . phys . * 145 * , 51 - 92 ( 2011 ) . p. erds and a. rnyi , publ . sci . * 5 * , 1761 ( 1960 ) . s. n. dorogovtsev , a. v. goltsev , j. f. f. mendes , and a. n. samukhin , phys . e * 68 * , 046109 ( 2003 ) . g. biroli and r. monasson , j. phys . a : math . gen . * 32 * , l255-l261(1999 ) . g. semerjian and l. f. cugliandolo , j. phys . a : math . gen . * 35 * , 4837 - 4851 ( 2002 ) . y. v. fyodorov and i. williams , j. stat . phys . * 129 * , 1081 - 1116 ( 2007 ) . a. khorunzny and g. j. rodgers , rep . 42 * , 297 - 319 ( 1998 ) . l. a. pastur , theor . * 10 * , 67 - 74 ( 1972 ) . i. oren , a. godel , and u. smilansky , j. phys . a : math . theor . * 42 * , 415 - 101 ( 2009 ) .

for random matrices with tree - like structure there exists a recursive relation for the local green functions whose solution permits to find directly many important quantities in the limit of infinite matrix dimensions . the purpose of this note is to investigate and compare expressions for the spectral density of random regular graphs , based on easy approximations for real solutions of the recursive relation valid for trees with large coordination number . the obtained formulas are in a good agreement with the results of numerical calculations even for small coordination number .

1310.1299

in the studies of spin glasses , much effort has been devoted either exprimentally or theoretically to the properties under magnetic fields . unfortunately , our understanding of them still has remained unsatisfactory@xcite . on theoretical side , most of the numerical studies have focused on the properties of the simple ising model , especially the three - dimensional ( 3d ) edwards - anderson ( ea ) model . while the existence of a true thermodynamic spin - glass ( sg ) transition has been established for this model in zero field , the question of its existence or nonexistence in magnetic fields has remained unsettled . this question is closely related to the hotly debated issue of whether the ordered state of the 3d ising sg in zero field exhibits a replica - symmetry breaking ( rsb ) or not . if one tries to understand real experimental sg ordering , one has to remember that many of real sg materials are more or less heisenberg - like rather than ising , in the sense that the random magnetic anisotropy is considerably weaker than the isotropic exchange interaction@xcite . for example , in widely studied canonical spin glasses , _ i.e. _ , dilute metallic alloys such as aufe , agmn and cumn , random magnetic anisotropy originated from the dzyaloshinski - moriya interaction or the dipolar interaction is often one or two magnitudes weaker than the isotropic rkky interaction . numerical simulations have indicated that the isotropic 3d heisenberg sg with finite - range interaction does not exhibit the conventional sg order at finite temperature in zero field @xcite . ( however , see also ref.@xcite . ) since applied fields generally tend to suppress the sg ordering , a true thermodynamic sg transition is even more unlikely under magnetic fields in case of the 3d heisenberg sg . experimentally , however , a rather sharp transition - like behavior has been observed under magnetic fields in typical heisenberg - like sg magnets , although it is not completely clear whether the observed anomaly corresponds to a true thermodynamic transition@xcite . the situation is in contrast to the zero - field case where the existence of a true thermodynamic sg transition has been established experimentally@xcite . set aside the question of the strict nature of the sg `` transition '' , it is experimentally observed that a weak applied field suppresses the zero - field sg transition temperature rather quickly . for higher fields , the sg `` transition '' becomes much more robust to fields , where the `` transition temperature '' shows much less field dependence @xcite . such behaviors of the sg transition temperature under magnetic fields @xmath0 were often interpreted in terms of the mean - field model @xcite . indeed , the mean - field sherrington - kirkpatrick ( sk ) model@xcite with an infinite - range heisenberg exchange interaction with weak random magnetic anisotropy exhibits a transition line similar to the experimental one @xcite , _ i.e. _ , the so - called de almeida - thouless ( at ) line@xcite @xmath1 in weak - field regime where the anisotropy is important , and the gabay - toulouse ( gt ) line@xcite @xmath2 in strong - field regime where the anisotropy is unimportant . nevertheless , if one notes that the true finite - temperature transition under magnetic fields , though possible in the infinite - range sk model , is unlikely to occur in a more realistic finite - range heisenberg model , an apparent success of the mean - field model in explaining the experimental phase diagram should be taken with strong reservation . thus , the question of the true nature of the experimentally observed sg `` transition '' under magnetic fields remains unsolved . recently , one of the present authors has proposed a scenario , the spin - chirality decoupling - recoupling scenario , aimed at explaining some of the puzzles concerning the experimentally observed sg transition@xcite . in this scenario , _ chirality _ , which is a multispin variable representing the sense or the handedness of local noncoplanar spin structures induced by spin frustration , plays an essential role . as illustrated in fig.[fig - chiral ] , locally noncoplanar spin structures inherent to the sg ordered state sustain two energetically degenerate `` chiral '' states , `` right - handed '' and `` left - handed '' states , characterized by mutually opposite signs of the `` chiralities '' . here , one may define the local chirality by _ three _ neighboring heisenberg spins @xmath3 , @xmath4 and @xmath5 by , @xmath6 this type of chirality is called `` scalar chirality '' , in distinction with `` vector chirality '' defined as a vector product of two neighboring heisenberg spins , @xmath7@xcite . note that the chirality defined by eq.([chidef ] ) is a pseudoscalar in the sense that it is invariant under global @xmath8 spin rotations but changes its sign under @xmath9 spin reflections ( or inversions which can be viewed as a combination of reflections and rotations ) . -@xmath10 denote four distinct heisenberg spins . ] for a fully isotropic heisenberg sg , in particular , the chirality scenario of ref.@xcite claims the occurrence of a novel _ chiral - glass _ ordered state in which only the chirality exhibits a glassy long - range order ( lro ) while the spin remains paramagnetic . at the chiral - glass transition , among the global symmetries of the hamiltonian , @xmath11 , only the @xmath9 spin reflection ( inversion ) symmetry is broken spontaneously with keeping the @xmath8 spin rotation symmetry preserved . note that this picture entails the spin - chirality ( or @xmath12 ) decoupling on long length and time scales : namely , although the chirality is not independent of the spin on microscopic length scale , it eventually exhibits a long - distance behavior entirely different from the spin . such a chiral - glass transition without the conventional spin - glass order was indeed observed in recent equilibrium and off - equilibrium monte carlo ( mc ) simulations in zero field performed by hukushima and one of the authors ( h.k.)@xcite . it was also found there that the critical properties associated with the chiral - glass transition were different from those of the ising sg , and that the chiral - glass ordered state exhibited a one - step - like peculiar rsb . in the chirality scenario of ref.@xcite , experimental sg transition in real heisneberg - like sg magnets is regarded essentially as a chiral - glass transition `` revealed '' via the random magnetic anisotropy . weak but finite random magnetic anisotropy inherent to real magnets `` recouples '' the spin to the chirality , and the chiral - glass transition shows up as an experimentally observable _ spin_-glass transition in real heisneberg - like sg magnets . an interesting outcome of this picture is that the experimental sg transition is dictated by the chiral - glass transition of the fully isotropic system , _ not by the spin - glass transition of the fully isotropic system _ , which has been separated from the chiral one . very recently , the present authors discussed some of the possible consequences of the chirality scenario of ref.@xcite on the finite - field properties of the fully isotropic 3d heisenberg sg@xcite . it was argued there that the chiral - glass transition , essentially of the same character as the zero - field one , occurred also in finite fields . in the weak field regime , the transition line was predicted to behave as @xmath13 where @xmath14 and @xmath10 are constants . generally , the coefficient @xmath14 could be either positive or negative . an interesting observation here is that the chiral - glass transition line ( [ eqn : phaseline ] ) aparrently has a form similar to the gt line of the mean - field model . we emphasize , however , that their physical origin is entirely different . the quadratic dependence of the chiral - glass transition line is simply of regular origin , whereas that of the gt - line in the sk model can not be regarded so . in the present paper , we report on our results of large - scale monte carlo simulations on the 3d isotropic heisenberg sg , performed with the aim to reexamine the sg ordering in magnetic fields in light of the chirality scenario . in particular , by means of extensive numerical simulations , we wish to clarify in detail how the spin and the chirality order in applied fields . part of the mc results have been reported in ref.@xcite . the present paper is organized as follows . in [ secmodel ] , we introduce our model and explain some of the details of our numerical method . various physical quantities calculated in our mc simulations are defined in [ secphysq ] . the results of mc simulations are presented in [ secresult ] . the results for the chirality- and spin - related quantities are presented in [ subsecchiral ] and [ subsecspin ] , respectively . it is found that the chiral - glass transition , essentially of the same character as the zero - field one , occurs under magnetic fields . the chiral - glass ordered state exhibits a one - step - like peculiar replica - symmetry breaking in the chiral sector , while it does not accompany the spin - glass order perpendicular to the applied field . critical properties of the chiral - glass transition are analyzed in [ subseccritical ] . the analysis suggests that the universality class of both the zero - field and finite - field chiral - glass transitions might be common , which , however , differs from that of the standard 3d ising sg . in [ subsecphase ] , we construct a magnetic phase diagram of the model . the chiral - glass ordered state remains quite robust against magnetic fields , while the chiral - glass transition line in applied fields has a character of the gt line of the mean - field model . section [ summary ] is devoted to summary and discussion . our numerical results are discussed in terms of the recent experimental result on canonical sg . in this section , we introduce our model and explain some of the details of our numerical method . the model we consider is the isotropic classical heisenberg model on a 3d simple cubic lattice defined by the hamiltonian , @xmath15 where @xmath16 is a three - component unit vector , and @xmath17 is the intensity of magnetic field applied along the @xmath18 direction . the nearest - neighbor coupling @xmath19 is assumed to take either the value @xmath20 or @xmath21 with equal probability ( @xmath22 distribution ) . we perform equilibrium mc simulations on this model . simulations are performed for a variety of fields @xmath23 , 0.1 , 0.5 , 2.0 , 3.0 , and 5.0 , while most extensive calculations are performed for @xmath24 and 0.5 . the lattices studied are simple - cubic lattices with @xmath25 sites with @xmath26 , 8 , 10 , 12 and 16 with periodic boundary conditions . sample average is taken over 128 - 1400 independent bond realizations , depending on the system size @xmath27 and the field intensity @xmath17 . limited amount of data is also taken for @xmath28 in some cases ( 30 samples only ) to check the size dependence of some physical quantities . to facilitate efficient thermalization , we combine the standard heat - bath method with the temperature - exchange technique@xcite . care is taken to be sure that the system is fully equilibrated . equilibration is checked by the following procedures : first , we monitor the system to travel back and forth many times during the the temperature - exchange process ( typically more than 10 times ) between the maximum and minimum temperature points , and at the same time check that the relaxation due to the standard heat - bath updating is reasonably fast at the highest temperature , whose relaxation time is of order @xmath29 monte carlo steps per spin ( mcs ) . this guarantees that different parts of the phase space are sampled in each `` cycle '' of the temperature - exchange run . second , we check the stability of the results against at least three times longer runs for a subset of samples . error bars of physical quantities are estimated by the sample - to - sample statistical fluctuation over bond realizations . further details of our monte carlo simulations are given in table [ table - condition ] . in this section , we define various physical quantities calculated in our simulations below . [ defquantity ] let us begin with the definition of the chirality . we define the local chirality at the @xmath30-th site and in the @xmath31-th direction , @xmath32 , for three neighboring heisenberg spins by the scalar @xmath33 where @xmath34 denotes a unit vector along the @xmath31-th axis . by this definition , there are in total @xmath35 local chiral variables in the system . the mean local amplitude of the chirality , @xmath36 , may be defined by @xmath37\ \ , \ ] ] where @xmath38 represents the thermal average and [ @xmath39 the average over the bond disorder . this quantity vanishes for coplanar spin structures , and its magnitude tells us the extent of the noncoplanarity of the local spin structures . by considering two independent systems ( `` replicas '' ) described by the same hamiltonian ( [ hamil ] ) , one can define an overlap of the chiral variable via the relation , @xmath40 where @xmath41 and @xmath42 represent the chiral variables of the replicas 1 and 2 , respectively . in our simulations , we prepare the two replicas 1 and 2 by running two independent sequences of systems in parallel with different spin initial conditions and different sequences of random numbers . in terms of this chiral overlap @xmath43 , the chiral - glass order parameter may be defined by @xmath44\ \ , \ ] ] while the associated chiral - glass susceptibility may be defined by @xmath45\ \ .\ ] ] unlike the spin variable , the local magnitude of the chirality is temperature dependent somewhat . in order to take account of this short - range order effect , we also consider the reduced chiral - glass order parameter @xmath46 and the reduced chiral - glass susceptibility @xmath47 by dividing @xmath48 and @xmath49 by appropriate powers of @xmath36 , @xmath50 the binder ratio of the chirality is defined by @xmath51 } { [ \langle q_{\chi}^2\rangle]^2}\right)\ \ .\ ] ] one may also define the distribution function of the chiral overlap @xmath43 by @xmath52\ \ .\ ] ] in order to study the equilibrium dynamics of the model , we also compute the autocorrelation function of the chirality defined by @xmath53\ \ . \label{cxt}\ ] ] where the `` time '' @xmath54 is measured in units of mcs . in computing ( [ cxt ] ) , simulation is performed according to the standard heat - bath updating without the temperature - exchange procedure , while the starting spin configuration at @xmath55 is taken from the equilibrium spin configurations generated in our temperature - exchange mc runs . we also calculate the so - called @xmath56 and @xmath57 parameters for the chirality , recently discussed in the literature@xcite , defined by , @xmath58-[\langle q_{\chi}^4\rangle ] } { [ \langle q_{\chi}^2\rangle]^2-[\langle q_{\chi}^4\rangle]},\ ] ] @xmath59-[\langle q_{\chi}^4\rangle ] } { [ \langle q_{\chi}^2\rangle]^2}\ \ .\ ] ] these @xmath60 and @xmath61 parameters are closely related to the sample - to - sample fluctuation of the chiral order parameter . the @xmath57 parameter is known to be an indicator of the non - self - averagingness of the order parameter , _ i.e. _ , it vanishes in the state where the order parameter is self - averaging and takes a nonzero value otherwise@xcite . by contrast , the @xmath56 parameter could take a nonzero value even in a self - averaging ordered state , and hence , can not be used as an unambiguous indicator of the non - self - averagingness@xcite . however , since in the thermodynamic limit it vanishes in the high - temperature phase and takes a nonzero value in the ordered state , it can still be used as an indicator of a phase transition . as in the case of the chirality , it is convenient to define an overlap variable for the heisenberg spin . in this case , the overlap might naturally be defined as a _ tensor _ variable @xmath62 between the @xmath31 and @xmath63 components ( @xmath31 , @xmath63=@xmath64 ) of the heisenberg spin , @xmath65 where @xmath66 and @xmath67 are the @xmath30-th heisneberg spins of the replicas 1 and 2 , respectively . in terms of these tensor overlaps , the `` longitudinal '' ( parallel to the applied field ) and `` transverse '' ( perpendicular to the applied field ) sg order parameters may be defined by @xmath68 , \ \ \ \ q_{\rm l}^2 = q_{zz}^2,\ ] ] @xmath69,\ \ \ \ q_{\rm t}^2 = \sum_{\mu,\nu = x , y}q_{\mu\nu}^2\ \ .\ ] ] the associated longitudinal and transverse binder ratios are defined by @xmath70 } { [ \langle q_{\rm l}^2\rangle]^2}\right ) , \label{binl}\ ] ] @xmath71 } { [ \langle q_{\rm t}^2\rangle]^2}\right)\ \ .\ ] ] here , @xmath72 and @xmath73 are normalized so that , in the thermodynamic limit , they vanish in the high - temperature phase and gives unity in the nondegenrate ordered state . since an odd quantity like @xmath74 does not vanish in applied fields , one can also define the `` connected binder ratio '' for the longitudinal component@xcite , @xmath75 } { [ \langle(q_{\rm l}-\langle q_{\rm l}\rangle)^2\rangle]^2}\right).\ ] ] in applied fields , @xmath76 might well behave differently from @xmath72 . the spin - overlap distribution function is generally defined in the tensor space . in the following , we pay particular attention to its transverse ( @xmath77 ) part . the relevant transverse overlap originally has @xmath78 independent components . for the convenience of illustration , we follow ref.@xcite here and introduce the projected transverse - spin - overlap distribution function @xmath79 defined in terms of the diagonal overlap @xmath80 which is the trace of the tensor overlap @xmath62 s , @xmath81 the distribution function @xmath79 is symmetric with respect to @xmath82 . in the high - temperature phase , each @xmath62 ( @xmath83 ) is expected to be gaussian - distributed around @xmath84 in the @xmath85 limit , and so is @xmath80 . let us hypothesize here that there exists a transverse _ spin_-glass ordered state characterized by a nonzero @xmath86 , or by a nonzero ea transverse sg order parameter @xmath87 . reflecting the fact that @xmath80 transforms nontrivially under independent @xmath88 rotations around the @xmath18-axis on the two replicas , which are the symmetries relevant to the transverse spin components in the presence of magnetic fields , even a self - overlap contributes nontrivial weights to @xmath79 other than at @xmath89 . in the @xmath85 limit , the self - overlap part of @xmath79 should be given by @xmath90 the derivation of eq.([ptform ] ) has been given in ref.@xcite in the context of the _ xy _ sg . if the transverse sg ordered state accompanies rsb , the associated nontrivial contribution would be added to the one given by eq.([ptform ] ) . in any case , an important observation here is that , as long as the ordered state possesses a finite transverse sg lro , the diverging peak should arise in @xmath79 at @xmath91 as illustrated in fig.[xylro ] , irrespective to the occurrence of the rsb . this section is the core part of the present paper . here , we present our mc results on the 3d @xmath22 heisenberg sg in magnetic fields . first , we begin with the chirality - related quantities . in fig.[fig - locx ] , we show the temperature and size dependence of the mean local amplitude of the chirality for various fields . as can clearly be seen from fig.[fig - locx](a ) , extrapolation of @xmath92 to @xmath93 gives non - zero values as long as the applied field intensity is not too large , _ i.e. _ , @xmath94 , 0.295 , 0.308 , 0.313 , 0.260 , and 0.100 for @xmath95 , 0.1 , 0.5 , 2.0 , 3.0 , and 5.0 , respectively . this indicates that the spin ordering of the 3d heisenberg sg is certainly noncoplanar , which guarantees that the system sustains the nontrivial chirality . meanwhile , a direct inspection of the spin pattern suggests that such noncoplanar spin configurations realized at low temperature in zero and weak fields is rather close to the coplanar one . indeed , for completely random configurations of heisenberg spins , @xmath36 should take a value @xmath96@xcite , a value considerably larger than the extrapolated @xmath97 values . this again suggests that the noncoplanar configuration realized in zero and weak fields is close to the coplanar one . interestingly , our mc data indicate that , in the weak field regime , @xmath98 slightly _ increases _ with increasing magnetic field at fixed temperatures . this observation could be understood if one notes that the zero - field noncoplanar spin configuration is close to the coplanar one , and that the application of a magnetic field to such nearly coplanar spin configuration tends to `` rise up '' the spins from this plane with keeping the plane orthogonal to the applied field . this gives rise to more `` three - dimensional '' local spin structures with larger @xmath36 . of course , when the field is further increased , @xmath98 eventually decreases simply because strong enough fields force spins to align along the field . in fig.[fig - locx](b ) , we show the size dependence of @xmath99 for the field @xmath24 . as can be seen from the figure , there is very little size dependence in @xmath99 . for @xmath95 , 0.1 and 0.5 , and is @xmath26 for other field values . for the case of @xmath24 , the size dependence of @xmath36 is shown in fig.(b ) . ] in fig.[fig - qx2 ] , we show the chiral - glass order parameter @xmath100 for the fields ( a ) @xmath24 , and ( b ) @xmath101 . for both fields , @xmath100 increases rather sharply at lower temperatures . , and ( b ) @xmath101 . ] in figs.[fig - gx](a ) and ( b ) , we show the binder ratio of the chirality @xmath102 for the fields ( a ) @xmath24 , and ( b ) @xmath101 . as can be seen from the figures , @xmath103 exhibits a negative dip which , with increasing @xmath27 , tends to deepen and shift toward lower temperature . furthermore , @xmath103 of various @xmath27 cross at a temperature slightly above the dip temperature @xmath104 _ on negative side of @xmath105 _ , eventually merging at temperatures lower than @xmath104 . the observed behavior of @xmath103 is similar to the one observed in zero field@xcite . as argued in ref.@xcite , the persistence of a negative dip and the crossing occurring at @xmath106 are strongly suggestive of the occurrence of a finite - temperature transition where @xmath107 takes a _ negative _ value in the @xmath108 limit . , and ( b ) @xmath101 . ] in fig.[fig - dip ] , we plot the negative - dip temperature @xmath109 versus @xmath110 for the fields @xmath24 and @xmath101 . for both fields , the data lie on a straight line fairly well . the linear extrapolation to @xmath111 , as shown by the solid lines in the figure , gives our first estimates of the bulk chiral - glass transition temperature , _ i.e. _ , @xmath112 for @xmath24 and @xmath113 for @xmath101 . more precisely , @xmath109 should scale with @xmath114 where @xmath115 is the chiral - glass correlation - length exponent . as shown below , our estimate of @xmath116 comes close to unity , more or less justifying the linear extrapolation employed here . indeed , extrapolation with respect to @xmath117 , shown by the dashed curve in fig.[fig - dip ] , yields @xmath118 for @xmath24 and @xmath112 for @xmath101 . as shall be argued below , we attribute several unusual features of @xmath105 , _ e.g. _ , the growing negative dip and the crossing occurring at @xmath106 , to the possible one - step - like peculiar rsb in the chiral - glass ordered state . in systems exhibiting the one - step rsb , _ e.g. _ , the mean - field three - state potts glass , the binder ratio is known to behave as illustrated in fig.[fig - binx ] , with a negative dip and the crossing occurring on the negative side@xcite . indeed , such a behavior is not dissimilar to the one we have observed in fig.[fig - gx ] . an independent estimate of @xmath119 can be obtained from the equilibrium dynamics of the model . thus , we also calculate the chirality autocorrelation function @xmath120 defined by eq.([cxt ] ) . to check the possible size dependence , we show in fig.[fig - auto ] the time dependence of @xmath120 for the field @xmath101 on a log - log plot , computed for ( a ) @xmath121 , and for ( b ) @xmath28 . as shown in the figures , @xmath120 shows either a downward curvature characteristic of the disordered phase , or an upward curvature characteristic of the long - range ordered phase , depending on whether the temperature is higher or lower than @xmath122 . just at @xmath122 , the linear behavior corresponding to the power - law decay is observed . hence , our data indicates that the chiral - glass transition takes place at @xmath123 , in agreement with our above estimate based on @xmath105 . from the slope of the data at @xmath124 , the exponent @xmath125 characterizing the power - law decay of @xmath126 is estimated to be @xmath127 . we note that both our data of @xmath121 shown in fig.[fig - auto](a ) and of @xmath28 shown in fig.[fig - auto](b ) give almost the same estimates of @xmath119 and of @xmath125 , even though the @xmath121 and @xmath28 data themselves do not completely overlap , particularly below @xmath128 . anyway , our observation that @xmath120 exhibits an upward curvature below @xmath119 , tending to a nonzero value corresponding to the static chiral ea parameter @xmath129 , indicates that the chiral - glass ordered state is `` rigid '' with a nonzero long - range order . the same analysis applied to the @xmath24 case yields @xmath130 and @xmath131 . for the sizes ( a ) @xmath121 , and ( b ) @xmath28 . temperatures correspond to @xmath132 , 0.18 , 0.19 , 0.20 , 0.21 , 0.22 , 0.23 , 0.24 , 0.25 , 0.26 from top to bottom . straight lines of power - decay fit are shown in both figures ( a ) and ( b ) at @xmath133 . ] in fig.[fig - pqxt016 ] , we show the chiral - overlap distribution function @xmath134 for the field @xmath101 at a temperature @xmath135 , well below @xmath136 . in addition to the standard `` side - peaks '' corresponding to the ea order parameter @xmath137 , which grow and sharpen with increasing @xmath27 , there appears a `` central peak '' at @xmath138 for larger @xmath27 , which also grows and sharpens with increasing @xmath27 . the shape of the calculated @xmath139 is very much similar to the one obtained in ref.@xcite in zero field , but is quite different from those observed in the standard ising - like models such as the 3d ea model@xcite or the mean - field sk model@xcite . as argued in ref.@xcite in case of zero field , such peculiar features of @xmath139 are likely to be related to the _ one - step_-like rsb . the existence of a negative dip in the binder ratio @xmath102 and the absence of the standard type of crossing of @xmath105 at @xmath140 are also consistent with the occurrence of such a one - step - like rsb @xcite . we note that our data of @xmath141 are also compatible with the existence of a continuous plateau between [ @xmath142 in addition to the delta - function peaks . at a temperature @xmath135 , well below the chiral - glass transition temperature , @xmath143 . ] in fig.[fig - gax ] , we show the the temperature and size dependence of the @xmath144 and @xmath145 parameters for the field @xmath101 . although error bars of the data are rather large here , the crossing occurs at temperatures somewhat higher than @xmath112 in both figures , while the crossing temperatures of neighboring sizes ( _ e.g. _ @xmath26 and @xmath146 _ etc _ ) gradually shift towards @xmath112 for larger @xmath27 . the data are consistent with our estimate of @xmath119 above based on the binder ratio and the autocorrelation . and @xmath57 parameters of the chirality for the field @xmath101 . ] in this subsection , we present our mc results of the spin - related quantities . in figs.[fig - gsl ] and [ fig - gst ] , we show the spin binder ratios for the longitudinal and transverse components , respectively , for the fields ( a ) @xmath24 , and ( b ) @xmath101 . for both fields , the _ longitudinal _ binder ratio @xmath72 increases monotonically toward unity with increasing @xmath27 at all temperatures studied : see fig.[fig - gsl ] . this observation reflects the fact that the longitudinal component of the spin exhibits a net magnetization induced by applied fields at any finite temperatures . by contrast , the binder ratio of the transverse component of the spin @xmath73 decreases toward zero with increasing @xmath27 , without a negative dip nor a crossing : see fig.[fig - gst ] . this suggests that the transverse component of spin remains disordered even below @xmath119 . , and ( b ) @xmath101 . for the field @xmath101 , magnified figure is shown in the inset . ] , and ( b ) @xmath101 . ] in fig.[fig - gs2 ] , we show the connected binder ratio of the longitudinal spin component for the fields ( a ) @xmath24 , and ( b ) @xmath101 . again , any anomalous behavior is not appreciable , no crossing nor extremum . instead , @xmath147 monotonously approaches zero with increasing @xmath27 , staying negative at any temperature . ( strictly speaking , the data of @xmath26 and @xmath146 for @xmath24 , exhibits a crossing - like behavior around @xmath148 , but this is limited to these smaller lattices . ) , and ( b ) @xmath101 . ] in fig.[fig - qsxy ] , we show the diagonal transverse - spin - overlap distribution function @xmath149 for the field @xmath24 at a temperature @xmath150 , well below the chiral - glass transition temperature @xmath151 . the calculated @xmath149 exhibits a symmetric `` shoulder '' at some nonzero value of @xmath80 , but as shown in the inset , this `` shoulder '' gets suppressed with increasing @xmath27 , _ not showing a divergent behavior_. such suppression of the shoulder indicates that the chiral - glass ordered state does not accompany the standard transverse sg order , at least up to temperatures @xmath152 . for @xmath101 , we have also observed similar suppression of the shoulder up to temperatures as low as around @xmath152 . hence , we conclude that the chiral - glass ordered state does not accompany the standard transverse sg order , at least just below the chiral - glass transition point . strictly speaking , the observed suppression of the shoulder is still not inconsistent with the kosterlitz - thouless(kt)-like critical sg ordered state . however , we note that such a critical sg ordered state appearing at @xmath153 is not supported by our data of @xmath73 shown in fig.[fig - gst ] . for the field @xmath24 at a temperature @xmath150 , well below the chiral - glass transition temperature @xmath118 . a magnified view of the shoulder part , indicated by the dashed circle in the main panel , is shown in the inset . ] in this subsection , we determine static and dynamical critical exponents associated with the chiral - glass transition . the analysis here is made for the two particular field values , @xmath24 and @xmath101 , where most extensive simulations have been performed . in the analysis below , we fix @xmath136 to be @xmath154 ( @xmath24 ) and @xmath155 ( @xmath101 ) , as determined above . we estimate first the chiral - glass susceptibility exponent @xmath156 from the asymptotic slope of the log - log plot of the reduced chiral - glass susceptibility @xmath47 versus the reduced temperature @xmath157 . an example is given in fig.[fig - getexponents](a ) for the case of @xmath101 , where an asymptotic slope @xmath158 is obtained . we then estimate the chiral - glass critical - point - decay exponent @xmath159 from the @xmath27-dependence of the chiral - glass order parameter @xmath160 at @xmath119 , according to the relation @xmath161 . an example for the @xmath101 case is shown in fig.[fig - getexponents](b ) , where we plot @xmath160 at @xmath162 versus @xmath27 on a log - log plot . as can be seen from the figure , the data lie on a straight line fairly well . from its slope @xmath163 , the exponent @xmath159 is estimated to be @xmath164 . the rest of the static exponents , @xmath165 , @xmath166 , and @xmath167 , can be estimated from @xmath168 and @xmath169 by using the standard scaling and hyperscaling relations as @xmath170 , @xmath171 and @xmath172 . versus the reduced temperature for the field @xmath101 . its slope @xmath173 determines the chiral - glass susceptibility exponent @xmath174 . the transition temperature is assumed here to be @xmath155 . ( b ) log - log plot of @xmath160 versus @xmath27 for the field @xmath101 at @xmath175 . its slope @xmath176 determines the chiral - glass critical - point - decay exponent to be @xmath177 . ] the dynamical exponent @xmath178 can be estimated from the exponent @xmath179 , via the relation @xmath180 . from our above estimate , @xmath127 , we get @xmath181 . the same procedure is repeated for the case of @xmath24 . we then get @xmath182 , @xmath183 , @xmath184 . these estimates for @xmath24 agree within errors with the corresponding estimates for @xmath101 . our estimates of the chiral - glass exponents are summarized in table [ table - criticalexp ] , and are compared with the corresponding zero - field exponents reported in ref.@xcite . the finite - field exponents turn out to agree within errors with the corresonding zero - field exponents , suggesting that the zero - field and finite - field chiral - glass transitions lie in a common universality class . we note that this observation is consistent with the chirality scenario of refs.@xcite . in table [ table - criticalexp ] , we also show the sg exponents of the 3d ising ea model@xcite together with typical experimental values ( in zero field ) of real heisenberg - like sg magnet agmn@xcite . the critical properties of the chiral - glass transition differ clearly from those of the 3d ising ea sg . by contrast , the chiral - glass exponents are close to the experimental exponent values for canonical sg agmn , giving further support to the spin - chirality decoupling - recoupling scenario . as a consistency check of our estimates of exponents and @xmath119 values , we have also done the following : we use the @xmath185 value determined above , @xmath186 , and extrapolate the dip temperature of @xmath105 , @xmath109 , to @xmath187 ( see the dashed lines of fig.[fig - dip ] ) . as mentioned , such an extrapolation yields the bulk chiral - glass transition temperature , @xmath151 ( @xmath24 ) and @xmath188 ( @xmath101 ) . these estimates of @xmath119 agree with those obtained from the chiral autocorrelation and employed in our scaling analysis . this guarantees that our analysis of exponents and @xmath119 is self consistent . in fig.[fig - scalingc ] , we show the the standard finite - size scaling plot for the chiral - glass order parameter @xmath48 based on the relation , @xmath189 where the @xmath119 , @xmath159 and @xmath167 values are set to the best values determined above . as can be seen from the figures , reasonable data collapsing are obtained , at least for larger lattices . at the same time , however , one sees that there exists a systematic deviation from the scaling for smaller lattices , particularly in the case of @xmath24 . such a deviation observed for smaller lattices suggests the existence of a significant finite - size correction . , and ( b ) @xmath101 . ] the existence of such significant finite - size effects has also been suggested from the behavior of the chiral - overlap distribution function @xmath190 and of the chiral binder ratio @xmath105 . in a truly asymptotic critical regime , @xmath134 itself should scale at @xmath124 with tuning one exponent @xmath159 . however , in the range of sizes studied here @xmath191 , we can not observe such a full scaling of @xmath134 . such a lack of complete scaling of @xmath134 gives rise to certain degrees of uncertainty in our estimate of @xmath159 : namely , if one tries to scale the width of the distribution such as its second moment @xmath48 , it yields @xmath192 as given above ( see fig . [ fig - getexponents](b ) ) , while if one tries to scale the height of @xmath190 , it instead yields @xmath193 , which is somewhat smaller than the above estimate , though still lying within the quoted error bar . lack of a complete scaling in @xmath134 is also reflected in the behavior of @xmath102 , which does not show a unique crossing at @xmath194 within the range of sizes studied : instead , as shown in fig.[fig - gx ] , the crossing occurs on the negative side of @xmath105 considerably above @xmath124 , while the crossing points tend to come down toward @xmath119 as @xmath27 increases . concerning the transverse spin order , from the behaviors of the binder ratio and of the diagonal transverse - spin - overlap distribution function , we have already found a strong numerical evidence that the chiral - glass transition does not accompany the transverse sg order , at least just below @xmath119 . in other words , the transverse component of the spin orders only at zero temperature , or else , if it orders at a finite temperature , the associated transverse sg transition temperature @xmath195 is significantly lower than @xmath119 , say , below @xmath152 . we warn the reader here that , so long as one looks at the sg correlation or the sg order parameter , a rather careful analysis is required to really see such a behavior . as an example , we show in fig.[scalesph01 ] the standard finite - size scaling plots of the transverse sg order parameter @xmath196 for @xmath24 ; ( a ) the one assuming @xmath197 , and ( b ) the other assuming @xmath198 . similar plots are given in fig.[scalesph05 ] for the field @xmath101 with assuming @xmath197 , assuming ( a ) @xmath197 and ( b ) @xmath199 . at a look , both fits seem equally acceptable without appreciable difference if the exponents are adjusted in appropriate ways . then , one may wonder if the transverse spin might order simultaneously with the chirality , with the associated sg exponents @xmath200 ) and @xmath201 : see fig.[scalesph01 ] . we believe , however , this not to be the case due to the following reasons . , assuming ( a ) @xmath202 , and ( b ) @xmath198 . ] , assuming ( a ) @xmath202 , and ( b ) @xmath199 . ] first , as shown above , such simultaneous chirality and transverse - spin ordering contradicts with our result of the binder ratio @xmath73 and the diagonal transverse - spin - overlap distribution function @xmath79 . second , a closer inspection of the data reveals that , at and below @xmath124 , there exists an important difference between the behaviors of the transverse spin @xmath86 and of the chirality @xmath100 . in fig.[fig - q2log ] , we show on a log - log plot the size dependence of the both order parameters , @xmath86 and @xmath100 , at several temperatures at and below @xmath119 . as can be seen from fig.[fig - q2log](a ) , below @xmath119 the chiral - glass order parameter exhibits a clear upbending for larger @xmath27 , indicating that @xmath100 tends to a nonzero value in the thermodynamic limit . in sharp contrast to this , such an upbending is never seen in the transverse sg order parameter @xmath86 : instead , @xmath86 shows a slight downbedning behavior at @xmath124 , which gradually shifts to the near linear behavior at lower temperatures . the observed behavior of @xmath86 is consistent with either , ( a ) the onset of the kosterlitz - thouless(kt)-like transition at a finite temperature below which the spin - glass correlations decay algebraically with a power - law or , ( b ) the gradual growth of the transverse sg correlation length @xmath203 which exceeds the investigated system size @xmath121 around a certain nonzero temperature close to @xmath119 . in the former case , there should exist a well - defined finite sg transition temperature with the critical sg ordered state , while , in the latter case , there need not be a thermodynamic sg transition at a finite temperature . generally speaking , it is difficult to discriminate between the above two possibilities only from the @xmath86 data of finite sizes with @xmath204 . nevertheless , we believe we can at least exclude here the possibility that the kt - like transverse sg transition occurs _ simultaneously _ with the chiral - glass transition at @xmath124 , accompanied by the critical sg ordered state at @xmath153 . first , we note that such a critical sg ordered state is not supported by our data of @xmath73 of fig.[fig - gst ] . second , the transverse _ spin_-glass correlation - length exponent estimated in figs.[scalesph01](b ) and [ scalesph05](b ) assuming the simultaneous spin and chiral transition , @xmath205 , is far from from the lower - critical - dimension ( lcd ) value , @xmath206 , generically expected for such a kt - like transition . in so far as one insists that the transverse sg order occurs simultaneously with the chiral - glass order , our numerical estimate of the transverse sg correlation - length exponent is not compatible with the lcd value @xmath207 , which is now hard to reconcile with the kt - like behavior observed in @xmath86 at @xmath208 . , at several temperatures at and below @xmath136 : ( a ) the transverse - spin - glass order parameter @xmath86 , and ( b ) the chiral - glass order parameter @xmath48 . the chiral - glass transition temperature at this field is @xmath118 . to emphasize the deviation from the linearity , lines connecting the two small - size data @xmath26 and @xmath209 are drawn at each temperature . ] in fact , as recently argued in ref.@xcite for the case of the _ xy_sg , the chirality scenario gives the possible cause why simultaneous spin and chiral orderings are apparently observed in the sg order parameter or the sg correlation function . this would closely be related to the length and time scales of the measurements . here , one should be aware of the fact that the spin - chirality decoupling is a _ long - scale _ phenomenon : at short scale , the chirality is never independent of the spin by its definition , roughly being its squared ( @xmath210 ) as expected from the naive power counting . hence , the behavior of the spin - correlation related quantities , including the sg order parameter which is a summed correlation , might well reflect the critical singularity associated with the _ chirality _ _ i.e. _ , the one of the chiral - glass transition , up to certain length and time scale . in such a scenario , apparent ( not true ) transverse `` spin - glass exponents '' expected would be @xmath211 and @xmath212 , the latter being derived from the short - scale relation , @xmath213 . note that these values are not very far from the ones we get from the finite - size scaling analysis of figs.[scalesph01](b ) and [ scalesph05](b ) , assuming the simultaneous occurrence of the spin and chiral transition . however , we stress again that such a disguised criticality in the spin sector is only a short - scale phenomenon , not a true critical one . by collecting our estimates of the @xmath119 values for various field values , as obtained by the extrapolation of @xmath109 to @xmath187 , we construct a phase diagram in the temperature vs. magnetic field plane . the result is shown in fig.[fig - phase ] . we have used here the zero - field estimate of ref.@xcite , @xmath214 . error bars are estimated here from the differences between the extrapolated @xmath119 values via the @xmath110 and @xmath215 fits . as is evident from fig.[fig - phase ] , the chiral - glass state remains quite robust against magnetic fields . this is most evident in fig.[fig - phase](b ) where we draw the same phase diagram on a plot where both the temperature and the magnetic - field axes have common energy scale . indeed , @xmath216 is not much reduced from the zero - field value even at a field as large as ten times of @xmath217 . at lower fields , the chiral - glass transition line is almost orthogonal to the @xmath218 axis , consistent with the behavior eq.([eqn : phaseline ] ) derived from the chirality scenario . our data are even not inconsistent with the coefficient @xmath14 in eq.([eqn : phaseline ] ) being slightly negative so that @xmath216 initially _ increases _ slightly with @xmath17 , though it is difficult to draw a definite conclusion due to the scatter of our estimate of @xmath216 . if one remembers here our mc observation that the application of a weak magnetic field tends to increase the mean local amplitude of the chirality , @xmath36 , from its zero - field value , such an initial increase of @xmath216 seems not totally unlikely . -@xmath219 phase diagram of the 3d @xmath22 heisenberg sg determined by the present simulation . note that the energy scales of the @xmath17 and of the @xmath219 axes are mutually different in fig.(a ) , while they are taken to be common in fig.(b ) . ] in summary , we have performed large - scale equilibrium monte carlo simulations on the 3d isotropic heisenberg sg in finite magnetic fields . we have confirmed that our mc results are consistent with the chirality scenario of ref.@xcite . among other things , we have verified the occurrence of a finite - temperature chiral - glass transition in applied fields , essentially of the same character as the zero - field one . the chiral - glass ordered state exhibits a one - step - like peculiar rsb , while it does not accompany the transverse sg order , at least up to temperatures around @xmath152 . the criticality of finite - field chiral - glass transitions seems to be common with that of the zero - field one , which , however , clearly differs from the criticality of the standard 3d ising ea model . meanwhile , the chiral - glass exponents turn out to be close to the experimental exponents determined for canonical sg such as agmn . we have also constructed a magnetic phase diagram of the 3d heisenberg sg model . the chiral - glass transition line in the @xmath17-@xmath219 plane is found to be almost vertical to the temperature axis , up to rather high fields of order @xmath220 , indicating that the chiral - glass ordered state is quite robust against magnetic fields . this somewhat surprising property probably arises from the fact that the magnetic field couples in the hamiltonian directly fo the spin , _ not to the chirality _ , and the effective coupling between the field and the chirality is rather weak . the chiral - glass transition line has a character of the gabay - toulouse line of the mean - field model , yet its physical origin being entirely different . . for comparison , we also show the present numerical result of the magnetic phase diagram of the 3d @xmath22 heisenberg sg model . the way how we scale the units of magnetic field and temperature in plotting the experimental data is explained in the text . ] it is not immediately possible to make a direct comparison of our results with experiments . this is mainly because the random magnetic anisotropy , which inevitably exists in real sg materials , is not introduced in our present model . furthermore , in real sg magnets , spins do not necessarily sit on a simple - cubic lattice , nor interact with other spins via the nearest - neighbor @xmath22 coupling , _ etc_. in spite of these obvious limitations , it might be interesting to try to compare our present magnetic phase diagram with the experimental one for heisenberg - like sg magnets . chirality scenario claims that , in the high - field region where the anisotropy is negligible relative to the applied magnetic field , the sg transition line should essentially be given by the chiral - glass transition line of the fully isotropic system . if so , our present result entails that the sg transition line of real heisenberg - like sg should be almost vertical against the temperature axis in the high - field regime where the magnetic field overwhelms the random magnetic anisotropy . in fig.[phasedg](a ) , we reproduce the experimental @xmath17-@xmath219 phase diagram of canonical sg aufe from ref.@xcite . in the same figure , we also show our present result of the chiral - glass transition line , scaled in the following way . we try to mimic the real system by the classical heisenberg hamiltonian with an effective coupling @xmath20 and an effective magnetic field @xmath17 , which is defined in terms of eq.([eqn : phaseline ] ) . first , we estimate the zero - field transition temperature of the hypothetical _ isotropic _ system to be @xmath221k , by extrapolating the high - field gt - like transition line of aufe to @xmath218 . then , with the knowledge of our present estimate of @xmath222 , we estimate the relevant @xmath20 roughly to be 50k . the field intensity @xmath17 is then translated into the field intensity in the standard unit @xmath223 by the relation @xmath224 , @xmath225 being the effective bohr number : in case of aufe , @xmath225 was experimentally estimated to be @xmath226 , where @xmath227 is the bohr magneton@xcite . thus , our fig.[fig - phase ] suggests that the sg phase boundary of aufe might stay nearly vertical up to the field as high as @xmath228[t ] . of course , considering the difference in microscopic details between the present model and real aufe , one can not expect a truly quantitative correspondence here . anyway , further high - field experiments on aufe and other heisenberg - like sg magnets might be worthwhile to determine the sg phase boundary in the high - field regime . in order to make further comparison with the experimental phase diagram in the low - field regime , it is essential to examine the effects of random magnetic anisotropy inherent to real sg materials . indeed , in the low - field regime where the applied field intensity is comparable to or weaker than the random magnetic anisotropy , the chirality scenario predicts the appearance of a singular crossover line which has some character of the at - line of the mean - field model@xcite . in order to make further insight into the spin - glass and the chiral - glass orderings in magnetic fields and to check further the validity of the chirality scenario , it would be interesting to make similar finite - field simulations for the _ anisotropic _ 3d heisenberg sg model . the numerial calculation was performed on the hitachi sr8000 at the supercomputer system , issp , university of tokyo . the authors are thankful to dr . k. hukushima , dr . h. yoshino and dr . i. a. campbell for useful discussion . for reviews on spin glasses , see _ e. g. , _ ( a ) k. binder and a. p. young : rev . * 58 * ( 1986 ) 801 ; ( b ) k. h. fischer and j. a. hertz : _ spin glasses _ cambridge university press ( 1991 ) ; ( c ) j. a. mydosh : _ spin glasses _ taylor & francis ( 1993 ) ; ( d ) a. p. young ( _ ed . _ ) : _ spin glasses and random fields _ world scientific , singapore ( 1997 ) . for the static critical exponents , see , for example , ( a ) n. kawashima and a. p. young : phys . rev . b*53 * ( 1996 ) , 484 ; ( b ) e. marinari , g. parisi and j. j. ruiz - lorenzo : phys . rev . b*58 * ( 1998 ) 14852 ; ( c ) b. a. berg and w. janke : phys . * 80 * ( 1998 ) 4771 ; ( d ) m. palassini and s. caracciolo : phys . * 82 * ( 1999 ) 5128 ; ( e ) h.g . ballesteros , a. cruz , l.a . fernndez , v. martn - mayor , j.j . ruiz - lorenzo , a. tarancn , p. tllez , c.l . ulod and c. ungil : phys . b*62 * ( 2000 ) 14237 . see , for example , ( a ) n. de cortenary , h. bouchiat , h. hurdequite and a. fert : j. physique * 47*(1986 ) 2659 ; ( b ) h. bouchiat : j. physique * 47*(1986 ) 71 ; ( c ) l. p. lvy and a. t. ogielski : phys . rev . lett . * 57*(1986 ) 3288 .

spin and chirality orderings of the three - dimensional heisenberg spin glass under magnetic fields are studied by large - scale equilibrium monte carlo simulations . it is found that the chiral - glass transition and the chiral - glass ordered state , which are essentially of the same character as their zero - field counterparts , occur under magnetic fields . the chiral - glass ordered state exhibits a one - step - like peculiar replica - symmetry breaking in the chiral sector , while it does not accompany the spin - glass order perpendicular to the applied field . critical perperties of the chiral - glass transition are different from those of the standard ising spin glass . magnetic phase diagram of the model is constructed , which reveals that the chiral - glass state is quite robust against magnetic fields . the chiral - glass transition line has a character of the gabay - toulouse line of the mean - field model , yet its physical origin being entirely different . these numerical results are discussed in light of the recently developed spin - chirality decoupling - recoupling scenario . implication to experimental phase diagram is also discussed .

cond-mat0110219

given competing mathematical models to describe a process , we wish to know whether our data is compatible with the candidate models . often comparing models requires optimization and fitting time course data to estimate parameter values and then applying an information criterion to select a ` best ' model @xcite . however sometimes it is not feasible to estimate the value of these unknown parameters ( e.g. large parameter space , nonlinear objective function , nonidentifiable etc ) . the parameter problem has motivated the growth of fields that embrace a parameter - free flavour such as chemical reaction network theory and stoichiometric theory @xcite . however many of these approaches are limited to comparing the behavior of models at steady - state @xcite . inspired by techniques commonly used in applied algebraic geometry @xcite and algebraic statistics @xcite , methods for discriminating between models without estimating parameters has been developed for steady - state data @xcite , applied to models in wnt signaling @xcite , and then generalized to only include one data point @xcite . briefly , these approaches characterize a model @xmath0 in only observable variables @xmath1 using techniques from computational algebraic geometry and tests whether the steady - state data are coplanar with this new characterization of the model , called a _ steady - state invariant _ @xcite . notably the method does nt require parameter estimation , and also includes a statistical cut - off for model compatibility with noisy data . here , we present a method for comparing models with _ time course data _ via computing a _ differential invariant_. we consider models of the form @xmath2 and @xmath3 where @xmath4 is a known input into the system , @xmath5 , @xmath6 is a known output ( measurement ) from the system , @xmath7 , @xmath8 are species variables , @xmath9 , @xmath10 is the unknown @xmath11dimensional parameter vector , and the functions @xmath12 are rational functions of their arguments . the dynamics of the model can be observed in terms of a time series where @xmath13 is the input at discrete points and @xmath14 is the output . in this setting , we aim to characterize our ode models by eliminating variables we can not measure using differential elimination from differential algebra . from the elimination , we form a differential invariant , where the differential monomials have coefficients that are functions of the parameters @xmath15 . we obtain a system of equations in 0,1 , and higher order derivatives and we write this implicit system of equations as @xmath16 , @xmath7 , and call these the input - output equations our _ differential invariants_. specifically , we have equations of the form : @xmath17 where @xmath18 are rational functions of the parameters and @xmath19 are differential monomials , i.e. monomials in @xmath20 . we will see shortly that in the linear case , @xmath21 is a linear differential equation . for non - linear models , @xmath21 is nonlinear . if we substitute into the differential invariant available data into the observable monomials for each of the time points , we can form a linear system of equations ( each row is a different time point ) . then we ask : does there exist a @xmath22 such that @xmath23 . if @xmath24 of course we are guaranteed a zero trivial solution and the non - trivial case can be determined via a rank test ( i.e. , svd ) and can perform the statistical criterion developed in @xcite with the bound improved in @xcite , but for @xmath23 there may be no solutions . thus , we must check if the linear system of equations @xmath23 is consistent , i.e. has one or infinitely many solutions . assuming measurement noise is known , we derive a statistical cut - off for when the model is incompatible with the data . however suppose that one does not have data points for the higher order derivative data , then these need to be estimated . we present a method using gaussian process regression ( gpr ) to estimate the time course data using a gpr . since the derivative of a gp is also gp , so we can estimate the higher order derivative of the data as well as the measurement noise introduced and estimate the error introduced during the gpr ( so we can discard points with too much gpr estimation error ) . this enables us to input derivative data into the differential invariant and test model compatibility using the solvability test with the statistical cut - off we present . we showcase our method throughout with examples from linear and nonlinear models . we now give some background on differential algebra since a crucial step in our algorithm is to perform differential elimination to obtain equations purely in terms of input variables , output variables , and parameters . for this reason , we will only give background on the ideas from differential algebra required to understand the differential elimination process . for a more detailed description of differential algebra and the algorithms listed below , see @xcite . in what follows , we assume the reader is familiar with concepts such as _ rings _ and _ ideals _ , which are covered in great detail in @xcite . a ring @xmath25 is said to be a _ differential ring _ if there is a derivative defined on @xmath25 and @xmath25 is closed under differentiation . differential ideal _ is an ideal which is closed under differentiation . a useful description of a differential ideal is called a _ differential characteristic set _ , which is a finite description of a possibly infinite set of differential polynomials . we give the technical definition from @xcite : let @xmath26 be a set of differential polynomials , not necessarily finite . if @xmath27 is an auto - reduced set , such that no lower ranked auto - reduced set can be formed in @xmath26 , then @xmath28 is called a _ differential characteristic set_. a well - known fact in differential algebra is that differential ideals need not be finitely generated @xcite . however , a radical differential ideal is finitely generated by the _ ritt - raudenbush basis theorem _ @xcite . this result gives rise to ritt s pseudodivision algorithm ( see below ) , allowing us to compute the differential characteristic set of a radical differential ideal . we now describe various methods to find a differential characteristic set and other related notions , and we describe why they are relevant to our problem , namely , they can be used to find the _ input - output equations_. consider an ode system of the form @xmath29 and @xmath30 for @xmath7 with @xmath31 and @xmath32 rational functions of their arguments . let our differential ideal be generated by the differential polynomials obtained by subtracting the right - hand - side from the ode system to obtain @xmath33 and @xmath34 for @xmath7 . then a differential characteristic set is of the form @xcite : @xmath35 the first @xmath36 terms of the differential characteristic set , @xmath37 , are those terms independent of the state variables and when set to zero form the _ input - output equations _ : @xmath38 specifically , the @xmath36 input - output equations @xmath39 are polynomial equations in the variables @xmath40 with rational coefficients in the parameter vector @xmath10 . note that the differential characteristic set is in general non - unique , but the coefficients of the input - output equations can be fixed uniquely by normalizing the equations to make them monic . we now discuss several methods to find the input - output equations . the first method ( ritt s pseudodivision algorithm ) can be used to find a differential characteristic set for a radical differential ideal . the second method ( rosenfeldgroebner ) gives a representation of the radical of the differential ideal as an intersection of regular differential ideals and can also be used to find a differential characteristic set under certain conditions @xcite . finally , we discuss grbner basis methods to find the _ input - output equations_. a differential characteristic set of a prime differential ideal is a set of generators for the ideal @xcite . an algorithm to find a differential characteristic set of a radical ( in particular , prime ) differential ideal generated by a finite set of differential polynomals is called ritt s pseudodivision algorithm . we describe the process in detail below , which comes from the description in @xcite . note that our differential ideal as described above is a prime differential ideal @xcite . let @xmath41 be the leader of a polynomial @xmath42 , which is the highest ranking derivative of the variables appearing in that polynomial . a polynomial @xmath43 is said to be of _ lower rank _ than @xmath42 if @xmath44 or , whenever @xmath45 , the algebraic degree of the leader of @xmath43 is less than the algebraic degree of the leader of @xmath42 . a polynomial @xmath43 is _ reduced with respect to a polynomial _ @xmath42 if @xmath43 contains neither the leader of @xmath42 with equal or greater algebraic degree , nor its derivatives . if @xmath43 is not reduced with respect to @xmath42 , it can be reduced by using the pseudodivision algorithm below . 1 . if @xmath43 contains the @xmath46 derivative @xmath47 of the leader of @xmath42 , @xmath42 is differentiated @xmath48 times so its leader becomes @xmath47 . 2 . multiply the polynomial @xmath43 by the coefficient of the highest power of @xmath47 ; let @xmath49 be the remainder of the division of this new polynomial by @xmath50 with respect to the variable @xmath47 . then @xmath49 is reduced with respect to @xmath50 . the polynomial @xmath49 is called the _ pseudoremainder _ of the pseudodivision . the polynomial @xmath43 is replaced by the pseudoremainder @xmath49 and the process is iterated using @xmath51 in place of @xmath50 and so on , until the pseudoremainder is reduced with respect to @xmath42 . this algorithm is applied to a set of differential polynomials , such that each polynomial is reduced with respect to each other , to form an auto - reduced set . the result is a differential characteristic set . using the differentialalgebra package in maple , one can find a representation of the radical of a differential ideal generated by some equations , as an intersection of radical differential ideals with respect to a given ranking and rewrites a prime differential ideal using a different ranking @xcite . specifically , the rosenfeldgroebner command in maple takes two arguments : sys and r , where sys is a list of set of differential equations or inequations which are all rational in the independent and dependent variables and their derivatives and r is a differential polynomial ring built by the command differentialring specifying the independent and dependent variables and a ranking for them @xcite . then rosenfeldgroebner returns a representation of the radical of the differential ideal generated by sys , as an intersection of radical differential ideals saturated by the multiplicative family generated by the inequations found in sys this representation consists of a list of regular differential chains with respect to the ranking of r. note that rosenfeldgroebner returns a differential characteristic set if the differential ideal is prime @xcite . finally , both algebraic and differential grbner bases can be employed to find the input - output equations . to use an algebraic grbner basis , one can take a sufficient number of derivatives of the model equations and then treat the derivatives of the variables as indeterminates in the polynomial ring in @xmath52 , @xmath53 , @xmath54 , ... , @xmath55 , @xmath56 , @xmath57 , ... , @xmath58 , @xmath59 , @xmath60 , ... , etc . then a grbner basis of the ideal generated by this full system of ( differential ) equations with an elimination ordering where the state variables and their derivatives are eliminated first can be found . details of this approach can be found in @xcite . differential grbner bases have been developed by carr ferro @xcite , ollivier @xcite , and mansfield @xcite , but currently there are no implementations in computer algebra systems @xcite . we now discuss how to use the differential invariants obtained from differential elimination ( using ritt s pseudodivision , differential groebner bases , or some other method ) for model selection / rejection . recall our input - output relations , or differential invariants , are of the form : @xmath17 the functions @xmath19 are differential monomials , i.e. monomials in the input / output variables @xmath61 , @xmath62 , @xmath63 , etc , and the functions @xmath18 are rational functions in the unknown parameter vector @xmath10 . in order to uniquely fix the rational coefficients @xmath18 to the differential monomials @xmath19 , we normalize each input / output equation to make it monic . in other words , we can re - write our input - output relations as : @xmath64 here @xmath65 is a differential polynomial in the input / output variables @xmath61 , @xmath62 , @xmath63 , etc . if the values of @xmath61,@xmath62 , @xmath63 , etc , were known at a sufficient number of time instances @xmath66 , then one could substitute in values of @xmath19 and @xmath65 at each of these time instances to obtain a linear system of equations in the variables @xmath67 . first consider the case of a single input - output equation . if there are @xmath68 unknown coefficients @xmath67 , we obtain the system : @xmath69 we write this linear system as @xmath23 , where @xmath28 is an @xmath70 by @xmath68 matrix of the form : @xmath71 @xmath22 is the vector of unknown coefficients @xmath72^t$ ] , and @xmath73 is of the form @xmath74^t$ ] . for the case of multiple input - output equations , we get the following block diagonal system of equations @xmath23 : @xmath75 where @xmath28 is a @xmath76 by @xmath77 matrix . for noise - free ( perfect ) data , this system @xmath23 should have a unique solution for @xmath22 @xcite . in other words , the coefficients @xmath67 of the input - output equations can be uniquely determined from enough input / output data @xcite . the main idea of this paper is the following . given a set of candidate models , we find their associated differential invariants and then substitute in values of @xmath20 , etc , at many time instances @xmath78 , thus setting up the linear system @xmath23 for each model . the solution to @xmath23 should be unique for the correct model , but there should be no solution for each of the incorrect models . thus under ideal circumstances , one should be able to select the correct model since the input / output data corresponding to that model should satisfy its differential invariant . likewise , one should be able to reject the incorrect models since the input / output data should not satisfy their differential invariants . however , with imperfect data , there could be no solution to @xmath23 even for the correct model . thus , with imperfect data , one may be unable to select the correct model . on the other hand , if there is no solution to @xmath23 for each of the candidate models , then the goal is to determine how `` badly '' each of the models fail and reject models accordingly . we now describe criteria to reject models . let @xmath80 and consider the linear system @xmath81 where @xmath82 . note , in our case , @xmath83 , so @xmath84 is just the vector @xmath73 . here , we study the solvability of under ( a specific form of ) perturbation of both @xmath28 and @xmath84 . let @xmath85 and @xmath86 denote the perturbed versions of @xmath28 and @xmath84 , respectively , and assume that @xmath87 and @xmath88 depend only on @xmath85 and @xmath86 , respectively . our goal is to infer the _ unsolvability _ of the unperturbed system from observation of @xmath85 and @xmath86 only . we will describe how to detect the rank of an augmented matrix , but first introduce notation . the singular values of a matrix @xmath80 will be denoted by @xmath89 ( note that we have trivially extended the number of singular values of @xmath28 from @xmath90 to @xmath68 . ) the rank of @xmath28 is written @xmath91 . the range of @xmath28 is denoted @xmath92 . throughout , @xmath93 refers to the euclidean norm . the basic strategy will be to assume as a null hypothesis that has a solution , i.e. , @xmath94 , and then to derive its consequences in terms of @xmath85 and @xmath86 . if these consequences are not met , then we conclude by contradiction that is unsolvable . in other words , we will provide _ sufficient but not necessary _ conditions for to have no solution , i.e. , we can only reject ( but not confirm ) the null hypothesis . we will refer to this procedure as _ testing _ the null hypothesis . we first collect some useful results . the first , weyl s inequality , is quite standard . let @xmath95 . then @xmath96 weyl s inequality can be used to test @xmath91 using knowledge of only @xmath85 . let @xmath97 and assume that @xmath98 . then @xmath99 [ cor : weyl - rank ] therefore , if is not satisfied , then @xmath100 . assume the null hypothesis . then @xmath94 , so @xmath101 ) = \operatorname{rank}(a ) \leq \min ( m , n)$ ] . therefore , @xmath102 ) = 0 $ ] . but we do not have access to @xmath103 $ ] and so must consider instead the perturbed augmented matrix @xmath104 $ ] . under the null hypothesis , @xmath105 ) \leq \| [ \tilde{a } - a , \tilde{b } - b ] \| \leq \| \tilde{a } - a \| + \| \tilde{b } - b \| . \label{eqn : augmented - sigma } \end{aligned}\ ] ] [ thm : augmented - matrix ] apply corollary [ cor : weyl - rank ] . in other words , if does not hold , then has no solution . this approach can fail to correctly reject the null hypothesis if @xmath28 is ( numerically ) low - rank . as an example , suppose that @xmath106 and let @xmath107 consist of a single vector ( @xmath108 ) . then @xmath101 ) \leq n$ ] , so @xmath102 ) = 0 $ ] ( or is small ) . assuming that @xmath109 and @xmath110 are small , @xmath111)$ ] will hence also be small . in principle , we should test directly the assertion that @xmath101 ) = \operatorname{rank}(a)$ ] . however , we can only establish lower bounds on the matrix rank ( we can only tell if a singular value is `` too large '' ) , so this is not feasible in practice . an alternative approach is to consider only _ numerical _ ranks obtained by thresholding . how to choose such a threshold , however , is not at all clear and can be a very delicate matter especially if the data have high dynamic range . the theorem is uninformative if @xmath112 since then @xmath102 ) = \sigma_{n + 1 } ( \tilde{a } , \tilde{b } ) = 0 $ ] trivially . however , this is not a significant disadvantage beyond that described above since if @xmath28 is full - rank , then it must be true that is solvable . as a proof of principle , we first apply theorem [ thm : augmented - matrix ] to a simple linear model . we start by taking perfect input and output data and then add a specific amount of noise to the output data and attempt to reject the incorrect model . in the subsequent sections , we will see how to interpret theorem [ thm : augmented - matrix ] statistically under a particular `` noise '' model for the perturbations . here , we take data from a linear 3-compartment model , add noise , and try to reject the general form of the linear 2-compartment model with the same input / output compartments . [ ex : mainex ] let our model be a 3-compartment model of the following form : @xmath113 @xmath114 here we have an input to the first compartment of the form @xmath115 and the first compartment is measured , so that @xmath116 represents the output . the solution to this system of odes can be easily found of the form : @xmath117 so that @xmath118 . the input - output equation for a @xmath119 compartment model with a single input / output to the first compartment has the form : @xmath120 where @xmath121 are the coefficients of the characteristic polynomial of the matrix @xmath28 and @xmath122 are the coefficients of the characteristic polynomial of the matrix @xmath123 which has the first row and first column of @xmath28 removed . we now substitute values of @xmath124 at time instances @xmath125 into our input - output equation and solved the resulting linear system of equations for @xmath126 . we get that @xmath127 , which agrees with the coefficients of the characteristic polynomials of @xmath28 and @xmath123 . we now attempt to reject the 2-compartment model using 3-compartment model data . we find the input - output equations for a @xmath128 compartment model with a single input / output to the first compartment , which has the form : @xmath129 where again @xmath130 are the coefficients of the characteristic polynomial of the matrix @xmath28 and @xmath131 is the coefficient of the characteristic polynomial of the matrix @xmath123 which has the first row and first column of @xmath28 removed . we substitute values of @xmath132 at time instances @xmath133 into our input - output equation and attempt to solve the resulting linear system of equations for @xmath134 . the singular values for the matrix @xmath28 with the substituted values of @xmath135 at time instances @xmath133 are : @xmath136 the singular values of the matrix @xmath137 with the substituted values of @xmath132 at time instances @xmath133 are : @xmath138 we add noise to our matrix a in the following way . to each entry @xmath139 , and @xmath140 , we add @xmath141 where @xmath142 is a random real number between @xmath143 and @xmath144 , and @xmath145 equals @xmath146 . then the noisy matrix @xmath85 has the following singular values : @xmath147 we now add noise to our vector @xmath73 in the following way . to each entry @xmath148 , we add @xmath141 where @xmath142 is a random real number between @xmath143 and @xmath144 , and @xmath145 equals @xmath146 . then the noisy matrix @xmath149 has the following singular values : @xmath150 we find the matrix @xmath151 and compare the norm of this matrix to the smallest singular value of @xmath149 . since the frobenius norm of @xmath151 is @xmath152 , which is _ less than _ the smallest singular value @xmath153 , we can reject this model . thus , using noisy 3-compartment model data , we are able to reject the 2-compartment model . we now consider the statistical inference of the solvability of . first , we need a noise model . if the perturbations @xmath109 and @xmath110 are bounded , e.g. , @xmath154 and @xmath155 for some @xmath156 ( representing a relative accuracy of @xmath145 in the `` measurements '' @xmath85 and @xmath86 ) , then theorem [ thm : augmented - matrix ] can be used at once . however , it is customary to model such perturbations as normal random variables , which are not bounded . here , we will assume a noise model of the form @xmath157 where @xmath158 is a ( computable ) matrix that depends on @xmath85 and similarly with @xmath159 , @xmath160 denotes the hadamard ( entrywise ) matrix product @xmath161 , and @xmath162 is a matrix - valued random variable whose entries @xmath163 are independent standard normals . in our application of interest , the entries of @xmath158 depend on those of @xmath85 as follows . let @xmath164 for some input vector @xmath165 but suppose that we can only observe the `` noisy '' vector @xmath166 . then the corresponding perturbed matrix entries are @xmath167 by the additivity formula @xmath168 for standard gaussians . however , the statistical conclusion is still valid since @xmath169 `` dominates '' @xmath170 in the sense that the former has variance @xmath171 , while the latter has variance only @xmath172 . in other words , we were wrong but in the conservative direction . this was taken into account in @xcite . ] @xmath173 therefore , @xmath174 so , to first order in @xmath145 , @xmath175 an analogous derivation holds for @xmath159 . each of the bounds in the theorems above are linear in @xmath109 and @xmath110 ( for theorem [ thm : augmented - matrix ] , the bound is simply the sum of these two ) and so may be written as @xmath176 by absorbing constants . the basic strategy is now as follows . let @xmath177 be a test statistic , i.e. , @xmath111)$ ] in [ sec : augmented - matrix ] . then since @xmath178 where we have made explicit the dependence of both sides on the same underlying random mechanism @xmath179 , the ( cumulative ) distribution function of @xmath177 must dominate that of @xmath176 , i.e. , @xmath180 thus , @xmath181 [ eqn : prob - tau ] note that if , e.g. , @xmath182 ( i.e. , if @xmath84 were known exactly ) , then simplifies to just @xmath183 . using , we can associate a @xmath184-value to any given realization of @xmath177 by referencing upper tail bounds for quantities of the form @xmath185 . recall that @xmath186 under the null hypothesis . in a classical statistical hypothesis testing framework , we may therefore reject the null hypothesis if is at most @xmath187 , where @xmath187 is the desired significance level ( e.g. , @xmath188 ) . we now turn to bounding @xmath189 , where we will assume that @xmath190 . this can be done in several ways . one easy way is to recognize that @xmath191 where @xmath192 is the frobenius norm , so @xmath193 but @xmath194 has a chi distribution ) . ] with @xmath195 degrees of freedom . therefore , @xmath196 however , each inequality in can be quite loose : the first is loose in the sense that @xmath197 while the second in that @xmath198 but @xmath199 a slightly better approach is to use the inequality @xcite @xmath200 where @xmath201 and @xmath202 denote the @xmath203th row and @xmath204th column , respectively , of @xmath205 . the @xmath206 term can then be handled using a chi distribution via @xmath207 as above or directly using a concentration bound ( see below ) . variations on this undoubtedly exist . here , we will appeal to a result by tropp @xcite . the following is from 4.3 in @xcite . let @xmath190 , where each @xmath163 . then for any @xmath208 , @xmath209 [ thm : hadamard - gaussian ] the bound for @xmath210 can then be computed as follows . let @xmath211 so that @xmath212 . then by theorem [ thm : hadamard - gaussian ] , @xmath213 \ , dt , \end{aligned}\ ] ] where @xmath214 and @xmath215 are the `` variance '' parameters in the theorem for @xmath158 and @xmath159 , respectively . the term in parentheses simplifies to @xmath216\\ & = \frac{1}{\sigma_{a}^{2 } \sigma_{b}^{2 } } \left [ ( \sigma_{a}^{2 } + \sigma_{b}^{2 } ) \left ( t - \frac{\sigma_{a}^{2}}{\sigma_{a}^{2 } + \sigma_{b}^{2 } } x \right)^{2 } + \sigma_{a}^{2 } \left ( 1 - \frac{\sigma_{a}^{2}}{\sigma_{a}^{2 } + \sigma_{b}^{2 } } \right ) x^{2 } \right]\\ & = \frac{1}{\sigma_{a}^{2 } \sigma_{b}^{2 } } \left [ ( \sigma_{a}^{2 } + \sigma_{b}^{2 } ) \left ( t - \frac{\sigma_{a}^{2}}{\sigma_{a}^{2 } + \sigma_{b}^{2 } } x \right)^{2 } + \frac{\sigma_{a}^{2 } \sigma_{b}^{2}}{\sigma_{a}^{2 } + \sigma_{b}^{2 } } x^{2 } \right]\\ & = \frac{\sigma_{a}^{2 } + \sigma_{b}^{2}}{\sigma_{a}^{2 } \sigma_{b}^{2 } } \left ( t - \frac{\sigma_{a}^{2}}{\sigma_{a}^{2 } + \sigma_{b}^{2 } } x \right)^{2 } + \frac{x^{2}}{\sigma_{a}^{2 } + \sigma_{b}^{2 } } \end{aligned}\ ] ] on completing the square . therefore , @xmath217 \int_{0}^{x } \exp \left [ -\frac{1}{2 } \left ( \frac{\sigma_{a}^{2 } + \sigma_{b}^{2}}{\sigma_{a}^{2 } \sigma_{b}^{2 } } \right ) \left ( t - \frac{\sigma_{a}^{2}}{\sigma_{a}^{2 } + \sigma_{b}^{2 } } x \right)^{2 } \right ] dt . \end{aligned}\ ] ] now set @xmath218 so that the integral becomes @xmath219 dt = \int_{0}^{x } \exp \left [ -\frac{(t - \alpha x)^{2}}{2 \sigma^{2 } } \right ] dt . \end{aligned}\ ] ] the variable substitution @xmath220 then gives @xmath221 dt = \sigma \int_{-\alpha x / \sigma}^{(1 - \alpha ) x / \sigma } e^{-u^{2}/2 } \ , du = \sqrt{2 \pi } \sigma \left [ \phi \left ( \frac{(1 - \alpha ) x}{\sigma } \right ) - \phi \left ( -\frac{\alpha x}{\sigma } \right ) \right ] , \end{aligned}\ ] ] where @xmath222 is the standard normal distribution function . thus , @xmath223 \exp \left [ -\frac{1}{2 } \left ( \frac{x^{2}}{\sigma_{a}^{2 } + \sigma_{b}^{2 } } \right ) \right ] . \label{eqn : p1 } \end{aligned}\ ] ] a similar ( but much simpler ) analysis yields @xmath224 we next present a method for estimating higher order derivatives and the estimation error using gaussian process regression and then apply the differential invariant method to both linear and nonlinear models in the subsequent sections . a gaussian process ( gp ) is a stochastic process @xmath225 , where @xmath226 is a mean function and @xmath227 a covariance function . gps are often used for regression / prediction as follows . suppose that there is an underlying deterministic function @xmath228 that we can only observe with some measurement noise as @xmath229 , where @xmath230 for @xmath231 the dirac delta . we consider the problem of finding @xmath228 in a bayesian setting by assuming it to be a gp with prior mean and covariance functions @xmath232 and @xmath233 , respectively . then the joint distribution of @xmath234^{{\mathsf{t}}}$ ] at the observation points @xmath235^{{\mathsf{t}}}$ ] and @xmath236^{{\mathsf{t}}}$ ] at the prediction points @xmath237^{{\mathsf{t}}}$ ] is @xmath238 the conditional distribution of @xmath239 given @xmath240 is also gaussian : @xmath241 where @xmath242 are the posterior mean and covariance , respectively . this allows us to infer @xmath239 on the basis of observing @xmath243 . the diagonal entries of @xmath244 are the posterior variances and quantify the uncertainty associated with this inference procedure . equation provides an estimate for the function values @xmath239 . what if we want to estimate its derivatives ? let @xmath245 for some covariance function @xmath48 . then @xmath246 by linearity of differentiation . thus , @xmath247 \hat{x } ( \boldsymbol{t } ) \cr\- x(\boldsymbol{s } ) \cr x'(\boldsymbol{s } ) \cr \vdots \cr x^{(n ) } ( \boldsymbol{s } ) \cr \end{pmat } \sim { \mathcal{n}}\left ( \begin{pmat}[{. } ] \mu_{{\text{prior } } } ( \boldsymbol{t } ) \cr\- \mu_{{\text{prior } } } ( \boldsymbol{s } ) \cr \mu_{{\text{prior}}}^{(1 ) } ( \boldsymbol{s } ) \cr \vdots \cr \mu_{{\text{prior}}}^{(n ) } ( \boldsymbol{s } ) \cr \end{pmat } , \begin{pmat}[{| ... } ] \sigma_{{\text{prior } } } ( \boldsymbol{t } , \boldsymbol{t } ) + \sigma^{2 } ( \boldsymbol{t } ) i & \sigma_{{\text{prior}}}^{{\mathsf{t } } } ( \boldsymbol{s } , \boldsymbol{t } ) & \sigma_{{\text{prior}}}^{(1,0),{\mathsf{t } } } ( \boldsymbol{s } , \boldsymbol{t } ) & \cdots & \sigma_{{\text{prior}}}^{(n,0 ) , { \mathsf{t } } } ( \boldsymbol{s } , \boldsymbol{t } ) \cr\- \sigma_{{\text{prior } } } ( \boldsymbol{s } , \boldsymbol{t } ) & \sigma_{{\text{prior } } } ( \boldsymbol{s } , \boldsymbol{s } ) & \sigma_{{\text{prior}}}^{(1,0 ) , { \mathsf{t } } } ( \boldsymbol{s } , \boldsymbol{s } ) & \cdots & \sigma_{{\text{prior}}}^{(n,0 ) , { \mathsf{t } } } ( \boldsymbol{s } , \boldsymbol{s } ) \cr \sigma_{{\text{prior}}}^{(1,0 ) } ( \boldsymbol{s } , \boldsymbol{t } ) & \sigma_{{\text{prior}}}^{(1,0 ) } ( \boldsymbol{s } , \boldsymbol{s } ) & \sigma_{{\text{prior}}}^{(1,1 ) } ( \boldsymbol{s } , \boldsymbol{s } ) & \cdots & \sigma_{{\text{prior}}}^{(n,1 ) , { \mathsf{t } } } ( \boldsymbol{s } , \boldsymbol{s } ) \cr \vdots & \vdots & \vdots & \ddots & \vdots \cr \sigma_{{\text{prior}}}^{(n,0 ) } ( \boldsymbol{s } , \boldsymbol{t } ) & \sigma_{{\text{prior}}}^{(n,0 ) } ( \boldsymbol{s } , \boldsymbol{s } ) & \sigma_{{\text{prior}}}^{(n,1 ) } ( \boldsymbol{s } , \boldsymbol{s } ) & \cdots & \sigma_{(n , n ) } ( \boldsymbol{s } , \boldsymbol{s } ) \cr \end{pmat } \right ) , \end{aligned}\ ] ] where @xmath248 is the prior mean for @xmath249 and @xmath250 . this joint distribution is exactly of the form . an analogous application of then yields the posterior estimate of @xmath251 for all @xmath252 . alternatively , if we are interested only in the posterior variances of each @xmath253 , then it suffices to consider each @xmath254 block independently : @xmath255 the cost of computing @xmath256 can clearly be amortized over all @xmath203 . we now consider the specific case of the squared exponential ( se ) covariance function @xmath257 , \end{aligned}\ ] ] where @xmath258 is the signal variance and @xmath90 is a length scale . the se function is one of the most widely used covariance functions in practice . its derivatives can be expressed in terms of the ( probabilists ) hermite polynomials @xmath259 ( these are also sometimes denoted @xmath260 ) . the first few hermite polynomials are @xmath261 , @xmath262 , and @xmath263 . we need to compute the derivatives @xmath264 . let @xmath265 so that @xmath266 . then @xmath267 and @xmath268 . therefore , @xmath269 the gp regression requires us to have the values of the hyperparameters @xmath270 , @xmath271 , and @xmath90 . in practice , however , these are hardly ever known . in the examples below , we deal with this by estimating the hyperparameters from the data by maximizing the likelihood . we do this by using a nonlinear conjugate gradient algorithm , which can be quite sensitive to the initial starting point , so we initialize multiple runs over a small grid in hyperparameter space and return the best estimate found . this increases the quality of the estimated hyperparameters but can still sometimes fail . we showcase our method on competing models : linear compartment models ( 2 and 3 species ) , lotka - volterra models ( 2 and 3 species ) and lorenz . as the linear compartment differential invariants were presented in an earlier section , we compute the differential invariants of the lotka - volterra and lorenz using rosenfeldgroebner . we simulate each of these models to generate time course data , add varying levels of noise , and estimate the necessary higher order derivatives using gp regression . as described in the earlier section , we require the estimation of the higher order derivatives to satisfy a negative log likelihood value , otherwise the gp fit is not ` good ' . in some cases , this can be remedied by increase the number of data points . using the estimated gp regression data , we test each of the models using the differential invariant method on other models . [ ex : lv2 ] the two species lotka - volterra model is : @xmath272 where @xmath273 and @xmath274 are variables , and @xmath275 are parameters . we assume only @xmath273 is observable and perform differential elimination and obtain our differential invariant in terms of only @xmath276 : @xmath277 [ ex : lv3 ] by including an additional variable @xmath278 , the three species lotka - volterra model is : @xmath279 assuming only @xmath116 is observable . after differential elimination , the differential invariant is : @xmath280 [ ex : lor ] another three species model , the lorenz model , is described by the system of equations : @xmath281 we assume only @xmath116 is observable , perform differential elimination , and obtain the following invariant : @xmath282 [ ex : lc2 ] a linear 2-compartment model without input can be written as : @xmath283 where @xmath273 and @xmath274 are variables , and @xmath284 are parameters . we assume only @xmath273 is observable and perform differential elimination and obtain our differential invariant in terms of only @xmath276 : @xmath285 [ ex : lc3 ] the linear 3-compartment model without input is : @xmath286 where @xmath287 are variables , and @xmath288 are parameters . we assume only @xmath273 is observable and perform differential elimination and obtain our differential invariant in terms of only @xmath276 : @xmath289 by assuming @xmath116 in examples 6.16.5 represents the same observable variable , we apply our method to data simulated from each model and perform model comparison . the models are simulated and 100 time points are obtained variable @xmath165 in each model . we add different levels of gaussian noise to the simulated data , and then estimate the higher order derivatives from the data . for example , during our study we found that for some parameters of the lotka - volterra three species model , e.g. @xmath290 $ ] , we obtained a positive log - likelihood , which meant that we could not estimate the higher order derivatives of the data . once the data is obtained and derivative data are estimated through the gp regression , each model data set is tested against the other differential invariants . results are shown in figure [ fig - four ] , where a value of 0 , means model rejected , and 1 means model is compatible . we find that we can reject the three species lotka - volterra model and lorenz model for data simulated from the lotka - volterra two species ; however both linear compartment models are compatible . for data from the three species lotka - volterra model , the linear compartment models and two - species lotka - volterra can be rejected until the noise increases and then the method can no longer reject any models . finally data generated from the lorenz model can only reject the two species linear compartment and two species lotka - volterra model . $ ] and initial condition @xmath291 $ ] . ( b ) data simulated from three species lotka - volterra model with parameter values @xmath292 $ ] and initial condition @xmath293 $ ] . ( c ) data simulated from the lorenz model with parameter values @xmath294 $ ] and initial condition @xmath293 $ ] . ( d ) data simulated from the linear compartment three species model with parameter values @xmath295 $ ] and initial condition @xmath296 $ ] . ] we have demonstrated our model discrimination algorithm on various models . in this section , we consider some other theoretical points regarding differential invariants . note that we have assumed that the parameters are all unknown and we have not taken any possible algebraic dependencies among the coefficients into account . this latter point is another reason our algorithm only concerns model rejection and not model selection . thus , each unknown coefficient is essential treated as an independent unknown variable in our linear system of equations . however , there may be instances where we d like to consider incorporating this additional information . we first consider the effect of incorporating known parameter values . in @xcite , an explicit formula for the input - output equations for linear models was derived . in particular , it was shown that all linear @xmath297compartment models corresponding to strongly connected graphs with at least one leak and having the same input and output compartments will have the same differential polynomial form of the input - output equations . for example , a linear 2-compartment model with a single input and output in the same compartment and corresponding to a strongly connected graph with at least one leak has the form : @xmath298 thus , our model discrimination method would not work for two distinct linear 2-compartment models with the above - mentioned form . in order to discriminate between two such models , we need to take other information into account , e.g. known parameter values . consider the following two linear 2-compartment models : @xmath299 @xmath300 whose corresponding input - output equations are of the form : @xmath301 notice that both of these equations are of the above - mentioned form , i.e. both 2-compartment models have a single input and output in the same compartment and correspond to strongly connected graphs with at least one leak . in the first model , there is a leak from the first compartment and an exchange between compartments @xmath144 and @xmath128 . in the second model , there is a leak from the second compartment and an exchange between compartments @xmath144 and @xmath128 . assume that the parameter @xmath302 is known . in the first model , this changes our invariant to : @xmath303 in the second model , our invariant is : @xmath304 in this case , the right - hand sides of the two equations are the same , but the first equation has two variables ( coefficients ) while the second equation has three variables ( coefficients ) . thus , if we had data from the second model , we could try to reject the first model ( much like the 3-compartment versus 2-compartment model discrimination in the examples below ) . in other words , a vector in the span of @xmath305 and @xmath306 for @xmath307 may not be in the span of @xmath139 and @xmath140 only . we next consider the effect of incorporating coefficient dependency relationships . while we can not incorporate the polynomial algebraic dependency relationships among the coefficients in our linear algebraic approach to model rejection , we can include certain dependency conditions , such as certain coefficients becoming known constants . we have already seen one way in which this can happen in the previous example ( from known nonzero parameter values ) . we now explore the case where certain coefficients go to zero . from the explicit formula for input - output equations from @xcite , we get that a linear model without any leaks has a zero term for the coefficient of @xmath140 . thus a linear 2-compartment model with a single input and output in the same compartment and corresponding to a strongly connected graph without any leaks has the form : @xmath308 thus to discriminate between two distinct linear 2-compartment models , one with leaks and one without any leaks , we should incorporate this zero coefficient into our invariant . consider the following two linear 2-compartment models : @xmath309 @xmath310 whose corresponding input - output equations are of the form : @xmath311 in the first model , there is a leak from the first compartment and an exchange between compartments @xmath144 and @xmath128 . in the second model , there is an exchange between compartments @xmath144 and @xmath128 and no leaks . thus , our invariants can be written as : @xmath312 again , the right - hand sides of the two equations are the same , but the first equation has three variables ( coefficients ) while the second equation has two variables ( coefficients ) . thus , if we had data from the first model , we could try to reject the second model . in other words , a vector in the span of @xmath305 and @xmath306 for @xmath307 may not be in the span of @xmath139 and @xmath306 only . finally , we consider the identifiability properties of our models . if the number of parameters is greater than the number of coefficients , then the model is unidentifiable . on the other hand , if the number of parameters is less than or equal to the number of coefficients , then the model could possibly be identifiable . clearly , an identifiable model is preferred over an unidentifiable model . we note that , in our approach of forming the linear system @xmath23 from the input - output equations , we could in theory solve for the coefficients @xmath22 and then solve for the parameters from these known coefficient values if the model is identifiable @xcite . however , this is not a commonly used method to estimate parameter values in practice . as noted above , the possible algebraic dependency relationships among the coefficients are not taken into account in our linear algebra approach . this means that there could be many different models with the same differential polynomial form of the input - output equations . if such a model can not be rejected , we note that an identifiable model satisfying a particular input - output relationship is preferred over an unidentifiable one satisying the same form of the input - output relations , as we see in the following example . consider the following two linear 2-compartment models : @xmath299 @xmath313 whose corresponding input - output equations are of the form : @xmath314 in the first model , there is a leak from the first compartment and an exchange between compartments @xmath144 and @xmath128 . in the second model , there are leaks from both compartments and an exchange between compartments @xmath144 and @xmath128 . thus , both models have invariants of the form : @xmath298 since the first model is identifiable and the second model is unidentifiable , we prefer to use the form of the first model if the model s invariant can not be rejected . after performing this differential algebraic statistics model rejection , one has already obtained the input - output equations to test structural identifiability @xcite . in a sense , our method extends the current spectrum of potential approaches for comparing models with time course data , in that one first can reject incompatible models , then test structural identifiability of compatible models using input - output equations obtained from the differential elimination , infer parameter values of the admissible models , and apply an information criterion model selection method to assert the best model . notably the presented differential algebraic statistics method does not penalize for model complexity , unlike traditional model selection techniques . rather , we reject when a model can not , for any parameter values , be compatible with the given data . we found that simpler models , such as the linear 2 compartment model could be rejected when data was generated from a more complex model , such as the three species lotka - volterra model , which elicits a wider range of behavior . on the other hand , more complex models , such as the lorenz model , were often not rejected , from data simulated from less complex models . in future it would be helpful to better understand the relationship between differential invariants and dynamics . we also think it would be beneficial to investigate algebraic properties of sloppiness @xcite . we believe there is large scope for additional parameter - free coplanarity model comparison methods . it would be beneficial to explore which algorithms for differential elimination can handle larger systems , and whether this area could be extended . the authors acknowledge funding from the american institute of mathematics ( aim ) where this research commenced . the authors thank mauricio barahona , mike osborne , and seth sullivant for helpful discussions . we are especially grateful to paul kirk for discussions on gps and providing his gp code , which served as an initial template to get started . nm was partially supported by the david and lucille packard foundation . hah acknowledges funding from ams simons travel grant , epsrc fellowship ep / k041096/1 and mph stumpf leverhulme trust grant . c. aistleitner , _ relations between grbner bases , differential grbner bases , and differential characteristic sets _ , masters thesis , johannes kepler universitt , 2010 . h. akaike , _ a new look at the statistical model identification _ , ieee trans . automat . control , * 19 * ( 1974 ) , pp . 716723 . f. boulier , _ differential elimination and biological modelling _ , radon series comp . math . , * 2 * ( 2007 ) , pp . 111 - 139 . f. boulier , d. lazard , f. ollivier , m. petitot , _ representation for the radical of a finitely generated differential ideal _ , in : issac 95 : proceedings of the 1995 international symposium on symbolic and algebraic computation , pp 158 - 166 . acm press , 1995 . g. carr ferro , em grbner bases and differential algebra , in l. huguet and a. poli , editors , proceedings of the 5th international symposium on applied algebra , algebraic algorithms and error - correcting codes , volume 356 of lecture notes in computer science , pp . 131 - 140 . springer , 1987 . clarke , _ stoichiometric network analysis _ , cell biophys . , 12 ( 1988 ) , pp . d. cox , j. little , and donal oshea , _ ideals , varieties , and algorithms _ , springer , new york , 2007 . c. conradi , j. saez - rodriguez , e.d . gilles , j. raisch , _ using chemical reaction network theory to discard a kinetic mechanism hypothesis _ , iee proc . 152 ( 2005 ) , pp . s. diop , _ differential algebraic decision methods and some applications to system theory _ , * 98 * ( 1992 ) , pp . 137 - 161 . m. drton , b. sturmfels , s. sullivant , _ lectures on algebraic statistics _ , oberwolfach seminars ( springer , basel ) vol . 39 . 2009 . m. feinberg , _ chemical reaction network structure and the stability of complex isothermal reactors i . the deficiency zero and deficiency one theorems _ , chem . , * 42 * ( 1987 ) , pp . 22292268 . m. feinberg , _ chemical reaction network structure and the stability of complex isothermal reactors ii . multiple steady states for networks of deficiency one _ , chem . , * 43 * ( 1988 ) , pp . 125 . k. forsman , _ constructive commutative algebra in nonlinear control theory _ , phd thesis , linkping university , 1991 . o. golubitsky , m. kondratieva , m. m. maza , and a. ovchinnikov , _ a bound for the rosenfeld - grbner algorithm _ , j. symbolic comput . , * 43 * ( 2008 ) , pp . 582 - 610 . e. gross , h.a . harrington , z. rosen , b. sturmfels , _ algebraic systems biology : a case study for the wnt pathway _ , bull . biol . , * 78*(1 ) ( 2016 ) , pp . 21 - 51 . e. gross , b. davis , k.l . ho , d. bates , h. harrington , _ numerical algebraic geometry for model selection _ , submitted . j. gunawardena , _ distributivity and processivity in multisite phosphorylation can be distinguished through steady - state invariants _ , biophys . j. , 93 ( 2007 ) , pp . gutenkunst , j.j . waterfall , f.p . casey , k.s . brown , c.r . myers , j.p . sethna , _ universally sloppy parameter sensitivities in systems biology models _ , plos comput . biol . , 3 ( 2007 ) , harrington , k.l . ho , t. thorne , m.p.h . stumpf , _ parameter - free model discrimination criterion based on steady - state coplanarity _ , proc . , * 109*(39 ) ( 2012 ) , pp . 1574615751 . i. kaplansky , _ an introduction to differential algebra _ , hermann , paris , 1957 . e. r. kolchin , _ differential algebra and algebraic groups _ , pure appl . math . , * 54 * ( 1973 ) . l. ljung and t. glad , _ on global identifiability for arbitrary model parameterization _ , automatica , * 30*(2 ) ( 1994 ) , pp . 265 - 276 . maclean , z. rosen , h.m . byrne , h.a . harrington , _ parameter - free methods distinguish wnt pathway models and guide design of experiments _ , proc . , * 112*(9 ) ( 2015 ) , pp . 26522657 . e. mansfield , _ differential grbner bases _ , phd thesis , university of sydney , 1991 . a.k . manrai , j. gunawardena , _ the geometry of multisite phosphorylation _ , j. , * 95 * ( 2008 ) , pp . 55335543 . maple documentation . url http://www.maplesoft.com/support/help/maple/view.aspx?path=differentialalgebra n. meshkat , c. anderson , and j. j. distefano iii , _ alternative to ritt s pseudodivision for finding the input - output equations of multi - output models _ , math biosci . , * 239 * ( 2012 ) , pp . 117 - 123 . n. meshkat , s. sullivant , and m. eisenberg , _ identifiability results for several classes of linear compartment models _ , bull . math . biol . , * 77 * ( 2015 ) , pp . 1620 - 1651 . f. ollivier , _ le probleme de lidentifiabilite structurelle globale : etude theoretique , methodes effectives and bornes de complexite _ , phd thesis , ecole polytechnique , 1990 . f. ollivier , _ standard bases of differential ideals_. in s. sakata , editor , proceedings of the 8th international symposium on applied algebra , algorithms , and error - 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we present a method for rejecting competing models from noisy time - course data that does not rely on parameter inference . first we characterize ordinary differential equation models in only measurable variables using differential algebra elimination . next we extract additional information from the given data using gaussian process regression ( gpr ) and then transform the differential invariants . we develop a test using linear algebra and statistics to reject transformed models with the given data in a parameter - free manner . this algorithm exploits the information about transients that is encoded in the model s structure . we demonstrate the power of this approach by discriminating between different models from mathematical biology . keywords : model selection , differential algebra , algebraic statistics , mathematical biology

1603.09730

there is recent observational evidence that a significant fraction of massive evolved spheroidal stellar systems is already in place at redshift @xmath4 . however , only a small percentage of these galaxies is fully assembled @xcite . the galaxies are smaller by a factor of three to five compared to present - day ellipticals at similar masses . their effective stellar mass densities are at least one order of magnitude higher @xcite with significantly higher surface brightnesses compared to their low redshift analogs . these observations are difficult to reconcile with some current idealized formation scenarios for elliptical galaxies . a simple conclusion from the data is that most early - type galaxies can neither have fully formed in a simple monolithic collapse nor a binary merger of gas - rich disks at high redshift , unless their increase in size can be explained by secular processes such as adiabatic expansion driven by stellar mass loss and/or strong feedback @xcite . additionally , simple passive evolution of the stellar population is in contradiction with observations of local ellipticals @xcite . + dry ( i.e. gas - poor , collisionless ) mergers and stellar accretion events are the prime candidates for the strong mass and size evolution of stellar spheroids at @xmath5 @xcite as the additional presence of a dissipative component in a major merger event would limit the size increase ( see e.g. @xcite ) . the observed ellipticals are already very massive at high redshift , thus we expect from the shape of the mass function that minor mergers should be much more common than major mergers until z=0 . massive early - type galaxies may undergo not more than one major merger ( with typically low cool gas content , see also @xcite ) since @xmath6 ( @xcite , see also @xcite ) with a significant contribution from minor mergers for the mass buildup @xcite . the low number of observed major early - type mergers is also supported by theoretical evidence that massive ( @xmath7 ) halos at @xmath8 typically experience only one major merger or less until @xmath9 and minor mergers are much more common @xcite . on average , this is not enough to account for the required mass and size growth ( see also @xcite ) as major dry mergers at most increase the size of a simple one component system by a factor of two and allowing for dark matter halos reduces the size growth further @xcite . in this letter we use , as a proof of principle , a very high resolution cosmological simulation of the formation of a spheroid with no major mergers below @xmath10 in combination with simple scaling relations to show that the observed rapid size growth and density evolution of spheroidal galaxies can be explained by minor mergers and small accretion events . the problem is computationally very expensive . at high redshift the observed ellipticals have half - mass sizes of @xmath11 with accreting subsystems of even smaller size . as we know from isolated merger simulations ( see e.g. @xcite ) , to resolve such a system reasonably well we require a force softening of 10% of the effective radius , which in our case is of the order of @xmath12 and we require particle numbers of @xmath13 to simulate the galaxy in a full cosmological context over a hubble time . finally , to accurately follow the kinematics high force and integration accuracy are required . using the virial theorem we make a simple estimate of how an initial one - component stellar systems evolves when mass in stellar systems is added . we assume that a compact initial stellar system has formed dissipatively from stars . this system has a total energy @xmath14 , a mass @xmath15 , a gravitational radius @xmath16 , and the mean square speed of the stars is @xmath17 . according to the virial theorem @xcite the total energy of the system is @xmath18 we then assume that systems are accreted with energies totaling @xmath19 , masses totaling @xmath20 , gravitational radii @xmath21 and mean square speeds averaging @xmath22 . we define the fractional mass increase from all the accreted material @xmath23 and the total kinetic energy of the material as @xmath24 , further defining @xmath25 . assuming energy conservation ( orbital parameters from cosmological simulations indicate that most halos merge on parabolic orbits ) , the total energy of the final system is @xmath26 the mass of the final system is @xmath27 . therefore the ratio of the final to initial mean square speeds is @xmath28 similarly , the ratio of the final to initial gravitational radius is @xmath29 and for the ratio of the densities we get @xmath30 if during one or more mergers the initial stellar system increases its mass by a factor of two then @xmath31 . this mass increase can be caused by one equal - mass merger in which case the mean square velocities of the two systems are identical and remain unchanged in the final system ( eqn . [ disp ] ) . the radius increases by a factor of two ( eqn . [ rg ] ) and the density drops by a factor of four ( eqn . [ dens])(see also @xcite ) . if , however , the total mass increase by a factor of two is caused by accretion of very small systems with @xmath32 or @xmath33 , then the mean square velocities are reduced by a factor two , the radius is four times larger and the density is reduced by a factor of 32 with respect to the initial system ( see also @xcite for a similar derivation of the scaling relations ) . we know from the shape of the schechter function for the distribution of stellar masses that a massive system ( @xmath34 ) accretes most of its mass from lower mass systems and thus the simple calculation above makes it very plausible that even though major mergers do occur minor mergers are the main driver for the evolution in size and density of massive galaxies . ) the system assembles by the formation of in - situ stars , at low redshift ( @xmath35 ) accretion is more dominant.,width=302 ] we have performed a cosmological n - body / sph high - resolution re - simulation of an individual galaxy halo . the process of setting up the initial conditions is described in detail in @xcite and is briefly reviewed . we have re - run galaxy a at @xmath36 particles resolution using a wmap-1 @xcite cosmology with a hubble parameter of @xmath37 ( @xmath38=100@xmath39 kms@xmath40mpc@xmath40 ) with @xmath41=0.86 , @xmath42=0.2 , @xmath43=0.3 , and @xmath44=0.7 . to re - simulate the target halo at high resolution we increased the particle number to @xmath45 gas and dark matter particles within a cubic volume at redshift @xmath46 containing all particles that end up within the virialized region ( we assumed a fixed radius of @xmath47 ) of the halo at @xmath9 . the tidal forces from particles outside the high resolution cube were approximated by increasingly massive dark matter particles in 5 nested layers of decreasing resolution . the galaxy was not contaminated by massive boundary particles within the virial radius . are indicated by the dashed vertical lines . the shaded area indicates the gravitational softening length.,width=302 ] the simulation was performed with gadget-2 @xcite on woodhen at the princeton picscie hpc center using a total of 177,000 cpu hours on 64 cpus . we used a fixed comoving softening until @xmath48 , and thereafter the softening , e.g. for the stars , remained fixed at physical @xmath49 . the mass of an individual stellar particle is @xmath50 and we spawn two stars per sph particle . star formation and feedback from supernovae was included using the sub - grid multiphase model of @xcite . we require an over - density contrast of @xmath51 for the onset of star formation to avoid spurious star formation at high redshift . the threshold number density for star formation is @xmath52 and the star formation time - scale is @xmath53 . we also included an uniform uv background radiation field peaking at at @xmath54 ( see @xcite ) . at present the galaxy has a total virial mass of @xmath55 and a total stellar mass of @xmath0 . the ratio of central stellar mass to halo mass is about a factor of two larger than predicted from gravitational lensing studies @xcite , however , it is comparable to some recent predictions derived for the milky way halo ( e.g. @xcite , see however @xcite ) . the central stellar component resembles an elliptical galaxy with properties very similar to the results presented in @xcite and in this letter we only focus on particular aspects of the assembly and size evolution of the stellar component . ( black diamonds ) . from @xmath56 to @xmath9 the size increases by a factor of @xmath57.we also show the evolution of the rest - frame k - band ( red cross ) and v - band ( green triangle ) half - light radius . outliers indicate minor merger events , e.g. the most massive ( 8:1 ) merger since z=3 at z=0.3.,width=302 ] during the assembly of the central galaxy we have separated the stars within a fiducial radius of @xmath58 in fixed physical coordinates into stars that have formed in - situ from gas within the galaxy and stars that have formed outside this radius and were accreted later - on . the mass assembly history of the in - situ and accreted components of the stellar system are shown in fig . the early mass evolution at @xmath59 is driven by the assembly of in - situ stars with a decreasing contribution towards @xmath60 . below this redshift only few stars are formed within @xmath58 . the final 20% of stars are added thereafter by accretion of systems formed outside the main stellar system at radii larger than @xmath58 . the upper panel of fig . [ rho_ins_acc_feed_200_comb ] shows the density profiles of the in - situ stars at redshifts z=5,3,2,1,0 . between z=5 and z=3 the central galaxy is still building up from gas flows feeding the central region of the galaxy directly , forming a concentrated stellar system . the in - situ central stellar densities decrease by more than an order of magnitude towards lower redshifts . the spherical half - mass radii of the in - situ stellar component , show that the in - situ system is very compact ( see also @xcite ) at @xmath10 @xmath61 and its size increases by about a factor of four ( @xmath62 ) until z=0 . in the bottom panel of fig . [ rho_ins_acc_feed_200_comb ] we show the density profiles for the stars that have formed outside @xmath58 and have been accreted later - on . this component is more extended at all redshifts and has a shallower density profile . its central density stays almost constant at @xmath63 while the density at larger radii subsequently increases towards @xmath9 . the half - mass radius of this component is significantly larger than for the in - situ stars ( @xmath64 ) . the central part of the galaxy is always dominated by in - situ stars whereas at redshifts below @xmath65 and at radii larger than @xmath66 the system is dominated by accreted stars . ( black dots ) and @xmath9 ( red dots ) . the black and red lines indicate the best fitting sersic profile . at high redshift the galaxy has a higher central surface brightness and is more compact.,width=302 ] in fig . [ re ] we show the time evolution of the edge - on projected half mass radius of stars in the central galaxy within the central physical @xmath58 as a function of time . at @xmath10 the stellar system resembles a compact disk - like or bar - like object with a peak ellipticity of @xmath67 and a size of @xmath68 at z=3 . thereafter its size increases by a factor of @xmath69 to its present value of @xmath70 . we also plot the projected half - light radii in the rest frame k- and v - band using the stellar population models of @xcite assuming solar metalicity . in general the half - mass radii trace the half - light radii even at larger redshifts reasonably well . the k - band rest - frame surface brightness profiles for edge - on projections at z=0 and z=3 are shown in fig . [ kmag_all ] in combination with the best fitting sersic profiles . using the fitting procedure of @xcite excluding the central three softening lengths . at high redshift the system is very compact , @xmath71 , and has a moderate sersic index of @xmath72 . this is in agreement with the system being flattened and disk - like . at low redshift the system is more extended @xmath73 , and its sersic index has increased to @xmath74 . the galaxy is slightly more compact than typical sdss early - type galaxies at this mass but lies within the observed distribution ( @xcite , see also @xcite ) . the errors given in the figure are bootstrap errors for a fixed projection . as we have shown before the evolution in surface brightness is mainly driven by an evolution in surface density and not by stellar evolution . at z=3 the system has a total stellar mass of @xmath75 with an effective radius @xmath76 and a corresponding effective density of @xmath77 . the projected stellar line - of - sight velocity dispersion is @xmath78 . the corresponding values at z=0 are @xmath79 , @xmath80 , @xmath81 and @xmath82 , which is a typical dispersion for early - type galaxies at this mass @xcite . from z=3 to z=0 the system accretes about @xmath83 ( see fig [ new ] ) and we can assume for the above scaling relations @xmath84 and @xmath33 ( which is a reasonable assumption as most mass is accreted in very small systems and we found the most massive merger since z=3 with a mass ratio of 8:1 ) . the z=0 values of the simulated galaxy are close to the simple prediction , however , the size increase as well as the decrease in density and dispersion are more moderate . this is however expected as the real evolution of the system is more complex and there is non - negligible in - situ star formation between z=3 and z=1 . still , the simple scaling relations for stellar accretion represent the evolution of the system from z=3 to z=0 very well . however , observations of more massive ellipticals than the one presented here indicate an even stronger size increase @xcite . this effect can be expected if the assembly of more massive galaxies is even more dominated by minor mergers and stellar accretion . in particular , the drop in velocity dispersion is in qualitative agreement with first direct observations by @xcite . recently , @xcite have reported a relatively weak evolution of the density within fixed 1kpc of only a factor 2 - 3 . this also is in qualitative agreement with our simulation which shows a decrease of only a factor 1.5 from z=2 to z=0 . we also note that the stellar population of the system is already evolved at high redshift . at @xmath8 the galaxy has a stellar mass of @xmath85 and a local star formation rate of only @xmath86 and an average stellar age of @xmath87 . in this paper we show that the observed size and density evolution of massive spheroids agrees with what is to be expected from a high resolution cosmological simulation of a system which grows at late times predominantly by minor mergers and accretion of stars . we can successfully apply simple scaling relations derived from the virial theorem to demonstrate that that the size increase and decrease in density and velocity dispersion is a natural consequence of mass assembly by much less massive stellar systems and accretion @xcite and can not be explained by mass assembly histories dominated by major stellar mergers . in the simulation , a first phase @xmath88 dominated by in - situ star formation from inflowing cold gas ( see e.g. @xcite and references therein ) produces a massive and dense stellar system with sizes @xmath89 . this phase of the formation of the cores of ellipticals is followed by an extended phase @xmath90 with little in - situ star formation but significant accretion of stellar material . this material can be stripped at larger radii and increases the size of the system with time . at the same time the central concentration is reduced by dynamical friction from the surviving cores ( see @xcite ) . the apparent size increase is caused by the initial dominance of the in - situ component being heated and , at larger radii , overshadowed ultimately by the accreted stars . from z=3 to z=0 the effective radius of the system increases by a factor of 3.5 with a decrease in the effective density of more than an order of magnitude and a decrease in velocity dispersion of 20% , in good agreement with predictions from simple scaling relations for the accretion of minor mergers . detailed investigations of dark matter simulations ( see e.g. @xcite ) as well as recent observations ( see e.g. @xcite ) on the mass assembly mechanisms of early - type galaxies and their dark matter halos demonstrate the significance of minor mergers . in addition , due to the shape of the mass function , massive systems at high redshift are more likely to experience minor mergers , than lower mass galaxies . if the size evolution is in general driven by minor mergers we would expect a differential size increase , e.g. more massive high redshift systems grow larger than lower mass systems . at the same time minor mergers do also play an important role for the gravitational heating of halo gas , thereby suppressing the formation of new stars @xcite . a picture of a two phase formation process for massive spheroidal galaxies has a number of virtues . in the first dissipative phase at high redshift stars form quickly and build the compact progenitor of present - day ellipticals . in fact , it seems of minor importance if the gas is funneled to the center through streams or mergers of extended gas dominated disks ( see e.g. @xcite ) as long as it happens on a short timescale . the stars formed at this early phase are expected to be significantly enriched in alpha - elements as expected from observations @xcite and form the compact core of the elliptical galaxy . later on metal - poor stars from smaller systems are accreted and form the halo of the galaxy resulting in the observed metalicity gradient . this inside out formation scenario is also made plausible by recent observations of @xcite . there is an important test of the picture presented in this paper . if correct , then the outer parts of massive giant ellipticals will tend to be old , blue , metal - poor and relatively uniform from galaxy to galaxy since they are all composed essentially of the debris from tidally destroyed accreted small systems . the simulation was performed at the princeton picscie hpc center . this research was supported by the dfg cluster of excellence origin and structure of the universe. we thank marijn franx , pieter van dokkum and ignacio trujillo for helpful comments on the manuscript . , e. f. , naab , t. , mcintosh , d. h. , somerville , r. s. , caldwell , j. a. r. , barden , m. , wolf , c. , rix , h .- w . , beckwith , s. v. , borch , a. , hussler , b. , heymans , c. , jahnke , k. , jogee , s. , koposov , s. , meisenheimer , k. , peng , c. y. , sanchez , s. f. , & wisotzki , l. 2006 , , 640 , 241 , j. & tremaine , s. 2008 , galactic dynamics : second edition ( galactic dynamics : second edition , by james binney and scott tremaine . isbn 978 - 0 - 691 - 13026 - 2 ( hb ) . published by princeton university press , princeton , nj usa , 2008 . ) , a. , cassata , p. , pozzetti , l. , kurk , j. , mignoli , m. , renzini , a. , daddi , e. , bolzonella , m. , brusa , m. , rodighiero , g. , dickinson , m. , franceschini , a. , zamorani , g. , berta , s. , rosati , p. , & halliday , c. 2008 , , 482 , 21 , e. , renzini , a. , pirzkal , n. , cimatti , a. , malhotra , s. , stiavelli , m. , xu , c. , pasquali , a. , rhoads , j. e. , brusa , m. , di serego alighieri , s. , ferguson , h. c. , koekemoer , a. m. , moustakas , l. a. , panagia , n. , & windhorst , r. a. 2005 , , 626 , 680 , i. , mccarthy , p. j. , abraham , r. g. , glazebrook , k. , yan , h. , mentuch , e. , leborgne , d. , savaglio , s. , crampton , d. , murowinski , r. , juneau , s. , carlberg , r. g. , jrgensen , i. , roth , k. , chen , h .- w . , & marzke , r. o. 2009 , , 695 , 101 , d. n. , verde , l. , peiris , h. v. , komatsu , e. , nolta , m. r. , bennett , c. l. , halpern , m. , hinshaw , g. , jarosik , n. , kogut , a. , limon , m. , meyer , s. s. , page , l. , tucker , g. s. , weiland , j. l. , wollack , e. , & wright , e. l. 2003 , , 148 , 175 , s. , van dokkum , p. , franx , m. , labbe , i. , frster schreiber , n. m. , wuyts , s. , webb , t. , rudnick , g. , zirm , a. , kriek , m. , van der werf , p. , blakeslee , j. p. , illingworth , g. , rix , h .- w . , papovich , c. , & moorwood , a. 2007 , , 671 , 285 , x. x. , rix , h. w. , zhao , g. , re fiorentin , p. , naab , t. , steinmetz , m. , van den bosch , f. c. , beers , t. c. , lee , y. s. , bell , e. f. , rockosi , c. , yanny , b. , newberg , h. , wilhelm , r. , kang , x. , smith , m. c. , & schneider , d. p. 2008 , , 684 , 1143

using a high resolution hydrodynamical cosmological simulation of the formation of a massive spheroidal galaxy we show that elliptical galaxies can be very compact and massive at high redshift in agreement with recent observations . accretion of stripped in - falling stellar material increases the size of the system with time and the central concentration is reduced by dynamical friction of the surviving stellar cores . in a specific case of a spheroidal galaxy with a final stellar mass of @xmath0 we find that the effective radius @xmath1 increases from @xmath2 at z = 3 to @xmath3 at z = 0 with a concomitant decrease in the effective density of an order of magnitude and a decrease of the central velocity dispersion by approximately 20% over this time interval . a simple argument based on the virial theorem shows that during the accretion of weakly bound material ( minor mergers ) the radius can increase as the square of the mass in contrast to the usual linear rate of increase for major mergers . by undergoing minor mergers compact high redshift spheroids can evolve into present - day systems with sizes and concentrations similar to observed local ellipticals . this indicates that minor mergers may be the main driver for the late evolution of sizes and densities of early - type galaxies .

0903.1636

there are basically two methods to determine the weak axial form factor of the nucleon . one set of experimental data comes from measurements of ( quasi)elastic ( anti)neutrino scattering on protons @xcite , deuterons @xcite and other nuclei ( al , fe ) @xcite or composite targets like freon @xcite and propane @xcite . in the ( quasi)elastic picture of ( anti)neutrino - nucleus scattering , the @xmath8 weak transition amplitude can be expressed in terms of the nucleon electromagnetic form factors @xmath9 and @xmath10 and the axial form factor @xmath11 . the axial form factor is then extracted by fitting the @xmath5-dependence of the ( anti)neutrino - nucleon cross section , @xmath12 in which @xmath13 is contained in the bilinear forms @xmath14 , @xmath15 and @xmath16 of the relevant form factors and is assumed to be the only unknown quantity . it can be parameterised in terms of an ` axial mass ' @xmath17 as @xmath18 . as extracted from ( quasi)elastic neutrino and antineutrino scattering experiments . the weighted average is @xmath19 , or @xmath20 using the scaled - error averaging recommended by ref . @xcite.,height=264 ] fig . [ fig : m_a_nu ] shows the available values for @xmath17 obtained from these studies . references @xcite reported severe uncertainties in either knowledge of the incident neutrino flux or reliability of the theoretical input needed to subtract the background from genuine elastic events ( both of which gradually improved in subsequent experiments ) . the values derived fall well outside the most probable range of values known today and exhibit very large statistical and systematical errors . following the data selection criteria of the particle data group @xcite , they were excluded from this compilation . another body of data comes from charged pion electroproduction on protons @xcite slightly above the pion production threshold . as opposed to neutrino scattering , which is described by the cabibbo - mixed @xmath21 theory , the extraction of the axial form factor from electroproduction requires a more involved theoretical picture . as extracted from charged pion electroproduction experiments . the weighted average ( excluding our result ) is @xmath22 , or @xmath23 using the scaled - error averaging @xcite . including our extracted value , the weighted scaled - error average becomes @xmath24 . note that our value contains both the statistical _ and _ systematical uncertainty ; for other values the systematical errors were not explicitly given . sp : soft - pion limit , dr : analysis using approach of ref . @xcite , fpv : ref . @xcite , bnr : ref . @xcite.,height=359 ] the basic result about low energy photoproduction of massless charged pions can be traced back to the kroll - ruderman theorem @xcite , extended to virtual photons by nambu , luri and shrauner @xcite , who obtained the @xmath25 result for the isospin @xmath26 ( see ref . @xcite , p. 29 for notation ) electric dipole amplitude at threshold @xmath27 + \mathcal{o}(q^3 ) \biggr\}\ > { , } \label{eq : nls}\ ] ] where @xmath28 is the nucleon isovector anomalous magnetic moment , @xmath29 is the axial coupling constant , and @xmath30 is the pion decay constant . in the following years , improved models were proposed @xcite , most of them including corrections due to the finite pion mass @xmath31 . the values of the axial mass were determined , within the framework of the respective model , from the slopes of the angle - integrated differential electroproduction cross sections at threshold , @xmath32 the results of various measurements and theoretical approaches are shown in fig . [ fig : m_a_ee ] . note again that references @xcite were omitted from the fit for lack of reasonable compatibility with the other results . although the results of these investigations deviate from each other by more than their claimed accuracy , the weighted averages from neutrino scattering and electroproduction give quite precise values of the axial mass . comparing the average values of the two methods , one observes a significant difference of @xmath33 , or @xmath34 using the scaled - error averaging . chiral perturbation theory ( @xmath35 ) has recently shown a remarkable and model - independent result that already at @xmath25 , the nls result of eq . ( [ eq : nls ] ) is strongly modified due to pion loop contributions @xcite . these contributions effectively reduce the mean - square axial radius , @xmath36 the loop correction in eq . ( [ eq : chiptcorr ] ) has a value of @xmath37 , which is a @xmath38 correction to a typical @xmath39 . correspondingly , the axial mass @xmath40 would appear to be about @xmath41 larger in electroproduction than in neutrino scattering , in agreement with the observed @xmath42 . the aim of the present investigation was to determine @xmath17 from new , high precision pion electroproduction data and thereby help verify whether this discrepancy was genuine . since both the energy and momentum transfers were too high to allow for a safe extraction of @xmath43 , these data were analysed in the framework of an effective lagrangian model with the electromagnetic nucleon form factors , the electric pion form factor and the axial nucleon form factor at the appropriate vertices @xcite . the differential cross sections for @xmath44 electroproduction on protons were measured at an invariant mass of @xmath1 and at four - momentum transfers of the virtual photon @xmath2 , @xmath3 and @xmath4 . for each value of @xmath5 , we measured the scattered electron and the outgoing pion in parallel kinematics at three different polarisations of the virtual photon , enabling us to separate the transverse and the longitudinal part of the cross section by the rosenbluth technique . table [ tab : settings ] shows the experimental settings . ccccrrc setting ( @xmath45 ) & @xmath5 & @xmath46 & @xmath47 & @xmath48 & @xmath49 & @xmath50 + & @xmath51 $ ] & @xmath52 $ ] & @xmath52 $ ] & @xmath53 $ ] & @xmath53 $ ] & @xmath54 $ ] + + @xmath55 & @xmath56 & @xmath57 & @xmath58 & @xmath59 & @xmath60 & @xmath61 + @xmath62 & & @xmath63 & @xmath64 & @xmath65 & @xmath66 & @xmath67 + @xmath68 & & @xmath69 & @xmath70 & @xmath71 & @xmath72 & @xmath67 + @xmath73 & @xmath3 & @xmath57 & @xmath74 & @xmath75 & @xmath76 & @xmath77 + @xmath78 & & @xmath79 & @xmath80 & @xmath81 & @xmath82 & @xmath67 + @xmath83 & & @xmath84 & @xmath85 & @xmath86 & @xmath87 & @xmath67 + @xmath88 & @xmath89 & @xmath57 & @xmath90 & @xmath91 & @xmath92 & @xmath93 + @xmath94 & & @xmath95 & @xmath96 & @xmath97 & @xmath98 & @xmath67 + @xmath99 & & @xmath79 & @xmath100 & @xmath101 & @xmath102 & @xmath67 + [ 5pt ] the measurements were performed at the institut fr kernphysik at the university of mainz , using the continuous - wave electron microtron mami @xcite . the energies of the incoming electron beam ranged from @xmath103 to @xmath104 , and the beam energy spread did not exceed @xmath105 . the @xmath106 to @xmath107 electron beam was scattered on a liquid hydrogen target cell ( @xmath108 with @xmath109 havar walls in settings with the photon polarisation parameter @xmath110 , @xmath88 , @xmath94 and @xmath99 , and on a @xmath111-diameter cylindrical target cell with @xmath112 havar walls in all other settings ) attached to a high power target cooling system . forced circulation and a beam wobbling system were used to avoid density fluctuations of the liquid hydrogen . with this system , luminosities of up to @xmath113 ( @xmath114 ) were attained . the scattered electrons and the produced pions were detected in coincidence by the high resolution ( @xmath115 ) magnetic spectrometers a ( speca ) and b ( specb ) of the a1 collaboration @xcite . in settings with @xmath116 , @xmath62 and @xmath73 , electrons were detected with specb and pions with speca , and vice versa in all other settings . the momentum acceptance @xmath117 was @xmath118 and @xmath119 in speca and specb , respectively . heavy - metal collimators ( @xmath120 in speca , @xmath121 or @xmath122 in specb ) were used to minimise angular acceptance uncertainties due to the relatively large target cells . a trigger detector system consisting of two planes of segmented plastic scintillators and a threshold erenkov detector were used in each spectrometer . the coincidence time resolution , taking into account the different times of flight of particles for different trajectory lengths through the spectrometer , was between @xmath123 and @xmath124 fwhm . in each spectrometer , four vertical drift chambers were used for particle tracking , measurement of momenta and target vertex reconstruction . back - tracing the particle trajectories from the drift chambers through the magnetic systems , an angular resolution ( all fwhm ) better than @xmath125 ( dispersive and non - dispersive angles ) and spatial ( vertex ) resolution better than @xmath126 ( non - dispersive direction , speca ) and @xmath127 ( specb ) were achieved on the target . a more detailed description of the apparatus can be found in ref . in the offline analysis , cuts in the corrected coincidence time spectrum were applied to identify real coincidences and to eliminate the background of accidental coincidences . a cut in the energy deposited in the first scintillator plane in the pion spectrometer was used to discriminate charged pions against protons . the erenkov signal was used to identify electrons in the electron spectrometer and to veto against positrons in the pion spectrometer . the true coincidences were observed in the peak of the accumulated missing mass distribution @xmath128 , using an event - by - event reconstruction of @xmath129 . the cross sections were subsequently corrected for detector and coincidence inefficiencies ( between @xmath130 and @xmath131 ) and dead - time losses ( between @xmath132 and @xmath133 ) . the detector efficiencies and their uncertainties were measured by the three - detector method , while the coincidence efficiency of the setup and the uncertainty of the dead - time measurement were determined in simultaneous single - arm and coincidence measurements of elastic @xmath134 , which was also used to check on the acceptance for the extended targets . the accepted phase space was determined by a monte carlo simulation which provided the ( event - wise ) lorentz transformation to the cm system and incorporated radiative corrections and ionisation losses of the incoming and scattered electrons and pions . full track was kept of the particles trajectories and their lengths in target and detector materials . since exact energy losses are not known event - wise , we used the most probable energy losses for the subsequent energy loss correction in both simulation and analysis programs , and compared the corresponding missing mass spectra . the uncertainty estimates were based on relative variations of their content in dependence of the cut - off energy along the radiative tail . the uncertainty of the integrated luminosity originates only in the target density changes due to temperature fluctuations within the target cell , while the electron beam current is virtually exactly known . finally , a computer simulation was used to determine the correction factors due to the pion decaying in flight from the interaction point to the scintillation detectors , taking into account the muon contamination at the target ( correction factors ranging from @xmath135 to @xmath136 in different settings ) . the systematical errors of the pion decay correction factors were estimated from the statistical fluctuations of the back - traced muon contamination at the target . in the born approximation , the coincidence cross section for pion electroproduction can be factorised as @xcite @xmath137 where @xmath138 is the virtual photon flux and @xmath139 is the virtual photon cross section in the cm frame of the final @xmath140 system . it can be further decomposed into transverse , longitudinal and two interference parts , @xmath141 with the transverse ( @xmath45 ) and longitudinal ( @xmath142 ) polarisations of the virtual photon fixed by the electron kinematics . the measured cross sections are listed in table [ tab : cross ] . crcc setting ( @xmath45 ) & @xmath143 & stat . error & syst . error + & @xmath144 $ ] & @xmath144 $ ] & @xmath144 $ ] + + @xmath55 & @xmath145 & @xmath146 ( @xmath147% ) & @xmath148 ( @xmath149% ) + @xmath62 & @xmath150 & @xmath151 ( @xmath152% ) & @xmath153 ( @xmath149% ) + @xmath68 & @xmath154 & @xmath155 ( @xmath156% ) & @xmath157 ( @xmath158% ) + @xmath73 & @xmath159 & @xmath151 ( @xmath160% ) & @xmath161 ( @xmath162% ) + @xmath78 & @xmath163 & @xmath164 ( @xmath165% ) & @xmath166 ( @xmath167% ) + @xmath83 & @xmath168 & @xmath166 ( @xmath169% ) & @xmath170 ( @xmath171% ) + @xmath88 & @xmath172 & @xmath173 ( @xmath174% ) & @xmath146 ( @xmath171% ) + @xmath94 & @xmath175 & @xmath176 ( @xmath152% ) & @xmath146 ( @xmath167% ) + @xmath99 & @xmath177 & @xmath173 ( @xmath178% ) & @xmath170 ( @xmath179% ) + [ 5pt ] in parallel kinematics ( @xmath180 ) the interference parts vanish due to their @xmath181 and @xmath182 dependence . at constant @xmath5 , the transverse and the longitudinal cross sections can therefore be separated using the rosenbluth method by varying @xmath45 , @xmath183 not only the statistical , but also the @xmath45-correlated systematical uncertainties of the data were considered in our fit , whereas the @xmath45-independent systematical errors were included in the final uncertainty of @xmath184 and @xmath185 . the results are shown in fig . [ fig:3lines ] and table [ tab : lt ] . the @xmath5-dependence of the separated transverse and longitudinal cross sections can be seen in fig . [ fig : wqs2 ] , together with the theoretical fits used to extract the form factors . at @xmath1 , for three values of @xmath5 . the smaller error bars correspond to statistical , the larger ones to the sum of statistical and systematical errors.,height=302 ] ccc setting ( @xmath5 ) & @xmath186 & @xmath187 + @xmath188 $ ] & @xmath144 $ ] & @xmath144 $ ] + + @xmath56 & @xmath189 & @xmath190 + @xmath3 & @xmath191 & @xmath192 + @xmath89 & @xmath193 & @xmath194 + [ 5pt ] fixed to @xmath195 ; the dotted line is the unconstrained fit . in the longitudinal part the fits are almost indistinguishable . the smaller error bars correspond to statistical , the larger ones to the sum of statistical and systematical errors.,title="fig:",height=283 ] fixed to @xmath195 ; the dotted line is the unconstrained fit . in the longitudinal part the fits are almost indistinguishable . the smaller error bars correspond to statistical , the larger ones to the sum of statistical and systematical errors.,title="fig:",height=283 ] since values of @xmath196 and @xmath197 in this experiment were too high for a direct application of @xmath35 , an effective lagrangian model @xcite was used to analyse the measured @xmath5-dependence of the cross section , and to extract the nucleon axial and pion charge form factors . in the energy region of our experiment , the pseudovector @xmath198 coupling evaluated at tree - level provided an adequate description of the reaction cross section . we included the @xmath199- and @xmath200-channel nucleon pole terms containing electric and magnetic sachs nucleon form factors of the well - known dipole form with a ` cut - off ' @xmath201 , the @xmath202-channel pion pole term with a monopole form factor @xmath203 , the contact ( seagull ) term with the axial dipole form factor @xmath204 , and the @xmath199-channel @xmath205-resonance term . vector meson exchange contributions in the @xmath202-channel were found to play a negligible role in the charged pion channel . due to cancellations between higher partial waves and interference terms with the @xmath199-wave , @xmath184 is predominantly sensitive to the @xmath206 amplitude and therefore to @xmath17 . on the other hand , the pion charge form factor appears in the longitudinal amplitude @xmath207 only at order @xmath208 , and the @xmath199-wave contribution to @xmath185 amounts to @xmath209 only . due to the contributions of the higher partial waves , however , the longitudinal cross section @xmath185 is quite sensitive to @xmath210 , which is a bonus ` by - product ' of the analysis . since current conservation is violated if arbitrary form factors are included , gauge invariance was imposed by additional gauge terms in the hadronic current . these terms modify the longitudinal part of the cross section and therefore influence the pion pole term and thus the extracted value of @xmath210 . however , this procedure does not affect the transverse part of the cross section . the axial mass can then be determined from the @xmath5-dependence of @xmath184 , and the result can be compared to the prediction of @xmath35 . in the theoretical fit , the transverse part is fitted first . we have used two different techniques to determine the axial mass : \(i ) only the axial mass was varied , whereas the value of @xmath184 at @xmath211 was fixed by extrapolating the transverse cross section ( i. e. the @xmath206 amplitude ) to the value of the photoproduction angular distribution at @xmath212 . a precise cross section at the photon point ( @xmath211 ) is very helpful for our analysis and can reduce the uncertainty in determining the axial mass considerably . unfortunately , there are no experimental data available around @xmath1 and forward angles . the only measurements in this energy region are performed at pion angles larger than @xmath213 with large deviations among the different data sets . therefore we have used a value at the photon point obtained from different partial - wave analyses of the vpi group @xcite and of the mainz dispersion analysis @xcite . this yields to a weighted - average cross section at the photon point of @xmath195 and this value was used as an additional data point . the corresponding value of @xmath206 is also well supported by the studies of the gdh sum rule @xcite and by the low energy theorem ( kroll - ruderman limit ) . \(ii ) the three data points alone were fitted , while the value of the transverse cross section at @xmath211 was taken as an additional parameter , with the result @xmath214 . the best - fit parameters for the transverse cross section were then used to fit the longitudinal part . using the first and preferred procedure , we find from the transverse cross section @xmath6 , corresponding to @xmath215 . from the longitudinal part , we obtain @xmath216 , corresponding to @xmath217 . the second procedure leads to the following results : @xmath218 or @xmath219 , and @xmath220 or @xmath221 . we have measured the electroproduction of positive pions on protons at the invariant mass of @xmath1 , and at four - momentum transfers of @xmath222 and @xmath4 . in conjunction with our previous measurement at @xmath223 @xcite , we were then able to study the @xmath5-dependence of the transverse and longitudinal cross sections , separated by the rosenbluth technique for each @xmath5 . the statistical uncertainties were between @xmath224 and @xmath225 , an improvement of an order of magnitude over the result of ref . the systematical uncertainties were estimated to be between @xmath226 and @xmath227 , and are expected to decrease significantly in the future experiments . we have extracted the axial mass parameter @xmath17 of the nucleon axial form factor from our pion electroproduction data using an effective lagrangian model with pseudovector @xmath198 coupling . our extracted value of @xmath6 is @xmath7 larger than the axial mass @xmath228 known from neutrino scattering experiments . our result essentially confirms with the scaled - error weighted average @xmath229 of older pion electroproduction experiments . if we include our value into the database , the weighted average increases to @xmath24 , and the ` axial mass discrepancy ' becomes @xmath230 . this value of @xmath42 is in agreement with the prediction derived from @xmath35 , @xmath231 . we conclude that the puzzle of seemingly different axial radii as extracted from pion electroproduction and neutrino scattering can be resolved by pion loop corrections to the former process , and that the size of the predicted corrections is confirmed by our experiment . theoretical input needed to extract the axial mass and the pion radius has naturally led to some model dependence of the results . the dominant contribution to pion electroproduction at @xmath1 is due to the born terms . these are based on very fundamental grounds and the couplings are very well known . for our purpose the electric and magnetic form factors of the nucleons are also accurately known whereas the remaining two ( the axial and the pion form factors ) are the subjects of our analysis . in parallel kinematics we are in the ideal situation where the pion form factor contributes only to the longitudinal cross section , therefore reducing very strongly the model dependence of the transverse cross section and consequently of the determination of the axial mass . furthermore , the @xmath205 resonance , which plays the second important role in our theoretical description , also contributes with a well - known m1 excitation to the transverse cross section , while the longitudinal c2 excitation gives rise to a larger uncertainty due to the less known c2 form factor , currently under investigation at different laboratories . altogether , the model dependence is smaller for extracting the axial mass than for the pion radius . this could partly explain the discrepancy in the different values of the pion radius between our analysis and the analysis of pion scattering off atomic electrons @xcite . however , as in the case of the axial mass , an additional correction of the pion radius obtained from electroproduction experiments is very likely . this should be investigated in future studies of chiral perturbation theory . 99 g. fanourakis et al . , phys . d * 21 * ( 1980 ) 562 ; + l. a. ahrens et al . d * 35 * ( 1987 ) 785 ; + l. a. ahrens et al . b * 202 * ( 1988 ) 284 . s. j. barish et al . , phys . d * 16 * ( 1977 ) 3103 ; + k. l. miller et al . d * 26 * ( 1982 ) 537 ; + w. a. mann et al . . lett . * 31 * ( 1973 ) 844 ; + n. j. baker et al . d * 23 * ( 1981 ) 2499 ; + t. kitagaki et al . d * 28 * ( 1983 ) 436 ; + t. kitagaki et al . d * 42 * ( 1990 ) 1331 . m. holder et al . , nuovo cim . a * lvii * ( 1968 ) 338 . r. l. kustom et al . , phys . rev . lett . * 22 * ( 1969 ) 1014 . d. perkins , in : _ proceedings of the _ @xmath232 _ international conference on high energy physics _ , j. d. jackson , a. roberts ( eds . ) , national accelerator laboratory , batavia , illinois , 1973 , vol . a. orkin - 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the reaction @xmath0 was measured at the mainz microtron mami at an invariant mass of @xmath1 and four - momentum transfers of @xmath2 , @xmath3 and @xmath4 . for each value of @xmath5 , a rosenbluth separation of the transverse and longitudinal cross sections was performed . an effective lagrangian model was used to extract the ` axial mass ' from experimental data . we find a value of @xmath6 which is @xmath7 larger than the axial mass known from neutrino scattering experiments . this is consistent with recent calculations in chiral perturbation theory . , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , and _ pacs : _ 13.60.le , 25.30.rw , 14.20.dh _ keywords : _ nucleon axial form factor , coincident pion electroproduction

nucl-ex9911003

both cosmic star formation rate and active galactic nucleus ( agn ) density have been found to reach their peaks at @xmath32 @xcite . in the local universe , a supermassive black hole ( smbh ) generically exists in the center of early - type galaxies with the black hole mass tightly correlating with that of the galaxy s stellar bulge @xcite . the connection and co - evolution between the central smbh and host galaxy have therefore been suggested @xcite . in one of the most popular co - evolution scenarios , galaxy mergers have been proposed to funnel gas into the center of galaxies , leading to a central starburst and rapid growth of a smbh @xcite . one promising approach to investigate the merger - driven co - evolution scenario is to study the merger features in agn host galaxies . however , previous studies have produced mixed results . on one side , the most moderate - luminosity x - ray selected agn hosts ( @xmath4erg s@xmath5 ) have similar disk - dominated morphologies as those of non - active galaxies , showing no significant difference in the distortion fraction , both at @xmath6 @xcite and at @xmath7 @xcite . on the other side , the high merger fraction ( @xmath8 ) has been found in a subsample of the bright ( @xmath9erg s@xmath5 ) , dust - reddened quasars @xcite . this may lead to an explanation that merger fraction is dependent on agn bolometric luminosity @xcite . there are also theoretical studies suggesting that galaxy mergers only trigger luminous agn activity while other internal mechanisms may be responsible in less luminous agns @xcite . therefore , it is crucial to examine the connection between the most luminous agns and merger fractions . however , host morphological studies of the most luminous agns ( @xmath9erg s@xmath5 ) at @xmath7 are rare in the literature . for the luminous blue agns , such studies have been challenged by the bright point source , even with the careful treatment of point source substraction @xcite . the present sampling in deep and narrow surveys has biased against the luminous x - ray selected agns . nasa s _ wide - field infrared survey explorer _ ( _ wise _ ; wright et al . 2010 ) all - sky survey provides an opportunity to search the most luminous galaxies at mid - infrared wavelengths . @xcite and @xcite discovered a new population of hyperluminous , hot dust - obscured galaxies ( thereafter hot dogs ) using a so - called `` w1w2 dropout '' selection criteria . follow - up studies have revealed several key aspects of this population : ( 1 ) spectroscopic follow - up studies show that they are mostly high - redshift objects , with redshift range from 1 to 4 @xcite . ( 2 ) most of hot dogs are extremely luminous with @xmath10 @xcite . ( 3 ) using x - ray observations @xcite and spectral energy distribution ( sed ) analysis ( assef et al . 2015a ; fan et al . 2016a ) , clear evidence has been shown that their luminous mid - ir emission is powered by a highly dust - obscured , possibly compton - thick agn . thanks to the heavy obscuration of circumnuclear dust , the host morphology of this population is easily observable . thus hot dogs are ideal objects for us to investigate the merger fraction in the most luminous agns . in this letter , we examine the host morphology of 18 hot dogs , which have _ hubble space telescope ( hst ) _ wide - field camera 3 ( wfc3 ) near - ir high - resolution imaging . our target is to provide some knowledge about the merger fraction in the most luminous agns . throughout this work we assume a flat @xmath11cdm cosmology with @xmath12 km s@xmath5 , @xmath13 , and @xmath14 . the hot dogs studied here are selected from the _ wise _ all - sky source catalog @xcite . in order to investigate the host morphology of hot dogs , we select a subsample of 18 objects ( table 1 ) with available _ hst _ wfc3 imaging . we also require that they have known spectroscopic redshift in the literature @xcite for calculating their bolometric luminosities . to investigate the host morphology of hot dogs , we use the high resolution _ hst _ wfc3 near - ir imaging . we search our targets and retrieve the calibrated images from mast . observations are from four different hst proposals with i d 12488 ( pi : m. negrello ) , 12585 ( pi : s. petty ) , 12481 and 12930 ( pi : c. bridge ) . we list hst proposal i d for each object in table 1 . all but one ( w0831 + 0140 ) use the _ f160w _ ( @xmath15-band ) imaging . only w0831 + 0140 use the _ f110w _ imaging . the wfc3 imaging has a pixel scale of 0.13 arcsec / pixel . .the sample of hot dogs[tbl - sample ] [ cols="<,^,^,^,^ " , ] we find a high merger fraction ( @xmath1 ) in the hot dog sample , using the visual classification described in section 3 . given the high agn bolometric luminosity ( @xmath16 ) in the hot dog sample , this finding provides a clear evidence that the most luminous agns have likely been related to galaxy mergers . in order to investigate the dependence of merger fraction on agn bolometric luminosity , we compile the data in the literature and plot them in figure 2 . gray diamonds are taken from the compilation of @xcite . red and purple symbols , which represent the samples at intermediate redshift @xmath6 and at @xmath7 , respectively , are taken from the recent works @xcite . our result for the hot dog sample has been shown as red asterisk . among all available data , our sample is the second brightest , which has the bolometric luminosity only lower than that of the dust - reddened qso sample in @xcite . an obvious trend can be found in figure 2 that the merger fraction increases with agn bolometric luminosity at high luminosity , while the merger fraction shows a weak dependence on agn bolometric luminosity for the less luminous agns . we compare the observed trend with the variability - driven model of @xcite ( red and blue lines in figure 2 ) . red and blue lines show the predictions of @xcite model at @xmath17 and @xmath18 , respectively . the merger fraction in our sample agrees well with the model prediction at @xmath7 . in other samples at @xmath7 of mid - ir luminous qsos ( red square , del moro et al . 2016 ) , compton - thick agns ( red triangle , lanzuisi et al . 2015 ) and dust - reddened qsos ( red point , glikman et al . 2015 ) , the merger fractions also show a good agreement with the model prediction . as suggested by @xcite , galaxy merger , especially major merger , may play the most significant role in triggering the most luminous agns at @xmath7 . in figure 3 , we show the distribution of srsic indices in our hot dog sample . the distribution peaks at @xmath19 , ranging from disk - dominated ( @xmath20 ) , disk with a prominent bulge component ( 1.5@[email protected] ) to bulge - dominated ( @xmath22 ) . the fractions of disk - dominated , intermediate and bulge - dominated morphologies are 22.2% , 61.1% and 16.7% , respectively , in our hot dog sample . we remind the reader that a single srsic model wo nt provide a good fitting for several heavily distorted hot dogs . for instance , both w0851 + 3148 and w2207 + 1939 have significantly morphological distortions ( see figure 1 ) and the results of a single srsic model fitting have a large value of the reduced @xmath23 ( > 5 ) . however , for most of our hot dogs , the single srsic model still provides a reasonable fitting . conservatively speaking , the srsic profile with @xmath24 can describe the distribution of most stars in hot dogs . our result suggests that most of our hot dogs have an intermediate morphology ( disk with a prominent bulge component ) . the intermediate morphology of hot dogs has been expected in the merger - driven scenario if hot dogs lie at the transient stage when major mergers with gas - rich disks are building galaxy bulge . automatically morphological classification provides an independent check on the reliability of our visual classification . the @xmath25 and @xmath26 values of our hot dogs range from 0.47 to 0.79 with a median value 0.70 , and from @xmath27 to @xmath28 with a median value of @xmath29 , respectively . we employ the @xmath30 parameters to locate major mergers using the merger criteria of lotz et al . 13 out of 18 hot dogs fulfill the relation @xmath31 , indicating them as mergers . the high merger fraction ( @xmath3272% ) derived here is well consistent with the result of our visual classification . in this letter , we use the high resolution _ hst _ wfc3 near - ir imaging to derive the merger fraction in a subsample of 18 hot dogs by visual classification . we find that the merger fraction is high ( @xmath1 ) for our hot dog sample . the high merger fraction in these most luminous , heavily - obscured agns are consistent with the predictions of hydrodynamical simulations @xcite and the variability - based model @xcite . we also derive the agn bolometric luminosity by using ir seds decomposition . the heavily obscured agns in hot dogs are among the most luminous agns in the universe ( @xmath2 ) . the cold dust emissions in hot dogs are also significant ( see table 2 ) , indicating that there resides intense starburst in the hosts . both intense agn and starburst activities are likely triggered by mergers as suggested by the high merger fraction . however , the lack of a comparison sample may weaken the merger - driven explanation of hot dogs . unfortunately , it is hard to select a suitable comparison sample , as both the stellar masses and black hole masses of hot dogs are not well known . the stellar mass upper bounds of hot dogs derived from sed modeling by @xcite are about @xmath33 . massive and inactive galaxies at @xmath34 show a relatively low fraction of strong distortion morphologies . for instance , only @xmath35 galaxies in an inactive galaxy sample with the median stellar mass @xmath36 show major merger features @xcite . this indirect evidence suggests that the high merger fraction in our hot dog sample is likely responsible for triggering the intense agn activities . according to fitting the surface brightness profiles , we find that the distribution of srsic indices of hot dogs peaks around 2 , which suggests that most of hot dogs have intermediate morphologies . from the point of view of morphology , this work confirms our previous result ( fan et al . 2016a ) , which suggests that hot dogs may represent a transit phase during the evolution of massive galaxies , transforming from the dusty starburst dominated phase to the optically bright qso phase . we can find that all samples with high merger fraction are dust - reddened @xcite and/or ir - luminous ( this work ) in figure 2 . while for the blue , luminous qsos , no high merger fraction or no enhancing merging feature has been found . for instance , @xcite recently studied an uv / optical - selected @xmath37 qso sample with high super - massive black hole masses ( @xmath38 ) , which are comparable to the estimated smbh masses of hot dogs @xcite . they found that the strong distortion fraction for qso hosts is low and comparable to that of inactive galaxies . they therefore concluded that major mergers are not the primary triggering mechanism for agn activity . the results between us look contradictory . however , the problem can be solved if considering that uv / optical - selected qsos and ir - selected obscured qsos represent distinct phases in the merger - driven evolutionary sequence @xcite . in the merger - driven evolutionary model , gas - rich galaxy mergers fuel an intense starburst and a phase of obscured bh growth , followed by an unobscured phase after the gas is consumed or expelled from the galaxy by qso feedback . dusty starburst like smgs and bump dogs may represent the early stage of the mergers . they are expected to have a high merger fraction , which has already been observed ( e.g. , bussmann et al . ir - luminous agns are in an intermediate stage of the mergers . thus the strong merger features still present in their host galaxies . as a result , a high merger fraction will be derived according to morphological classification . while for unobscured qsos whose hosts are found to be largely elliptical galaxies , they are in a late stage of mergers , just before the host galaxies settling into passive elliptical galaxies . we suggest that both high luminosity and obscuration should be essential for observing the high merger fraction in agn hosts . we emphasize that this conclusion is only valid for radio - quiet agns while it may be another story for radio - loud agns @xcite . a similar evolutionary sequence has also been proposed by @xcite involving x - ray selected agn , infrared selected obscured agn , dust - reddened quasars and unobscured agn ( see their figure 10 ) . we thank the referee for the careful reading and the valuable comments that helped improving our paper . this work is supported by the national natural science foundation of china ( nsfc , nos . 11203023 , 11303084 , 11303002 and 11433005 ) , the fundamental research funds for the central universities ( wk3440000001 ) . lf acknowledges the support by qilu young researcher project of shandong university . yh thanks the support from the western light youth project . gf acknowledges the support by the yunnan applied basic research projects ( 2014fb155 ) . abraham , r. g. , tanvir , n. r. , santiago , b. x. , et al . 1996 , , 279 , l47 alexander , d. m. , & hickox , r. c. 2012 , newar , 56 , 93 assef , r. j. , eisenhardt , p. r. m. , stern , d. , et al . 2015 , , 804 , 27 assef , r. j. , walton , d. j. , brightman , m. , et al . 2015 , arxiv:1511.05155 brandt , w. n. , & alexander , d. m. 2015 , , 23 , 1 bussmann , r. s. , dey , a. , lotz , j. , et al . 2011 , , 733 , 21 chiaberge , m. , gilli , r. , lotz , j. m. , & norman , c. 2015 , , 806 , 147 cisternas , m. , jahnke , k. , inskip , k. j. , et al . 2011 , , 726 , 57 cutri , r. m. , & et al . 2013 , vizier online data catalog , 2328 , 0 del moro , a. , alexander , d. m. , bauer , f. e. , et al . 2016 , , 456 , 2105 draper , a. r. , & ballantyne , d. r. 2012 , , 751 , 72 eisenhardt , p. r. m. , wu , j. , tsai , c .- w . , et al . 2012 , , 755 , 173 fan , l. , fang , g. , chen , y. , et al . 2014 , , 784 , l9 fan , l. , han , y. , nikkuta , r. , et al . 2016 , arxiv:1604.01467 ferrarese , l. , & ford , h. 2005 , , 116 , 523 glikman , e. , simmons , b. , mailly , m. , et al . 2015 , , 806 , 218 griffin , m. j. , abergel , a. , abreu , a. , et al . 2010 , , 518 , l3 , y. , & han , z. 2012 , , 749 , 123 , y. , & han , z. 2014 , , 215 , 2 hickox , r. c. , mullaney , j. r. , alexander , d. m. , et al . 2014 , , 782 , 9 hong , j. , i m , m. , kim , m. , & ho , l. c. 2015 , , 804 , 34 hopkins , p. f. , hernquist , l. , cox , t. j. , & kere , d. 2008 , , 175 , 356 hopkins , p. f. , & hernquist , l. 2009 , , 694 , 599 jones , s. f. , blain , a. w. , stern , d. , et al . 2014 , , 443 , 146 kartaltepe , j. s. , mozena , m. , kocevski , d. , et al . 2015 , , 221 , 11 kocevski , d. d. , faber , s. m. , mozena , m. , et al . 2012 , , 744 , 148 kocevski , d. d. , brightman , m. , nandra , k. , et al . 2015 , , 814 , 104 kormendy , j. , & ho , l. c. 2013 , , 51 , 511 lanzuisi , g. , perna , m. , delvecchio , i. , et al . 2015 , , 578 , a120 lotz , j. m. , davis , m. , faber , s. m. , et al . 2008 , , 672 , 177 - 197 magorrian , j. , tremaine , s. , richstone , d. , et al . 1998 , , 115 , 2285 mechtley , m. , jahnke , k. , windhorst , r. a. , et al . 2015 , arxiv:1510.08461 nenkova , m. , sirocky , m. m. , ivezi , . , & elitzur , m. 2008 , , 685 , 147 peng , c. y. , ho , l. c. , impey , c. d. , & rix , h .- w . 2002 , , 124 , 266 piconcelli , e. , vignali , c. , bianchi , s. , et al . 2015 , , 574 , l9 poglitsch , a. , waelkens , c. , geis , n. , et al . 2010 , , 518 , l2 reddy , n. a. , steidel , c. c. , pettini , m. , et al . 2008 , , 175 , 48 rosario , d. j. , mcintosh , d. h. , van der wel , a. , et al . 2015 , , 573 , a85 sanders , d. b. , soifer , b. t. , elias , j. h. , et al . 1988 , , 325 , 74 schawinski , k. , treister , e. , urry , c. m. , et al . 2011 , , 727 , l31 schawinski , k. , simmons , b. d. , urry , c. m. , et al . 2012 , , 425 , l61 stern , d. , lansbury , g. b. , assef , r. j. , et al . 2014 , , 794 , 102 tsai , c .- w . , eisenhardt , p. r. m. , wu , j. , et al . 2015 , , 805 , 90 treister , e. , schawinski , k. , urry , c. m. , et al . 2012 , , 758 , l39 urrutia , t. , lacy , m. , & becker , r. h. 2008 , , 674 , 80 villforth , c. , hamann , f. , rosario , d. j. , et al . 2014 , , 439 , 3342 wright , e. l. , eisenhardt , p. r. m. , mainzer , a. k. , et al . 2010 , , 140 , 1868 wu , j. , tsai , c .- w . , sayers , j. , et al . 2012 , , 756 , 96 wu , j. , bussmann , r. s. , tsai , c .- w . , et al . 2014 , , 793 , 8 wylezalek , d. , zakamska , n. l. , liu , g. , & obied , g. 2016 , , 457 , 745

previous studies have shown that _ wise_-selected hyperluminous , hot dust - obscured galaxies ( hot dogs ) are powered by highly dust - obscured , possibly compton - thick agns . high obscuration provides us a good chance to study the host morphology of the most luminous agns directly . we analyze the host morphology of 18 hot dogs at @xmath0 using hubble space telescope / wfc3 imaging . we find that hot dogs have a high merger fraction ( @xmath1 ) . by fitting the surface brightness profiles , we find that the distribution of srsic indices in our hot dog sample peaks around 2 , which suggests that most of hot dogs have transforming morphologies . we also derive the agn bolometric luminosity ( @xmath2 ) of our hot dog sample by using ir seds decomposition . the derived merger fraction and agn bolometric luminosity relation is well consistent with the variability - based model prediction @xcite . both the high merger fraction in ir - luminous agn sample and relatively low merger fraction in uv / optical - selected , unobscured agn sample can be expected in the merger - driven evolutionary model . finally , we conclude that hot dogs are merger - driven and may represent a transit phase during the evolution of massive galaxies , transforming from the dusty starburst dominated phase to the unobscured qso phase .

1605.00661

the photo - nuclear cross sections parameterized in the geant4 covers all incident photon energies from hadron production threshold up - wards . the parameterization is subdivided into five energy regions , each corresponding to the physical process that dominates . these are the giant dipole resonance ( gdr ) region , the `` quasi - deuteron '' region , the @xmath3 isobar region characterized by the dominant peak in the cross section which extends from the pion threshold to 450 mev , the roper resonance region that extends from roughly 450 mev to 1.2 gev , and the reggeon - pomeron region above 1.2 gev . from the geant4@xcite photo - nuclear data base currently 14 nuclei are used in defining the parameterization : @xmath4h , @xmath5h , @xmath6he , @xmath7li , @xmath8li , @xmath9be , @xmath10c , @xmath11o , @xmath12al , @xmath13ca , cu , sn , pb , and u. the result is a function of @xmath0 and @xmath1 , where @xmath2 is the incident photon energy . the cross - section is the sum of the components which parameterize each energy region . + the cross section in the gdr region can be described as the sum of two peaks , @xmath14 the exponential describes the falling edge of the resonance which has power law behavior . the function @xmath15 describes the rising edge of the resonance . it is the nuclear - barrier - reflection function and behaves like a threshold , cutting off the exponential . the exponential powers @xmath16 and @xmath17 are @xmath18 the @xmath0-dependent parameters @xmath19 , @xmath20 and @xmath21 were found for each of the 14 nuclei and are interpolated for other nuclei . + the @xmath3 isobar region was parameterized as @xmath22 where @xmath23 is an overall normalization factor . @xmath24 can be interpreted as the energy of the @xmath3 isobar and @xmath25 as the inverse @xmath3 width . @xmath26 is the threshold function . the @xmath0-dependence of these parameters is as follows : * @xmath27 ( for @xmath4h it is 0.55 , for @xmath5h it is 0.88 ) , i.e. the @xmath3 yield is proportional to @xmath0 ; * @xmath28 . @xmath29 shows how the pion threshold depends on @xmath0 . * @xmath30 for @xmath31 and 0.04 for @xmath32 ; * @xmath33 , which means that the `` mass '' of the @xmath3 isobar moves to lower energies ; * @xmath34 . @xmath25 is 18.0 for @xmath4h . the @xmath0-dependence of the @xmath35 , @xmath24 and @xmath25 parameters is due to the @xmath36 reaction , which can take place in the nuclear medium below the pion threshold . + the quasi - deuteron contribution was parameterized with the same form as the @xmath3 contribution but without the threshold function : @xmath37 for @xmath4h and @xmath5h the quasi - deuteron contribution is almost zero . for these nuclei the third baryonic resonance was used instead , so the parameters for these two nuclei are quite different , but trivial . the parameter values are given below . * @xmath38 , where @xmath39 . the @xmath0-dependence in the quasi - deuteron region is stronger than @xmath40 it contributes only little for light nuclei . for @xmath4h it is 0.078 and for @xmath5h it is 0.08 . * @xmath41 and @xmath42 . experimental information is insufficient to fix their @xmath0-dependence . for both @xmath4h and @xmath5h we have @xmath43 and @xmath44 . the roper contribution was parameterized using the same form as the quasi - deuteron contribution : @xmath45 using @xmath39 , the values of the parameters are * @xmath46 . for @xmath4h it is 0.22 and for @xmath5h it is 0.34 . * @xmath47 ( for @xmath4h and for @xmath5h it is 6.57 ) , so the roper mass increases with @xmath0 . * @xmath48 . for @xmath4h it is 20.0 and for @xmath5h it is 15.0 . the regge - pomeron contribution was parametrized in terms of two exponentials describing the pomeron and higher reggeon contributions respectively : @xmath49 with @xmath50 . for simulating final states for gamma - nuclear reactions , we are using the chiral invariant phase - space ( chips ) approach@xcite@xcite@xcite . the chips model uses a set of simple rules which govern microscopic quark - level behavior to model macroscopic hadronic systems . the invariant phase space distribution as a paradigm of thermalized chaos is applied to quarks , and simple kinematic mechanisms are used to model the hadronization of quarks into hadrons . along with relativistic kinematics and the conservation of quantum numbers , the following concepts are introduced : the quasmon is any excited hadronic system , and can be viewed as a generalized hadron . at the constituent level , a quasmon may be thought of as a bubble of quark - parton plasma in which the quarks are massless . the quark - partons in the quasmon are massless and homogeneously distributed over the invariant phase space . it may also be considered as a bubble of the three - dimensional feynman - wilson @xcite parton gas . the traditional hadron is a particle defined by quantum numbers and a fixed mass or width . the quark content of the hadron is a secondary concept constrained by the quantum numbers . the quasmon , however , is defined by its quark content and mass , and the concept of a well defined particle with quantum numbers is of secondary importance . the quark fusion hypothesis determines the rules of final state hadron production , with energy spectra reflecting the momentum distribution of the quarks in the system . fusion occurs when a quark - partons in a quasmon join to form a hadron . in cases of multiple quasmon , quark - partons may be exchanged between the two quasmons . resulting hadrons are constrained to be produced on mass shell . the type of the outgoing hadron is selected using combinatoric and kinematic factors consistent with conservation laws . the only non - kinematic concept is the hypothesis of a critical temperature of the quasmon . this has a 35-year history , starting with ref . @xcite and is based on the experimental observation of regularities in the inclusive spectra of hadrons produced in different reactions at high energies . qualitatively , the critical temperature hypothesis assumes that the quasmon can not be heated above a certain temperature . adding more energy to the system increases only the number of constituent quark - partons while the temperature remains constant . the critical temperature @xmath51 mev is the principal parameter of the model . the choice of this parameter is motivated from the results shown in fig.[apcmul ] . for the sake of briefness , we will only include the solution of the vacuum problem in this paper , and refer for the solution of the in - medium equations to the chips publications@xcite,@xcite . to generate hadron spectra from free quasmons , the number of partons in the system must be found . for a finite system of @xmath52 partons with a total invariant mass @xmath53 , the invariant phase space integral , @xmath54 , is proportional to @xmath55 . at a temperature @xmath56 the statistical density of states is proportional to @xmath57 and the probability to find a system of @xmath52 quark - partons in a state with mass @xmath53 is @xmath58 . for this kind of probability distribution the mean value of @xmath59 is @xmath60 for large @xmath52 we obtain for massless particles the well - known @xmath61 result . after a nucleon absorbs an incident quark - parton , such as a real or virtual photon , for example , the newly formed quasmon has a total of @xmath52 quark - partons , where @xmath52 is determined by eq . [ temperature ] . choosing one of these quark - partons with energy @xmath62 in the center of mass system ( cms ) of @xmath52 partons , the spectrum of the remaining @xmath63 quark - partons is given by @xmath64 where @xmath65 is the effective mass of the @xmath63 quark - partons . the effective mass is a function of the total mass @xmath53 , @xmath66 so that the resulting equation for the quark - parton spectrum is : @xmath67 in order to decompose a quasmon into a hadron and a residual quasmon , one needs the probability of two quark - partons to produce the effective mass of the hadron . we calculate the spectrum of the second quark - parton by following the same argument used to determine eq . [ spectrum_1 ] . one quark - parton is chosen from the residual @xmath63 . it has an energy @xmath24 in the cms of the @xmath63 quark - partons . the spectrum is obtained by substituting @xmath63 for @xmath52 and @xmath65 for @xmath53 in eq . [ spectrum_1 ] and then using eq . [ m_n-1 ] to get @xmath68 to ensure that the fusion will result in a hadron of mass @xmath69 , we apply the mass shell constraint for the outgoing hadron , @xmath70 here @xmath71 is the angle between the momenta * k * and * q * of the two quark - partons in the cms of @xmath63 quarks . the kinematic quark fusion probability for any primary quark - parton with energy @xmath62 is then : @xmath72 using the @xmath73-function to perform the integration over @xmath24 one gets : @xmath74 or @xmath75 after the substitution @xmath76 , this becomes @xmath77 where the limits of integration are @xmath78 when @xmath79 , and @xmath80 when @xmath81 . the resulting range of @xmath71 is therefore @xmath82 . integrating from @xmath78 to @xmath83 yields @xmath84 and integrating from @xmath78 to @xmath85 yields the total kinematic probability for hadronization of a quark - parton with energy @xmath62 into a hadron with mass @xmath69 : @xmath86 the ratio of expressions [ z_probab ] and [ tot_kin_probab ] can be treated as a random number , @xmath87 , uniformly distributed on the interval [ 0,1 ] . solving for @xmath83 then gives @xmath88{r}\cdot z_{\max } . \label{z_random}\ ] ] in addition to the kinematic selection of the two quark - partons in the fusion process , the quark content of the quasmon and the spin of the candidate final hadron are used to determine the probability that a given type of hadron is produced . because only the relative hadron formation probabilities are necessary , overall normalization factors can be dropped . hence the relative probability can be written as @xmath89 here , only the factor @xmath90 is used since the other factors in equation [ tot_kin_probab ] are constant for all candidates for the outgoing hadron . the factor @xmath91 counts the spin states of a candidate hadron of spin @xmath92 , and @xmath93 is the number of ways the candidate hadron can be formed from combinations of the quarks within the quasmon . in making these combinations , the standard quark wave functions for pions and kaons were used . for @xmath94 and @xmath95 mesons the quark wave functions @xmath96 and @xmath97 were used . no mixing was assumed for the @xmath98 and @xmath99 meson states , hence @xmath100 and @xmath101 . at high energies we use quark - gluon string model and a diffractive ansatz for string excitation to describe the interactions of real and virtual photons with nuclei . a description of the means of doing this can be found in a separate paper in the present proceedings@xcite . electro - nuclear reactions are very connected with photo - nuclear reactions . they are sometimes called `` photo - nuclear '' because the one - photon exchange mechanism dominates the reaction . in this sense electrons can be replaced by a flux of equivalent photons . this is not completely true , because at high energies diffractive mechanisms are possible , but these types of reactions are beyond the scope of this discussion . the equivalent photon approximation ( epa ) was proposed by e. fermi @xcite and developed by c. weizsacker and e. williams @xcite and by l. landau and e. lifshitz @xcite . the covariant form of the epa method was developed in refs . @xcite and @xcite . when using this method it is necessary to take into account that real photons are always transversely polarized while virtual photons may be longitudinally polarized . in general the differential cross section of the electro - nuclear interaction can be written as @xmath102 where @xmath103 the differential cross section of the electro - nuclear scattering can be rewritten as @xmath104 where @xmath105 for small @xmath106 and is written as a function of @xmath107 , @xmath108 , and @xmath106 for large @xmath106 . interactions of longitudinal photons are normally included in the effective @xmath109 cross section through the @xmath107 factor , but in the present method , the cross section of virtual photons is considered to be @xmath107-independent . the electro - nuclear problem , with respect to the interaction of virtual photons with nuclei , can thus be split in two . at small @xmath106 it is possible to use the @xmath110 cross section . in the @xmath111 region it is necessary to calculate the effective @xmath112 cross section . + following the epa notation , the differential cross section of electro - nuclear scattering can be related to the number of equivalent photons @xmath113 . for @xmath114 and @xmath115 the canonical method @xcite leads to @xmath116 in @xcite , integration over @xmath106 for @xmath117 leads to @xmath118 in the @xmath114 limit this formula converges to eq.([neq ] ) . but the correspondence with eq.([neq ] ) can be made more explicit if the exact integral @xmath119 where @xmath120 , @xmath121 , @xmath122 , @xmath123 , is calculated for @xmath124 . the factor @xmath125 is used arbitrarily to keep @xmath126 , which can be considered as a boundary between the low and high @xmath106 regions . the transverse photon flux can be calculated as an integral of eq.([diff ] ) with the maximum possible upper limit @xmath127 it can be approximated by @xmath128 where @xmath129 . it must be pointed out that neither this approximation nor eq.([diff ] ) works at @xmath130 . the formal limit of the method is @xmath131 . + to the `` photon flux '' for ( a ) @xmath132 electrons and ( b ) @xmath133 electrons . in figures ( c ) and ( d ) the photon distribution @xmath134 is multiplied by the photo - nuclear cross section and integrated over @xmath106 . the dashed lines are integrals over low @xmath106 ( under the dashed line in the first two figures ) , the solid lines are integrals over high-@xmath106 ( above the dashed lines in the first two figures).,width=321 ] fig . [ nsigma](a , b ) shows the energy distribution for the equivalent photons . the low-@xmath106 flux is calculated using eq.([neq ] ) ( dashed lines ) and eq.([diff ] ) ( dotted lines ) . the total flux is calculated using eq.([neqhq ] ) ( the solid lines ) and using eq.([diff ] ) with the upper limit defined by eq.([q2max ] ) ( dash - dotted lines visible only around @xmath135 ) . we find that to calculate the number of low-@xmath106 equivalent photons or the total number of equivalent photons one can use the approximations given by eq.([neq ] ) and eq.([neqhq ] ) , respectively , instead of using eq.([diff ] ) . comparing the low-@xmath106 photon flux and the total photon flux we find that the low-@xmath106 photon flux is about half of the the total . from the interaction point of view the decrease of @xmath136 with increasing @xmath106 must be considered . the cross section reduction for the virtual photons with large @xmath106 is governed by two factors . the cross section drops with @xmath106 as the squared dipole nucleon form - factor @xmath137 and the thresholds the @xmath138 reactions are shifted to higher @xmath108 by a factor @xmath139 , which is the difference between the @xmath140 and @xmath108 values . following the method proposed in @xcite , @xmath141 at large @xmath106 can be approximated as @xmath142 where @xmath143 . the @xmath107-dependence of the @xmath144 and @xmath145 functions is weak , so for simplicity the @xmath146 and @xmath147 functions are averaged over @xmath107 . they can be approximated as @xmath148 the integrated photon flux folded with the cross section approximated by eq.([abc ] ) is shown in fig . [ nsigma](c , d ) . we show separately the low-@xmath106 region ( @xmath149 , dashed lines ) , and the high-@xmath106 region ( @xmath150 , solid lines ) . these functions will be integrated over @xmath151 , hence because of the giant dipole resonance contribution , the low-@xmath106 part covers more than half the total @xmath152 cross section . but at @xmath153 , where the hadron multiplicity increases , the large @xmath106 part dominates . in this sense , for a better simulation of the production of hadrons by electrons , it is necessary to simulate the high-@xmath106 part as well as the low-@xmath106 part . + taking into account the contribution of high-@xmath106 photons it is possible to use eq.([neqhq ] ) with the over - estimated @xmath105 cross section . the slightly over - estimated electro - nuclear cross section is @xmath154 where @xmath155 @xmath156 @xmath157 the equivalent photon energy @xmath158 can be obtained for a particular random number @xmath87 from the equation @xmath159 eq.([diff ] ) is too complicated for the randomization of @xmath106 but there is an easily randomized formula which approximates eq.([diff ] ) above the hadronic threshold ( @xmath160 ) . it reads @xmath161 where @xmath162 @xmath163 @xmath164 @xmath165 @xmath166 the @xmath106 value can then be calculated as @xmath167 where @xmath87 is a random number . in fig . [ q2dep ] , eq.([diff ] ) ( solid curve ) is compared to eq.([rq2hh ] ) ( dashed curve ) . because the two curves are almost indistinguishable in the figure , this can be used as an illustration of the @xmath106 spectrum of virtual photons , which is the derivative of these curves . an alternative approach is to use eq.([diff ] ) for the randomization with a three dimensional table @xmath168 . spectra of virtual photons for three energies @xmath169 , @xmath170 , and @xmath132 at @xmath171 , @xmath172 , and @xmath173 . the solid line corresponds to eq.([diff ] ) and the dashed line ( which almost everywhere coincides with the solid line ) corresponds to eq.([diff]).,width=321 ] after the @xmath108 and @xmath106 values have been found , the value of @xmath174 is calculated using eq.([abc ] ) . note that if @xmath175 , no interaction occurs and the electron keeps going . final states are generated using the single photon exchange assumption . sampling the equivalent photon distribution described in the previous section allows to construct an exchange particle that in turn can be treated by the mechanisms used for gamma nuclear scattering . the question to be answered is that of the absorption mechanisms of these particles in the context of parton exchange diagrams . in the example of the photo - nuclear reaction discussed in the comparison section , namely the description of @xmath176 proton and deuteron spectra in @xmath177 reactions at @xmath178 mev , the assumption on the initial quasmon excitation mechanism was the same . the description of the @xmath176 data was satisfactory , but the generated data showed very little angular dependence , as the velocity of quasmons produced in the initial state was small , and the fragmentation process was almost isotropic . experimentally , the angular dependence of secondary protons in photo - nuclear reactions is quite strong even at low energies ( see , for example , ref . this is a challenging experimental fact which is difficult to explain in any model . it s enough to say that if the angular dependence of secondary protons in the @xmath179ca interaction at 60 mev is analyzed in terms of relativistic boost , then the velocity of the source should reach @xmath180 ; hence the mass of the source should be less than pion mass . the main subject of the present publication is to show that the quark - exchange mechanism used in the chips model can not only model the clusterization of nucleons in nuclei and hadronization of intra - nuclear excitations into nuclear fragments , but can also model complicated mechanisms of interaction of photons and hadrons in nuclear matter . quark - exchange diagrams help to keep track of the kinematics of the quark - exchange process to apply the mechanism to the first interaction of a photon with a nucleus , it is necessary to assume that the quark - exchange process takes place in nuclei continuously , even without any external interaction . nucleons with high momenta do not leave the nucleus because of the lack of excess energy . the hypothesis of the chips model is that the quark - exchange forces between nucleons @xcite continuously create clusters in normal nuclei . since a low - energy photon ( below the pion production threshold ) can not be absorbed by a free nucleon , other absorption mechanisms involving more than one nucleon have to be used . the simplest scenario is photon absorption by a quark - parton in the nucleon . at low energies and in vacuum this does not work because there is no corresponding excited baryonic state . but in nuclear matter there is a possibility to exchange this quark with a neighboring nucleon or a nuclear cluster . the diagram for the process is shown in the upper part of fig . [ diagram1 ] . in this case the photon is absorbed by a quark - parton from the parent cluster @xmath181 , and then the secondary nucleon or cluster @xmath182 absorbs the entire momentum of the quark and photon . the exchange quark - parton @xmath24 restores the balance of color , producing the final - state hadron f and the residual quasmon rq . the process looks like a knockout of a quasi - free nucleon or cluster out of the nucleus . it should be emphasized that in this scenario the chips event generator produces not only `` quasi - free '' nucleons but `` quasi - free '' fragments too . the yield of these quasi - free nucleons or fragments is concentrated in the forward direction . the second scenario , shown in the lower part of fig.[diagram1 ] which provides for an angular dependence is the absorption of the photon by a colored fragment ( @xmath183 in fig . [ diagram1 ] ) . in this scenario , both the primary quark - parton with momentum @xmath62 and the photon with momentum @xmath184 are absorbed by a parent cluster ( @xmath182 in the lower part of fig . [ diagram1 ] ) , and the recoil quark - parton with momentum @xmath24 can not fully compensate the momentum @xmath185 . as a result the radiation of the secondary fragment in the forward direction becomes more probable . in both cases the angular dependence is defined by the first act of hadronization . the further fragmentation of the residual quasmon is almost isotropic . . , width=302 ] it was shown in above that the energy spectrum of quark partons in a quasmon can be calculated as @xmath186 where @xmath187 is the energy of the primary quark - parton in the center of mass system ( cms ) of the quasmon , @xmath53 is the mass of the quasmon , and @xmath52 , the number of quark - partons in the quasmon , can be calculated from the equation @xmath188 here @xmath56 is the temperature of the system . in the first scenario of the @xmath138 interaction ( fig . [ diagram1 ] ) , as both interacting particles are massless , we assumed that the cross section for the interaction of the photon with a particular quark - parton is proportional to the charge of the quark - parton squared , and inversely proportional to the mass of the photon - parton system @xmath189 , which can be calculated as @xmath190 here @xmath98 is the energy of the photon , and @xmath62 is the energy of the quark - parton in the laboratory system ( ls ) : @xmath191 in the case of a virtual photon , equation ( [ s ] ) can be written as @xmath192 where @xmath184 is the momentum of the virtual photon . in both cases equation ( [ spectrum_1iii ] ) transforms into @xmath193 and the angular distribution in @xmath194 converges to a @xmath195-function : in the case of a real photon @xmath196 , and in the case of a virtual photon @xmath197 . in the second scenario for the photon interaction ( lower part of fig . [ diagram1 ] ) we assumed that both the photon and the primary quark - parton , randomized according to equation ( [ spectrum_1iii ] ) , enter the parent cluster @xmath182 , and after that the normal procedure of quark exchange continues , in which the recoiling quark - parton @xmath24 returns to the first cluster . an additional parameter in the model is the relative contribution of both mechanisms . as a first approximation we assumed equal probability , but in the future , when more detailed data are obtained , this parameter can be adjusted . we begin the comparison with the data on proton production in the @xmath13ca@xmath198 reaction at @xmath176 at 5965 mev @xcite , and at @xmath199 and @xmath200 at 60 mev @xcite . we analyzed these data together to compare the angular dependence generated by chips with experimental data . the data are presented as a function of the invariant inclusive cross section @xmath201 depending on the variable @xmath202 , where @xmath203 and @xmath204 are the kinetic energy and the momentum of the secondary proton . as one can see from fig . [ gam62iii ] , the angular dependence of the proton yield in photo - production on @xmath13ca at @xmath205 mev is reproduced quite well by the chips event generator . the second set of measurements that we use for the benchmark comparison deals with the secondary proton yields in @xmath10c@xmath198 reactions at 123 and 151 mev @xcite , which is still below the pion production threshold on a free nucleon . inclusive spectra of protons have been measured in @xmath206c reactions at @xmath207 , @xmath208 , @xmath209 , @xmath210 , and @xmath211 . originally , these data were presented as a function of the missing energy . we present the data in figs . [ gam_123 ] and [ gam_151 ] together with chips calculations in the form of the invariant inclusive cross section dependent on @xmath62 . the agreement between the experimental data and the chips model results is quite remarkable . both data and calculations show significant strength in the proton yield cross section up to the kinematic limits of the reaction . the angular distribution in the model is not as prominent as in the experimental data , but agrees well qualitatively . using the same parameters , we applied the chips event generator to the @xmath10c(e , e@xmath212p ) reaction measured in ref.@xcite . the proton spectra were measured in parallel kinematics in the interaction of virtual photons with energy @xmath213 mev and momentum @xmath214 mev/@xmath215 . to account for the experimental conditions in the chips event generator , we have selected protons generated in the forward direction with respect to the direction of the virtual photon , with the relative angle @xmath216 . the chips generated distribution and the experimental data are shown in fig . [ vgam ] in the form of the invariant inclusive cross section as a function of @xmath62 . the chips event generator works only with ground states of nuclei so we did not expect any narrow peaks for @xmath217-shell knockout or for other shells . nevertheless we found that the chips event generator fills in the so - called `` @xmath218-shell knockout '' region , which is usually artificially smeared by a lorentzian @xcite . in the regular fragmentation scenario the spectrum of protons below @xmath219 mev is normal ; it falls down to the kinematic limit . the additional yield at @xmath220 mev is a reflection of the specific first act of hadronization with the quark exchange kinematics . the slope increase with momentum is approximated well by the model , but it is obvious that the yield close to the kinematic limit of the @xmath221 reaction can only be described in detail if the excited states of the residual nucleus are taken into account . the angular dependence of the proton yield in low - energy photo - nuclear reactions is described in the chips model and event generator . the most important assumption in the description is the hypothesis of a direct interaction of the photon with an asymptotically free quark in the nucleus , even at low energies . this means that asymptotic freedom of qcd and dispersion sum rules @xcite can in some way be generalized for low energies . the knockout of a proton from a nuclear shell or the homogeneous distributions of nuclear evaporation can not explain significant angular dependences at low energies . the same mechanism appears to be capable of modeling proton yields in such reactions as the @xmath11c(e , e@xmath212p ) reaction measured at mit bates @xcite , where it was shown that the region of missing energy above 50 mev reflects `` two - or - more - particle knockout '' ( or the `` continuum '' in terms of the shell model ) . the chips model may help to understand and model such phenomena .

adequate description of electro and gamma nuclear physics is of utmost importance in studies of electron beam - dumps and intense electron beam accelerators . i also is mandatory to describe neutron backgrounds and activation in linear colliders . this physics was elaborated in geant4 over the last year , and now entered into the stage of practical application . in the geant4 photo - nuclear data base there are at present about 50 nuclei for which the photo - nuclear absorption cross sections have been measured . of these , data on 14 nuclei are used to parametrize the gamma nuclear reaction cross - section the resulting cross section is a complex , factorized function of @xmath0 and @xmath1 , where @xmath2 is the energy of the incident photon . electro - nuclear reactions are so closely connected with photo - nuclear reactions that sometimes they are often called `` photo - nuclear '' . the one - photon exchange mechanism dominates in electro - nuclear reactions , and the electron can be substituted by a flux of photons . folding this flux with the gamma - nuclear cross - section , we arrive at an acceptable description of the electro - nuclear physics . final states in gamma and electro nuclear physics are described using chiral invariant phase - space decay at low gamma or equivalent photon energies , and quark gluon string model at high energies . we will present the modeling of this physics in geant4 , and show results from practical applications .

nucl-th0306012

believed to be the main origin of the jet quenching phenomena observed @xcite in nucleus nucleus collisions at rhic energy @xmath2@xmath3 , parton energy loss via gluon - radiation is expected to depend on the properties ( gluon density and volume ) of the ` medium ' formed in the collision and on the properties ( color charge and mass ) of the ` probe ' parton @xcite . hard gluons would lose more energy than hard quarks due to the stronger color coupling with the medium . in addition , charm and beauty quarks are qualitatively different probes with respect to light partons , since their energy loss is expected to be reduced , as a consequence of a mass - dependent restriction in the phase - space into which gluon radiation can occur @xcite . we study quenching effects for heavy quarks by supplementing perturbative qcd calculations of the baseline @xmath4 distributions with in - medium energy loss , included via the bdmps quenching weights . the quenching weights , computed for light quarks and gluons in @xcite and for heavy quarks in @xcite , depend on the transport coefficient @xmath5 , a measure of the medium density , and on the in - medium path length . these inputs are evaluated on a parton - by - parton level , using a glauber - model based description of the local @xmath5 profile in the transverse direction @xcite . the @xmath5 value is chosen in order to reproduce the light - flavor particles nuclear modification factor @xmath6 measured in central collisions at @xmath7 ( fig . [ fig : rhic ] , left ) : the range favored by the data for the parton - averaged transport coefficient is @xmath8@xmath9 . [ cols="<,^ " , ] heavy - quark energy loss is presently studied at rhic using measurements of the nuclear modification factor @xmath10 of ` non - photonic ' ( @xmath11-conversion- and @xmath12-dalitz - subtracted ) single electrons . the most recent data by phenix @xcite and star @xcite , reaching out to 5 and 9 gev , respectively , are shown in fig . [ fig : rhic ] ( right ) . since this is an inclusive measurement , with charm decays dominating at low @xmath4 and beauty decays dominating at high @xmath4 , the comparison with mass - dependent energy loss predictions should rely on a solid and data - validated pp baseline . such baseline is still lacking at the moment , as we explain in the following . the state - of - the - art perturbative predictions ( fonll ) , that we use as a baseline , indicate that , in pp collisions , charm decays dominate the electron @xmath4 spectrum up to about 5 gev @xcite . however , there is a large perturbative uncertainty on position in @xmath4 of the @xmath13-decay/@xmath14-decay crossing point : depending on the choice of the factorization and renormalization scales this position can vary from 3 to 9 gev @xcite . in addition , the calculation tends to underpredict the non - photonic electron spectrum measured in pp collisions @xcite . for our electron @xmath10 results ( fig . [ fig : rhic ] , right ) , in addition to the uncertainty on the medium density ( curves for @xmath8 , 10 , @xmath9 ) , we also account for the perturbative uncertainty by varying the values of the scales and of the @xmath13 and @xmath14 quark masses ( shaded band associated to the @xmath15 curve ) @xcite . we find that the nuclear modification factor of single electrons is about 0.2 larger than that of light - flavor hadrons . thus , electrons are in principle sensitive to the mass hierarchy of parton energy loss . the available data neither allow us to support claims of inconsistency between theory and experiment , nor do they support yet the expected mass hierarchy . it is important to note that , in general , the perturbative uncertainty in calculating the partonic baseline spectrum is comparable to the model - intrinsic uncertainty in determining @xmath5 . if future experimental studies at rhic succeeded in disentangling the charm and beauty contributions to single electrons , the sensitivity in the theory - data comparison would be largely improved . ( left ) and @xmath1 ( right ) mesons for the case of realistic heavy - quark masses and for a case study in which the quark mass dependence of parton energy loss is neglected @xcite , scaledwidth=85.0% ] heavy quarks will be produced with large cross sections at lhc energy and the experiments will be equipped with detectors optimized for the separation of charm and beauty decay vertices . thus , it should be possible to carry out a direct comparison of the attenuation of light - flavor hadrons , @xmath0 mesons , and @xmath1 mesons . we calculate the expected nuclear modification factors @xmath10 exploring a conservatively - large range in the medium density for central collisions at @xmath16 : @xmath17 . we use standard nlo perturbative predictions for the @xmath13 and @xmath14 @xmath4-differential cross sections @xcite . figure [ fig : lhc ] ( thick lines ) shows our results for the heavy - to - light ratios of @xmath0 and @xmath1 mesons @xcite , defined as the ratios of the nuclear modification factors of @xmath18 mesons to that of light - flavor hadrons ( @xmath19 ) : @xmath20 . we illustrate the effect of the mass by artificially neglecting the mass dependence of parton energy loss ( thin curves ) . the enhancement above unity that persists in the @xmath21 cases is mainly due to the color - charge dependence of energy loss , since at lhc energy most of the light - flavor hadrons will originate from a gluon parent . our results indicate that , for @xmath0 mesons , the mass effect is small and limited the region @xmath22 , while for @xmath1 mesons a large enhancement can be expected up to @xmath23 . therefore , the comparison of the high-@xmath4 suppression for @xmath0 mesons and for light - flavor hadrons will test the color - charge dependence ( quark parent vs. gluon parent ) of parton energy loss , while the comparison for @xmath1 mesons and for light - flavor hadrons will test its mass dependence @xcite .

the attenuation of heavy - flavored particles in nucleus nucleus collisions tests the microscopic dynamics of medium - induced parton energy loss and , in particular , its expected dependence on the identity ( color charge and mass ) of the parent parton . we discuss the comparison of theoretical calculations with recent single - electron data from rhic experiments . then , we present predictions for the heavy - to - light ratios of @xmath0 and @xmath1 mesons at lhc energy . address = universit degli studi di padova and infn , padova , italy address = dep . de fsica de partculas and igfae , universidade de santiago de compostela , spain address = lpthe , universit pierre et marie curie ( paris 6 ) , france address = department of physics , cern , theory division , genve , switzerland address = department of physics and astronomy , university of stony brook , ny , usa

hep-ph0601107

frustration is one of the interesting subjects in statistical physics , mainly because it can induce additional symmetry and lead the system to display rich low - temperature structures . the so - called two - dimensional ( 2d ) fully frustrated xy models have attracted an extensive investigation in the last two decades @xcite . due to the frustration the systems possess additional discrete reflection symmetry @xmath7 , apart from the global spin rotation symmetry @xmath8 . the breakdown of these symmetries are the onset of two types of phase transitions , namely one corresponding to the magnetic transition of kosterlitz - thouless ( kt ) type @xcite and the other to the chiral transition . whether these transitions are decoupled or occur at the same temperature has long been a controversy @xcite . another debated issue is whether the universality class of the chiral ordering belongs to the ising universality class or not @xcite . the system has a corresponding physical realization on a planar arrays of coupled josephson junctions in a transverse magnetic field @xcite and discotic liquid crystals @xcite . as a 2d frustrated xy system , two lattice systems are frequently studied numerically . the first one is the square lattice where the interactions can be a regular mixture of ferromagnetic ( f ) and antiferromagnetic ( af ) couplings ( villain model ) @xcite . the second one is the af xy model on the triangular lattice @xcite . as for the 2d xy model , the effect of the @xmath9-fold symmetry - breaking fields is an interesting subject @xcite ; that is essentially the same as treating the @xmath9-state clock model , where only the discrete values are allowed for the angle of the xy spins . the @xmath8 symmetry of the xy model is replaced by the discrete @xmath10 symmetry in the @xmath9-state clock model . it was shown @xcite that the 2d @xmath9-state clock model has two phase transitions of kt type at @xmath11 and @xmath0 ( @xmath12 ) for @xmath13 . there is an intermediate xy - like phase between a low - temperature ordered phase ( @xmath14 ) and a high - temperature disordered phase ( @xmath15 ) . it is quite interesting to investigate the effect of the @xmath9-fold symmetry - breaking fields in the case of the fully frustrated xy model . quite recently , noh _ et al . _ @xcite studied the af six - state clock model on the triangular lattice using the metropolis monte carlo simulation because of the experimental relevance to cf@xmath16br monolayers physisorbed on graphite @xcite . however , they did not pay attention to the lower temperature phase transition of kt type . it is to be noticed that the existing controversy involves very fine values . most studies claiming single transition scenario still do not exclude the possibility of two very close critical temperatures . meanwhile , the studies in favor of double transition scenario always found that two critical phase transitions occur at slightly different temperatures . therefore , it is desirable to obtain precise numerical information . recently , much progress has been made in the development of efficient algorithms of monte carlo simulation . especially , several attempts have been proposed for the monte carlo algorithms to calculate the energy density of states ( dos ) directly . examples are the multicanonical method @xcite , the broad histogram method @xcite , the flat histogram method @xcite , and the wang and landau method @xcite . all of these algorithms use the random walk in the energy space . in this paper we report our monte carlo study on the af six - state clock model on the triangular lattice . the ground state ( gs ) of the af six - state clock model on the triangular lattice has the same structure as the af xy model ; therefore this model is regarded as a commensurate discrete model for the fully frustrated xy model . on the other hand , the six - state clock model on the square lattice ( villain model ) has different gs configurations since there exist extra degeneracies . the presence of such extra degeneracy may bring about another interest in the fully frustrated six - state clock model . however , we will not cover such possibility in the present study . the xy villain and the eight - state clock villain models are commensurate because they have the same gs configuration . for the monte carlo method , we employ the wang - landau algorithm @xcite , and the energy dos is refined by the use of the broad histogram relation @xcite . the fact that the energy of the six - state clock model is represented by the multiple of @xmath17 , where @xmath18 is the coupling constant , is another supporting factor for the study of the six - state clock model ; it is convenient to treat discrete energy in the monte carlo simulation of calculating the dos directly . the rest of the present paper is organized as follows : in the next section we define the model and briefly explain the simulation method . details of the calculation and results will be presented in sec . the last section is devoted to the concluding remarks . the xy spin model is written with the hamiltonian @xmath19 where @xmath20 denotes the summation over nearest neighbor interactions , @xmath21 a unit planar spin vector occupying the @xmath22-th site , and @xmath23 the angle associated with the @xmath22-th spin . here , we mainly study the six - state clock model ; therefore the angle takes discrete values , @xmath24 with @xmath25 . the frustration is conveyed by @xmath26 . for the villain model on the square lattice this can be set by taking regular mixture of f and af couplings . for the triangular lattice on the other hand , @xmath26 are simply set to be uniform af couplings , @xmath27 , so that the system becomes fully frustrated . the hamiltonian ( [ ham ] ) is invariant under the symmetries of the global spin rotation @xmath8 and the global spin reflection @xmath7 . the breaking of these symmetries is expected to cause two kinds of ordering , which respectively correspond to magnetic ordering and chiral ordering . the gs configuration is well known as @xmath28-configuration , where two neighboring spins align in @xmath28 difference in angle , which is shown in fig . [ gsconf ] . we decompose the lattice into three interpenetrating sublattices for studying magnetic order . a site in a triangle belongs to one of the sublattices , @xmath29 or @xmath30 . we assign the magnetic order parameter as @xmath31 where @xmath32 is the magnetization of sublattice @xmath33 , and @xmath34 is the number of spins ; @xmath35 and @xmath36 follow the same definitions for the sublattices @xmath37 and @xmath30 . . spins are represented by arrows . sites belonging to the same sublattice have the same orientation of spins . the @xmath38 and @xmath39 signs indicate the handedness of the local chiralities . the ground state has 12-fold degeneracy . ] to discuss the global spin reflection @xmath7 , we deal with the chirality . the local chirality on the elementary triangle is defined as @xmath40}_z = \frac{2}{3\sqrt 3 } \sum_{j , k \in { \triangle}}\sin(\theta_k-\theta_j),\ ] ] where the @xmath41 component of the vector chirality is considered . the numerical factor in eq . ( [ chiral ] ) is chosen such that the maximum of the absolute value is one . in the gs configuration depicted in fig . [ gsconf ] , the local chirality takes a checkerboard pattern of the right - handed ( positive ) orientation and the left - handed ( negative ) orientation . then , the staggered chirality @xmath42 becomes the order parameter for the @xmath7 symmetry breaking transition . the gs configuration has @xmath43-fold degeneracy which is induced by the discrete global spin rotation symmetry @xmath44 with @xmath45-fold and by @xmath7 symmetry with @xmath46-fold . the number of this degeneracy is used as one of the check conditions in the calculation of energy dos . we use the monte carlo method to calculate the energy dos directly to obtain precise numerical information . first , we briefly describe the wang - landau algorithm @xcite . this algorithm is similar to the multicanonical method ( entropic sampling ) of lee @xcite , the broad histogram method @xcite and the flat histogram method @xcite ; the idea is based on the observation that performing a random walk in energy space with a probability proportional to the reciprocal of the dos , @xmath47 , will result in a flat histogram of energy distribution . the wang - landau method introduces a modification factor to accelerate the diffusion of the random walk in the early stage of the simulation . since the dos is not known at the beginning , it is simply set @xmath48 for all energy @xmath49 . the transition probability from energy @xmath50 to @xmath51 reads @xmath52 , \label{trans}\ ] ] and the dos @xmath53 is iteratively updated as @xmath54 every time the state is visited . the modification factor @xmath55 is gradually reduced to unity by checking the ` flatness ' of the energy histogram ; the histogram for all possible @xmath49 is not less than some value of the average histogram , say , 0.80 . we also use the broad histogram relation for getting a refined dos . in proposing the broad histogram method , oliveira _ et al . _ @xcite paid attention to the number of potential moves , or the number of the possible energy change , @xmath56 , for a given state @xmath57 . the total number of moves is @xmath58 for a single spin flip process of the ising model simulation . the energy dos is related to the number of potential moves as @xmath59 where @xmath60 denotes the microcanonical average with fixed @xmath49 . this relation is shown to be valid on general grounds @xcite , and we call eq . ( [ bhr ] ) the broad histogram relation . we measure the average of the potential move , @xmath61 , and use this information for getting a better estimate of the energy dos . it was stressed @xcite that @xmath56 is a macroscopic quantity , which is the advantage of using the number of potential moves . we should also note that the broad histogram relation does not depend on the particular dynamic rule one adopts , and the microcanonical averages of the potential moves can be obtained by any rule of monte carlo dynamics . in order to reduce calculation time for larger system sizes , we break simulation into several energy windows and perform random walk in each different range of energy . the resultant pieces of the dos are joined together and used to produce the thermal average with the inverse temperature @xmath62 through the standard relation @xmath63 using the parallel machine , we perform the measurements of the physical quantity @xmath64 up to @xmath65 monte carlo steps . also , we perform 10 independent runs for each system size in order to get better statistics and to evaluate statistical errors . here we present the results for the af six - state clock model on the triangular lattice . we have treated the system with the linear sizes @xmath66 = 24 , 36 , 48 , 60 , and 72 . we apply the periodic boundary conditions . we normalize the dos by using the condition @xmath67 , and the degeneracy in the gs energy , @xmath68 , is checked in order to confirm the accuracy of the calculation . in fig . [ ds48 ] , we show the energy dos of system size @xmath69 as an example . here , the energy is represented in units of @xmath17 , and the gs energy is given by @xmath70 . . the energy is represented in units of @xmath17 . ] the energy - dependent data of quantity @xmath71 are used to calculate the thermal average @xmath72 by using eq . ( [ ave ] ) . we calculate the specific heat per spin through the relation @xmath73,\ ] ] where @xmath74 is the boltzmann constant . = 24 , 36 , 48 , 60 , and 72 . ] we show the temperature dependence of specific heat for different lattice sizes in fig . the divergent peak around @xmath75 in units of @xmath76 gives a clear sign of the existence of second - order phase transition . we also observe a hump on the lower temperature side around @xmath77 , which may be related to the transition of kt type . however , we should study the magnetic and chiral orders for the detailed analysis of the phase transition . the critical behavior and the transition temperature can be investigated more precisely from the evaluation of the order parameter or its corresponding correlation function . the magnetic and chirality correlation functions are defined as the following : @xmath78 where @xmath79 is the fixed distance between spins . precisely , the distance @xmath79 is a vector , but we have used a simplified notation . two of the present authors @xcite showed that the ratio of the correlation functions with different distances is a useful estimator for the analysis of the second - order phase transition as well as for the kt transition , and this correlation ratio can be used for the generalization of the probability - changing cluster algorithm @xcite . at the critical point or on the critical line , the correlation function @xmath80 for an infinite system decays as a power of @xmath79 , @xmath81 where @xmath82 is the spatial dimension and @xmath83 the decay exponent . for a finite system in the critical region , the correlation function depends on two length ratios , @xmath84 where @xmath85 is the correlation length . then , the ratio of the correlation functions with different distances has a finite - size scaling ( fss ) form with a single scaling variable , @xmath86 if we fix two ratios , @xmath87 and @xmath88 . in the present work , we set @xmath89 and @xmath90 for two distances . thus , we evaluate the correlation ratios @xmath91 and @xmath92 , where @xmath93 and @xmath94 are referred respectively to eqs . ( [ magchi1 ] ) and ( [ magchi2 ] ) . it is important for two correlated spins to belong to the same sublattice ; since the fixed distances are set as @xmath89 and @xmath95 , we choose the system size as a multiple of @xmath43 . we show the correlation ratios both for the ( a ) magnetic and ( b ) chiral correlations in fig . [ corf ] . from the temperature dependence of the magnetic correlation ratio plotted in fig . [ corf](a ) , we observe that the curves of different sizes merge in the intermediate temperature range ( @xmath96 ) , and spray out for the low - temperature and high - temperature ranges . this behavior is the same as that for the unfrustrated six - state clock model @xcite , which suggests that there are two phase transitions of kt type at @xmath11 and @xmath0 . the hump on the lower temperature side in the specific heat , fig . [ spht ] , may correspond to the lower temperature kt transition at @xmath11 . the higher temperature kt transition at @xmath0 is not obvious from the specific heat plot as it is veiled by the divergent peak due to the chiral transition . we can make a fss analysis based on the kt form of the correlation length , @xmath97 , where @xmath98 . the @xmath66 dependence of @xmath99 is given by @xmath100 using the data of the magnetic correlation ratio @xmath101 for different sizes , we estimate two kt transition temperatures . we consider the size - dependent temperature that gives the constant @xmath102 . in fig . [ t12 ] , we plot @xmath99 as a function of @xmath103 with @xmath104 for the best - fitted parameters in eq . ( [ t_kt ] ) . for a fitting function we have used a quadratic function in @xmath103 to include correction terms . the value of @xmath102 has been set to be 0.86 , 0.88 and 0.90 for the determination of @xmath0 , whereas @xmath102 has been set to be 0.99 , 0.985 and 0.98 for @xmath11 . the data with different @xmath102 are represented by different marks in fig . [ t12 ] , but they are collapsed on a single curve in this plot , which means that @xmath105 depends on @xmath102 in eq . ( [ t_kt ] ) and the difference of @xmath102 can be absorbed in the @xmath102 dependence of @xmath105 . we estimate the kt temperatures of the magnetic order using eq . ( [ t_kt ] ) as @xmath106 where the numbers in the parentheses denote the uncertainty in the last digits . the estimate of @xmath0 is slightly lower than the estimate by noh _ @xcite , @xmath107 . it is due to the fact that the moment ratio was used in ref . @xcite , and the estimate of the kt temperature becomes higher because of large corrections to fss @xcite . and ( b ) @xmath108 of the af six - state clock model on the triangular lattice for @xmath66 = 24 , 36 , 48 , 60 , and 72 , where @xmath109 . the data for @xmath102=0.86 , 0.88 and 0.90 are shown by different marks in ( a ) , and those for @xmath102=0.99 , 0.985 and 0.98 in ( b ) . , title="fig : " ] + and ( b ) @xmath108 of the af six - state clock model on the triangular lattice for @xmath66 = 24 , 36 , 48 , 60 , and 72 , where @xmath109 . the data for @xmath102=0.86 , 0.88 and 0.90 are shown by different marks in ( a ) , and those for @xmath102=0.99 , 0.985 and 0.98 in ( b ) . , title="fig : " ] next we consider the decay exponent @xmath83 . we first look at the constant value of correlation ratio @xmath102 for different sizes and find the associate correlation function @xmath110 . we give attention to the power - law dependence of the correlation function on the system size , @xmath111 , which can be seen from eq . ( [ corr_finite ] ) and @xmath82 is set to be 2 . we plot @xmath110 versus @xmath66 for various @xmath102 s in logarithmic scale in fig . [ slope ] . the value of @xmath83 is obtained as the slope of the best - fitted line for each constant of correlation ratio . the multiplicative logarithmic corrections for the kt transition @xcite were shown to be small compared to statistical errors . we plot @xmath83 thus determined with respect to the fixed correlation ratio @xmath102 in fig . [ eta ] . in the kt phase , @xmath102 is directly related to the temperature . we should note that the exponent @xmath83 is meaningful only in the temperature range @xmath112 on the fixed line . we show the values of @xmath102 s which give @xmath11 and @xmath0 by arrows in fig . as can be seen , the decay exponent @xmath83 behaves like a typical kt transition ; that is , the exponent @xmath83 continuously changes with the temperature in the kt phase . since @xmath83 is almost constant for larger @xmath102 in fig . [ eta ] , the exponent at the lower kt temperature @xmath11 is estimated as @xmath113 for smaller @xmath102 ( higher temperature ) side , @xmath83 depends on @xmath102 due to corrections in fig . [ eta ] . using the fact that the fitted value of @xmath105 in eq . ( [ t_kt ] ) reflects on the difference from the transition point , that is , @xmath114 , we estimate the exponent @xmath83 at the higher kt temperature @xmath0 by extrapolation . the obtained @xmath83 and @xmath105 are , for example , 0.310 and 1.76 for @xmath115 , 0.298 and 2.18 for @xmath116 , and 0.284 and 3.25 for @xmath117 . plotting the @xmath83 s as a function of @xmath118 , and extrapolating to @xmath119 , we obtain @xmath120 of course , other dependences such as @xmath121 are possible ; such an ambiguity is included in the error . in fig . [ eta ] we show the value of @xmath102 which gives @xmath0 by the arrow . the @xmath83 at this @xmath102 is consistent with the estimated value , @xmath122 . for the unfrustrated six - state clock model , the exponents @xmath123 and @xmath124 were predicted as 1/4 and 1/9 respectively @xcite , and they were confirmed numerically @xcite . the present results suggest that the exponents associated with the kt transitions are universal even for the frustrated model , which the previous work @xcite failed to show . versus @xmath66 . here the slope of the best - fit straight line of each corresponding @xmath102 is the value of exponent @xmath83 . ] of kt phase as a function of magnetic coefficient ratio @xmath102 . line is just guide to the eyes . ] the temperature dependence of the chiral correlation ratio was also plotted in fig . [ corf](b ) . the existence of chiral phase transition can be clearly observed . in the figure , there is a single crossing point which indicates the second - order phase transition ; it corresponds to the divergent peak in the specific heat plot . by using the fss plot of chirality correlation ratio , as shown in fig . [ nu1 ] , we can estimate the critical temperature and exponent @xmath125 of chiral ordering . the estimates are @xmath126 our result exhibiting that the chiral transition occurs at slightly higher than @xmath0 of kt transition is consistent with most studies in favor of double transition scenario . quite recently , korshunov @xcite has discussed that the phase transition associated with the unbinding of vortex pairs takes place at a lower temperature than the other phase transition associated with proliferation of the ising - type domain walls . our estimate for the exponent @xmath125 is consistent with the results by lee and lee @xcite and by ozeki and ito @xcite , but contradicts with the result by olsson @xcite ; that is , the critical phenomena are not governed by the ising universality class . we have not observed an appreciable size dependence of the estimated @xmath125 up to our maximum system size , @xmath127 . olsson @xcite argued that corrections to the scaling are important in the fully frustrated xy model , and the data are consistent with @xmath128 . et al . _ also postulated that only for large enough system the ising - like behavior is observed . however , using nonequilibrium relaxation study for large enough systems up to @xmath129 , ozeki and ito @xcite recently obtained the @xmath130 , which suggests that the corrections to fss are not so serious . thus , more careful calculations will be needed for the critical phenomena of chiral transition . . in summary , we have investigated the af six - state clock model on the triangular lattice using the wang - landau method combined with the broad histogram relation . the model is closely related to the 2d fully frustrated xy model . we have found that the system possesses two orderings , spin ordering and chiral ordering . the former undergoes the kt transition while the latter indicates the second - order transition . we have also observed the lower temperature kt transition due to the discrete symmetry of the clock model . our estimates of the higher kt temperature @xmath0 and the critical temperature of chiral ordering @xmath1 , that is , @xmath2 and @xmath3 , support the double transition scenario . the lower kt temperature is estimated as @xmath4 . two decay exponents of kt transitions are estimated as @xmath5 and @xmath6 , which suggests that the exponents associated with the kt transitions are universal even for the frustrated model . for the critical phenomena of the chiral transition , our estimate of the exponent @xmath125 , that is , @xmath131 , suggests that the model does not belong to the ising universality class , but more detailed study is still required . the authors wish to thank n. kawashima , h. otsuka , c. yamaguchi , m. suzuki , and y. ozeki for valuable discussions . one of the authors ( ts ) gratefully acknowledges the fellowship provided by the ministry of education , science , sports and culture , japan . this work was supported by a grant - in - aid for scientific research from the japan society for the promotion of science . the computation of this work has been done using computer facilities of tokyo metropolitan university and those of the supercomputer center , institute for solid state physics , university of tokyo . s. teitel and c. jayaprakash , phys . b * 27 * , 598 ( 1983 ) . e. granato , j. m. kosterlitz , j. lee and m. p. nightingale , phys . lett . * 66 * , 1090 ( 1991 ) . g. ramirez - santiago and j. v. jose , phys . b * 49 * , 9567 ( 1994 ) . s. lee and k - c . lee , phys . b * 49 * , 15184 ( 1994 ) . p. olsson , phys . lett . * 75 * , 2758 ( 1995 ) ; phys . lett . * 77 * , 4850 ( 1996 ) ; phys . b * 55 * , 3585 ( 1997 ) . m. benakli , h. zheng and g. gabay , phys . rev . b * 55 * , 278 ( 1997 ) . h. j. luo , l. sch@xmath132lke , and b. zheng , phys . lett . * 81 * , 180 ( 1998 ) . e. h. boubcheur and h. t. diep , phys . b * 58 * , 5163 ( 1998 ) . y. ozeki and n. ito , phys . b * 68 * , 054414 ( 2003 ) . s. miyashita and h. shiba , j. phys . soc . jpn . * 53 * , 1145 ( 1984 ) . d. h. lee and j. d. joannopoulous , j. w. negele and d. p. landau , phys . lett . * 52 * , 433 ( 1984 ) ; phys . b * 33 * , 450 ( 1986 ) . e. van himbergen , phys . b * 33 * , 7857 ( 1986 ) . j. lee , e. granato and j. m. kosterlitz , phys . b * 44 * , 4819 ( 1991 ) ; phys . rev . b * 43 * , 11531 ( 1991 ) . xu and b. w. southern , j. phys . a * 29 * l133 ( 1996 ) . s. lee and k - c . lee , phys . b * 57 * , 8472 ( 1998 ) . d. j. resnick , j. c. garland , j. t. boyd , s. shoemaker , r. s. newrock , phys . lett . * 47 * , 1542 ( 1981 ) . b. j. van wees , h. s. j. van der zant , and j. e. mooij , phys . b * 35 * , 7291 ( 1987 ) . h. eikmans , j. e. van himbergen , h. j. f. knops and j. m. thijssen , phys . b * 39 * , 11759 ( 1989 ) . v. i. marconi and d. dominguez , phys . . lett . * 87 * , 017004 ( 2001 ) . b. a. berg and t. neuhaus , phys b * 267 * , 249 ( 1991 ) ; phys . lett . * 68 * , 9 ( 1992 ) . j. lee , phys . lett . * 71 * , 211 ( 1993 ) . m. c. de oliveira , t. j. p. penna , and h. j. herrmann , braz . * 26 * , 677 ( 1996 ) ; eur . j. b * 1 * , 205 ( 1998 ) . j. s. wang , eur . j. b. * 8 * , 287 ( 1998 ) . j. s. wang and l. w. lee , comput . phys . commun . * 127 * , 131 ( 2000 ) ; j. s. wang , physica a * 281 * , 147 ( 2000 ) . f. wang and d. p. landau , phys . lett . * 86 * , 2050 ( 2001 ) ; phys . e * 64 * , 056101 ( 2001 ) . p. m. c. de oliveira , eur . j. b * 6 * , 111 ( 1998 ) . b. a. berg and u. h. e. hansmann , eur . j. b * 6 * , 395 ( 1998 ) .

monte carlo simulations using the newly proposed wang - landau algorithm together with the broad histogram relation are performed to study the antiferromagnetic six - state clock model on the triangular lattice , which is fully frustrated . we confirm the existence of the magnetic ordering belonging to the kosterlitz - thouless ( kt ) type phase transition followed by the chiral ordering which occurs at slightly higher temperature . we also observe the lower temperature phase transition of kt type due to the discrete symmetry of the clock model . by using finite - size scaling analysis , the higher kt temperature @xmath0 and the chiral critical temperature @xmath1 are respectively estimated as @xmath2 and @xmath3 . the results are in favor of the double transition scenario . the lower kt temperature is estimated as @xmath4 . two decay exponents of kt transitions corresponding to higher and lower temperatures are respectively estimated as @xmath5 and @xmath6 , which suggests that the exponents associated with the kt transitions are universal even for the frustrated model .

cond-mat0402613

hamilton equations of motion constitute a system of ordinary first order differential equations , @xmath3 where the @xmath4 denotes differentiation with respect to time @xmath5 , and @xmath6 . they can be viewed as the characteristic equations of the partial differential equation @xmath7 with @xmath8 the first order differential operator @xmath9 generating a flow on phase space . if @xmath10 does not depend explicitly on @xmath5 , a formal solution of ( [ hamiltonianflow ] ) is @xmath11 in most cases this expression remain just formal , but one may often split the hamiltonian into two parts , @xmath12 , with a corresponding splitting @xmath13 such that the flows generated by @xmath14 and @xmath15 separately are integrable . one may then use the cambell - baker - hausdorff formula to approximate the flow generated by @xmath8 . one obtains the strang splitting formula @xcite @xmath16\right]+\cdots},\ ] ] which shows that time stepping this expression with a timestep @xmath1 provides an approximation with relative accuracy of order @xmath0 , exactly preserving the symplectic property of the flow . this corresponds to the symplectic splitting scheme of iterating the process of solving @xmath17 here the last part of one iteration may be combined with the first part of the next , unless one deals with time dependent systems or wants to register the state of the system at the intermediate times . from a practical point of view the most interesting property of this formulation is that it can be interpreted directly in terms of physical processes . for instance , for hamiltonians @xmath18 , a standard splitting scheme is to choose @xmath19 and @xmath20 . in that case ( [ symplecticsplitting ] ) corresponds to a collection of freely streaming particles receiving kicks at regular time intervals @xmath1 , these kicks being dependent of the positions @xmath21 of the particles . i.e , we may think of the evolution as a collection of _ kicks _ and _ moves _ @xcite . it is not clear that this is the best way to approximate or model the exact dynamics of the real system . for instance , why should the motion between kicks be the free streaming generated by @xmath22 ? there are more ways to split the hamiltonian into two integrable parts @xcite ; the best splitting is most likely the one which best mimics the physics of equation ( [ hamiltonequation ] ) . further , since this equation is not solved exactly by ( [ symplecticsplitting ] ) for any finite value of @xmath1 we need not necessarily choose @xmath23 to be _ exactly _ @xmath24 as long as it approaches this quantity sufficiently fast as @xmath25 . we will exploit this observation to improve the accuracy of the splitting scheme ( [ symplecticsplitting ] ) in a systematic manner . we are , of course , not the first trying to improve on the strmer - verlet splitting scheme . an accessible review of several earlier approaches can be found in reference @xcite . neri @xcite has provided the general idea to construct symplectic integrators for hamiltonian systems . forest and ruth @xcite discussed the explicit fouth order method for the integration of hamiltonian equations for the simplest non - trivial case . yoshida @xcite worked out a symplectic integrator for any even order , and suzuki @xcite presented the idea of how recursive construction of successive approximants may be extended to other methods . for a simple illustration of our idea consider the hamiltonian @xmath26 whose exact evolution over a time interval @xmath1 is @xmath27 p^{\text{e } } \end{pmatrix } = \begin{pmatrix*}[r ] \cos \tau & \sin \tau \\ - \sin \tau & \cos\tau \end{pmatrix * } \begin{pmatrix } q\\p\end{pmatrix}. \label{h_0a}\ ] ] compare this with a _ kick - move - kick _ splitting scheme over the same time interval , with @xmath28 and @xmath29 , where @xmath30 and @xmath31 may depend on @xmath1 . one full iteration gives @xmath32 1-\frac{1}{2}m k\tau^2 & m \tau \\[0.4ex ] -(1-\frac{1}{4 } km \tau^ 2 ) k\tau & 1-\frac{1}{2 } k m\tau^2 \end{pmatrix * } \begin{pmatrix}q \\ p \end{pmatrix}.\ ] ] we note that by choosing @xmath33 \label{harmonicoscillatorcorrection}\\[-1ex ] k & = \frac{2}{\tau } \tan \frac{\tau}{2 } = 1+\frac{1}{12}\tau^2+\frac{1}{120}\tau^4 + \frac{17}{20160}\tau^6 + \cdots,\nonumber\end{aligned}\ ] ] the exact evolution is reproduced . if we instead choose a _ move - kick - move _ splitting scheme , with @xmath34 and @xmath35 , one iteration gives @xmath36 p^{\text{s}}\end{pmatrix } = \begin{pmatrix*}[c ] 1-\frac{1}{2}\bar{m } \bar{k}\tau^2 & ( 1-\frac{1}{4}\bar{m}\bar{k})\bar{m } \tau \\[0.5ex ] -\bar{k}\tau & 1-\frac{1}{2 } \bar{k}\bar{m}\tau^2 \end{pmatrix * } \begin{pmatrix}q \\ p \end{pmatrix},\ ] ] which becomes exact if we choose @xmath37 it should be clear that this idea works for systems of harmonic oscillators in general , i.e. for quadratic hamiltonians of the form @xmath38 where @xmath39 and @xmath40 are symmetric matrices . for a choosen splitting scheme and step interval @xmath1 there are always modified matrices @xmath41 and @xmath42 which reproduces the exact time evolution . for systems where @xmath39 and @xmath40 are too large for exact diagonalization , but sparse , a systematic expansion of @xmath43 and @xmath44 in powers of @xmath0 could be an efficient way to improve the standard splitting schemes . for a more general treatment we consider hamiltonians of the form @xmath45 a series solution of the hamilton equations in powers of @xmath1 is @xmath46 here we have introduced notation to shorten expressions , @xmath47 \\[-1.5ex ] & d \equiv p_a \partial^a,\quad \bar{d } \equiv ( \partial_a v)\partial^a,\nonumber\end{aligned}\ ] ] where we employ the _ einstein summation convention _ : an index which occur twice , once in lower position and once in upper position , are implicitly summed over all available values . i.e , @xmath48 ( we will generally use the matrix @xmath39 to rise an index from lower to upper position ) . the corresponding result for the _ kick - move - kick _ splitting scheme is @xmath49 as expected it differs from the exact result in the third order , but the difference can be corrected by introducing second order generators @xmath50 to be used in respectively the _ move _ and _ kick _ steps . specialized to a one - dimensional system with potential @xmath51 this agrees with equation ( [ harmonicoscillatorcorrection ] ) . with this correction the _ kick - move - kick _ splitting scheme agrees with the exact solution to @xmath52 order in @xmath1 , but differ in the @xmath53-terms . we may correct the difference by introducing fourth order generators , @xmath54 \\[-1ex ] v_4 & = \frac{1}{480 } \bar{d}^2 v\tau^4.\nonumber\end{aligned}\ ] ] specialized to a one - dimensional system with potential @xmath51 this agrees with equation ( [ harmonicoscillatorcorrection ] ) . with this correction the _ kick - move - kick _ splitting scheme agrees with the exact solution to @xmath55 order in @xmath1 , but differ in the @xmath56-terms . we may correct the difference by introducing sixth order generators , @xmath57 \\[-1ex ] v_6 & = \frac{1}{161280}\left ( 17\ , \bar{d}^3 - 10\,\bar{d}_3\right)v\tau^6,\nonumber\end{aligned}\ ] ] where we have introduced @xmath58 specialized to a one - dimensional system with potential @xmath51 this agrees with equation ( [ harmonicoscillatorcorrection ] ) . with this correction the _ kick - move - kick _ splitting scheme agrees with the exact solution to @xmath59 order in @xmath1 , but differ in the @xmath60-terms . one may continue the correction process , but this is probably well beyond the limit of practical use already . addition of extra potential terms @xmath61 is in principle unproblematic for solution of the _ kick _ steps . the equations , @xmath62 can still be integrated exactly , preserving the symplectic structure . the situation is different for the kinectic term @xmath63 , since it now leads to equations @xmath64 which is no longer straightforward to integrate exactly . although the problematic terms are small one should make sure that the _ move _ steps preserve the symplectic structure exactly . let @xmath65 denote the positions and momenta just before the _ move _ step , and @xmath66 the positions and momenta just after . we construct a generating function @xcite @xmath67 , with @xmath68 this preserves the symplectic structure ; we just have to construct @xmath69 to represent the _ move _ step sufficiently accurately . consider first the case without the correction terms . the choice @xmath70 gives @xmath71 which is the correct relation . now add the @xmath72-term to the _ move _ step . to order @xmath73 the exact solution of equation ( [ movesteps ] ) becomes @xmath74 \label{exactmove } \\[-1.5ex ] p_a & = p_a + \frac{1}{12 } \partial_a \ , d^2 v\,\tau^3 + \frac{1}{24}\partial_a d^3 v\,\tau^4.\nonumber\end{aligned}\ ] ] compare this with the result of changing @xmath75 where @xmath76 . the solution of equation ( [ canonicaltransformation ] ) change from the relations ( [ simplekick ] ) to @xmath77 since @xmath78 is linear in @xmath79 , equation ( [ pequation ] ) constitute a system of third order algebraic equation which in general must be solved numerically . this should usually be a fast process for small @xmath1 . an exact solution of this equation is required to preserve the symplectic structure , but this solution should also agree with the exact solution of ( [ movesteps ] ) to order @xmath73 . this may be verified by perturbation expansion in @xmath1 . a perturbative solution of equation ( [ pequation ] ) is @xmath80 which inserted into ( [ qequation ] ) reproduces the full solution ( [ exactmove ] ) to order @xmath73 . this process can be systematically continued to higher orders . we write the transformation function as @xmath81 and find the first terms in the expansion to be @xmath82 it remains to demonstrate that our algorithms can be applied to real examples . we have considered the hamiltonian @xmath83 with initial condition @xmath84 , @xmath85 . the exact motion is a nonlinear oscillation with @xmath10 constant equal to @xmath86 , and period @xmath87 here @xmath88 is the beta function . in figure [ energypreservation ] we plot the behaviour of @xmath89 during the last half of the @xmath90 oscillation , for various values of @xmath1 and corrected generators up to order @xmath91 ( corresponding to @xmath92 ) . we have shown that it is possible to systematically improve the accuracy of the usual symplectic integration schemes for a rather general class of hamilton equations . the process is quite simple for linear equations , where it may be useful for sparse systems . for general systems the method requires the solution of a set of nonlinear algebraic equations at each _ move _ step . to which extent an higher - order method is advantageous or not will depend on the system under analysis , and the wanted accuracy . as always with higher order methods the increased accuracy per step may be countered by the higher computational cost per step @xcite . d. cohen , t. jahnke , k. lorenz , and c. lubich , _ numerical integrator for highly oscillatory hamiltonian systems : a review _ , in analysis , modelling and simulations of multiscale problems , springer - verlag , ( 2006 ) 553576 .

we show how the standard ( strmer - verlet ) splitting method for differential equations of hamiltonian mechanics ( with accuracy of order @xmath0 for a timestep of length @xmath1 ) can be improved in a systematic manner without using the composition method . we give the explicit expressions which increase the accuracy to order @xmath2 , and demonstrate that the method work on a simple anharmonic oscillator . splitting - method , hamilton - equations , higher - order - accuracy , symplecticity

1204.4117

in our experiment , we begin with an optical characterization of the qds and observe a significantly reduced spectral linewidth of the emitted photons from a resonantly driven single qd compared with incoherent excitation methods including via above - bandgap and p - shell using cw lasers . figure@xmath3s2(a - c ) present a direct comparison of the spectral linewidth of the emitted photons from a single qd ( qd2 ) neutral exciton for different cw - laser excitation methods . at moderate power regime ( around saturation ) , the cw photoluminescence spectra arising from above band - gap and _ p_-shell excitation yields a linewidth of @xmath52.5ghz ( see fig.@xmath3s2a ) and @xmath51.5ghz ( fig.@xmath3s2b ) , respectively . on the other hand , cw rf photons ( see fig.@xmath3s2c ) exhibit a significantly narrower linewidth of @xmath50.48ghz even at high power regime well above saturation ( 32@xmath46 ) where a mollow triplet arises @xcite . figure@xmath3s2d shows a series of cw rf spectra at different laser power . the coherence time @xmath47 fitted ( using the corrected eqn.(1 ) from ref.@xcite ) from the cw rf spectra at @xmath46 is closest to being radiative lifetime limited : @xmath47/@xmath48=0.93(6 ) , where @xmath49 is the exciton lifetime which is measured separately to be of 390(10)ps using time - resolved pulsed rf . this is consistent with the prediction that the pure _ s_-shell resonant excitation can eliminate dephasings associated with the incoherent excitation methods @xcite . a high - resolution pulsed rf spectrum from qd2 is shown in fig.@xmath3s3 . for a range of laser power from 0.2@xmath0 to @xmath0 pulse , we fit the rf spectra with the voigt profile to extract the inhomogeneous ( gaussian ) linewidth ( @xmath50 ) . as plotted in the inset of fig.@xmath3s3 , the @xmath50 shows a increase at larger excitation power , which is in qualitative agreement with previous investigations of light - induced spectral diffusion @xcite . figure@xmath3s4a - b show the data of full histogram obtained on qd2 . clusters of five peaks appear periodically with repetition period of @xmath1112.2 ns . the central cluster shows an overall reduced photon counts compared to the side clusters due to the single - photon nature of the source . the hom interference are tested with @xmath0 , 0.72@xmath0 and 0.41@xmath0 pulse excitation where the rf counts reach @xmath6@xmath51 , @xmath6@xmath52 and @xmath6@xmath53 saturation level , and show raw visibilities of 0.903(55 ) , 0.912(56 ) and 0.934(39 ) , respectively ( see fig.@xmath34(c - h ) ) . taking into account of the residual two - photon emission probability for this qd , @xmath54 , and the optical imperfections of our interferometric setup ( same as in the main text ) , we obtain corrected degrees of indistinguishability to be 0.956(58 ) , 0.966(59 ) , 0.989(41 ) for the @xmath0 , 0.72@xmath0 and 0.41@xmath0 pulses , respectively .

single photon sources based on semiconductor quantum dots offer distinct advantages for quantum information , including a scalable solid - state platform , ultrabrightness , and interconnectivity with matter qubits . a key prerequisite for their use in optical quantum computing and solid - state networks is a high level of efficiency and indistinguishability . pulsed resonance fluorescence ( rf ) has been anticipated as the optimum condition for the deterministic generation of high - quality photons with vanishing effects of dephasing . here , we generate pulsed rf single photons on demand from a single , microcavity - embedded quantum dot under _ s_-shell excitation with 3-ps laser pulses . the @xmath0-pulse excited rf photons have less than 0.3@xmath1 background contributions and a vanishing two - photon emission probability . non - postselective hong - ou - mandel interference between two successively emitted photons is observed with a visibility of 0.97(2 ) , comparable to trapped atoms and ions . two single photons are further used to implement a high - fidelity quantum controlled - not gate . single photons have been proposed as promising quantum bits ( qubits ) for quantum communication @xcite , linear optical quantum computing @xcite and as messengers in quantum networks @xcite . these proposals primarily rely upon a high degree of indistinguishability between individual photons to obtain the hong - ou - mandel ( hom ) type interference @xcite which is at the heart of photonic controlled logic gates and photon - interference - mediated quantum networking @xcite . among different types of single - photon emitters @xcite , quantum dots ( qds ) are attractive solid - state devices since they can be embedded in high - quality nanostructure cavities and waveguides to generate ultra - bright sources of single and entangled photons @xcite . qds also provide a light - matter interface @xcite and can in principle be scaled to large quantum networks @xcite . two - photon hom interference experiments using photons from a single qd @xcite , as well as from independent sources @xcite , have not only demonstrated the potential of qds as single - photon sources , but also revealed the level of dephasing arising from incoherent excitation . the method of incoherent pumping ( via above band - gap or _ p_-shell excitation ) typically causes reduced photon coherence times due to homogeneous broadening of the excited state @xcite and uncontrolled emission time jitter from the nonradiative high - level to _ s_-shell relaxation @xcite , leading to a decrease of photon indistinguishability . to eliminate these dephasings , an increasing effort has been devoted to _ s_-shell resonant optical excitation of qds . the mollow triplet spectra and photon correlations of the resonance fluorescence ( rf ) have been measured @xcite . under continuous - wave ( cw ) laser excitation , a high degree of indistinguishability for continuously generated rf photons has been demonstrated through post - selective hom interference @xcite . however , in the cw regime , as the emission time of the rf photons is uncontrolled , the hom interference relies on the finite single - photon detection time resolution to discriminate and post - select a small fraction of photons that overlapped on the beam - splitter at the same time @xcite . therefore , the obtained interference visibility needs to be convoluted with and is thus limited by the realistic detection time response . this limitation , together with the low efficiency of two - photon interference owing to the unsynchronized photon arrival time , prohibits the direct application of cw rf photons in many quantum information protocols @xcite . more recent experiments operating on the low excitation regime have showed that the coherent scattering part of the rf could have coherence comparable to the excitation laser @xcite . however , such a single - photon source would suffer an intrinsically low efficiency . it has been anticipated @xcite that _ pulsed _ and resonant _ s_-shell excitation could remedy the above problems and be used for deterministic generation of time - tagged , highly indistinguishable single photons . in addition , the pulsed and transition - selective rf single photons are also a prerequisite for the much sought - after goal of entangling distant qd spins through photon interference @xcite , as well as for the scheme of generating on - demand multi - photon cluster states @xcite . earlier experiments @xcite have used pulsed resonant excitation to demonstrate rabi oscillation , a hallmark for quantum optics . yet , access to a background - free on - demand single - photon source with near - unity indistinguishability proved elusive @xcite . in this article , by applying resonant _ s_-shell optical excitation with picosecond laser pulses , we generate pulsed rf single photons on demand from a single qd embedded in a planar microcavity . rabi oscillations are visible from the variation of the rf intensity as a function of pump pulse area . under deterministic @xmath0-pulse excitations , the rf photons have less than 0.3@xmath1 background contributions and show an anti - bunching of @xmath2 . we observe non - postselective hom interference with a raw visibility of 0.91(2 ) and corrected visibility of 0.97(2 ) for two rf photons excited by two successive @xmath0 pulses separated by 2ns . finally , the highly indistinguishable rf photons are utilized to demonstrate a quantum controlled - not gate . our experiments are performed on self - assembled ingaas qds which are embedded in a planar microcavity and cooled in a cryogen - free bath cryostat at 4.2k ( see fig.@xmath3s1 ) . laser excitation of a single qd and collection of the emitted fluorescence are carried out with a confocal microscope . the excitation laser is pulsed with nominal pulse width of 3ps . the microscope is operated in a cross - polarization configuration , whereby a polarizer is placed in the collection arm with its polarization perpendicular to the excitation light , extinguishing the scattered laser by a factor exceeding @xmath4 . the microcavity has a quality factor of @xmath5200 which increases the fluorescence collection efficiency and reduces the laser power required for excitation of the qds . figure@xmath31a shows the detected rf photon counts as a function of the square root of the excitation laser power . the oscillation of the rf intensity is due to the well - known rabi rotation between the ground and the excitonic state . it has been demonstrated previously by quasi - resonant @xcite or resonant driving @xcite . the rf intensity reaches its first peak at the @xmath0 pulse . we excite the qd with @xmath0 pulses at a repetition rate of @xmath682mhz and observed @xmath6230,000 photon counts on a single - photon detector ( with an efficiency of 22@xmath1 ) . the overall rf collection efficiency is @xmath61.3@xmath1 . after correcting for the fibre coupling efficiency ( @xmath645@xmath1 ) , polarizer ( @xmath650@xmath1 ) and beam splitter ( @xmath695@xmath1 ) , we estimate that @xmath66@xmath1 of the photons emitted by the qd are collected into the first lens , which is in good agreement with numerical simulations ( see supplementary information ) . to verify that it is indeed a single - photon source , figure@xmath31b shows the second - order correlation measurement of the @xmath0-pulse driven rf photons . at zero delay , it shows a clear anti - bunching with a vanishing multi - photon probability of @xmath2 . thus it can be concluded that one and only one rf photon is generated on demand from every @xmath0-pulse excitation . however , the photon extraction efficiency needs to be drastically improved for it to become a deterministic single - photon source . figure@xmath32a shows a linear - log plot of the pulsed rf ( the sharp central line ) together with the residual laser leakage ( the broadband feature fitted by the red line ) monitored on a spectrometer . taking advantage of the huge linewidth mismatch between the rf signal and the laser background , we pass the rf through an etalon which has a bandwidth of @xmath620ghz much wider than that of the rf photons and much narrower than that of the pulsed laser to further suppress the excitation laser background . this results in a clean rf spectrum as shown in the inset of fig.@xmath32a , with an improvement of the signal to background ( including the detector dark counts ) ratio from 20 to 357 at @xmath0-pulse excitation . for a range of laser powers , the signal to background ratio is extracted and plotted in fig.@xmath32b . -pulse excitation is obtained with reasonable quality . our current work only focuses on the @xmath0-pulse regime . ( b ) intensity - correlation histogram of the rf emission from the qd under pulsed _ s_-shell excitation obtained using a hanbury brown and twiss - type setup . the second - order correlation @xmath2 is calculated from the integrated photons counts in the zero time delay peak divided by the average of the adjacent six peaks , and its error ( 0.002 ) which denotes one standard deviation , is deduced from propagated poissonian counting statistics of the raw detection events.,scaledwidth=49.0% ] a typical example of high - resolution spectra of the pulsed rf measured using a fabry - prot scanning cavity is shown in fig.@xmath32c . it shows a pronounced deviation from the lorentzian lineshape obtained from cw excitation as shown in fig.@xmath3s2 , and can be fitted with a voigt profile with a homogeneous linewidth of 0.4(1)ghz ( corresponding to t@xmath7=0.7(2)ns ) and an inhomogeneous linewidth of 1.0(1)ghz . the spontaneous emission lifetime for this qd is measured to be t@xmath8=0.41(2)ps ( see fig.@xmath3s3 ) , and we estimate the pure dephasing time t@xmath9=5.7@xmath10ns . the gaussian component in this voigt profile could potentially be caused by spectral diffusion owing to pulsed - laser - induced charge fluctuations in the vicinity of the qd ( trapping and untrapping of charges in nearby defects and impurities ) @xcite . the inhomogeneous linewidth varies for different qds and typically shows an increase at larger excitation power ( see fig.@xmath3s3 ) , which is in qualitative agreement with previous investigations of light - induced spectral diffusion @xcite . to perform pulsed two - photon interference , we adopt a similar experimental configuration ( see fig.@xmath33a ) as in ref.@xmath3@xcite . each excitation laser pulse , originally separated by @xmath1112.5ns , is further split into two pulses with a 2-ns delay . thus , every @xmath1112.5ns , the qd is excited twice , generating two successive single rf photons . the output rf photons are then fed into an unbalanced mach - zehnder interferometer with a 2-ns path - length difference ( fig.@xmath33a ) . the two outputs of this interferometer are detected by single - mode fiber - coupled single - photon counters , and a record of coincidence events is kept to build up a time - delayed histogram ( for more details see fig.@xmath3s4 ) . figure@xmath33(b ) and ( c ) show the central cluster of the histogram when the two @xmath0-pulse excited single photons , before recombining in the last beam splitter , are prepared in cross and parallel polarization states respectively . the five peaks , from left to right , corresponds to the cases where the two photon arrives at the beam splitter with a time delay of -4ns , -2ns , 0ns , 2ns , and 4ns , respectively . for distinguishable photons with different polarization , the expected peak - area ratio equals 1:2:2:2:1 , which is in good agreement with fig.@xmath33b . if two perfectly indistinguishable photons are superposed on a beam splitter , they will always exit the beam splitter together through the same output port , leading to a zero coincidence rate the hom dip @xcite which can not explained by classical optics . figure@xmath33c shows a strong suppression of the coincidence counts at zero delay when the two incoming photons are prepared in the same polarization state . quantitative evaluation ( see the caption of fig.@xmath33 for details ) shows that the probability of the two photons to exit the same channel in a 2-photon fock state ( bunching ) is 95.4@xmath1 . this corresponds to a raw two - photon hom interference visibility of 0.91(2 ) . taking into account the residual two - photon emission probability @xmath2 , and the optical imperfections of our interferometric setup which are independently measured , @xmath12 and @xmath13 , where @xmath14 , @xmath15 are the reflectivity and transmitivity of the beam splitter and @xmath16 is the first - order interference visibility of the mach - zehnder interferometer tested with a cw laser , we obtain corrected degrees of indistinguishability to be 0.97(2 ) . the visibility can be further increased slightly by decreasing the excitation laser power . on another qd , we test the hom interference with @xmath0 , 0.72@xmath0 and 0.41@xmath0 pulse excitation and observe visibilities of 0.96(6 ) , 0.97(6 ) and 0.99(4 ) , respectively ( see the data in fig.@xmath3s4 ) . taken together , these are to date the highest visibilities reported for qd - based single - photon sources . these results demonstrate that the solid - state pulsed rf single photons in quick succession are highly indistinguishable to a level comparable to the best results from those well - developed systems such as parametric down - conversion @xcite , trapped atoms and ions @xcite . the high - visibility results indicate a reduction of the fast dephasing and an elimination of the emission time jitter associated with the pulsed rf , compared to the previous incoherent excitation methods . the pure dephasing time t@xmath9=5.7@xmath17ns is considerably larger than the 2ns and thus should have little effect on the visibility . the spectral diffusion ( as shown in fig.@xmath32c ) should also happen at a time scale much longer than the 2-ns separation , which is consistent with previous experiments @xcite . we now demonstrate how the on - demand rf single photons can be utilized to implement a quantum controlled - not ( cnot ) gate . the quantum cnot gate is a fundamental two - qubit logic gate . if the control qubit is in logic @xmath18 , nothing happens to the target qubit , whereas if the control qubit is in logic @xmath19 , the target qubit will flip ( @xmath20 , @xmath21 ) . the photonic cnot gate is a basic building block for quantum computing and has been demonstrated many times with down - converted photons @xcite , and very recently , with _ p_-shell excited single photons from qds @xcite . we prepare two input qubits encoded in the polarization states of the pulsed rf single photons @xmath22 and @xmath23 , where @xmath24(@xmath25 ) denotes horizontal@xmath3(vertical ) polarization and is used to encode @xmath26 . the two inputs are then fed into the optical circuit for the cnot operation as shown in fig.@xmath34a . the key element in this optical network is a partial polarizing beam splitter ( _ p_-pbs ) which has a transmission of 1(1/3 ) and a reflectivity of 0(2/3 ) for the @xmath24(@xmath25 ) photons . when the two single photons are superimposed on the _ p_-pbs as shown in fig.@xmath34a , and if one and only one photon leaves through each output channel , the composite state of the two output photons can be written as : @xmath27 the first term corresponds to the case in which both input photons are @xmath28 and fully transmitted . the second and third terms correspond to the cases where one photon is in @xmath28 and fully transmitted while the other photon is in @xmath29 and partially ( 1/3 ) transmitted . it is most important to note the last term @xmath30 , where the resulting minus sign of the probability amplitude ( @xmath311/3 ) is due to the quantum interference between two indistinguishable paths , both photons are transmitted ( @xmath32 ) or reflected ( @xmath33 ) , which requires the indistinguishability of the single photons . next , we swap the @xmath24 and @xmath25 polarizations in eqn.[[1 ] ] using half - wave plates and pass the two photons through two other _ p_-pbss to compensate the unbalanced coefficient ( see fig.@xmath34a ) , and we can obtain @xmath34 this effectively realizes a controlled phase - flip gate with a success probability of 1/9 . finally , after applying two additional hadamard rotations , it can be transformed into the cnot gate ( see the caption of fig.@xmath34a and ref . @xcite for more details ) . we experimentally evaluate the performance of the quantum cnot gate using an efficient method proposed by hofmann @xcite . to show the quantum behaviour of the cnot gate , it is tested for different combinations of input - output states using complementary bases , that is , in both the computational basis ( @xmath35 ) and their linear superpositions ( @xmath36 ) , which are refereed to as the @xmath37 and @xmath38 basis using the pauli matrix language respectively . in the @xmath37 basis , the cnot is expected to flip the target qubit if the control qubit is in logic 1 . interestingly , in the @xmath38 basis , the target and control qubits are reversed : the control qubit will flip if the target qubit is logic 1 . the measurement results of the input - output probabilities of the cnot gate in the @xmath37 basis and in the @xmath38 basis are shown in fig.@xmath34b and fig.@xmath34c respectively . the fidelity of the cnot operation , defined as the probability of obtaining the correct output averaged over all four possible inputs , is in the @xmath37 basis : @xmath39 , and in the @xmath38 basis : @xmath40 . these two complementary fidelities , @xmath41 and @xmath42 , are sufficient to give an upper and a lower bound for the full quantum process fidelity @xmath43 of the gate by @xmath44 . thus , here we have @xmath45 . the process fidelity is directly related to the quantum entangling capability of the cnot gate , that is , the cnot gate can produce entangled states from unentangled input states @xcite . here , the @xmath43 well surpasses the the threshold of 0.5 , which is sufficient to confirm the entangling capability of our cnot gate . in this work , we have demonstrated the on - demand generation of near background - free ( @xmath1199.7@xmath1 purity ) and highly indistinguishable rf single photons , from a quantum dot in a planar microcavity driven by resonant @xmath0 pulses . using two rf photons emitted in 2-ns succession , non - postselective hom two - photon interference has revealed near - unity visibilities ( @xmath1197% ) , and a quantum cnot gate with entangling capability has been successfully demonstrated . such a pulsed rf single - photon source may open the way to new interesting experiments in quantum optics and quantum information . with the high degree of indistinguishability of the rf photons shown here , they can be used to realize various optical quantum computing algorithms @xcite , interference of multiple photons @xcite , and the on - demand generation of photonic cluster state from a single qd @xcite . in parallel , the rf spectra of a two - level system under strong pulsed laser excitation which are expected to exhibit novel features beyond the mollow triplet @xcite is in itself a subject worth studying . a natural extension is to realize non - postselective high - visibility quantum interference between two pulsed rf single photons from separate qds @xcite . based on this , it is possible to entangle remote , independent qd spins @xcite . we note that although the relatively slow spectral diffusion and pure dephasing does not affect the two - photon interference in our present work due to the 2-ns time separation of the photons , it will limit the degree of indistinguishability for photons from independent qds . for future experiments , gate - controlled qds could be used to reduce the spectral diffusion . alternatively , spectral filtering at the expense of photon rate may be needed . for quantum information applications , the photon extraction efficiency is a critical issue . so far , we have obtained @xmath0-pulse excited single photons with an overall collection efficiency of 1.3% reaching the single - photon detector . the photon extraction efficiency can be improved , for example , by embedding the qds in micropillars or photonic crystal cavities @xcite . large purcell effects from these microcavities can be helpful to efficiently funnel the spontaneous emission into a guided mode , to further mitigate the dephasings @xcite , and increase the pulse repetition rate to tens of ghz . lastly , it is important to note that in the previous pulsed above - bandgap or _ p_-shell excitation experiment , the photon coherence time had to be much larger than the incoherent carrier relaxation time jitter ( about tens of ps ) in order to obtain a good two - photon interference visibility @xcite , which fundamentally put a limit on the radiative lifetime shortening through the purcell effect . we emphasize that the true resonant , time - jitter - free , pulsed rf technique developed here has no such limitation and can be fully compatible with large purcell factors to be implemented in the future . 10 [ 1]`#1 ` [ 2][]#2 pan , j .- w . _ et al . _ multi - photon entanglement and interferometry . _ rev . mod . phys . _ * 84 * , 777 - 838 ( 2012 ) . kok , p. _ et al . _ linear optical quantum computing with photonic qubits . _ rev . mod . phys . _ * 79 * , 135 - 174 ( 2007 ) . obrien , j. l. , furusawa , a. & vuckovic , j. , photonic quantum technologies . _ nature photonics _ * 3 * , 687 - 695 ( 2009 ) . kimble , h. j. the quantum internet . _ nature _ * 453 * , 1023 - 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photon source for distributed quantum networking . _ phys . rev . lett . _ * 89 * , 067901 ( 2002 ) . mckeever , j. _ et al . _ single photons from one atom trapped in a cavity . _ science _ * 303 * , 1992 - 1994 ( 2004 ) . beugnon , j. _ et al . _ quantum interference between two single photons emitted by independently trapped atoms . _ nature _ * 440 * , 779 - 782 ( 2006 ) . maunz , p. _ et al . _ quantum interference of photon pairs from two remote trapped atomic ions . _ nature phys . _ * 3 * , 538 - 541 ( 2007 ) . obrien , j. l. , pryde , g. j. , white , a. g. , ralph , t. c. & branning , d. demonstration of an all - optical quantum controlled - not gate . _ nature _ * 426 * , 264 - 267 ( 2003 ) . langford , n. k. _ et al . _ demonstration of a simple entangling optical gate and its use in bell - state analysis . _ phys . rev . lett . _ * 95 * , 210504 ( 2005 ) . kiesel , n. _ et al . _ linear optics controlled - phase gate made simple . _ phys . rev . lett . _ * 95 * , 210505 ( 2005 ) . okamoto , r. _ et al . _ demonstration of an optical quantum controlled - not gate without path interference . _ phys . rev . lett . _ * 95 * , 210506 ( 2005 ) . pooley , m. a. _ et al . _ controlled - not gate operating with single photons . _ appl . . lett _ * 100 * , 21103 ( 2012 ) . hofmann , h. f. complementary classical fidelities as an efficient criterion for the evaluation of experimentally realized quantum operations . _ phys . rev . lett . _ * 94 * , 160504 ( 2005 ) . fattal , d. , diamanti , e. , inoue , k. & yamamoto , y. quantum teleportation with a quantum dot single photon source . _ phys . rev . lett . _ * 92 * , 037904 ( 2004 ) ; scholz , m. , aichlele , t. , ramelow , s. & benson , o. deutsch - jozsa algorithm using triggered single photons from a single quantum dot . _ phys . rev . lett . _ * 96 * , 180501 ( 2006 ) . moelbjerg , a. , kaer , p. , lorke , m. & mrk , j. resonance fluorescence from semiconductor quantum dots : beyond the mollow triplet . _ phys . rev . lett . _ * 108 * , 017401 ( 2012 ) . _ acknowledgement _ : we thank y. yu , z. xi , j. bowles , k. chen , c. matthiesen , x .- l . wang , l .- j . wang , n. vamivakas , and y. zhao for helpful discussions . this work was supported by the national natural science foundation of china , the chinese academy of sciences and the national fundamental research program ( under grant no : 2011cb921300 , 2013cb933300 ) , and the state of bavaria . m.a . acknowledges the cas visiting professorship . c .- y.l acknowledges the anhui nsf and youth qianren program . * author contributions * : m.a . , c .- y.l . and j .- w.p . conceived and designed the experiments , c.s . , s.h . , and m.k . grew and fabricated the sample , y .- m.h . , y.h . , y .- j.w . , d.w . , m.a . , and c .- y.l . carried out the optical experiments , y .- m.h . , s.h . , c .- y.l . , and j .- w.p . analyzed the data , c .- y.l . wrote the manuscript with input from all authors , s.h . , c .- y.l . and j .- w.p . guided the project . * additional information : * the authors declare no competing financial interests . correspondence and requests for materials should be addressed to c .- y.l . ( [email protected] ) or s.h . ( [email protected] ) or j .- w.p . ( [email protected] ) .

1303.4058

efficient mechanisms to generate coherent superpositions of quantum states are central to a rich variety of applications in modern quantum physics . quantum logic gates , i.e. the key components of a quantum computer @xcite , rely on superpositions of two degenerate quantum states as qubits @xcite . quantum tunnelling and localization in a double well potential can be controlled @xcite by techniques that require preparation of coherent superpositions . numerous authors have noted that nonlinear optical processes , e.g. resonantly enhanced frequency mixing in atomic vapours , can be significantly improved @xcite by preparing the nonlinear optical medium in a coherent superposition of nondegenerate quantum states . whereas numerous techniques exist to prepare coherent superpositions of degenerate states , as needed for qubits in quantum computing , nonlinear optics still hold challenges . for example , frequency conversion to short wavelength radiation involves high - lying states , requiring excitation by multi - photon transitions . procedures based on adiabatic passage driven by coherent interactions @xcite provide reliable and robust tools for the creation of superpositions . unlike diabatic techniques , which rely on precise control of individual pulses , adiabatic processes are insensitive to small variations of pulse duration and peak intensity . high - intensity laser systems , commonly used for efficient frequency conversion , exhibit fluctuations in intensity and other parameters . thus robust techniques are most appropriate to support nonlinear optical processes driven by the high - intensity lasers . in what follows we will discuss adiabatic passage in a three - state system involving two near - resonant laser pulses . we assume that initially the atom is in the ground state 1 of energy @xmath0 and that the first laser field , of frequency @xmath1 , links this with an excited state 2 of energy @xmath2 while the second field , of frequency @xmath3 , links this state with a final target state 3 of energy @xmath4 we assume that the two fields , though near resonant with the specified transitions , are far from resonance with any other transition . the relative ordering of the third - state energy @xmath5 is not significant ; we shall assume that it lies above @xmath2 as is appropriate for application to nonlinear optics , so that the linkages form a ladder . figure [ scheme_config ] shows this linkage pattern , along with spectroscopic labels appropriate to implementation in mercury . , the rabi frequencies @xmath6 and the stark shift @xmath7 . dotted lines show transitions for four - wave mixing , through a nonresonant state 4 . to the left are the spectroscopic labels appropriate to implementation with mercury . ] our modeling is based upon the three - state time - dependent schrdinger equation in the rotating wave approximation ( rwa ) @xcite , for which the hamiltonian is @xmath8.\ ] ] the elements of this rwa hamiltonian are the detunings _ 2 = ( e_2 - e_1)/- _ 1 , _ 3 = ( e_3 - e_2)/- _ 2 . and the pulsed rabi frequencies @xmath9 and @xmath10 that parametrize the laser - induced excitation interaction with the dipole - transition moments @xmath11 of the transition between states @xmath12 and @xmath13 and the electric field envelopes @xmath14 and @xmath15 . a third laser , nonresonant with either of these transitions , induces dynamic stark shifts of the energies . for the ladder - like linkage the shift is expected to be largest for state 3 ; we denote that shift as @xmath7 and we neglect shifts of the other states . in the rwa the statevector has the expansion ( t ) = c_1(t ) _ 1 + c_2(t ) e^-i_1 t _ 2 + c_3(t ) e^-i(_1+_2 ) t _ 3 where @xmath16 is a unit vector representing quantum state @xmath12 . our principal objective is to create , for times @xmath17 later than the conclusion of a pulse sequence at @xmath18 , an equal - probability superposition of states @xmath19 and @xmath20 , specifically @xmath21 , \qquad t > t_f\ ] ] where @xmath22 is a time independent phase defining the relative sign of the superposition . this particular superposition , with equal probabilities @xmath23 of the two constituent nondegenerate states , provides the basis of a technique referred to as `` nonlinear optics at maximum coherence '' @xcite , that substantially improves the efficiency of nonlinear frequency conversion processes . the following section reviews how the efficiency of four - wave mixing is enhanced by preparing the atomic medium in the coherent superposition of eqn . ( [ final_super ] ) with equal probability amplitudes . we review in section [ sec - background ] two techniques recently suggested to generate coherent superpositions of this form : ( i ) fractionally - completed stimulated raman adiabatic passage ( f - stirap ) and ( ) half - completed stark - chirped rapid adiabatic passage ( half - scrap ) . section [ sec - sacs ] describes our proposed method , stark - assisted coherent superposition ( sacs ) , in which two near - resonant fields , accompanied by a laser - induced ac stark shift , produce adiabatic passage into the desired superposition . section [ sec - simulation ] illustrates different techniques by presenting simulations of excitation of mercury vapor , a medium of significant interest for applications in frequency conversion . the starting point for nonlinear optical phenomena is the wave equation describing the propagation of the electric field @xmath24 through matter . the effects of matter are incorporated into a polarization @xmath25 that serves as an inhomogeneous term in the wave equation , & & ( t , ) = _ 0 ^2 t^2 ( t , ) . [ 12.2 - 25 ] we idealize the medium as a uniform distribution of identical motionless atoms , of number density @xmath26 . then the polarization is the @xmath26 time the expectation value @xmath27 of the single - atom dipole moment : ( t , ) = ( t , ) . in the following we idealize the laser fields as plane waves traveling along the @xmath28 axis . we suppose that there are several such fields , each characterized by a common polarization vector @xmath29 but different carrier frequency @xmath30 . we express the resulting electric vector as ( t , ) = _ k _ k(t ) ( _ k t + _ k ) where @xmath31 the presence of the laser fields causes changes of the material . with these changes come changes in the individual dipole expectation values and hence of the polarization @xmath32 we will discuss a four - wave mixing process in an atomic gas , as are typical for the generation of short - wavelength radiation , i.e. vacuum- or extreme - ultraviolet . for application to four - wave mixing in mercury , the three states depicted in fig . [ scheme_config ] are : ( 1 ) , ( 2 ) and ( 3 ) . for the four - wave mixing processes , two fields at frequencies @xmath1 and @xmath3 create an induced dipole moment . this dipole moment , together with a third field of frequency @xmath33 far from resonance with any state @xmath34 combine to produce a dipole moment that varies at the frequency @xmath35 . following the definition of the expectation value of the dipole moment @xmath36 it can be expressed as a function of the probability amplitude ( t ) & = & 2 . the last term serves as the source of an electric field at the frequency @xmath37 for a detuning @xmath38 much larger than the rabi frequency @xmath39 one can adiabatically eliminate the state @xmath40 @xcite . the amplitude @xmath41 is then given by _ 4(t)= _ 3(t)+ _ 1(t ) . when @xmath42 is small compared to @xmath43 , as it is initially , the source term of the electric field at frequency @xmath44 is [ source_term4 ] _ 4(t ) = c^_3(t ) _ 1(t ) _ 3(t ) e^i_4 t where @xmath45 , for @xmath46=(1,2,3,4 ) , is defined from the expression ( t)=2 . these successive equations show that the induced polarization , and hence the efficiency of any subsequent nonlinear optical process , depends on the product @xmath47 , i.e. on the coherence , established between the ground and the excited state . laser fields typically exhibit inhomogeneities in time , space or phase . if there is a dynamical contribution to the phase , these inhomogeneities will , when averaged , tend to reduce any coherence to nearly zero . to generate the desired radiation at the frequency @xmath44 it is thus crucial that the relative phase of the components @xmath48 and @xmath49 in the superposition state of eqn . ( [ final_super ] ) does not depend on the time integrated rabi frequencies , as dynamical phases do . if this condition is fulfilled , the source term of eqn . ( [ source_term4 ] ) reaches a maximum value when the atomic coherence is maximum , i.e. when @xmath50 . the polarization , the efficiency of four - wave mixing and thus the intensity of the generated radiation at frequency @xmath44 will be enhanced , if the atomic medium is prepared in maximum coherence ( [ final_super ] ) . as will be discussed below , to produce this superposition we make use of adiabatic states @xmath51 and adiabatic eigenvalues , @xmath52 , defined as instantaneous solutions to the eigenvalue equation of the rwa hamiltonian h(t ) _ k(t ) = _ k(t ) _ k(t ) . prior to describing the proposed technique , we comment on two common adiabatic techniques , first intended for complete population transfer but later applied to the preparation of superposition states . figure [ scheme_comparison ] illustrates the three schemes . stimulated raman adiabatic passage ( stirap ) @xcite is well established as an appropriate tool to drive complete population transfer in a three - state lambda - type system @xcite . with suitable modification @xcite , the technique provides a means of creating superpositions of initial and final states . essentially the evolution is adiabatically stopped just when half of the population is transferred . we refer to this as fractional stirap ( f - stirap ) . the basic stirap procedure is based on a lambda linkage in which @xmath53 , i.e. a stimulated raman process ( see fig . [ scheme_comparison]a ) . it works also in a ladder linkage if the lifetime of the uppermost state is much longer than the interaction time , i.e. the pulse duration which we will assume in the following . the process relies on interaction of the system with two laser pulses , termed stokes and pump , which produce overlapping but not coincident interactions , parameterized by rabi frequencies , @xmath54 and @xmath55 . the stokes pulse , linking states 2 and 3 , precedes the pump pulse , linking states 1 and 2 . adiabatic evolution requires that the time integral of the rabi frequencies is much larger than 1 . under these conditions ( and two - photon resonance , @xmath56 ) the statevector remains at all times aligned with one particular adiabatic state , the dark state or population trapping state , @xmath57\ ] ] where @xmath58 is the difference between stokes and pump phases . at the beginning of the interaction in f - stirap , i.e. when the stokes is strong and the pump pulse is negligible , the adiabatic state @xmath59 coincides with the initial state @xmath19 . stirap proceeds to completion when the pump field is strong and the stokes field is negligible , i.e. at the end of the interaction . the adiabatic state @xmath59 is then aligned with @xmath20 , and complete population transfer has occurred . by contrast , in f - stirap the pulses need to be terminated simultaneously ( see fig . [ scheme_comparison]b ) . once the pump pulse has produced the desired fractional transfer , both pulses must diminish simultaneously while maintaining a fixed ratio of the two rabi frequencies , @xmath60 as with stirap , it is necessary to maintain the two - photon resonance condition @xmath61 . when the mixing angle @xmath62 remains constant as the pulses diminish , the result is the statevector ^()(t ) = _ 1- e^-ie^-i(_p+_s)t _ 3 , t > t_f . ( note the phases that appear here , a consequence of the steady rotation of the coordinates of the adiabatic states . ) the desired superposition of the ground and target states , eqn . ( [ final_super ] ) , requires @xmath63 i.e. equal rabi frequencies as the pulse sequence concludes . because f - stirap , in contrast to regular stirap , demands laser pulses of specific temporal shape and duration , it is difficult to implement experimentally in a system having a ladder - type configuration . two - photon excitations in ladder systems , involving absorption of two photons from one radiation field , permit population transfer to high - lying excited states without the need for lasers with very short wavelength . however , because such two - photon interactions involve nonresonant couplings to intermediate states ( which can be summarized in a virtual state ) , strong two - photon excitation is inevitably accompanied by dynamic stark shifts . such shifts , when produced by nonresonant fields , can be put to good use as a means of sweeping the two - photon detuning through resonance and thereby inducing stark chirped rapid adiabatic passage ( scrap ) @xcite . like stirap , scrap was initially developed as a technique for complete population transfer . subsequently , it has been used to create transient superpositions for use in enhancing frequency conversion @xcite . a modification of scrap has been suggested for the preparation of persistant superposition states , the so - called half - scrap process @xcite . the half - scrap technique ( see fig . [ scheme_comparison]c ) employs an intense pump laser field that couples , e.g. via a two - photon transition , a ground state @xmath19 and a target state @xmath20 . there is no resonance with an intermediate state . the interaction that links states @xmath19 and @xmath20 is represented by an effective two - photon rabi frequency @xmath64 proportional to the intensity of the pump laser . the pump field induces dynamic stark shifts of the energies @xmath65 . an additional stark - shifting laser pulse induces further energy shifts , which should be larger than the stark shifts induced by the pump laser . experimentally this is easy to implement , required a strong , fixed - frequency stark - shifting laser is available . typically the polarizability of the upper state greatly exceeds the polarizability of the ground state , thus only the stark shifts of the energy @xmath5 need be considered . the resulting detuning of the pump laser from two - photon resonance is _ ( t ) = ( e_3 - e_1)/- 2 _ p - _ s(t ) - _ p(t ) where @xmath66 is the pump - field carrier frequency , @xmath67 is the ( small ) energy shift induced by the pump field and @xmath68 the ( large ) shift induced by the stark - shifting field . under appropriate conditions the combination of the two pulses will produce , apart from an overall phase factor , the superposition ^(t ) = _ 1-e^-i e^-2i_p t _ 3 , t > t_f where the mixing angle @xmath69 , defined by the equation @xmath70 , is here evaluated at the time @xmath18 , and where @xmath58 is the phase of the pump field . the half - scrap process requires that the following conditions be met : ( i ) the stark - shifting pulse must precede the pump pulse , ( ) the adiabatic condition , @xmath71 must be fulfilled and ( iii ) at the end of the interaction the effective two - photon rabi frequency @xmath72 must be much larger than the effective detuning @xmath73 although the half - scrap technique relies on an adiabatic process , and hence offers the robustness of any adiabatic process , it has two practical disadvantages . as noted , the pump - induced two - photon coupling @xmath64 is accompanied by a dynamical stark shift included in @xmath74 these two interactions are both proportional to the intensity of the pump pulse . the pump laser - induced dynamic stark shift has to be compensated by an additional static two - photon detuning , such that at the end of the interaction @xmath75 in contrast to techniques utilizing one - photon transitions , half - scrap as described above requires a pump laser of large intensity to drive a two - photon transition sufficiently strongly . we propose an alternative technique to prepare the coherent superposition of eqn . ( [ final_super ] ) . the mechanism , termed stark - assisted coherent superposition ( sacs ) , is based on adiabatic passage and uses the dark state of eqn . ( [ dark_state_stirap ] ) . in contrast to stirap and f - stirap two simultaneous , rather than delayed radiation fields with a similar temporal shape are applied . as in scrap , a non - resonant stark - shifting pulse varies the energy @xmath5 by @xmath76 ( see fig . [ scheme_comparison]d ) . our starting point is the rwa hamiltonian of eqn . ( [ hamiltonian-3state ] ) . in the following we assume the two rabi frequencies to be equal , @xmath77 and we require two - photon resonance , @xmath78 without loss of generality we assume that the stark shift is negative @xmath79 thus we deal with the rwa hamiltonian @xmath80.\ ] ] because we focus on adiabatic evolution , the statevector @xmath81 for this system should at all times be aligned with an adiabatic state @xmath51 of this hamiltonian . the eigenvalues @xmath52 and eigenvectors @xmath51 are readily obtained for any time @xmath17 by numerical means . in two important limiting cases they have simple properties that underly our proposed method . when the interaction is absent , i.e. @xmath82 , but a stark shift ( and positive @xmath83 ) is present , the three adiabatic states align with the diabatic states @xmath16 . the eigenvalues and eigenstates are then & _ - = -_s , & _ -(t ) = = _ 3 , + & _ 0 = 0 , & _ 0(t ) = = _ 1 , + & _ + = _ 2 , & _ + ( t ) = = _ 2 . when , in addition , no stark shift is present , then the adiabatic states @xmath84 and @xmath85 are degenerate and can be taken as any superposition of states @xmath19 and @xmath20 ; we take these to be the choices presented here . in the other limiting case , when no stark shift is present , @xmath86 , the interaction @xmath87 moves the adiabatic states away from alignment with the diabatic states . the eigenvalues are then _ - & = & , + _ 0 & = & 0 , + _ + & = & . of particular interest is the null - eigenvalue adiabatic state ; i.e. the `` dark state '' superposition _ 0(t ) = = . [ dark_state_sacs ] the proposed excitation process can be understood by observing the three adiabatic eigenvalues @xmath52 as they vary due to changes of the rwa hamiltonian . figure [ surface_starap ] displays these three energies , evaluated numerically , as a function of the two parameters @xmath88 and @xmath7 that define this hamiltonian . similar figures have earlier been used to explain adiabatic processes ; the topology of the surfaces can exhibit conical intersections and avoided crossings @xcite . and of the stark shift @xmath7 ( in units of @xmath83 ) . the path represented by the ( blue ) curve on the intermediate surface gives values of the couple ( @xmath87 , @xmath7 ) that connect the initial state @xmath19 to the superposition of eqn . ( [ final_super ] ) . ] the two - photon resonance condition ensures that initially , when no interaction is present , two of the adiabatic states , @xmath85 and @xmath84 share the eigenvalue zero . a large dot on the figure indicates this point in parameter space ( the initial system point that characterizes the system in the parameter space as the initial statevector characterizes it in the time space ) . at this time the remaining eigenstate , @xmath89 , aligned with the bare state @xmath90 , has the eigenvalue @xmath83 , also denoted by a dot in the figure . when @xmath82 the three adiabatic states align uniquely with the diabatic basis states , @xmath16 . the sacs process can be described by a path on a surface in the parametric space of fig . [ surface_starap ] . at the beginning no fields are present ; the statevector is aligned with state @xmath19 and with the adiabatic state @xmath85 . subsequently the detuning @xmath7 grows , while the rabi frequency remains null . in the vertical plane at @xmath91 the energy line with constant value 0 is associated with the adiabatic state @xmath85 , while the constant value @xmath83 belongs to adiabatic state @xmath89 . the adiabatic state @xmath84 has an energy that varies linearly with the stark shift @xmath7 . because there is no interaction to alter the composition of the adiabatic states , the statevector remains aligned with the initial state @xmath19 during the growth of the stark - shifting pulse . once @xmath7 has reached a satisfactory value , the interaction fields @xmath87 are applied . thereafter the system point moves , on the energy surface of @xmath85 , away from the plane at @xmath82 , along the path , shown as a dark ( blue ) line in fig . [ surface_starap ] . eventually the stark shift ceases , and the system point evolves in the vertical plane of @xmath86 . as the driving field diminishes , the system point moves towards the origin . along this path there is an interaction @xmath87 , and thus the adiabatic state @xmath85 is a superposition of the two diabatic states [ see eqn . ( [ dark_state_sacs ] ) ] . the population is thus shared between the states @xmath19 and @xmath20 . we will show , in section [ numerical_simulation_starkstirap ] , that the relative composition of this superposition can be altered by changing either the detuning @xmath83 or the ratio of the two rabi frequencies @xmath92 and @xmath93 at the conclusion of the pulse sequence . an earlier paper @xcite discussed similar adiabatic evolution , but with a delay between the two driving laser pulses . in that work the stark - shifting pulse coincided with one of the driving fields @xcite . under these conditions , the system is divided between two adiabatic states of the hamiltonian of eqn . ( [ hamiltonian_starap ] ) . the relative phase of the resulting superposition depends on the time integral of the difference between the two associated eigenvalues . many applications require the control of this dynamical phase , i.e. the time integral of the rabi frequency must then be controlled . we have here considered the case where the driving pulses are synchronized and delayed with respect to the stark pulse . in contrast to the sequence discussed in ref . @xcite , the dynamics of sacs follows a unique eigenstate and the relative phase of the superposition is reduced to the sum of the two driving frequencies . as can be seen from viewing the energy surfaces of fig . [ surface_starap ] , many paths in the two - dimensional parameter space link the initial state with the desired final - state superposition . it is only required that the evolution be adiabatic and that the system remains on the surface associated with the adiabatic state @xmath85 . in general , the adiabaticity requirement implies that the time integrated rabi frequencies are much larger than 1 . this , together with the requirement that the transition to the intermediate state is near resonant , @xmath94 , leads to the condition @xmath95 where @xmath96 is the duration ( fwhm ) of the driving and stark pulse , assuming gaussian temporal shapes . as was done earlier @xcite , we fix the detuning @xmath83 and minimize the non - adiabatic losses by adjusting the peak values of the rabi frequency @xmath97 , the stark - shift @xmath98 and the delay between the pulses . that earlier work , modeling a two - level system driven by a chirped pulse , found that the non - adiabatic losses are minimized ( for fixed pulse shape and peak rabi frequency ) when the system point follows a trajectory of constant energy ( i.e. a level line ) in the adiabatic energy space generated by the rabi frequency and the detuning . in a temporal representation , the non - adiabatic couplings between two adiabatic eigenstates are minimized when the associated eigenvalues exhibit parallel temporal evolution . following ref . @xcite , we used the criteria of parallel evolution to find laser parameters that minimized the non - adiabatic couplings between the energy of the adiabatic state followed by the system and the closest alternative adiabatic energy . ) as a function of @xmath88 and @xmath7 ( in units of @xmath83 ) . the white line is an ideal trajectory that would optimize adiabaticity during population transfer . the dark trajectories are associated with the realisation of the transfer with sine - squared pulses . in contrast to the dotted ( red ) trajectory , the dashed ( blue ) trajectory follows approximately the optimal level line . ] figure [ contour_starap_adia ] shows the difference of the two lowest energy surfaces of fig . [ surface_starap ] . this contour plot exhibits level lines continuously connected to state @xmath19 and to the desired superposition state . a path in the parameter space is represented by a closed loop starting and ending at @xmath99 , @xmath86 . the white trajectory follows a level line . it is thus associated with a population transfer process which minimizes non - adiabatic losses . this trajectory will lead to parallel lines in the temporal evolution of eigenenergies for @xmath100 and @xmath101 an inspection of the ideal loop gives the optimized peak amplitudes @xmath102 for a given temporal duration @xmath103 we first chose the value of @xmath83 satisfying ( [ cond_adia_starap_delta ] ) . the conditions ( [ cond_mini_adiab ] ) give the values of the parameters @xmath104 and @xmath105 which minimize the non - adiabatic losses . the delay between the pulses , and their shapes , have to be chosen such that the system follows the ideal trajectory as closely as possible . we suggest a possible implementation of the proposed technique to the preparation of a coherent superposition that could enhance four - wave mixing in mercury vapor . the three states of interest are chosen from the degenerate energy levels , and , as indicated in fig . [ scheme_config ] . when the laser fields have definite polarizations , as we assume , the resulting magnetic - sublevel selection rule makes the excitation ladder a simple three - state system . table [ table_hg ] lists the relevant properties of these states . .spectroscopic values of the considered transitions in mercury . @xmath106 is the wavelength of the transition . here @xmath107 is the einstein coefficient for the spontaneous emission and @xmath108 is the dipole - transition moment . [ cols="^,^,^,^",options="header " , ] in the following simulations we assume laser pulses whose temporal shape follows a sine - squared pattern within one period . such pulses are very similar to the gaussian pulses often found in experiments , and they have the mathematical advantage of vanishing identically outside a finite pulse duration ; they have finite support . the use of such pulses simplifies the identification of temporal regions where the two interactions vanish , and for which the adiabatic states have exact analytic expressions . specifically , we used the profiles , @xmath109 , \quad \omega(t)=0 \quad \hbox{elsewhere } , \\ & & \nonumber \delta_s(t)=\delta_s^{\max } \sin\left(t / t\right)^2 \quad \hbox{for } \quad t\in[0 , \pi t ] , \quad \delta_s(t)=0 \quad \hbox{elsewhere},\end{aligned}\ ] ] with equal widths of @xmath110ns . from table [ table_hg ] we deduce the relation between the rabi frequencies and the laser intensities as _ 1^ [ ^-1 ] = 5.610 ^ 7 [ ^2 ] , _ 2^ [ ^-1 ] = 3.810 ^ 7 [ ^2 ] . for our simulations we chose the peak intensities as @xmath111 mw/@xmath112 and @xmath113 mw/@xmath112 for the transitions 1 and 2 . these choices give equal peak rabi frequencies of @xmath114 ns@xmath115 for both transitions . the delay between the driving pulses and the stark pulse is @xmath116 ns . the stark shift of state 3 is evaluated from the expression _ j summed over all the non - resonant intermediate states j. here @xmath117 and @xmath118 are the electric field envelope and the carrier frequency of the stark pulse respectively . the energy of state 3 shifts by a maximum of @xmath119 ns@xmath120 this shift can be achieve using a stark laser with the wavelength @xmath121 nm and with the intensity @xmath122 mw/@xmath123 we evaluate the stark shift , taking into account the relevant states @xmath124 and @xmath125 , as _ s^ [ ^-1]=-1885i_s [ ^2 ] . we neglect the shifts of states 1 and 2 , as is usually the case for lower - lying states . this approximation was confirmed from estimations of the stark shifts . these parameters have been chosen based on the following criteria . the pulse durations were fixed at @xmath110 ns . the fields 1 and 2 were detuned by @xmath126 to satisfy the condition ( [ cond_adia_starap_delta ] ) . the intensities were chosen such that @xmath127 . the stark shift satisfies @xmath128 to fulfill eqn . ( [ cond_mini_adiab ] ) . the delay between the stark and driving pulses was adjusted such that , in fig . [ contour_starap_adia ] , the trajectory of the system point in the parameter space ( blue dashed line ) follows as closely as possible the ideal level line ( white level line ) . at the beginning of the sequence , when the driving pulses 1 and 2 are still absent the populated adiabatic state coincides with the initial state @xmath129 at the end of the process , when the stark pulse is absent , this adiabatic state corresponds with the dark state of eqn . ( [ dark_state_stirap ] ) . because the rabi frequencies @xmath130 remain equal as they diminish , the populations @xmath131 and @xmath132 are both equal to 0.50 and the state of the system , for @xmath133 , is ( t ) with @xmath134s@xmath115 , corresponding to a wavelength of 157 nm . the optimization of the pulse parameters to minimize the non - adiabatic losses can produce a variety of superpositions . by changing the relative intensity of the pulses , and hence the relative peak rabi frequencies @xmath135 and @xmath136 ) , we cause the trajectory of the system point to deviate from a contour line ( red dotted line in fig . [ contour_starap_adia ] ) and the populations become @xmath137 and @xmath138 figure [ dyn_starap_hg ] shows results of numerical simulation of sacs in mercury . the upper frames show the pulse sequence , while the middle frames show the time varying adiabatic eigenvalues . the thick line marks the eigenvalue associated with the adiabatic state that coincides with @xmath19 before the pump pulses are present and with the dark adiabatic state @xmath85 , eqn . ( [ dark_state_stirap ] ) , after the stark pulse vanishes . the lower frame shows the populations . in this example , they evolve into a 50:50 superposition of states 1 and 3 . as a function of time . middle frame : eigenvalues versus time . lower frame : populations @xmath139 as a function of time ( @xmath140 , dotted line ; @xmath141 full thin line and @xmath142 full thick line),title="fig : " ] ( -80,75 ) ( -60,34 ) ( -104,28 ) ( -124,170 ) ( -40,185 ) a maximum coherence , once formed , can be used in any subsequent frequency mixing process . the atoms act like a local oscillator at the two - photon transition frequency @xmath143 . this can beat with an additional radiation field of frequency @xmath33 to generate new radiation fields , e.g. at the sum of all the contributing frequencies @xmath144 . when applied to four - wave mixing in mercury , the introduction of a tunable probe laser field with wavelength in the visible regime will generate radiation deep in the vacuum - ultraviolet spectrum . in contrast to conventional frequency mixing techniques , the maximum coherence maximizes the induced polarization in the medium , and thereby enhances the efficiency of the frequency mixing process . moreover , the coherence in the medium decays only as slowly as the excited state lifetime , and so the mixing process is also possible when the probe laser is delayed with respect to the driving fields . this feature has no counterpart in conventional nonlinear optics , which requires coincident radiation fields . the proposed sacs technique offers the possibility of constructing any superposition of states 1 and 3 with controllable weights by changing the two - photon detuning @xmath145 our analytical analysis and concomitant discussion assumed two - photon resonance . it is instructive to see the consequences of relaxing this condition . figure [ ctrl_weight_super_hg ] shows the population of states 1 and 3 after the interaction with the pulse sequence for various detunings @xmath145 the pulse parameters are identical to the ones of fig . [ dyn_starap_hg ] . when the two - photon detuning is far from zero and negative ( @xmath146 ) , the population is left in the ground state . in the opposite case , ( @xmath147 ) , there is complete transfer into state 3 . for intermediate detunings the populations are shared between states 1 and 3 with various weigths . when two - photon detuning is present additional phases appear in the expression of the superposition ( see @xcite ) . ( dotted line ) , and @xmath142 ( full line ) at the end of the pulse sequence as a function of the two - photon detuning @xmath145,title="fig : " ] ( -140,70 ) ( -140,34 ) the relative weights of the components @xmath19 and @xmath20 can be altered by changing either the two - photon detuning or the ratio of the rabi frequencies @xmath148 at the end of the pulse sequence . indeed , the superposition state created by the two pulses is exactly the stirap dark state , eqn . ( [ dark_state_stirap ] ) . from the latter expression , it is obvious that the rabi frequencies at the end of the interaction can be used as control parameters for the superposition of states @xmath19 and @xmath20 . the stark pulse of sacs allows successful implementation with a large variety of pulse shapes . the dark state in f - stirap , eqn . ( [ dark_state_stirap ] ) , connects state @xmath19 to the target superposition of eqn . ( [ final_super ] ) , as long as the pump and stokes pulses maintain a constant ratio of rabi frequencies as they vanish . in sacs , the stark pulse lifts the degeneracy of the adiabatic states , thereby permitting implementation with simultaneous driving pulses 1 and 2 of similar shape and width . figures [ contour_stirap_delay ] and [ ctrl_weight_super_hg ] illustrates this flexibility with simulations using gaussian pulses , _ p(t)&=&_p^e^-(t / t)^2 + _ s(t)&=&_s^e^-((t-)/t ) , + _ s(t)&=&_s^e^-((t--1)/t)^2 . [ gausspulses ] each of these figures depicts the populations of states 1 and 3 at the end of the sacs process as a function of the stokes peak amplitude and of the delay between the stokes and pump pulses . for fig . [ contour_stirap_delay ] the stark shifting pulse is absent ( the process thus corresponds to stirap ) . population transfer exhibits very clear rabi oscillations , as indicated by the curved bands . by contrast , the simulations of fig . [ ctrl_weight_super_hg ] include a stark shift . the contour patterns here are indicative of adiabatic passage , as contrasted with rabi oscillations . and of the delay @xmath149 between pump and stokes pulses for the f - stirap process . the stark field is here absent . the parameters are the following : the detunings @xmath150 are equal to @xmath151 . the pulse shapes are gaussian , as specified in eqn . ( [ gausspulses ] ) , with @xmath152 the four outset plots show pulse sequences for four couples of values ( @xmath153 , @xmath154 ) correspond to the dots ( a ) , ( b ) , ( c ) , and ( d ) . in those plots the pump field @xmath155 is a full line and the stokes field @xmath156 is a dashed line . ] . parameters are as in that figure , but there is a stark shift , @xmath157 in the outset frames the pump pulse @xmath155 is a full line , the stokes pulse @xmath156 is a dashed line and the stark shift @xmath7 is a dotted line . ] viewing fig . [ contour_stirap_delay ] one can identify two distinct regimes for the f - stirap process . for a small delay between the pump and stokes pulses , rabi oscillations of the population occur between the states 1 and 3 . for sufficiently large delay , the dynamics becomes adiabatic and there occurs an equal - weight superposition of states 1 and 3 . viewing fig . [ contour_starap_delay ] we see that when the stark shift is sufficiently large the rabi oscillations disappear and the dynamics becomes adiabatic even if there is no delay between pulses 1 and 2 . the weights of the superposition components then depend on the ratio of rabi frequencies @xmath148 at the end of the sequence . when a stark shift of the energy of state 3 occurs the two - photon resonance condition is generally not fulfilled , and the adiabatic state followed by the statevector does not coincide with the dark state of stirap , eqn . ( [ dark_state_stirap ] ) . under these circumstances the intermediate state @xmath90 is populated when both stark and driving pulses are present . for adiabatic transfer to succeed it is necessary that the lifetime of the intermediate state be much longer than the duration of the overlapping between pulses . the scrap technique , applied to excite high - lying states , is usually based on a two - photon transition with no resonantly driven intermediate state . thus it requires higher intensities than processes that rely on a combination of single - photon near - resonant excitations , as sacs does . to illustrate the main differences between sacs and half - scrap , we calculate the intensity of the driving field we would need to excite the transition 6@xmath158 - 7@xmath158 by half - scrap with a similar rabi frequency to the one used in the sacs technique . as above , the states 1 and 3 are taken to be 6@xmath158 and 7@xmath158 . a pump laser at wavelength 313 nm drives a two - photon transition between these two states . the effective two - photon rabi frequency is calculated as = _ j . in this sum we included only the intermediate states 6@xmath159 6@xmath160 7@xmath161 and 9@xmath161 . other states give only minor contributions . we calculate the effective two - photon rabi frequency to be ^ [ ^-1]= 37 i_p [ ^2 ] . the adiabatic condition @xmath162 can be fulfilled for a pulse duration of @xmath110 ns with a pump intensity of @xmath1631.3 gw/@xmath112 . the pump intensity needed for half - scrap is thus three orders of magnitude larger than the intensities of the pulses 1 and 2 used in the sacs technique for the same adiabatic condition @xmath164 on the other hand , with the scrap technique the intermediate states are far from resonance , and so they are never populated during the dynamics . four - wave mixing , and other nonlinear optics processes , are enhanced by preparing the nonlinear medium in a maximally coherent quantum state , thereby maximizing the induced polarization field . for such purposes it is necessary to create superpositions of nondegenerate quantum states . we have introduced an alternative adiabatic technique , sacs , for the preparation of coherent superpositions of two nondegenerate states , 1 and 3 , coupled by a two - photon process . the technique uses two pulses , each near resonance with an intermediate state 2 . the time dependent rabi frequencies of these two pulses are assumed to have identical pulse shapes . additionally , a stark - shifting pulse manipulates the energy of state 3 . the weights of the components of the superposition can be controlled by adjusting the two - photon detuning or by changing the ratio of the driving pulses at the end of the sequence . we gave a set of optimal parameters that minimize the non - adiabatic losses . the sacs technique has potential advantages over alternative techniques for preparing such nondegenerate superpositions . the sacs method requires less laser power than half - scrap . it also offers a larger choice than does f - stirap for the pulse shape of the fields and for the delay between pulses . we acknowledge s. gurin and h.r . jauslin for preliminary discussions . n.s . acknowledges financial supports from the eu network quacs under contract no . hprn - ct-2002 - 0039 and from la fondation carnot . bws is grateful to prof . k. bergmann for hospitality under the max planck forschungspreis 2003 .

we propose a technique to prepare coherent superpositions of two nondegenerate quantum states in a three - state ladder system , driven by two simultaneous fields near resonance with an intermediate state . the technique , of potential application to enhancement of nonlinear processes , uses adiabatic passage assisted by dynamic stark shifts induced by a third laser field . the method offers significant advantages over alternative techniques : ( i ) it does not require laser pulses of specific shape and duration and ( ) it requires less intense fields than schemes based on two - photon excitation with non - resonant intermediate states . we discuss possible experimental implementation for enhancement of frequency conversion in mercury atoms .

quant-ph0601028

we are interested in the transitions from a single - electron bound state @xmath0 ( with a wave function @xmath1 and energy @xmath2 ) to a continuum state @xmath3 ( with a wave function @xmath4 and energy @xmath5 ) ( fig . [ fig : sketch ] ) . ( a ) sketch of a qd chain , ( b ) energy diagram of a qd chain with an electron transition from the bound state @xmath6 in the intermediate band to the state @xmath3 in the conduction band . , width=321 ] we model the confined electron states @xmath0 as superpositions of the ground states @xmath7 confined in the individual dots ( where @xmath8 numbers the dots ) . for simplicity , we assume that each of these single dot states has an identical wave function , @xmath9 where @xmath10 is the position of the @xmath8th dot ( we assume that the dots are stacked along the growth direction @xmath11 ) . the ground state electron energies in the dots , @xmath12 , may differ . the states @xmath7 are coupled by nearest neighbor couplings . the eigenstates @xmath0 and the corresponding energies @xmath2 are thus obtained as the eigenstates of the effective chain hamiltonian ( assuming a single confined state in each dot ) @xcite , @xmath13 where @xmath14 is the coupling constant . this coupling constant is determined by the barrier between the neighboring qds . the height of the barrier depends on the band edge mismatch between the qds and on the host materials whereas the barrier width is set in the process of growing of the qd stack . since the stacks of self - organized qds are produced using molecular beam epitaxy @xcite or metal organic chemical vapor deposition @xcite the barrier width ( i.e. inter - dot distance @xmath15 ) is controlled with a high precision up to a single monolayer , so the coupling constant @xmath14 can be assumed to be the same for all pairs of neighboring qds . we assume the overlap between the wave functions localized in different dots to be negligible , so that @xmath16 . the inhomogeneity of the qd stack is taken into account by choosing the energies @xmath12 from the gaussian distribution with the mean @xmath17 and variance @xmath18 . we assume that the wave function for the electron in the @xmath8th dot has the gaussian form , @xmath19 } , \ ] ] where @xmath20 is the position of the @xmath8th dot and @xmath21 are the extensions of the wave function in the @xmath22 plane and along @xmath11 , respectively . our choice to use the same wave function for all qds which have not necessarily the same ground energy levels can be argued as follows . using the model of quantum harmonic oscillator we can estimate that small differences of the confined energy levels in a qd ( of the order of a few mev ) correspond to very small changes of the parameters of the wave function ( of the order of a few percent ) , so we can approximate wave function of each qd by a gaussian function with constant parameters @xmath23 and @xmath24 . on the other hand , when the differences of the qd confined level energies are larger strong localization of an electron on the qd with the lowest energy level occurs , which means that the exact form of the wave functions ( i.e. knowledge of the precise values of parameters ) of other qds become irrelevant , so that in this case we also can use the same parameters @xmath23 and @xmath24 for all qds of the chain . for the bulk electron states , we assume plane waves @xcite orthogonalized to the localized states , as previously proposed for calculating carrier capture rates @xcite . these states are labeled by the wave vector @xmath25 describing the plane wave far away from the qd structure . thus , we have @xmath26,\ ] ] where @xmath27 is the appropriate normalization constant , we assume normalization in a box of volume @xmath28 with periodic boundary conditions , and the orthogonalization coefficients @xmath29 are given by @xmath30 where @xmath31 } .\ ] ] the coupling of carriers to the incident light is described by the dipole hamiltonian @xmath32 where @xmath33 is the elementary charge and @xmath34 is the electric field . we will consider two cases : a monochromatic laser light will be described as a classical plane wave field @xmath35 where @xmath36 is the vacuum permittivity , @xmath37 is the high - frequency dielectric constant of the semiconductor , @xmath38 is the amplitude of the electric field of the electromagnetic wave , @xmath39 is a unit vector defining its polarization , @xmath40 is its wave vector ( inside the dielectric medium ) , and @xmath41 is its frequency , where @xmath42 is the refractive index of the semiconductor . on the other hand , for thermal radiation , corresponding to the natural working conditions of a solar cell , the field is @xmath43 where @xmath44 is the annihilation operator for a photon with the wave vector @xmath40 , @xmath28 is the formal normalization volume , and we take into account that the incident solar radiation is propagating into a specific direction , hence its wave vectors are distributed over a very small solid angle around its direction of propagation @xmath45 ( which is represented by the prime at the summation sign ) . for more flexibility of the modeling , we assume also that the radiation is polarized ( the effects of unpolarized radiation can be modeled by averaging over the directions of polarization ) . in the description of light induced transitions from the confined states to the extended states we assume that the occupation of the latter is negligible , which in a solar cell corresponds to assuming efficient carrier collection . in the case of classical ( coherent ) monochromatic light with frequency @xmath46 , propagation direction @xmath47 , and polarization @xmath39 , the transition rate from a state @xmath0 to the continuum of extended states is obtained in the usual way from the fermi golden rule @xcite using the interaction hamiltonian with the field given by eq . , @xmath48 where @xmath49 . this can be written in the form @xmath50 where @xmath51 is the energy density of the electromagnetic wave and @xmath52 in the case of illumination by broad band thermal radiation , we first use the hamiltonian with the quantum field given by eq . to calculate the fermi golden rule probability of absorption of a photon with a wave vector @xmath40 , @xmath53 where @xmath54 is the bose distribution of photon occupations at the temperature of solar black body radiation and the transition matrix elements @xmath55 are given by eq . . we note that for radiation propagating in a fixed direction @xmath45 , the overlap integral @xmath56 actually depends only on the frequency , @xmath57 . then , the total photon absorption rate for an initial confined state @xmath0 per unit frequency interval is @xmath58 the final sum in this expression is the spectral distribution of the energy density of radiation @xmath59 which , for solar light , is approximately described by the planck law . hence , the absorption rate under illumination by thermal radiation can be written in the form analogous to eq . , @xmath60 eqs . and , along with the effective chain hamiltonian , are the basis for numerical calculations the results of which are presented in the next section . in this section , we present results of numerical calculations of the intraband absorption in qd chains using the approach described in sec . [ sec : model ] and sec . [ sec : absorption ] . in all our simulations we focus on an inas / gaas structure ( not optimal for a solar cell @xcite but important for the current laboratory scale investigations @xcite ) . we assume the wave function confinement sizes @xmath61 nm , @xmath62 nm ( see ref . for the discussion of the dependence on these parameters ) , @xmath63 mev . based on earlier @xmath64 calculations @xcite , the tunnel coupling @xmath14 is given by the formula @xmath65 where @xmath66 , @xmath67 ev ( @xmath68 and @xmath69 mev for @xmath70 and @xmath71 nm , respectively ) . ( a ) low temperature intraband absorption spectra for a chain of identical dots illuminated by coherent monochromatic unpolarized light incident along and perpendicular to the stacking direction @xmath11 ( as indicated by @xmath72 and @xmath73 , respectively ) . ( b ) the dependence of the spectrum on the angle of incidence for 10 qds.,width=321 ] in fig . [ fig : length](a ) , we show the intraband absorption spectra of short chains of identical qds compared to the spectrum of a single dot for light incident along the chain ( in the @xmath11 direction ) and perpendicular to the chain . both spectra are calculated assuming that only the lowest confined state is considerably occupied ( which corresponds to low temperatures ) and for unpolarized light ( for perpendicular incidence , the contribution from light polarized along @xmath11 is small ) . in both cases , the absorption spectrum for a single dot shows only a single maximum with a long tail on the high energy side but already for a few dots it develops a series of additional maxima . as can be seen in fig . [ fig : length](b ) , the spectra for a chain of 10 qds smoothly evolve between the two limiting geometries as the incidence angle is tilted from normal to in - plane incidence . low temperature intraband absorption spectrum for a chain of non - identical dots illuminated by coherent monochromatic unpolarized light incident along the stacking direction @xmath11 . ( a , b ) as a function of the degree of inhomogeneity ; ( c , d ) as a function of the chain length . each spectrum is an average of 1000 realization with randomly chosen values of @xmath12.,width=321 ] the additional absorption peaks that can be seen in fig . [ fig : length ] result from interference effects which are due to delocalization of the electron state and lead to preferred transitions to states with @xmath74 , where @xmath75 is an integer @xcite . disorder , which in our case has the form of inhomogeneity of the `` on - site '' energies @xmath12 , destroys the coherently delocalized electron states and leads to localization . as a result , the additional peaks are suppressed and the absorption spectrum becomes more similar to that of a single dot ( see fig . [ fig : inhom](a , b ) ) . as we show in fig . [ fig : inhom](c , d ) , while the spectra for qd stacks differ considerably from those for a single dot , the extension of the stack above @xmath76 dots has little effect . note that the effect of interference of transition amplitudes depends to some extent on whether their magnitudes are equal or not . hence , if one allows different geometries of wave functions in non - identical qds then additional suppression of the interference - related additional peaks may occur . however , this effect is expected to be small compared to the suppression due to localization . intraband absorption spectrum for a chain of 10 non - identical dots with @xmath77 mev , illuminated by coherent monochromatic unpolarized light incident along the stacking direction @xmath11 as a function of temperature for two values of the inter - dot coupling : ( a , b ) @xmath68 mev ( corresponding to @xmath70 nm ) , ( c , d ) @xmath78 mev ( corresponding to @xmath79 nm).,width=321 ] for a realistic device , absorption at elevated temperatures is relevant . at non - zero temperatures , the occupation of excited states in the qd - related pseudo - band is non - negligible , which leads to a reconstruction of the spectrum , as shown in fig . [ fig : temp ] . for chains with @xmath80 , the structure of the spectrum is washed away with increasing temperature ( fig . [ fig : temp](a , b ) ) . however , as can be seen from fig . [ fig : temp](c , d ) strong coupling broadens the pseudo - band beyond the thermal energies and stabilizes the chain absorption against temperature . ( a , b ) electron intraband transition rates under illumination by one sun thermal radiation as a function of the degree of inhomogeneity ( a ) and as a function of the number of qds in the chain ( b ) . each spectrum is an average of 1000 realization with randomly chosen values of @xmath12 . ( c , d ) integrated single - electron transition rate under illumination by one sun thermal radiation as a function of the degree of inhomogeneity ( c ) and as a function of the number of qds in the chain(d).,width=321 ] in order to study the properties of the device under realistic working conditions we calculate also the electron intraband transition rate @xmath81 ( from the ground state ) for a chain of non - identical qds illuminated by black body radiation . the energy density of radiation with @xmath82 k was normalized to the incident flux of 1 kw / m@xmath83 ( one sun ) . in fig . [ fig : beta](a , b ) we show the spectral distribution of the transition rates as a function of the energy of the absorbed photon . comparison with the absorption spectra shown in fig . [ fig : length ] shows that the qualitative form of these two spectral characteristics is similar but quantitatively the transition rate at higher photon energies is enhanced relative to that at lower energies . this is due to the very quickly growing spectral densities of the thermal photon field at higher energies ( according to the the planck law ) which enhances the role of higher energy photons ( and , in consequence , the overall absorption of energy in the corresponding spectral range ) . one should notice that inhomogeneity ( higher values of @xmath84 ) leads to a less structured spectrum with non - vanishing absorption over all the spectral range . in fig fig . [ fig : beta](c ) , we show the total transition rate , integrated over photon energies . the result at @xmath85 k includes the thermal distribution of the initial states of the electron . as one can see , the total rate increases as the inhomogeneity grows . this increase is due to the increasing probability that the electron will localize in a dot with much lower `` on - site '' energy as the qd energy distribution broadens . this enhances the transition rate because of the growing spectral density of radiation for higher energy differences between the initial and final states . as a function of the chain length ( fig . [ fig : beta](d ) ) , the total transition rate increases slightly ( up to several % ) for strongly coupled chains at low temperatures and then saturates for @xmath86 . this increase is mostly due to high - frequency contributions corresponding to the enhanced high - energy absorption features in a qd stack ( cf . [ fig : length ] ) . one can notice in fig . [ fig : beta](d ) that the gain in the total transition rate for strongly coupled dots not only is much larger than for weaker coupling but it is also much less sensitive to temperature . we have proposed a model which describes electron states and intraband absorption spectra which correspond to the transitions from the qd pseudo - band to the conduction band . we have studied the absorption spectra as a function of the degree of inhomogeneity , the number of qds in the chain , the inter - dot separation , the temperature and the illumination conditions . we have shown that the absorption spectrum even of short qd chains ( a few qds ) is dominated by interference effects that leads to the appearance of additional absorption maxima and does not considerably evolve further if the chain length exceeds approximately 10 dots . the overall transition rate under illumination with black body radiation increases by up to several per cent with the number of qds but also saturates already for a few qds in the chain . the structured absorption spectrum persists when the inhomogeneity of the qd transition energies is increased up to the values comparable with the magnitude of the inter - dot coupling . strong coupling between the dots is essential for maintaining the chain absorption features up to high temperatures . on practical side , our results show that already a stack of a few qds manifests the absorption features characteristic of qd chains and that these features are stable against temperatures and energy inhomogeneities if the dots are sufficiently strongly coupled . however , a qd chain shows only a slightly increases absorption as compared to a single dot and the increase of absorption is dominated by high - energy photons , which may lead to competition with the interband absorption . moreover , a trade - off has to be found between the absorption enhancement in strongly coupled chains and the need to reduce tunneling in order to avoid carrier escape from the intermediate bands @xcite . s. a. blokhin , a. v. sakharov , a. m. nadtochy , a. s. pauysov , m. v. maximov , n. n. ledentsov , a. r. kovsh , s. s. mikhrin , v. m. lantratov , s. a. mintairov , n. a. kaluzhniy , and m. z. shvarts , fiz . . poluprovodn . * 43 * , 537 ( 2009 ) , [ semiconductors * 43 * , 514 ( 2009 ) ] .

we present a theoretical analysis of intraband optical transitions from the intermediate pseudo - band of confined states to the conduction band in a finite , inhomogeneous stack of self - assembled semiconductor quantum dots . the chain is modeled with an effective hamiltonian including nearest - neighbor tunnel couplings and the absorption under illumination with both coherent ( laser ) and thermal radiation is discussed . we show that the absorption spectrum already for a few coupled dots differs from that of a single dot and develops a structure with additional maxima at higher energies . we find out that this leads to an enhancement of the overall transition rate under solar illumination by up to several per cent which grows with the number of qds but saturates already for a few qds in the chain . the decisive role of the strength of inter - dot coupling for the stability of this enhancement against qd stack inhomogeneity and temperature is revealed . one of the ways to improve the efficiency of solar cells is to introduce an intermediate band in the energy spectrum of a photovoltaic structure @xcite . in this way , electrons can be sequentially promoted from the valence band to the intermediate band and then to the conduction band by absorbing photons with energies below the band gap which are not converted into useful electrochemical energy in a standard structure . as an implementation of this concept , a stack of quantum dots ( qds ) in the intrinsic region of a p - i - n junction solar cell has been proposed @xcite . this idea has indeed gained some experimental support in recent years @xcite . quantum - dot - embedded p - i - n solar cells show higher quantum efficiency in near infrared range but their overall efficiency still is lower than the efficiency of similar devices without qds @xcite . on the theory side , models involving a single qd were formulated to describe the kinetics of transitions from and into the intermediate levels @xcite . on the other hand , modeling of the electron states and optical absorption in chains and arrays of qds has been mostly limited to infinite , periodic superlattices of identical dots @xcite . as we have shown recently @xcite , enhanced absorption can appear also in finite chains of non - identical qds but it is suppressed if the inhomogeneity of the qd chain ( leading to non - identical electron ground state energies in the individual qds ) becomes too large . since the actual qd chains are always finite ( usually built of several to a few tens of qds ) @xcite and unavoidably inhomogeneous it is of large practical importance for the optimal design of intermediate band photovoltaic devices to extend the theoretical analysis to such more realistic structures . in this paper , we study the intraband optical absorption associated with the electron transition from the states confined in a finite stack of quantum dots to the conduction band . ( fig . [ fig : sketch ] ) we propose a relatively simple and computation - effective model which , however , includes all the essential features of the system , in particular the inhomogeneity of the energetic parameters of the dots forming the stack and the coupling between them . we show that the enhanced absorption features appear already for a few dots and lead to enhanced transition rate from the pseudo - band of confined states to the bulk continuum . while this effect is relatively stable with respect to inhomogeneity , it turns out to be associated with the transitions from the ground state of the stack and is washed out as soon as the temperature becomes comparable to the width of the pseudo - band of confined states . our modeling results suggest that this detrimental temperature effect can be to a large extent overcome by increasing the tunnel coupling between the dots . the paper is organized as follows . sec . [ sec : model ] defines the model of the system . in sec . [ sec : absorption ] , we briefly discuss the theoretical description of intraband transitions in the cases of coherent and thermal radiation . next , in sec . [ sec : results ] , we present the results of our calculations . sec . [ sec : conclusions ] concludes the paper .

1207.6999

the standard model of elementary particles ( sm ) has been very successful in describing the nature at the electroweak ( ew ) scale . recently , the atlas and cms collaborations at the large hadron collider ( lhc ) have discovered a new particle @xcite , which is consistent with the sm higgs boson . this discovery also strengthens the correctness of the sm . so far , no explicit evidence of physics beyond the sm has been reported from the lhc . several groups , however , have reported an anomaly of the muon anomalous magnetic moment @xmath9 ( muon g-2 ) , which has been precisely measured experimentally @xcite and compared with state - of - the - art theoretical predictions ( for example , see @xcite and references therein ) . the estimated discrepancies between the sm predictions and the measured value are consistently more than @xmath10 , as listed in table [ tab : g-2 ] . although it is too early to conclude that this anomaly is evidence of new physics beyond the sm , we expect new particles and interactions related with the muon sector once we regard it as a hint of new physics . gauge interactions have been playing a central role to construct fundamental models in particle physics history . following this line , in this paper , we purse the possibility that the muon has a new gauge interaction beyond the sm . .measured muon g-2 ( @xmath11 ) and the estimated differences ( @xmath12 ) from the recent sm predictions in several references . [ cols="^,^",options="header " , ] from the charge assignments , renormalizable terms in a lagrangian which contribute to the lepton masses are given by @xmath13 here , @xmath14 are the yukawa couplings of the charged leptons and not related to the neutrino mass . the neutrino mass is determined from yukawa couplings @xmath15 , majorana masses @xmath16 and @xmath17 , and yukawa couplings @xmath18 and @xmath19 . note that the mass terms between the left and right - handed neutrinos are diagonal . therefore , the neutrino mixing is obtained by mixing among the right - handed neutrinos . if the majorana masses @xmath20 and @xmath21 are of the same order , the seesaw mechanism @xcite provides the observed order one neutrino mixing . from the seesaw formula , a relation between the parameters is given by @xmath22 where @xmath23 gev is the vacuum expectation value of the sm higgs , and @xmath24 is the difference between the mass squared of the left - handed neutrinos . @xmath25 and @xmath26 denote @xmath27 and @xmath20 and @xmath21 collectively . interactions given by eq . ( [ eq : mass terms ] ) break the lepton symmetry . on the other hand , @xmath28 symmetry is broken by the anomaly against the @xmath29 gauge interaction , whose effect is efficient at the early universe by the sphaleron process in the finite temperature @xcite . therefore , the baryon asymmetry is washed out if both effects are important simultaneously . let us calculate a condition such that the washout does not occur . first of all , the sphaleron process is efficient only at the temperature above the ew scale . therefore , if the baryon asymmetry is generated below the ew scale , the washout does not occur . in the following , we assume that the baryon asymmetry is produced above the ew scale and calculate the constraint on the parameters in eq . ( [ eq : mass terms ] ) . let us consider two possibilities in which the washout does not occur . 1 . @xmath30 is small , 2 . @xmath16 and @xmath17 are small . if any of the two conditions are satisfied , the lepton number is effectively conserved . therefore , one should adopt the weakest condition among them . let us discuss the two cases in detail . in the limit @xmath31 , the lepton symmetry is restored for each flavors . therefore , if the interaction by @xmath30 is inefficient , the washout of the baryon asymmetry does not occur . the most efficient interaction is shown in figure [ fig : lambda ] and its rate is given by @xmath32 where @xmath33 , @xmath34 , @xmath35 , @xmath36 are the cross section of the process , the number density of related particles , the velocity of related particles , and the yukawa coupling of the top quark , respectively . @xmath37 denotes the thermal average . by requiring that the rate is smaller than the hubble scale for @xmath38 gev , we obtain the bound @xmath39 .,width=264 ] if both majorana masses vanish , @xmath40 symmetry is restored . the most efficient interaction which induces the symmetry violation by the majorana masses is shown in figure [ fig : m ] . its rate is given by @xmath41 this rate is smaller than the hubble scale for @xmath38 gev if @xmath42 from the relation ( [ eq : seesaw ] ) , one can see that the condition ( [ eq : constraint - m ] ) is severer than the condition ( [ eq : constraint - l ] ) . therefore , it is enough to satisfy the condition ( [ eq : constraint - l ] ) in order for the washout not to occur . with the relation ( [ eq : seesaw ] ) , the condition is interpreted as @xmath43 since the right - handed neutrinos are light and weakly coupled , it is necessary to consider whether they are long - lived . if they are long - lived , they might over - close the universe , or destroy the success of the big - bang nucleosynthesis ( bbn ) . the most important decay channel is given by the diagram shown in figure . [ fig : decay of n ] . here , we have assumed that @xmath33 is heavier than the right - handed neutrinos and hence the decay mode @xmath44 is closed . the decay rate is given by @xmath45 the decay of the right - handed neutrinos is efficient around the temperature @xmath46 therefore , the right - handed neutrinos decay before the bbn begins and does not affect it . passarino - veltman functions @xcite are defined by @xmath47 \left[(k+p)^2-m_b^2+i \epsilon\right]},\nonumber \\ p^\mu b_1(a , b;p ) & = & 16\pi^2 \mu^{2\epsilon } \int \frac{d^n k}{i(2\pi)^n}\frac{k^\mu } { \left[k^2-m_a^2+i\epsilon\right]\left[(k+p)^2-m_b^2+i\epsilon\right ] } , \nonumber \\ p^\mu p^\nu b_{21}(a , b;p)&+&g^{\mu \nu}b_{22}(a , b;p ) \nonumber \\ & = & 16\pi^2 \mu^{2\epsilon } \int \frac{d^n k}{i(2\pi)^n } \frac{k^\mu k^\nu } { \left[k^2-m_a^2+i\epsilon\right]\left[(k+p)^2-m_b^2+i\epsilon\right ] } , \ ] ] @xmath48[(k+p_1)^2-m_b^2+i\epsilon][(k+p_1+p_2)^2-m_c^2+i\epsilon ] } , \nonumber \\ & & \left ( p_1^\mu c_{11}+p_2^\mu c_{12 } \right)(a , b , c;p_1,p_2 ) \nonumber \\ & = & 16\pi^2 \mu^{2\epsilon } \int \frac{d^n k}{i(2\pi)^n}\frac{k^\mu } { [ k^2-m_a^2+i\epsilon][(k+p_1)^2-m_b^2+i\epsilon][(k+p_1+p_2)^2-m_c^2+i\epsilon ] } , \nonumber \\ & & \left\{(p_1^\mu p_1^\nu c_{21 } + p_2^\mu p_2^\nu c_{22}+(p_1^\mu p_2^\nu+p_1^\nu p_2^\mu)c_{23 } + g^{\mu \nu}c_{24 } \right\ } ( a , b , c;p_1,p_2 ) \nonumber \\ & = & 16\pi^2 \mu^{2\epsilon } \int \frac{d^n k}{i(2\pi)^n}\frac{k^\mu k^\nu } { [ k^2-m_a^2+i\epsilon][(k+p_1)^2-m_b^2+i\epsilon][(k+p_1+p_2)^2-m_c^2+i\epsilon ] } , \nonumber \\\end{aligned}\ ] ] where we use dimensional regularization in @xmath49 dimensions , and @xmath50 is a renormalization scale . 99 g. aad _ et al . _ [ atlas collaboration ] , phys . b * 716 * , 1 ( 2012 ) . s. chatrchyan _ et al . _ [ cms collaboration ] , phys . b * 716 * , 30 ( 2012 ) . j. beringer _ [ particle data group collaboration ] , phys . d * 86 * , 010001 ( 2012 ) and 2013 partial update for the 2014 edition . t. aoyama , m. hayakawa , t. kinoshita and m. nio , phys . rev . lett . * 109 * , 111808 ( 2012 ) [ arxiv:1205.5370 [ hep - ph ] ] . a. czarnecki , w. j. marciano and a. vainshtein , phys . d * 67 * , 073006 ( 2003 ) [ erratum - ibid . d * 73 * , 119901 ( 2006 ) ] . k. hagiwara , r. liao , a. d. martin , d. nomura and t. teubner , j. phys . g * 38 * , 085003 ( 2011 ) [ arxiv:1105.3149 [ hep - ph ] ] . t. teubner , k. hagiwara , r. liao , a. d. martin and d. nomura , chin . c * 34 * , 728 ( 2010 ) [ arxiv:1001.5401 [ hep - ph ] ] . m. benayoun , p. david , l. delbuono and f. jegerlehner , eur . j. c * 72 * , 1848 ( 2012 ) [ arxiv:1106.1315 [ hep - ph ] ] . f. jegerlehner and r. szafron , eur . j. c * 71 * , 1632 ( 2011 ) [ arxiv:1101.2872 [ hep - ph ] ] . f. jegerlehner and a. nyffeler , phys . * 477 * , 1 ( 2009 ) [ arxiv:0902.3360 [ hep - ph ] ] . m. davier , a. hoecker , b. malaescu and z. zhang , eur . j. c * 71 * , 1515 ( 2011 ) [ erratum - ibid . c * 72 * , 1874 ( 2012 ) ] . t. moroi , phys . rev . d * 53 * , 6565 ( 1996 ) [ erratum - ibid . d * 56 * , 4424 ( 1997 ) ] . for example , see t. hambye , k. kannike , e. ma and m. raidal , phys . d * 75 * , 095003 ( 2007 ) [ hep - ph/0609228 ] ; 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in this paper , we consider phenomenology of a model with an @xmath0 gauge symmetry . since the muon couples to the @xmath0 gauge boson ( called @xmath1 boson ) , its contribution to the muon anomalous magnetic moment ( muon g-2 ) can account for the discrepancy between the standard model prediction and the experimental measurements . on the other hand , the @xmath1 boson does not interact with the electron and quarks , and hence there are no strong constraints from collider experiments even if the @xmath1 boson mass is of the order of the electroweak scale . we show an allowed region of a parameter space in the @xmath0 symmetric model , taking into account consistency with the electroweak precision measurements as well as the muon g-2 . we study the large hadron collider ( lhc ) phenomenology , and show that the current and future data would probe the interesting parameter space for this model . # 1 .75 in * muon g-2 and lhc phenomenology in the @xmath0 gauge symmetric model * .75 in keisuke harigaya@xmath2 , takafumi igari@xmath3 , mihoko m. nojiri@xmath4 , + michihisa takeuchi@xmath5 , and kazuhiro tobe@xmath6 0.25 in _ @xmath2kavli ipmu ( wpi ) , todias , university of tokyo , chiba , kashiwa , 277 - 8583 , japan _ _ @xmath3department of physics , nagoya university , aichi , nagoya 464 - 8602 , japan _ _ @xmath7theory center , kek , tsukuba , ibaraki 305 - 0801 , japan _ _ @xmath5theoretical particle physics and cosmology group , department of physics , king s college london , london wc2r 2ls , uk _ _ @xmath8kobayashi - maskawa institute for the origin of particles and the universe , + nagoya university , aichi , nagoya 464 - 8602 , japan _ .5 in kcl - ph - th/2013-*37 * kek - th-1684 ipmu 13 - 0213

1311.0870

let @xmath5 is an @xmath0-cobordism between smooth manifolds @xmath6 and @xmath7 . since pioneering work in the 1980 s it has been known that @xmath6 and @xmath7 are not necessarily diffeomorphic @xcite , but they are homeomorphic @xcite when the fundamental group is of a certain type , called `` good '' ( a kind of time - dependent definition ) by practitioners . good groups @xcite were built from the class of polynomial growth groups and later the more general subexponential class by closure under four operations : extension , quotient , direct limit , and subgroup . it is tempting to guess that good groups can be extended to all amenable groups , but this is not presently known . though the question of any classification up to diffeomorphism seems completely intractable at this point , it was noticed by quinn @xcite in the 1970 s that these subtleties disappear under stabilization . that is , there exists a natural number @xmath1 so that @xmath8 is a product cobordism , where for a @xmath4-manifold @xmath6 ( or @xmath7 ) @xmath9 and for a @xmath10-dimensional cobordism @xmath5 , @xmath11 denotes the `` connected sum @xmath12 $ ] '' with @xmath13 summed parametrically along a vertical arc in @xmath5 . for the remainder of the paper we denote @xmath14 by @xmath15 , and `` connected sum @xmath12 $ ] '' by @xmath16 . this paper is concerned with how small @xmath1 can be . when @xmath6 and @xmath7 are simply connected complete intersections , it follows from @xcite ( and a little five - dimensional surgery ) that @xmath17 suffices . beyond this , little seems to be known : no argument from gauge theory or any other direction excludes ( as far as we know ) the possibility that @xmath17 suffices universally . suppose @xmath18 and @xmath19 is an infinite collection of finite index subgroups of @xmath20 with @xmath21 . we let @xmath22 denote the index @xmath23 $ ] . consider the corresponding covering spaces @xmath24 with @xmath25 and define @xmath26 to be the minimal @xmath1 so that @xmath27 is a product . it is clear that @xmath28 , since @xmath29 , i.e. the covering space of @xmath30 corresponding to @xmath31 is @xmath32 . the main theorem of this paper is : [ thm : main ] if @xmath33 is an amenable group , then for any sequence of subgroups @xmath34 with @xmath35 we have @xmath36 more generally the theorem holds if the maximal residually finite quotient of @xmath33 is amenable . recall that the maximal residually finite quotient of a group @xmath20 is the quotient group @xmath37 , where the intersection ranges over all finite index subgroups . the main theorem is actually a combination of two theorems , one in smooth topology and one in coarse geometry . before stating these we discuss the notion of sweepout width of a coset space . we consider a finitely generated group @xmath20 as a discrete metric space by choosing a finite generating set and building its cayley graph . the distance between two group elements is then defined to be the minimal number of edges needed to join them in the cayley graph . let @xmath38 be a finite index subgroup , and let @xmath39 be the space of cosets , with the induced metric . given a set @xmath40 , define @xmath41 to be the set of all vertices in @xmath42 to points in the complement of @xmath42 . sweep out _ @xmath43 of @xmath44 is a sequence of subsets @xmath45 , @xmath46 , @xmath47 , with @xmath48 . define the _ width _ of @xmath43 by @xmath49 . we say that @xmath50 is the _ sweepout width of @xmath44_. theorem [ thm : main ] follows immediately from the following two theorems . [ thm : main amenable ] let @xmath20 be an amenable group , let @xmath31 be any sequence of finite index subgroups with @xmath35 , and let @xmath51 . then @xmath52 in fact , the conclusion of the theorem holds exactly when the maximally residually finite quotient of @xmath20 is amenable . [ thm : main topology ] let @xmath53 be a smooth @xmath10-dimensional @xmath0-cobordism , let @xmath34 be a sequence of finite index subgroups and let @xmath24 be the covering space of @xmath5 corresponding to @xmath54 ( thus @xmath24 is an @xmath0-cobordism between the corresponding covering spaces of @xmath6 and @xmath7 ) . let @xmath55 and let @xmath26 be the minimal number so that @xmath56 is a trivial product . then there is a constant @xmath57 not depending on @xmath58 so that @xmath59 . often theorem [ thm : main topology ] gives better bounds than theorem [ thm : main ] , if we have additional knowledge of the group @xmath33 or the sequence @xmath31 . for example : [ cor : pi1 z ] let @xmath53 be a smooth @xmath10-dimensional @xmath0-cobordism as above , suppose @xmath60 , and let @xmath24 be the covering space of @xmath5 corresponding to the subgroup @xmath61 . then there is a constant @xmath57 so that for all @xmath58 @xmath62 is a trivial product . define @xmath63 by @xmath64 . then @xmath65 is a scale-@xmath66 sweepout of @xmath67 with width @xmath68 . suppose @xmath6 and @xmath7 are closed @xmath4-manifolds and there is a homeomorphism @xmath69 with vanishing kirby - siebenmann invariant . then @xmath70 has a controlled isotopy to a diffeomorphism , and using this diffeomorphism one can build a smooth @xmath0-cobordism @xmath5 with @xmath71 . therefore if @xmath72 is amenable theorem [ thm : main ] tells us that the number of @xmath73 connect sums required to make @xmath6 diffeomorphic to @xmath7 is subextensive in covers . does the same hold true if @xmath74 ? gompf @xcite and kreck @xcite independently proved that if @xmath74 then there is a homeomorphism @xmath75 so that @xmath76 . but even a single copy of @xmath73 connect summed before construction of @xmath5 leads to an extensive number of @xmath73 s in covers , so this fact can not be directly applied . the remainder of the paper is divided into two sections . theorem [ thm : main amenable ] is proved in section [ sec : coarse ] and theorem [ thm : main topology ] is proved in section [ sec : smooth ] . the authors are grateful to pierre pansu , richard schwartz , and romain tessera for conversations about polynomial growth groups which was very useful for developing intuition . suppose @xmath77 is a graph . we will also write @xmath77 for the set of vertices in @xmath77 and @xmath78 for the set of edges in @xmath77 . in this section , if @xmath79 is a set of vertices , we define @xmath80 to be set of edges from @xmath42 to the complement of @xmath42 . ( this is slightly different from the definition from the previous section , where we defined @xmath41 to be the set of vertices in @xmath42 which are adjacent to the complement of @xmath42 . since we always work with graphs with uniformly bounded valance , @xmath81 is the same with either definition , up to a constant . ) the folner profile of @xmath77 records some information about the sizes of sets and their boundaries in @xmath77 . we define the folner profile of @xmath77 , @xmath82 by @xmath83 we note that @xmath84 is non - increasing . we will be mainly interested in cayley graphs of groups and their quotients . suppose that @xmath20 is a finitely generated group with generating set @xmath85 . ( we assume that @xmath85 is closed under taking inverses : @xmath86 if and only if @xmath87 . ) we denote the cayley graph of @xmath20 by @xmath88 . note that @xmath20 is amenable if and only if @xmath89 we often denote @xmath90 when the generating set if clear from context . we are interested in coset spaces of @xmath20 . suppose that @xmath38 is a subgroup of @xmath20 . given a generating set @xmath85 for @xmath20 , there is a natural graph structure on the quotient @xmath91 called the _ schreier graph _ , which we denote @xmath92 . the set @xmath93 is the set of right cosets @xmath94 . there is an edge between @xmath95 and @xmath96 if and only if @xmath97 for some @xmath86 , if and only if @xmath98 for some @xmath86 . here is an equivalent description of @xmath92 . the group @xmath99 acts on @xmath20 on the right , and so @xmath99 acts by isomorphisms on the graph @xmath88 . the graph @xmath100 is the quotient graph for this action . again , if the generating set @xmath85 is clear from the context , we abbreviate @xmath101 . [ prop : folquot ] if @xmath20 is a finitely generated group with generating set @xmath85 and @xmath38 is a subgroup , then for all @xmath102 , @xmath103 we briefly discuss why this proposition is tricky and the main idea of the proof . suppose that @xmath104 with @xmath105 , and @xmath106 . we want to find a set @xmath107 with @xmath108 and @xmath109 . let @xmath110 be the quotient map , the most obvious set to look at is @xmath111 . clearly @xmath112 , but it seems hard to control the ratio @xmath113 . part of the problem is that some points in @xmath111 may have many preimages in @xmath42 while other points in @xmath111 may have few preimages in @xmath42 . instead of working just with @xmath111 , it turns out to be more natural to consider a different type of pushforward from @xmath20 to @xmath114 which takes into account the number of preimages . suppose that @xmath115 is a function with finite support . then we define the pushforward @xmath116 by @xmath117 to make good use of the pushforward for functions , we recall the notion of the gradient of a function on a graph . if @xmath77 is a graph , @xmath118 , and @xmath119 is an oriented edge of @xmath77 from @xmath120 to @xmath121 , then @xmath122 . if @xmath119 is an unoriented edge with boundary vertices @xmath120 and @xmath121 , then we can still define @xmath123 : @xmath124 this gradient is closely related to the size of the boundary of a set : for any set @xmath79 , we have @xmath125 the set @xmath126 that we construct comes from @xmath127 , where @xmath128 denotes the characteristic function of @xmath42 . more precisely , @xmath126 is a super - level set of the pushforward , @xmath129 for some @xmath130 . the first step in the proof of proposition [ prop : folquot ] is [ lem : pushdown ] if @xmath115 has finite support , then @xmath131 for each @xmath132 and each @xmath86 , either @xmath133 or there is an edge from @xmath134 to @xmath135 in the graph @xmath136 . when we sum over all pairs @xmath137 , we get @xmath138 ( each edge @xmath139 is counted twice on the right - hand side because it has two endpoints . on the other hand , if @xmath140 , then @xmath141 and so it does nt contribute anything . ) the right - hand side is @xmath142 pick @xmath104 with @xmath105 and with @xmath143 we note that @xmath144 , and so @xmath145 we also note that @xmath146 , and so by lemma [ lem : pushdown ] @xmath147 next we rewrite these two equations in terms of the super - level sets @xmath148 , for integers @xmath130 . the first equation becomes : @xmath149 the second equation becomes : @xmath150 therefore , there exists some integer @xmath151 so that @xmath152 we let @xmath153 . we note that @xmath154 , and so @xmath155 . therefore we have shown @xmath156 we use the folner profile to control the size of approximate bisections in @xmath114 . if @xmath42 is a subset of a graph @xmath77 , we consider writing @xmath42 as a disjoint union @xmath157 , where each of the sets @xmath158 obeys @xmath159 . we call such a decomposition an approximate bisection of @xmath42 . among all the approximate bisections , we are interested in minimizing the number of edges between @xmath160 and @xmath161 . @xmath162 we write @xmath163 for the set of edges from @xmath160 to @xmath161 . [ folbisec ] suppose that @xmath20 is a finitely generated group and @xmath38 is a cofinite subgroup . if @xmath164 , and @xmath165 , then @xmath166 in particular , if @xmath20 is an amenable group , then @xmath167 and so @xmath168 for any subset @xmath42 of any finite quotient @xmath114 . informally , if @xmath20 is amenable , any large subset of any finite quotient can be bisected efficiently . we start by proving the following weaker lemma : [ folbite ] suppose that @xmath20 is a finitely generated group and @xmath38 is a cofinite subgroup . if @xmath164 , with @xmath169 , then there exists a subset @xmath170 with @xmath171 so that @xmath172 if @xmath173 is small , then this lemma says that we can cut off a piece @xmath170 in an efficient way efficient in the sense that the number of edges we need to cut to separate @xmath126 from the rest of @xmath42 is much smaller than the number of vertices in @xmath126 . let @xmath174 be a finite set that realizes @xmath175 . in other words , @xmath176 , and @xmath177 we consider the translations of @xmath178 by the left action of @xmath20 on @xmath114 . the set @xmath126 will be @xmath179 for a group element @xmath180 chosen by an averaging argument . we can not exactly average over @xmath20 because @xmath20 is infinite . however , we have assumed that @xmath114 is finite , and so the elements of @xmath20 act on @xmath114 in only finitely many ways . consider the homomorphism @xmath181 , sending each element @xmath182 to the corresponding isomorphism of the graph @xmath114 . the image of this map is a finite subgroup @xmath183 . the group @xmath20 acts transitively on @xmath114 , and so @xmath184 acts transitively on @xmath114 . therefore , we see that for any @xmath185 , @xmath186 with this fact we can get good estimates about the average behavior of @xmath187 . first we compute @xmath188 . given a point @xmath189 for some @xmath190 , we associate the point @xmath191 and the point @xmath192 . now for any points @xmath193 and @xmath192 , there are @xmath194 choices of @xmath195 so that @xmath196 . there are @xmath197 choices of @xmath198 and @xmath199 choices of @xmath134 . therefore , we see that @xmath200 next we would like to upper bound @xmath201 . if @xmath119 is an edge from @xmath202 to @xmath203 , then we see that @xmath204 , @xmath205 , and so @xmath206 . we associate to an edge @xmath207 the point @xmath208 and the edge @xmath209 . given an edge @xmath209 from @xmath210 to @xmath211 , and an element @xmath208 , there are @xmath194 elements @xmath195 taking @xmath212 to @xmath213 . therefore , @xmath214 therefore , there exists some @xmath195 so that @xmath215 we set @xmath216 . the last equation tells us that @xmath217 we also know that @xmath218 . we now find an efficient bisection of @xmath42 by using lemma [ folbite ] repeatedly . we keep efficiently cutting off small pieces of @xmath42 until we have approximately bisected @xmath42 . let @xmath164 . let @xmath170 be as in lemma [ folbite ] . lemma [ folbite ] tells us that @xmath219 we know that @xmath171 . if @xmath220 , then we set @xmath221 and @xmath222 , and we have the desired bisection of @xmath42 . if @xmath223 , then we set @xmath224 , and we apply lemma [ folbite ] to @xmath225 . ( we also let @xmath226 and @xmath227 . ) lemma [ folbite ] tells us that there is a subset @xmath228 with @xmath229 , and @xmath230 ( the last step came from noting that @xmath231 . ) we repeat this process until the first step @xmath0 where @xmath232 . for @xmath233 , we see that @xmath234 , and so @xmath235 then we set @xmath236 , and @xmath237 . we first check that @xmath238 . note that @xmath239 . on the other hand , @xmath240 , and @xmath241 , and so @xmath242 . next we estimate @xmath163 . @xmath243 suppose that @xmath42 is a finite subset of a graph @xmath77 . a sweepout @xmath43 of @xmath42 is a nested sequence of subsets @xmath244 with @xmath48 . the width of a sweepout is defined by @xmath245 in the special case of most interest to us , the set @xmath42 may be the entire graph @xmath77 , in which case @xmath246 . the sweepout width of a set @xmath42 is the minimal width of any sweepout of @xmath42 : @xmath247 in @xcite , balacheff and sabourau introduced an inductive procedure to construct sweepouts from bisections . they worked on riemannian manifolds instead of graphs ( which is harder ! ) , and in the case of graphs their ideas easily imply the following lemma . [ lembalsab ] suppose that @xmath42 is a finite subset of a graph @xmath77 . then there exists a subset @xmath248 with @xmath249 , so that @xmath250 by the definition of @xmath251 , we can decompose @xmath42 into two disjoint pieces , @xmath160 and @xmath161 , so that @xmath159 and @xmath252 let @xmath253 be a sweepout of @xmath160 and let @xmath254 be a sweepout of @xmath161 . we will assemble @xmath255 and @xmath254 to get a sweepout @xmath43 of @xmath42 obeying @xmath256 then choosing @xmath255 and @xmath254 so that @xmath257 and @xmath258 gives the conclusion . we let @xmath259 for @xmath260 . then we let @xmath261 for @xmath262 . it is straightforward to check that @xmath263 is a sweepout of @xmath42 , and we just have to estimate @xmath264 . if @xmath265 we have that @xmath266 , and so @xmath267 if @xmath268 we have that @xmath269 for @xmath270 , and so @xmath271 using the bounds for @xmath251 from proposition [ folbisec ] , and iterating lemma [ lembalsab ] , we get the following estimate which implies theorem [ thm : main amenable ] . [ folsw ] suppose that @xmath20 is a finitely generated group with generating set @xmath85 , and @xmath272 is a cofinite subgroup . for any subset @xmath164 , @xmath273 iterating lemma [ lembalsab ] , we see that there is a sequence of subsets @xmath274 so that @xmath275 , @xmath276 and @xmath277 applying proposition [ folbisec ] to estimate @xmath278 , we see that @xmath279 clearly @xmath280 . also , there is at most one @xmath281 for which @xmath282 lies in any interval @xmath283 of the form @xmath284.\ ] ] if @xmath285 , then , since @xmath286 is non - increasing , @xmath287 therefore , @xmath288 to complete the proof of theorem [ thm : main amenable ] , we just take @xmath289 and note the above bound implies @xmath290 as long as @xmath291 . the following bounds may also be useful in practice . suppose @xmath20 is a finitely generated group , and let @xmath31 be any sequence of finite index subgroups with @xmath292 . if @xmath293 for some @xmath294 , then @xmath295 . if @xmath296 , then @xmath297 . plug the given bound on @xmath286 into equation [ folswest ] and then note that the series decays exponentially , so it is dominated by its first term . the first step in our discussion of @xmath0-cobordism theory is to review ideas of smale , casson , and quinn . @xmath5 may be presented as @xmath298 . the all - important middle level @xmath299 is the upper boundary of @xmath300-handles . inside @xmath299 there are two collections of disjointly ( normally framed ) embedded @xmath301-spheres the @xmath302 ascending spheres of the @xmath301-handles and the @xmath303 descending spheres of the @xmath304-handles . because @xmath5 is an @xmath0-cobordism , the intersections are algebraically : @xmath305 over the group ring @xmath306 $ ] . this is where smale took us for @xmath307 , and for general @xmath308 the result was adapted by mazur , stallings , and barden ( independently ) giving the @xmath0-cobordism theorem for @xmath309 . for @xmath310 casson showed that at the expense of increasing the _ geometric _ number of intersections @xmath311 , we could arrange : @xmath312 is an isomorphism . it is said `` @xmath313 is @xmath314-negligible . '' now , correctly framed , disjointly immersed whitney disks @xmath315 can be added into the middle level @xmath299 so that @xmath316 consist only of the whitney circles which pair the excess geometric double points , as in figure [ fig:1 ] . finally , quinn observed that if there were @xmath317 whitney disk making up @xmath5 and if there happened to be @xmath317 disjoint copies : @xmath318 , then each whitney disk could be replaced by a ( disjointly ) embedded one , @xmath319 , by tubing the @xmath320 whitney disk to the first @xmath321 factor of the @xmath320 @xmath15-point and then `` piping off '' ( the norman trick @xcite ) its intersection to the copies of the second factor sphere . to relate the topology of @xmath24 to the graph theory of @xmath323 , we make the following definition . let @xmath299 be the middle level of an @xmath0-cobordism @xmath5 as above . we define a graph @xmath324 , the _ geometric intersection graph _ of @xmath5 as follows . @xmath324 has a vertex for each red and blue sphere in @xmath299 , and for each geometric intersection of a red and blue sphere we have an edge in @xmath324 . ( in particular , @xmath324 is bipartite , since all spheres of the same color are disjoint from each other . we could also assign @xmath325 signs to edges based on the sign of the intersections , and note that the `` signed reduction '' of @xmath324 is the @xmath326 graph . however we will not need this additional data , so we ignore it . ) let @xmath329 be the universal cover of @xmath5 . @xmath20 acts on @xmath330 , and the quotient graph @xmath331 , furthermore @xmath332 . for each vertex @xmath333 , choose a lift @xmath334 . we can then define maps @xmath335 by @xmath336 @xmath337 is a bijection , though clearly it is not an isometry . let @xmath338 and let @xmath339 be the cardinality of a generating set of @xmath20 . let @xmath340 be a set , and define @xmath341 we claim @xmath342 ( here we define @xmath41 as the subset of vertices in @xmath42 adjacent to the complement of @xmath42 ) . let @xmath343 be the set of all vertices in @xmath42 which are distance no more than @xmath344 away from the complement of @xmath42 , then @xmath345 . but by definition of @xmath344 we see that @xmath346 . we then take @xmath347 to complete the proposition . [ prop : resolve via sw ] let @xmath24 be a sequence of covering spaces of an @xmath0-cobordism @xmath5 , and let @xmath26 be the minimal integer so that @xmath348 is a trivial product . then there is a constant @xmath57 so that @xmath349 . let @xmath350 be the maximal number of whitney disks attached to any one red or blue sphere in @xmath299 . for a set @xmath351 , let @xmath41 be the set of vertices in @xmath42 which are adjacent to the complement of @xmath42 , and let @xmath352 be the set of vertices in @xmath42 which are distance at most @xmath301 from the complement of @xmath42 . it follows that @xmath353 since @xmath350 is the maximal degree of any vertex in @xmath354 . for each @xmath58 we choose a sweepout @xmath355 of @xmath354 so that @xmath356 . let @xmath357 . we show that @xmath348 is a trivial product , which shows that @xmath358 suffices for the conclusion of the proposition . by convention we think of all red spheres in @xmath359 as fixed , and gradually isotope the blue spheres so that they are in geometrically canceling position , i.e. , they intersect only one red sphere , and intersect it exactly once . we trivialize @xmath360 inductively in @xmath281 . more precisely , we prove the following statement : suppose we can find an isotopy of the blue spheres @xmath361 so that all blue spheres in @xmath362 are in geometrically canceling position . suppose further that there are @xmath363 disjoint copies of @xmath364 in @xmath365 , so that each copy intersects only a single blue sphere in @xmath366 , and each such blue sphere intersects at most @xmath350 copies of @xmath364 . then we can find an isotopy satisfying the same property for @xmath367 . @xmath367 differs from @xmath362 by addition of a single vertex . if that vertex is a red sphere , then we do nothing : the blue spheres in @xmath362 are exactly the same as the blue spheres in @xmath367 , and also the blue spheres in @xmath366 are the same as the blue spheres in @xmath367 since @xmath354 is bipartite . so suppose @xmath368 is a blue sphere , @xmath369 . after the given isotopy there are at most @xmath370 copies of @xmath15 which intersect some blue sphere , therefore there are at least @xmath371 copies of @xmath15 which are disjoint from all blue spheres . since @xmath372 is adjacent to at most @xmath350 whitney disks , we can use these @xmath350 copies of @xmath15 to replace these whitney disks with disjointly embedded whitney disks . we can then apply the whitney move to these disks , which brings @xmath372 into algebraically canceling position . the price we pay is that @xmath372 now intersects these copies of @xmath15 . it remains to show that any blue spheres which are in @xmath373 can be made disjoint from all copies of @xmath15 . let @xmath374 be such a blue sphere , and let @xmath375 be its paired red sphere . since @xmath376 it is in canceling position , meaning it intersects @xmath375 in a single point and intersects no other red spheres . also @xmath375 intersects no other blue spheres : if @xmath377 , then @xmath378 is not in geometrically canceling position , which means @xmath379 , contradicting the fact that @xmath380 . let @xmath382 be diffeomorphic to @xmath364 , so that @xmath383 . let @xmath384 . each intersection @xmath385 can be resolved by pushing @xmath386 along an arc in @xmath374 and piping off the intersection to parallel copies of @xmath375 . see figure [ fig : remove ] . after this process we have constructed a new zero framed copy of @xmath387 , and its neighborhood is a new copy of @xmath364 which is disjoint from all blue spheres and all other copies of @xmath364 . this completes the induction . f. balacheff , s. sabourau _ diastolic and isoperimetric inequalities on surfaces _ , annales scientifiques de lcole normale suprieure , vol . 43 ( 2010 ) , no . 4 , 579605 . s. k. donaldson , _ irrationality and the h - cobordism conjecture _ , journal of differential geometry 26 ( 1987 ) no . 1 141-168 . m. h. freedman , _ the disk theorem for four - dimensional manifolds _ , proceedings of the international congress of mathematicians , vol . 1 , 2 ( warsaw , 1983 ) , 647663 , pwn , warsaw , 1984 . m. h. freedman , f. quinn , * the topology of 4-manifolds * , princeton mathematical series , vol . 39 princeton university press princeton , nj , 1990 . m. h. freedman , p. teichner , _ 4-manifold topology i : subexponential groups _ invent . ( 1995 ) , no . 3 , 509529 . r. gompf , _ stable diffeomorphism of compact 4-manifolds _ , topology and its applications , 18 ( 1984 ) 115120 . m. kreck , _ surgery and duality _ , ann . of math . 149 ( 1999 ) 707754 . v. krushkal , f. quinn , _ subexponential groups in 4-manifold topology _ , geometry & topology , vol . 4 ( 2000 ) 407430 . b. moishezon , _ complex surfaces and connect sums of complex projective planes _ , springer - verlag , lecture notes in math . 603 ( 1977 ) r. a. norman , _ dehn s lemma for certain 4-manifolds _ , invent . ( 1969 ) 143-147 . f. quinn , _ the stable topology of 4-manifolds _ , topology appl . 15 ( 1983 ) 7177 .

a smooth five - dimensional @xmath0-cobordism becomes a smooth product if stabilized by a finite number @xmath1 of @xmath2)$ ] s . we show that for amenable fundamental groups , the minimal @xmath1 is subextensive in covers , i.e. , @xmath3 . we focus on the notion of sweepout width , which is a bridge between @xmath4-dimensional topology and coarse geometry .

1503.05497

competing electronic phases underlie a number of the most unconventional phenomena in condensed matter systems . when this competition is sufficiently strong , the usual outcome is a phase separation . one outstanding example of this situation is provided by materials showing colossal magnetoresistance , where competing magnetic interactions lead to phase separation between conducting ferromagnetic and insulating antiferromagnetic ( afm ) regions @xcite . as a consequence , an external magnetic field can be used to control the resistance over many orders of magnitude , offering possible applications in electronic devices . in cuprate superconductors , the competition between antiferromagnetism and superconductivity forms the basis for the majority of the observed phenomena and for several classes of materials the debate can be phrased in terms of the extent to which phase separation is the outcome . the stripe phase , which has been the object of heated research interest for two decades , can be considered as a form of atomic - scale phase separation between afm and superconducting ( sc ) regions , and such self - organizing heterostructures are a direct reflection of the electronic correlations whose effects are essential to understanding the mechanism of high - temperature superconductivity @xcite . the competition between afm and sc phases also forms the foundation for the physics of iron - based superconductors @xcite , where it is manifest in the emergence of a tetragonal sc phase upon doping- or pressure - induced suppression of an orthorhombic afm phase . iron - based superconductors have in common a quasi - two - dimensional atomic structure of weakly coupled feas or fese planes , although the exact crystal structure varies somewhat among the 1111 , 122 , 111 , and 11 families of materials @xcite ; in fact the phenomena we report here will highlight some of the important differences arising between families as a consequence of the strength of their interplane coupling . currently , the detailed phase diagram close to optimal doping ( the concentration giving the maximum sc transition temperature , @xmath8 ) remains hotly debated , with evidence cited in favor of phase coexistence , of a possible afm quantum critical point , or of heterostructures of afm and sc phases . the levels of doping and disorder , and their impact on the phases and their competition , seem to vary between structural families , defying any search for universal properties . however , this variety does open additional avenues in the search for novel forms of phase separation or heterostructure formation , and with them the scope for obtaining further clues to the mechanism of high-@xmath8 superconductivity . the 111 family is based on the materials lifeas and nafeas , with doping effected most easily as nafe@xmath2co@xmath3as . the parent compound nafeas has a separate structural transition ( @xmath9 55 k ) and magnetic transition ( @xmath10 41 k ) @xcite , the latter to an afm phase with small ordered moments ( @xmath11 0.32@xmath12/fe ) @xcite . here we choose to use the notation @xmath13 , rather than @xmath14 , to reflect the strong local - moment character of the magnetic phase , an issue to which we return in sec . the separation of @xmath15 and @xmath13 , and also the relatively low @xmath13 values , count among the initial pieces of evidence for a rather weak interlayer coupling @xcite between feas planes in the nafeas system . the crystal quality , particularly the homogeneity of dopant distribution , is thought to be among the best in any iron - based superconductors , as measured in transition widths and observed by scanning tunneling microscopy ( stm ) . nevertheless , for underdoped 111 compounds a coexistence of inhomogeneous antiferromagnetism and superconductivity has been suggested by transport @xcite , angle - resolved photoemission spectroscopy ( arpes ) @xcite , and stm measurements @xcite . by contrast , the `` coexistence '' of a strongly ordered afm phase ( _ s_-afm ) and a weakly ordered one ( _ w_-afm ) on different spatial sites ( this situation may be denoted more specifically as a `` cohabitation '' ) has been reported from nmr measurements @xcite . clearly a phase inhomogeneity is observed in all of these studies . however , a detailed analysis of the intrinsic properties of the primary phases , of the exact phase diagram around optimal doping , and of the different phase volume fractions , is still required . in this paper , we exploit the power of nmr as a completely local probe to resolve the appearance and properties of the different afm and sc phases in nafe@xmath2co@xmath3as . for low dopings , we confirm the cohabitation of two regimes , _ s_-afm and _ w_-afm , finding that the _ w_-afm phase has a constant volume fraction of order @xmath16 ; this indicates an intrinsic effect unrelated to the doping concentration and we suggest that the _ w_-afm phenomenon is actually a proximity - induced moment distribution in a paramagnetic ( pm ) phase . for dopings around optimal , we find at @xmath17 0.0175 ( @xmath18 20 k ) the onset of regions of antiferromagnetism below 25 k , where the afm volume grows with cooling but the application of a magnetic field suppresses both @xmath13 and the magnetic volume fraction . at lower temperatures , superconductivity enters in the pm phase and its volume fraction increases at the expense of the afm region both on cooling and ( somewhat paradoxically ) with increasing field . for @xmath19 ( @xmath18 22 k ) , superconductivity suppresses not only afm order but also the critical afm fluctuations below @xmath8 , forming in the terminology of some authors the mechanism by which the afm quantum critical point is `` avoided . '' our results present direct evidence for the mutual exclusion of antiferromagnetism and superconductivity , which leads to a `` volume competition '' between regions of established ( finite - order - parameter ) phases replacing each other in space in a first - order manner . this volume competition can be controlled systematically by the temperature and magnetic field , and we suggest that it exists in many other iron - based superconductors . our detailed studies of the spin - lattice relaxation rate across the phase diagram indicate the importance of both itinerant ( conduction - electron ) and local - moment ( valence - electron ) contributions to both types of order . a theoretical interpretation of the strong competition points to the key role of the very two - dimensional ( 2d ) fermi surfaces in 111 systems and to orbital - selective phenomena depending on the specific bands involved at the different fermi surfaces . the structure of this paper is as follows . in sec . ii we summarize our basic sample properties and measurement procedures . we begin the presentation of our results in sec . iii by considering the nature of the afm phases in the underdoped regime . in sec . iv we focus on our samples close to optimal doping to elucidate the nature of phase cohabitation and volume competition . with these results in hand , in sec . v we complete a detailed phase diagram for the nafe@xmath2co@xmath3as system . section vi contains an interpretation of our results and a discussion of their implications for the understanding of superconductivity in iron - based materials , concluded by a short summary . our nafe@xmath2co@xmath3as single crystals are synthesized by the flux - grown method with naas as the flux @xcite . the co doping levels are monitored by the inductively coupled plasma ( icp ) technique . however , icp measurements are subject to significant inaccuracies and are by no means appropriate to establish the doping concentrations to the degree of precision required to study the nafe@xmath2co@xmath3as system , where all doping levels are anomalously low ( optimal doping @xmath20% ) . here we report the nominal stoichiometries of the different crystals and establish their relative doping values from our physical measurements by seeking continuity and possible irregularities ( sec . the sc transition temperature @xmath8 was determined _ in situ _ by the sudden decrease in inductance of the nmr coil . the zero - field @xmath8 values at different dopings are consistent with earlier reports @xcite . we have performed nmr measurements on both the @xmath0na and @xmath1as nuclei , with the field applied both within the crystalline @xmath21-plane and along the @xmath7-axis . we use a tecmag spectrometer and obtain the nmr spectra from the fourier transform of the spin - echo signal . the spin - lattice relaxation rates @xmath22 and @xmath23 were measured by the inversion method and all magnetization recovery rates could be fitted well with the function @xmath24 ( appropriate for @xmath25 nuclei ) . for detecting the magnitude of the different ordered magnetic moments in the nafe@xmath2co@xmath3as system , it is important to be able to use both the @xmath0na and @xmath1as spectra , in order to exploit their very different relative hyperfine coupling strengths , @xmath26 @xcite . because @xmath0na has a rather weak hyperfine coupling , the resulting narrow line width makes it a very accurate probe of magnetically ordered states with inhomogeneous moment distributions . by contrast , @xmath1as has a strong hyperfine coupling , making it sensitive to very weak ordered moments and ideal for proving the absence of magnetic order in a true pm phase . ( color online ) ( a ) spectral weight of the @xmath1as pm signals of nafe@xmath2co@xmath3as samples with five different dopings @xmath27 , shown as a function of temperature and measured with a field of 11.5 t applied along the @xmath7-axis . the sharp loss of spectral weight indicates the onset of afm ordering . ( b ) spectral weight of the @xmath0na signal at the center frequency as a function of temperature for four sample dopings . the inset shows the residual spectral weight of the center line at low temperatures . ( c ) @xmath0na spectra for the parent compound ( @xmath28 ) at temperatures above and far below @xmath13 . ( d ) spin - lattice relaxation rate @xmath29 for @xmath30 0.013 ; dotted lines mark the onset temperatures @xmath13 and @xmath8 for afm and sc order at the measurement field of 11.5 t.,width=321,height=302 ] we begin by considering our underdoped samples to investigate the potentially inhomogeneous afm phases reported previously @xcite . the onset temperature for the transition to magnetic order can be determined from the @xmath1as and @xmath0na nmr spectra . figure [ tn1](a ) shows the spectral weight of the center peak of the @xmath1as spectrum as a function of temperature for five underdoped samples . this quantity is the pm signal and it drops sharply at the onset of magnetic order , as the character of the magnetic environment is altered and spectral weight is transferred away from the center . we determine the nel temperature @xmath13 at each doping and we note that , far below @xmath13 , the spectral weight appears to decrease to zero , indicating no residual pm phase at any doping . in fig . [ tn1](b ) we show the spectral weight of @xmath0na at the center frequency for four dopings up to @xmath31 . again the sharp drop of spectral weight indicates the onset of afm order . however , a residual 8@xmath32 spectral weight persists far below @xmath13 for all dopings below @xmath33 [ inset , fig . [ tn1](b ) the spectra for the parent phase ( @xmath28 ) , shown in fig . [ tn1](c ) , contain a pm signal with one center peak and two satellites above @xmath13 . far below @xmath13 , there is a clear spectral splitting due to strong magnetic order , accompanied by a residual peak in the center . this result provides a good example of the sensitivity of @xmath0na measurements : our data demonstrate that a very weak afm order must be present to account for the residual spectral weight and line - width broadening . this is consistent with the absence of the @xmath1as pm signal , although the strong hyperfine coupling of @xmath1as makes it difficult to discern the nature of the magnetic state . below @xmath13 , the line width of the @xmath0na spectrum is approximately 35 khz , and therefore the upper bound on the ordered moments is only @xmath34 of that in the parent compound , where the @xmath0na spectrum is split by 540 khz . the nmr study of oh _ et al . _ @xcite reports two species of antiferromagnetism in a sample with @xmath35 0.017 , one with a large ordered moment ( _ s_-afm ) and the other one with a small moment ( _ w_-afm ) . our data from lower dopings are consistent with the finding of a small volume fraction of a _ w_-afm phase @xcite , but we find [ inset , fig , [ tn1](b ) ] that this volume fraction does not change with doping . this result indicates that the appearance of the _ w_-afm phase is not an intrinsic consequence of co doping ; if it is a disorder effect then it must be of a different type , perhaps with its origin in a strain or chemical inhomogeneity . in fig . [ tn1](d ) , we present the spin - lattice relaxation rate , @xmath29 , measured at the peak frequencies of both the _ s_-afm and the _ w_-afm signals . as noted in sec . ii the magnetization recoveries at the two separate frequencies each follow the single - component function expected for @xmath25 nuclei . the relaxation rates for both signals fall at the same ordering temperature @xmath13 . for the _ w_-afm component , a phase having such a small ordered moment but a high onset temperature @xmath13 is generically very unlikely , and we suggest that the consistent explanation is a microscopic phase separation into afm and pm regimes , but with weak magnetic order ( appearing as the _ w_-afm phase ) induced in the pm phase by its proximity to the _ s_-afm one . we note in addition that the significantly faster @xmath36 below @xmath13 in the _ w_-afm phase [ fig . [ tn1](d ) ] is also consistent with spin fluctuations being only partially suppressed by a weak proximity effect . the concept of `` nanoscale phase separation '' in iron - based superconductors is known from the depleted iron selenide materials @xmath37fe@xmath38se@xmath39 ( a = k , rb , cs , tl ; `` 245 '' ) @xcite , which appear to show a robust afm phase accompanied by an equally robust but quite separate pm phase ; the latter is the only part of the system to turn sc at @xmath8 , forming a percolating sc phase despite having a volume fraction below 10% . however , it is generally thought that this phase separation is primarily a consequence of vacancy - induced structural inhomogeneity , causing a clear doping inhomogeneity , whereas our results ( previous paragraph ) appear to exclude this in nafe@xmath2co@xmath3as . here we observe for @xmath31 [ fig . [ tn1](d ) ] that @xmath29 shows a similar fall in both the _ s_- and _ w_-afm signals at @xmath8 , which is approximately 10 k in a field of 11.5 t. although this result implies that the same type of sc state sets in at all lattice sites , we caution that the drop in @xmath29 is not at all sharp and the superconductivity is weak at best . we find only a very modest decrease in @xmath29 , by a factor of two from 10 k down to 2 k , whereas the data at @xmath40 , which we present in sec . iv , show a much larger drop of @xmath29 below @xmath8 for the pm phase . for completeness , we comment here that we also did not find an appreciable decrease in the knight shifts ( @xmath41 or @xmath42 ) below @xmath8 in the _ w_-afm phase ( data not shown ) . however , it is worth noting that nmr results showing similar drops in @xmath43 far below @xmath13 for undoped nafeas and cafe@xmath44as@xmath44 systems have been interpreted as a type of activated behavior of magnetic domain - wall motion @xcite . we leave to a future study the investigation of whether the _ w_-afm phase may in fact arise from magnetic domain walls , whose characteristic width gives the small but doping - insensitive volume fraction we observe . in this regime we can only report that our current data are not sufficient to differentiate between a scenario of microscopic coexistence , which would be expected to show a far clearer signal , and a scenario where the pm regime is a domain - boundary phase whose sc coherence length , unlike 245 , exceeds the domain size ( the length scale of the nanoscopic phase separation ) , causing proximity superconductivity to pervade the entire magnetic regime . we can state that our results are fully consistent with stm data for a similarly underdoped system ( @xmath30 0.014 ) @xcite , which show phase inhomogeneity , a sc gap on all sites , and a strong anticorrelation ( competition ) between the afm and sc order , a topic we discuss next ( sec . iv ) . to summarize our analysis of the underdoped regime , our data for the constant _ w_-afm volume fraction and the common magnetic onset temperatures for _ s- _ and _ w_-afm are strong evidence in support of proximity magnetism in the _ w_-afm / pm phase . we return in sec . vi to a detailed discussion of the phase separation between the pm and _ s_-afm regions , and of its implications for iron - based superconductivity . ( color online ) @xmath0na central line for the naco175 crystal at selected temperatures in fields of ( a ) 1 t and ( b ) 11.5 t applied along the @xmath7-axis . solid curves are fits to the spectrum with one or more gaussian functions . ( c ) temperature dependence of the magnetic ( mvf ) and superconducting ( svf ) volume fractions at each field , deduced from the gaussian fits . ( d ) field dependence of the onset temperatures of static antiferromagnetism , @xmath45 , and superconductivity , @xmath46 . , width=321 ] we turn next to our results for crystals with slightly higher doping levels , @xmath6 ( which we label naco175 ) and @xmath47 ( naco190 ) . the naco175 sample has a lower onset nel temperature and smaller ordered moment , while naco190 shows no long - ranged magnetic order , and so these samples represent the evolution of the system to optimal doping . figure [ spec2](a ) shows the @xmath0na spectra at selected temperatures in a field of 1 t oriented along the crystal @xmath7-axis . the spectrum at 30 k shows a narrow line with a fwhm of approximately 5 khz . on cooling to 25 k , a shoulder feature develops at both sides of the peak , which on further cooling grows in weight , whereas the weight of the center peak decreases . this feature indicates the development of two phases at low temperature , with pm sites giving the sharp center peak and magnetic sites giving the broad shoulders , which in contrast to undoped nafeas @xcite show a wide distribution of local fields . here @xmath48 25 k is the onset temperature of antiferromagnetism . by measuring the rf inductance , we find that the onset temperature of superconductivity is @xmath49 20 k , and below 16 k the spectrum becomes too small to detect because of strong rf screening in the sc phase . as shown in fig . [ spec2](a ) , the spectra below @xmath13 can be fitted by two superposed gaussian functions . we deduce the magnetic volume fraction ( mvf ) from the ratio of the spectral weight , taken from the gaussian fit , of the magnetic ( shoulder ) feature to the total weight . the mvf at a field of 1 t is shown as a function of temperature in fig . [ spec2](c ) , where it clearly starts to develop at @xmath48 25 k , and increases with cooling . at @xmath50 k , the mvf reaches 90@xmath32 , indicating that the sample is almost entirely magnetic . the average ordered moment for the magnetic part can also be estimated from the nmr spectrum , as the extension of the shoulder away from the central peak reflects the increase of internal static field ( ordered moment ) . at @xmath50 k , the fwhm of the magnetic part of the spectrum is approximately 150 khz , which corresponds to @xmath51 of the moment in the parent compound nafeas @xcite , or in other words an average moment of @xmath52/fe with a spatially inhomogeneous distribution . this behavior suggests that afm order develops in islands below @xmath45 and enlarges on cooling both in moment size and especially in volume fraction . at higher magnetic fields , both @xmath45 and the mvf are suppressed . figure [ spec2](b ) shows @xmath0na spectra in a field of 11.5 t applied along @xmath53 . the spectrum is single - peaked and sharp above 20 k , with no shoulder feature and hence no static magnetism . below this , the shoulder appears and we fit the spectrum with three gaussian functions to account for the center ( pm ) and shoulder ( magnetic ) components . in fig . [ spec2](c ) , the mvf at 11.5 t is seen to increase on cooling , similar to the low - field data , but with a lower onset temperature ( @xmath48 20 k ) and a lower mvf @xmath54 at 12 k. similar results for intermediate fields , also shown in fig . [ spec2](c ) , demonstrate the continuous nature of these effects . the most striking feature of fig . [ spec2](c ) occurs at the onset of superconductivity ( @xmath46 ) . above @xmath46 , the mvf increases monotonically on cooling at a fixed field , but below @xmath8 it falls away ; at 11.5 t ( @xmath8 = 18 k ) , the mvf decreases from 50@xmath32 at 12 k to 18@xmath32 at 2 k. this behavior demonstrates a direct competition for volume fraction between antiferromagnetism and superconductivity , which is also visible in the spectra shown in fig . [ spec2](b ) . [ spec2](d ) , we show both @xmath45 and @xmath46 as functions of field . @xmath13 shows a quite significant decrease with field , which can be fitted by the functional form @xmath55 , producing an estimate of the critical field for @xmath56 to occur at @xmath57 t. by contrast , the field - induced decrease of @xmath46 is slower , consistent with the critical field for superconductivity being located at @xmath58 t in this system @xcite , and suggesting that antiferromagnetism is suppressed at far lower fields than superconductivity . we comment here that such strong field effects on @xmath13 are highly unusual in iron - based sc materials , where the in - plane magnetic interactions are normally many tens of mev , and we stress that this result is obtained only for our samples close to optimal doping . for the underdoped samples discussed in sec . iii , we found no significant field - induced changes either to @xmath13 or to the _ s_-afm volume fraction up to 12 t ( data not shown ) . we return to this issue in sec . indeed our mvf data demonstrate that superconductivity is more stable than antiferromagnetism , replacing it at low temperatures for all fields . we will show later that for @xmath6 , superconductivity occupies the pm phase , but not the afm phase , during the replacement process . if the rf screening is non - uniform and strong in the sc regions , then our data provide an upper bound for the mvf below @xmath8 . the drop of mvf on cooling indicates that the sc volume fraction , meaning the fraction of a percolating sc state , increases . however , the average moment in the magnetic regions remains large , @xmath59/fe , when @xmath60 , and therefore we observe that afm and sc order compete over the system volume , excluding each other in a first - order manner rather than coexisting with a reduced order parameter . although competitive behavior of antiferromagnetism and superconductivty has been reported by neutron scattering studies of the ba(fe@xmath2co@xmath3)@xmath44as@xmath44 system @xcite , these can not distinguish whether the phases compete by order - parameter suppression ( second order ) or volume suppression ( first order ) . our unambiguous demonstration of volume competition in nafe@xmath2co@xmath3as is a key result whose origin and implications we discuss in sec . ( color online ) ( a ) spin - lattice relaxation rate @xmath29 measured on @xmath0na for naco175 , shown as a function of temperature with a field of 11.5 t applied along @xmath53 . dotted lines denote @xmath13 and @xmath8 , circles are data measured at the peak of the spectrum ( pm phase ) , and diamonds are measured in the broad shoulder ( afm phase ) . solid lines are guides to the eye . ( b ) @xmath61 measured on @xmath1as for naco175 as a function of temperature with the same field . inset : @xmath1as nmr spectra at selected temperatures . solid lines are gaussian fits to the data . , width=321,height=226 ] turning now to further details of magnetism in the naco175 sample , we note that the observed shoulder spectrum of @xmath0na is consistent with the _ s_-afm phase reported in sec . iii , but with the lower @xmath13 value expected at a higher doping . however , there is no longer any evidence for ordered moments in the @xmath0na pm signal , suggesting that the _ w_-afm phase is absent at this doping . we confirm that the absence of ordered moments is not a resolution issue by comparing with the @xmath1as nmr spectrum . as noted in sec . ii , @xmath1as has a much stronger hyperfine coupling than @xmath0na , and therefore its strong sensitivity to any weak magnetic order makes it the optimal probe for excluding a _ w_-afm component in the pm signal . as shown in the inset of fig . [ invt1t3](b ) , the @xmath1as spectrum is single - peaked at all temperatures . a narrow line with fwhm @xmath62 14 khz is observed above @xmath63 k , and below this its spectral weight begins to decrease due to the increasing afm volume fraction . the afm signal lies outside our @xmath1as measurement window , because its fwhm is very large ( it can be estimated from the @xmath0na data to be around 2.4 mhz ) . the single - peaked form of the @xmath1as spectrum below @xmath45 is consistent with the pm @xmath0na signal , which remains sharp on cooling . below @xmath64 k , the spectra shift downward , as expected for singlet superconductivity , and at this point the @xmath0na spectrum does become broadened ; at @xmath65 6 k , far below @xmath8 , we observe a fwhm @xmath62 24 khz as a consequence of the vortex structure in the sc phase . this value of the fwhm sets a strict limit on the ordered moment of the pm phase , which should be less than @xmath66 of the moment in nafeas ( 0.32@xmath12/fe ) , and thus effectively excludes any possibility of _ w_-afm character at @xmath6 . next we focus on the sc state of the naco175 sample . at low temperatures , the pm phase is found to be purely superconducting by inspection of the spin - lattice relaxation rates for both @xmath0na and @xmath1as . figure [ invt1t3](a ) shows @xmath29 in a field applied along the @xmath7-axis ; above @xmath45 we observe a decrease on cooling down to 40 k , followed by an increase on further cooling below 40 k. the high - temperature behavior is consistent with local - moment fluctuations @xcite and the low - temperature upturn with the spin fluctuations of itinerant electrons at the fermi surface @xcite . below @xmath45 , @xmath29 is no longer uniform , showing a different form if taken at different parts of the spectrum . for the shoulder , @xmath29 drops quickly to a small value below @xmath45 , as a consequence of the onset of static afm order . for the peak , @xmath29 continues to increase on cooling , falling only when the sc state is reached . figure [ invt1t3](b ) shows the corresponding results taken from @xmath1as , which are naturally uniform because the spectra [ inset , fig . [ invt1t3](b ) ] have only a pm peak and no shoulders . from @xmath67 k down to 80 k , @xmath61 decreases linearly with temperature due to thermal excitation of local spin fluctuations in 2d @xcite . below 80 k , the relaxation rate increases strongly with the @xmath68 form characteristic of low - energy itinerant spin fluctuations @xcite . the uniform sharp drop of @xmath43 at @xmath8 for the pm signal of both nuclei indicates that the pm state becomes fully sc and it is believed from arpes measurements that a full gap opens at all points on the fermi surface ; however , we comment that nmr data have not been able to verify this second point directly ( the apparent linear form of @xmath29 visible around @xmath69 in fig . [ invt1t3](a ) , which may be of extrinsic or intrinsic origin ) . ( color online ) ( a ) @xmath0na and ( b ) @xmath1as nmr spectra for naco190 , shown at selected temperatures a field of 11.5 t applied along @xmath53.,width=321,height=226 ] the spin - lattice relaxation rates in fig . [ invt1t3](a ) suggest further that the _ s_-afm phase is not strongly coupled to superconductivity . measurements of @xmath29 for magnetic sites ( in the shoulder of the spectrum ) show a drop at @xmath45 , where magnetic order sets in , but there is no discernible drop at @xmath8 . thus there is no evidence that the _ s_-afm phase supports even weak or proximity superconductivity . we stress that @xmath29 at 12 k and 11.5 t , deep within the ordered phases , reaches a similar value for sites in both the sc and the _ s_-afm regions . this appears to be a clear statement that electrons on the fermi surface are gapped by either type of order , and , taken together with fig . [ spec2](c ) , that relaxation contributions become dominated by sc electrons at low temperatures . this again reflects the fact that the competition between antiferromagnetism and superconductivity for electrons on the fermi surface ( i.e. in reciprocal space ) , and its apparent first - order nature , results in the volume competition ( in real space ) we observe in the nmr spectra . finally , we comment that further information concerning the volume - competition effect can be gained by investigating samples with higher doping , namely @xmath70 . our naco190 sample shows a structural transition at @xmath71 k , which we discuss in detail in sec . v. figures [ spec4](a ) and ( b ) show respectively the @xmath0na and the @xmath1as spectra for different temperatures . for @xmath0na , the spectrum is single - peaked and no shoulder feature develops on cooling , even down to 2 k. this observation excludes the existence not only of a possible _ w_-afm component but also of the _ s_-afm phase . for @xmath1as , the spectrum also has a single peak , with fwhm @xmath62 40 khz at the lowest temperatures , which also excludes any type of afm order . below 18 k , the spectra shift to lower frequencies and a line broadening is clearly visible for @xmath1as , which is the hallmark of the onset of singlet - pairing superconductivity . thus the effect of doping on volume competition is to terminate the battle in favor of superconductivity at @xmath19 , where one finds a single , uniform phase with only structural and sc transitions , but a complete absence of afm order . we now compile all of our results , from samples across the full range of doping , to prepare a definitive @xmath72 phase diagram . first , the structural transition can be detected from the frequency of a chosen satellite line in the @xmath1as spectrum @xcite , which is shown in fig . [ sus5](a ) as a frequency shift relative to the center line . when the field is applied in the @xmath21-plane , cooling from the tetragonal to the orthorhombic phase causes each satellite to shift and to split into two due to sample twinning . the sudden change of the satellite frequency as a function of temperature , clearly visible in fig . [ sus5](a ) , determines @xmath15 for the structural transition at each doping . we comment here that above @xmath15 this satellite frequency , which for perfect field alignment is the quadrupole frequency @xmath73 , is generally expected to show a systematic increase with sample doping @xcite ; however , such a dependence is barely discernible in our data [ fig . [ sus5](a ) ] due to the fact that the maximum doping we studied in the nafe@xmath2co@xmath3as system is so low ( @xmath74 ) . the variation in our measured values of @xmath75 above @xmath15 for the different samples is in fact dominated by the small but finite misalignment of the magnetic field , whose exact orientation with respect to the crystalline @xmath76-and @xmath77-axes was not determined . nmr provides an accurate measurement of @xmath15 up to @xmath40 , beyond which the sample remains tetragonal at all temperatures . ( color online ) ( a ) determination of the structural transition temperature @xmath15 from the frequency shift @xmath75 of the satellite line in the @xmath1as spectrum measured with an in - plane field , shown for a range of sample dopings . ( b ) determination of the sc onset temperature @xmath8 from measurements of the inductance change @xmath78 of the nmr coil at zero field , also for a range of dopings . ( c ) phase diagram showing the values of @xmath13 determined from the nmr spectra of figs . [ tn1](a ) and [ tn1](b ) , @xmath15 determined from the satellite line shift , and @xmath8 determined from the rf inductance . , width=321,height=302 ] next , the transition to static magnetic order can be detected at all dopings by the decrease of the pm spectral weight , as shown in figs . [ tn1](a ) and ( b ) . finally , the sc transition is detected by the relative inductance change of the nmr coil , as noted in sec . ii and shown in fig . [ sus5](b ) ; these measurements can also be benchmarked from the drop of the nmr knight shift and the spin - lattice relaxation rate , at least for @xmath79 ( see sec . iii ) . in fig . [ sus5](c ) we present the hierarchy of deduced transition temperatures , which show the clear evolution characteristic of iron - based superconductors . in the parent compound , nafeas , the system becomes fully magnetic below @xmath80 k , but as electron doping is induced by co substitution , the antiferromagnetism is gradually suppressed and superconductivity develops . ( color online ) ( a ) temperature dependence of @xmath61 for nafe@xmath2co@xmath3as crystals under a field of 11.5 t applied in the @xmath21-plane of the crystal . solid lines are fits to the function @xmath81 . inset : @xmath82 as a function of temperature . @xmath15 and @xmath83 denote respectively the temperatures where @xmath82 changes slope and where it goes to zero . ( b ) phase diagram of nafe@xmath2co@xmath3as established by nmr : @xmath15 , @xmath8 , @xmath83 , and @xmath13 denote respectively the structural transition , the sc transition temperature , and the curie - weiss temperature , all obtained from the fit to @xmath61 above @xmath8 , and the nel temperature obtained from the pm spectral weight [ figs . [ tn1](a ) and ( b ) ] . data for @xmath8 below @xmath6 are taken only from inductance measurements [ fig . [ sus5](b ) ] . @xmath84 denotes the approximate doping where the _ s_-afm+_w_-afm region and the _ s_-afm+pm region meet and @xmath85 denotes the doping where @xmath86 and beyond which afm order is absent.,width=321,height=415 ] however , in nafe@xmath2co@xmath3as there is a complementary approach to obtaining the phase diagram , which is that all three phase transitions can be detected directly and accurately in the spin - lattice relaxation rate . this is not possible in other pnictide materials , and as we explain below it is also a consequence of the strongly 2d fluctuations acting in the 111 system . in fig . [ pd6](a ) we show @xmath61 , obtained over the full range of available dopings with the field applied in the @xmath21-plane , a geometry known @xcite to maximize the sensitivity of the measurement to the dominant in - plane spin fluctuations in pnictides as the @xmath87 magnetically ordered state is approached . at high temperatures , the @xmath61 values are similar for all dopings other than the heavily doped @xmath88 sample , and fall slowly on cooling down to 100 k. below 100 k , a curie - type upturn develops for all dopings ( other than @xmath88 ) , becoming progressively stronger for lower doping values . clearly the low - energy spin fluctuations responsible for this behavior , which have an itinerant origin , are suppressed strongly by increasing doping , reflecting a very high sensitivity to details of the fermi surface . by contrast , the very small change above 100 k suggests an origin in local - moment ( valence - electron ) fluctuations , consistent with a general two - component interpretation of the spin response in pnictides @xcite . if the low - temperature @xmath61 data are fitted with the function @xmath68 , one may extract a curie temperature @xmath89 . the structural transition can be deduced by considering the quantity @xmath82 , as shown in the inset of fig . [ pd6](a ) for different underdoped and near - optimal dopings . to a good approximation , @xmath90 can be fitted with two straight lines of different gradients , one ( which we denote @xmath91 ) corresponding to the tetragonal structure and the other , @xmath83 , to the orthorhombic one . the change in this gradient gives the structural transition temperature @xmath15 . the values of @xmath91 and @xmath83 can be obtained from the intercept of the fitting line with @xmath92 . we find that the values of @xmath91 ( obtained by extrapolation from above @xmath15 ) are all negative and decrease with doping , but these are of limited physical meaning because the tetragonal structure is replaced at @xmath15 , perhaps precisely because it does not allow the magnetic transition from which the orthorhombic phase can profit . by contrast , @xmath83 , which is determined from the structure and data below @xmath15 , is positive and decreases with doping up to @xmath40 . the fact that @xmath83 is far larger than @xmath91 reinforces the evidence that the low - energy spin fluctuations are strongly enhanced below @xmath15 , suggesting a clear role for the magnetic sector in driving the structural transition . although there is already significant evidence for a coupling between the lattice structure and the magnetism in 111 materials @xcite , this coupling is manifestly not strong enough that @xmath13 coincides with @xmath15 , as it does in the 122 system . as noted in sec . i , some authors @xcite have taken the lack of coincidence between @xmath13 and @xmath15 as a hallmark of strong two - dimensionality in some iron pnictide materials . as noted above , the nel temperature @xmath13 is determined directly from the decrease of the pm spectral weight [ figs . [ tn1](a ) , [ tn1](b ) , [ sus5](c ) ] . for nafe@xmath2co@xmath3as , the fitting parameter @xmath83 , which we obtain from our data below @xmath15 , has exactly the same value as @xmath13 for @xmath93 . this extremely significant result has not been reported in any other iron pnictide systems . @xmath13 is a true measure of static afm order , setting in due to all of the couplings in the system . by contrast , @xmath83 is a measure of the 2d spin fluctuations of the feas planes . for quasi-2d systems , the energy scale of in - plane fluctuations ( @xmath94 ) sets a characteristic temperature @xmath95 in the spin response , but true long - range order is forbidden by the mermin - wagner theorem . however , a weak coupling @xmath96 between these planes is sufficient to create long - range order in three dimensions and the transition temperature will be @xmath97 . only when @xmath96 becomes a significant fraction of @xmath94 will the planar physics be supplemented by conventional three - dimensional fluctuations and @xmath13 will exceed @xmath95 by an amount depending on @xmath96 . the magnetic interactions in the parent compound nafeas have been measured directly in a very recent study of spin - wave dispersion relations by inelastic neutron scattering @xcite . these authors find in - plane couplings @xmath98 mev , @xmath99 mev , and @xmath100 mev , respectively for superexchange processes in the @xmath76 , @xmath77 , and diagonal ( @xmath101 ) directions in the feas plane , where @xmath102 is the effective spin ( moment ) on the fe ions , but an interplanar coupling of only @xmath103 mev . thus our observation of perfect coincidence between @xmath83 and @xmath13 is completely consistent with the measured value @xmath104 , reflecting the minimal contributions from interplanar coupling , and these results form the best evidence yet available for the extremely 2d nature of the nafeas system . we defer a discussion of the microscopic implications of this result to sec . we are now in a position to present the complete phase diagram , which is shown in fig . [ pd6](b ) . we begin by drawing attention to the error bars on the doping ( @xmath27 ) axis , which given the extreme sensitivity of the system to small changes in @xmath27 is critical information . as noted in sec . ii , we do not possess probes capable of determining the doping level to the 0.1% accuracies mandated by the physics of nafe@xmath2co@xmath3as , and so we have taken the nominal doping in every case . however , we are able to benchmark our samples against each other and the remarkably smooth evolution in properties , as well as the perfect match between the @xmath15 datasets , shown in figs . [ sus5](c ) and [ pd6](b ) , indicates that our conservative estimate of the doping error as @xmath105 0.2% is reasonable . this smooth evolution over such a fine range of doping also supports the suggestion that nafe@xmath2co@xmath3as provides some of the most homogeneous and highest - quality crystals of any iron - based sc material . returning to the matching of datasets between figs . [ sus5](c ) and [ pd6](b ) , the agreement in @xmath15 benchmarks the value of the analysis of fig . [ pd6](a ) ; the agreement in @xmath8 is perfect for @xmath79 , but the values extracted from @xmath61 at lower dopings are inaccurate for the reasons discussed in sec . iii , and so the data shown in fig . [ pd6](b ) for the underdoped samples are those of fig . [ sus5](c ) ; the agreement between @xmath13 and @xmath83 is remarkable for the reasons discussed above . our nmr data [ figs . [ sus5](c ) and [ pd6](b ) ] provide direct evidence for a complete separation of @xmath15 and @xmath13 at all dopings in nafe@xmath2co@xmath3as , with the structural transition always occurring some 12@xmath10615 k above the magnetic one . thus , in common with the majority of iron - based sc systems , the coupling between the lattice and the magnetism is `` subcritical '' in the sense of a combined transition . the @xmath15 line terminates at a slightly higher doping than @xmath13 , and in fact at a value coinciding with optimal doping , as indicated also in transport studies @xcite . the @xmath8 line is quite flat in the doping range @xmath107 , whereas all of @xmath15 , @xmath83 , and @xmath13 change dramatically . in the zero - temperature limit , antiferromagnetism and superconductivity cohabit for @xmath108 , as shown in sec . iv [ fig . [ spec2](c ) ] , but within the phase - separated _ s_-afm and pm ( _ w_-afm ) regions ( sec . we comment again here that the onset of the _ w_-afm phase at the same temperature , @xmath13 , as the _ s_-afm one suggests that the former is actually a set of narrow pm regions ordered by a proximity effect . above @xmath8 , the _ s_-afm and _ w_-afm phases `` coexist , '' by which we mean `` cohabit as phase - separated regions , '' in the doping range @xmath109 and the _ s_-afm and pm phases cohabit over the range @xmath110 , as represented in fig . [ pd6](b ) . our data fix the values @xmath111 and @xmath112 . to determine the detailed structure of the phase diagram close to optimal doping , we show in fig . [ pd6](b ) all of the values of @xmath83 determined from the curie - weiss fit below @xmath15 . as noted above , @xmath83 coincides with @xmath13 for @xmath113 0.0175 . however , for doping @xmath40 , @xmath83 determined from the @xmath61 curve above @xmath8 predicts a finite nel temperature ( 13.5 k ) below @xmath8 ( 22 k ) . in a scenario where antiferromagnetism and superconductivity coexist , one might expect a magnetic quantum critical point ( @xmath114 ) to occur at @xmath35 0.02 . however , we demonstrated at the end of sec . iv that afm order is completely absent for @xmath40 . thus we conclude that in nafe@xmath2co@xmath3as , the appropriate scenario for the regime @xmath115 is a complete suppression of long - range afm order by volume competition ( sec . iv ) . given that @xmath83 @xmath116 25 k at @xmath6 and @xmath117 13.5 k at @xmath40 , it is reasonable to estimate that the afm transition line touches the sc dome at @xmath35 0.018 , although we do not have samples with this doping . because superconductivity suppresses the mvf below @xmath8 , we expect that the _ s_-afm transition line is pushed to lower dopings below @xmath8 , as also represented in the slope of the line to ( @xmath118 ) in fig . [ pd6 ] . for all dopings ( @xmath27 ) beyond this line , our experiments show unequivocally that the system is single - phased with only one transition , the onset of superconductivity in a structurally tetragonal system . our nmr measurements across the phase diagram in nafe@xmath2co@xmath3as reveal a number of key features , including the first - order volume competition , the microscopic phase separation , and the dominance of superconductivity at low temperatures . we discuss these points in turn , finding that their common denominator is the two - dimensional nature of the nafeas system . beginning with the volume competition effect , the large changes of the mvf we observe ( sec . iv ) as functions of both temperature and field suggest a mutual exclusion of afm and sc order in real space . such a real - space competition implies a first - order transition between two states with finite order parameters , which would appear to reflect a strong competition in reciprocal space , meaning for electrons at the fermi surface . certainly the fact that the low - energy spin fluctuations , which are due to itinerant ( fermi - surface ) contributions @xcite , are fully gapped below both @xmath13 and @xmath8 for the @xmath30 0.0175 sample suggests that both types of order , stabilized by their own particular fermi - surface electronic order parameter ( @xmath119 or @xmath120 ) , compete for the same electrons . such a temperature- and field - controlled magnetic and sc volume fraction has not been reported in any other iron pnictides . we suggest that there are two reasons why this highly unconventional phenomenon has been observed ( to date ) only in nafe@xmath2co@xmath3as . one is the extreme 2d nature of the fermi surfaces and the other is the very fine control of the doping level , which is obtained in nafeas systems with no loss of chemical homogeneity . addressing first the two - dimensionality , the question of whether electronic correlation effects result in competition or coexistence between antiferromagnetism and superconductivity has been fraught with contradictions in the iron - based sc systems . microscopic coexistence of afm and sc order has been reported in several compounds with the 122 structure , including by some of us @xcite . afm and sc states in iron - based sc materials depend rather sensitively on the interactions of quasiparticles at the fermi surface , and as a result it was suggested in ref . @xcite that the ability of an iron - based sc system to host both types of order may be dictated by the variety of projected 2d fermi surfaces available . the @xmath7-axis band dispersion in bafe@xmath44as@xmath44 is quite significant @xcite , making it possible that the 122 structure may allow the coexistence of the two types of order for electrons on different parts of the fermi surface . in ba(fe@xmath2ru@xmath3)@xmath44as@xmath44 , the fact that @xmath61 drops due to gap formation at @xmath13 and again at @xmath8 demonstrates that additional sc electrons are present in the afm phase @xcite . by contrast , this dispersion is much weaker in the nafe@xmath2co@xmath3as system , as shown both by arpes studies @xcite and by the present results ( sec . v ) , indicating a highly 2d system with little flexibility in fermi - surface sizes and spanning wave vectors . this leaves little option for the afm and sc order but to `` fight it out '' for the available electrons , leading to the strong competition we observe ( sec . iv ) : in nafe@xmath2co@xmath3as , @xmath29 drops only at @xmath13 or at @xmath8 , and to the same low - temperature value , showing directly that antiferromagnetism and superconductivity compete for the same electrons . our results suggest strongly that the microscopic coexistence of afm and sc order is not possible in nafe@xmath2co@xmath3as , which among other things should exclude any intrinsic superconductivity on sites with _ s_-afm order in the underdoped region ( sec . iii ) . on this note , we turn next to the issue of phase separation , for which we find evidence throughout the phase diagram . clearly distinguishable _ s_-afm and _ w_-afm regions are present in the underdoped system ( sec . iii ) , although the volume fraction of the _ w_-afm phase barely changes with either doping or temperature and there is only one magnetic transition . at near - optimal underdopings , there is clear phase separation between the afm and pm / sc regions , demonstrated very explicitly by the large changes of afm and sc volume fractions we observe in our naco175 sample as functions of both temperature and field ( sec . iv ) , which strongly suggest a first - order transition between the two phases . the phase diagram of fig . [ pd6](b ) shows directly that the optimal @xmath8 is achieved when both the afm and orthorhombic phases are suppressed . although this type of phase diagram has been interpreted as an incipient afm quantum critical point of the orthorhombic system , in reality its suppression by the onset of the mutually exclusive ( volume - competitive ) sc phase is the dominant physics . such a complex phase diagram may also exist in other iron - based sc systems , although , for reasons of the fine doping control mentioned above , none has yet been resolved in the kind of detail possible in nafe@xmath2co@xmath3as . within this intricate phase structure , the inevitable presence of weak disorder could certainly provide an extrinsic origin for the phase separation of _ s_-afm and _ w_-afm regions in the underdoped regime . however , we appeal again to the evidence for remarkably high sample homogeneity in nafe@xmath2co@xmath3as and to the obvious fact of very low carrier densities . if the origin of phase separation between the _ s_-afm and _ w_-afm regions is the same as that between the _ s_-afm and pm / sc regions near optimal doping , then such behavior may in fact be intrinsic in systems with strong electronic correlations . in cuprate materials , the stripe phase @xcite may be considered as an atomic - scale phase separation of afm and pm / sc regions . similar phase separation has been reported recently close to the phase boundary both in an organic superconductor @xcite and in heavy - fermion superconductors tuned by pressure and magnetic field @xcite . complete nanoscale phase separation is familiar in iron - based sc materials from the case of the 245 iron selenides . however , as noted in sec . iii , the situation in nafe@xmath2co@xmath3as does not appear to be the same , first in that structural and hence doping inhomogeneity is significant in 245 materials and second in that weak superconductivity can be observed throughout the sample below @xmath8 . in sec . iii we outlined two scenarios for the present result , a microscopic coexistence or a microscopic phase separation , and , as noted above , the results of sec . iv make a coexistence appear extremely unlikely . although we can not claim evidence for a nanoscale phase separation from our data , all of our results are consistent with such a scenario , under the proviso that the length scale of the phase - separation phenomenon be extremely short . we have observed a phase susceptible to an apparent bulk magnetic order by proximity effects , which is a definite statement that the phase - separation length scale should be short compared to the magnetic correlation length . the weakness and feeble onset of the apparent bulk sc regime could be the fingerprints of proximity superconductivity originating in narrow pm regions but spreading throughout the very small magnetic domains . microscopically , in the absence of doping inhomogeneity effects , the scale of the phenomenon is expected to result from a subtle interplay between electronic correlations and lattice or charge inhomogeneities ( independent of co doping ) , and could indeed be on the nanometer scale . first - order phase separation close to the afm quantum phase transition has certainly been suggested in other iron - based sc materials , most notably bafe@xmath121ni@xmath3as@xmath44 , although the fact that @xmath15 and @xmath13 merge in the 122 systems may cause qualitative differences in the phase diagram . one key proposal from these neutron scattering studies is an incommensurate nature of the resulting afm phase @xcite . if the direct volume competition of the afm and sc phases we observe below @xmath46 takes place in a nanoscale lamellar structure , then an incommensurate afm signal may indeed be observed . a different interpretation , namely a double quantum critical point , has been offered from combined transport and nmr studies in bafe@xmath121ni@xmath3as@xmath44 @xcite , but the magnetic structure below @xmath8 was not resolved . nmr measurements on ba(fe@xmath2co@xmath3)@xmath44as@xmath44 suggest a cluster spin - glass phase close to optimal doping @xcite ; in nafe@xmath2co@xmath3as this type of physics can be excluded explicitly from our data , which do not contain the stretched spin recovery of a spin glass , indicating again the high quality of our samples . one of the fundamental questions in formulating a microscopic model for the iron - based sc materials is the equitable treatment of local - moment ( valence - electron ) and itinerant ( conduction - electron , or fermi - surface ) contributions to the macroscopic properties of magnetism and pairing . returning again to the key observation ( sec . iv ) of large changes in the mvf with both temperature and field , these suggest a microscopic phase separation , which one expects to be driven by a first - order magnetic quantum phase transition . because the field is applied along the @xmath7-axis , which is perpendicular to the direction of the ordered moments , a spin - flop transition can not account for the observed field effect . rather , the strong suppression of @xmath45 by an applied field near optimal doping ( sec . iii ; the rate is approximately @xmath122 k / t close to @xmath123 ) , the classical critical behavior with increasing field [ fig . [ spec2](d ) ] , and the extreme field sensitivity to the competing sc phase close to the putative magnetic quantum critical point , all suggest that the interactions causing afm order are largely local in nature . in this context it is worth remarking that for overdoped systems up to @xmath30 0.07 , where low - energy spin fluctuations are entirely absent [ fig . [ pd6](a ) ] but @xmath8 is still approximately 10 k , it is local - moment spin fluctuations originating in the valence electrons that provide the pairing interactions for superconductivity . high - pressure nmr studies in overdoped nafe@xmath2co@xmath3as samples demonstrate direct contributions to superconductivity from both itinerant - electron spin fluctuations ( which are pressure - dependent ) and local - moment fluctuations ( which are largely pressure - independent ) @xcite . thus it is clear that both local - moment magnetism and itinerant electrons are required for a complete understanding of the nature of antiferromagnetism and superconductivity , and in this light we revisit the key result of sec . v that @xmath124 . the low - energy spin fluctuations causing the curie - weiss upturn in @xmath43 , and hence determining @xmath83 , are due only to fermi - surface electrons , so the equality with @xmath13 indicates only very weak contributions to long - range order from an inter - plane interaction @xmath96 ( sec . v ) , which from the previous paragraph we conclude is mediated by local - moment fluctuations . this result , demonstrating clearly the extremely 2d nature of the nafeas system , suggests that the cornerstone of a microscopic model should be the itinerant contribution to ( planar ) magnetic order , without which the local - moment interactions appear quite unable to order alone . finally , we comment that the suppression of antiferromagnetism by superconductivity at low temperatures for the @xmath6 sample , even though @xmath13 exceeds @xmath8 , indicates that the sc phase possesses additional electronic or magnetic channels with which it can `` win '' against afm order . a full microscopic model of the band structure , fermi surfaces , and correlation effects due to valence - electron contributions is required to account for this effect , and here we can only present the possibilities suggested by our data . one may be that superconductivity competes only with the afm tendencies of the itinerant electrons , which in other iron - based superconductors are reinforced by sizeable local - moment contributions , but that ( previous paragraph ) the extremely 2d nature of nafeas weakens this link to the point that the afm phase is disrupted completely . the microscopic physics underlying such behavior will be found naturally in an orbital - specific model . even in a fully 2d system , the electronic bands of iron pnictides contain five different @xmath125-orbitals , and hence many degrees of freedom in orbital symmetries and admixtures . antiferromagnetism and superconductivity are favored by itinerant electrons in bands of different orbital content , and the unique feature of nafe@xmath2co@xmath3as is that the band occupations , and thus their fermi surfaces , are inordinately sensitive to the doping because of the highly 2d nature of the system . a recent arpes study of nafe@xmath2co@xmath3as samples with small and large @xmath27 @xcite has provided some indications for orbital - selective connections between the competing afm and sc phases , specifically concerning the relative @xmath126 and @xmath127 content of the bands near the fermi surface . we suggest that similarly detailed studies of samples near optimal doping have the potential to reveal the underlying physics of nafeas . in summary , by using nmr as a local probe sensitive to both antiferromagnetism and superconductivity , we observe a strong volume competition between the two phases at the boundary of the antiferromagnetic phase transition in nafe@xmath2co@xmath3as . the volume fractions of the two phases can be controlled by varying both the temperature and the applied magnetic field , and show a complete mutual exclusion in real space . thus our nmr data support a first - order phase transition between antiferromagnetism and superconductivity , which is driven by the competition between their electronic order parameters in reciprocal space . as striking as the volume competition effect is the exquisite sensitivity of the competition to doping , with optimal doping and all of the phase - separation effects occuring between 0 and 2% . these phenomena have their origin in the extremely weak interplane coupling in the nafeas materials , resulting in a very two - dimensional nature of the electronic band structure , and hence of the fermi surfaces . one key anomaly compared to other iron - based superconductors is the winning of superconductivity over antiferromagnetism in real space at moderate underdopings , even where the magnetic transition temperature is higher , suggesting that the generic behavior of a two - dimensional iron pnictide may be for the electronic or magnetic channels of the fermi - surface electrons to favor superconductivity . further , because the very weak co doping also appears to be remarkably homogeneous , nafe@xmath2co@xmath3as is an excellent system in which to seek evidence of unconventional phases arising purely due to intrinsic electronic correlations . for this we obtain additional information concerning the _ w_-afm minority phase , which may be a paramagnetic regime occurring as thin lamellae due to nanoscale phase separation , but appears antiferromagnetic by proximity for underdoped samples and is the first region to turn superconducting on the approach to optimal doping . we are grateful to t. giamarchi , s. p. kou , h. h. wen , z. y. weng , and r. yu for helpful discussions . work at renmin university of china was supported by the national basic research program of china under grant nos . 2010cb923004 , 2011cba00112 , and 2012cb921704 and by the nsf of china under grant nos . 11174365 , 11222433 , and 11374364 . work at rice university and the university of tennessee in knoxville was supported by the us doe office of basic energy sciences through contracts de - sc0012311 ( p.d . ) and de - fg02 - 08er46528 ( c.l.z ) . r. m. fernandes , d. k. pratt , w. tian , j. zarestky , a. kreyssig , s. nandi , m. g. kim , a. thaler , n. ni , p. c. canfield , r. j. mcqueeney , jrg schmalian , and a. i. goldman , phys . b * 81 * , 140501(r ) ( 2010 ) . p. vilmercati , a. fedorov , i. vobornik , u. manju , g. panaccione , a. goldoni , a. s. sefat , m. a. mcguire , b. c. sales , r. jin , d. mandrus , d. j. singh , and n. mannella , phys . b * 79 * , 220503(r ) ( 2009 ) . c. he , y. zhang , x. f. wang , j. jiang , f. chen , l. x. yang , z. r. ye , fan wu , m. arita , k. shimada , h. namatame , m. taniguchi , x. h. chen , b. p. xie , and d. l. feng , j. phys . chem . solids * 72 * , 479 ( 2011 ) . liu , p. richard , y. li , l .- l . jia , g .- f . chen , t .- l . wang , j .- b . he , h .- b . yang , z .- h . pan , t. valla , p. d. johnson , n. xu , h. ding , and s .- c . wang , appl . * 101 * , 202601 ( 2012 ) . a. p. dioguardi , j. crocker , a. c. shockley , c. h. lin , k. r. shirer , d. m. nisson , m. m. lawson , n. aproberts - warren , p. c. canfield , s. l. budko , s. ran , and n. j. curro , phys . lett . * 111 * , 207201 ( 2013 ) .

we report a detailed nuclear magnetic resonance ( nmr ) study by combined @xmath0na and @xmath1as measurements over a broad range of doping to map the phase diagram of nafe@xmath2co@xmath3as . in the underdoped regime ( @xmath4 0.017 ) , we find a magnetic phase with robust antiferromagnetic ( afm ) order , which we denote the _ s_-afm phase , cohabiting with a phase of weak and possibly proximity - induced afm order ( _ w_-afm ) whose volume fraction @xmath5% is approximately constant . near optimal doping , at @xmath6 , we observe a phase separation between static antiferromagnetism related to the _ s_-afm phase and a paramagnetic ( pm ) phase related to _ w_-afm . the volume fraction of afm phase increases upon cooling , but both the nel temperature and the volume fraction can be suppressed systematically by applying a @xmath7-axis magnetic field . on cooling below @xmath8 , superconductivity occupies the pm region and its volume fraction grows at the expense of the afm phase , demonstrating a phase separation of the two types of order based on volume exclusion . at higher dopings , static antiferromagnetism and even critical afm fluctuations are completely suppressed by superconductivity . thus the phase diagram we establish contains two distinct types of phase separation and reflects a strong competition between afm and superconducting phases both in real space and in momentum space . we suggest that both this strict mutual exclusion and the robustness of superconductivity against magnetism are consequences of the extreme two - dimensionality of nafeas .

1406.2412

these notes are an abbreviated version of lectures given at the 1997 zakopane school . they contain two topics . the first is a description in elementary terms of the basic ideas underlying the speculative hypothesis that pieces of strong - interaction vacuum with a rotated chiral order parameter , disoriented chiral condensate or dcc , might be produced in high energy elementary particle collisions . the second topic is a discussion of the phenomenological techniques which may be applied to data in order to experimentally search for the existence of dcc . two other topics were discussed in the lectures but will not be mentioned in these notes other than in this introduction . one was a review of the experimental situation regarding dcc searches . there are so far only two such attempts . one has been carried out at the fermilab tevatron collider by test / experiment t864 ( minimax ) . preliminary results , all negative , have been presented at a variety of conferences@xcite . no new information is available now , and the interested reader is invited to consult the references@xcite . the other experiment , wa98 , is in the fixed - target heavy - ion beam at cern . again there is no evidence of dcc production@xcite . the analysis methods are at present being refined by that group and are different than for minimax , because they are blessed with a relatively large phase - space acceptance . a recent status report is given by nayak@xcite . the other omitted topic is quite separate , and has to do with the initiative called felix . it is a proposal for a full - acceptance detector and experimental program for the lhc dedicated to the study of qcd in all its aspects hard and soft , perturbative and non - perturbative . much has happened since zakopane with respect to felix . especially noteworthy is the production of a lengthy and detailed letter of intent@xcite , which provides much more detail than was possible in the lectures on what it is about , and in any case provides guidelines for all lhc experiments on interesting issues and opportunities in qcd worthy of study . unfortunately , at this time of writing , the initiative has run into difficulties with the cern committees and management , with its future at present uncertain . we accept without question that the usual qcd lagrangian provides a correct description of the strong interactions . nevertheless , at large distances the spectrum of the theory is that of colorless hadrons , not colorful quarks and gluons . the confinement mechanism responsible for this situation is only part of the story . in addition there is an approximate chiral @xmath0 flavor symmetry which is spontaneously broken . the pions are the collective modes , or goldstone bosons , associated with this phenomenon of spontaneous symmetry breaking . in addition , in this low - energy world where hadron resonances are a dominant feature , the constituent quark model works quite well , with an interaction potential which does not involve in any explicit way the gluons ; direct gluonic effects seem to be quite muted@xcite . there are a variety of low - energy effective lagrangians which are in use , associated with this regime . and they are quite well - motivated , with a starting point being the fundamental qcd short - distance lagrangian . the procedure of starting at short distances and ending up with a large - distance effective theory depends strongly upon taking into consideration the effects of instantons@xcite . these lectures are not the place to go into what instantons are , and it has to be assumed that the uneducated reader will search elsewhere to find out@xcite . it is rather clear on phenomenological grounds , and is supported by detailed variational calculations , that the important instantons have a size characterized by a momentum scale of about 600 mev and that the size distribution is quite sharply peaked about this value@xcite . the instantons form a dilute liquid in ( euclidean ) space - time , with a packing fraction of only 10% . nevertheless , the interactions between them , and the effects of the instantons on the fermi sea of light quarks , are very important . there are light - fermion zero modes " surrounding the instantons , and their correlations are such as to rearrange the fermi sea of light quarks in just the right way to create the chiral symmetry breaking . assuming that these instantons are indeed the most important gluonic configurations at the 600 mev scale , their main effect when integrated out " of the qcd path integrals , is to leave behind an effective action of the nambu - jona - lasinio type between the light quarks . this effective action , to be applied at momentum scales below 600 mev , does indeed imply spontaneous chiral symmetry breaking and the existence of the pionic goldstone bosons , which emerge as composites of the quark - antiquark degrees of freedom . it also constitutes a definite starting point in general for constituent - quark spectroscopy . an extensive amount of work utilizing this effective action is quite successful phenomenologically@xcite . at still lower momentum scales , or larger distance scales , the constituent - quarks themselves can be integrated out " of the effective action . they are replaced by the pionic degrees of freedom , comprising the lowest mass scale , or largest distance scale , in the strong interactions . the effective action looks then very much like the one used for the higgs sector . however the action of this effective theory need not be restricted to be renormalizable . there will be , in addition to the quadratic free - particle term and quartic interaction , terms of higher polynomial order , some with derivative couplings depending upon the choice of description . this is just the purely chiral effective action studied in great detail by gasser and leutwyler , among others@xcite . simplified versions of the chiral effective theory are the linear and nonlinear sigma models . the linear sigma model is what is isomorphic to the usual higgs theory ; the nonlinear sigma model essentially `` integrates out '' the massive sigma degree of freedom ( analogous to the massive higgs boson of standard - model electroweak theory ) , leaving only the pionic degrees of freedom in the effective action . in what follows , these effective actions are what are relevant for the description of dcc because , as we shall see , the space - time scale for the dcc system is assumed to be macroscopic " , _ i.e. _ have dimensions large compared to 1 fermi . however , we should emphasize here that the use of sigma models , be they linear or nonlinear , is only a rough approximation to the true situation at best , because the yukawa couplings of the goldstone system of pions and sigma to constituent quarks are large and should not be ignored , except perhaps at the very lowest momentum scales . as we have mentioned in the introduction , dcc is by assumption a macroscopic " region of space - time within which the chiral order parameter is not oriented in the same direction in the internal @xmath1 space as the ordinary vacuum . if we use for simplicity the language of the linear sigma model , the sigma and pion fields form a four - vector @xmath2 in the @xmath3 space , and in the vacuum there is a nonvanishing value of @xmath4 in the @xmath5 direction . what we shall assume is that in a high energy collision , there are regions of space - time where @xmath6 , essentially a classical quantity , is rotated away from the @xmath5 direction . at late times , of course , this chirally rotated vacuum " must relax back to ordinary vacuum . the mechanism will clearly be the radiation of the collective modes , the pions , via a semiclassical mechanism . it should be clear that , if indeed this happens , it is quite an interesting phenomenon , because it would provide a rather direct look into the properties of the qcd vacuum itself . how might the dcc be produced ? in hadron - hadron collisions , most models of particle production , be they stringy or partonic , put the bulk of the space - time activity near the light cone . that is , the flow of produced quanta is concentrated in a rather thin shell expanding from the collision point at the speed of light . but what happens within the interior of this shell ? if it rapidly relaxes to vacuum , then it is hard to see why the normal vacuum must be chosen , because the interior region is separated from the exterior true vacuum by a hot " shell of expanding partons . and as we shall see , the time scale for the true vacuum to be selected via the small chiral - symmetry - breaking effects associated with nonvanishing pion mass is quite long . it is this baked - alaska " scenario which will be the main thrust of this discussion@xcite . however , there is a second space - time scenario relevant to heavy ion collisions . the idealized case is that of infinite pancakes colliding with each other at the speed of light , with quark - gluon plasma produced in between the pancakes at early times . but at late times , after this plasma goes through the deconfining / chiral phase transition , the possibility of dcc formation also exists@xcite and has been rather extensively studied@xcite . the geometry in this idealized situation is that of a boost - invariant 1 + 1 dimensional expanding system , and is the most tractable case to consider from a calculational point of view . we shall review a simple example studied by blaizot and krzywicki in the next section . can one actually predict that dcc should be produced under these circumstances ? certainly not . on the other hand can the possibility of dcc production be excluded theoretically ? again , certainly not . the subject needs to be data driven . this is the reason that both cyrus taylor and i decided to go into experimental physics and make an experimental search at the tevatron collider . while the minimax experiment which emerged is very modest , and while the results so far are negative , the experience gained has been invaluable in learning how to construct good search strategies , and in determining what is necessary in order to do a better experimental job in the future . the space - time scenario for dcc production begins at an early proper time , _ i.e. _ near the light cone , when the energy density becomes low enough that the chiral order parameter is nonvanishing . the proper time scale here is plausibly somewhere between 0.3 and 0.8 fermi . at this point one must assume initial conditions for the chiral field , which then evolves , at the simplest level of approximation , according to the classical equations of motion . a natural hypothesis is , in the context of the linear sigma model , that the chiral field initially vanishes ( at least on average ) . this means it is initially on top of the mexican hat " , and rolls off the hat into the region of the minimum . during this initial roll , the linear sigma model , at the least , should be used for the proper - time evolution of the vacuum condensate . as we shall see , this evolution can go on for anywhere from one to five fermi of proper time . the formalism is as follows . the chiral fields can be written as @xmath7 or alternatively as a @xmath8 matrix of fields @xmath9 with the lagrangian for the former case being @xmath10 at late times , when the chiral field is near the minimum of the mexican - hat potential radially , but is still rolling around azimuthally , the nonlinear sigma model can be used . one writes , in the @xmath11 notation @xmath12 and freezes out the radial degree of freedom @xmath13 if one further writes as a special case @xmath14 it follows that the pion field which appears in the exponential can be shown to obey free - field equations of motion . @xmath15 this can be extended using a general , global @xmath16 rotation to a class of solutions ( anselm class"@xcite ) : @xmath17 it is important to keep in mind in what follows that the linear sigma model leads to nonlinear equations of motion , while the nonlinear sigma model here leads to linear equations of motion . while this discussion has been only at the classical level , there has been a lot of work , especially by those interested in the heavy - ion dcc , in going beyond this approximation . the state of the art is to incorporate quantum corrections to the linear sigma model in the mean - field ( or hartree , or large - n , or random - phase " ) approximation@xcite . this amounts essentially to inclusion of tadpole contributions to self - energy parts or geometrical - series bubble summations to appropriate propagators . while this level of computation is simple in translationally - invariant systems , it is not at all simple in cases like this , where the space - time geometry of sources is quite nontrivial . so far , the inclusion of quantum effects is important quantitatively but has not seriously changed the qualitative properties . in this discussion we shall remain within the classical approximation . a very easy application can be made of the above considerations@xcite . consider boost - invariant solutions of the nonlinear sigma - model equations within infinitely large heavy - ion disks receding from each other at the speed of light after a central collision . the equation of motion @xmath18 leads to the solution @xmath19 with @xmath20 and @xmath21 we see that in general the chiral angle in isospin space precesses with increase of proper time , provided the left - handed global rotation differs from the right - handed one . this curious feature depends on the choice of geometry , however , and does not generalize to the baked - alaska scenario . these calculations have been extended to the case of the linear sigma model , with qualitatively similar results . the simplest case of the baked - alaska scenario again utilizes the nonlinear sigma model . inside the future light cone one assumes spherical symmetry . in addition , it is assumed that for @xmath22 , with @xmath23 some decoupling time " , there is chirally rotated vacuum within the light cone , _ i.e. _ @xmath24 we have taken , without much loss of generality , the chiral orientation to be purely in the @xmath25 direction , because other cases can be obtained via global chiral rotations . with this hypothesis , the free equation of motion for the pion field can be used for times @xmath26 , using the boundary conditions for the field at @xmath27 . when the pion mass is neglected , the form of the solution must be that of a right - moving plus left - moving pulse : @xmath28 and the shape of the function @xmath29 is easily found to be a triangular pulse of width @xmath30 and maximum height @xmath31 . @xmath32 \ . \label{eq : o}\ ] ] thus for times @xmath33 there is only a triangular pulse of pion field ( actually @xmath34 ) radiated outward at the speed of light , which comprises the decay products of the initially formed dcc . the calculations and formalism for this example have been laid down in considerable detail in a recent paper@xcite , to which the interested reader is referred for details . in particular , the calculations have been redone for nonvanishing pion mass , as well as for the linear sigma model . an important feature of this class of solutions is that , when one generalizes them away from the production of @xmath25 field by applying a global @xmath16 rotation , the right - handed and left - handed @xmath35 rotations ( _ cf . ( 8) ) must be the same in order that the vacuum at large times inside the light cone is the same as the ordinary vacuum . another way of looking at all this is in terms of sources . the free wave equation does not hold on the light cone for times @xmath36 . it is in that region that the hot shell of expanding partons or other quanta separates the dcc vacuum on the inside from the true vacuum on the outside . so one has , everywhere in spacetime @xmath37 with the source @xmath38 having support only on the light cone . a careful look at the discontinuity of the vacuum fields across the light cone leads to the expression @xmath39 now there is a relation , well - known from the bloch - nordsieck treatment of semiclassical electromagnetic radiation , between the source which creates a classical field and the spectrum of quanta produced by the source . in this case@xcite it is @xmath40 where @xmath41 is the fourier transform of @xmath38 , put on mass shell . @xmath42 it follows that , with pion mass neglected , the expression for @xmath41 is @xmath43 one readily checks that this agrees with the spectrum calculated explicitly from the equations of motion . note that the spectrum has a high - momentum tail coming from the delta - function support on the light cone , but that the typical momentum scale is of order @xmath44 . it is an important feature of this radiation that it is coherent , which means that for a specific choice of source the multiplicity distribution of produced particles is poisson - distributed , and that there is no bose - einstein enhancement . however , upon averaging over sources , in particular their chiral orientations , these simple features undergo essential complications . another simple and interesting example of dcc production has been considered by lampert , dawson , and cooper@xcite ( hereafter ldc ) . they assume a baked - alaska scenario , within the linear sigma model , which has full boost invariance . in other words , the chiral fields existing within the light cone depend only upon the proper time . this scenario can not be regarded as realistic , because the particle distribution must look the same in all reference frames , and hence can not have a limited spectrum of energies . they argue that one can extract the physics by sampling the distribution at large proper times within a small space - time region . however , this is not a faithful description of what a real piece of experimental apparatus would see . a better strategy in my opinion is to use their solutions up to time t , and then assume , as was done in the previous section , that there is at later times no explicit source on the light cone . in other words the ldc solution at time t is evolved via the source - free linear sigma - model equations of motion , in order to generate the asymptotic fields . some work has been done along these lines@xcite , but not enough to report here . however , let us return to the simplified version of ldc . at the classical level the field equations become ordinary coupled differential equations , and can be easily solved numerically : @xmath45\ , { \sigma\choose\pi } = { f_\pi m^2_\pi\choose 0 } \ . \label{eq : u } \ ] ] however ldc do better , and include the mean - field quantum corrections . it has been found , however , that the quantum effects do not qualitatively or even quantitatively make a big difference@xcite . what is noteworthy about the solutions is that it takes a rather long proper time for the initial chiral fields to roll " into the minimum of the potential ; the time taken for the sigma field to settle down to near its vacuum value is about 5 fermi . this long proper - time interval , even in the presence of pion mass , would indicate the credibility of scenarios which create dcc from deep within the light cone . however , the finite extent of the source on the light cone needs to be investigated , as well as the effect of considering a statistical ensemble of initial conditions for the initial roll " , before drawing serious conclusions . all the theoretical attempts we have described are extremely idealized . there does exist some numerical simulation work which attempts less idealized scenarios@xcite , but i think it fair to say that none of it is yet very near to what is needed for , say , monte - carlo input appropriate to real experimental searches . there are many issues to be addressed . perhaps the most important deficiency is that the idealized cases very likely have far too much symmetry . looking at the baked - alaska scenario from the source point of view , it is probably the case that in a given event not only the source strength but also the chiral orientation of the source depends upon where in the lego phase - space one is . since the chiral dynamics is spin - zero , the correlations in the lego phase - space are most probably short - range , of order one to two units . this means that on the sphere , near 90 degrees relative to the beam , only a steradian at a time may have the rather symmetrical structure of the classical solutions we have considered . there is another way of looking at this . suppose dcc is produced and observed . it almost by definition will consist of a cluster of pions of almost identical momenta . this cluster will have a rest frame , and in that rest frame the classical radiation field associated with this cluster will in fact have approximate spherical symmetry . however in general , this frame of reference is related to the observer s frame of reference by not only a longitudinal lorentz boost , but probably also by a transverse boost . it is in fact reasonable to assume a distribution of transverse velocities of dcc such that the mean is semirelativistic , say somewhere between 0.4 and 0.8 . if this is the case , and the internal relative velocities of the pions within the dcc cluster are smaller or at least no larger , then it will often be the case that the dcc in the laboratory frame will look like a coreless minijet . in fact dcc searches within minijets , using the techniques sketched in the next section , might be very fruitful . in any case , these pieces of transversely boosted dcc , in reference frames where the longitudinal velocity is zero or small , will typically occupy of order a steradian of solid angle , indicating again that perhaps the natural correlation length for dcc is of order 12 . however , there is an unsolved theoretical issue here . suppose one has a piece of dcc centered at rapidity of + 1 , and another centered at @xmath46 1 with a different chiral order parameter . they will have some small overlap at rapidity 0 . how do the two pieces interact ? will there be a tendency to create a common alignment , or will they form independent domains ? i believe this to be an important fundamental question ; it has not yet received serious attention by theorists . the classical linear sigma model should be sufficient as the fundamental theoretical tool ; the problem is the introduction of a realistic collision - geometry scenario which is still computationally tractable . there is even a remote possibility that the nonlinearities of the linear sigma model are strong enough to promote long - range correlations in rapidity , _ i.e. _ to create a semiclassical structure which produces the phenomenology of a soft pomeron . for example it looks not at all out of the question that a ladder built from pions on the sides and sigmas on the rungs , with couplings dictated by the linear sigma model , could produce a regge intercept of unity or larger for forward hadron - hadron scattering . but i do not know how such a reggeon ladder could be related to the classical dcc scenarios we have discussed . peter lipa asked this question and , together with brigitte buschbeck , we have begun to look into this issue@xcite . the idea is even more speculative than ordinary dcc . but if it makes any sense at all , it has the advantage that one can make the search using data sets containing charged - particle information only . this is not the case for ordinary dcc , where , as discussed in the next section , the relationship between charged and neutral pion production is what is examined . for charged dcc it is the relationship between positive and negative pion production which is the object of study . charged pion fields are to the real , cartesian pion degrees of freedom @xmath47 and @xmath48 as circularly polarized light is to linearly polarized light . and certainly there are classical sources of circularly polarized light , either associated with vorticity of the source , or with a 90@xmath49 phase difference of the sources of the two cartesian components ( or both ) . so from this point of view it seems not totally out of the question to imagine a similar possibility for the pions . we are presently modeling a space - time scenario where dcc in the @xmath47 direction is produced at positive eta , and dcc in the @xmath48 direction is produced at negative eta , with a domain wall " in between . we find , in the spectrum of produced dcc pions , a dipole layer in the lego plot , with an average positive charge per pion on one side of the domain wall , compensated by negative charge on the other . it is to be emphasized that this is an average charge _ per pion _ at the quantum level . the width of the dipole layer is 12 units of rapidity , as might be anticipated on general grounds . we need to do a little more work before reporting on its strength and momentum dependence . a basic signature of dcc production is the presence of very large event - by - event fluctuations in the fraction of produced pions which are neutral@xcite . generic production models will give a distribution in the neutral fraction , defined as @xmath50 which is binomial and peaked at @xmath51 . there will be in the classical large @xmath52 limit a very small probability that , for example , all the pions are neutral . but if the pions are dcc decay products this probability is not at all so small . the simplest estimate for the distribution of neutral fraction in dcc production assumes that the chiral orientation in ( cartesian ) isospin space is random . then a totally elementary calculation gives the result that the distribution of neutral fraction is inverse - square - root . with @xmath53 it follows that @xmath54 it is probable however that this component is to some extent immersed in generic background . cuts in @xmath55 , for example , may be useful in enhancing the signal . it is also the case that it appears to be better to use an indirect technique to test for the presence of the inverse - square - root component@xcite . this utilizes the machinery of multiparticle production dynamics and multiplicity distributions , to which we now turn . in the minimax experiment , the phase - space coverage is small , about 1.0 lego - area units . for that analysis it is reasonable to assume that within the acceptance the chiral order parameter takes a fixed value . the raw information relevant to dcc physics is the multiplicity distribution of produced pions . ideally they should be momentum analyzed , with the data binned in intervals of @xmath55 . however this so far has been not done experimentally , and we simplify by ignoring the momentum degree of freedom . let the probability of producing @xmath52 pions be @xmath56 , and introduce the generating function @xmath57 which contains all information about the multiplicity distribution . for a poisson distribution , the generating function is an exponential @xmath58 in the general case it is a superposition of poisson distributions with positive semidefinite weight function @xmath59 @xmath60 for the two species of charged and neutral pions , the generalization is a generating function of two variables , again a superposition of poisson distributions . the definition of generic pion production is that the only correlation is produced by the aforementioned @xmath59 for the total pion multiplicity distribution , so that @xmath61 } \label{eq : aa}\end{aligned}\ ] ] with the neutral fraction @xmath29 approximately 1/3 . by expanding things out , one sees that the partition into charged and neutral pions is governed by a binomial distribution ; indeed this could have been the common - sense starting point . it is also the case that existing monte - carlo codes have the property that the distribution of the neutral fraction is approximately binomial . for pure dcc , all one needs do is introduce the inverse square - root distribution in @xmath29 as another weight factor : @xmath62 one sees that the basic difference between generic production and dcc production is that in the former case the generating function depends only upon one variable , while for dcc it depends nontrivially upon two . a very good way of testing for the distinction is via factorial moments@xcite . these are just the derivatives of the generating function with respect to the @xmath63 s at @xmath64 . in the case of only one variable , one has @xmath65 the normalized factorial moments are @xmath66 in the case of interest , there is a two - dimensional array of normalized factorial moments which captures the information contained in the joint multiplicity distribution : @xmath67 however for generic production there are many relations between these , because the generating function depends upon only one variable : @xmath68 therefore many ratios of the @xmath69 are expected to be unity @xmath70 while for pure dcc they can also be explicitly computed , and are far from unity : @xmath71 there are also experimental reasons why these ratios are useful . first of all , one can go from @xmath25 production to photon production via convolution , and the factorial moment method remains robust : one simply replaces the neutral - pion fugacity @xmath72 by the generating function for the gammas , a second - order polynomial in the photon fugacity . in addition , if efficiencies are not 100 percent , but are uncorrelated with total multiplicity or other global parameters , they can be incorporated in terms of modified fugacities . the properties of the resultant factorial moments , which are the direct observables experimentally , essentially do not change . that is , the bivariate factorial moment ratios directly extracted from data on production of charged hadrons and of gammas , as in eqs.(33 ) and ( 34 ) , still will be unity for generic production and far from unity in the case of dcc production . once the experimental acceptance becomes large , or if one investigates , _ e.g. _ the @xmath55 dependence of the presumed dcc fraction ( large at low @xmath55 , small at high @xmath55 ? ) , then the correlation structure of the dcc order parameter , and even the generic particle distributions themselves , becomes of paramount importance . clearly the formalism of generating functions should be preserved as closely as possible , and this is in principle a straightforward matter of replacing generating functions with generating functionals . in the case of dcc production , the bridge from the classical calculations to phenomenology is fairly clear . for a given classical solution of , say , the linear sigma model there will be a source function @xmath73 as we discussed , the multiplicity distribution is built from the squared fourier transform of the source function , put on mass shell , @xmath74 the fluctuations for the classical case are poissonian , and so the generating functional is again just an exponential in all the continuous number of fugacity variables , now parametrized by the particle momenta . finally one should average over the choice of source function , which at the least is parametrized by choices of initial conditions for the classical field configurations ( including chiral orientation ) . this leads to a dcc generating functional of the form @xmath75 a generalized dcc distribution of the inverse square root type follows if @xmath76 thus far this generalization is reasonably straightforward , and leaves the unknown issues mostly at the level of the choice of boundary conditions and structure of the classical field equations considered in the previous section . however it is not reasonable to assume that the totality of particle production originates as dcc . and there is no consensus on what generating functional to use to describe generic particle production . many correlation phenomena exist , some from minijets and perturbative qcd , undoubtedly some associated with the fluctuations in the number of wounded " constituent quarks per collision , still others associated with impact parameter dependence , and more associated with resonance production . in addition , one can consider various ways of combining the generic particle production with dcc production . three extreme cases are as follows : \1 ) @xmath77 this means that in a given event either generic particles are produced or dcc , but not both . \2 ) @xmath78 in this case the amount of dcc produced in a given event is not correlated at all with the amount of generic particles . \3 ) @xmath79 in this case the amount of dcc produced is , up to binomial - distribution fluctuations , in proportion to the amount of generic particle production . the phenomenological consequences of this distinction are very strong ; a small dcc admixture of the first type is much easier to isolate experimentally than a small admixture of the third type . the case of independent production is intermediate . there is an interesting question of whether observation of the inverse square - root distribution implies observation of baked - alaska dcc . this is not at all clear . the inverse square - root distribution was discovered in a different context , namely the production of a cluster of pions in a maximally symmetric state . the inverse square - root behavior is also a consequence of the well - known andreev , plumer , weiner ( apw ) description@xcite of bose - einstein correlations . in addition , it has been shown that in the tree - level expansion of the chiral effective theory , the inverse - square root distribution emerges@xcite . but none of these descriptions follows the same line as the dcc description above . in particular , the part of the apw scenario which leads to dcc - like behavior assumes a random gaussian distribution of source functions , except with respect to the isospin degrees of freedom . in the dcc description , one computes the sources via the sigma - model equations of motion and ( most probably ) a random set of initial conditions . this would for sure make the distribution of dcc sources non - random functionals . and the tree - level chiral expansion does not seem to have the possibility of incorporating the baked - alaska physics . it is probably the case that one should view dcc as a specific mechanism of bose - einstein enhancement . there seems to me nothing wrong with this point of view , because there really is an ongoing problem of describing bose - einstein effects from the most general point of view . in any case , for me this is not a very important issue until there is concrete experimental evidence for dcc - like behavior . first of all something needs to be seen . thereafter there will be plenty of opportunity for figuring out what it means via further interaction between follow - on observations and the theory . the phenomenological description of the charged dcc production discussed in section 2.8 is similar to that of the conventional dcc . it is being worked out by peter lipa , brigitte buschbeck@xcite , and myself . the major change is that the source function in momentum space is taken to be complex @xmath80 and the real and imaginary parts @xmath81 and @xmath82 are each taken as random variables . in the usual case there is assumed to be complete correlation between @xmath81 and @xmath82 . one finds before : @xmath83 after : @xmath84 with @xmath85 for the `` action '' of the generating function one has @xmath86 \ . \label{eq : tt}\ ] ] again the minimax - like ratios of factorial moments can be constructed , and compared with charged particle data . it appears from a cursory first look that when averaged over all @xmath55 the data will be of opposite sign to the charged - dcc expectation . charge tends to be locally neutralized in the lego plot , more than it would from a random throwing of charge into phase space . this to be expected in both the string and parton cascade pictures ; there is very little charge separation and flow in the space - time evolution . however when the @xmath55 s are low , the experimental situation changes . it is here that the charged - dcc hypothesis is not ruled out , and the detailed phenomenology of this region of phase - space may turn out to be interesting . the assumption of complete randomness of @xmath81 and @xmath82 is an extreme one . there are evidently interpolations between the extremes of no correlation and complete correlation which can be easily constructed . study of the structure of the classical models as outlined in section 2.8 will help in choosing a reasonable starting hypothesis . thanks go to jan czyzewski , jacek wosiek , and all the organizers of the zakopane school for a most productive and pleasant meeting . it is also a great pleasure to dedicate these notes to my good friend and colleague wieslaw czyz on the occasion of his seventieth birthday . t. k. nayak in _ proceedings of the international conference on physics and astrophysics of the quark - gluon plasma _ , d. k. srivastava , b. c. sinha , and y. p. viyogi , editors , icpa - qgp97 , new delhi , 1007 , narosa publishing house ( in press ) . k. l. kowalski , j. d. bjorken , and c. c. taylor , slac - pub-6109 ( 1993 ) ; les rencontres de physique de la vallee daoste ; _ results and perspectives in particle physics _ , la thuile , aoste valley , italy , march 713 , 1993 ; ed . m. greco ( editions frontieres ) . e.g. _ the review by k. rajagopal , hep - ph/9703258 and references therein . in addition there exists a dcc web page set up by peter steinberg ( wa98 ) : http://www.cern.ch/wa98/dcc , where more information can be obtained . s. gavin , a. gocksch , and r. d. pisarski , _ phys . lett . _ * 72 * , 2143 ( 1994 ) . d. boyanovsky , h. j. de vega , and r. holman , _ phys . rev . _ * d51 * , 734 ( 1995 ) . f. cooper , y. kluger , e. mottola , and j. p. paz , _ phys . * d51 * , 2377 ( 1995 ) ; f. cooper , y. kluger , and e. mottola , hep - ph/9604284 . see also j. randrup , _ nucl . phys . _ * a616 * , 531 ( 1997 ) ; hep - ph/9612453 for further references .

the basic ideas underlying the production dynamics and search techniques for disoriented chiral condensate are described .

hep-ph9712434

in 2011 , i described a timing sequencer and related laser lab instrumentation based on 16-bit microcontrollers and a homemade custom keypad / display unit.@xcite since then , two new developments have enabled a far more powerful approach : the availability of high - performance 32-bit microcontrollers in low - pin - count packages suitable for hand assembly , and the near - ubiquitous availability of tablets with high - resolution touch - screen interfaces and open development platforms . this article describes several new instrument designs tailored for research in atomic physics and laser spectroscopy . each utilizes a 32-bit microcontroller in conjunction with a usb interface to an android tablet , which serves as an interactive user interface and graphical display . these instruments are suitable for construction by students with some experience in soldering small chips , and are programmed using standard c code that can easily be modified . this offers both flexibility and educational opportunities . the instruments can meet many of the needs of a typical optical research lab : event sequencing , ramp and waveform generation , precise temperature control , high - voltage pzt control for micron - scale optical alignment , diode laser current control , rf frequency synthesis for modulator drivers , and dedicated phase - sensitive lock - in detection for frequency locking of lasers and optical cavities . the 32-bit processors have sufficient memory and processing power to allow interrupt - driven instrument operation concurrent with usage of a real - time graphical user interface . the central principle in designing these instruments has been to keep them as simple and self - contained as possible , but without sacrificing performance . with simplicity comes small size , allowing control instrumentation to be co - located with optical devices for example , an arbitrary waveform synthesizer could be housed directly in a diode laser head , or a lock - in amplifier could fit in a small box together with a detector . as indicated in fig . [ systemoverview ] , each instrument is based on a commodity - type 32-bit microcontroller in the microchip pic32 series , and can be controlled by an android app designed for a 7 `` or 8 '' tablet . an unusual feature is that the tablet interface is fully interchangeable , using a single app to communicate with any of a diverse family of instruments as described in sec . [ subsec : usb ] . further , all of the instruments are fully functional even when the external interface is removed . when the operating parameters are modified , the values are stored in the microcontroller program memory , so that these new values will be used even after power has been disconnected and reconnected . the usb interface also allows connection to an external pc to provide centralized control . ( color online ) block diagram of a microcontroller - based instrument communicating with an android tablet via usb . a tablet app , microcontroller , uploads parameter values and their ranges from the instrument each time the usb interface cable is connected . ] four printed - circuit boards ( pcbs ) have so far been designed . one , the labint32 board described in section [ sec : labint ] , is a general - purpose laboratory interface specifically designed for versatility . the others are optimized for special purposes , as described in section [ sec : specialpurpose ] . the pcbs use a modular layout based in part on the daughter boards `` described in sec . [ subsec : daughterboards ] . they range from simple interface circuits with just a handful of components to the relatively sophisticated wvfm32 board , which uses the new analog devices ad9102 or ad9106 waveform generation chips to support a flexible voltage - output arbitrary waveform generator and direct digital synthesizer ( dds ) . it measures 1.5''@xmath00.8 " , much smaller than any comparable device known to the author . further details on these designs , including circuit board layout files and full source code for the software , are available on my web page at the university of connecticut.@xcite in designing the new instrumentation i considered several design approaches . one obvious method is to use a central data bus , facilitating inter - process communication and central control . apart from commercial systems using labview and similar products , some excellent homemade systems of this type have been developed , including an open - source project supported by groups at innsbruck and texas.@xcite this approach is best suited to labs that maintain a stable long - term experimental configurations of considerable complexity , such as the apparatus for bose - einstein condensation that motivated the innsbruck / texas designs . as already mentioned , the approach used here is quite different , intended primarily for smaller - scale experiments or setups that evolve rapidly , where a flexible configuration is more important than providing full central control from a single console . the intent is that most lab instruments will operate as autonomous devices , although a few external synchronization and control signals are obviously needed to set the overall sequence of an experiment . these can come either from a central lab computer or , for simple setups , from one of the boards described here , set up as an event sequencer and analog control generator . this approach is consistent with our own previous work and with recent designs from other small laser - based labs.@xcite once having decided on decentralized designs using microcontrollers , there are still at least three approaches : organized development platforms , compact development boards , or direct incorporation of microcontroller chips into custom designs . numerous development platforms are now available , ranging from the hobbyist - oriented arduino and raspberry pi to more engineering - based solutions.@xcite however , these approaches were ruled out because they increase the cost , size , and complexity of an instrument . for simple hardware - oriented tasks requiring rapid and repeatable responses , a predefined hardware interfacing configuration and the presence of an operating system can be more of a hindrance than a help . initially it seemed attractive to use a compact development card to simplify design and construction . my initial design efforts used the simple and affordable mini-32 development card from mikroelektronika,@xcite which combines an 80 mhz microchip pic32mx534f064h processor with basic support circuitry and a usb connector . this board was used to construct a ramp generator and event sequencer very similar in design to an earlier 16-bit version.@xcite while successful , this approach entailed numerous inconveniences : the microcontroller program and ram memories are too small at 64 kb and 16 kb , the oscillator crystal is not a thermally stabilized txco type , the usb interface requires extensive modification to allow host - mode operation , and the 80 mhz instruction rate is somewhat compromised by mandatory wait states and interrupt latency . finally , certain microcontroller pins that are essential for research lab use , such as the asynchronous timing input t1ck , are assigned for other purposes on the mini-32 , requiring laborious cutting and resoldering of traces . tests of the event sequencer yielded reasonably good results : the maximum interrupt event rate of 1.5 mhz is about twice as fast as the 16-bit design operating at 20 mhz , although the typical interrupt latency of 400 ns is not very different . nevertheless , it became evident that the effort in using preassembled development boards outweighs the advantages . ( color online ) photograph of the 5``@xmath02.25 '' labint32 pcb . it includes a wvfm32 daughter board with required support circuitry , and a dual 16-bit dac with one output connected to a card - edge sma connector . ] instead , the designs described here use low - pin - count chips in the microchip pic32mx250 series that are directly soldered to the circuit boards , as can be seen in fig . [ labintphoto ] . these microcontrollers , even though they are positioned as basic commodity - type devices by the manufacturer , have twice the memory of the mini-32 processor and can operate at 40 mhz without wait states.@xcite they feature software - reassignable pins that increase interfacing flexibility , as described in sec . [ subsec : modular ] . while the reduced 40 mhz speed is a consideration for event sequencing , it does not impact the performance of any of the other instruments described here , and the absence of wait states during memory access is partially compensatory . the processor clock and other timing references are derived from miniature temperature - compensated crystal oscillators in the fox electronics fox924b series , which are small , inexpensive , and accurate within 2.5 parts per million . ease of construction is a major consideration for circuits used in an academic research lab . to facilitate this , the easily - mounted 28-pin pic32mx250f128b microcontroller is used where possible , and a 44-pin variant when more extensive interfacing is needed . the basic support circuitry for the controller is laid out to allow hand soldering , as is other low - frequency interface circuitry . nevertheless , all of the pcbs include at least a few surface - mounted chips that are more easily mounted using hot - air soldering methods . we have obtained very good results using solder paste and a light - duty hot - air station.@xcite for rf circuits the hot - air method is unfortunately a necessity , because modern rf chips commonly use compact flat packages such as the qfn-32 , with closely - spaced pins located underneath the chip . construction can also be made easier by including a full solder mask on the pcb , greatly reducing the incidence of accidental solder bridges between adjacent pins . these masks are available for a modest extra fee from most pcb fabricators , and their additional services usually also include printed legends that can conveniently label the component layout . ( color online ) typical screen view of the microcontroller app on a google nexus 7 tablet . when a parameter is selected on the scrollable list at the upper left , its value can be adjusted either with a pop - up keypad or with the two slider bars . the strip - chart graph shows in yellow the output voltage produced by a temperature controller card , and in blue the temperature offset from the set point ( 25 units @xmath1 1 mk ) . ] as previously described , a user interface to a commodity - type tablet is very appealing because it offers a fast , responsive high - resolution graphical touch - screen interface that requires no specialized instrumentation or construction . although rf communication with a tablet is possible using bluetooth or wi - fi protocols , a usb interface is a better choice for lab instruments because it avoids the need for extra circuitry , and it avoids the proliferation of multiple rf - based devices operating in a limited space . an interface based on an open - source development environment is important , so that programs on both the tablet and the microcontroller can be freely modified for individual research needs . fortunately the android operating system provides such a resource , the android open accessory ( aoa ) protocol.@xcite for this reason , the programs described here were developed for the widely available google nexus 7 android tablet , which offers a 1280@xmath0800 display and up to 32 gb of memory , with a fast quad - core processor . the microcontroller programs use the aoa protocol mainly to transfer five - byte data packets consisting of a command byte plus a 32-bit integer . they also support longer data packets in the microcontroller - to - tablet direction for displaying text strings and graphics . an important consideration is that the usb interface at the microcontroller end of the link must operate in host mode because many tablets , including the google nexus 7 , support only device - mode operation . an additional consideration is that for extended operation of the graphical display , a continuous charging current must be provided . the only way to charge most tablets is via the usb connector , and charging concurrent with communication is only possible if the tablet operates in usb device mode . on the other hand , it is important that the microcontroller usb interface also be capable of device - mode operation , because when control by an external personal computer is desired , the pc will support only host - mode operation . for this reason , the full usb on - the - go ( otg ) protocol has been implemented in hardware , allowing dynamic host - vs - device switching . presently the microcontroller software supports only host - mode operation with a tablet interface , but extension to a pc interface would require only full incorporation of the usb otg sample code available from microchip.@xcite a more subtle hardware consideration is that both of the tablets i have so far examined , the google nexus 7 and archos 80 g9 , use internal switching power supplies that present a rapidly shifting load to the 5 v charging supply . in initial designs , the 5 v power supply on the microcontroller pcb was unable to accommodate the rapidly switched load , causing fluctuations of @xmath2 mv which then propagated to some of the analog signal lines . a good solution is to provide a separate regulator for the usb charging supply , operating directly from the same 6v input power that powers the overall circuit card . with this design there is no measurable effect on the 5 v and 3.3 v power supplies used to power chips on the main circuit board . a single android app , microcontroller , supports all of the instruments described here by using a flexible user interface based on a scrolling parameter list that is updated each time a new usb connection is established . it was developed in java using the android software development kit , for which extensive documentation is available.@xcite the app is available on my web page,@xcite both as java source code and in compiled form . as shown in figs . [ systemoverview ] and [ screenshot ] , the app displays a parameter list with labels and ranges specific to the application . several check boxes and status indicators are also available , also with application - specific labels . once the user selects a parameter by touching it , its value can be changed using either a pop - up keypad or the coarse and fine sliders visible in fig . [ screenshot ] . the remainder of the display screen is reserved for real - time graphics displayed using the open - source achartengine package,@xcite , and can show plots of data values , error voltages from locking circuits , and similar information . the graphics area can be fully updated at rates up to about 15 hz . while certain tasks will eventually require their own specialized android apps to offer full control , particularly arbitrary waveform generation and diode laser frequency locking , the one - size - fits all solution offered by the microcontroller app still works surprisingly well as a starting point . for a majority of the instruments described here , it is also quite satisfactory as a permanent user interface . although this paper mentions seven distinct instruments , they are accommodated using only four pcbs , all of which share numerous design elements as well as a common usb tablet interface . multiple instruments can also share a single tablet for user interfacing because it needs to be connected only when user interaction is needed , a major advantage of this design approach . another common design element is a 5-pin programming header included on each pcb that allows a full program to be loaded in approximately 10 - 20 seconds using an inexpensive microchip pickit 3 programmer . the programs are written in c and are compiled and loaded to the programmer using the free version of the microchip xc32 compiler and the mplab x environment.@xcite the pic32mx250 processor family further enhances design flexibility by providing numerous software - reassignable i / o pins , so that a given pin on a card - edge interface terminal might be used as a timer output by one program , a digital input line by another , and a serial communication output by a third . to avoid repetitive layout work and to further enhance flexibility , several commonly used circuit functions have been implemented on small daughter boards " as described in sec . [ subsec : daughterboards ] . two of these daughter boards are visible on the general - purpose lab interface shown in fig . [ labintphoto ] , as is an unpopulated additional slot . some of these daughter boards simply offer routine general - purpose functionality , such as usb power switching , while others offer powerful signal generation and processing capabilities . with the exception of the 1``@xmath00.8 '' usb interface board , the daughter boards measure 1.5``@xmath00.8 '' , and share a common 20-pin dip connector formed by two rows of square - pin headers . the power supply and spi lines are the same for all of the boards , while the other pins are allocated as needed . these connectors can be used as a convenient prototyping area for customizing interface designs after the circuit boards have been constructed , by wire - wrapping connections to the square pins . as already mentioned , the lab interface ( labint32 ) pcb was designed to allow a multitude of differing applications by providing hardware support for up to two interchangeable daughter boards , as well as powerful on - board interfacing capabilities . as shown in figs . [ labintphoto ] and [ labintschematic ] , the core of the design is a pic32mx250f128d microcontroller in a 44-pin package . this provides enough interface pins to handle a wide variety of needs , particularly considering that many of them are software - assignable . several card - edge connectors and jacks provide access to numerous interface pins and signals , including an 8-bit digital i / o interface , of which six bits are tolerant of 5 v logic levels . two of the connectors are designed to support an optional rotary shaft encoder and serial interface as described in sec . [ subsec : currentctrl ] . the board operates from a single 6 v , 0.5 a power module but contains several on - board supplies and regulators . these provide the 3.3 v and 5 v power required for basic operation , as well as optional supplies at -5 v and @xmath312 v for op amps , analog conversion , and rf signal generation . these optional supplies are small switching power supplies that operate directly from the 6 v input power , so that they do not impose switching transients on the 5 v supply as mentioned in sec . [ subsec : usb ] . there are also provisions on the main board for three particularly useful interface components : a dual 16-bit voltage - output dac with a buffered precision 2.5 v reference ( analog devices ad5689r ) , a robust instrumentation amplifier useful for input signal amplification or level shifting ( ad8226 ) , and a 1024-position digital potentiometer ( ad5293 - 20 ) that can provide computer - based adjustment of any signal controllable by a 20 k@xmath4 resistor , up to a bandwidth limit of about 100 khz . presently there are two demonstration - type programs available for the microcontroller on the labint card . one uses the on - board 16-bit dac to provide a high - resolution analog ramp with parameters supplied by the tablet interface . the other operates with the wvfm32 daughter board to provide a synthesized complex waveform with data output rates up to 96 mhz , as described in the next section . the simplest of the daughter boards , the tiny 1``@xmath00.6 '' usb32 board , is used on all of the pcbs . it simply provides the power and switching logic for a usb otg host / device interface , by use of a 0.5 a regulator and a tps2051b power switch . it includes a micro usb a / b connector , which is inconveniently small for soldering but is necessary because it is the only connector type that is approved both for host - mode connections to tablets and device - mode connection to external computers.@xcite as part of the usb otg standard , an internal connection in the usb cable is used to distinguish the a ( host ) end from the b ( device ) end . the remainder of the daughter boards are slightly larger at 1.5``@xmath00.8 '' , and they all share a common 20-pin dip connector as described in sec . [ subsec : modular ] . they can be used interchangeably on the labint32 pcb or for specific purposes on other pcbs , as for the dac32 daughter board needed by the tempctrl card . this section describes the wvfm32 daughter board in detail , and briefly describes three others . * wvfm32 * the wvfm32 daughter board , whose schematic is shown in fig . [ wvfm32 ] , benefits from the simplicity of a direct spi interface and provides an extremely small but highly capable instrument . it combines the remarkable analog devices ad9102 ( or ad9106 ) waveform generation chip with a fast dc - coupled differential amplifier ( two for the ad9106 ) , along with a voltage regulator and numerous decoupling capacitors necessitated by the bandwidth of about 150 - 200 mhz . the ad9102/06 provides both arbitrary waveform generation from a 4096-word internal memory and direct digital synthesis ( dds ) of sine waves , with clock speeds that can range from single - step to 160 mhz . when it is used on the labint32 pcb , a fast complementary clock generator is not available , but the programmable refclko output of the pic32 microcontroller works very well for moderate - frequency output waveforms after it is conditioned by the simple passive network shown near the center of fig . [ labintschematic ] . the refclko output can be clocked at up to 40 mhz using the pic32 system clock or at up to 96 mhz using the internal usb pll clock.@xcite even though the pic32 output pins are not specified for operation above 40 mhz , the 96 mhz clock seems to work well . the differential buffer amplifiers , ad8129 or ad8130 , can drive a terminated 50 ohm line with an amplitude of @xmath32.5 v. at full bandwidth the rms output noise level is approximately 1 mv , or 1 part in 5000 of the full - scale output range . the dac switching transients were initially very large at @xmath580 mv , but after improving the ground connection between the complementary output sampling resistors ( r5 and r6 in fig . [ wvfm32 ] ) , the transients were reduced to 6 mv pulses about 60 ns in duration , and they alternate in sign so that the average pulse area is nearly zero . the large - signal impulse response was measured using an ad8129 to drive a 1 m , 50 @xmath4 cable , by setting up the ad9102 waveform generator to produce a step function . the shape of the response function is nearly independent of the step size for 15 v steps . the output reaches 0.82 v after 4 ns , approaching the limits of the 100 mhz oscilloscope used for the measurement , demonstrating that the circuit approaches its design bandwidth of @xmath6 mhz . however , it exhibits a slight shoulder after 4 ns , taking nearly 8 ns to reach 90% of full output and then reaching 100% at @xmath511 ns . after this initial rise , the output exhibits slight ringing at the @xmath51% level with a period of about 100 ns , damping out in about three cycles to reach the noise level . this ringing is caused at least in part by the response of the @xmath35 v regulators to the sudden change in current on the output line , and is not observed with smaller steps of @xmath50.1 v. \2 . * dac32 * the dac32 daughter board is a straightforward design that includes one or two of the same ad5689r dac chips described in section [ sec : labint ] , providing up to four 16-bit dac outputs , together with an uncommitted dual op amp . the op amps have inputs and outputs accessible on the 20-pin dip connector , and can be used in combination with the dacs or separately . \3 . * lockin * the lockin daughter board does just what its name implies . it realizes a simple but complete lock - in amplifier , with a robust adjustable - gain instrumentation amplifier ( ad8226 or ad8422 ) driving an ad630 single - chip lock - in amplifier that works well up to about 100 khz . the output is amplified and filtered , then digitized by an ad7940 14-bit adc . the performance is determined mainly by the ad630 specifications , except that the instrumentation amplifier determines the input noise level and the common - mode rejection ratio . when used on the labint32 pcb , the digital potentiometer on the main board can be tied to this daughter board to allow computer - adjustable gain on the input amplifier . \4 . * adc32 * the adc32 board is still in the design stage . it will use an ad7687b 16-bit adc , together with a robust adg5409b multiplexer and an intersil isl28617fvz differential amplifier , to provide a flexible high - resolution analog - to - digital converter supporting four fully differential inputs . in addition to offering higher resolution than the built - in 10-bit adcs on the pic32 microcontroller , it offers a much wider input voltage range and considerable protection against over - voltage conditions . ( color online ) major elements of the precision temperature controller . high - voltage pa340cc op amps can be substituted for the opa548 high - current drivers for use as a dual pzt driver . ] the temp32 pcb , shown in block form in fig . [ tempctrl ] , uses a 28-pin pic32mx250f128b on a card optimized specifically for low - bandwidth analog control , with three separate ground planes for digital logic , signal ground , and analog power ground . while simple enough to be used for general - purpose temperature control , the board was designed to allow the very tight control needed for single - mode distributed bragg reflector ( dbr ) lasers , for which a typical temperature tuning coefficient of 25 ghz / c necessitates mk - level control for mhz - level laser stability.@xcite as shown in fig . [ tempctrl ] , a divider formed by a thermistor and a 5 ppm / c precision resistor provides the input to a 22-bit adc . the microchip mcp3550 - 60 is a low - cost sigma - delta " adc that provides very high accuracy and excellent rejection of 60 hz noise at low data rates ( 15 hz ) . a 2.5 v precision reference is used both for the thermistor divider and to set the full - scale conversion range of the adc , making the results immune to small reference fluctuations . no buffering is required for the thermistor , although if a different sensor were used a low - noise differential amplifier might be desirable.@xcite the microcontroller program implements a pid ( proportional - integral - differential ) controller using integer arithmetic , with several defining and constraining parameters that can be optimized via the tablet interface . the output @xmath7 after iteration @xmath8 is determined by the error @xmath9 , gain factors @xmath10 , @xmath11 and @xmath12 , the sampling frequency @xmath13 , and a scale factor @xmath14 that allows the full 16-bit range of the output dac to be used : @xmath15 this output is sent a dac32 daughter board , then amplified by an opa548 power op amp capable of driving 60 v or 3 a. the separate analog power ground plane for the output section of the pcb is connected to the analog signal ground plane only at a single point . with a conventional 10 k@xmath4 thermistor , the single - measurement rms noise level is approximately 7 adc units , corresponding to about 0.3 mk near room temperature . assuming a bandwidth of about 1 hz for heating or cooling a laser diode or optical crystal , the time - averaged noise level and accuracy can exceed 0.1 mk , adequate for most purposes in a typical laser - based research lab . the temp32 circuit board can alternatively be used as a dual 350-v pzt controller for laser spectrum analyzers or other micron - scale adjustments . to accomplish this , the adc is omitted and the output op amps are substituted with apex pa340cc high - voltage op amps , using a simple adapter pcb that accommodates the changed pin - out . the freqsynth32 pcb , presently in the testing phase , is intended to provide accurate high - frequency rf signals for applications such as driving acousto - optic modulators . it supports up to two adf4351 ultra - broadband frequency synthesizers , which can produce far higher frequencies than dds synthesizers . these pll - based devices include internal voltage - controlled oscillators and output dividers , allowing self - contained rf generation from 35 - 4000 mhz . as shown in fig . [ freqsynth ] , an rf switch allows ns - timescale switching between the two synthesizers , or if one is turned off , it allows fast on - off switching . signal conditioning includes a low - pass filter to eliminate harmonics from the adf4351 output dividers , as well as a broadband amplifier and digital attenuator that provide an output level adjustable from about -15 dbm to + 16 dbm . the output can drive higher - power amplifier modules such as the rfhic rfc1g21h4 - 24 , which provides up to 4w in the range 20 - 1000 mhz . it is a challenge to work over such a broad frequency range . although an impedance - matched stripline design was not attempted because this would require a pcb substrate thinner than the 0.062 " norm , considerable attention has been paid to keeping the rf transmission path short , wide , and guarded " from radiative loss by numerous vias connecting the front and back ground planes on the pcb . the mpl_interface pcb , described more fully on my web page,@xcite is designed as a single - purpose interface to a laser diode current driver compatible with the mpl series from wavelength electronics . however , this circuit may be of more general interest for two reasons . first , it allows control and readout of devices with a ground reference level that can float in a range of @xmath16 v , with 13 - 16 bit accuracy . second , its control program includes full support for a rotary shaft encoder ( bourns em14a0d - c24-l064s ) and a simple serial lcd display ( sparkfun lcd-09067 ) , allowing the laser current to be adjusted and displayed without the usb tablet interface . the same encoder and display could also be attached to the labint32 pcb using jacks provided for this purpose . a significant portion of the research needs of a typical laser spectroscopy or atomic physics laboratory can be met by the four pcbs described in secs . [ sec : labint ] and [ sec : specialpurpose ] , together with appropriate software and the daughter boards in sec . [ subsec : daughterboards ] . the advantages of this approach include simple and accessible modular designs , a user interface to an android tablet with interactive high - resolution graphics , and easily reconfigurable software . the circuit designs are intended for in - house construction , reducing expenses and allowing valuable educational opportunities for students , while still offering the high performance expected of a specialized research instrument . most of the pcbs can be hand - soldered , although a hot - air soldering station is required for the two rf circuits ( wvfm32 and freqsynth32 ) . full design information and software listings are available at my website.@xcite apart from these general considerations , these instruments offer some unusual and valuable capabilities . one is the single shared android app that provides a full graphical interface to numerous different devices . when the tablet is removed after adjusting the operating parameters , the microcontroller stores the updated parameter values and the instrument will continue to use them indefinitely . another is the very small size of the wvfm32 waveform generator , which takes advantage of a simple direct interface connection to a microcontroller to provide voltage - output arbitrary waveform generation and dds on a 1.5``@xmath00.8 '' pcb . up to two of these pcbs can be mounted on a labint32 general - purpose interface card , itself measuring only 5``@xmath02.25 '' , and only a single semi - regulated 6v , 0.5a power supply is required . similarly , the dac32 and lockin daughter boards share the same small footprint , facilitating control instrumentation that can fit inside the device being controlled . the primary usage of these instruments in our own laboratory is to control several diode lasers and to provide flexible control of numerous frequency modulators needed for research on optical polychromatic forces on atoms and molecules.@xcite although the available circuits and software reflect this focus , most of these instruments can be used for diverse applications in their present form , and all can be modified readily for special needs . + e. e. eyler , rev . instrum . * 82 * , 013105 ( 2011 ) . e. e. eyler , web page at http://www.phys.uconn.edu/~eyler/microcontrollers/ , _ microcontroller designs for atomic , molecular , and optical physics laboratories _ , university of connecticut physics dept . p. e. gaskell , j. j. thorn , s. alba , and d. a. steck , rev . instrum . * 80 * , 115103 ( 2009 ) . t. meyrath and f. schreck , a laboratory control system for cold atom experiments , atom optics laboratory , center for nonlinear dynamics and department of physics , university of texas at austin , http://www.strontiumbec.com/control/control.html , 2013 . see , for example , m. ugray , j. e. atfield , t. g. mccarthy , and r. c. shiell , rev . instrum . * 77 * , 113109 ( 2006 ) . d. sandys , life after pi , digi - key corporation , june 14 , 2013 , available at http://www.digikey.com/us/en/techzone/microcontroller/resources/articles/life-after-pi.html . mikroelektronika corp . viegradska 1a , 11000 belgrade , serbia . see http://www.mikroe.com / mini / pic32/. faster 50 mhz versions are also described in the data sheets , but were not yet widely available as of july 2013 . see pic32mx1xx/2xx data sheet , document ds61168e , 2012 , microchip technology inc . , 2355 west chandler blvd . , chandler , arizona , http://www.microchip.com . aoyue968a+ , available from sra soldering products , foxboro , ma . to mount surface - mount chips a thin layer of solder paste with flux should be applied to the pcb pads ; we have obtained good results using chipquik model smd291ax . links to information and software for usb interfacing with the aoa protocol can be found at http://source.android.com/accessories/custom.html . part of the microchip application libraries , microchip technology inc . , 2355 west chandler blvd . , chandler , arizona . see http://www.microchip.com/pagehandler/en-us/technology/usb/gettingstarted.html ( 2013 ) . w - m lee , _ beginning android application development _ , wiley publishing , indianapolis , 2011 . z. mednieks , l. dornin , g. blake meike , and m. nakamura , _ programming android _ , oreilly media , sebastopol , ca , 2011 . android software development kit ( sdk ) , http://developer.android.com/sdk/index.html , 2013 . the free achartengine library for android , written in java , is available at http://code.google.com / p / achartengine/. mplab xc32 compiler , free version , microchip technology inc . , 2355 west chandler blvd . , chandler , arizona , http://www.microchip.com / pagehandler / en_us / devtools / mplabxc/. version 1.20 was used for this work . on - the - go and embedded host supplement to the usb revision 2.0 specification , rev . 2.0 version 1.1a , july 27 , 2012 , available for download from http://www.usb.org / developers / docs/. _ pic32 family reference manual _ , microchip technology inc . available for download ( by chapter ) at http://www.microchip.com ) . j. spencer , photodigm corp . , _ tunable laser diode absorption spectroscopy ( tldas ) with dbr lasers _ , aug . 4 , 2011 , available at http://photodigm.com/blog/bid/62359 . j. horn and g. gleason , weigh scale applications for the mcp3551 , _ microchip application note an1030 _ , microchip technology inc . m. a. chieda and e. e. eyler , phys . rev . a * 86 * , 053415 ( 2012 ) . s. e. galica , l. aldridge , and e. e. eyler , to be published .

several high - performance lab instruments suitable for manual assembly have been developed using low - pin - count 32-bit microcontrollers that communicate with an android tablet via a usb interface . a single android tablet app accommodates multiple interface needs by uploading parameter lists and graphical data from the microcontrollers , which are themselves programmed with easily - modified c code . the hardware design of the instruments emphasizes low chip counts and is highly modular , relying on small daughter boards `` for special functions such as usb power management , waveform generation , and phase - sensitive signal detection . in one example , a daughter board provides a complete waveform generator and direct digital synthesizer that fits on a 1.5''@xmath00.8 " circuit card .

1307.7160

entanglement in quantum multipartite systems is a unique property in quantum world . it plays an important role in quantum information processing @xcite . therefore , the study of its essential features and dynamical behavior under the ubiquitous decoherence of relevant quantum system has attracted much attention in recent years @xcite . for example , it was found that the entanglement of qubits under the markovian decoherence can be terminated in a finite time despite the coherence of single qubit losing in an asymptotical manner @xcite . the phenomenon called as entanglement sudden death ( esd ) @xcite has been observed experimentally @xcite . this is detrimental to the practical realization of quantum information processing using entanglement . surprisingly , some further studies indicated that esd is not always the eventual fate of the qubit entanglement . it was found that the entanglement can revive again after some time of esd @xcite , which has been observed in optical system @xcite . it has been proven that this revived entanglement plays a constructive role in quantum information protocols @xcite . even in some occasions , esd does not happen at all , instead finite residual entanglement can be preserved in the long time limit @xcite . this can be due to the structured environment and physically it results from the formation of a bound state between the qubit and its amplitude damping reservoir @xcite . these results show rich dynamical behaviors of the entanglement and its characters actually have not been clearly identified . recently , lpez _ et al . _ asked a question about where the lost entanglement of the qubits goes @xcite . interestingly , they found that the lost entanglement of the qubits is exclusively transferred to the reservoirs under the markovian amplitude - damping decoherence dynamics and esd of the qubits is always accompanied with the entanglement sudden birth ( esb ) of the reservoirs . a similar situation happens for the spin entanglement when the spin degree of freedom for one of the two particles interacts with its momentum degree of freedom @xcite . all these results mean that the entanglement does not go away , it is still there but just changes the location . this is reminiscent of the work of yonac _ et al . _ @xcite , in which the entanglement dynamics has been studied in a double jaynes - cummings ( j - c ) model . they found that the entanglement is transferred periodically among all the bipartite partitions of the whole system but an identity ( see below ) has been satisfied at any time . this may be not surprising since the double j - c model has no decoherence and any initial information can be preserved in the time evolution . however , it would be surprising if the identity is still valid in the presence of the decoherence , in which a non - equilibrium relaxation process is involved . in this paper , we show that it is indeed true for such a system consisted of two qubits locally interacting with two amplitude - damping reservoirs . it is noted that although the infinite degrees of freedom of the reserviors introduce the irreversibility to the subsystems , this result is still reasonable based on the fact that the global system evolves in a unitary way . furthermore , we find that the distribution of the entanglement among the bipartite subsystems is dependent of the explicit property of the reservoir and its coupling to the qubit . the rich dynamical behaviors obtained previously in the literature can be regarded as the special cases of our present result or markovian approximation . particularly , we find that , instead of entirely transferred to the reservoirs , the entanglement can be stably distributed among all the bipartite subsystems if the qubit and its reservoir can form a bound state and the non - markovian effect is important , and the esd of the qubits is not always accompanied with the occurrence of esb of reservoirs . irrespective of how the entanglement distributes , it is found that the identity about the entanglement in the whole system can be satisfied at any time , which reveals the profound physics of the entanglement dynamics under decoherence . this paper is organized as follows . in sec . [ model ] , the model of two independent qubits in two local reservoirs is given . and the dynamical entanglement invariance is obtained based on the exact solution of the non - markovian decoherence dynamics of the qubit system . in sec . [ edd ] , the entanglement distribution over the subsystems when the reservoirs are pbg mediums is studied explicitly . a stable entanglement - distribution configuration is found in the non - markovian dynamics . finally , a brief discussion and summary are given in sec . we consider two qubits interacting with two uncorrelated vacuum reservoirs . due to the dynamical independence between the two local subsystems , we can firstly solve the single subsystem , then apply the result obtained to the double - qubit case . the hamiltonian of each local subsystem is @xcite @xmath0 where @xmath1 and @xmath2 are the inversion operators and transition frequency of the qubit , @xmath3 and @xmath4 are the creation and annihilation operators of the @xmath5-th mode with frequency @xmath6 of the radiation field . the coupling strength between the qubit and the reservoir is denoted by @xmath7 , where @xmath8 and @xmath9 are the unit polarization vector and the normalization volume of the radiation field , @xmath10 is the dipole moment of the qubit , and @xmath11 is the free space permittivity . for such a system , if the qubit is in its ground state @xmath12 and the reservoir is in vacuum state at the initial time , then the system does not evolve to other states . when the qubit is in its excited state @xmath13 , the system evolves as @xmath14 here @xmath15 denotes that the qubit jumps to its ground state and one photon is excited in the @xmath5-th mode of the reservoir . @xmath16 satisfies an integro - differential equation @xmath17 where the kernel function @xmath18 is dependent of the spectral density @xmath19 . introducing the normalized collective state of the reservoir with one excitation as @xmath20 and with zero excitation as @xmath21 @xcite , eq . ( [ t9 ] ) can be written as @xmath22 , where @xmath23 . it should be emphasized that the introducing of normalized collective state is not a reduction of present model to the j - c model @xcite , as noted in @xcite . the dynamics is given by eq . ( [ t7 ] ) , which is difficult to obtain analytically since its non - markovian nature . in general the numerical integration should be used . it is emphasized that our treatment to the dynamics of the system is exact without resorting to the widely used born - markovian approximation . to compare with the conventional approximate result , we may derive straightforwardly the master equation from eq . ( [ t9 ] ) after tracing over the degree of freedom of the reservoir @xcite , @xmath24+\gamma ( t)[2\sigma _ { -}\rho ( t)\sigma _ { + } \nonumber \\ & & -\sigma _ { + } \sigma _ { -}\rho ( t)-\rho ( t)\sigma _ { + } \sigma _ { - } ] , \label{mstt}\end{aligned}\]]where the time - dependent parameters are given by @xmath25,~\gamma ( t)=-\text{re}[\frac{% \dot{b}(t)}{b(t)}]$ ] . the time - dependent parameters @xmath26 and @xmath27 play the roles of lamb shifted frequency and decay rate of the qubit , respectively . the integro - differential equation ( [ t7 ] ) contains the memory effect of the reservoir registered in the time - nonlocal kernel function and thus the dynamics of qubit displays non - markovian effect . if the time - nonlocal kernel function is replaced by a time - local one , then eq . ( [ mstt ] ) recovers the conventional master equation under born - markovian approximation @xcite . according to the above results , the time evolution of a system consisted of two such subsystems with the initial state @xmath28 is given by @xmath29 where @xmath30 and @xmath31 are the coefficients to determine the initial entanglement in the system . from @xmath32 , one can obtain the time - dependent reduced density matrix of the bipartite subsystem qubit1-qubit2 ( @xmath33 ) by tracing over the reservoir variables . it reads @xmath34 where @xmath35 and @xmath36 . similarly , one can obtain the corresponding reduced density matrices for other subsystems like reservoir1-reservoir2 ( @xmath37 ) and qubit - reservoir ( @xmath38 , @xmath39 , @xmath40 , @xmath41 ) . using the concurrence @xcite to quantify entanglement , we can calculate the entanglement of each subsystem as @xmath42 with @xmath43 for different bipartite partitions labeled by @xmath44 as @xmath45 one can verify that @xmath46 in eqs . ( [ qq])-([qr2 ] ) satisfy an identity @xmath47 where @xmath48 is just the initial entanglement present in @xmath33 . ( [ t15 ] ) recovers the explicit form derived in a double j - c model @xcite when each of the reservoirs contains only one mode , i.e. @xmath49 , where the decoherence is absent and the dynamics is reversible . it is interesting that this identity is still valid in the present model because the reservoirs containing infinite degrees of freedom here lead to a completely out - of - phase interaction with qubit and an irreversibility . furthermore , one notes that the identity is not dependent of any detail about @xmath16 , which only determines the detailed dynamical behavior of each components in eq . ( [ t15 ] ) . this result manifests certain kind of invariant nature of the entanglement . ( [ t15 ] ) can be intuitively understood by the global multipartite entanglement of the whole system . the global entanglement carried by the subsystem @xmath50 can be straightforwardly calculated from eq . ( [ phit ] ) by generalized concurrence @xcite as @xmath48 , which , coinciding with the bipartite entanglement initially present in @xmath33 , just is the right hand side of eq . ( [ t15 ] ) . since there is no direct interaction between @xmath51 and @xmath52 , this global entanglement is conserved during the time evolution . from this point , our result is consistent with the one in refs . @xcite and @xcite . another observation of eq . ( [ t15 ] ) is that the different coefficients in the left hand side are essentially determined by the energy / information transfer among the local subsystems . explicitly , in our model the total excitation number is conserved , so the energy degradation in @xmath53 with factor @xmath16 is compensated by the energy enhancement in @xmath54 with factor @xmath55 . this causes that @xmath56 , @xmath57 , and @xmath58 , in all of which the double excitation is involved , have similar form except for the different combinations of @xmath16 and @xmath55 in eqs . ( [ qq ] ) , ( [ rr ] ) , and ( [ qr2 ] ) . the dynamical consequence of the competition of the two terms in these equations causes the sudden death / birth of entanglement characterized by the presence of negative @xmath59 . a different case happens for @xmath60 , where only single excitation is involved and no sudden death is present . with these observation , one can roughly understand why such combination in left hand side of eq . ( [ t15 ] ) gives the global entanglement . the significance of eq . ( [ t15 ] ) is that it gives us a guideline to judge how the entanglement spreads out over all the bipartite partitions . it implies that entanglement is not destroyed but re - distributed among all the bipartite subsystems and this re - distribution behavior is not irregular but in certain kind of invariant manner . the similar invariant property of entanglement evolution has also been studied in ref . @xcite . in the following we explicitly discuss the entanglement distribution , especially in the steady state , by taking the reservoir as a photonic band gap ( pbg ) medium @xcite and compare it with the previous results . we will pay our attention mainly on the consequence of the non - markovian effect on the entanglement distribution and its differences to the results in refs . @xcite and @xcite . for the pbg medium , the dispersion relation near the upper band - edge is given by @xcite @xmath61 where @xmath62 , @xmath63 is the upper band - edge frequency and @xmath64 is the corresponding characteristic wave vector . in this case , the kernel function has the form @xmath65 where @xmath66 is a dimensionless constant . in solving eq . ( [ t7 ] ) for @xmath16 , eq . ( [ kn ] ) is evaluated numerically . here we do not assume that @xmath5 is replaced by @xmath64 outside of the exponential @xcite . so our result is numerically exact . in the following we take @xmath67 as the unit of frequency . [ cols="^,^ " , ] in figs . [ bst ] and [ nbst ] , we show the entanglement evolutions of each subsystem for two typical cases of @xmath68 and @xmath69 , which correspond to the atomic frequency being located at the band gap and at the upper band of the pbg medium , respectively . in the both cases the initial entanglement in @xmath33 begins to transfer to other bipartite partitions with time but their explicit evolutions , in particular the long time behaviors , are quite different . in the former case , the entanglement could be distributed stably among all possible bipartite partitions . [ bst](a ) shows that after some oscillations , a sizeable entanglement of @xmath33 is preserved for the parameter regime of @xmath70 . remarkably , the entanglement in @xmath71 forms quickly in the full range of @xmath30 [ fig . [ bst](c ) ] and dominates the distribution . on the contrary , only slight entanglement of @xmath37 is formed in a very narrow parameter regime @xmath72 , as shown in fig . [ bst](b ) . however , when @xmath73 is located at the upper band of the pbg medium , the initial entanglement in @xmath33 is transferred completely to the @xmath37 in the long - time limit , as shown in fig . [ nbst ] . at the initial stage , @xmath74 and @xmath75 are entangled transiently , but there is no stable entanglement distribution . this result is consistent with that in refs . it is noted that the entanglement in @xmath76 comes from two parts : one is transferred from @xmath33 , the other is created by the direct interaction between @xmath53 and @xmath54 . this can be seen clearly from fig . [ bst](c ) when @xmath30 is very small , the initial entanglement of @xmath33 is very small , while that of @xmath76 is rather large , which just results from the interaction between @xmath53 and @xmath54 . another interesting point is a stable entanglement can even be formed for the non - interacting bipartite system @xmath39 [ see [ bst](d ) when @xmath77 . this entanglement transfer also results from the local interaction between @xmath53 and @xmath54 @xcite . it is not difficult to understand these rich behaviors of entanglement distribution according to eqs . ( [ qq])-([qr2 ] ) and its invariance ( [ t15 ] ) . from these equations , one can clearly see that the entanglement dynamics and its distributions in the bipartite partitions are completely determined by the time - dependent factor @xmath78 of single - qubit excited - state population . [ tong0 ] shows its time evolutions for the corresponding parameter regimes presented above . we notice that @xmath79 when @xmath73 is located at the band gap , which means that there is some excited - state population in the long - time limit . this phenomenon known as population trapping @xcite is responsible for the suppression of the spontaneous emission of two - level system in pbg reservoir and has been experimentally observed @xcite . such population trapping just manifests the formation of bound states between @xmath53 and @xmath54 @xcite , which has been experimentally verified in @xcite . consequently , @xmath53 and @xmath54 are so correlated in the bound states that the initial entanglement in @xmath33 can not be fully transferred to @xmath37 . the oscillation during the evolution is just the manifestation of the strong non - markovian effect induced by the reservoirs . on the contrary , if @xmath73 is located in the upper band , then @xmath80 and the qubits decay completely to their ground states . in this case the bound states between @xmath53 and @xmath54 are absent and , according to eq . ( [ t15 ] ) , the initial entanglement in @xmath33 is completely transferred to @xmath37 , as clearly shown in eq . ( [ rr ] ) . in addition , in refs . @xcite it was emphasized that esd of @xmath33 is always accompanied with esb of @xmath37 . however , this is not always true . to clarify this , we examine the condition to obtain esd of the qubits and the companying esb of the reservoirs . from eqs . ( [ qq ] ) and ( [ rr ] ) it is obvious that the condition is @xmath81 and @xmath82 at any @xmath83 and @xmath84 , which means @xmath85 when the bound states is absent , @xmath86 , the condition ( [ dd ] ) can be satisfied when @xmath87 . so one can always expect esd of the qubits and the companying esb of the reservoirs in the region @xmath88 , as shown in fig . [ nbst ] and refs . however , when the bound states are available , the situation changes . in particular , when @xmath89 in the full range of time evolution , no region of @xmath90 can make the condition ( [ dd ] ) to be satisfied anymore . for clarification , we present three typical behaviors of the entanglement distribution in fig . [ td ] . in all these cases the bound states are available . [ td](a ) shows the situation where the entanglement is stably distributed among all of the bipartite subsystems . [ td](b ) indicates that the entanglement of @xmath91 shows esb and revival , while the entanglement of @xmath33 does not exhibit esd . [ td](c ) shows another example that while the entanglement of @xmath33 has esd and revival @xcite , the entanglement of @xmath37 does not show esb but remains to be zero . [ td](b , c ) reveal that esd in @xmath33 has no direct relationship with esb in @xmath37 . the above discussion is not dependent of the explicit spectral density of the individual reservoir . to confirm this , we consider the reservoir in free space . the spectral density has the ohmic form @xmath92 , which can be obtained from the free - space dispersion relation @xmath93 . one can verify that the condition for the formation of bound states is : @xmath94 @xcite . in fig . [ osd ] , we plot the results in this situation . the previous results can be recovered when the bound states are absent @xcite . on the contrary , when the bound states are available , a stable entanglement is established among all the bipartite partitions . therefore , we argue that the stable entanglement distribution resulted from the bound states is a general phenomenon in open quantum system when the non - markovian effect is taken into account . in summary , we have studied the entanglement distribution among all the bipartite subsystems of two qubits embedded into two independent amplitude damping reservoirs . it is found that the entanglement can be stably distributed in all the bipartite subsystems , which is much different no matter to the markovian approximate result @xcite or to the decoherenceless double j - c model result @xcite , and an identity about the entanglement in all subsystems is always satisfied . this identity is shown to be independent of any detail of the reservoirs and their coupling to the qubit , which affect only the explicit time evolution behavior and the final distribution . the result is significant to the study of the physical nature of entanglement under decoherence . it implies an active way to protect entanglement from decoherence by modifying the properties of the reservoir via the potential usage of the newly emerged technique , i.e. quantum reservoir engineering @xcite . this work is supported by the fundamental research funds for the central universities under grant no . lzujbky-2010 - 72 , gansu provincial nsf under grant no . 0803rjza095 , the national nsf of china , the program for ncet , and the cqt wbs grant no . r-710 - 000 - 008 - 271 .

we study the entanglement dynamics of two qubits , each of which is embedded into its local amplitude - damping reservoir , and the entanglement distribution among all the bipartite subsystems including qubit - qubit , qubit - reservoir , and reservoir - reservoir . it is found that the entanglement can be stably distributed among all components , which is much different to the result obtained under the born - markovian approximation by c. e. lpez _ et al . _ [ phys . rev . lett . * 101 * , 080503 ( 2008 ) ] , and particularly it also satisfies an identity . our unified treatment includes the previous results as special cases . the result may give help to understand the physical nature of entanglement under decoherence .

1005.1001

herbig - haro ( hh ) objects immersed in an ultraviolet ( uv ) radiation field can be photoionized externally @xcite . the photoionized jets / outflows of hh objects become optically visible , and thus their detailed physical properties can be studied . such photoionized hh jet systems have been identified in the orion nebula and in the reflection nebula ngc 1333 @xcite . recently , two such photoionized jet systems , the rosette hh1 and hh2 jets , were discovered within the central cavity of the rosette nebula @xcite . the rosette nebula is a spectacular region excavated by strong stellar winds from dozens of ob stars at the center of the young open cluster ngc 2244 , the primary component of a possible twin cluster recently identified using the 2mass ( two micron all sky survey ) database @xcite . at a distance of @xmath2 1.39 kpc @xcite , this emerging young open cluster is found to have a main sequence turn - off age of about 1.9 myr @xcite . the photoionized jets discovered in the rosette nebula @xcite and their counterparts found in the vicinity of @xmath8 orionis @xcite are both bathed in harsh uv radiation from massive ob stars within a few parsecs , and thus share many similar properties consistent with an irradiated origin of the jet systems : ( 1 ) their jet - driving sources are visible and show spectral characteristics of t tauri stars . ( 2 ) these sources were not detected by _ iras _ ( _ infrared astronomical satellite _ ) , indicating a lack of circumstellar material such as extended disks and/or envelopes . ( 3 ) the jets show [ ] /h@xmath5 line ratio decreasing from the base outward , indicating that the dominant excitation mechanism changes from shocks at the base to photoionization at the end of the jet . ( 4 ) the jet systems all have a highly asymmetric or even unipolar morphology , indicating perhaps different jet forming conditions in the launch and collimation regions . the rosette hh jets show subtle differences from other externally photoionized hh jets because of different degrees of hardness in the uv radiation field or strength of fast stellar winds . both the rosette hh1 and hh2 jets show high excitation @xcite , as the rosette nebula contains an o4 star and an o5 star @xcite . in the orion nebula , hh jets with [ ] emission are found only within 30 , or @xmath20.06 pc , from @xmath9oric , an o4 - 6 star , the earliest o star in the orion nebula @xcite . the high excitation of these hh jets results from both the harsh uv radiation and strong fast stellar wind of @xmath9oric @xcite . @xcite propose that the rosette jets provide evidence for efficient dissipation of circumstellar disks and envelopes in the close vicinity of massive ob stars . this uv dissipation of pre - existing protostellar systems may lead to the formation of isolated brown dwarfs ( bds ) and free - floating giant planets . such a formation mechanism for single sub - stellar objects has indeed been shown to be effective by theoretical studies @xcite . it is therefore important to explore the nature of jet formation and disk dissipation of low - mass ysos in close vicinity of massive ionizing ob stars , as the occurrence of such ob clusters and associations is common in the galaxy , and the solar system may have been formed in such environments @xcite . furthermore , there has been an on - going debate whether weak - lined ttauri stars ( wttss ) evolve from classical ttauri stars ( cttss ) through gradual dissipation of circumstellar material , or wttss are formed through rapid disk dissipation due to external forces after the formation of the protostar . a detailed study of the rosette jet systems may provide insight on the rapid evolution of cttss to wttss due to external photoionization of their protostellar disks in massive star forming regions . wttss formed in this way have indistinguishable evolutionary ages from those of ctts that originated from the same episode of star formation . @xcite presented a kinematical study of the rosette hh1 jet and confirmed the jet nature of the system . here we investigate in detail the physical nature of the jet system using high - resolution imaging and echelle spectroscopy , as well as data from a simultaneous photometric and spectroscopic monitoring of the jet - driving source . narrow - band h@xmath5 images of the rosette nebula were obtained with the 8k@xmath108k mosaic ccd camera on the mayall 4 m telescope at the kitt peak national observatory on 2001 october 13 . a set of five 600 s exposures was taken , with each image slightly offset to fill in physical gaps between the mosaic ccds . the pixel scale is 0258 pixel@xmath0 , resulting in roughly a 36@xmath11 field of view . we obtained high - dispersion spectroscopic observations of rosette hh1 with the echelle spectrograph on the blanco 4 m telescope at the cerro tololo inter - american observatory on 2004 january 9 and 12 . in each observation a 79 line mm@xmath0 echelle grating was used . the observations on 2004 january 9 were made in a multi - order mode , using a 226 line mm@xmath0 cross - disperser and a broad - band blocking filter ( gg385 ) . the spectral coverage is roughly 40007000 , so that nebular lines of a range of excitation can be examined . in the case of the [ ] @xmath126717 , 6731 doublet , the line ratio has been used to estimate the electron densities within the jet . the observations on 2004 january 12 were made in a single - order mode , using a flat mirror and a broad h@xmath5 filter ( central wavelength 6563 with 75 fwhm ) to isolate the order containing the h@xmath5 and [ ] @xmath126548 , 6583 lines . the exposure time used for both instrumental setups was 1,200 s. for each observation the long - focus red camera was used to obtain a reciprocal dispersion of 3.5 mm@xmath0 at h@xmath5 . the spectra were imaged using the site2k # 6 ccd detector . the 24 @xmath13 m pixel size corresponds to 026 pixel@xmath0 along the slit and @xmath20.08 pixel@xmath0 along the dispersion axis . both observations used a 16 slit oriented roughly along the jet direction , at position angles of 312@xmath14 ( multi - order ) and 318@xmath14 ( single - order ) . the resultant instrumental resolution , as measured by the fwhm of the unresolved telluric emission lines , was 0.29 or 13 km s@xmath0 at h@xmath5 . the observations were reduced following standard procedures in the iraf ( ver . 2.12 ) software package . this included bias correction , flat - fielding and gain - jump removal between the chips . cosmic - ray hits were manually rejected from the 2d spectrograms . wavelength calibration of the data was carried out based on th - ar lamp exposures and further improved by comparison with night sky emission lines , which resulted in an accuracy of @xmath21 km s@xmath0 before converting to the heliocentric frame . we have carried out a simultaneous photometric and spectroscopic monitoring campaign of the rosette hh1 source between 2004 december 31 and 2005 january 7 . the time - series photometric observations , unaccompanied by spectroscopy , were further extended from january 8 to january 13 . the photometric observations were made in @xmath15 and @xmath16 filters with the 0.8 m telescope of the hsing - hua university , located at the xing - long station of the national astronomical observatory of the chinese academy of sciences ( naoc ) . differential photometry of the jet - driving source was obtained through comparisons with two slightly brighter stars in the same field at @xmath5(j2000 ) = 06@xmath17 , @xmath18(j2000 ) = @xmath19 and @xmath5(j2000 ) = @xmath20 , @xmath18(j2000 ) = @xmath21 . the @xmath16 band photometry of the reference stars is found to be constant within 0.04 mag throughout the monitoring campaign . many of the @xmath15 band exposures were affected by charge bleeding from the saturated o9.5 star hd46241 . these unreliable data are not presented here . low - resolution spectroscopy of the jet source was obtained with the 2.16 m telescope of naoc during this monitoring campaign . two different spectrographs were used . from 2004 december 31 to 2005 january 4 , the beijing faint object spectrograph and camera ( bfosc ) , a copy of efosc in service at the european southern observatory , and a thinned back - illuminated orbit 2k @xmath10 2k ccd were used . the g4 grating was employed , which gave a two - pixel resolution of 8.3 . on 2005 january 57 , an omr ( optomechanics research inc . ) spectrograph and a tecktronix 1024 @xmath10 1024 ccd were used . these spectroscopic data have a higher resolution , with a 100 mm@xmath0 reciprocal dispersion and a two - pixel resolution of 4.8 . both sets of observations used a 2@xmath22 slit . the spectroscopic data were reduced using standard procedures and packages in iraf . the ccd reductions included bias and flat - field correction , nebular background subtraction , and cosmic rays removal . wavelength calibration was performed using he - ar lamp exposures at both the beginning and the end of the observations every night . flux calibration of each spectrum was based on observations of at least 2 of the kpno spectral standards @xcite per night . figure 1 presents our new h@xmath5 image of the rosette hh1 jet . this high - quality image reveals that the jet does not trace back through the exact center of the jet - driving source . the morphology of the jet appears to indicate episodic or nonsteady mass ejection . a close inspection shows a split at the end of the collimated jet , with one branch remaining straight while the other bending north possibly as a result of an interaction with the stellar wind of the o4 star , hd46223 . in figure 2 , we present the echelle spectrograms of the rosette hh1 jet . the continuum emission at the origin of each spectrogram is from the jet - driving source . the single - order observations cover only the h@xmath5 and [ ] lines ( the two panels to the left in fig . 2 ) . the multi - order observations detected the h@xmath5 , h@xmath23 , h@xmath24 , @xmath255876 , [ ] @xmath126548 , 6583 , [ ] @xmath124959 , 5007 , and [ ] @xmath126716 , 6731 lines . the three panels to the right in figure 2 show the [ ] @xmath256583 , [ ] @xmath256731 , and [ ] @xmath255007 lines ( the brighter component of each doublet ) . these lines have different thermal widths and require different excitation energies , and thus appear different and can be intercompared to gain physical insight . the [ ] lines have smaller thermal widths than the h@xmath5 line and thus resolve the velocity structures of the hh jet and the superposed nebula more clearly . the [ ] lines detect a prominent irregular component at the location of the jet . this jet component is blue - shifted with respect to two nebular components that have nearly uniform velocity and surface brightness throughout the slit . these two nebular components , at heliocentric velocities ( @xmath26 ) of @xmath213 and 40 km s@xmath0 , arise from the approaching and receding sides of the rosette nebula s expanding shell . these velocities imply a systemic velocity of @xmath27 27 km s@xmath0 and an expansion velocity of @xmath214 km s@xmath0 . the extreme velocity of the jet reaches @xmath26 = @xmath130 km s@xmath0 , which is blue - shifted from the rosette s sytemic velocity by 57 km s@xmath0 . these results are consistent with those reported by @xcite . in the single - order observation along pa = 318@xmath14 , the [ ] emission of the jet shows two bright knots and two faint knots , with the outermost knot being the faintest . no emission from a counterjet is detected . the multi - order observation has a shorter slit along a slightly different position angle , pa = 312@xmath14 , and thus shows a slighly different velocity structure in the [ ] line . the [ ] emission shows velocity structure and surface brightness similar to those of the [ ] emission . the [ ] emission , on the other hand , shows a smooth surface brightness distribution , in contrast to the knots seen in the other lines . the observed velocity fwhm of the [ ] line ranges from @xmath216 to @xmath223 km s@xmath0 , with the fainter knots showing broader velocity widths . these widths are not much larger than the observed fwhm of 18 - 20 km s@xmath0 in the rosette expanding shell components . for an instrumental fwhm of 12 km s@xmath0 and a thermal width of 5.7 km s@xmath0 for [ ] at 10@xmath7 k , the observed fwhm of the jet implies an intrinsic turbulent fwhm of 9 to 19 km s@xmath0 . one interesting spectral feature suggested by @xcite for the jet - driving source is an inverse p cygni profile at the h@xmath5 line based on a low - dispersion spectrum . an inverse p cygni profile , if confirmed , indicates that material is being accreted onto the star . our new high - dispersion echelle observations clearly resolve both spatially and spectrally the nebular and stellar components of the h@xmath5 line profile , and thus allow a critical assessment of this suggested inverse p cygni profile . as seen in figure 2 and shown below , the bright nebular emission from the rosette nebula makes it difficult to accurately extract a clean stellar spectrum . the nebular spectrum varies along the slit . to assess the nebular contribution , we have extracted seven h@xmath5 line profiles using 1@xmath28-wide windows and 05 intervals stepping across the stellar spectrum . these h@xmath5 profiles are shown in figure 3 . the spectrum j is extracted from the jet side of the star , and the blue - shifted jet component is clearly seen . seeing spreads the jet emission into the stellar spectra s1s4 . the contribution from the rosette nebula is better represented by the spectra n1 and n2 , extracted outside the star on the side opposite to the jet . the average of these nebular spectra is subtracted from the four stellar spectra s1s4 . the nebula - subtracted stellar spectra s1@xmath29s4@xmath29 , displayed in figure 4 , show a narrow , blue - shifted emission component and a broad , red - shifted absorption component superposed on a continuum . to determine the origin of the blue - shifted emission component , we use the [ ] @xmath256583 forbidden line that is expected only from low - density gas , such as the rosette nebula and the hh1 jet . the seven [ ] line profiles extracted in a similar manner are displayed in the right panel of figure 3 , and they indeed show the two components from the rosette nebula throughout the slit . the [ ] profiles in the four nebula - subtracted stellar spectra , displayed in the right panel of figure 4 , show that the nebula - subtraction satisfactorily removes the emission from the rosette nebula and that the remaining [ ] emission is from the hh1 jet . the comparison between the nebula - subtracted h@xmath5 and [ ] profiles suggests that the blue - shifted h@xmath5 emission in the stellar spectra predominantly arises from the jet . to obtain a clean stellar spectrum , we scale and subtract the nebula - subtracted jet spectrum ( j@xmath29 ) from the nebula - subtracted stellar spectra ( s1@xmath29s4@xmath29 ) by trial and error until the [ ] emission is minimized . to better show the stellar continuum , the spectra from each step of this procedure are shown in figure 5 over a larger wavelength range . the final clean stellar spectra s2@xmath28 and s3@xmath28 show h@xmath5 absorption with little or no blue - shifted emission . the lack of strong stellar h@xmath5 emission implies the absence of a significant disk , making it difficult to be associated with a jet . this will be discussed further in section 7 . our final clean stellar h@xmath5 line profile is quite different from the previously reported inverse p - cygni profile @xcite . as we have illustrated above , the h@xmath5 emission is dominated by contributions from the rosette nebula and the hh1 jet . these emission components can be resolved and subtracted accurately only if high - dispersion spectra are used . the apparent difference between the final clean stellar h@xmath5 profiles s2@xmath28 and s3@xmath28 probably results from imperfect subtraction of the jet component , as the [ ] /h@xmath5 ratio may vary along the jet . the previously reported inverse p - cygni profile is most likely an artifact caused by difficulties in subtracting the nebular background and jet contribution using low - dispersion spectra . the uv radiation in the rosette nebula is predominantly provided by the massive stars hd 46223 and hd 46150 . hd 46223 , of spectral type o4v(f ) , is the hottest star in ngc 2244 , and produces lyman photons at a rate of 10@xmath30 s@xmath0 @xcite . it is located at 277@xmath22 , or 2.0 pc for a distance of 1.5 kpc @xcite , from the hh1 jet source . hd 46150 is an o5v star projected at 433@xmath22 , or 3.1 pc , from the jet source . it produces ionizing photons at a rate of 10@xmath31 s@xmath0 @xcite . the combined lyman continuum emission from these two exciting stars renders 1 - 2 orders of magnitudes higher impact on the rosette hh1 jet than that on similar jets discovered in the vicinity of @xmath8 orionis @xcite and the trapezium stars @xcite . the rosette nebula is therefore among the most extreme environments in which photoionized jets are found . although immersed in a photoionized medium , the presence of highly collimated jets strongly suggests the existence of at least a relic disk as a sustained feed to the surviving jet . in the case of the rosette hh1 jet , we expect a photoevaporating disk with a configuration resembling that of hh527 in the orion nebula , as resolved by the _ hubble space telescope _ @xcite . this could serve as a schematic impression of the appearance of the disk - jet system . their configuration of the disk subject to photoevaporation induced dissipation is believed to be similar , although the jet associated with hh527 may be oriented at a different direction with respect to the incident uv radiation and has a low excitation , being located in the outskirts of the orion nebula . the electron density of the hh1 jet was derived from the [ ] doublet ratios of @xmath256716/@xmath256731 measured with our multi - order echelle observation along the jet . the @xmath256716/@xmath256731 ratios are [email protected] in the jet and [email protected] in the rosette nebula . the corresponding electron densities are @xmath21000 @xmath3 in the jet and @xmath4100 @xmath3 in the backgound region . if the hh1 jet is indeed within the cavity of the rosette nebula , the medium between the ionizing stars and the hh1 jet is hot and ionized with a density of @xmath20.1 h - atom @xmath3 @xcite . the stellar ionizing flux at the hh1 jet would be nearly unattenuated , at a level of @xmath33 photons @xmath34 s@xmath0 . for a medium of 1000 @xmath3 density , this flux can ionize gas to a thickness of 0.37 pc . the width of the hh1 jet ( measured from fig . 1 ) is @xmath351.5@xmath28 , or @xmath35 0.01 pc ; thus the hh1 jet can be fully photoionized by the radiation from hd 46223 and hd 46150 . using the h@xmath5 surface brightness of the hh1 jet , @xmath36 ergs s@xmath0 @xmath34 sr@xmath0 , and a density of 1000 @xmath3 , we find that the width ( or the depth for a cylindrical geometry ) of the jet is @xmath20.8@xmath28 , or 0.0056 pc . if we assume a flow velocity of 200 km s@xmath0 , as did @xcite , the mass loss rate would be @xmath37 yr@xmath0 . it ought to be noted that when the disk - jet systems are exposed to photoionizing environments , the jet production is probably no longer a dominant role of mass loss from the circumstellar disk . photoionization and dissipation of the disk then takes place , or at least consumes the circumstellar materials at a comparable rate as the mass ejection in the form of a jet . if we assume that evaporated flows associated with the dissipating disk of the jet source govern a comparably effective mass erosion as those of the proplyds in the orion nebula @xcite , then for a mean mass loss rate of 4.1 x 10@xmath38 @xmath39 yr@xmath0 @xcite and a disk mass of 0.006 @xmath39 associated with the rosette hh1 source @xcite , the estimated photodissipation timescale of the relic disk is @xmath2 10@xmath40 yrs . given the more extreme environment the rosette hh1 source faces , a mass loss rate an order of magnitude higher may be more likely and the disk dissipation time would be reduced to @xmath41 yrs . photometric results of the jet driving source are presented in figure 6 , which shows irregular variations around the mean with an amplitude as large as @xmath20.2 mag in the r band . this amplitude of variation is about one magnitude lower than that detected for the energy source of the rosette hh2 jet , which amounts to as large as 1.4 mag in r @xcite . variations in the r band are primarily attributed to erratic fluctuations of the h@xmath5 emission of the source , which relies on a time - variable mass accretion rate , disk inhomogeneity , or otherwise chromospheric activity of the central yso . the rosette hh1 source s low amplitude of variation is believed to be due to a lack of circumstellar material and subsequently a subtle mass accretion rate , as shown by the nearly absent h@xmath5 emission from the jet source , although the irregular variation itself is consistent with a young status of evolution of the central source . we thus suggest that the rosette hh1 source may well represent a transient phase of ysos evolving rapidly from a ctts to a wtts by fast photodissipation of their circumstellar disks . based on the time series photometric data achieved , we find no evidence of a binary origin of the jet source , which otherwise could imply a different mechanism of jet production . as noted in section 2.3 , the b band observations do not give very good results , although reminiscent irregular variations with a comparable magnitude of up to @xmath20.25 mag are indeed indicated . the spectral monitoring observations are not very useful for the h@xmath5 line profile because of the difficulty in background nebular subtraction , as discussed in section 4 . nevertheless , the overall spectral characteristics of the star can be determined . we find that the spectral type appears to vary between f8v and f9v during the period of observations . this spectral change may be related to photo - erosion of the rotational disk with an inhomogeneous configuration . being immersed in the fierce uv radiation field of the rosette , the optical jets associated with ysos indicate either a jet production timescale of as long as 1 - 2 myr , comparable to the evolutionary age of the main cluster ngc 2244 , or that the ysos have a much younger age and the cocoons associated with their protostars had , in some way , been successfully shielded from the strong ionization fields . @xcite investigated the ysos with near infrared excesses , an indicator of the existence of circumstellar disks , of the young open cluster ngc 2244 based on the 2mass database . the jet - driving sources in the rosette , however , show infrared colors commensurate with those of wttss , which have spectral energy distributions indistinguishable from main - sequence dwarfs . see the color - color and color - magnitude diagrams in figures 4 and 7 of @xcite . this , along with the fact that none of the rosette jet sources and their only rivals found near @xmath8 orionis were detected by iras , suggests a lack of circumstellar material as compared to conventional ysos driving outflows . this is in agreement with the estimated mass of 0.006 @xmath39 for the relic disk associated with the rosette hh1 source @xcite , far below the typical value of @xmath20.1 @xmath39 around ctts . given the emerging nature of the young open cluster with a turnoff age of 1.9 myr @xcite , fast disk dissipation is suggested . @xcite suggest that this provides indirect observational evidence for the formation of isolated bds and free - floating giant planets , as discovered in orion by @xcite , by uv dissipation of unshielded protostellar systems . this can be very important to our understanding of the formation of such sub - stellar and planetary mass objects , particularly in regions of massive star formation . such uv dissipation could , on the other hand , impose strong effects on the formation of and hence the search for extra - solar planets around low - mass stars , the circumstellar disks of which could otherwise be potential sites of terrestrial planet formation . this alternatively introduces a viable solution to the long puzzle of how wtts were formed as a consequence of fast ctts evolution and the rapid dissipation of circumstellar disks under particular forming conditions near massive ob stars or in cluster environments . the spatial distribution of the extreme jets with respect to the dozens of exciting ob stars of the spectacular region is presented in figure 7 , superimposed on which is the relic shell structure as delineated by the apparent congregation of excessive emission sources in the near infrared @xcite . this suggests the existence of a former working interface layer of the region with its ambient molecular clouds . the projected location of the energy source of the rosette hh1 jet near the relic arc provides evidence of a triggered origin of its formation in or near the swept - up layer . the rosette hh2 source has a similar radial distance from the statistical center of ngc 2244 @xcite and introduces a similar origin . in this scenario , the jet sources should have a much younger age than the main cluster ngc 2244 . molecular gas and dust in the shell could have played an important role in shielding new generation protostellar objects from the harsh uv evaporation and ionization from the massive ob stars . it is therefore reasonable to infer that the jet sources have been directly exposed to the harsh photoionization fields recently . @xcite first noted the existence of ionized knots and filaments in the southeastern quadrant of the rosette s central cavity . the rosette hh jets are also located in this region . all are prominent features in narrowband [ ] images , indicating a high excitation . among these , knot c shows high - velocity components and is in association with a high excitation bow - shock at its tip @xcite . this feature is believed to be a hh flow , though no apparent energy source has yet been identified . however , a giant shock - like structure to the west of the rosette hh1 jet can be easily identified ( figure 8) . at a distance of @xmath21.5 kpc , its large - scale appearance and lack of a potential exciting source seem to exclude the possibility of a herbig - haro origin . the preferential distribution of these high - excitation structures to the southeast edge of the region suggests a possible association with the large monoceros loop supernova remnant ( snr ) projected to the northeast of the rosette nebula . while there is no morphological evidence for dynamical interactions between these two objects , it is possible that some of the high - excitation structures in the rosette nebula are caused by the monoceros loop snr s ballistic ejecta that proceeds ahead of the snr shock front . to test this scenario , proper motion or abundance measurements of the high - excitation structures in the rosette nebula are needed . alternatively , we propose that these structures are globules or former dust pillars , similar to those around the working surface of the region , that have been overrun by the ionization front and are now in the process of photodissipation , as is the fate of the hh jets in this region . at least one high - excitation structure is likely associated with a neutral cometary knot ( see figure 8) , the tip of which is highly ionized and has an appearance resembling those in the orion nebula @xcite , but with a physical size roughly 4 times larger . we present follow - up high - qaulity imaging and echelle spectroscopic observations of the rosette hh1 jet . the high angular and spectral resolution allow us to determine accurately the contributions from the region , jet , and star . the expansion of the region and the kinematics of the jet are consistent with the previous measurements by @xcite . using the [ ] doublet ratios , we further determined the electron density of the jet , @xmath21000 @xmath3 . with a careful subtraction of the nebular and jet components , we find the stellar h@xmath5 line is dominated by a broad absorption profile with little or no emission component , indicating a lack of substantial circumstellar material . the circumstellar material has most likely been photo - evaporated by the strong uv radiation field in the rosette nebula . the evaporation time scale is 10@xmath6 10@xmath7 yr . the rosette hh1 jet source provides evidence for an accelerated evolution from a ctts to a wtts due to the strong uv radiation field ; therefore , both cttss and wttss can be spatially mixed in regions with massive star formation . finally , we suggest that the giant high - excitation structures residing at the center of the rosette nebula may be globules or former dust pillars in the midst of uv dissipation . further observations of the nebular kinematics are needed to determine whether these are dissipating interstellar structures or related to the supernova ejecta associated with the monoceros loop snr . we greatly appreciate the helpful comments and suggestions from the referee of the paper , john meaburn . thanks to the team working with the hsing - hua 80 cm telescope for their help on coordinating the photometric observations . this project is supported by the national natural science foundation of china through grant no.10503006 . andrews , s. m. , reipurth , b. , bally , j. , & heathcote , s. r. 2004 , , 606 , 353 bally , j. , odell , c. r. , & mccaughrean , m. j. 2000 , , 119 , 2919 bally , j. & reipurth , b. 2001 , , 546 , 299 bally , j. , sutherland , r. s. , devine , d. , & johnstone , d. 1998 , , 116 , 293 clayton , c. a. , & meaburn , j. 1995 , , 302 , 202 clayton , c. a. , meaburn , j. , lopez , j. a. , & christopoulou , p. e. , & goudis , c. d. 1998 , , 334 , 264 dorland , h. , & montmerle , t. 1987 , , 177 , 243 henney , w. j. , & odell , c. r. 1999 , , 118 , 2350 hensberge , h. , pavlovski , k. , & verschueren , w. , 2000 , , 358 , 553 johnstone , d. , hollenbach , d. , & bally , j. 1998 , , 499 , 758 li , j. z. 2003 , chinese j. astron . astrophys . , 3 , 495 li , j. z. 2005 , , 625 , 242 li , j. z. , chu , y .- h . , & gruendl , r. a. 2007 , , submitted li , j. z. , & rector , t. a. 2004 , , 600 , l67 looney , l. w. , tobin , j. j. , & fields , b. d. 2006 , , 652,1755 ( astro - ph/0608411 ) maz - apellniz , j. , walborn , n. r. , galu , h. . , & wei , l. h. 2004 , , 151 , 103 massey , p. , strobel , k. , barnes , j. v. , & anderson , e. 1988 , , 328 , 315 meaburn , j. , lopez , j. a. , richer , m. g. , riesgo , h. , & dyson , j. e. 2005 , , 130 , 730 meaburn , j. , & walsh , j. r. 1996 , , 220 , 745 osterbrock , d. e. 1989 , astrophysics of gaseous nebulae and active galactic nuclei ( mill valley : university science books ) panagia , n. 1973 , , 78 , 929 park , b .- g . , & sung , h. 2002 , , 123 , 892 prez , m. r. , th , p. s. , & westerlund , b. e. 1987 , , 99 , 1050 reipurth , b. , bally , j. , fesen , r. a. , & devine d. 1998 , nature , 396 , 343 townsley , l. k. , feigelson , e. d. , montmerle , t. , broos , p. s. , chu , y .- h . , & garmire , g. p. 2003 , , 593 , 874 welsh , b. y. , sfeir , d. m. , sallman , s. , & lallement , r. 2001 , , 372 , 516 whitworth , a. p. , & zinnecker , h. 2004 , , 427 , 299 zapatero osorio , m. r. , bejar , v. j. s. , martin , e. l. , rebolo , r. , barrado y navascus , d. , bailer - jones , c. a. l. , & munct , r. 2000 , science , 290 , 103

the rosette hh1 jet is a collimated flow immersed in the strong uv radiation field of the rosette nebula . we investigate the physical properties of the rosette hh1 jet using high - quality narrow - band images and high - dispersion spectroscopy . the new images show that the axis of the jet is not precisely aligned with the star near the base of the jet . the high resolution of the spectra allows us to accurately determine the contributions from the region , jet , and star . the appoaching and receding sides of the expanding shell of the rosette nebula are at heliocentric velocities of 13 and 40 km s@xmath0 , while the jet reaches a maximum velocity offset at a heliocentric velocity of @xmath130 km s@xmath0 . the [ ] doublet ratios indicate an electron density of @xmath21000 @xmath3 in the jet and @xmath4100 @xmath3 in the region . with a careful subtraction of the nebular and jet components , we find the stellar h@xmath5 line is dominated by a broad absorption profile with little or no emission component , indicating a lack of substantial circumstellar material . the circumstellar material has most likely been photo - evaporated by the strong uv radiation field in the rosette nebula . the evaporation time scale is 10@xmath6 10@xmath7 yr . the rosette hh1 jet source provides evidence for an accelerated evolution from a ctts to a wtts due to the strong uv radiation field ; therefore , both cttss and wttss can be spatially mixed in regions with massive star formation .

astro-ph0603119

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

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

1106.4784

where river water meets the sea , an enormous amount of energy is dissipated as a result of the irreversible mixing of fresh and salt water . the dissipated energy is about 2 kj per liter of river water , _ i.e. _ equivalent to a waterfall of 200 m @xcite . it is estimated that the combined power from all large estuaries in the world could take care of approximately 20% of today s worldwide energy demand @xcite . extracting or storing this energy is therefore a potentially serious option that our fossil - fuel burning society may have to embrace in order to become sustainable . however , interesting scientific and technical challenges are to be faced . so far pressure - retarded osmosis ( pro ) @xcite and reverse electrodialysis ( red ) @xcite have been the two main and best - investigated techniques in this field of so - called `` blue energy '' , or salinity - gradient energy . in pro the osmotic pressure difference across a semi - permeable membrane is used to create a pressurised solution from incoming fresh and salt water , which is able to drive a turbine @xcite . in red stacks of alternating cation- and anion - exchange membranes are used to generate an electric potential difference out of a salinity gradient @xcite . these techniques enable the generation of ( electrical ) work at the expense of the mixing of streams with different salinity . actually , pro and red can be thought of as the inverse processes of reverse osmosis and electrodialyses , where one has to supply ( electrical ) work in order to separate an incoming salt - water stream in a saltier and a fresher stream . + the applicability of pro and red are currently being explored : a 1 - 2 kw prototype plant based on pro was started up in 2009 in norway @xcite , and a 5 kw red device is planned to be upscaled to a 50 kw demonstration project in the netherlands @xcite . interestingly , the bottleneck to large - scale applications of both these techniques is often _ not _ the available fuel there is a lot of fresh and salt water but rather the very large membranes that are required to operate at commercially interesting power outputs . tailoring such membranes with a very high transport capacity and minimal efficiency losses due to biofouling requires advanced membrane technology . recently , however , a solid - state device _ without _ membranes was constructed by brogioli @xcite , who directly extracts energy from salinity differences using porous carbon electrodes immersed in an aqueous electrolyte . due to the huge internal surface of porous carbon , of the order of @xmath0 m@xmath1 per gram of carbon , the capacitance of a pair of electrolyte - immersed porous carbon electrodes can be very large , allowing for large amounts of ionic charge to be stored in the diffuse part of the double layers of the electrolytic medium inside the pores @xcite . in fact , although the energy that is stored in the charged state of such large - area electrodes is somewhat lower than that in modern chargeable batteries , the power uptake and power delivery of these ultracapacitors is comparable or even larger @xcite . the capacitance of these devices not only scales with the contact area between the electrode and the electrolyte , but also with the inverse distance between the electronic charge on the electrode and the ionic charge in the diffuse part of the double layer , i.e. the capacitance increases with the inverse of the thickness of the ionic double layer . as a consequence , the capacitance increases with increasing salinity , or , in other words , the potential increases at fixed electrode charge upon changing the medium from salt to fresh water . this variability of the capacity was used by brogioli @xcite , and also more recently by brogioli _ _ et al.__@xcite , to extract electric work from salinity gradients without membranes . although sales _ et al . _ showed that the combination of membranes and porous electrodes has some desirable advantages @xcite , we will focus here on brogioli s experiment . the key concept of ref.@xcite is a four - stage cycle abcda of a pair of porous electrodes , together forming a capacitor , such that 1 . the two electrodes , immersed in sea water , are charged up from an initial state a with low initial charges @xmath2 to a state b with higher charges @xmath3 ; 2 . the salt water environment of the two electrodes is replaced by fresh water at fixed electrode charges @xmath3 , thereby increasing the electrostatic potential of the electrodes from @xmath4 to @xmath5 ; 3 . the two highly charged electrodes , now immersed in fresh water in state c , are discharged back to @xmath2 in state d , and finally 4 . the fresh water environment of the electrodes is replaced by salt water again , at fixed electrode charges @xmath2 , thereby lowering the electrode potentials to their initial values @xmath6 in state a. this cycle , during which a net transport of ions from salt to fresh water takes place , renders the salt water fresher and the fresh water saltier although only infinitessimally so if the reservoir volumes are infinitely large . as a consequence , the ionic entropy has increased after a cycle has been completed , and the associated free - energy reduction of the combined device and the two electrolyte reservoirs equals the electric work done by the device during the cycle , as we will see in more detail below . brogioli extrapolates an energy production of 1.6 kj per liter of fresh water in his device @xcite , equivalent to a waterfall of 160 m , quite comparable to current membrane - based techniques . these figures are promising in the light of possible future large - scale blue - energy extraction . together with the large volume of fresh and salt water at the river mouths of this planet , they also put an interesting and blessing twist to bob evans quotes at the beginning of this article . below we investigate the ( free ) energy and the performed work of electrolyte - immersed supercapacitors within a simple density functional that gives rise to a modified poisson - boltzmann ( pb ) equation for the ionic double layers . by seeking analogies with the classic carnot cycle for heat engines with their maximum efficiency to convert heat into mechanical work given the two temperatures of the heat baths , we consider modifications of brogioli s cycle that may maximise the conversion efficiency of ionic entropy into electric work given the two reservoir salt concentrations . our modification does _ not _ involve the trajectories ab and cd of the cycle where the ( dis)charging electrodes are in diffusive contact with an electrolytic reservoir with the inhomogeneously distributed salt ions `` properly '' treated grand - canonically as often advocated by bob evans @xcite . in fact , we will argue that the grand - canonical trajectories ab and cd at constant ionic chemical potential are the analogue of the isotherms in the carnot cycle . rather we consider to modify the constant - charge trajectories bc and da ( which correspond to isochores in a heat - engine as we will argue ) by a continued ( dis)charging process of the electrodes at a constant number of ions ( which corresponds to an adiabatic ( de)compression in the heat engine ) . in other words , we propose to disconnect the immersed electrodes from the ion reservoirs in bc and da , treating the salt ions canonically while ( dis)charging the electrodes , thereby affecting the ion adsorption and hence the bulk concentration from salty to fresh ( bc ) and _ vice versa _ ( da ) . finally , we will consider a ( dis)charging cycle in the ( realistic ) case of a finite volume of available fresh water , such that the ion exchange process renders this water brackish ; the heat - engine analogue is a temperature rise of the cold bath due to the uptake of heat . + similar cycles were already studied theoretically by biesheuvel @xcite , although not in this context of osmotic power but its reverse , capacitive desalination . the `` switching step '' in biesheuvel s cycle , where the system switches from an electrolyte with a low salt concentration to an electrolyte with a higher salt concentration , appears to be somewhat different from our proposal here , e.g. without a direct heat - engine analogue . we consider two electrodes , one carrying a charge @xmath7 and the other a charge @xmath8 . the electrodes , which can charge and discharge by applying an external electric force that transports electrons from one to the other , are both immersed in an aqueous monovalent electrolyte of volume @xmath9 at temperature @xmath10 . we denote the number of cations and anions in the volume @xmath9 by @xmath11 and @xmath12 , respectively . global charge neutrality of the two electrodes and the electrolyte in the volume @xmath9 is guaranteed if @xmath13 . if the two electrodes are separated by a distance much larger than the debye screening length a condition that is easily met in the experiments of ref.@xcite then each electrode and its surrounding electrolyte will be separately electrically neutral such that @xmath14 , where @xmath15 is the proton charge and where we assume @xmath16 without loss of generality . note that this `` local neutrality '' can only be achieved provided @xmath17 , where the extreme case @xmath18 corresponds to an electrode charge that is so high that all @xmath12 anions in the volume @xmath9 are needed to screen the positive electrode and all @xmath11 cations to screen the negative one . for @xmath19 , which we assume from now on , we can use @xmath7 and @xmath20 as independent variables of a neutral system of the positive electrode immersed in an electrolyte of volume @xmath21 at temperature @xmath10 , the helmholtz free energy of which is denoted by @xmath22 . at fixed volume and temperature we can write the differential of the free energy of the positive electrode and its electrolyte environment as @xmath23 with @xmath24 the average of the ionic chemical potentials @xmath25 and @xmath26 the electrostatic potential of the electrode . the last term of eq.([df ] ) is the electric work _ done on _ the system if the electrode charge is increased by @xmath27 at fixed @xmath20 , and hence the electrostatic work _ done by _ the electrode system is @xmath28 . given that @xmath29 is a state function , such that @xmath30 for any cycle , the total work _ done by _ the system during a ( reversible ) cycle equals @xmath31 in order to be able to _ calculate _ we thus need explicit cycles _ and _ the explicit equations - of - state @xmath33 and/or @xmath34 , for which we will use a simple density functional theory to be discussed below . however , before performing these explicit calculations a general statement can be made , because there is an interesting analogy to be made with mechanical work @xmath35 _ done by _ a fixed amount of gas at pressure @xmath36 that cyclically changes its volume and entropy ( by exchanging heat ) . in that case the differential of the thermodynamic potential reads @xmath37 with @xmath38 a state function denoting the internal energy . since @xmath39 we then find @xmath40 . if the exchange of heat takes place between two heat baths at given high and low temperatures @xmath41 and @xmath42 , it is well known that the most - efficient cycle the cycle that produces the maximum work per adsorbed amount of heat from the hotter bath is the carnot cycle with its two isothermal and two adiabatic ( de-)compressions @xcite . if we transpose all the variables from the gas performing mechanical work to the immersed electrodes performing electric work , we find @xmath43 , @xmath44 , @xmath45 , @xmath46 , and @xmath47 , where all pairs preserve the symmetry of being both extensive or both intensive . the analogue of high and low temperatures are thus high and low ionic chemical potentials @xmath48 and @xmath49 ( corresponding to sea and river water , respectively ) , the analogue of the isothermal volume change is thus the ( dis)charging at constant @xmath50 , and the analogue of an adiabatic volume change is ( dis)charging at constant @xmath20 . therefore , the analogue of the most efficient gas cycle is the electric cycle consisting of ( grand)canonical ( dis)charging processes . indeed , the trajectories ( ab ) and ( cd ) of the experimental cycle of ref.@xcite , as discussed in section i , are of a grand - canonical nature with the electrode in contact with a salt reservoir during the ( dis)charging . however , the processes ( bc ) and ( da ) take place at constant @xmath7 , i.e. they are equivalent to isochores in a gas cycle , instead of adiabats . efficiency is thus to be gained , at least in principle , by changing bc and da into canonical charging processes . whether this is experimentally easily implementable is , at this stage for us , an open question that we will not answer here . for the most efficient cycles , which are schematically shown in fig.[fig : carnotcompare ] in the @xmath51 and the @xmath52 representation , we can easily calculate the work performed during a cycle . for the mechanical work of the gas one finds @xmath53 , with @xmath54 the temperature difference and @xmath55 the entropy that is exchanged between the heat baths during the isothermal compression and decompression . the analogue for the work @xmath32 delivered by the electrode is given by @xmath56 , with @xmath57 and @xmath58 the number of exchanged ions between the reservoirs during the grand - canonical ( dis)charging processes . this result also follows directly from eq.([w ] ) . below we will calculate @xmath58 and hence @xmath32 from a microscopic theory moreover , we will also consider several other types of cycles . in the context of the thermodynamics that we discuss here , it is also of interest to analyse the `` global '' energy flow that gives rise to the work @xmath32 that the immersed porous electrodes deliver per ( reversible ) cycle . for this analysis it is crucial to realise that the device and the two salt reservoirs at chemical potentials @xmath48 and @xmath49 are considered to be at constant temperature @xmath10 throughout , which implies that they are thermally coupled to a heat bath ( that we call the `` atmosphere '' here for convenience ) at temperature @xmath10 . we will show that with every completed cycle , during which @xmath59 ions are being transported from the sea to the river water , a net amount of heat @xmath60 flows from the atmosphere to the two salt reservoirs , and that @xmath61 in the limit that the ion clouds do not store potential energy due to multi - particle interactions . this may at first sight contradict kelvin s statement of the second law ( `` no process is possible whose sole result is the complete conversion of heat into work '' @xcite ) , but one should realise that the cycle _ also _ involves the transport of ions from the sea to the river ; the word `` sole '' in kelvin s statement is thus crucial , of course . the analysis is based on the entropy changes @xmath62 , @xmath63 and @xmath64 of the device , the highly - concentrated salt reservoir and the one with low salt concentration , respectively , upon the completion of a cycle . given that the device returns to its initial state after a complete cycle , its entropy change vanishes and @xmath65 . this implies that the device , at its fixed temperature , does not adsorb or desorb any net amount of heat . during a cycle the `` river '' gains @xmath58 ions , and hence its ( helmholtz or gibbs ) free energy changes by @xmath66 , while the `` sea '' loses @xmath58 ions such that @xmath67 . now the basic identity @xmath68 implies that @xmath69 and @xmath70 , where @xmath71 is the average energy ( or enthalpy if @xmath29 denotes the gibbs free energy ) per particle . we assume @xmath72 to be independent of density , which physically corresponds to the case that there are no multi - particle contributions to the internal energy of the reservoirs , as is the case for hard - core systems or ions treated within poisson - boltzmann theory as ideal gases in a self - consistent field . the total energy in the reservoirs therefore remains constant during mixing , such that the entropy changes of the salt reservoirs are @xmath73 and @xmath74 . as a consequence of the global preservation of entropy in the reversible cycle , the ion exchange actually drives a heat exchange whereby the sea extracts a net amount of heat @xmath75 from the atmosphere , while the river dumps a net amount of heat @xmath76 into the atmosphere . of course the transport of ions itself is also accompanied with a heat exchange in between the reservoirs , the only relevant flow is therefore the net flow of heat out of the atmosphere , which is @xmath77 . the energy flow and the particle flow of the device and reservoirs are tentatively illustrated in fig . [ fig : flows ] , where one should realise that the distribution of the heat flow from the atmosphere into the sea ( @xmath78 ) and the river ( @xmath79 ) depends on the heat - flow from river to sea or _ vice versa _ , which we have not considered here in any detail ; _ only _ the net heat flow @xmath80 is fixed by global thermodynamic arguments . this identification of @xmath81 with @xmath32 would have the interesting implication that the conversion of this work into heat again , e.g. by using it to power a laptop , would _ not _ contribute to ( direct ) global warming since the released heat has previously been taken out of the atmosphere@xcite . it is not clear to us , however , to what extent this scenario is truly realistic and relevant , given that rivers , seas , and the atmosphere are generally _ not _ in thermal equilibrium such that other heat flows are to be considered . in this study we do not consider the heat fluxes at all , and just consider systems that are small enough for the temperature to be fixed . in order to calculate @xmath33 and @xmath34 of a charged electrode immersed in an electrolyte of volume @xmath21 , we need a microscopic model of the electrode and the electrolyte . we consider a positively charged porous electrode with a total pore volume @xmath82 , total surface area @xmath83 , and typical pore size @xmath84 . we write the total charge of the positive electrode as @xmath85 with @xmath86 the number of elementary charges per unit area . the negative electrode is the mirror image with an overall minus sign for charge and potential , see also fig.[fig : electrodes ] . the volume of the electrolyte surrounding this electrode is @xmath87 , with @xmath88 the volume of the electrolyte outside the electrode . the electrolyte consists of ( i ) water , viewed as a dielectric fluid with dielectric constant @xmath72 at temperature @xmath10 , ( ii ) an ( average ) number @xmath89 of anions with a charge @xmath90 and ( iii ) an ( average ) number @xmath91 of cations with a charge @xmath92 . the finite pore size @xmath84 inside the electrodes is taken into account here only qualitatively by regarding a geometry of two laterally unbounded parallel half - spaces representing the solid electrode , both with surface charge density @xmath93 , separated by a gap of thickness @xmath84 filled with the dielectric solvent and an inhomogeneous electrolyte characterised by concentration profile @xmath94 . here @xmath95 is the cartesian coordinate such that the charged planes are at @xmath96 and @xmath97 . the water density profile @xmath98 is then , within a simple incompressibility approximation @xmath99 with @xmath100 a molecular volume that is equal for water and the ions , given by @xmath101 . if the electrolyte in the gap is in diffusive contact with a bulk electrolyte with chemical potentials @xmath102 and @xmath103 of the cations and anions , we can write the variational grand - potential as a functional @xmath104 $ ] given by @xmath105}{ak_bt}&=&\int_0^{l/2 } \mathrm{d}z \left[\rho_+(z)\big(-1+\ln \rho_+(z ) \lambda_+^3 - \frac{\mu_+}{k_bt}\big)\right.\nonumber\\ & & \,\,\,\,\,+\rho_-(z)\big(-1+\ln \rho_-(z)\lambda_-^3 - \frac{\mu_-}{k_bt}\big)\nonumber\\ & & \,\,\,\,\ , + \rho_w(z)\big(-1+\ln\rho_w(z)v\big ) \nonumber\\ & & \,\,\,\,\,+\left.\frac{\phi(z)q(z)}{2}\right].\label{func}\end{aligned}\ ] ] here the first two lines denote the ideal - gas grand potential of the two ionic species , with @xmath106 the ionic thermal wavelengths . the third line is the ideal water - entropy , which effectively accounts for ionic excluded volume interactions as it restricts the total local ion concentration to a maximum equal to @xmath107 of course we could have taken the much more accurate hard - sphere functionals closer to bob evans heart to account for steric repulsions @xcite , but for now we are satisfied with the more qualitative lattice - gas - like description of packing @xcite . the last line of eq.([func ] ) denotes the mean - field approximation of the electrostatic energy in terms of the total charge number density @xmath108 and the electrostatic potential @xmath109 . note that @xmath110 is a functional of @xmath94 through the poisson equation @xmath111 with @xmath112 the bjerrum length of water , and that @xmath113 is the electrode potential . a prime denotes a derivative with respect to @xmath95 . the euler - lagrange equations @xmath114 that describe the equilibrium concentration profiles yield @xmath115 , with @xmath116 the bulk reservoir salt concentration and @xmath117 the ionic packing fraction in the reservoir , where @xmath118 . when the euler - lagrange equations are combined with the poisson equation , the modified poisson - boltzmann ( pb ) equation with boundary conditions ( bcs ) @xmath119 with @xmath120 the debye screening length , is found . bc ( [ bc1 ] ) follows from gauss law on the surface of the electrode , and bc ( [ bc2 ] ) from charge - neutrality and the symmetry with respect to the midplane of the gap . note that this equation with accompanying bc s was already studied in ref.@xcite . eq.([modpb ] ) reduces to the standard pb equation if @xmath121 , and the large - gap case @xmath122 was studied in ref.@xcite . ( [ modpb ] ) with its bc s ( [ bc1 ] ) and ( [ bc2 ] ) forms a closed set , and once its solution is found , numerically in general or analytically in the special case that @xmath123 and @xmath124 , the required equation of state of the electrode potential follows from @xmath125 . moreover , the equilibrium density profiles can be used to calculate the cationic and anionic adsorption , i.e. the excess number of ions per unit surface area , defined by @xmath126 note that we integrate the profile up to @xmath127 as required , and that our `` local charge neutrality '' assumption implies that @xmath128 . interestingly , the total surface excess of ions , defined by @xmath129 is related to the total number of ions in the volume @xmath130 by @xmath131 below we will use expression ( [ gampm ] ) and ( [ gam ] ) for @xmath132 to calculate @xmath133 from eq.([nrions ] ) , _ or _ to calculate @xmath134 by solving eq.([nrions ] ) for @xmath116 at given @xmath20 and @xmath86 . before discussing our numerical results , it is useful to consider the limiting case @xmath135 and @xmath136 , which is in fact the classic gouy - chapman ( gc ) problem of a single , planar , charged wall in contact with a bulk electrolyte of point ions . in this case the pb equation can be solved analytically @xcite , and the resulting total adsorption is given by @xmath137 with the crossover surface charge @xmath138 . the crossover behavior from @xmath139 at low @xmath86 to @xmath140 at high @xmath86 signifies a qualitative change from the linear screening regime , where the double layers exchange co- for counter ions keeping the total ion concentration fixed ( such that @xmath141 is small ) , to the nonlinear screening regime where counterion condensation takes place . for the alleged most - efficient ( dis)charging cycle of current interest , operating between two ionic reservoirs with a high salt concentration @xmath142 and a low one @xmath143 such that @xmath144 , and for which we argued already that the electric work per cycle reads @xmath145 , the gc result ( [ gc ] ) allows for the calculation of the ionic uptake @xmath146 during the grand - canonical charging from a low charge density @xmath147 to a high one @xmath148 . in the limit of highly charged surfaces we thus find @xmath149 , and hence the optimal work per unit area within the gc limit reads @xmath150 with the typical numbers @xmath151 of the order of nm@xmath152 , @xmath153 m@xmath1 per gram of porous carbon , and @xmath154 one arrives at @xmath155 j per gram of carbon . interestingly , this is substantially higher than brogioli s experimental findings of only @xmath156 mj / gram per cycle . we will discuss this difference below . the gc limit also yields an analytic expression for the surface potential @xmath157 given by @xmath158 with typical debye lengths @xmath159 nm and @xmath160 nm we find for the typical crossover surface charge density @xmath161 nm@xmath152 which corresponds to a surface potential @xmath162 mv . for completeness we also mention the differential capacitance @xmath163 of an immersed electrode , which within the gc limit was already written by chapman @xcite as @xmath164,\ ] ] which corresponds to a two - plate capacitor with spacing @xmath165 in a dielectric medium characterised by its relative dielectric constant @xmath72 . this result shows that the capacity indeed increases with the salt concentration , in agreement with observations that the electrode potential rises at fixed charge upon sweetening the surrounding water @xcite . useful insights can be obtained from these analytic gc expressions . moreover , practical linear approximations @xcite and even ( almost ) analytical solutions @xcite for the pb equation exist in the case of small pore size @xmath84 . nevertheless , the pointlike nature of the ions gives rise to surface concentrations of counterions that easily become unphysically large , e.g. far beyond 10 m for the parameters of interest here . for this reason we consider the steric effects through the finite ionic volume @xmath100 and the finite pore size @xmath84 below , at the expense of some numerical effort . the starting point of the explicit calculations is the numerical solution of eq . ( [ modpb ] ) with bc s ( [ bc1 ] ) and ( [ bc2 ] ) on a discrete @xmath95-grid of @xmath166 equidistant points on the interval @xmath167 $ ] , which we checked to be sufficient for all values of @xmath168 that we considered . throughout the remainder of this text we set @xmath169 with @xmath170 nm , which restricts the total local ion concentration @xmath171 to a physically reasonable maximum of 10 m. the bjerrum length of water is set to @xmath160 nm . we first consider a positive electrode immersed in a huge ( @xmath172 ) ionic bath at a fixed salt concentration @xmath116 , such that the ions can be treated grand - canonically . in fig . [ fig : potchargerel ] we plot ( a ) the electrode potential @xmath26 and ( b ) the total ion adsorption @xmath141 , both as a function of the electrode charge number density @xmath86 , for three reservoir salt concentrations @xmath1731 , 10 , and 100 mm from top to bottom , where the full curves represent the full theory with pore size @xmath174 nm , the dashed curves the infinite pore limit @xmath135 , and the dotted curve the analytic gouy - chapman expressions ( for @xmath135 and @xmath121 ) of eqs.([gc ] ) and ( [ psigc ] ) . the first observation in fig . [ fig : potchargerel](a ) is that gc theory breaks down at surface charge densities beyond @xmath175 nm@xmath152 , where steric effects prevent too dense a packing of condensing counterions such that the actual surface potential rises much more strongly with @xmath86 than the logarithmic increase of gc theory ( see eq.([psigc ] ) ) . this rise of the potential towards @xmath176 v may induce ( unwanted ) electrolysis in experiments , so charge densities exceeding , say , 5@xmath15 nm@xmath152 should perhaps be avoided . a second observation is that the finite pore size @xmath84 hardly affects the @xmath177 relation for @xmath178 nm@xmath152 , provided the steric effects are taken into account . the reason is that the effective screening length is substantially smaller than @xmath84 in these cases due to the large adsorption of counterions in the vicinity of the electrode . a third observation is that the full theory predicts , for the lower salt concentrations @xmath179 and 10 mm , a substantially larger @xmath26 at low @xmath86 , the more so for lower @xmath116 . this is due to the finite pores size , which is _ not _ much larger than @xmath180 in these cases , such that the ionic double layers must be distorted : by increasing @xmath26 a donnan - like potential is generated in the pore that attracts enough counterions to compensate for the electrode charge in the small available volume . interestingly , steric effects do _ not _ play a large role for @xmath181 in fig . [ fig : potchargerel](b ) , as the full curves of the full theory with @xmath169 are indistinguishable from the full theory with @xmath121 . the finite pore size appears to be more important for @xmath181 , at least at first sight , at low @xmath86 , where @xmath182 appears substantially lower than @xmath141 from the full calculation in the finite pore . however , this is in the linear regime where the adsorption is so small that only the logarithmic scale reveals any difference ; in the nonlinear regime at high @xmath86 all curves for @xmath141 coincide and hence the gc theory is accurate to describe the adsorption . we now consider the ( reversible ) @xmath26-@xmath86 cycle abcda shown in fig.[fig : cyclus](a ) , for an electrode with pore sizes @xmath183 nm that operates between two salt reservoirs at high and low salt salt concentrations @xmath184 m ( sea water ) and @xmath185 m ( river water ) , respectively , such that @xmath186 . for simplicity we set @xmath187 such that the total electrolyte volume equals the pore volume @xmath188 . the trajectory ab represents the charging of the electrode from an initial charge density @xmath189 nm@xmath152 to a final charge density @xmath190 nm@xmath152 at @xmath142 , which involves an increase in the number of ions per unit area @xmath191 nm@xmath152 using eqs.([gam ] ) and ( [ nrions ] ) which we calculate numerically with ( [ gampm ] ) . the trajectory bc is calculated using the fixed number of particles in state @xmath192 , @xmath193 , calculating a lower and lower value for @xmath116 for increasing @xmath86 s using eq.([nrions ] ) until @xmath143 at @xmath194 nm@xmath152 . then the discharging curve cd , at fixed @xmath143 is traced from surface charges @xmath195 down to @xmath196 nm@xmath152 for which @xmath197 , i.e. the discharging continues until the number of expelled ions equals their uptake during the charging process ab . the final trajectory , da , is characterised by the fixed number of particles in state d ( which equals that in a ) , and is calculated by numerically finding higher and higher @xmath116-values from eq.([nrions ] ) for surface charges @xmath86 decreasing from @xmath198 to @xmath147 , where @xmath142 at @xmath199 such that the loop is closed . note that all four trajectories involve numerical solutions of the modified poisson - boltzmann problem and some root - finding to find the state points of interest , and that the loop is completely characterised by @xmath200 , @xmath201 , @xmath147 , and @xmath148 . fig.[fig : cyclus](b ) shows the concentration profiles of the anions ( full curve ) and cations ( dashed curves ) in the states a , b , c , and d , ( i ) showing an almost undisturbed double layer in a and b that reaches local charge neutrality and a reservoir concentration @xmath202 in the center of the pore , ( ii ) an increase of counterions at the expense of a decrease of coions in going from b to c by a trade off with the negative electrode , accompanied by the saturation of counterion concentration at 10 m close to the electrode in state c and the ( almost ) complete absence of co - ions in the low - salt states c and d , and ( iii ) the trading of counterions for coions from d to a at fixed overall ion concentration . the work done during the cycle abcda follows from either the third or the fourth term of eq.([w ] ) , yielding @xmath203 nm@xmath152 or , equivalently , @xmath204 for the present set of parameters . the enclosed area of the cycle abcda in fig . [ fig : cyclus ] corresponds to the amount of extracted work ( up to a factor @xmath205 ) , and equals the net decrease of free energy of the reservoirs . in order to compare the presently proposed type of cycle abcda with the type used in the experiments of brogioli @xcite , where `` isochores '' at constant @xmath86 rather than `` adiabats '' at constant @xmath20 were used to transit between the two salt baths , we also numerically study the dashed cycle abcda of fig.[fig : cyclus](a ) . this cycle has exactly the same trajectory ab characterised by @xmath206 as before . state point c at @xmath201 and @xmath207 has , however , a much smaller number of ions @xmath208 than @xmath209 in state b and c , because its surface charge @xmath210 . trajectory cd at fixed @xmath201 is quite similar to cd but extends much further down to @xmath211 , where the number of ions in d is even further reduced to the minimum value in the cycle @xmath212 nm@xmath152 . finally , at fixed @xmath213 the number of ions increases up to @xmath214 by gradually increasing @xmath116 from @xmath201 to @xmath200 . so also this cycle is completely determined by @xmath200 , @xmath201 , @xmath147 , and @xmath148 . the electric work @xmath215 done during the cycle abcda follows from eq.(2 ) and reads @xmath216 nm@xmath152 , which is equivalent to @xmath217 where @xmath218 is the number of ions that was exchanged between the two reservoirs during the cycle . clearly , @xmath219 , i.e. the brogioli - type cycle with the `` isocharges '' bc and da produces more work than the presently proposed abcda cycle with canonical trajectories bc and da . however , the efficiency of abcda , defined as @xmath220 , indeed exceeds the efficiency @xmath221 of the abcda cycle . this is also illustrated in fig.[fig : cyclus2 ] , where the two cycles abcda ( a ) and abcda ( b ) are shown in the @xmath20-@xmath50 representation . whereas the total area of ( b ) is larger than that of ( a ) , so @xmath219 according to eq.([w ] ) , the larger spread in @xmath222 compared to @xmath58 renders the efficiency of ( b ) smaller . the work @xmath215 is therefore less than the decrease of the free energy of the reservoirs combined . the hatched area of fig.[fig : cyclus2](b ) denotes the work that could have been done with the number @xmath222 of exchanged ions , if a cycle of the type abcda had been used . the fact that @xmath219 while @xmath223 proves to be the case for all charge densities @xmath147 and @xmath148 for which we calculated @xmath32 ( of an abcda - type cycle ) and @xmath215 ( of an abcda - type cycle ) , at the same reservoirs @xmath200 and @xmath201 and the same pore size @xmath84 as above . this is illustrated in table 1 , which lists @xmath32 and @xmath215 per unit area and per transported ion for several choices of @xmath147 and @xmath148 . the data of table 1 shows that @xmath219 by up to a factor 2 , while @xmath223 by up to a factor of three for @xmath187 , and a factor 8 for @xmath224 . we thus conclude that the choice for a particular cycle to generate electric work depends on optimization considerations ; our results show that maximum work or maximum efficiency do not necessarily coincide . table 1 not only shows the work per area and per ion , but in the last column also @xmath225 with @xmath226 , _ i.e. _ the work per charge that is put on the electrode during the charging of trajectory ab . interestingly , in these units the work is comparable to @xmath227 provided @xmath228 , as also follows from gouy - chapman theory for highly charged surfaces . note that the work per transported charge does _ not _ equal the amount of performed work per transported ion as @xmath229 is typically much larger than @xmath230 . nevertheless , the fact that @xmath231 gives us a handle to link our results with the experiments of brogioli @xcite . during the experiment , the charge on the electrodes varies by @xmath232 @xmath233c , such that one arrives at an expected work of 6 @xmath50j per electrode . this agrees reasonably well with the obtained value of 5 @xmath234 out of the entire system . unfortunately , the relation between the electrostatic potential and the charge in the experiments differs significantly from that of our theory by at least hundreds of millivolts ; at comparable electrostatic potentials the charge density in brogioli s experiments is almost two orders of magnitude smaller than our theoretical estimates . therefore a qualitative comparison with the brogioli - cycle is at this point very hard . the relatively low experimental charge densities clarify the lower amount of work produced per gram of electrode , which was noted earlier in the text . including the stern layer may be a key ingredient that is missing in the present analysis @xcite . .the work @xmath32 and @xmath215 of cycles abcda and abcda , respectively , as illustrated in figs.[fig : cyclus](a ) and [ fig : cyclus2 ] , for several choices of surface charges @xmath147 and @xmath148 in states a and b , for systems operating between electrolytes with high hand low salt concentrations @xmath235 m and @xmath236 m , for electrodes with pore size @xmath183 nm . we converted @xmath32 and @xmath215 to room temperature thermal energy units @xmath237 , and not only express them per unit electrode area @xmath83 but also per exchanged number of ions @xmath58 and @xmath222 during the two cycles , respectively . note that @xmath220 is a property of the two reservoirs , not of the charge densities of the cycle . also note that @xmath221 depends on the volume @xmath88 of electrolyte outside the electrodes , here we successively give values for the optimal situation @xmath187 as well as for the situation @xmath238 . [ cols="^,^,^,^,^,^,^",options="header " , ] of course many more cycles are possible . the two cycles abcda and abcda considered so far generate electric work out of the mixing of two very large reservoirs of salt and fresh water , taking up ions from high - salt water and releasing them in fresh water . due to the large volume of the two reservoirs the ionic chemical potentials @xmath48 and @xmath49 , and hence the bulk salt concentrations @xmath200 and @xmath201 in the reservoirs , do not change during this transfer of a finite number of ions during a cycle . however , there could be relevant cases where the power output of an osmo - electric device is limited by the finite inflow of fresh water , which then becomes brackish due to the mixing process ; usually there is enough sea water to ignore the opposite effect that the sea would become less salty because of ion drainage by a cycle . in other words , the volume of fresh water can not always be regarded as infinitely large while the salt water reservoir is still a genuine and infinitely large ion bath . the cycle with a limited fresh water supply is equivalent to a heat - engine that causes the temperature of its cold `` bath '' to rise due to the release of rest heat from a cycle , while the hot heat bath does not cool down due to its large volume or heat capacity . here we describe and quantify a cycle abca that produces electric work by reversibly mixing a finite volume of fresh water with a reservoir of salt water . we consider a finite volume @xmath239 of fresh water with a low salt concentration @xmath236 m , such that the number of ions in this compartment equals @xmath240 . this fresh water is assumed to be available at the beginning of a ( new ) cycle ; its fate at the end of the cycle is to be as salty as the sea by having taken up @xmath241 ions from the electrode ( which received them from the sea ) , with @xmath235 m the salt concentration in the sea . the cycle , which is represented in fig . [ fig : newcycle ] , starts with the electrodes connected to a large volume of sea water at concentration @xmath200 , charged up in state a at a charge density @xmath242 nm@xmath152 . during the first part ab of the cycle , the electrodes are further charged up until the positive one has taken up @xmath241 ions in its pores , which fixes the surface charge @xmath243 nm@xmath152 in state b. then the electrodes are to be disconnected from the sea , after which the charging proceeds in trajectory bc such that the increasing ion adsorption at a fixed total ion number reduces the salt chemical potential down to @xmath49 ( and hence the salt concentration far from the electrode surface down to @xmath201 ) at @xmath244 nm@xmath152 in state c. the system can then be reversibly coupled to the finite compartment of initially fresh water , after which the discharging process ca takes place such that the released ions cause the fresh water to become more salty , reaching a charge density @xmath147 when the salt concentration in the compartment of volume @xmath245 equals @xmath200 . the cycle can then be repeated by replacing the compartment @xmath245 by fresh water again . the relation between the surface potential @xmath26 , the charge density on the electrodes @xmath86 , and the ion reservoir concentration @xmath116 or the ion number @xmath20 , was numerically calculated using the modified pb - equation ( [ modpb ] ) with bc s ( [ bc1 ] ) and ( [ bc2 ] ) , combined with the adsorption relation ( [ gam ] ) , with the same parameters @xmath169 , @xmath170 nm , @xmath183 nm , and @xmath246 as before . the enclosed area in fig . [ fig : newcycle ] gives , using eq.([w ] ) , the net amount of ( reversible ) work @xmath32 performed during a cycle , which again equals the decrease of the free energy of the salt water reservoir and fresh water volume combined . in fact , this work can be calculated analytically as @xmath247 , \label{newwork}\ ] ] where @xmath248 . this result agrees with the prediction by pattle @xcite for very small @xmath201 . for the parameters of the cycle discussed here , we find @xmath249 kj per liter of fresh water , or @xmath250 nm@xmath152 . the figures show that the amount of work per ion that is transported is typically smaller than what we found for the carnot - like cycle , of course . + we may compare this reversible cycle with the one proposed by biesheuvel for the reverse process , which is called desalination . this cycle is very similar to ours , except that biesheuvel s switching step from sea - to river water and v.v . is actually an iso-@xmath141 trajectory instead of our iso-@xmath20 tracjectory . this iso - adsorption trajectory does not seem to have a reversible heat engine analogue , as the degree of reversibility depends on the extent to which the electrolyte can be drained out of the micropores before . nevertheless , we find agreement with the work that must be provided in the case of only a relatively small output volume of fresh water , and the expression found by biesheuvel exactly equals eq . ( [ newwork ] ) . the point we would like to stress is that irreversible mixing during the switching step can be prevented by introducing a canonical(iso-@xmath20 ) part into the cycle which enables the system to adapt to a new salt concentration in a time - reversible fashion , such that maximal efficiency is preserved . although substantial attempts to extract renewable energy from salinity gradients go back to the 1970 s , there is considerable recent progress in this field stemming from the availability of high - quality membranes @xcite and large - area nanoporous electrodes @xcite with which economically interesting yields of the order of 1 kj per liter of fresh water can be obtained equivalent to a waterfall of one hundred meter . the key concept in the recent experiments of brogioli @xcite is to cyclically ( dis)charge a supercapacitor composed of two porous electrodes immersed in sea ( river ) water . in this article we have used a relatively simple density functional , based on mean - field electrostatics and a lattice - gas type description of ionic steric repulsions , to study the relation between the electrode potential @xmath26 , the electrode surface charge density @xmath86 , the ion adsorption @xmath141 , the ion chemical potential @xmath50 , and the total number of ions @xmath20 in a ( slit - like ) pore of width @xmath84 that should mimic the finite pores of the electrodes . with this microscopic information at hand , we have analysed several cycles of charging and discharging electrodes in sea and river water . by making an analogy with heat engines , for which the most - efficient cycle between two heat baths at fixed temperatures is the carnot cycle with isothermal and adiabatic ( de)compressions , we considered cycles composed of iso@xmath251 and iso@xmath252 ( dis)charging processes of the electrodes . we indeed found that these cycles are maximally efficient in the sense that the work per ` consumed ' ion that is transported from the sea to the river water during this cycle is optimal , given the salt concentrations in the river- and sea water . however , although the cycles used by brogioli , with two iso@xmath251 and two iso@xmath253 trajectories ( where the latter are analogous to isochores in the heat - engine ) are less efficient per transported ion , the total work of a `` brogioli - cycle '' is larger , at least when comparing cycles that share the iso@xmath251 charging in the sea water trajectory . we find , for electrode potentials @xmath254 mv and electrode charge densities @xmath255 nm@xmath256 in electrolytes with salt concentrations @xmath235 m ( sea water ) and @xmath236 m ( river water ) , typical amounts of delivered work of the order of several @xmath237 per transported ion , which is equivalent to several @xmath237 per nm@xmath1 of electrode area or several kj per liter of consumed fresh water . + our calculations on the brogioli type of cycle agree with experiments regarding the amount of performed work per cycle with respect to the variance in the electrode charge during ( dis- ) charging ; each unit charge is responsible for an amount of work that is given by the difference in chemical potential between the two reservoirs . however , the experimental data concerning the electrostatic potential could _ not _ be mapped onto our numerical data . this could very well be due to the fact that the pore size in the experiments by brogioli is very small such that ion desolvation , ion polarisability , and image charge effects may be determining the relation between the surface charge and electrostatic potential . models which go beyond the present mean - field description are probably required for a quantitative description of this regime . another ingredient in a more detailed description must involve the finite size of the ions combined with the microscopic roughness of the carbon . the ions in the solvent and the electrons ( holes ) in the electrode material can not approach infinitely close , and the resulting charge free zone can be modeled by a stern capacitance . standard gouy - chapman - stern ( gcs ) theory has successfully been applied to fit charge - voltage curves for porous carbon capacitive cells @xcite within the context of osmo - electrical and capacitive desalination devices . extensions to gcs theory are currently being developed which include finite pore sizes , in order to obtain a physically realistic and simultaneously accurate model of the stern layer within this geometry . + throughout this work we ( implicitly ) assumed the cycles to be reversible , which implies that the electrode ( dis)charging is carried out sufficiently slowly for the ions to be in thermodynamic equilibrium with the instantaneous external potential imposed by the electrodes . this reversibility due to the slowness of the charging process has the advantage of giving rise to optimal conversion from ionic entropy to electric work in a given cycle . however , if one is interested in optimizing the _ power _ of a cycle , _ i.e. _ the performed work per unit time , then quasistatic processes are certainly not optimal because of their inherent slowness . heuristically one expects that the optimal power would result from the trade - off between reversibility ( slowness ) to optimize the work per cycle on the one hand , and fast electronic ( dis)charging processes of the electrodes and fast fluid exchanges on time scales below the relaxation time of the ionic double layers on the other . an interesting issue is the diffusion of ions into ( or out of ) the porous electrode after switching on ( or off ) the electrode potential @xcite . ongoing work in our group employs dynamic density functional theory @xcite to find optimal - power conditions for the devices and cycles studied in this paper , e.g. focussing on the delay times between the electrode potential and the ionic charge cloud upon voltage ramps . + the recovery of useful energy from the otherwise definite entropy increase at estuaries , which may be relevant because our planet is so full of water , is just one example where one can directly build on bob evans fundamental work on ( dynamic ) density functional theory , inhomogeneous liquids , electrolytes , interfaces , and adsorption . it is a great pleasure to dedicate this paper to bob evans on the occasion of his 65th birthday . rvr had the privilege of being a postdoctoral fellow in bob s group in the years 1997 - 1999 in bristol , where he experienced an unsurpassed combination of warm hospitality , unlimited scientific freedom , and superb guidance on _ any _ aspect of scientific and ( british ) daily life . bob s words `` ren , you look like a man who needs a beer ! '' when entering the post - doc office in the late afternoon , which usually meant that he was ready to discuss physics after his long day of lecturing and administration , trigger memories of evening - long pub - discussions and actual pencil - and - paper calculations on hard - sphere demixing , like - charge attraction , liquid - crystal wetting , poles in the complex plane , or hydrophobic interactions , with ( long ) intermezzos of analyses of , say , bergkamp s qualities versus those of beckham . even though not all of this ended up in publications , bob s input , explanations , historic perspective , and style contained invaluable career - determining elements for a young postdoc working with him . rvr is very grateful for all this and more . we wish bob , and of course also margaret , many happy and healthy years to come . + we thank marleen kooiman and maarten biesheuvel for useful discussions . this work was financially supported by an nwo - echo grant .

a huge amount of entropy is produced at places where fresh water and seawater mix , for example at river mouths . this mixing process is a potentially enormous source of sustainable energy , provided it is harnessed properly , for instance by a cyclic charging and discharging process of porous electrodes immersed in salt and fresh water , respectively [ d. brogioli , phys . . lett . * 103 * , 058501 ( 2009 ) ] . here we employ a modified poisson - boltzmann free - energy density functional to calculate the ionic adsorption and desorption onto and from the charged electrodes , from which the electric work of a cycle is deduced . we propose optimal ( most efficient ) cycles for two given salt baths involving two canonical and two grand - canonical ( dis)charging paths , in analogy to the well - known carnot cycle for heat - to - work conversion from two heat baths involving two isothermal and two adiabatic paths . we also suggest a slightly modified cycle which can be applied in cases that the stream of fresh water is limited . + bob evans about water ( 1998 ) . + + bob evans about ionic criticality ( 1998 ) .

1012.4946

the problem of boundary lubrication is very interesting from the physical point of view and important for practical applications , but it is not fully understood yet @xcite . conventional lubricants belong to the type of liquid ( `` soft '' ) lubricants , where the amplitude of molecular interactions within the lubricant , @xmath0 , is smaller than the lubricant - substrate interaction , @xmath1 . due to strong coupling with the substrates , lubricant monolayers cover the surfaces , and protect them from wear . a thin lubricant film , when its thickness is lower than about six molecular layers , typically solidifies even if the conditions ( temperature and pressure ) are those corresponding to the bulk liquid state . as a result , the static friction force is nonzero , @xmath2 , and the system exhibits stick - slip motion , when the top substrate is driven through an attached spring ( which also may model the slider elasticity ) . in detail , at the beginning of motion the spring elongates , the driving force increases till it reaches the static threshold @xmath3 . then a fast sliding event takes place , the spring relaxes , the surfaces stick again , and the whole process repeats itself . this stick - slip regime occurs at low driving velocities , while at high velocities it turns into smooth sliding . since the pioneering work by thompson and robbins @xcite , who studied the lubricated system by molecular dynamics ( md ) , the stick - slip is associated with the melting - freezing mechanism : the lubricant film melts during slip and solidifies again at stick . such a sliding may be named the `` liquid sliding '' ( ls ) regime . however , at low velocities the `` layer over layer sliding '' ( lols ) regime sometimes occurs , where the lubricant keeps well ordered layered structure , and the sliding occurs between these layers @xcite . in real systems the substrates are often made of the same material and may even slide along the same crystallographic face , but can hardly be assumed to be perfectly aligned , especially if the substrates have polycrystalline structure . in the majority of md simulations , however , both substrates are modelled identically , i.e. , they have the same structure and are perfectly aligned . this fact may affect strongly the simulation results , as became clear after predicting the so - called superlubricity , or structural lubricity @xcite . for example , the `` dry '' contact ( no lubricant ) of two incommensurate rigid infinite surfaces , produces null static friction , @xmath4 @xcite . if the surfaces are deformable , an analog of the aubry transition should occur with the change of stiffness of the substrates ( or the change of load @xcite ) : the surfaces are locked together for a weak stiffness , and slide freely over each other for sufficiently high stiffness ( this effect was observed in simulation @xcite ) . in a real - world 3d contact , incommensurability can occur even for two identical surfaces , if the 2d surfaces are rotated with respect to each other . simulations @xcite do show a large variation of friction with relative orientation of the two bare substrates . similarly to the 1d frenkel - kontorova system , where the amplitude of the peierls nabarro barrier is a nonanalytic function of the misfit parameter , in the 2d system the static frictional force should be a nonanalytic function of the misfit angle between the two substrates . this was pointed out by gyalog and thomas in their study of the 2d fk tomlinson model @xcite . however , surface irregularities as well as fluctuations of atomic positions at nonzero temperature makes this dependence smooth and less pronounced . for example , md simulations @xcite of the ni(100)/ni(100 ) interface at @xmath5 k showed that for the case of perfectly smooth surfaces , a @xmath6 rotation leads to a decrease in static friction by a factor of @xmath7 . however , if one of the surfaces is roughened with an amplitude 0.8 , this factor reduces to @xmath8 only , which is close to values observed experimentally . mser and robbins @xcite noted that for a contact of atomically smooth and chemically passivated surfaces , realistic values of the stiffness usually exceed the aubry threshold , thus one should expect @xmath4 for such a contact . an approximately null static frictional force was indeed observed experimentally in the contact of tungsten and silicon crystals @xcite . more recently the friction - force microscopy experiment made by dienwiebel _ et al . _ @xcite demonstrated a strong dependence of the friction force on the rotation angle for a tungsten tip with an attached graphene flake sliding over a graphite surface , where sliding occurs between the graphene layers as relative rotation makes them incommensurate . the case of lubricated friction was investigated by he and robbins @xcite for a very thin lubricant film ( one monolayer or less ) . the dependences of the static @xcite and kinetic @xcite friction on the rotation angle were calculated . the authors considered the rigid substrates of fcc crystal with the ( 111 ) surface and rotated the top substrate from @xmath9 to @xmath10 . it was found that static friction exhibits a peak at the commensurate angle ( @xmath9 ) and then is approximately constant ; the peak / plateau ratio is about 7 ( for the monolayer lubricant film , where the variation is the strongest ) . the kinetic friction varies slowly with a minimum at the commensurate angle and a smooth maximum at @xmath11 , changing by a factor near two . also , the kinetic friction decreases with velocity at @xmath9 , while it increases at the other angles . _ the goal of our work is a detailed md study of stick - slip and smooth sliding for lubricated system with rotated surfaces . _ compared to the work by he and robbins @xcite , we study thicker lubricant films , up to five atomic layers thick . we explore a fairly basic model ( see sec . [ model ] and ref . interactions are of simple lennard - jones type , typically each substrate consists of two layers with 12@xmath1211 atoms , arranged as a square lattice , and the lubricant has 80 atoms per layer ) which allows us a rather detailed study of the system dynamics for long simulation times . this model attempts to address the effects of relative crystal rotations in generic lubricated sliding , without focusing on a specific system . while the microscopic interactions are treated at a minimal level of sophistication , we describe energy dissipation by means of a `` realistic '' damping scheme , with a damping coefficient in langevin equation , which mimics the energy exchange between the lubricant atoms and the substrate . this is certainly important for smooth sliding , the kinetics of melting and freezing processes and the ensuing metastable configurations which emerge at stick during stick - slip regime . section [ simulation ] presents typical simulation results . the model exhibits stick - slip at a low driving speed , which changes to smooth sliding with increasing speed . in the smooth sliding regime , as well as during slips in the stick - slip regime , the system indeed exhibits either the ls or lols regime , depending on the simulation parameters , and in particular the rotation angle . a new important result of the present study is that the lols regime should be observed much more often than the ls regime . section [ discussion ] discusses and summarizes the obtained results . \(a ) ( b ) ( c ) as we address rather general properties of lubricated friction , we explore a relatively simple model in simulation . this allows us to span a wider range of sliding velocities and longer simulation times as well as to analyze the atomic trajectories in greater detail than by simulating fully realistic force fields appropriate to a specific system . the md model was introduced and described in previous work @xcite ; therefore , we only discuss here its main features briefly . we study a film composed by few atomic layers and confined between two substrates , `` bottom '' and `` top '' . as illustrated in fig . [ b06 ] , each substrate consists of two layers . the external one is rigid , while the dynamics of the atoms belonging to the inner layer , the one directly in contact with the lubricant , is fully included in the model . the rigid part of the bottom substrate is held fixed , while the top substrate is mobile in the three space directions @xmath13 . all atoms interact with pairwise lennard - jones potentials @xmath14 , \label{m0}\ ] ] where @xmath15 or @xmath16 for the substrate or lubricant atoms respectively , so that the interaction parameters @xmath17 and @xmath18 depend on the type of atoms . between two substrate atoms we use @xmath19 and an equilibrium distance @xmath20 . the interaction between the substrate and the lubricant is much weaker , @xmath21 . for the `` soft '' lubricant itself , we consider @xmath22 and an equilibrium spacing @xmath23 , which is poorly commensurate with @xmath24 . the equilibrium distance between the substrate and the lubricant is @xmath25 . for the long - range tail of all potentials we adopt a standard truncation to @xmath26 . the atomic masses are @xmath27 . the two substrates are pressed together by a loading force @xmath28 per substrate atom ( typically we used the value @xmath29 ) . all parameters are given in dimensionless units defined in ref . @xcite , for example the model units for force is @xmath30 . the chosen parameters correspond roughly to a typical system where energy is measured in electronvolts and distances in ngstrms , so that forces are in the nanonewton range . the main difference between our technique and other simulations of confined systems lies in the dissipative coupling with the heat bath , representing the bulk of the substrates . we use langevin dynamics with a position- and velocity - dependent damping coefficient @xmath31 , which is designed to mimic realistic dissipation , as discussed in refs . @xcite . in a driven system the energy pumped into the system must be removed from it . in reality energy losses occurs through the excitation of degrees of freedom not included in the calculation , namely energy transfer into the bulk of the substrates . to model this fact , damping should occur primarily when a moving atom comes at a small distance @xmath32 from either substrate . moreover , the efficiency of the energy transfer depends on the velocity @xmath33 of the atom because @xmath33 affects the frequencies of the motions that it excites within the substrates . the damping is written as @xmath34 with @xmath35 $ ] , where @xmath36 is a characteristic distance of the order of one lattice spacing . the expression of @xmath37 is deduced from the results known for the damping of vibrations of an atom adsorbed on a crystal surface ( see ref . @xcite and references therein ) . in simulations we explore the `` spring '' algorithm , where a spring is attached to the rigid top layer , and the spring end is driven at a constant velocity . we apply periodic boundary conditions ( pbc ) in the @xmath38 and @xmath39 directions . the geometric construction of the rotated substrate is explained in the appendix [ appendix ] . in simulation , it is simpler to rotate the bottom substrate only . the initial configuration of the lubricant is prepared as a set of @xmath40 ( @xmath41 , 2 , 3 , and 5 ) closely packed atomic layers . most simulations include @xmath42 atoms in each lubricant layer , although we increased the system size by up to 16 times to check for size effects . the system is then annealed , i.e. , the temperature is raised adiabatically to @xmath43 , which exceeds the melting temperature @xmath44 ( @xmath45 for our lubricant @xcite ) and then decreased back to the desired value . after preparation of the annealed configuration , we perform a standard protocol of runs : starting from the high - speed @xmath46 ls regime , corresponding to a sliding speed comparable to the sound velocity of the lubricant in its solid state , we reduce the driving velocity @xmath47 in steps down to the value @xmath48 ( which produces stick - slip motion for most employed simulation parameters ) , and then increase @xmath47 , up to @xmath49 . typical md snapshots are shown in fig . [ b06 ] . to estimate @xmath3 in the stick - slip regime we select the @xmath50 runs and take an average of the peak spring force immediately prior to slip . a lower sliding speed would lead to slightly larger friction , due to longer aging of pinning contacts , but would require longer simulation times to record the same number of stick - slip events . the moderate speed @xmath50 realizes a fair compromise , which in practical simulation times produces a @xmath51 underestimate of @xmath3 with respect to its value at adiabatically slow sliding . to find the kinetic friction force @xmath52 in the smooth sliding regime , we average the spring force over the whole run . the simulation results are summarized in fig . [ c17all ] . qualitatively , the results agree with those of he and robbins @xcite , with stick - slip motion at low driving velocities and smooth sliding for @xmath53 . at zero substrate temperature , the static friction can vary with the misfit angle @xmath54 by two orders of magnitude , and the kinetic friction for smooth sliding at low velocity ( e.g. as shown for @xmath55 in fig . [ c17all]b ) by more than one order of magnitude . for the thin lubricant films , @xmath56 , the static friction peaks for perfectly aligned substrates , @xmath57 . this does not occur any more for thicker films . the friction achieves sharp minima at the angles @xmath58 , @xmath59 and @xmath60 as will be discussed below . at large driving velocities , @xmath61 ( which is in fact huge , comparable to the solid - lubricant sound velocity ) the lubricant film is completely molten , and friction becomes almost independent of @xmath54 . the variation of friction with @xmath54 is the most pronounced for the one - layer lubricant film . the simulation results for this thinnest film are presented in fig . the static and low - speed ( @xmath62 ) kinetic friction force displays sharp minima for the angles @xmath58 , @xmath59 and @xmath60 . for these `` special '' angles , the lubricant film remains ordered and slides together either with the top or the bottom substrate both during slips and in smooth sliding ( we call this regime as the `` solid sliding '' , or ss ) . of course the motion is not rigid but corresponds to a `` solitonic '' sliding mechanism @xcite . for the other angles studied , at stick configurations the film orders locally , with a structure adjusted partly to the bottom and partly to the top substrates , while during slips , as well as at smooth sliding , the film is 2d - melted ( ls regime ) . the dependence of the kinetic friction force on temperature is shown in fig . [ fw1kt ] . for all `` non - special '' angles , when the static friction is relatively high , the kinetic friction decreases with temperature ( e.g. , see the dependence for @xmath63 in fig . [ fw1kt ] ) , reflecting the standard thermolubric effect @xcite due to temperature - assisted barrier overcoming . however , for all special angles producing the ss regime , the behavior is different : thermal fluctuations perturb a rather delicate solitonic motion , leading to an initial increase of friction with @xmath64 . such a behavior occurs also for thicker films . for the misfit angle @xmath65 we observe that at @xmath66 the one - layer film reaches an exceptional sliding state characterized by very low friction due to ss ; this type of superlubricity is not typical , and it is quickly destroyed with the increase of either temperature ( fig . [ fw1kt ] ) or velocity ( fig . [ c17c ] ) . note also that the superlubric ss regime is recovered significantly after it is abandoned as temperature is lowered ( dotted line , open symbols ) , thus opening a nontrivial hysteretic loop in the thermal cycle . consider now a thicker lubricant film with @xmath67 ( see fig . [ c17b ] ) . for @xmath68 we observe the ls regime , where the lubricant is 3d molten ( except the `` special '' angles @xmath69 and @xmath60 at @xmath46 , where we have lols between the two attached lubricant layers ) . at lower velocities , @xmath70 , the behavior is as follows . for @xmath57 at stick , the two ordered lubricant layers are ordered and attached to the corresponding substrates , but are 3d melted at slips . for all other angles , the lols regime operates during slips : for @xmath71 the attached layers are 2d molten , while for @xmath72 the attached layers remain ordered . finally , for the angles @xmath58 , @xmath59 and @xmath60 the friction forces exhibit deep minima produced by the ss solitonic mechanism . we come now to describe the results for the @xmath73 system as a prototypical lubricant of mesoscale thickness . the static friction , as well as the kinetic friction in the lols regime , can change by more than one order in magnitude when the misfit angle varies , as illustrated in fig . @xmath9 produces the highest friction like for thinner lubricants . the smooth sliding , as well as slips during stick - slip , correspond to either the lols or the ls regime . for all other angles @xmath74 , _ sliding always corresponds to the lols regime _ at low driving velocities @xmath70 . contrary to the @xmath9 case , now sliding is typically asymmetric and takes place at one interface only , between the middle layer and one of the attached layers , so that the middle lubricant layer sticks with either the top or the bottom substrate . the middle layer may remain ordered during sliding or , for some values of @xmath54 , it is 2d - melted ; in the latter case , the friction force is higher . for the special misfit angles @xmath58 , @xmath59 and @xmath60 identified by stars in fig . [ c15 ] , we again observe the `` superlubricity '' characterized by the very low friction . in these cases , the lubricant film remains solid and ordered during sliding , and moves as a whole with the top or bottom substrate , in a ss sliding . however , the lubricant is not rigid during motion , thus enhancing the `` solitonic '' mechanism . the results described above , remain qualitatively the same at nonzero temperatures @xmath75 . for example , fig . [ c15]b shows the dependence of friction force on @xmath54 for the `` room '' temperature @xmath76 . both the static and kinetic ( for the lols regime ) friction forces decrease when @xmath64 increases . however , for the ss regime , the behavior is different fluctuations due to temperature perturb the solitonic motion , leading to an increase of friction . the thicker lubricant film , @xmath77 , behaves similarly to @xmath73 , as illustrated in fig . the main difference is the lack of a maximum in @xmath78 at @xmath9 . again , for the misfit angles @xmath58 and @xmath59 we observe `` superlubric '' sliding . for smooth sliding with @xmath70 as well for slips during stick - slip motion , we observe either the lols regime , where the three central layers are 2d melted and sliding occurs between the middle layers ( e.g. , between layers 1 - 2 or 2 - 3 ) , or the ls regime , where all three middle layers are 3d - molten . the lols regime marks the dips in @xmath52 , while ls generates larger @xmath52 , as occurs in the angular intervals @xmath79 and @xmath80 . for larger velocities , @xmath68 , all five lubricant layers are melted and the ls regime operates in full . figure [ load1 ] reports the friction force for 4 values of the applied load . these calculations demonstrate that the dependency of friction on the substrate rotation is very similar for different loads , with a general increase of friction with load . we obtain very similar results for @xmath67 and 3 . however , for thicker films , @xmath81 , the situation changes dramatically at high load ( @xmath82 ) : the film rearranges into a closely packed four - layer configuration which remains solid under sliding . as a consequence , the peak structure of friction as a function of @xmath54 changes as well , because the lubricant structure acquires more atoms per layer and changes symmetry . because the static friction varies over such a large interval , its specific value for a given misfit angle @xmath54 has little importance . indeed one could hardly control the misfit angle in a real system , except in especially favorable situations @xcite . moreover , a system where the sliding surfaces have an ideal crystalline structure oriented with a controllable @xmath54 is exceptional . real surfaces usually have areas ( domains ) with different orientation . for polycrystalline substrates , it is reasonable to assume that all angles are presented with equal likeliness . it may then be more interesting to examine the probability distribution of @xmath3 values , regardless of @xmath54 . the insets of fig . [ static1235 ] report the histograms of forces as resulting from our simulations of different thicknesses . besides , if we also assume that the lubricant film is not uniform but has different thickness at different places ( it is certainly so if the surfaces have some roughness ) , then we can average over different thicknesses ; the resulting distribution @xmath83 is shown fig . [ static1235 ] . it is precisely this distribution which represents the main output of our md simulations , as it then allows us to predict tribological behavior of the system with the help of a master - equation approach @xcite . the calculation summarized in fig . [ static1235 ] are carried out at @xmath84 . the effect of a load increase would be mainly to scale the @xmath83 distribution , so that it would peak at larger friction . the resulting distribution displays several spikes which are likely due to the fixed size of the simulated contact . we will investigate the role of contact size on the @xmath83 distribution using a simplified model , in a separate publication @xcite . the main results of the present work can be summarized as follows : ( _ i _ ) the relative rotation of sliding contacts in a lubricated context promotes lols more frequently then the standard ls . ( _ ii _ ) for a few special angles lols leads to superlubric sliding completely analogous of the unlubricated sliding of misaligned perfect crystalline substrates . this superlubric regime is however delicate and can be suppressed by small relative rotations of the substrates , by temperature , or by velocity - induced local heating . ( _ iii _ ) in a regime of boundary lubrication , friction forces do vary quite significantly with the relative substrate relative orientation @xmath54 , even when the lubricant film becomes several atomic layers thick . ( _ iv _ ) to describe macroscopic friction in a context of multiasperity contact , where relative orientation is not really under control , the most important information to be extracted from md simulations is a probability distribution @xmath83 rather than specific values of the static friction force @xmath3 . the present calculation is consistent with a rapidly ( approximately exponentially ) decaying distribution @xmath83 , up to a cutoff force related to the average contact size . when temperature can promote rotations , beside the standard reduction of friction at low speed due to thermal crossing of barriers , thermal fluctuations could affect the barrier distribution itself by suppressing small barriers in favor of higher ones @xcite . we wish to express our gratitude to b.n.j . persson for helpful discussions . this research was supported in part by a grant from the cariplo foundation managed by the landau network centro volta , and by the italian national research council ( cnr , contract esf / eurocores / fanas / afri ) , whose contributions are gratefully acknowledged . in this appendix we report the construction of the substrate rotated by a given misfit angle @xmath54 . the main problem here is to obtain numerous misfit angles @xmath54 which satisfy pbc in @xmath38 and @xmath39 directions simultaneously , while maintaining a constant simulation size , and rectangular pbc ( for the square shape of the simulation cell the construction is much simpler , e.g. , see ref . the idea of the construction is demonstrated in fig . the substrate is arranged according to a square lattice with lattice constant @xmath85 , and the simulation cell area is @xmath86 . the rotated bottom lattice is constructed as a set of parallelograms , so that the elementary cell of the new lattice has size @xmath87 with a base angle @xmath88 . in the perfect case we would have @xmath89 and @xmath90 . however , to satisfy the pbc , the rotated lattice has to be distorted from the ideal square shape , and the idea is to reduce this distortion to a minimum . the rotated lattice is defined by two integers @xmath91 and @xmath92 ( see fig . [ c16 ] ) which determine the rotation angle @xmath93 . for example , the choice @xmath94 and @xmath95 or @xmath96 and @xmath97 gives the original square lattice , while the sets @xmath94 and @xmath98 or @xmath99 and @xmath97 lead to the minimally allowed misfit angle for the given size of the simulation cell . let us draw two lines ( see fig . [ c16 ] ) , the first line starts at the point o@xmath100 and has the length @xmath101 , while the second line starts at the point o@xmath102 and has the length @xmath103 . these lines intersect at point o@xmath104 forming an angle @xmath105 . the rotated substrate atoms are placed along these lines , and then periodic shifts by multiples of @xmath106 and @xmath107 in the directions defined by these two lines will generate the rotated oblique lattice . the oblique lattice constants @xmath106 and @xmath107 are determined by two constrains . first , we must preserve the area of the elementary cell : @xmath108 second , from the triangle o@xmath109o@xmath110o@xmath104 we have @xmath111 from eqs . ( [ axay ] ) and ( [ lx2 ] ) we obtain @xmath112 where @xmath113 and @xmath114 . the signs @xmath115 in eq . ( [ axfin ] ) yield two possible solutions which we call as `` left '' and `` right '' ; typically we use the `` right '' variant when @xmath116 and the `` left '' one for @xmath117 . finally , the actual misfit angle @xmath54 is given by @xmath118 , where @xmath119 the construction described above guarantees the perfect pbc in the @xmath38 direction , but not in the @xmath39 one . therefore , a next step in construction is to characterize this distortion . considering the triangle o@xmath120o@xmath110o@xmath121 in fig . [ c16 ] ( the line o@xmath120o@xmath121 is parallel to o@xmath109o@xmath104 ) , the lengths of its short sides are @xmath122 and @xmath123 , where @xmath124 . the distortion of pbc is determined by two parameters @xmath125 ^ 2 $ ] and @xmath126 ^ 2 $ ] . in the ideal case , it should be @xmath127 . a `` quality '' of the rotated lattice can be characterized by two parameters : the first parameter @xmath128 describes the distortion of periodic boundary conditions , while the second parameter @xmath129 indicates how close @xmath106 and @xmath107 are to the original square - lattice constant @xmath85 . perfect rotations are realized for @xmath130 and @xmath131 simultaneously at @xmath90 . thus , for given integers @xmath91 and @xmath92 , we plot @xmath132 and @xmath133 as functions of @xmath134 , to choose an appropriate minimum of @xmath135 close to the point @xmath90 , and to check that @xmath133 is not too large at that point . note that in the non - ideal case , some atoms within the @xmath136 simulation cell may be missing or , for other sets of parameters , some atoms may overlap with their `` image '' atoms generated by pbc . to overcome this problem , we shifted slightly the bottom boundary of the selected area . as a result , in the rotated lattice the number of substrate atoms may differ from the original one by a value @xmath137 . finally , because different sets of parameters may provide approximately the same misfit angle , we can choose the best set , the one which minimizes @xmath138 and @xmath134 . two typical examples of the construction described above are shown in fig . [ c17 ] . 99 b.n.j . persson , _ `` sliding friction : physical principles and applications '' _ ( springer - verlag , berlin , 1998 ) . o.m . braun and a.g . naumovets , surf . reports * 60 * , 79 ( 2006 ) . thompson and m.o . robbins , , 6830 ( 1990 ) . thompson and m.o . robbins , science * 250 * , 792 ( 1990 ) ; m.o . robbins and p.a . thompson , science * 253 * , 916 ( 1991 ) . m. hirano and k. shinjo , , 11837 ( 1990 ) . mser and m.o . robbins , , 2335 ( 2000 ) . g. he , m.h . mser , and m.o . robbins , science * 284 * , 1650 ( 1999 ) . mser , l. wenning , and m.o . robbins , , 1295 ( 2001 ) . f. lancon , europhys . lett . * 57 * , 74 ( 2002 ) . m. hirano and k. shinjo , wear * 168 * , 121 ( 1993 ) . robbins and e.d . smith , langmuir * 12 * , 4543 ( 1996 ) . verhoeven , m. dienwiebel , and j.w.m . frenken , * 70 * , 165418 ( 2004 ) . f. bonelli , n. manini , e. cadelano , and l. colombo , eur . j. b * 70 * , 449 ( 2009 ) . de wijn , c. fusco , and a. fasolino , , 046105 ( 2010 ) . t. gyalog and h. thomas , europhys . lett . * 37 * , 195 ( 1997 ) . y. qi , y .- t . cheng , t. cagin , and w.a . goddard iii , , 085420 ( 2002 ) . m. hirano , k. shinjo , r. kaneko , and y. murata , , 1448 ( 1997 ) . m. dienwiebel , g.s . verhoeven , n. pradeep , j.w.m . frenken , j.a . heimberg , and h.w . zandbergen , , 126101 ( 2004 ) . g. he and m.o . robbins , , 035413 ( 2001 ) . g. he and m.o . robbins , tribology letters * 10 * , 7 ( 2001 ) . braun and m. peyrard , , 046110 ( 2001 ) . braun and r. ferrando , , 061107 ( 2002 ) . braun and m. peyrard , , 011506 ( 2003 ) . o.m . braun and yu.s . kivshar , _ `` the frenkel - kontorova model : concepts , methods , and applications '' _ ( springer - verlag , berlin , 2004 ) . a. vanossi , n. manini , g. divitini , g.e . santoro , and e. tosatti , , 056101 ( 2006 ) . a. vanossi , n. manini , f. caruso , g.e . santoro , and e. tosatti , , 206101 ( 2007 ) .

using molecular dynamics based on langevin equations with a coordinate- and velocity - dependent damping coefficient , we study the frictional properties of a thin layer of `` soft '' lubricant ( where the interaction within the lubricant is weaker than the lubricant - substrate interaction ) confined between two solids . at low driving velocities the system demonstrates stick - slip motion . the lubricant may or may not be melted during sliding , thus exhibiting either the `` liquid sliding '' ( ls ) or the `` layer over layer sliding '' ( lols ) regimes . the lols regime mainly operates at low sliding velocities . we investigate the dependence of friction properties on the misfit angle between the sliding surfaces and calculate the distribution of static frictional thresholds for a contact of polycrystalline surfaces .

1012.5922

we consider the one dimensional semilinear wave equation : @xmath2 where @xmath3 and @xmath4 . we may also add more restriction on initial data by assuming that @xmath5 the cauchy problem for equation ( [ waveq ] ) in the space @xmath6 follows from fixed point techniques ( see section [ cauchy - problem ] below ) . + if the solution is not global in time , we show in this paper that it blows up ( see theorems [ th ] and [ new ] below ) . for that reason , we call it a blow - up solution . the existence of blow - up solutions is guaranteed by ode techniques and the finite speed of propagation . more blow - up results can be found in kichenassamy and littman @xcite , @xcite , where the authors introduce a systematic procedure for reducing nonlinear wave equations to characteristic problems of fuchsian type and construct singular solutions of general semilinear equations which blow up on a non characteristic surface , provided that the first term of an expansion of such solutions can be found . the case of the power nonlinearity has been understood completely in a series of papers , in the real case ( in one space dimension ) by merle and zaag @xcite , @xcite , @xcite and @xcite and in cte and zaag @xcite ( see also the note @xcite ) , in the complex case by azaiez . some of those results have been extended to higher dimensions for conformal or subconformal @xmath7 : @xmath8 under radial symmetry outside the origin in @xcite . for non radial solutions , we would like to mention @xcite and @xcite where the blow - up rate was obtained . we also mention the recent contribution of @xcite and @xcite where the blow - up behavior is given , together with some stability results . in @xcite and @xcite , caffarelli and friedman considered semilinear wave equations with a nonlinearity of power type . if the space dimension @xmath9 is at most @xmath10 , they showed in @xcite the existence of solutions of cauchy problems which blow up on a @xmath11 spacelike hypersurface . if @xmath12 and under suitable assumptions , they obtained in @xcite a very general result which shows that solutions of cauchy problems either are global or blow up on a @xmath11 spacelike curve . in @xcite and @xcite , godin shows that the solutions of cauchy problems either are global or blow up on a @xmath11 spacelike curve for the following mixed problem ( @xmath13 , @xmath14 ) @xmath15 in @xcite , godin gives sharp upper and lower bounds on the blow - up rate for initial data in @xmath16 . it happens that his proof can be extended for initial data @xmath17 ( see proposition [ p ] below ) . let us consider _ u _ a blow - up solution of ( [ waveq ] ) . our aim in this paper is to derive upper and lower estimates on the blow - up rate of @xmath18 . in particular , we first give general results ( see theorem [ th ] below ) , then , considering only non - characteristic points , we give better estimates in theorem [ new ] . from alinhac @xcite , we define a continuous curve @xmath19 as the graph of a function @xmath20 such that the domain of definition of @xmath21 ( or the maximal influence domain of @xmath21 ) is @xmath22 from the finite speed of propagation , @xmath23 is a 1-lipschitz function . the graph @xmath19 is called the blow - up graph of @xmath21 . let us introduce the following non - degeneracy condition for @xmath19 . if we introduce for all @xmath24 @xmath25 and @xmath26 , the cone @xmath27 then our non - degeneracy condition is the following : @xmath28 is a non - characteristic point if @xmath29 if condition ( [ 4 ] ) is not true , then we call @xmath28 a characteristic point . we denote by @xmath30 ( resp . @xmath31 ) the set of non - characteristic ( resp . characteristic ) points . + we also introduce for each @xmath32 and @xmath33 the following similarity variables : @xmath34 + if @xmath35 , we write @xmath36 instead of @xmath37 . + from equation ( [ waveq ] ) , we see that @xmath38 ( or @xmath39 for simplicity ) satisfies , for all @xmath40 , and @xmath41 , @xmath42 + in the new set of variables @xmath43 , deriving the behavior of @xmath21 as @xmath44 is equivalent to studying the behavior of _ w _ as s @xmath45 . + our first result gives rough blow - up estimates . introducing the following set @xmath46 where @xmath47 , we have the following result * ( blow - up estimates near any point)*[th ] we claim the following : * \i ) _ _ * ( upper bound ) * _ _ for all @xmath47 and @xmath48 such that @xmath49 , it holds : + @xmath50 + @xmath51 + where @xmath52 is the ( euclidean ) distance from @xmath53 to @xmath19 . * \ii ) _ _ * ( lower bound ) * _ _ for all @xmath47 and @xmath54 such that @xmath49 , it holds that + @xmath55 + if in addition , @xmath56 then @xmath57 * \iii ) _ _ * ( lower bound on the local energy `` norm '' ) * _ _ there exists @xmath58 such that for all @xmath59 , and @xmath60 @xmath61 where @xmath62 * remark * : the upper bound in item @xmath63 was already proved by godin @xcite , for more regular initial data . here , we show that godin s strategy works even for less regular data . we refer to the integral in ( [ 10,1 ] ) as the local energy `` norm '' , since it is like the local energy as in shatah and struwe , though with the `` @xmath64 '' sign in front of the nonlinear term . note that the lower bound in item @xmath65 is given by the solution of the associated ode @xmath66 . however the lower bound in @xmath67 does nt seem to be optimal , since it does not obey the ode behavior . indeed , we expect the blow - up for equation ( [ waveq ] ) in the `` ode style '' , in the sense that the solution is comparable to the solution of the ode @xmath68 at blow - up . this is in fact the case with regular data , as shown by godin @xcite . + if in addition @xmath69 , we have optimal blow - up estimates : * ( an optimal bound on the blow - up rate near a non - characteristic point in a smaller space)*[new ] assume that @xmath56 . then , for all @xmath47 , for any @xmath69 such that @xmath49 , we have the following * \i ) _ _ * ( uniform bounds on @xmath39 ) * _ _ for all @xmath70 @xmath71 where @xmath36 is defined in ( [ trans_auto ] ) . * \ii ) _ _ * ( uniform bounds on @xmath21 ) * _ _ for all @xmath72 @xmath73 in particular , we have @xmath74 * remark * : this result implies that the solution indeed blows up on the curve @xmath19 . * remark * : note that when @xmath75 , theorem [ th ] already holds and directly follows from theorem [ new ] . accordingly , theorem [ th ] is completely meaningful when @xmath76 . + following antonini , merle and zaag in @xcite and @xcite , we would like to mention the existence of a lyapunov functional in similarity variables . more precisely , let us define @xmath77 we claim that the functional @xmath78 defined by ( [ lyapunov ] ) is a decreasing function of time for solutions of ( [ equaw ] ) on ( -1,1 ) . * ( a lyapunov functional for equation ( [ waveq]))*[2.1 ] for all @xmath79 , the following identities hold for @xmath80 : @xmath81 * remark * : the existence of such an energy in the context of the nonlinear heat equation has been introduced by giga and kohn in @xcite , @xcite and @xcite . * remark * : as for the semilinear wave equation with conformal power nonlinearity , the dissipation of the energy @xmath82 degenerates to the boundary @xmath83 . this paper is organized as follows : + in section [ cauchy - problem ] , we solve the local in time cauchy problem . + section [ sec2 ] is devoted to some energy estimates . + in section [ sec3 ] , we give and prove upper and lower bounds , following the strategy of godin @xcite . + finally , section [ sec4 ] is devoted to the proofs of theorem [ th ] , theorem [ new ] and proposition [ 2.1 ] . in this section , we solve the local cauchy problem associated to ( [ waveq ] ) in the space @xmath6 . in order to do so , we will proceed in two steps : + @xmath84 in step 1 , we solve the problem in @xmath6 , for some uniform @xmath85 small enough . + @xmath86 in step 2 , we consider @xmath87 , and use step 1 and a truncation to find a local solution defined in some cone @xmath88 for some @xmath89 . then , by a covering argument , the maximal domain of definition is given by @xmath90 . + @xmath91 in step 3 , we consider some approximation of equation ( [ waveq ] ) , and discuss the convergence of the approximating sequence . * step 1 : the cauchy problem in @xmath6 * + in this step , we will solve the local cauchy problem associated to ( [ waveq ] ) in the space @xmath92 . in order to do so , we will apply a fixed point technique . we first introduce the wave group in one space dimension : @xmath93 @xmath94 clearly , @xmath95 is well defined in @xmath96 , for all @xmath97 , and more precisely , there is a universal constant @xmath98 such that @xmath99 this is the aim of the step : * ( cauchy problem in @xmath6)*[a1 ] for all @xmath100 , there exists @xmath85 such that there exists a unique solution of the problem ( [ waveq ] ) in @xmath101,h).$ ] consider @xmath85 to be chosen later small enough in terms of @xmath102 . we first write the duhamel formulation for our equation @xmath103 introducing @xmath104 we will work in the banach space @xmath105 , h)$ ] equipped with the norm @xmath106 then , we introduce @xmath107 and the ball @xmath108 . + we will show that for @xmath85 small enough , @xmath109 has a unique fixed point in @xmath108 . to do so , we have to check 2 points : 1 . @xmath109 maps @xmath108 to itself . @xmath109 is k - lipschitz with @xmath110 for @xmath23 small enough . * proof of 1 : let @xmath111 , this means that : @xmath112 , \ ; v(t ) \in h_{loc , u}^{1}(\bbb r)\subset l^{\infty}(\bbb r)\ ] ] and that @xmath113 + therefore @xmath114 this means that @xmath115\ ; ( 0 , e^{v(\tau ) } ) \in h,\ ] ] hence @xmath116 is well defined from ( [ contins ] ) and so is its integral between @xmath117 and @xmath118 . so @xmath109 is well defined from @xmath78 to @xmath78 . + let us compute @xmath119 : + using ( [ contins ] ) , ( [ rayon ] ) and ( [ carreaux ] ) we write for all @xmath120 $ ] , @xmath121 choosing @xmath23 small enough so that @xmath122 or @xmath123 guarantees that @xmath109 goes from @xmath108 to @xmath108 . * proof of 2 : let @xmath124 we have @xmath125 since @xmath126 and the same for @xmath127 , we write @xmath128 + hence @xmath129 applying @xmath130 we write from ( [ contins ] ) , for all @xmath131 @xmath132 integrating we end - up with @xmath133 can be made @xmath134 if @xmath23 is small . + _ conclusion _ : from points 1 and 2 , @xmath109 has a unique fixed point @xmath135 in @xmath136 . this fixed point is the solution of the duhamel formulation ( [ duhamel ] ) and of our equation ( [ waveq ] ) . this concludes the proof of lemma [ a1 ] . * step 2 : the cauchy problem in a larger region * + let @xmath137 initial data for the problem ( [ waveq ] ) . using the finite speed of propagation , we will localize the problem and reduces it to the case of initial data in @xmath6 already treated in step 1 . for @xmath138 , we will check the existence of the solution in the cone @xmath139 . in order to do so , we introduce @xmath140 a @xmath141 function with compact support such that @xmath142 if @xmath143 , let also @xmath144 ( note that @xmath145 and @xmath146 depend on @xmath147 but we omit this dependence in the indices for simplicity ) . so , @xmath148 . from step 1 , if @xmath149 is the corresponding solution of equation ( [ waveq ] ) , then , by the finite speed of propagation , @xmath150 in the intersection of their domains of definition with the cone @xmath139 . as @xmath149 is defined for all @xmath53 in @xmath151 from step 1 for some @xmath152 , we get the existence of @xmath21 locally in @xmath153 . varying @xmath147 and covering @xmath154 by an infinite number of cones , we prove the existence and the uniqueness of the solution in a union of backward light cones , which is either the whole half - space @xmath155 , or the subgraph of a 1-lipschitz function @xmath156 . we have just proved the following : * ( the cauchy problem in a larger region ) * consider @xmath137 . then , there exists a unique solution defined in @xmath157 , a subdomain of @xmath158 , such that for any @xmath159 , with @xmath160 . moreover , * either @xmath161 , * or @xmath162 for some 1-lipschitz function @xmath156 . * step 3 : regular approximations for equation ( [ waveq ] ) * + consider @xmath163 , @xmath21 its solution constructed in step 2 , and assume that it is non global , hence defined under the graph of a 1-lipschitz function @xmath164 consider for any @xmath165 , a regularized increasing truncation of @xmath166 satisfying @xmath167 and @xmath168 . consider also a sequence @xmath169 such that @xmath170 in @xmath171 as @xmath172 , for any @xmath47 . + then , we consider the problem @xmath173 since steps 1 and 2 clearly extend to locally lipschitz nonlinearities , we get a unique solution @xmath174 defined in the half - space @xmath155 , or in the subgraph of a 1-lipschitz function . since @xmath175 , for all @xmath176 , it is easy to see that in fact : @xmath174 is defined for all @xmath177 . from the regularity of @xmath178 , @xmath179 and @xmath180 , it is clear that @xmath174 is a strong solution in @xmath181 . introducing the following sets : @xmath182 @xmath183 and @xmath184 we claim the following * ( uniform bounds on variations of @xmath174 in cones)*[lab ] consider @xmath47 , one can find @xmath185 such that if @xmath186 then @xmath187 : @xmath188 @xmath189 * remark : * of course @xmath190 depends also on initial data , but we omit that dependence , since we never change initial data in this setting . note that since @xmath191 , it follows that @xmath192 . we will prove the first inequality , the second one can be proved by the same way . for more details see @xcite page @xmath193 . + let @xmath47 , consider @xmath53 fixed in @xmath194 and @xmath43 in @xmath195 . we introduce the following change of variables : @xmath196 from ( [ 16.5 ] ) , we see that @xmath197 satisfies : @xmath198 let @xmath199 the new coordinates of @xmath43 in the new set of variables . note that @xmath200 and @xmath201 . we note that there exists @xmath202 and @xmath203 such that the points @xmath204 and @xmath205 lay on the horizontal line @xmath206 and have as original coordinates respectively @xmath207 and @xmath208 for some @xmath209 and @xmath210 in @xmath211 $ ] . we note also that in the new set of variables , we have : @xmath212 from ( [ xieta ] ) , @xmath213 is monotonic in @xmath214 . so , for example for @xmath215 , as @xmath216 , we have : @xmath217 similarly , for any @xmath218 , we can bound from above the function @xmath219 by its value at the point @xmath220 , which is the projection of @xmath221 on the axis @xmath206 in parallel to the axis @xmath214 ( as @xmath222 ) . by the same way , from ( [ xieta ] ) , @xmath223 is monotonic in @xmath224 . as @xmath225 , we can bound , for @xmath226 , @xmath227 by its value at the point @xmath228 , which is the projection of @xmath229 on the axis @xmath206 in parallel to the axis @xmath224 ( @xmath230 ) . so it follows that : @xmath231 by a straightforward geometrical construction , we see that the coordinates of @xmath220 and @xmath228 , in the original set of variables @xmath232 , are respectively @xmath233 and @xmath234 . both points are in @xmath211 $ ] . furthermore , we have from ( [ 21,5 ] ) : @xmath235 using ( [ equadi3 ] ) , the cauchy - schwarz inequality and the fact that @xmath180 and @xmath236 are uniformly bounded in @xmath237 since they are convergent , we have : @xmath238 using ( [ equadi1 ] ) , ( [ equadi2 ] ) and ( [ equadi4 ] ) , we reach to conclusion of lemma [ lab ] . let us show the following : * ( convergence of @xmath174 as @xmath172)*[xt ] consider @xmath177 . we have the following : * if @xmath239 , then @xmath240 , * if @xmath241 , then @xmath242 . we claim that it is enough to show the convergence for a subsequence . indeed , this is clear from the fact that the limit is explicit and does nt depend on the subsequence . consider @xmath243 up to extracting a subsequence , there is @xmath244 such that @xmath245 as @xmath172 . + let us show that @xmath246 . since @xmath247 , it follows that @xmath248 where @xmath249 since @xmath250 , for any @xmath47 , from the fact that @xmath251 is convergent in @xmath252 , it follows that @xmath253 . + note from the fact that @xmath254 that we have @xmath255 introducing @xmath256 , we see by definition ( [ 6.5 ] ) of @xmath257 that @xmath258 . let us handle two cases in the following : * case 1 : * @xmath259 + let us introduce @xmath260 the solution of @xmath261 from the local cauchy theory in @xmath262 and the sobolev embedding , we know that @xmath263 let us consider @xmath264 and @xmath265 , since @xmath266 is a compact set in @xmath157 . + from ( [ triangle1 ] ) , we may assume @xmath267 large enough , so that @xmath268 , + @xmath269 and @xmath270 in particular , @xmath271 we claim that @xmath272 indeed , arguing by contradiction , we may assume from ( [ dd ] ) and continuity of @xmath174 that @xmath273,\ , ||u_n(s)||_{l^{\infty}(\tilde k\cap \{t = s\})}\le \tilde m+2,\end{aligned}\ ] ] and @xmath274 . + from ( [ 4d ] ) , ( [ 3d ] ) and the definition ( [ 16.3 ] ) of @xmath178 , we see that @xmath275 therefore , @xmath174 and @xmath260 satisfy the same equation with the same initial data on @xmath276 . from uniqueness of the solution to the cauchy problem , we see that @xmath277 a contradiction then follows from ( [ 5d ] ) and ( [ 6d ] ) . thus , ( [ 7d ] ) holds . + again from the choice of @xmath267 in ( [ 4d ] ) , we see that @xmath278 hence , from uniqueness , @xmath279 from ( [ triangle1 ] ) , and since @xmath280 , it follows that @xmath242 as @xmath172 . * case 2 : * @xmath281 + assume by contradiction that @xmath282 from lemma [ lab ] , it follows that @xmath283 for @xmath284 large enough , this gives @xmath285 + if @xmath286 then @xmath287 and @xmath174 satisfies ( [ waveq ] ) in @xmath288 with initial data @xmath289 . from the finite speed of propagation and the continuity of solutions to the cauchy problem with respect of initial data , it follows that @xmath174 and @xmath21 are both defined in @xmath290 for @xmath267 large enough , in particular @xmath21 is defined at @xmath291 with @xmath292 with @xmath293 in @xmath288 . contradiction with the expression of the domain of definition ( [ domaine - de - definition ] ) of @xmath21 . in this section , we use some localized energy techniques from shatah and struwe @xcite to derive a non - blow - up criterion which will give the lower bound in theorem [ th ] . more precisely , we give the following : _ * ( non blow - up criterion for a semilinear wave equation)*_[theor ] @xmath294 , there exist @xmath295 and @xmath296 such that , if @xmath297 then equation ( [ waveq ] ) with initial data @xmath298 has a unique solution @xmath299 such that for all @xmath300 we have : + @xmath301 and + @xmath302 note that here , we work in the space @xmath303 which is larger than the space @xmath6 which is adopted elsewhere for equation ( [ waveq ] ) . before giving the proof of this result , let us first give the following corollary , which is a direct consequence of proposition [ theor ] . [ pro ] there exists @xmath304 such that if @xmath305 then the solution @xmath21 of equation ( [ waveq ] ) with initial data @xmath298 does nt blow up in the cone @xmath306 . let us first derive corollary [ pro ] from proposition [ theor ] . : from ( [ soleil ] ) , if @xmath307 we see that @xmath308 therefore , for some @xmath309 , we have @xmath310 , hence @xmath311 . using ( [ solei ] ) , we see that for all @xmath312 @xmath313 defined in proposition [ theor ] , provided that @xmath314 is small enough . therefore , the hypothesis ( h ) of proposition [ theor ] holds with @xmath315 , and so does its conclusion . this concludes the proof of corollary [ pro ] , assuming that proposition [ theor ] holds . now , we give the proof of proposition [ theor ] . consider @xmath316 and introduce @xmath317 then , we consider @xmath298 satisfying hypothesis ( h ) . from the solution of the cauchy problem in @xmath252 , which follows exactly by the same argument as in the space @xmath6 presented in section [ cauchy - problem ] , there exists @xmath318 $ ] such that equation ( [ waveq ] ) has a unique solution with @xmath319 . our aim is to show that @xmath320 and that ( [ p1 ] ) and ( [ p2 ] ) hold for all @xmath321 . clearly , from the solution of the cauchy problem , it is enough to show that ( [ p1 ] ) and ( [ p2 ] ) hold for all @xmath322 , so we only do that in the following : arguing by contradiction , we assume that there exists at least some time @xmath323 such that either ( [ p1 ] ) or ( [ p2 ] ) does nt hold . if @xmath324 is the lowest possible @xmath118 , then , we have from continuity , either @xmath325 or @xmath326 note that since ( [ p1 ] ) holds for all @xmath327 , it follows that @xmath328 following the alternative on @xmath324 , two cases arise in the following : _ * case 1 * _ : @xmath329 . + referring to shatah and struwe @xcite , we see that : @xmath330 where @xmath331\ ] ] using ( [ to ] ) , it follows that @xmath332 @xmath333 therefore , from ( [ equashatah ] ) and ( [ h ] ) , we write @xmath334 which is a contradiction . + _ * case 2*_:@xmath335 + recall duhamel s formula : @xmath336 from ( h ) , we write , @xmath337 from ( [ to ] ) , we write @xmath338 since @xmath339 , it follows from ( [ duhamel1 ] ) that @xmath340 and a contradiction follows . this concludes the proof of proposition [ theor ] . since we have already derived corollary [ pro ] from proposition [ theor ] , this is also the conclusion of the proof of corollary [ pro ] . in this section , we extend the work of godin in @xcite . in fact , we show that his estimates holds for more general initial data . as in the introduction , we consider @xmath18 a non global solution of equation ( [ waveq ] ) with initial data @xmath341 . this section is organized as follows : + -in the first subsection , we give some preliminary results and we show that the solution goes to @xmath342 on the graph @xmath343 . + -in the second subsection , we give and prove upper and lower bounds on the blow - up rate . in this subsection , we first give some geometrical estimates on the blow - up curve ( see lemmas [ co ] , [ so ] and [ fo ] ) . then , we use equation ( [ waveq ] ) to derive a kind of maximum principle in light cones ( see lemma [ 3.2 ] ) , then , a lower bound on the blow - up rate ( see proposition [ ll ] ) . we first we give the following geometrical property concerning the distance to + @xmath344 the boundary of the domain of definition of @xmath18 . * ( estimate for the distance to the blow - up boundary).*[co ] for all @xmath345 , we have @xmath346 where @xmath52 is the distance from @xmath53 to @xmath19 . note first by definition that @xmath347 then , from the finite speed of propagation , @xmath19 is above @xmath348 , the backward light cone with vertex @xmath349 . since @xmath350 , it follows that @xmath351 this concludes the proof of lemma [ co ] . then , we give a geometrical property concerning distances , specific for non - characteristic points . * ( a geometrical property for non - characteristic points)*[so ] let @xmath352 . there exists @xmath353 , where @xmath354 is given by ( [ 4 ] ) , such that for all @xmath355 , @xmath356 * remark * : from lemma [ co ] , it follows that @xmath357 whenever @xmath358 and @xmath355 . let @xmath359 be a non - characteristic point . we recall form condition ( [ 4 ] ) that @xmath360 let @xmath53 be in the light cone with vertex @xmath361 . using the fact that the blow - up graph is above the cone @xmath362 and the fact that @xmath363 we see that @xmath364 in addition , as @xmath19 is a 1-lipchitz graph , we have @xmath365 so , for all @xmath366 @xmath367 from ( [ soo ] ) and ( [ sooo ] ) , there exists @xmath368 such that @xmath369 this concludes the proof of lemma [ so ] . finally , we give the following coercivity estimate on the distance to the blow - up curve , still specific for non - characteristic points . [ fo ] let @xmath370 and @xmath371 . for all @xmath372 and @xmath373 we have @xmath374 where @xmath375 and @xmath376 . * remark : * note that @xmath377 for @xmath373 lay on the backward light cone with vertex @xmath53 . consider @xmath378 , @xmath379 . by definition , there exists @xmath380 such that @xmath381 . we will prove the estimate for @xmath382 and @xmath383 , since the the estimate for @xmath384 follows by symmetry . in order to do so , we introduce the following notations , as illustrated in figure 1 : @xmath385 , @xmath386 and @xmath387 , which is on the left boundary of the backward light cone @xmath388 , @xmath389 the orthogonal projection of @xmath390 on the left boundary of the cone @xmath391 , @xmath392 the orthogonal projection of @xmath393 on @xmath394 $ ] . note that the quadrangle @xmath395 is a rectangle . if @xmath396 is such that @xmath397 and @xmath398 , then we see from elementary considerations on angles that @xmath399 and @xmath400 . + therefore , using lemma [ co ] , and the angles on the triangle @xmath401 , we see that : @xmath402 moreover , since the blow - up graph is above the cone @xmath391 , it follows that @xmath403 in particular , @xmath404 since @xmath405 , hence @xmath406 , it follows that @xmath407 . since @xmath408 , the result follows from ( [ 49.5 ] ) and ( [ ppf ] ) . by the same way , we can prove this for the other point @xmath409 , which gives ( [ c ] ) . + now , we give the following corollary from the approximation procedure in lemmas [ lab ] and [ xt ] . [ 3.2]*(uniform bounds on variations of @xmath21 in cones ) . * for any @xmath47 , there exists a constant @xmath185 such that if @xmath410 then @xmath411 @xmath412 where the cones @xmath413 and @xmath414 are defined in ( [ 16.55 ] ) . * remark * : the constant @xmath415 depends also on @xmath416 and @xmath417 , but we omit this dependence in the sequel . + in the following , we give a lower bound on the blow - up rate and we show that @xmath418 as @xmath419 . * ( a general lower bound on the blow - up rate)*[ll ] \(i ) if @xmath56 , then for all @xmath47 , there exists @xmath185 such that for all @xmath420 @xmath421 in particular , for all @xmath410 , @xmath422 as @xmath423 . \(ii ) if , we only have @xmath341 , then for all @xmath47 , there exists @xmath185 such that for all @xmath424 @xmath425 in particular , @xmath426 converges to @xmath117 in average over slices of the light cone , as @xmath427 . * remark * : near non characteristic points , we are able to derive the optimal lower bound on the blow - up rate . see item @xmath428 of proposition [ p ] . \(i ) clearly , the last sentence in item @xmath429 follows from the first , hence , we only prove the first . let @xmath47 and @xmath430 using the approximation procedure defined in ( [ 16.5 ] ) , we write @xmath431 with : @xmath432 @xmath433 ( note that @xmath434 was already defined in ( [ 28.5 ] ) ) . since @xmath435 from ( [ 16.3 ] ) , it follows that @xmath436 differentiating @xmath434 , we see that @xmath437 differentiating @xmath438 , we get @xmath439 since @xmath440 . since @xmath441 and @xmath442 from lemma [ lab ] , it follows that @xmath443 therefore , using ( [ carreau1 ] ) we see that @xmath444 hence , @xmath445 integrating ( [ 1 ] ) on any interval @xmath446 $ ] with @xmath447 , we get @xmath448 . making @xmath449 and using lemma @xmath450 we see that @xmath451 . + taking @xmath452 and making @xmath453 , we get @xmath454 . using lemma [ co ] concludes the proof of item @xmath455 of proposition [ ll ] . \(ii ) if @xmath341 , then small modification in the argument of item @xmath429 gives the result . indeed , if @xmath456 , @xmath457 and @xmath458 , we write from ( [ carreau1 ] ) and ( [ 50.5 ] ) @xmath459 furthermore , from ( [ 50.6 ] ) we write @xmath460 therefore , it follows that @xmath461 integrating ( [ 50.7 ] ) on interval @xmath462 , where @xmath463 we get @xmath464 since @xmath156 is 1-lipschitz and @xmath465 is the middle of @xmath466 $ ] , we clearly see that the segment @xmath467\times \{t_0'\}$ ] lays outside the domain of definition of @xmath18 , using lemma [ xt ] , we see that @xmath468 on the one hand ( we use lebesgue lemma together with the bound ( [ carreau1 ] ) ) . on the other hand , similarly , we see that @xmath469 thus , the conclusion follows from ( [ 50.8 ] ) , together with lemma [ co ] , this concludes the proof of proposition [ ll ] . this subsection is devoted to bound the solution @xmath21 . we have obtained the following result . [ p ] for any @xmath47 , there exists @xmath185 , such that : \(i ) * ( upper bound on @xmath21 ) * for all @xmath470 we have @xmath471 \(ii ) * ( lower bound on @xmath21 ) * if in addition @xmath56 and @xmath472 is a non - characteristic point , then for all @xmath473 @xmath474 * remark * : in @xcite , godin did nt use the notion of characteristic point , but the regularity of initial data was fundamental to have the result . in this work , our initial data are less regular , so we focused on the case of non - characteristic point in order to get his result . @xmath475 + @xmath429 consider @xmath47 . we will show the existence of some @xmath185 such that for any @xmath476 we have @xmath477 consider then @xmath478 since @xmath156 is 1-lipschitz , we clearly see that @xmath479 consider now @xmath480 to be fixed later . we introduce the square domain with vertices @xmath481 . let @xmath482 @xmath483 respectively the upper and lower half of the considered square . from duhamel s formula , we write : @xmath484 and , @xmath485 so , @xmath486 since the square @xmath487 from ( [ * ] ) , applying lemma [ 3.2 ] , we have for all @xmath488 and for some @xmath185 : @xmath489 applying this to ( [ t ] ) , we get @xmath490 now , choosing @xmath491 where @xmath492 , we see that + -either @xmath493 + -or @xmath494 and @xmath495 + by the above - given analysis . in the second case , we may proceed similarly and define for @xmath496 a sequence @xmath497 as long as @xmath498 clearly , the sequence @xmath499 is increasing whenever it exists . repeating between @xmath500 and @xmath501 , for @xmath496 , the argument we first wrote for @xmath502 and @xmath503 , we see that @xmath504 as long as @xmath505 . two cases arise then : + -*case 1 * : the sequence @xmath499 can be defined for all @xmath506 , which means that @xmath507 in particular , ( [ triangle ] ) and ( [ carreau ] ) hold for all @xmath508 . if @xmath509 , then , from ( [ trian ] ) , we see that @xmath510 . since @xmath511 as @xmath172 from ( [ carreau ] ) , we need to have @xmath512 from the cauchy theory . therefore , using lemma [ co ] , ( [ triangle ] ) and ( [ carreau ] ) , we see that @xmath513 which is the desired estimate . + -*case 2 * : the sequence @xmath499 exists only for all @xmath514 $ ] for some @xmath515 . this means that @xmath516 , that is @xmath517 . + moreover , ( [ triangle ] ) holds for all @xmath518 $ ] , and ( [ carreau ] ) holds for all @xmath519 $ ] ( in particular , it is never true if @xmath520 ) . as for case 1 , we use lemma [ co ] , ( [ triangle ] ) and ( [ carreau ] ) to write @xmath521 which is the desired estimate . this concludes the proof of item @xmath63 of proposition [ p ] . @xmath428 consider @xmath47 and @xmath472 a non - characteristic point such that @xmath522 . we dissociate @xmath21 into two parts @xmath523 with : @xmath524 @xmath525 differentiating @xmath149 , we see that @xmath526 since @xmath258 . consider now an arbitrary @xmath527 . since @xmath418 as @xmath423 ( see proposition [ ll ] above ) , it follows that @xmath528 now , we will prove a similar inequality for @xmath529 . differentiating @xmath529 , we see that @xmath530 using the upper bound in proposition [ p ] , part @xmath429 , which is already proved and lemma [ 3.2 ] , we see that for all @xmath531 @xmath532 so , @xmath533 since @xmath534 is a non - characteristic point , there exists @xmath535 such that the cone @xmath536 is below the blow - up graph @xmath19 . applying @xmath537 and lemma [ fo ] to @xmath538 , and using the fact that @xmath539 , we write ( recall that @xmath540 ) : @xmath541 which yields @xmath542 in conclusion , we have from ( [ cp ] ) , ( [ ccp ] ) and lemma [ co ] @xmath543 since @xmath544 as @xmath423 from proposition [ ll ] , integrating ( [ a9 ] ) between @xmath118 and @xmath545 , we see that @xmath546 using again lemma [ co ] , we complete the proof of part @xmath428 of proposition [ p ] . in this section , we prove the three results of our paper : theorem [ th ] , theorem [ new ] and proposition [ 2.1 ] . each proof is given in a separate subsection . @xmath429 let @xmath47 , @xmath48 such that @xmath49 and @xmath355 . consider @xmath547 the closest point of @xmath548 to @xmath53 . this means that @xmath549 by a simple geometrical construction , we see that it satisfies the following : @xmath550 hence , @xmath551 , so @xmath552 using the second equation of ( [ ss ] ) and ( [ sss ] ) we see that : @xmath553 from proposition [ p ] , ( [ ssss ] ) and the similarity transformation ( [ trans_auto ] ) we have : @xmath554 which gives the first inequality of @xmath429 . the second one is given by proposition [ p ] and lemma [ co ] . + we introduce the following change of variables : @xmath559 note that @xmath560 satisfies equation ( [ waveq ] ) . for @xmath561 , @xmath560 satisfies ( @xmath562 ) , so , by corollary [ pro ] , @xmath563 does nt blow - up in @xmath564 thus , @xmath21 does nt blow - up in @xmath565 , which is a contradiction . this concludes the proof of theorem [ th ] . + assume that @xmath56 and consider @xmath47 and @xmath358 such that @xmath49 . on the one hand , we recall from proposition [ p ] that @xmath569 using ( [ trans_auto ] ) , we see that @xmath570 since @xmath571 from lemmas [ co ] and [ so ] , this yields to the conclusion of corollary [ cou ] . consider @xmath47 and @xmath358 such that @xmath49 . we note first that the fact that @xmath572 for all @xmath567 and @xmath573 follows from the corollary [ cou ] . it remains only to show that @xmath574 . from proposition [ p ] , lemmas [ co ] and [ so ] , we have : @xmath575 we define @xmath576 the energy of equation ( [ waveq ] ) by from shatah - struwe @xcite , we have @xmath578 integrating it over @xmath579 and using ( [ l ] ) , we see that @xmath580 thus , @xmath581 now , using ( [ l ] ) and ( [ k ] ) to bound the two first terms in the definition of @xmath582 ( [ f ] ) , we get @xmath583 multiplying ( [ equaw ] ) by @xmath585 , and integrating over ( -1,1 ) , we see that @xmath586 thus , @xmath587 where , @xmath588and @xmath589 thus , @xmath590\ , dy\right ) = -({\partial_s w(-1,s)})^2- ( { \partial_s w(1,s)})^2\end{aligned}\ ] ] + which yields the conclusion of proposition [ 2.1 ] by integration in time . * address : * + universit de cergy - pontoise , laboratoire analyse gometrie modlisation , + cnrs - umr 8088 , 2 avenue adolphe chauvin 95302 , cergy - pontoise , france . + ` e - mail : [email protected] ` + courant institute , nyu , 251 mercer street , ny 10012 , new york . + ` e - mail : [email protected] ` + universit paris 13 , institut galile , laboratoire analyse gometrie et applications , + cnrs - umr 7539 , 99 avenue j.b . clment 93430 , villetaneuse , france . + ` e - mail : [email protected] `

we consider in this paper blow - up solutions of the semilinear wave equation in one space dimension , with an exponential source term . assuming that initial data are in @xmath0 or some times in @xmath1 , we derive the blow - up rate near a non - characteristic point in the smaller space , and give some bounds near other points . our result generalize those proved by godin under high regularity assumptions on initial data .

1601.04007

let @xmath0 denote a solution of the second painlev equation @xmath1 it is known that for special values of the parameter @xmath2 the equation admits rational solutions . in fact vorobev and yablonski @xcite showed that for @xmath3 , the equation has a unique rational solution of the form @xmath4 which is constructed in terms of the vorobev - yablonski polynomials @xmath5 . these special polynomials can be defined via a differential - difference equation @xmath6 where @xmath7 , or equivalently @xcite in determinantal form : with @xmath8 for @xmath9 , @xmath10_{\ell , j=0}^{n-1},\ \ n\in\mathbb{z}_{\geq 1};\ \ \ \ \ \sum_{k=0}^{\infty}q_k(x)w^k=\exp\left[-\frac{4}{3}w^3+wx\right].\ ] ] for our purposes , it will prove useful to rewrite in terms of schur polynomials . in general ( cf . @xcite ) , the schur polynomial @xmath11 $ ] in the variable @xmath12 associated to the partition @xmath13 with @xmath14 is determined by the jacobi - trudi determinant , [ jtrudi ] s _ ( ) = _ j , k=1^ ( ) . here , @xmath15 for @xmath16 is defined by the generating series [ hdef ] _ k=0^h_k()z^k=(_j=1^t_j z^j ) ; and h_k()=0 , k<0 . from it follows immediately that @xmath17 is a weighted - homogeneous function , h_k ( ) = ^k h_k ( ^-1 t_1 , ^-2 t_2 , ^-3 t_3 , ) , \\{0 } , and hence also [ homogschur ] s_ ( ) = ^|| s_(^-1 t_1 , ^-2 t_2 , ^-3 t_3 , ) , ||=_j=1^()_j . for the special choice of a staircase partition , @xmath18 the identities , and lead to the representation of @xmath19 in terms of schur polynomials , @xmath20 it is well known that equation admits higher order generalizations and itself forms the first member of a full hierarchy . to be more precise , let @xmath21 denote the following quantities expressed in terms of the lenard recursion operator , @xmath22=\left(\frac{{{\mathrm d}}^3}{{{\mathrm d}}x^3}+4u\frac{{{\mathrm d}}}{{{\mathrm d}}x}+2u_x\right)\mathcal{l}_n[u],\ \ n\in\mathbb{z}_{\geq 0};\ \ \ \mathcal{l}_0[u]=\frac{1}{2 } , \ ] ] and with the integration constant determined uniquely by the requirement @xmath23=0,\ n\geq 1 $ ] . the recursion gives , for instance , @xmath24=u,\ \ \ \ \mathcal{l}_2[u]=u_{xx}+3u^2,\ \ \ \ \mathcal{l}_3[u]=u_{xxxx}+5(u_x)^2 + 10uu_{xx}+10u^3.\ ] ] the @xmath25-th member of the painlev ii hierarchy is subsequently defined as the ordinary differential equation @xmath26=xu+\alpha_n,\ \ \ x\in\mathbb{c},\ \ \alpha_n\in\mathbb{c};\ \ \ \ u = u(x;\alpha_n , n).\ ] ] hence , the first member @xmath27 is painlev ii itself , and more generally , the @xmath25-th member is an ordinary differential equation of order @xmath28 . besides , we shall also consider a case which involves additional complex parameters @xmath29 . with @xmath30 for @xmath31 and @xmath32 , [ genpiihier ] ( + 2u)_n= _ k=1^n-1 ( 2k+1 ) t_2k+1 ( + 2u ) _k+ xu+ _ n. for and , it is known @xcite that rational solutions exist if and only if @xmath33 . moreover , clarkson and mansfield in @xcite introduced generalizations of the vorobev - yablonski polynomials for @xmath34 which allow to compute the rational solutions of once more in terms of logarithmic derivatives , @xmath35}(x)}\right\},\ n\in\mathbb{z}_{\geq 1};\hspace{0.5cm}u(x;0,n)=0,\ \ \ u(x ;- n , n)=-u(x;n , n),\ \ n\in\mathbb{z}_{\geq 1}.\ ] ] this approach has been extended to for general @xmath36 by demina and kudryashov @xcite who found in particular the analogues of for , what we shall call _ generalized vorobev - yablonski polynomials _ t)$ ] , @xmath38}(x;\un t)\mathcal{q}_{n-1}^{[n]}(x;\un t)&=&\big(\mathcal{q}_n^{[n]}(x;\un t)\big)^2\bigg\{x-2\mathcal{l}_n\left[2\frac{{{\mathrm d}}^2}{{{\mathrm d}}x^2}\ln\mathcal{q}_n^{[n]}(x;\un t)\right]\label{diffrel}\\ & & \hspace{0.5cm}+2\sum_{k=1}^{n-1}(2k+1)t_{2k+1}\mathcal{l}_k\left[2\frac{{{\mathrm d}}^2}{{{\mathrm d}}x^2}\ln\mathcal{q}_n^{[n]}(x;\un t)\right]\bigg\},\ \ n\in\mathbb{z}_{\geq 1}\nonumber\end{aligned}\ ] ] with @xmath39}(x;\un t)=1 $ ] and @xmath40}(x;\un t)=x$ ] . for fixed @xmath41 and @xmath42 these special polynomials are then used in the construction of the unique rational solutions of , @xmath43}(x;\un t)}{\mathcal{q}_n^{[n]}(x;\un t)}\right\};\hspace{0.5cm}u(x;0,\un t , n)=0,\ \ \ n,\un t , n)=-u(x;n,\un t , n).\ ] ] it is mentioned in @xcite , but not proven , that also @xmath37}(x;\un t)$ ] can be expressed as a schur polynomial . in our first theorem below we shall close this small gap . } _ { 60}$ ] , @xmath44}_{70}$ ] , @xmath45}_{72}$ ] , @xmath46}_{77}$ ] , @xmath47}_{78}$ ] , @xmath48}_{75}$ ] ( from left to right and top to bottom ) . see , for their definition . the symmetry of the pattern is easily explained from the definition of the polynomials . the locations of the outer vertices of the star shaped regions are given in . the various lines that appear in the figures are not straight lines but real analytic arcs defined by the implicit equation . it is quite evident that for @xmath49 there are further subdivisions of the star - shaped region into subregions . , title="fig:",scaledwidth=32.9% ] } _ { 60}$ ] , @xmath44}_{70}$ ] , @xmath45}_{72}$ ] , @xmath46}_{77}$ ] , @xmath47}_{78}$ ] , @xmath48}_{75}$ ] ( from left to right and top to bottom ) . see , for their definition . the symmetry of the pattern is easily explained from the definition of the polynomials . the locations of the outer vertices of the star shaped regions are given in . the various lines that appear in the figures are not straight lines but real analytic arcs defined by the implicit equation . it is quite evident that for @xmath49 there are further subdivisions of the star - shaped region into subregions . , title="fig:",scaledwidth=32.9% ] } _ { 60}$ ] , @xmath44}_{70}$ ] , @xmath45}_{72}$ ] , @xmath46}_{77}$ ] , @xmath47}_{78}$ ] , @xmath48}_{75}$ ] ( from left to right and top to bottom ) . see , for their definition . the symmetry of the pattern is easily explained from the definition of the polynomials . the locations of the outer vertices of the star shaped regions are given in . the various lines that appear in the figures are not straight lines but real analytic arcs defined by the implicit equation . it is quite evident that for @xmath49 there are further subdivisions of the star - shaped region into subregions . , title="fig:",scaledwidth=32.9% ] } _ { 60}$ ] , @xmath44}_{70}$ ] , @xmath45}_{72}$ ] , @xmath46}_{77}$ ] , @xmath47}_{78}$ ] , @xmath48}_{75}$ ] ( from left to right and top to bottom ) . see , for their definition . the symmetry of the pattern is easily explained from the definition of the polynomials . the locations of the outer vertices of the star shaped regions are given in . the various lines that appear in the figures are not straight lines but real analytic arcs defined by the implicit equation . it is quite evident that for @xmath49 there are further subdivisions of the star - shaped region into subregions . , title="fig:",scaledwidth=32.9% ] } _ { 60}$ ] , @xmath44}_{70}$ ] , @xmath45}_{72}$ ] , @xmath46}_{77}$ ] , @xmath47}_{78}$ ] , @xmath48}_{75}$ ] ( from left to right and top to bottom ) . see , for their definition . the symmetry of the pattern is easily explained from the definition of the polynomials . the locations of the outer vertices of the star shaped regions are given in . the various lines that appear in the figures are not straight lines but real analytic arcs defined by the implicit equation . it is quite evident that for @xmath49 there are further subdivisions of the star - shaped region into subregions . , title="fig:",scaledwidth=32.9% ] } _ { 60}$ ] , @xmath44}_{70}$ ] , @xmath45}_{72}$ ] , @xmath46}_{77}$ ] , @xmath47}_{78}$ ] , @xmath48}_{75}$ ] ( from left to right and top to bottom ) . see , for their definition . the symmetry of the pattern is easily explained from the definition of the polynomials . the locations of the outer vertices of the star shaped regions are given in . the various lines that appear in the figures are not straight lines but real analytic arcs defined by the implicit equation . it is quite evident that for @xmath49 there are further subdivisions of the star - shaped region into subregions . , title="fig:",scaledwidth=32.9% ] [ small ] let @xmath50 denote the staircase partition of length @xmath51 . for any @xmath52 the generalized vorobev - yablonski polynomial @xmath37}(x;\un t),x\in\mathbb{c}$ ] defined in equals @xmath53}(x;\un t)=\prod_{k=1}^n\frac{(2k)!}{2^kk!}\,s_{\delta_n}\left(x,0,2 ^ 2t_3,0,2 ^ 4t_5,\ldots,2^{2n}t_{2n+1},0,0,0,\ldots\right),\ \ \ \ t_{2n+1}\equiv -\frac{1}{2n+1}.\ ] ] besides the jacobi - trudi type identity , vorobev - yablonski polynomials can also be expressed as hankel determinants , in fact in @xcite the following hankel determinant representation for the squared polynomial @xmath54 was obtained , [ the1 ] _ n^2(x ) = ( -1)^_k=1^n^2_,j=1^n+1 , x with @xmath55 defined by the generating function @xmath56 = \sum_{j=0}^\infty \mu_j(x ) w^j.\ ] ] in our second theorem we present the analogue of for the generalized vorobev - yablonski polynomial @xmath37}(x;\un t)$ ] . [ fesq ] let @xmath32 and @xmath51 . for any @xmath57 we have the hankel determinant representation @xmath58}(x;\un t)\big)^2=(-1)^{\binom{n+1}{2}}\frac{1}{2^{n}}\prod_{k=1}^{n}\left[\frac{(2k)!}{k!}\right]^2\,\det\big[\mu_{\ell+j-2}^{[n]}(\bt_o)\big]_{\ell , j=1}^{n+1}\ ] ] where we use the abbreviation @xmath59 and the coefficients @xmath60}(\bt_o)\}_{j\in\mathbb{z}_{\geq 0}}$ ] are defined by the generating function @xmath61=\sum_{k=0}^{\infty}\mu_k^{[n]}(\bt_o)w^k,\ \ \ \ \ \ t_j\equiv 0,\ j>2n+1.\ ] ] in fact , the statement of theorem [ fesq ] is the specialization of a more general identity for schur functions ( compare lemma [ lemmadouble ] below ) which in our case reads s__n^2(t_1,0,2 ^ 2t_3,0,2 ^ 4t_5,0, )= 2^n^2s_(n+1)^n(t_1,0,t_3,0,t_5,0,t_7 , ) . here , @xmath62 denotes the rectangular partition with @xmath63 rows of length @xmath64 and the specialization consists in simply setting @xmath65 in analogy to @xcite , we provide a direct application of theorem [ fesq ] . numerical studies carried out in @xcite show that the zeros of generalized vorobev - yablonski polynomials form highly regular and symmetric patterns as can be clearly seen in figure [ stars ] . these patterns in case of the painlev ii equation itself have been first analyzed in @xcite . however , the approach outlined in @xcite starts directly from and not from a lax pair associated with . to be more precise , the identity allows us to localize the roots of the generalized vorobev - yablonski polynomials as @xmath66 by analyzing associated pseudo - orthogonal polynomials . of course in the generalized case these patterns depend on the parameters @xmath67 ( compare @xcite ) ; we shall confine ourselves here to the case of _ higher vorobev - yablonski polynomials _ , namely the case @xmath68 more specifically , we are considering the roots of the rescaled higher vorobev - yablonski polynomials [ roots ] _ n^[n ] = \ { x:_n^[n](n^ x ) = 0 } . these sets admit a discrete @xmath69 rotational symmetry , which follows immediately from theorem [ small ] and the homogeneity ; _ n^[n ] ( x)= ^(n+1 ) _ n^[n](x),= e^. we can provide a partial analytic description for the boundary of the polygons @xmath70 seen in figure [ stars ] which asymptotically contain the sets as @xmath66 . more precisely we have first the following theorem . there exists a compact region @xmath70 in the complex @xmath71plane such that for any @xmath72 the root sets @xmath73}$ ] are contained in a @xmath74-neighborhood @xmath75 of @xmath70 as @xmath76 . the description of the regions @xmath70 is provided in part by theorem [ thm : boundary ] below . first we require given @xmath36 , let @xmath77 denote the unique solution of the algebraic equation @xmath78 which is analytic in the domain x\_k=0 ^ 2nand behaves near @xmath79 as @xmath80 here , the points @xmath81},k=0,\ldots,2n$ ] are the solutions of @xmath82 and form the outer vertices of the regular star - shaped regions shown in figure [ stars ] . [ thm : boundary ] the regions @xmath70 are compact , invariant under the rotations of angle @xmath83 , contain the origin and their boundary @xmath84 consists of branches of the locus in the complex @xmath71-plane described by @xmath85}}=0\ri\}.\ ] ] here @xmath86},k=1,\ldots,2n$ ] are the solutions of the equation @xmath87 where @xmath88 denotes the maclaurin polynomial of degree @xmath89 of the function @xmath90 . moreover @xmath91 is defined in and , and the function @xmath92 is defined by @xmath93 with principal branches for fractional exponents and logarithms . the branches of the real analytic curves specified by @xmath94 of theorem [ thm : boundary ] are plotted as the arcs in figure [ stars ] . perhaps more important than what theorem [ thm : boundary ] above says , is what it does not say . in fact of all the branches of curves defined by we are not able to effectively discern which ones actually form the boundary of @xmath70 . in particular we can not conclude in general that the points belong to @xmath95 . a local analysis ( which we do not propose here but is essentially identical to @xcite ) shows that the angles between consecutive arcs emanating from the points @xmath96}$ ] is @xmath97 . inspection of figure [ stars ] clearly shows that the pattern of roots within @xmath70 is subdivided in subregions . this can be easily _ qualitatively _ understood in terms of the steepest descent analysis ; the so called @xmath98-function of the problem ( see section [ character ] below ) is an abelian integral on a riemann surface of genus @xmath99 on the outside of @xmath70 and of genera @xmath100 inside . in fact we can show that @xmath101 belongs to a region where the genus is @xmath28 and thus it is reasonable to deduce that there are nested regions of higher and higher genus , until the maximum is reached ( @xmath28 ) . these regions are quite evident in figure [ stars ] . in principle the boundaries between these nested regions could be described as well in terms of abelian integrals , but it is beyond the scope of this paper to attempt any such detailed description . we conclude the introduction with a short outline of the upcoming sections . first theorem [ small ] is derived in section [ piihier ] by referring to the kdv and mkdv hierarchies for which we construct a rational tau function in terms of schur polynomials . subsequently an explicit scaling reduction brings us back to the painlev ii hierarchy and theorem [ small ] follows . after that we turn towards theorem [ fesq ] , but opposed to the proof of in @xcite which relied on , identity will follow from schur function identities and theorem [ small ] . in the final section [ character ] we follow largely the logic outlined in @xcite . however we choose not to present any details on the nonlinear steepest descent analysis for the underlying orthogonal polynomials . once the correct inequalities for the @xmath102-function have been verified the asymptotic analysis outside of @xmath70 is almost identical to @xcite , see section [ character ] for further details . all authors are grateful to p. clarkson for useful discussions about this project . is supported in part by the natural sciences and engineering research council of canada . f. b. and m. b. are supported in part by the fonds de recherche nature et technology du qubec . t.b . acknowledges hospitality of sissa , trieste in february 2015 . early stages of the manuscript were carried out while f.b . was a postdoctoral fellow at sissa . the goal of this section is to remind the reader very briefly of the construction of the painlev ii hierarchy as a scaling reduction of the modified korteweg - de vries ( mkdv ) hierarchy , cf . @xcite . in doing so we will en route derive theorem [ small ] . the kdv hierarchy involves the lenard recursion operator @xmath103 = \le(\frac{\pa^3 } { \pa x^3 } + 4u(x)\frac{\pa}{\pa x } + 2u_x(x)\ri ) \l_n[u],\ \ \ \l_0[u]=\frac{1}{2},\ \ \ \l_n[0]=0\ ] ] and its equations are written as the partial differential equations @xmath104,\ \ \ n\in\mathbb{z}_{\geq 0};\ \ \ u = u(\bt_o),\ \ \bt_o=(t_1,0,t_3,0,t_5,\ldots).\ ] ] it is customary , and we will adhere to the custom , to denote the variable @xmath105 by @xmath71 since @xmath106 = u$ ] and hence the first member of the hierarchy above reads simply @xmath107 . in general , the equations of the hierarchy should be viewed as an infinite set of compatible evolution equations for a single function @xmath108 . solution _ of the hierarchy is then a function @xmath109 . a function @xmath110 is called a tau function for the kdv hierarchy if the function @xmath111 solves the hierarchy . we note that multiplication by an arbitrary constant ( in @xmath71 ) of @xmath112 gives another tau function . the solutions of the kdv equation rational in @xmath71 for all values of @xmath113 ( and for all higher times @xmath114 ) and vanishing at @xmath79 were completely characterized in @xcite ; they all belong to the countable union of orbits flowing out of initial data of the form u_n(x,0,0, )= , n_0 . the corresponding tau functions @xmath115 were obtained explicitly in @xcite in terms of wronskians of certain polynomials in @xmath116 . up to normalization and re - parametrization these wronskians coincide with schur polynomials associated to staircase partitions evaluated at the odd times , namely [ adler_moser ] _ n(_o ) = s__n(t_1,0,2 ^ 2t_3,0,2 ^ 4 t_5 , ) , u_n(_o ) = 2 _ n(_o ) , where @xmath50 denotes the staircase partition of length @xmath51 . moreover , it can be shown ( cf . @xcite ) that these are the only schur polynomials that give kdv tau functions when all even times are set to zero . the particular rescaling @xmath117 is used in in order to correct the normalization so that the coefficients in are as indicated . the _ modified _ kdv ( mkdv ) hierarchy is defined in terms of a new dependent variable @xmath118 which is related to @xmath119 via the miura transformation @xmath120 where the choice of signs is arbitrary . more is true : if @xmath121 satisfies @xmath122 , then the new function @xmath123 is a _ different _ solution of the kdv hierarchy ( and vice versa ) ; this is an example of a bcklund transformation . inserting into yields a new set of evolution equations @xmath124 \stackrel{\eqref{lenard}}{= } \le(\frac{\pa^3 } { \pa x^3 } - 4(\pm v_x+v^2 ) \frac{\pa}{\pa x } - 2(\pm v_{xx}+2vv_x)\ri ) \l_n\big[\mp v_x - v^2\big]\\ & = \le(\frac \pa{\pa x } \pm 2v\ri ) \frac \pa{\pa x}\le(\frac \pa{\pa x } \mp 2v\ri ) \l_n\big[\mp v_x - v^2\big].\end{aligned}\ ] ] this can be rewritten as follows @xmath125\ ] ] or equivalently @xmath126}_{\mathfrak q_{n}^{(\pm)}[v ] } \bigg\ } = 0.\ ] ] we now notice that the two expressions @xmath127=\frac{1}{2 } \le ( \mp \frac \pa{\pa x } + 2v\ri ) \le[\int { { \mathrm d}}x \le(\frac \pa{\pa x}\pm 2v\ri ) \frac \pa{\pa x}\le(\frac \pa{\pa x } \mp 2v\ri)\ri]^{n}\ ] ] define _ the same _ differential polynomial in @xmath121 since the right hand side is clearly invariant under the map @xmath128 . thus we can simply write @xmath129= \mathfrak q_{n}^{(-)}[v]=\mathfrak q_{n}[v],\ ] ] omitting the reference to the choice of sign . we now want to conclude that the expression @xmath130 $ ] vanishes identically ; the two equations in below are simply stating that @xmath131 $ ] is a joint solution of the two ordinary differential equations @xmath132 . thus @xmath130 $ ] should be the in the null - space of both equations @xmath133 ; as long as @xmath121 is not identically zero ( which is an un - interesting situation ) , the only function in both null - spaces is the null function and hence @xmath130\equiv 0 $ ] . thus we have concluded that if @xmath119 is a solution of the kdv hierarchy and @xmath121 is related to @xmath119 by , then @xmath121 must solve the hierarchy of equations indicated below and named _ mkdv hierarchy _ , @xmath134,\ \ n\in\mathbb{z}_{\geq 0};\ \ \ \ v = v(\bt_o).\ ] ] the choice of signs is irrelevant , since the right hand side ( as noted above ) yields the same differential polynomial in @xmath121 . let us now return to our special situation for which we fix @xmath135 [ propmiura ] for @xmath42 define the two functions @xmath136 with some fixed branch for the logarithm . we then have the miura relation @xmath137 a proof of can be found in appendix [ appmiura ] . in view of proposition [ propmiura ] we note that the two functions @xmath138 satisfy precisely the miura relation with the choice of the minus sign , namely @xmath139 . since @xmath140 gives a tau function for the kdv hierarchy it follows that @xmath121 satisfies the hierarchy for @xmath141 . summarizing the function @xmath142 satisfies the mkdv hierarchy in the form @xmath143,\ \ 0\leq n\leq n.\ ] ] recalling the homogeneity property we see that @xmath144 obeys a simple scaling invariance which will allow us to reduce the partial differential equations to an ordinary differential equation ; we carry out a _ scaling reduction _ : view @xmath145 as a function in the variables @xmath146 and @xmath147 . 2 . by homogeneity , it follows that @xmath148 is a function of the form @xmath149 and @xmath150 depends on the new " variables @xmath151 3 . substituting into the left hand side of with @xmath152 , we find @xmath153.\ ] ] 4 . next we evaluate , at @xmath154 and compare the result to , @xmath155=v+\sum_{j=0}^{n-1}(2j+1)t_{2j+1}\frac{\partial v}{\partial t_{2j+1}}.\ ] ] 5 . since @xmath156 and @xmath157 , can be rewritten with the help of , @xmath158-xv-\sum_{j=1}^{n-1}(2j+1)t_{2j+1}\left(\frac{\partial}{\partial x}+2v\right)\l_j\big[v_x - v^2\big]\right\}=0.\ ] ] equation is an ordinary differential equation for the function @xmath159 in which @xmath160 appear as _ parameters_. since @xmath161 integration in yields with @xmath162 . recall @xcite that @xmath163 is necessary to have a rational solution to and for all integer values of @xmath164 there exists a unique rational solution which can be obtained from the trivial solution for @xmath165 by bcklund transformations . therefore we have the following for @xmath42 the unique rational solution of the painlev ii hierarchy is @xmath166 and we have the identity @xmath167}(x;\un t)=\prod_{k=1}^n\frac{(2k)!}{2^kk!}s_{\delta_n}\big(x,0,2 ^ 2t_3,0,2 ^ 4t_5,\ldots,2^{2n}t_{2n+1},0,0,0,\ldots),\ \ x\in\mathbb{c}.\ ] ] it is easy to see that the lhs of is a rational solution to by the scaling reduction - . by the uniqueness of the rational solutions of the painlev ii hierarchy we have @xmath167}(x;\un t ) = c_{n , n}(\un t)s_{\delta_n}\big(x,0,2 ^ 2t_3,0,2 ^ 4t_5,\ldots,2^{2n}t_{2n+1},0,0,0,\ldots\big),\ ] ] with an @xmath71-independent factor @xmath168 . however , to leading order , @xmath169 where @xmath170 denotes the product of the hook - lengths of @xmath171 ( cf . since @xmath172 and @xmath37}(x;\un t)$ ] is a monic polynomial of degree @xmath173 , the claim follows . we will appeal to certain identities satisfied by symmetric functions which can be found , for instance , in @xcite . first let us start with the following lemma . [ lemmadouble ] the symmetric polynomial identity @xmath174 holds , where @xmath175 stands for the rectangular partition with @xmath63 rows of length @xmath64 and @xmath50 is the staircase partition . the schur polynomial @xmath140 can be written in terms of the projective schur polynomial @xmath176 labeled by the same partition @xmath177 for a proof of this identity see @xcite , @xmath178 3.8 , example @xmath179 , page @xmath180 , and also @xcite , lemma v.4 . second , for a strict partition @xmath171 , i.e. @xmath181 , we have @xcite , theorem @xmath182 , @xmath183 with @xmath184 denoting the double of the partition @xmath171 which is defined via its frobenius characteristics , @xmath185 combining and , @xmath186 where we have used homogeneity in the last step . this concludes the proof . we are now ready to derive theorem [ fesq ] by referring to and lemma [ lemmadouble ] . let @xmath41 and @xmath187 this gives us @xmath188}(x;\un t)\big)^2&\stackrel{\eqref{schurid}}{=}&\prod_{k=1}^n\left[\frac{(2k)!}{2^kk!}\right]^2\,s_{\delta_n}^2\big(2 ^ 0t_1,0,2 ^ 2t_3,0,2 ^ 4t_5,\ldots,2^{2n}t_{2n+1},0,0,0,\ldots)\\ & = & \frac{1}{2^n}\prod_{k=1}^n\left[\frac{(2k)!}{k!}\right]^2s_{(n+1)^n}(t_1,0,t_3,0,t_5,\ldots , t_{2n+1},0,0,0,\ldots)\\ & = & \frac{1}{2^n}\prod_{k=1}^n\left[\frac{(2k)!}{k!}\right]^2s_{(n)^{n+1}}(\bt_o)=\frac{1}{2^n}\prod_{k=1}^n\left[\frac{(2k)!}{k!}\right]^2\det\big[\mu_{n-\ell+j}^{[n]}(\bt_o)\big]_{\ell , j=1}^{n+1}\\ & = & ( -1)^{\binom{n+1}{2}}\frac{1}{2^n}\prod_{k=1}^n\left[\frac{(2k)!}{k!}\right]^2\det\big[\mu_{\ell+j-2}^{[n]}(\bt_o)\big]_{\ell , j=1}^{n+1},\end{aligned}\ ] ] where we used that for the transposed partition @xmath189 , @xmath190 and that the schur polynomials of rectangular partitions are hankel determinants . [ corfreak ] let @xmath191 and @xmath192 as in . introducing the notation @xmath193_{j , k=1}^{n+1},\ \ n,\ell\in\mathbb{z}_{\geq 0},\ ] ] we have the hankel determinant identity @xmath194 note that @xmath195_{j , k=1}^n=(-1)^{n-1}\det\big[h_{j+k}(\bt_{o})\big]_{j , k=1}^n,\\ s_{(n)^{n+1}}(\bt_{o})&=&\det\big[h_{n - j+k}(\bt_{o})\big]_{j , k=1}^{n+1}=(-1)^n\det\big[\mu_{j+k-2}(\bt_{o})\big]_{j , k=1}^{n+1},\end{aligned}\ ] ] and since @xmath196 , the stated identity follows from . identity in corollary [ corfreak ] _ does not hold _ if any of the even - index times is nonzero . the logic we are following here is identical to @xcite . the square of the polynomials @xmath37}(x)$ ] is proportional to a hankel determinant @xmath197}(\bt_o)\big]_{j , k=1}^{n+1}\ ] ] of the moments @xmath198}(\bt_o)$ ] , which can alternatively be written as _ k^[n](_o ) = _ sz^ke^- ; _ o=(x,0,0, ,0,t_2n+1,0,0,0 , ) , t_2n+1=- where @xmath199 denotes the unit circle traversed in counterclockwise direction . it is then a well - known fact that @xmath200 if and only if the riemann hilbert problem [ master ] has no solution , or equivalently , if and only if the @xmath64-th monic orthogonal polynomial for the weight @xmath201 does not exist . in view of the scaling @xmath202 in we also perform a scaling @xmath203 so that we arrive at the following riemann hilbert problem with a varying exponential weight . [ master ] suppose @xmath204 is a smooth jordan curve which encircles the origin counterclockwise . let @xmath205 denote the @xmath206 matrix - valued piecewise analytic function which is uniquely characterized by the following three properties . 1 . @xmath207 is analytic for @xmath208 2 . given the orientation of @xmath209 , the limiting values @xmath210 from the @xmath211 and @xmath212 side of the contour exist and are related via the jump condition @xmath213 3 . the function @xmath207 is normalized as @xmath214 , @xmath215 then we have , compare @xcite , the zeros of the scaled vorobev - yablonski polynomials @xmath216}(n^{\frac { 2n}{2n+1 } } x)$ ] coincide with the values of @xmath71 for which the problem [ master ] is _ not _ solvable . in principle an asymptotic analysis of the problem [ master ] as @xmath66 is possible using the deift - zhou steepest descent analysis @xcite , and the zeros will be located asymptotically in terms of appropriate theta functions as in @xcite . however here we simply want to prove the _ absence _ of zeros outside of a certain compact region @xmath70 and , _ en route _ , give a partial characterization of the boundary @xmath95 . for a more comprehensive analysis which is only marginally different from the present situation we refer to @xcite ; here we shall just remind the reader that the method requires the construction of an appropriate function , called customarily `` the @xmath98-function '' . in case of the problem [ master ] the @xmath98-function is a priori expressible in the form @xmath217 where , in general , @xmath218 is an appropriate polynomial of the indicated degree . the ansatz is explained in the paragraph " construction of the @xmath98-function of @xcite and the discussion there can be applied almost verbatim here . from we see that the @xmath98-function is an abelian integral on the riemann surface of the square root of @xmath218 ; depending on the number of odd roots , this surface has a genus that can range from a minimum of @xmath99 ( if there are only two simple roots in @xmath218 ) to a maximum of @xmath28 ( if all the roots are simple ) . subsequently the deift - zhou analysis shows that _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ if @xmath71 is such that the genus of the above riemann surface is zero and the @xmath98-function satisfies the appropriate inequalities ( recalled below ) , then the riemann - hilbert problem [ master ] is eventually _ solvable _ for sufficiently large @xmath64 . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ therefore our strategy is as follows ; we postulate a genus zero ansatz for the @xmath98-function in ; the algebraic requirements are easily verified , but the required inequalities are not always verified . we shall then find where the inequalities fail , and hence where the roots are asymptotically confined . for the concrete construction of the @xmath98-function in the genus zero region we follow the logic outlined in @xcite . we seek a function @xmath219 of the form @xmath220\ \ \ \ \textnormal{deg}(p)=2n , \label{y1}\ ] ] where @xmath221 is defined and analytic off the oriented branch cut @xmath222 which connects the points @xmath223 . the precise location of @xmath224 shall be discussed in section [ bcut ] below , for now we require that @xmath225 satisfies the two conditions @xmath226 using simple algebra , we directly obtain the conditions imply that @xmath227 and @xmath71 are related via @xmath228 and the polynomial @xmath229 is uniquely determined as @xmath230 where @xmath231 is the maclaurin polynomial of degree @xmath89 of the function @xmath232 . observe that the condition @xmath233 implies that @xmath229 is monic and from the behavior at @xmath234 we find that @xmath235 writing @xmath236 with a polynomial @xmath237 of degree at most @xmath238 and reading at @xmath239 , we get @xmath240 and thus @xmath241 which gives . in order to deduce , we recall that @xmath229 is monic , i.e. we must have @xmath242 that is . [ propstarcorners ] the branch points of the map @xmath91 , defined implicitly by , coincide with the values of @xmath243 for which two zeros of @xmath244 overlap with the branch points @xmath245 . we have to evaluate the condition @xmath246 ; using this amounts to @xmath247 thus the condition determining the coincidence of a zero of @xmath244 with @xmath248 is [ ss1 ] a^2n+1= ( 2nn ) . on the other hand the map has a branch point where @xmath249 , which gives exactly . for further steps it will prove useful to define the effective potential , @xmath250 which in the given situation can be evaluated explicitly , @xmath251 and all branches in are principal ones such that @xmath252 as @xmath214 . [ lemmader ] given , we have for @xmath36 , ( z;a ) = - , z\. the jump of @xmath253 equals @xmath254 on a contour that extends to infinity . hence @xmath255 has no jump on a contour which extends to infinity . along @xmath256 we have @xmath257 . since @xmath253 vanishes at the branch point @xmath258 ( and is constant @xmath259 at @xmath260 on the two sides ) we deduce that @xmath255 must be zero at @xmath261 . also ( compare section [ bcut ] below ) , ( z;a ) = -+(1),&z0 + ( 1),&z . thus the ratio of the proposed expression for @xmath255 is bounded at @xmath248 , analytic across the cut and bounded at @xmath262 with limit @xmath263 at @xmath234 . the proposition now follows from liouville s theorem . the potential is related to the @xmath98-function by [ phitog ] ( z)= 1 2 ( ( z;x , n)-(z;a)+ ) , z\ where the constant @xmath264 ( _ modified robin constant _ ) is defined by the requirement that @xmath265 as @xmath266 . the relevant inequalities for @xmath267 are more conveniently expressed directly as inequalities for the effective potential . in terms of the latter , the following properties of the effective potential are equivalent to the existence of the @xmath98-function and characterize @xmath253 ( the proof of these statements is simple if not already obvious ) 1 . near @xmath234 the effective potential has the behavior @xmath268 while near @xmath262 it behaves as ( z ) = -2z + o(1).[phiatinfty ] 2 . analytic continuation of @xmath253 in the domain @xmath269 yields the same function up to addition of _ imaginary _ constants ; in particular , the analytic continuation of @xmath253 around a large circle yields @xmath270 ; 3 . for each component @xmath271 of @xmath272 we have that , @xmath273 4 . the effective potential @xmath274 with @xmath91 as in and is a harmonic function in @xmath269 . . 5 . * inequality 1 . * the sign of @xmath276 on the left and right of @xmath272 is _ negative_. 6 . * inequality 2 . * we can continuously deform the contour of integration @xmath209 to a simple jordan curve ( still denoted by @xmath209 ) such that @xmath277 and such that @xmath278 . note that @xmath253 and @xmath267 are both related to the antiderivative of the differential @xmath279 which is defined on a riemann surface @xmath280 of genus between @xmath99 and @xmath28 . since @xmath281 vanishes along @xmath272 , it also follows that @xmath272 is a subset of its zero level set ; therefore , @xmath272 consists of an union of arcs defined locally by the differential equation @xmath282 . the following proposition appeared in @xcite but applies also to the present situation . [ cutbg0 ] the effective potential @xmath283 has the following properties : 1 . the function @xmath284 is defined modulo a sign depending on the determination of @xmath221 . the zero - level set @xmath285 is well defined independent of the determination of the square root in ( 1 ) and invariant under the reflection @xmath286 . 3 . for @xmath287 sufficiently large there are two smooth branches of the zero - level set @xmath288 which connect @xmath223 and which are symmetric under @xmath286 . statements @xmath289 and @xmath290 follow just as in ( @xcite proposition 3.5 ) , for @xmath291 we note that , as @xmath292 @xmath293 and the limit is uniform on compact subsets of the riemann sphere not containing @xmath234 . the remaining logic is now as in @xcite . suppose that @xmath294 is sufficiently large and thus proposition [ cutbg0 ] applies . we claim that @xmath222 is the branch in point @xmath291 above which intersects the positive half ray @xmath295 ( by deformation this fixes the branch cut for all @xmath296 ) . in order to see this , recall from , as @xmath297 , @xmath298 but this requires in that @xmath299 as @xmath300 . simultaneously requires @xmath301 near @xmath262 and hence @xmath252 as @xmath302 . hence , the determination of the square root in has to change on the positive half ray , i.e. @xmath222 is as claimed . since the quadratic differentials @xmath303 are of the type studied by jenkins and spencer @xcite , that is , of the form @xmath304 with @xmath305 a rational function , we can follow some of the reasoning which was already explained in @xcite . * preliminaries . * define the set @xmath306 to consist of the union of the second order poles and all _ critical trajectories _ , i.e. , all solutions of @xmath307 that issue from each of the zeros and simple poles of @xmath305 ; the latter are absent in our case . the zeros are at @xmath261 and at the @xmath25 pairs @xmath308 which are the roots of the even polynomial @xmath229 . also @xcite , there are _ @xmath309 _ branches of @xmath306 issuing from each of the points of order @xmath310 of @xmath305 , @xmath311 ( the case @xmath312 corresponds to simple poles , and all others to zeroes ) . we are interested in the connected components of @xmath313 and a simple argument in analytic function theory ( see @xcite ) shows that each _ simply _ connected component @xmath314 is conformally mapped by @xmath253 into a half - plane or a vertical strip @xmath315 ; each _ doubly _ connected component @xmath316 is mapped to an annulus ( or a punctured disk ) @xmath317 by @xmath318 where @xmath319 and @xmath209 is a closed simple contour separating the two boundary components of @xmath316 . it is also shown in @xcite that there are no other possibilities for the topology of the connected components @xmath314 . moreover , there is a one - to - one correspondence between annular domains ( including the degenerate case of a punctured disk ) and free homotopy classes of simple closed contours @xmath209 for which @xmath320 . in our case there is only one such class corresponding to a loop encircling the origin , and hence only one annular domain which we denote by @xmath321 ( which is actually a punctured disk ) . by construction , @xmath322 which shows that @xmath262 is at the center of a conformal punctured disk via the conformal map @xmath323 . moreover the level sets @xmath324 are foliating a region around @xmath262 in topological circles if @xmath325 is sufficiently large . thus none of the hyperelliptic trajectories issuing from @xmath326 can `` escape '' to infinity ; they either connect to @xmath234 or amongst each other . suppose @xmath327 is the infimum of the @xmath328 for which @xmath329 is smooth ; this means that @xmath330 contains at least one zero of @xmath331 ( by symmetry , it contains then two zeros in our situation ) . the annular ( punctured disk ) domain @xmath321 is then ( see figure [ kinfty ] ) k_= . we denote also @xmath332 , which is a simply connected , symmetric region containing the origin . * necessary and sufficient condition for the correct inequalities in genus zero . * we argue that we need to have @xmath333 . to put it differently , the `` first encounter '' of the level sets @xmath329 as @xmath334 decreases must be with the two branch points @xmath261 rather than any of the zeros @xmath335 . we shall then verify that this occurs for @xmath336 large enough . + * sufficiency*. suppose now that @xmath333 and thus @xmath337 and @xmath338 . then the simple , closed loop @xmath339 is separated into two components by @xmath261 and each of them is an hyperelliptic trajectory . we know that there must be three trajectories from each @xmath261 and two of them are already accounted for and form the boundary of @xmath340 ( see fig . [ kinfty ] ) ; thus the third trajectory is entirely contained in @xmath340 , which is compact . now let us turn our attention to @xmath340 ; the points @xmath341 for @xmath342 . in @xmath340 each branch of @xmath343 is single valued ( the branch points are on the boundary of @xmath340 ) . only one of the two branches of @xmath343 has the behavior @xmath344 ; integrating this branch from @xmath345 coincides with @xmath346 in @xmath340 . the value of the sign of @xmath347 in the interior of @xmath340 close to the boundary @xmath340 determines which of the two parts of @xmath348 is the branch cut @xmath272 : this is the part which has @xmath349 on _ both _ sides ( ie . in @xmath340 and @xmath321 ) . thus @xmath347 is continuous but not harmonic on @xmath272 , while on @xmath350 it is continuous and harmonic . we still need to show that there is a path connecting @xmath261 and which lies within the region @xmath351 . [ kinfty ] this follows from the topological description of the possible regions @xmath314 discussed in the paragraph `` preliminaries '' . indeed let @xmath352 be the region containing the arc @xmath353 where @xmath253 is conformally one - to - one . from the discussion of signs thus far , this is either a half - plane @xmath354 or a strip @xmath355 ( the only annular domain is @xmath321 ) . the two points @xmath261 are mapped on the imaginary axis @xmath356 ; thus there is a path connecting @xmath357 to @xmath358 in the @xmath359-plane which lies in the left half plane . the pre - image of this path in the @xmath360 plane connects thus @xmath261 and @xmath361 restricted to the interior points of this path is strictly negative . * necessity . * if @xmath362 then @xmath363 . the trajectories issuing from @xmath261 all belong to the zero level set of @xmath347 . none of them can connect to any of the zeros @xmath364 , and thus they either connect to each other or to the origin . since the sign of @xmath361 changes @xmath365 times around @xmath234 , they all must go to the origin and thus there is no possibility of deforming the contour of integration so that it contains the branch cut @xmath272 and avoids the origin . * sufficient condition for the correct inequalities in highest genus . * we work with the same general setup as in the previous case . now the quadratic differential is of the form on the right in . suppose that @xmath218 there has all simple roots @xmath366 ( the roots come clearly in pairs of opposite signs ) . we claim that a sufficient condition for the fulfillment of the inequalities is that * all * branch points @xmath367 lie on @xmath368 . in this case @xmath339 is broken into @xmath369 arcs ( see for example figure [ maxgenus ] ) . there is only one branch of @xmath343 that behaves as @xmath370 near @xmath234 ; the integral of this branch with base point @xmath371 is single valued in @xmath372 because the region contains no branch points and the residue of @xmath343 at @xmath234 vanishes ; this integral then defines @xmath253 ( and @xmath347 ) within @xmath340 . the level curves of @xmath373 that issue from @xmath374 and do not connect to other branch points must connect to the origin because @xmath373 changes sign exactly @xmath369 times when going around the origin . the regions where @xmath253 is now one - to - one within @xmath340 are @xmath369 half - planes because their boundary has only one connected component . necessarily in @xmath365 of them @xmath375 and @xmath365 of them @xmath376 . the arcs of @xmath377 bounding the three regions where @xmath375 are the cuts and the other are simply zero level sets separating regions where @xmath373 has opposite signs . the possibility of connecting two branch points that are connected by an arc of these level sets follows exactly by the same argument used in the previous paragraph . * occurrence of the necessary / sufficient conditions . * by point * ( 3 ) * of proposition [ cutbg0 ] , for @xmath227 ( hence , @xmath71 ) large enough there is a smooth branch of the zero level set of @xmath373 that connects @xmath261 ; by symmetry , there is another one and thus the third branch of the level set that issue from @xmath261 must go to zero ( we have seen that there is no branch that extends to infinity ) . the remaining roots of @xmath229 all tend to zero as @xmath378 ( which is easily seen from the explicit expression ) . thus they must fall within the region @xmath340 . then the necessary condition in genus @xmath99 is fulfilled . for the case of maximal genus , the occurrence of the sufficient condition is contained in proposition [ propgmax ] . + * the discriminant locus*. by discriminant locus we refer to the boundary of the locus in the @xmath71plane where the inequalities for the genus zero ansatz fail . this is the boundary of a region @xmath70 ; from the discussion above it follows that the inequalities are preserved under a deformation in @xmath71 , until failure occurs exactly when one of the zeros of @xmath229 falls on the branch cut @xmath256 connecting @xmath261 , and hence , by symmetry , one also intersecting the opposite arc . we know that this does not happen for @xmath379 sufficiently large and hence the discriminant must be a bounded set . in order to detect the occurrence of the situation above it is _ necessary _ ( but not sufficient ) that @xmath380 for some @xmath381 , i.e. , one of the saddle points of @xmath347 lies on the zero - level set ; p_n z_n= \ { x : z : ( ( z;a(x ) ) ) = 0 , _ zz(z;a(x ) ) = 0 } [ condzero ] . the set @xmath94 is clearly closed , and thus @xmath95 must be compact ( since we know already it is bounded ) . however , the set @xmath94 is strictly larger than @xmath95 ; indeed it describes the situation where any of the saddle points of @xmath373 intersect any branch of the zero level set ; the zero level set contains several branches besides the branch cut and hence the set @xmath94 in describes also all these `` fake '' situations . a detailed analysis for arbitrary @xmath25 seems unwieldy . we shall attempt below only a partial study of the case @xmath34 in appendix [ outer23 ] , where we show that the points do indeed belong to the boundary of the regions @xmath70 for @xmath34 . however the set @xmath94 is easily drawn and the results are displayed in figure [ stars ] together with the roots of some higher polynomials . the result of this discussion is the following theorem ; the roots @xmath382}$ ] of the polynomials @xmath216 } \le ( n^{\frac { 2n}{2n+1 } } x\ri)$ ] lie all within an arbitrarily small neighborhood of a compact region @xmath70 as @xmath383 ; the boundary of this region consists entirely of a finite union of real - analytic arcs in the @xmath71 plane satisfying the condition . the condition is spelt out in more detail in the statement of theorem [ thm : boundary ] , which is henceforth proved as well . a careful consideration should allow also to prove that the region is simply connected . it is also relatively simple to show that @xmath101 belongs to the interior of this region ( see proposition [ propgmax ] below ) . the set @xmath94 in contains the points @xmath71 for which one pair of roots of @xmath229 coincides with the branch points @xmath261 ; these points are easily computed and are precisely the @xmath365 points in . however we can not positively conclude for general @xmath25 that they are on the boundary of @xmath70 , although this is quite evident from the numerics . also , the detailed shape of @xmath95 , beyond the easily established discrete @xmath69 symmetry , is hard to describe in more detail ; for example it is not obvious how to conclude that it consists of @xmath369 smooth arcs for @xmath384 , as the figure [ stars ] clearly shows . we find it however already sufficiently interesting that we can narrow down the boundary of @xmath385 as a subset of a simple set of equations , although we can not completely describe it . in a small vicinity of @xmath101 , we have @xmath386 with @xmath387 for @xmath388 . as in @xcite , the branch points @xmath389 are partially determined through , and in addition through boutroux type conditions @xmath390 the latter are imposed on the hyper elliptic curve @xmath280 which is obtained by crosswise gluing together two copies of @xmath391 with @xmath392\cup[a_{2n+1}^+,a_1 ^ -]\cup\bigcup_{k=1}^n[a_{2k}^-,a_{2k+1}^-]$ ] . solvability of the resulting system for @xmath393 would now follow as in @xcite , but here we are only interested in the case @xmath101 . [ propgmax ] for @xmath101 the @xmath98function is obtained from , using @xmath394 which is defined and analytic off @xmath395 with the branch points @xmath396 . * local behavior . * near @xmath234 we have y(z ) = -1 2 ( 1 + o(z^4n+2 ) ) and near infinity clearly @xmath397 . note that the determination of the root near @xmath234 is the opposite . * boutroux condition . * we have _ a_j,0^a_j+1,0 ( r(z))^ = _ a_j,0^a_j+1,0 ^-2n-1 ( r(z))^ = - _ a_j+1,0^a_j+2,0 ( r(z))^ , and thus it is sufficient to verify the boutroux condition @xmath398 for a specific @xmath399 . this condition guarantees that all branch points lie in the zero level set of @xmath347 . but for @xmath400 it follows immediately that the integral is imaginary using the schwartz symmetry . * connectedness of the level curves . * first of all the set @xmath401 in @xmath402 consists of one connected component alone ; this is so because there are no saddle points and if there were two or more connected components , there would have to be a saddle point in the region bounded by them . we shall now verify that the level curves satisfy the necessary and sufficient conditions specified in section [ gfunctdelta ] . the critical trajectories must 1 . connect all @xmath369 branch points 2 . obey the @xmath403 symmetry because of obvious symmetry . a simple counting then shows that the only possibility is that exactly one trajectory from each branch point ( in fact a straight segment ) connects the branch points to @xmath99 because the sign of @xmath347 changes @xmath369 times around a small circle surrounding the origin . the other two trajectories must then connect the branch points . this is depicted in figure [ maxgenus ] . the discussion on the necessary and sufficient condition for the correct inequalities is now as explained in ( @xcite , section 3.1 ) . in the case of @xmath101 ; the red arcs are the branch cuts of the the @xmath98function . the cases presented correspond to @xmath34 ( left / right ) . also indicated is the foliation by the trajectories @xmath329 of the region @xmath321 ( see section [ gfunctdelta ] ) . the shaded ( cyan ) regions indicate where @xmath349 ; all the trajectories that issue from the branch points constitute the set @xmath404 . , title="fig:",scaledwidth=30.0% ] in the case of @xmath101 ; the red arcs are the branch cuts of the the @xmath98function . the cases presented correspond to @xmath34 ( left / right ) . also indicated is the foliation by the trajectories @xmath329 of the region @xmath321 ( see section [ gfunctdelta ] ) . the shaded ( cyan ) regions indicate where @xmath349 ; all the trajectories that issue from the branch points constitute the set @xmath404 . , title="fig:",scaledwidth=30.0% ] we draw the reader s attention to the various notations used in this section , = ( t_1,t_2,t_3,t_4 , ) , _ o= ( t_1,0,t_3,0,t_5 , ) , _ o= ( t_1 , 0,2 ^ 2 t_3 , 0 , 2 ^ 4 t_5 , ) , t_j . let @xmath405 and introduce @xmath406_{j , k=1}^n,\ \ \delta_{0,\ell}(\bt)\equiv 1\ ] ] where @xmath199 denotes the unit circle traversed in counterclockwise orientation . recalling we see that @xmath407\bigg|_{w=0}=h_k(\bt)\ ] ] and thus with , @xmath408 in particular , by lemma [ lemmadouble ] , we know that for the special value @xmath409 we have the identity @xmath410 next , let @xmath411 be the monic orthogonal polynomials associated with the measure @xmath412 it is well known @xcite that the matrix @xmath413 satisfies a riemann hilbert problem , i.e. @xmath414 is analytic for @xmath415 and we have the conditions @xmath416 [ satoid ] the following identities hold for the hankel determinants @xmath417 . @xmath418)}{\delta_{n , \ell}(\bt ) } = ( -1)^n\big(\gamma_{n,\ell}(z)\big)_{11 } \ \qquad \ \ \frac{\delta_{n,\ell-1 } ( \bt+[z])}{\delta_{n , \ell}(\bt ) } = ( -1)^n\big(\gamma_{n,\ell}(z)\big ) _ { 22 } \label{satodelta}\ ] ] and @xmath419 where @xmath420 $ ] denotes the infinite vector of components @xmath421 , i.e. @xmath422=\left(t_1\mp z , t_2\mp\frac{z^2}{2},t_3\mp\frac{z^3}{3},\ldots\right).\ ] ] the two identities in follow simply by inspection of the expression . as for the identities , the proof follows from heine s formula for the orthogonal polynomials and the observation that @xmath423)\big ] = w^{\ell-1 } ( w - z ) \exp\big[\vartheta(w ; \bt)\big],\ \ \ \ w^{\ell } \exp\big[\vartheta(w ; \bt + [ z])\big ] = \frac{w^{\ell+1}}{w - z } \exp\big[\vartheta(w ; \bt)\big].\ ] ] indeed , we have @xmath424)=\det\big[\mu_{\ell+j+k-1}(\bt-[z])\big ] _ { j , k=1}^n= \frac 1{n ! } \oint_{s^n } \prod _ { j < k } ( w_j - w_k)^2 \prod_{j=1}^n w_j^{\ell } \exp\big [ \vartheta(w_j ; \bt -[z])\big]\frac{{{\mathrm d}}w_j}{2\pi{\mathrm{i}}}\\ = \frac 1{n ! } \oint_{s^n } \prod_{j < k } ( w_j - w_k)^2 \prod_{j=1}^n ( w_j - z ) { { \mathrm d}}\nu_{\ell}(w_j)= ( -1)^n\det\begin{bmatrix } \mu_{\ell } & \cdots & \mu_{\ell+n}\\ \vdots & & \vdots\\ \mu_{\ell+n-1 } & & \mu_{\ell+2n-1}\\ 1 & \cdots & z^n \end{bmatrix}= ( -1)^np_{n,\ell}(z ) \delta_{n , \ell}(\bt ) \end{aligned}\ ] ] where we used the well - known representation of orthogonal polynomials in terms of moment determinants ( see , e.g. proposition 3.8 in @xcite ) . the second identity can be found in @xcite , but we can give here a direct derivation using andreief s identity @xcite . recall that @xmath425 . then @xmath426_{j , k=1}^n \det\big[w_j^{k-1}\big]_{j , k=1}^n \prod_{j=1}^{n } \frac { { { \mathrm d}}\nu_{\ell}(w_j)}{w_j - z}\nonumber\\ = \oint_{s^n } \det \big[w_j^{k-1 } \big]_{j , k=1}^n\det \bigg[\frac{w_j^{k-1}}{w_j - z}\bigg]_{j , k=1}^n\prod_{j=1}^{n } { { { \mathrm d}}\nu_{\ell}(w_j ) } \label{132}\end{aligned}\ ] ] multi - linearity allows us to replace the monic powers in the first determinant by the monic orthogonal polynomials @xmath427 , so that we obtain @xmath428_{j , k=1}^n\det \bigg[\frac{w_j^{k-1}}{w_j - z } \bigg]_{j , k=1}^n\prod_{j=1}^{n } { { { \mathrm d}}\nu_{\ell}(w_j)}.\ ] ] now , in the second determinant we can subtract to the columns @xmath429 the multiple @xmath430 of the first column , thus obtaining @xmath431_{j , k=1}^n \det\begin{bmatrix } \frac{1}{w_1-z } & \frac{w_1-z}{w_1-z } & \cdots & \frac{w_1^{n-1}-z^{n-1}}{w_1-z}\\ \vdots & \vdots & & \vdots\\ \frac{1}{w_n - z } & \frac{w_n - z}{w_n - z } & \cdots & \frac{w_n^{n-1}-z^{n-1}}{w_n - z } \end{bmatrix } \prod_{j=1}^n{{\mathrm d}}\nu_{\ell}(w_j ) \ ] ] using now andreief s identity we obtain @xmath432 but due to orthogonality the matrix above has the following structure @xmath433 however @xmath434 and therefore @xmath435 ) = ( -1)^{n+1 } \frac{\delta_{n-1,\ell}(\bt)}{\delta_{0,\ell}(\bt)}\oint_s p_{n-1,\ell}(w)\frac{{{\mathrm d}}\nu_{\ell}(w ) } { w - z}=(-1)^n\delta_{n,\ell}(\bt)\big(\gamma_{n,\ell}(z)\big)_{22}.\ ] ] consider the following matrix valued function @xmath436 a direct inspection using the jumps of @xmath437 shows that this matrix has no jumps on the contour @xmath438 , however an essential singularity at @xmath234 due to the presence of the exponentials . we can thus compute the contour integral below in two ways . first by evaluation as a residue at infinity ; @xmath439 where the @xmath440 indicates expressions which are not relevant to the steps below . secondly we evaluate the left hand side in as a residue at @xmath234 , but we are only interested in the @xmath441-entry , @xmath442 hence with and proposition , @xmath443)\delta_{n,\ell}({\bf s}+[z])}{\delta_{n,\ell-1}(\bt)\delta_{n,\ell+1}({\bf s})}= 1-\frac{\delta_{n+1,\ell-1}(\bt)\delta_{n-1,\ell+1}({\bf s})}{\delta_{n,\ell-1}(\bt)\delta_{n,\ell+1}({\bf s})},\ ] ] or equivalently @xmath444)\delta_{n,\ell}({\bf s}+[z])=\delta_{n,\ell-1}(\bt)\delta_{n,\ell+1}({\bf s})-\delta_{n+1,\ell-1}(\bt)\delta_{n-1,\ell+1}({\bf s}).\ ] ] identity closely resembles a hirota " version of the classical dodgson determinantal identity , for if we set @xmath445 then reduces to the dodgson identity for hankel determinants , @xmath446 we now rewrite equation with the substitution @xmath447 and define @xmath448 ) \delta_{n , \ell}(\bt-{\bf h } + [ z])\ri ) \\ & - \delta_{n,\ell-1}(\bt+{\bf h } ) \delta_{n,\ell+1}(\bt-{\bf h})- \delta_{n+1,\ell-1}(\bt+{\bf h } ) \delta_{n-1,\ell+1}(\bt-{\bf h})\end{aligned}\ ] ] so that can be written in the compact form [ hdnl ] hd_n , ( , * h * ) 0 , , * h * , n,_1 . for the rest of this section we shall set all even times to zero , i.e. we choose @xmath449 . now use corollary [ corfreak ] in conjunction with , @xmath450 and recall lemma [ lemmadouble ] , @xmath451 hence with for @xmath452 and @xmath156 , @xmath453 differentiating with respect to @xmath454 we can derive a whole hierarchy of equations , however we are only interested in one particular identity : hd_n , ( _ o,*h*)|_*h*=*0 * = - _ n-1 , + 1 - _ n+1 , -1 + 2 + + _ n , + 1 -2 + _ n , -1 + 2 ( _ n , ) ^2 = 0 [ premiura ] and the argument of all determinants in the right hand side equals @xmath449 . for @xmath455 , with and , this leads to @xmath456 which can be rewritten as @xmath457 and after simplification with , @xmath458 which completes the proof of . in this section we offer a proof that the points belong to the boundary of @xmath70 . the proof is a verification that the inequalities for the effective potential are fulfilled at the particular values of @xmath243 determined in . these correspond in the @xmath227-plane to the points in the @xmath71-plane . the proof is a simple deformation argument starting from large @xmath287 ( and hence also large @xmath71 ) . observing various panes in figure [ stars ] and using the @xmath69 symmetry of the region , it is sufficient to show that the point a_0^[n]= 2 ( -1)^n ( 2n(2nn))^12n+1 x_0^[n ] = ( -1)^n((2n+1)()^2n)^12n+1 ( or rather its @xmath71image ) belongs to the boundary of @xmath70 . this point is alternatively positive or negative , depending on the parity of @xmath25 . consider now in some detail the case @xmath459 ; then @xmath460 } \simeq 0.944 $ ] ( @xmath461 } \simeq 2.36021 $ ] ) . in this case the polynomial @xmath229 equals + let @xmath463 denote the roots of @xmath464 . we know from the argument in section [ gfunctdelta ] that for @xmath287 large the inequalities are fulfilled ; as we deform @xmath227 from larger absolute values to smaller ones , these inequalities can fail only if the sign of @xmath465 changes . we now simply have to verify that the sign of @xmath466 remains constant as @xmath227 decreases from @xmath467 to the critical value @xmath460}$ ] ( corresponding to @xmath71 decreasing from @xmath467 to the rightmost corner @xmath461}$ ] ) . since the four roots admit an explicit expression in terms of @xmath227 , this verification is a simple exercise in calculus . to be more precise , one pair that we denote @xmath468 is purely imaginary and lies on the zero level set of @xmath284 identically for @xmath469 } , \infty ) $ ] ; this is not a cause of concern because it belongs to the level curve ( in fact a straight line ) joining @xmath248 to @xmath234 . the other pair @xmath470 is real for @xmath469 } , \infty ) $ ] . then one can easily verify that @xmath471 is , depending on which of the two member of the pair , a monotone increasing / decreasing function of @xmath469 } , \infty ) $ ] and not changing sign . this verification uses lemma [ lemmader ] and the explicit expression for the roots , so that ( for @xmath227 real ) [ signphi ] ( z_j(a);a ) = - ( _ x(z;a ) |_z = z_j(a ) ) = ( |_z = z_j(a ) ) in figure [ rootdescent ] we display the graph of @xmath472 in the range @xmath473},\infty)$ ] ; the monotonicity can be shown by inspecting the sign of ; we leave the detail to the reader . the argument above can be repeated for @xmath474 , but for larger @xmath25 we were not able to find a unifying argument . 100 m. adler , j. moser , on a class of polynomials connected with the korteweg - de vries equation , _ comm . _ , * 61*(1 ) : 1 - 30 , ( 1978 ) h. airault , h. p. mckean , j. moser , rational and elliptic solutions of the korteweg - de vries equation and a related many - body problem , _ comm . pure appl . _ , * 30*(1 ) : 95 - 148 ( 1997 ) c. andreief , note sur une relation entre les intgrales dfinies des produits des fonctions , _ mm . de la soc . sci . bordeaux _ , * 3*(2):1 - 14 , ( 1883 ) j. baik , p. deift , e. strahov , products and ratios of characteristic polynomials of random hermitian matrices , _ j. math . phys . _ * 44*(8):3657 - 3670 , ( 2003 ) m. bertola , t. bothner , zeros of large degree vorobev - yablonski polynomials via a hankel determinant identity , _ international mathematics research notices _ doi : 10.1093/imrn / rnu239 ( 2014 ) r. buckingham , p. miller , large - degree asymptotics of rational painlev - ii functions . noncritical behavior , _ nonlinearity _ * 27*(10 ) : 2489 - 2578 , ( 2014 ) r. buckingham , p. miller , large - degree asymptotics of rational painlev - ii functions . critical behavior , preprint arxiv:1406.0826 p. clarkson , e. mansfield , the second painlev equation , its hierarchy and associated special polynomials , _ nonlinearity _ * 16*(3 ) : r1-r26 , ( 2003 ) p. deift , x. zhou , a steepest descent method for oscillatory riemann - hilbert problems . asymptotics for the mkdv equation , _ ann . of math . _ * 137 * , 296 - 368 ( 1993 ) p. deift , _ orthogonal polynomials and random matrices : a riemann - hilbert approach _ , volume 3 of _ courant lecture notes in mathematics_. new york university courant institute of mathematical sciences , new york , 1999 . p. deift , s. venakides , x. zhou , new results in small dispersion kdv by an extension of the steepest descent method for riemann - hilbert problems , _ int . . notices _ * 6 * ( 1997 ) , 286 - 299 p. deift , t. kriecherbauer , k. mclaughlin , s. venakides , x. zhou , uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory , _ comm . pure and appl . _ * 52 * , no.11 : 1335 - 425 ( 1999 ) m. demina , a. kudryashov , special polynomials and rational solutions of the hierarchy of the second painlev equation , _ teoret . fiz _ * 153*(1):58 - 67 , ( 2007 ) m. demina , a. kudryashov , the generalized yablonskii - vorobev polynomials and their properties , _ phys . * 372*(29):4885 - 4890 , ( 2008 ) e. dewitt , _ idenities relating schur @xmath475-functions and @xmath476-functions _ , proquest llc , ann arbor , mi , 2012 . v. gromak , bcklund transformations of the higher order painlev equations . in _ bcklund and darboux transformations . the geometry of solitons ( halifax , ns , 1999 ) _ , volume 29 of _ crm proc . lecture notes _ , pages 3 - 28 , ( 2001 ) j. jenkins , d. spencer , hyperelliptic trajectories , _ ann . of math . _ * 2 * , 53:4 - 35 , ( 1951 ) k. kajiwara , y. ohta , determinant structure of the rational solutions for the painlev ii equation , _ j. math . _ , * 37*(9):4693 - 4704 , ( 1996 ) y. kametaka , m. noda , y. fukui , s. hirano , a numerical approach to toda equation and painlev ii equation , _ mem . ehime univ . _ * 9 * 1 - 24 . i. macdonald , _ symmetric functions and hall polynomials _ , oxford mathematical monographs . the clarendon press oxford university press , new york , second edition , 1995 . g. segal , g. wilson , loop groups and equations of kdv type , _ inst . hautes tudes sci . publ . math . _ * 61*:5 - 65 , ( 1985 ) k. strebel , _ quadratic differentials _ , volume @xmath477 of _ ergebnisse der mathematik und ihrer grenzgebiete _ , springer - verlag , berlin , 1984 . a. vorobev , on rational solutions of the second painlev equation , _ differential equations _ * 1*:58 - 9 , ( 1965 ) a. yablonskii , on rational solutions of the second painlev equation , _ vestsi akademii navuk bssr . seryya fizika - matematychnykh navuk _ * 3*:30 - 5 ( 1959 ) y. you , polynomial solutions of the bkp hierarchy and projective representations of symmetric groups . in _ infinite - dimensional lie algebras and groups ( luminy - marseille , 1988 ) _ , volume 7 of _ adv . ser . math . _ , pages 449 - 464 ( 1989 )

generalized vorobev - yablonski polynomials have been introduced by clarkson and mansfield in their study of rational solutions of the second painlev hierarchy . we present new hankel determinant identities for the squares of these special polynomials in terms of schur polynomials . as an application of the identities , we analyze the roots of generalized vorobev - yablonski polynomials and provide formul for the boundary curves of the highly regular patterns observed numerically in @xcite .

1504.00440

we consider the following sequence space model @xmath4 where @xmath5 are the coefficients of a signal and the noise @xmath6 has a diagonal covariance matrix @xmath7 . this heterogeneous model may appear in several frameworks where the variance is fluctuating , for example in heterogeneous regression , coloured noise , fractional brownian motion models or statistical inverse problems , for which the general literature is quite exhaustive @xcite . the goal is to estimate the unknown parameter @xmath8 by using the observations @xmath9 . model selection is a core problem in statistics . one of the main reference in the field dates back to the aic criterion @xcite , but there has been a huge amount of papers on this subject ( e.g. , @xcite ) . model selection is usually linked to the choice of a penalty and its precise choice is the main difficulty in model selection both from a theoretical and a practical perspective . there is a close relationship between model selection and thresholding procedures , which is addressed e.g. in @xcite . the idea is that the search for a `` good penalty '' in model selection is indeed very much related to the choice of a `` good threshold '' in wavelet procedures . there exists also a fascinating connection between the false discovery rate control ( fdr ) and both thresholding and model selection , as studied in @xcite , which will become apparent later in our paper . our main modeling assumption is that the parameter @xmath3 of interest is sparse . sparsity is one of the leading paradigms nowadays and signals with a sparse representation in some basis ( for example wavelets ) or functions with sparse coefficients appear in many scientific fields ( see @xcite among many others ) . in this paper , we consider the sequence space model with heterogeneous errors . our goal is then to select among a family of models the best possible one , by use of a data - driven selection rule . in particular , one has to deal with the special heterogeneous nature of the observations , and the choice of the penalty must reflect this . the heterogenous case is much more involved than the direct ( homogeneous ) model . indeed , there is no more symmetry inside the stochastic process that one needs to control , since each empirical coefficient has its own variance . the problem and the penalty do not only depend on the number of coefficients that one selects , but also on their position . this also appears in the minimax bounds where the coefficients in the least favourable model will go to the larger variances . by a careful and explicit choice of the penalty , however , we are able to select the correct coefficients and get a sharp non - asymptotic control of the risk of our procedure . results are also obtained for full model selection and a fdr - type control on a family of thresholds . in the case of known sparsity @xmath10 , we consider a non - adaptive threshold estimator and obtain a minimax upper bound . this estimator exactly attains the lower bound and is then minimax . using our model selection approach , the procedure is almost minimax ( up to a factor 2 ) . moreover , the procedure is fully adaptive . indeed , the sparsity @xmath10 is unknown and we obtain an explicit penalty , valid in the mathematical proofs and directly applicable in simulations . the paper is organized as follows . in the following subsection [ sec : exa ] , we give examples of problems where our heterogeneous model appears . section [ sec : sel ] contains the data - driven procedure and a general result . in section [ sec : spa ] , we consider the sparsity assumptions and obtain theorems for the full subset selection and thresholding procedures . section [ sec : low ] and [ sec : upp ] are concerned with minimax lower and upper bounds . in section [ sec : num ] , we present numerical results for the finite - sample properties of the methods . consider first a model of heterogeneous regression @xmath11 where @xmath12 are i.i.d . standard gaussian , but their variance are fluctuating depending on the design points @xmath13 and @xmath14 is some spiky unknown function . in this model @xmath15 . by spiky function we mean that @xmath16 is zero apart from a small subset of all design points @xmath13 . these signals are frequently encountered in applications ( though rarely modeled in theoretical statistics ) , e.g. when measuring absorption spectra in physical chemistry ( i.e. rare well - localised and strong signals ) or jumps in log returns of asset prices ( i.e. log - price increments which fluctuate at low levels except when larger shocks occur ) . often in applications coloured noise models are adequate . let us consider here the problem of estimating an unknown function observed with a noise defined by some fractional brownian motion , @xmath17,\ ] ] where @xmath14 is an unknown @xmath18periodic function in @xmath19 , @xmath20=0 , @xmath21 is the noise level and @xmath22 is a fractional brownian motion , defined by ( see @xcite ) , @xmath23 where @xmath24 is a brownian motion , @xmath25 , @xmath26 is the gamma function . the fractional brownian motion also appears in econometric applications to model the long - memory phenomena , e.g. in @xcite . the model ( [ mod ] ) is close to the standard gaussian white noise model , which corresponds to the case @xmath27 . here , the behaviour of the noise is different . we are not interested in the fractional brownian motion itself , but we want to estimate the unknown function @xmath14 based on the noisy data @xmath28 , as in @xcite . a very important point is linked with the definition of the fractional integration operator . in this framework , if the function @xmath14 is supposed to be @xmath18periodic , then the natural way is to consider the periodic version of fractional integration ( given in ( [ frac ] ) ) , such that @xmath29 and thus ( see p.135 in @xcite ) , @xmath30 by integration and projection on the cosine ( or sine ) basis and using ( [ eigen ] ) , one obtains the sequence space model ( as in @xcite ) , @xmath31 where @xmath32 are independent with @xmath33 , where @xmath34 and @xmath35 . consider the following framework of a general inverse problem @xmath36 where @xmath37 is a known injective compact linear bounded operator , @xmath14 an unknown @xmath38-dimensional function , @xmath39 is a gaussian white noise and @xmath40 the noise level . we will use here the framework of singular values decomposition ( svd ) , see e.g. @xcite . denote by @xmath41 the eigenfunctions of the operator @xmath42 associated with the strictly positive eigenvalues @xmath43 . remark that any function @xmath14 may be decomposed in this orthonormal basis as @xmath44 , where @xmath45 . let @xmath46 be the normalized image basis @xmath47 by projection and division by the singular values , we may obtain the empirical coefficients @xmath48 we then obtain a model in the sequence space ( see @xcite ) @xmath49 with @xmath33 and @xmath50 . we consider the sequence space model for coefficients of an unknown @xmath51-function @xmath14 with respect to an orthornormal system @xmath52 . the estimator over an arbitrary large , but finite index set @xmath53 is then defined by @xmath54 where @xmath55 the empirical version of @xmath14 is defined as @xmath56 we write @xmath57 and @xmath58 for the cardinality of @xmath53 . let us write @xmath59 for the covariance matrix of the @xmath60 restricted to the indices @xmath61 for which @xmath62 , i.e. @xmath63 with @xmath64 . by @xmath65 we denote the operator norm , i.e. the largest absolute eigenvalue . the random elements @xmath66 take values in the sample space @xmath67 . we now consider an arbitrary family @xmath68 of borel - measurable data - driven subset selection rules . define an estimator by minimizing in the family @xmath69 the penalized empirical risk : @xmath70 with the penalty @xmath71 where @xmath72 denotes the @xmath73-th largest value among @xmath74 and @xmath75 . remark that @xmath76 is defined in an equivalent way by @xmath77 where @xmath78 then , define the data - driven estimator @xmath79 the next lemma shows that one has an explicit risk hull , a concept introduced in full detail in @xcite . [ th : hull ] the function @xmath80 with the penalty from is a risk hull , i.e. we have @xmath81 recall @xmath58 and introduce the stochastic term @xmath82 remark that @xmath83 such that @xmath84 follows from @xmath85 let us write @xmath86 and let @xmath87 denote the inverse rank of @xmath88 in @xmath89 ( e.g. , @xmath90 if @xmath91 such that @xmath92 note that for any enumeration @xmath93 of @xmath94 by monotonicity : @xmath95 holds . we therefore obtain with the inverse order statistics @xmath96 and @xmath97 ( i.e. @xmath98 etc . ) of @xmath99 and @xmath100 , respectively , @xmath101 { \leqslant}\operatorname{{\mathbf e}}\big[\sum_{j=1}^n\sigma_{(j)}^2\big(\zeta_{(j)}^2 - 2(\log(ne / j)+j^{-1}\log_+(n{\lvert \sigma \rvert}))\big)_+ \big].\ ] ] it remains to evaluate @xmath102 $ ] . we obtain by independence , @xmath103 and by the mill ratio inequality @xmath104 @xmath105 this implies for any @xmath106 @xmath107=\int_p^\infty p(\zeta_{(j)}^2>\kappa)\,d\kappa{\leqslant}2j^{-1}p^{-j/2}\exp(j\log(ne / j)-jp/2).\ ] ] we conclude @xmath108\ ] ] @xmath109 @xmath110 where @xmath111 and the supremum is attained at @xmath112 with value @xmath113 . [ th : oracle ] let @xmath76 be the data - driven rule defined in ( [ hstar ] ) . for any @xmath114 , we have @xmath115+\omega_\delta,\ ] ] where @xmath116 in view of lemma [ th : hull ] , @xmath117 is a risk hull , and therefore we have @xmath118 on the other hand , since @xmath76 minimizes @xmath119 we have @xmath120.\ ] ] in order to combine the inequalities ( [ ep1 ] ) and ( [ ep2 ] ) , we rewrite @xmath121 in terms of @xmath122 @xmath123 therefore , using this equation and ( [ ep1 ] , [ ep2 ] ) , we obtain @xmath124+{\lvert f \rvert}^2 + \sqrt{2}\min\big(\tfrac1n,{\lvert \sigma \rvert}\big ) + 2\mathbf{e}_f \ , \sum_{\lambda \in \lambda } h_\lambda^\star f_\lambda \xi_\lambda \\ & + \mathbf{e}_f\,\biggl[\sum_{\lambda \in \lambda } h_\lambda^\star \xi_\lambda^2-pen(h^\star)\biggr ] . \end{split}\ ] ] remark now that for any deterministic index set @xmath125 @xmath126 this implies for @xmath127 @xmath128 then , by the general inequality @xmath129 for @xmath130 we obtain @xmath131 note that @xmath132 since @xmath133 . by ( [ cauch ] ) and ( [ vari_1 ] ) we obtain @xmath134 in a similar way , we obtain @xmath135 note that @xmath136 since @xmath137 . using ( [ cauch2 ] ) and ( [ vari_2 ] ) one has @xmath138 note also that , since @xmath139 , we have @xmath140 insertion of ( [ var - theta ] ) and ( [ var - theta2 ] ) into yields @xmath141 by using the risk hull as in lemma [ th : hull ] , one obtains @xmath142 { \leqslant}\sqrt{2}\min\big(\tfrac1n,{\lvert \sigma \rvert}\big).\ ] ] inserting ( [ var - theta ] ) , ( [ var - theta2 ] ) and ( [ hull2 ] ) into ( [ ep3 ] ) yields @xmath143+{\lvert f \rvert}^2 + \sqrt{2}\min\big(\tfrac1n,{\lvert \sigma \rvert}\big)+ \tfrac{2}{\delta}\sum_{\lambda \in \lambda}\min(f_\lambda^2,\sigma^2_{\lambda})\\ & \quad + \sqrt{2}\min\big(\tfrac1n,{\lvert \sigma \rvert}\big)+ \frac{\delta}{2 } \mathbf{e}_f\|\hat { f}(h^\star ) -f\|^2.\end{aligned}\ ] ] using ( [ fin1 ] ) we obtain , @xmath144 + 2\sqrt{2}\min\big(\tfrac1n,{\lvert \sigma \rvert}\big)+\frac{2}{\delta}\sum_{\lambda \in \lambda}\min(f_\lambda^2,\sigma^2_{\lambda}).\ ] ] finally , we let the bias explicitly appear in @xmath145 and the result follows from @xmath146 for @xmath147 $ ] . let us consider the intuitive version of sparsity by assuming a small proportion of nonzero coefficients ( cf . @xcite ) , i.e. the family @xmath148 where @xmath149 denotes the maximal proportion of nonzero coefficients . throughout , we assume that this proportion @xmath10 is such that asymptotically @xmath150 the goal here is to study the accuracy of the full model selection over the whole family of estimators . each coefficient may be chosen to be inside or outside the model . let us consider the case where @xmath151 denotes all deterministic subset selections , @xmath152 [ th : ms ] let @xmath76 be the data - driven rule defined in ( [ hstar ] ) with @xmath151 as in ( [ hms ] ) . we have , for @xmath153 , uniformly over @xmath154 , @xmath155 in particular , if @xmath156 ( i.e. , any polynomial growth for @xmath157 is admissible ) and @xmath158 , then we obtain @xmath159 for @xmath154 the right - hand side in theorem [ th : oracle ] can be bounded by considering the oracle @xmath160 such that @xmath161+\omega_\delta & { \leqslant}(1+\delta)2pen(h^f)+\omega_\delta.\end{aligned}\ ] ] we will use the following inequality , as @xmath162 , @xmath163 by comparison with the integral . since @xmath164 , we obtain that @xmath165 @xmath166 as @xmath153 . on the other hand , we have @xmath167 we use @xmath168 which shows @xmath169 choosing @xmath170 such that @xmath171 , e.g. @xmath172 , we thus find , as @xmath173 , @xmath174 using theorem [ th : oracle ] , equation ( [ omega_del ] ) we have ( [ bound1 ] ) . moreover , using the bounds on @xmath175 and @xmath157 we obtain ( [ bound1b ] ) . consider now a family of threshold estimators . the problem is to study the data - driven selection of the threshold . let us consider the case where @xmath151 denotes the threshold selection rules with arbitrary threshold values @xmath176 @xmath177 note that @xmath69 consists of @xmath58 different subset selection rules only and can be implemented efficiently using the order statistics of @xmath178 . [ th : tr ] let @xmath76 be the data - driven rules defined in ( [ hstar ] ) with @xmath151 as in ( [ htr ] ) . if @xmath179 , then we have , for @xmath153 , uniformly over @xmath154 @xmath180 assuming for @xmath181 the growth bounds @xmath182 with a second condition always checked if @xmath183 , this inequality simplifies to @xmath184 let us now evaluate the right - hand side of the oracle inequality in theorem [ th : oracle ] for the threshold selection rules with arbitrary threshold values @xmath176 defined in ( [ htr ] ) . given an oracle parameter @xmath185 ( to be determined below ) , we set @xmath186 . we obtain with @xmath187 denoting the ( inverse ) rank of the coefficient with index @xmath188 among @xmath189 @xmath190\\ & { \leqslant}\mathbf{e}_f\big [ \sum_{\lambda\in\lambda}\big({\bf 1}({\lvert x_\lambda \rvert}{\leqslant}\tau_\lambda)f_\lambda^2- { \bf 1}({\lvert x_\lambda \rvert}>\tau_\lambda ) ( x_\lambda^2-f_\lambda^2)\\ & \qquad\qquad + 4\sigma_\lambda^2 { \bf 1}({\lvert x_\lambda \rvert}>\tau_\lambda)(\log(en / r_\lambda ) + r_\lambda^{-1}\log_+(n{\lvert \sigma \rvert}))\big)\big].\end{aligned}\ ] ] let us first show that @xmath191 $ ] is always non - negative . by symmetry @xmath192 has the same law as @xmath193 . defining the function @xmath194 , we check by considering the different cases that @xmath195 holds . we conclude @xmath196 & = \tfrac12 \operatorname{{\mathbf e}}_f[g(\xi_\lambda)+g(-\xi_\lambda)]{\geqslant}0.\end{aligned}\ ] ] hence , the term with a minus sign in can be discarded for an upper bound . let us now consider the coefficients that contain a signal part ( i.e. with @xmath197 ) . the following inequality will be helpful to obtain a bound independent of the size of @xmath198 . let us denote by @xmath199 the corresponding inverse rank within @xmath200 . with @xmath201 on the event @xmath202 we obtain @xmath203 where for the last inequality we have used that for @xmath204 distinct values @xmath205 the expression is maximal in the case @xmath206 . the general identity @xmath207=c+\int_c^\infty p(z{\geqslant}z)dz$ ] applied to @xmath208 and deterministic @xmath209 yields @xmath210&{\leqslant}c_\lambda+\int_{c_\lambda}^\infty p({\lvert \xi_\lambda \rvert}{\geqslant}\sqrt{z}-\tau_\lambda)\,dz { \leqslant}c_\lambda+2e^{-(\sqrt{c_\lambda}-\tau_\lambda)^2/(2\sigma_\lambda^2)}.\end{aligned}\ ] ] in order to ensure @xmath211 whenever @xmath197 , we are lead to choose @xmath212 in the sequel we bound @xmath213 simply by @xmath214 in the case @xmath197 . then using again the bound on sums of logarithms ( [ sumlog ] ) and @xmath215 as well as the concavity of @xmath216 for bounding the sum of exponentials , we obtain that over the signal part satisfies @xmath217\\ & { \leqslant}\sum_{\lambda\in\lambda , f_\lambda\not=0}(c_\lambda+2e^{-(\sqrt{c_\lambda}-\tau_\lambda)^2/(2\sigma_\lambda^2 ) } ) { \leqslant}n\gamma_n(c_n{\lvert \sigma_{h_f } \rvert}+2 e^{-(c_n-(t^0)^2)/2}),\,\end{aligned}\ ] ] where @xmath218 owing to @xmath219 we even have @xmath217 \nonumber\\ & { \leqslant}{\lvert \sigma_{h_f } \rvert}n\gamma_n c_n(1+o(1)).\label{sig2}\end{aligned}\ ] ] on the other hand , for the non - signal part @xmath220 , we introduce @xmath221 and we use the large deviation bound : @xmath222=n p({\lvert \xi_\lambda \rvert}>\tau_\lambda){\leqslant}2n(t^0)^{-1}e^{-(t^0)^2/2}. \ ] ] again by considering worst case permutations instead of the ranks , using ( [ sumlog ] ) and by jensen s inequality for the concave functions @xmath223 we infer : @xmath224\\ & { \leqslant}4{\lvert \sigma \rvert}\mathbf{e}_f\left[\sum_{\lambda\in\lambda } { \bf 1}({\lvert \xi_\lambda \rvert}>\tau_\lambda)(\log(en / r_\lambda)+r_\lambda^{-1}\log_+(n{\lvert \sigma \rvert}))\right]\\ & { \leqslant}4{\lvert \sigma \rvert}\mathbf{e}\left[\sum_{j=1}^{n_\tau}(\log(en / j)+j^{-1}\log_+(n{\lvert \sigma \rvert}))\right]\\ & { \leqslant}4{\lvert \sigma \rvert}\mathbf{e}\left[(n_\tau\log(en / n_\tau)+\log(n_\tau)\log_+(n{\lvert \sigma \rvert}))\right ] ( 1+o(1))\\ & { \leqslant}4{\lvert \sigma \rvert } ( 2n ( t^0)^{-1 } e^{-(t^0)^2/2}(1+t_0 ^ 2/2)+(\log n-(t^0)^2/2)\log_+(n{\lvert \sigma \rvert}))(1+o(1))\\ & { \leqslant}2{\lvert \sigma \rvert}(2n e^{-(t^0)^2/2}t^0+(2\log n-(t^0)^2)\log_+(n{\lvert \sigma \rvert}))(1+o(1)).\end{aligned}\ ] ] for the @xmath225 chosen , the total bound over is thus , by ( [ sig2 ] ) , ( [ eq42 ] ) and by definition of @xmath226 in ( [ c_n ] ) , @xmath227 this yields the asserted general bound and inserting the bound for @xmath228 gives directly the second bound . _ heterogeneous case . _ one may compare the method and its accuracy with other results in related frameworks . for example , @xcite considers a very close framework of model selection in inverse problems by using the svd approach . this results in a noise @xmath229 which is heterogeneous and diagonal . @xcite study the related topic of inverse problems and wavelet vaguelette decomposition ( wvd ) , built on @xcite . the framework in @xcite is more general than ours . however , this leads to less precise results . in all their results @xcite , there exist universal constants which are not really controlled . this is even more important for the constants inside the method , for example in the penalty . our method contains an explicit penalty . it is used in the mathematical results and also in simulations without additional tuning . a possible extension of our method to the dependent wvd case does not seem straight - forward . _ homogeneous case . _ let us compare with other work for the homogeneous setting @xmath230 . there exist a lot of results in this framework , see e.g. @xcite . again those results contain universal constants , not only in the mathematical results , but even inside the methods . for example , constants in front of the penalty , but also inside the fdr technique , with an hyper - parameter @xmath231 which has to be tuned . the perhaps closest paper to our work is @xcite in the homogeneous case . our penalty is analogous to `` twice the optimal '' penalty considered in @xcite . this is due to difficulties in the heterogenous case , where the stochastic process that one needs to control is much more involved in this setting . indeed , there is no more symmetry inside this stochastic process , since each empirical coefficient has its own variance . the problem and the penalty do not only depend on the number of coefficients that one selects , but also on their position . this leads to a result @xmath232 , where one gets a constant @xmath233 in @xcite . the potential loss of the factor 2 in the heterogeneous framework might possibly be avoidable in theory , but in simulations the results seem comparably less sensitive to this factor than to other modifications , e.g. to how many data points , among the @xmath204 non - zero coefficients , are close to the critical threshold level , which defines some kind of effective sparsity of the problem ( often muss less than @xmath204 ) . this effect is not treated in the theoretical setup in all of the fdr - related studies , where implicitly a worst case scenario of the coefficients magnitude is understood . [ th : lower ] for any estimator @xmath234 based on @xmath235 observations we have the minimax lower bound @xmath236 { \geqslant}\sup_{\alpha_n\in s_\lambda(n\gamma_n , c_n ) } 2\big(1+o(1)\big)\big(\sum_{\lambda\in\lambda } \sigma_\lambda ^2\alpha_{\lambda , n}\log(\alpha_{\lambda , n}^{-1})\big)\ ] ] for some @xmath237 where @xmath238^\lambda\,|\,\sum_\lambda\alpha_\lambda{\leqslant}r(1-c)\}$ ] denotes the intersection of @xmath239-times the @xmath235-dimensional unit cube with @xmath240-times the @xmath235-simplex and where @xmath241 as @xmath153 . distributing mass uniformly over the @xmath242 indices with largest values @xmath243 yields the lower bound , as @xmath153 , @xmath236 { \geqslant}2n\gamma_n\log(\gamma_n^{-1})\big(1+o(1)\big)\frac 1{r_n}\sum_{i=1}^{r_n } \sigma_{(i ) } ^2\ ] ] in terms of the inverse order statistics @xmath244 , provided @xmath245 ( i.e. , @xmath242 must be somewhat larger than @xmath204 ) . note that for polynomial growth @xmath246 , @xmath247 , the lower bound is , as @xmath153 , @xmath236 { \geqslant}2\big(1+o(1)\big){\lvert \sigma \rvert}n\gamma_n\log(\gamma_n^{-1}).\ ] ] the lower bound is a kind of weighted entropy . in contrast to the upper bounds above the minimax ( and the bayes ) lower bound does not involve the quantity @xmath214 , individual to each unknown @xmath14 . in the proof for this heterogeneous model , conceptually we need to allow for a high complexity of the class @xmath248 , leading to the entropy factor @xmath249 , and to put more prior probability on coefficients with larger variance , which explains the abstract weighted entropy expression . consider for each coefficient @xmath250 the following bayesian prior , which turns out to be asymptotically least favorable : @xmath251 with some @xmath252 . without loss of generality we may assume @xmath253 so slowly that @xmath254 . introducing the number of non - zero entries @xmath255 and writing @xmath256 for the joint law of prior and observations , we deduce by chebyshev inequality @xmath257 the property @xmath258 then implies that the bayes - optimal risk , derived below , will be an asymptotic minimax lower bound over @xmath248 . we need to calculate the bayes risk and find the posterior law of @xmath259 for each coordinate @xmath61 : @xmath260 since we deal with quadratic loss , the bayes estimator @xmath261 equals the conditional expectation @xmath262 $ ] and the bayes risk the expectation of the conditional variance , which is calculated as @xmath263=\operatorname{{\mathbf e}}[f_\lambda ^2]-\operatorname{{\mathbf e}}[\operatorname{{\mathbf e}}[f_\lambda |x_\lambda ] ^2 ] = \mu_{\lambda , n}^2\big(\alpha_{\lambda , n}-\int \frac{\alpha_{\lambda , n}^2{\varphi}_{\mu_{\lambda , n},\sigma_\lambda ^2}(x)^2 } { ( 1-\alpha_{\lambda , n}){\varphi}_{0,\sigma_\lambda ^2}(x)+ \alpha_{\lambda , n}{\varphi}_{\mu_{\lambda , n},\sigma_\lambda ^2}(x)}\,dx\big).\ ] ] the integral can be transformed into an expectation with respect to @xmath264 and bounded by jensen s inequality : @xmath265\\ & \qquad { \leqslant}\alpha_{\lambda , n } \big(1+\alpha_{\lambda , n}^{-1}(1-\alpha_{\lambda , n})\operatorname{{\mathbf e}}[\exp(\sigma_\lambda ^{-1}z-\mu_{\lambda , n}^2/(2\sigma_\lambda ^2))]\big)^{-1 } \\ & \qquad = \alpha_{\lambda , n } \big(1+\alpha_{\lambda , n}^{-1}(1-\alpha_{\lambda , n})\exp((1-\mu_{\lambda , n}^2)/(2\sigma_\lambda ^2))\big)^{-1}.\end{aligned}\ ] ] since @xmath266 uniformly , we just select @xmath267 such that @xmath263{\geqslant}2\sigma_\lambda ^2\alpha_{\lambda , n}(1-(\log c_n^{-1})^{-1/2})\log(\alpha_{\lambda , n}^{-1 } ) ( 1-((1+(1-\alpha_{\lambda , n})\alpha_{\lambda , n}^{-(\log c_n^{-1})^{-1/2}}e^{1/(2\sigma_\lambda ^2)}))^{-1}).\ ] ] noting @xmath268 uniformly over @xmath188 , the overall bayes risk is hence uniformly lower bounded by @xmath269 the supremum at @xmath235 is attained for @xmath270 where @xmath271 is such that @xmath272 holds , provided @xmath273 for all @xmath188 . the latter condition is fulfilled if @xmath274 . alternatively , we may write @xmath275 and the entropy expression becomes @xmath276 where the @xmath277 $ ] sum up to one : @xmath278 . from this representation we immediately infer the lower bound @xmath279 using the uniform weights @xmath280 . note that for polynomial growth @xmath246 , @xmath247 , and for @xmath281 , we have @xmath282 and the lower bound is indeed @xmath236 { \geqslant}2\big(1+o(1)\big){\lvert \sigma \rvert}n\gamma_n\log(\gamma_n^{-1}).\ ] ] consider now the setting where the sparsity @xmath10 is known and a correctly tuned threshold estimator is applied in order to identify the unknown positions of the significant non - zero coefficients @xmath283 . [ th : upper ] consider the threshold estimator defined coordinate - wise by @xmath284 and @xmath285 chosen such that @xmath286 . then , as @xmath153 , @xmath236{\leqslant}2n\gamma_n\beta_n(1+o(1))\ ] ] holds . this implies that , as @xmath153 , @xmath236{\leqslant}2n\gamma_n\log(\gamma_n^{-1}){\lvert \sigma \rvert}(1+o(1)),\ ] ] which is minimax optimal for at most polynomial growth in @xmath287 by the lower bound in theorem [ th : lower ] . for faster growth than polynomial , we might well have @xmath288 . so , in general the upper bound matches exactly the lower bound with respect to the term @xmath289 , while the influence of the heterogeneous noise depends on the specific case . however , this procedure is non - adaptive since the threshold relies on the knowledge of the sparsity @xmath10 . introduce the threshold value @xmath290 and note @xmath291 . we can split the error as follows : @xmath292=f_\lambda^2\operatorname{{\mathbb p}}((\xi_\lambda+f_\lambda/\sigma_\lambda)^2{\leqslant}\tau_{\lambda , n}^2)+\operatorname{{\mathbf e}}[\sigma_\lambda^2\xi_\lambda^2{\bf 1}_{\{(\xi_\lambda+f_\lambda/\sigma_\lambda)^2>\tau_{\lambda , n}^2\}}]=:i+ii.\ ] ] for @xmath293 term i is estimated by @xmath294 together with a symmetric argument for @xmath295 and a direct bound for @xmath296 , we thus obtain a bound for general @xmath283 : @xmath297 since for @xmath298 we have @xmath299 , we consider @xmath300 and infer @xmath301 inserting the choice of the thresholds , we conclude @xmath302 for term ii and @xmath197 the immediate estimate @xmath303 suffices , while for @xmath220 we integrate out explicitly and obtain : @xmath304=\sigma_\lambda^22(\tau_{\lambda , n}+1)e^{-\tau_{\lambda , n}^2/2 } = 2\sigma_\lambda^2\sqrt{2\log(\alpha_{\lambda , n}^{-1})}\alpha_{\lambda , n}(1+\tau_{\lambda , n}^{-1}).\ ] ] the overall risk of our estimator is therefore bounded by @xmath305{\leqslant}\sum_{\lambda : f_\lambda\not=0}\big(2\sigma_\lambda^2\log(\alpha_{\lambda , n}^{-1})(1+o(1 ) ) + \sigma_\lambda^2\big)+\sum_{\lambda : f_\lambda=0 } 2\sigma_\lambda^2\sqrt{2\log(\alpha_{\lambda , n}^{-1})}\alpha_{\lambda , n}(1+o(1))\\ & { \leqslant}(2+o(1 ) ) \big(\sum_{\lambda : f_\lambda\not=0}\log(\alpha_{\lambda , n}^{-1})\sigma_\lambda^2 + \sqrt 2\max_\lambda\big((\log(\alpha_{\lambda , n}^{-1}))^{-1/2}\alpha_{\lambda , n}\big)\sum_{\lambda : f_\lambda=0 } \sigma_\lambda^2\log(\alpha_{\lambda , n}^{-1})\big).\end{aligned}\ ] ] choosing @xmath306 , with @xmath285 satisfying @xmath286 , minimises the last bound ( asymptotically ) and yields @xmath307{\leqslant}(2+o(1))n\gamma_n\beta_n\ ] ] because by @xmath308 the second term is of smaller order . the last result is a direct consequence . indeed , we always have @xmath309 by bounding @xmath310 , which is minimax optimal for at most polynomial growth in @xmath287 by the lower bound in theorem [ th : lower ] . ( blue ) , observations @xmath311 ( green in full subset , green / yellow in adaptive threshold , magenta not taken ) and universal / sparse thresholds ( black ) ( parameter values : @xmath312 , @xmath313 , @xmath314 for @xmath315 ) . , width=529,height=264 ] in figure [ fig1 ] a typical realisation of the coefficients @xmath283 is shown in blue with 50 non - zero coefficients chosen uniformly on @xmath316 $ ] and increasing noise level @xmath314 for @xmath317 . the inner black diagonal lines indicate the sparse threshold ( with oracle value of @xmath10 ) and the outer diagonal lines the universal threshold . the non - blue points depict noisy observations @xmath193 . observations included in the adaptive full subset selection estimator are coloured green , while those included for the adaptive threshold estimator are the union of green and yellow points ( in fact , for this sample the adaptive thresholding selects all full subset selected points ) , the discarded observations are in magenta . we have run 1000 monte carlo experiments for the parameters @xmath312 , @xmath314 in the sparse ( @xmath318 ) and dense ( @xmath313 ) case . in figure [ fig2 ] the first 100 relative errors are plotted for the different estimation procedures in the dense case . the errors are taken as a quotient with the sample - wise oracle threshold value applied to the renormalised @xmath319 . therefore only the full subset selection can sometimes have relative errors less than one . table [ tab1 ] lists the relative monte carlo errors for the two cases . the last column reports the relative error of the oracle procedure with @xmath320 that discards all observations @xmath193 with @xmath220 ( not noticing the model selection complexity ) . the simulation results are quite stable for variations of the setup . altogether the thresholding works globally well . the ( approximate ) full subset selection procedure ( see below for the greedy algorithm used ) is slightly worse and exhibits a higher variability , but is still pretty good . by construction , in the dense case the oracle sparse threshold works better than the universal threshold , while the universal threshold works better in very sparse situations . the reason why the sparse threshold even with a theoretical oracle choice of @xmath10 does not work so well is that the entire theoretical analysis is based upon potentially most difficult signal - to - noise ratios , that is coefficients @xmath283 of the size of the threshold or the noise level . here , however , the effective sparsity is larger ( i.e. , effective @xmath10 is smaller ) because the uniformly generated non - zero coefficients can be relatively small especially at indices with high noise level , see also figure [ fig1 ] . let us briefly describe how the adaptive full subset selection procedure has been implemented . the formula attributes to each selected coefficient @xmath193 the individual penalty @xmath321 with the inverse rank @xmath322 of @xmath323 . due to @xmath324 all coefficients with @xmath325 are included into @xmath326 in an initial step . then , iteratively @xmath327 is extended to @xmath328 by including all coefficients with @xmath329 the iteration stops when no further coefficients can be included . the estimator @xmath330 at this stage definitely contains all coefficients also taken by @xmath331 . in a second iteration we now add in a more greedy way coefficients that will decrease the total penalized empirical risk . including a new coefficient @xmath332 , adds to the penalized empirical risk the ( positive or negative ) value @xmath333 here , @xmath334 is to be understood as the rank at @xmath335 when setting @xmath336 . consequently , the second iteration extends @xmath330 each time by one coefficient @xmath332 for which the displayed formula gives a negative value until no further reduction of the total penalized empirical risk is obtainable . this second greedy optimisation does not necessarily yield the optimal full subset selection solution , but most often in practice it yields a coefficient selection @xmath331 with a significantly smaller penalized empirical risk than the adaptive threshold procedure . the numerical complexity of the algorithm is of order @xmath337 due to the second iteration in contrast to the exponential order @xmath338 when scanning all possible subsets . a more refined analysis of our procedure would be interesting , but might have minor statistical impact in view of the good results for the straight - forward adaptive thresholding scheme . the authors would like to thank iain johnstone , debashis paul and thorsten dickhaus for interesting discussions . m. rei gratefully acknowledges financial support from the dfg via research unit for1735 _ structural inference in statistics_. massart p. ( 2007 ) . _ concentration inequalities and model selection . _ lectures from the 33rd summer school on probability theory held in saint - flour , july 6 - 23 , 2003 . lecture notes in mathematics , springer , berlin .

we consider a gaussian sequence space model @xmath0 where @xmath1 has a diagonal covariance matrix @xmath2 . we consider the situation where the parameter vector @xmath3 is sparse . our goal is to estimate the unknown parameter by a model selection approach . the heterogenous case is much more involved than the direct model . indeed , there is no more symmetry inside the stochastic process that one needs to control since each empirical coefficient has its own variance . the problem and the penalty do not only depend on the number of coefficients that one selects , but also on their position . this appears also in the minimax bounds where the worst coefficients will go to the larger variances . however , with a careful and explicit choice of the penalty we are able to select the correct coefficients and get a sharp non - asymptotic control of the risk of our procedure . some simulation results are provided .

1312.5839

one of the most important astrophysical phenomena still lacking an explanation is the origin of the celestial gamma - ray bursts ( grb ) . these are powerful flashes of gamma - rays lasting from less than one second to tens of seconds , with isotropic distribution in the sky . they are observed above the terrestrial atmosphere with x gamma ray detectors aboard satellites @xcite . thanks to the bepposax satellite @xcite , afterglow emission at lower wavelengths has been discovered @xcite and we now know that at least long ( @xmath17s ) grb s are at cosmological distances , with measured red shifts up to 4.5 ( see , e.g. , review by djorgovski @xcite and references therein ) . among the possible explanations of these events , which involve huge energy releases ( up to @xmath18 erg , assuming isotropic emission ) , the most likely candidates are the collapse of a very massive star ( hypernova ) and the coalescence of one compact binary system ( see , e.g. , reviews by piran @xcite and mszros @xcite and references therein ) . in both cases emission of gravitational waves ( gw ) is expected to be associated with them ( e.g. ref . @xcite ) . according to several models , the duration of a gw burst is predicted to be of the order of a few milliseconds for a variety of sources , including the coalescing and merging black holes and/or neutron star binaries . therefore gw bursts can be detected by the present resonant detectors , designed to detect gw through the excitation of the quadrupole modes of massive cylinders , resonating at frequencies near 1 khz . at the distances of the grb sources ( @xmath19 gpc ) , the gw burst associated with a total conversion of 1 - 2 solar masses should have amplitude of the order of @xmath20 . the present sensitivity for 1 ms gw pulses of the best gw antennas with signal to noise ratio ( snr ) equal to unity is @xmath21 ( see e.g. ref . @xcite ) , which requires a total conversion of one million solar masses at 1 gpc . however , although detection of a gravitational signal associated with a single grb appears hopeless , detection of a signal associated with the sum of many events could be more realistic . thus we launched a program devoted to studying the presence of correlations between grb events detected with bepposax and the output signals from gravitational antennas nautilus and explorer . searching for correlation between grb and gw signals means dealing with the difference between the emission times for the two types of phenomena . furthermore , there is also the fact to consider that the time difference can vary from burst to burst . in the present analysis we use an algorithm based on cross - correlating the outputs of two gw detectors ( see @xcite ) , thus coping with the problem of the unknown possible time difference between grb and gw bursts , and also of the unmodelled noise . the rome group operates two resonant bar detectors : explorer @xcite , since 1990 , at the cern laboratories , and nautilus @xcite , since 1995 , at the infn laboratories in frascati . 0.1 in . main characteristics of the two detectors . @xmath22 indicates , for each detector , the two resonant frequencies and @xmath23 indicates the bandwidth . the relatively larger bandwidth of explorer is due to an improved readout system . [ cols="^,^,^,^,^,^,^,^ " , ] [ finale ] the agreement between the values of the simulated input signals and the values calculated using eq . ( [ uppere ] ) shows that our model is correct . having presented the experimental method and the model for the averaged correlation at zero delay time @xmath24 , we can infer the values of gw amplitude @xmath25 consistent with the observation . we note that , using eqs . [ uppere ] and [ snrc1 ] , energy @xmath26 is related to the measured cross - correlation @xmath24 by e_0=t_eff ( ) ^1/4 ( ) ^1/2 [ eq : relazione ] hence , the data are summarized by an observed average squared energy @xmath27 , at @xmath28 standard deviation from the expected value in the case of noise alone , as calculated with the aid of eq . ( [ eq : sigmar_e ] ) where we put @xmath29 . the standard deviation , expressed in terms of squared energy , is obtained from eq . ( [ eq : sigmar_e ] ) , in the case @xmath30 , which gives @xmath31 . according to the model discussed above , in the case of gw signals of energy @xmath32 , we expect @xmath33 to be a random number , modeled with a gaussian probability density function around @xmath34 with a standard deviation @xmath35 : f(e_0 ^ 2|e^2 ) , where @xmath32 is the unknown quantity we wish to infer from the observed value of @xmath26 , given in eq . [ eq : relazione ] . this probability inversion is obtained using bayes theorem ( see , e.g. , @xcite for a physics oriented introduction ) : f(e^2|e_0 ^ 2 ) f(e_0 ^ 2|e^2 ) f_(e^2 ) [ eq : bayes ] where @xmath36 is the prior probability density function of observing gw signals of squared energy @xmath34 . in fact , we are eventually interested in inferring the gw s amplitude @xmath25 , related to the energy @xmath32 by eq . ( [ upperh ] ) . therefore we have a similar equation : f(h|e_0 ^ 2 ) f(e_0 ^ 2| h ) f_(h ) [ eq : bayesh ] where @xmath37 is obtained by a transformation of @xmath38 . as prior for @xmath25 we considered a uniform distribution , bounded to non negative values of @xmath25 , obtained from eq . [ eq : bayesh ] , i.e. @xmath39 is a step function @xmath40 . this seems to us a reasonable choice and it is stable , as long as other priors can be conceived which model the positive attitude of reasonable scientists ( see ref . @xcite ) . , but rather the probability per decade of @xmath25 , i.e. researchers may feel equally uncertain about the orders of magnitudes of @xmath25 . this prior is known as jeffreys prior , but , in our case , it produces a divergence for @xmath41 in eq . [ eq : bayesh ] , a direct consequence of the infinite orders of magnitudes which are equally believed . to get a finite result we need to put a cut - off at a given value of @xmath25 . this problem is described in depth , for example , in @xcite and in @xcite . [ foot:1 ] ] the probability density function of @xmath25 is plotted in fig . [ finaleh ] . the highest beliefs are for very small values , while values above @xmath42 are practically ruled out . from fig . [ finaleh ] we obtain an expected value and standard deviation for @xmath25 of @xmath43 and @xmath44 , respectively , which fully account for what is perceived as a null result . in these circumstances , we can provide an upper limit , defined as value @xmath45 , such that there is a given probability for the amplitude of gw s to be below it , i.e. _ 0^h(ul)f(h |e_0 ^ 2)h = p_l , with @xmath46 the chosen probability level . results are plotted in fig.[fighh ] . for example , we can exclude the presence of signals of amplitudes @xmath47 with 95% probability . probabilistic results depend necessarily on the choice of prior probability density function of @xmath25 . for example , those firmly convinced that gw burst intensities should be in the @xmath48 region would never allow a 5% chance to @xmath25 above @xmath49 . therefore , in frontier research particular care has to be used , before stating probabilistic results . the bayesian approach , thanks to the factorization between likelihood and prior , offers natural ways to a present prior - independent result . the simple idea would be just to provide the likelihood for each hypothesis under investigation , in our case @xmath50 . more conveniently , it has been proposed ( @xcite-@xcite ) to publish the likelihood rescaled to the asymptotic limit , where experimental sensitivity is lost completely ; @xmath51 , in our case . indicating with @xmath52 this rescaled likelihood , we have ( h ) = . [ eq : rbur_def ] in statistics jargon , this function gives the bayes factor of all @xmath25 hypotheses with respect to @xmath53 . in intuitive terms , it can be interpreted as a `` relative belief updating ratio '' or a `` probability density function shape distortion function '' , since from eq . ( [ eq : bayesh ] ) we have = . [ eq : bayesf ] in the present case we get , numerically ( h ) = a h 0 , where @xmath54 is the rescaling factor , @xmath55 and @xmath56 the result is given in fig . [ loglinrh ] , where the choice of the log scale for @xmath25 is to remember that there are infinite orders of magnitudes where the value could be located ( and hence the problem discussed in footnote [ foot:1 ] ) . interpretation of fig . [ loglinrh ] , in the light of eqs . ( [ eq : rbur_def])-([eq : bayesf ] ) , is straightforward : up to a fraction of @xmath57 the experimental evidence does not produce any change in our belief , while values much larger than @xmath57 are completely ruled out . the region of transition from @xmath52 from 1 to zero identifies a _ sensitivity bound _ for the experiment . the exact value of this bound is a matter of convention , and could be , for example , at @xmath58 , or @xmath59 . we have @xmath60 and @xmath61 , respectively . note that these bounds have no probabilistic meaning . in any case , the full result should be considered to be the @xmath52 function , which , being proportional to the likelihood , can easily be used to combine results ( for independent datasets the global likelihood is the product of the likelihoods , and proportional constants can be included in the normalization factor ) . note that the result given in terms of scaled likelihood and sensitivity bound can not be misleading . in fact , these results are not probabilistic statements about @xmath25 and no one would imagine they were . on the other hand , confidence limits , which are not probabilistic statements on the quantity of interest , tend to be perceived as such ( see e.g. @xcite and references therein ) . using for the first time a cross - correlation method applied to the data of two gw detectors , explorer and nautilus , new experimental upper limits have been determined for the burst intensity causing correlations of gw s with grb s . analyzing the data over 47 grbs , we exclude the presence of signals of amplitude @xmath62 , with 95% probability , with a time window of @xmath63 . with the time window of @xmath64 s , we improve the previous gw upper limit to about @xmath65 . the result is also given in terms of scaled likelihood and sensitivity bound , which we consider the most complete and unbiased way of providing the experimental information . in a previous paper @xcite we had given more stringent upper limits , but this was under the hypothesis that the gw signals always occur at the same time with respect to the grb arrival time . here , instead , we only require that the time gap between the grb and the gw burst be within a given time window . similar comparison can be made with the auriga / batse result @xcite , where an upper limit `` @xmath66 with c.l . 95% '' is estimated under the assumption that gw s arrive at the grb time within a time window of @xmath3 s. finally , we remark that this method can be applied for any expected delay between grb and gw , with appropriate time shifting of the integration window with respect to the gbr arrival time , according to the prediction of the chosen model . we thank f. campolungo , r. lenci , g. martinelli , e. serrani , r. simonetti and f. tabacchioni for precious technical assistance . we also thank dr . r. elia for her contribution . given two independent detectors , let @xmath67 and @xmath68 be the measured quantities , which are the ( filtered ) data in our case . we introduce the variables @xmath69 and @xmath70 where @xmath71=t_{eff}$ ] and @xmath72=t_{eff}$ ] . we recall that the cross - correlation function is r()= [ crossa ] with summation extended up to the number of independent samples @xmath73 . we calculate @xmath74 . thus @xmath75 . we easily verify that @xmath76=0 $ ] . in the absence of any correlated signal we also have @xmath77=0 $ ] . let us calculate the variance of @xmath78 . we notice that when squaring the numerator of eq.[crossa ] and taking the average , the cross - terms vanish if the @xmath73 data are independent from each other . then , since also @xmath79 and @xmath80 are independent variables , we obtain _ r^2== = [ siga ] the previous considerations still apply to the cumulative cross - correlation @xmath81 , obtained by averaging @xmath82 independent @xmath78 . the final variance for the cross - correlation @xmath81 is _ r^2= [ sigma ] let us now consider an energy signal @xmath83 on both detectors at the same time . the expected value of the cross - correlation @xmath78 at @xmath84 will be positive . in the case of @xmath73 independent data the signal @xmath83 will appear in one datum only and we have : e[r(=0)-r(0)]== [ segnalea ] where we put @xmath85 . we note that , in eq.[segnalea ] , @xmath86=0 $ ] also when a signal is present . p. astone , c. buttiglione , s. frasca , g.v . pallottino and g. pizzella , il nuovo cimento 20,9 ( 1997 ) p.astone et al . `` gravitational astronomy '' pag 189 + ed . d.e.mcclelland and h.a.bachor , world scientific 1991 g. d agostini _ bayesian reasoning in high - energy physics : principles and applications _ cern 99 - 03 ( 1999 ) this report and related references can be found at the url : http://www-zeus.roma1.infn.it/ agostini / prob+stat.html g. dagostini , _ overcoming priors anxiety _ , invited contribution to the monographic issue of the revista de la real academia de ciencias on bayesian methods in the sciences , ed . j.m . bernardo ( 1999 ) physics/9906048 . g. dagostini , proc . xviii international workshop on maximum entropy and bayesian methods , garching ( germany ) , july 1998 , v. dose , w. von der linden , r. fischer , and r. preuss , eds , kluwer academic publishers , dordrecht , 1999 , physics/9811046 .

data obtained during five months of 2001 with the gravitational wave ( gw ) detectors explorer and nautilus were studied in correlation with the gamma ray burst data ( grb ) obtained with the bepposax satellite . during this period bepposax was the only grb satellite in operation , while explorer and nautilus were the only gw detectors in operation . no correlation between the gw data and the grb bursts was found . the analysis , performed over 47 grb s , excludes the presence of signals of amplitude @xmath0 , with 95% probability , if we allow a time delay between gw bursts and grb within @xmath1 s , and @xmath2 , if the time delay is within @xmath3 s. the result is also provided in form of scaled likelihood for unbiased interpretation and easier use for further analysis . _ @xmath4 istituto nazionale di fisica nucleare infn , rome _ + _ @xmath5 university of rome `` tor vergata '' and infn , rome 2 _ + _ @xmath6 ifsi - cnr and infn , rome _ + _ @xmath7 university of laquila and infn , rome 2 _ + _ @xmath8 iess - cnr and infn , rome _ + _ @xmath9 university of rome `` la sapienza '' and infn , rome _ + _ @xmath10 istituto nazionale di fisica nucleare infn , frascati _ + _ @xmath11 university of ferrara and iasf - cnr , bologna _ + _ @xmath12 university of ferrara , ferrara _ + _ @xmath13 university of ferrara , ferrara and ita `` i. calvi '' , finale emilia , modena _ + _ @xmath14 university of rome `` tor vergata '' and infn , frascati _ + _ @xmath15 ifsi - cnr and infn , rome 2 _ + _ @xmath16 ifsi - cnr and infn , frascati _ + +

astro-ph0206431

in order to formally handle ( specify and prove ) some properties of prolog execution , we needed above all a definition of a port . a port is perhaps the single most popular notion in prolog debugging , but theoretically it appears still rather elusive . the notion stems from the seminal article of l.byrd @xcite which identifies four different types of control flow in a prolog execution , as movements in and out of procedure _ boxes _ via the four _ ports _ of these boxes : * _ call _ , entering the procedure in order to solve a goal , * _ exit _ , leaving the procedure after a success , i.e. a solution for the goal is found , * _ fail _ , leaving the procedure after the failure , i.e. there are no ( more ) solutions , * _ redo _ , re - entering the procedure , i.e. another solution is sought for . in this work , we present a formal definition of ports , which is a calculus of execution states , and hence provide a formal model of pure prolog execution , s : pp . our approach is to define ports by virtue of their effect , as _ port transitions_. a port transition relates two _ events_. an event is a state in the execution of a given query @xmath0 with respect to a given prolog program @xmath1 . there are two restrictions we make : 1 . the program @xmath1 has to be pure 2 . the program @xmath1 shall first be transformed into a canonical form . the first restriction concerns only the presentation in this paper , since our model has been prototypically extended to cover the control flow of full standard prolog , as given in @xcite . the canonical form we use is the common single - clause representation . this representation is arguably ` near enough ' to the original program , the only differences concern the head - unification ( which is now delegated to the body ) and the choices ( which are now uniformly expressed as disjunction ) . first we define the canonical form , into which the original program has to be transformed . such a syntactic form appears as an intermediate stage in defining the clark s completion of a logic program , and is used in logic program analysis . however , we are not aware of any consensus upon the name for this form . some of the names in the literature are _ single - clausal form _ @xcite and _ normalisation of a logic program _ @xcite . here we use the name _ canonical form _ , partly on the grounds of our imposing a transformation on if - then as well ( this additional transformation is of no interest in the present paper , which has to do only with pure prolog , but we state it for completeness ) . [ def : canon ] we say that a predicate @xmath2 is in the canonical form , if its definition consists of a single clause @xmath3 here @xmath4 is a `` canonical body '' , of the form @xmath5 , and @xmath6 is a `` canonical head '' , i.e. @xmath7 are distinct variables not appearing in @xmath8 . further , @xmath9 is a disjunction of canonical bodies ( possibly empty ) , @xmath10 is a conjunction of goals ( possibly empty ) , and @xmath11 is a goal ( for facts : @xmath12 ) . additionally , each if - then goal @xmath13 must be part of an if - then - else ( like @xmath14 ) . for the following program q(a , q(z , c ) @xmath15r(z ) . r(c ) . we obtain as canonical form q(x , y ) @xmath15x = a , y = b , true ; x = z , y = c , r(z ) . r(x ) @xmath15x = c , true . @xmath16 having each predicate represented as one clause , and bearing in mind the box metaphor above , we identified some elementary execution steps . for simplicity we first disregard variables . the following table should give some intuition about the idea . the symbols @xmath17 , @xmath18 in this table serve to identify the appropriate redo - transition , depending on the exit - transition . transitions are deterministic , since the rules do not overlap . [ fig : port : intuit ] [ cols="^ , < , < , < , < " , ] [ def : rules ] @xmath19 \intertext{conjunction } { \ensuremath{{{\ensuremath{\mathit{call}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathit{a}}}}},{{{\ensuremath{\mathit{b}}}}}}}}}\ifempty{{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } & { \ensuremath{\,\rightarrowtriangle\,}}{\ensuremath{{{\ensuremath{\mathit{call}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathit{a}}}}}}}}}\ifempty{{{{\ensuremath{\mathsf{1}}}}}/{{{\ensuremath{\mathit{a}}}}},{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{{\ensuremath{\mathsf{1}}}}}/{{{\ensuremath{\mathit{a}}}}},{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } { \label{spec : conj:1}{\tag{s : conj:1}}}\\ { \ensuremath{{{\ensuremath{\mathit{exit}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathit{a}}}}}'}}}}\ifempty{{{{\ensuremath{\mathsf{1}}}}}/{{{\ensuremath{\mathit{a}}}}},{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{{\ensuremath{\mathsf{1}}}}}/{{{\ensuremath{\mathit{a}}}}},{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } & { \ensuremath{\,\rightarrowtriangle\,}}{\ensuremath{{{\ensuremath{\mathit{call}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathit{b}}}}}''}}}}\ifempty{{{{\ensuremath{\mathsf{2}}}}}/{{{\ensuremath{\mathit{a}}}}},{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{{\ensuremath{\mathsf{2}}}}}/{{{\ensuremath{\mathit{a}}}}},{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } , \text { with } { { { \ensuremath{\mathit{b}}}}}''{\ensuremath{\mathrel{\joinrel{:=}}}}{{\ensuremath{{\ensuremath{{\rm{\text{substof}}}{\ensuremath{\boldsymbol{(}}}{{\ensuremath{\mathbb{\sigma}}}}{\ensuremath{\boldsymbol{)}}}}}{\ensuremath{\boldsymbol{(}}}{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\boldsymbol { ) } } } } } } { \label{spec : conj:2}{\tag{s : conj:2}}}\\ { \ensuremath{{{\ensuremath{\mathit{fail}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathit{a}}}}}'}}}}\ifempty{{{{\ensuremath{\mathsf{1}}}}}/{{{\ensuremath{\mathit{a}}}}},{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{{\ensuremath{\mathsf{1}}}}}/{{{\ensuremath{\mathit{a}}}}},{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } & { \ensuremath{\,\rightarrowtriangle\,}}{\ensuremath{{{\ensuremath{\mathit{fail}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathit{a}}}}},{{{\ensuremath{\mathit{b}}}}}}}}}\ifempty{{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } { \label{spec : conj:3}{\tag{s : conj:3}}}\\ { \ensuremath{{{\ensuremath{\mathit{exit}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathit{b}}}}}'}}}}\ifempty{{{{\ensuremath{\mathsf{2}}}}}/{{{\ensuremath{\mathit{a}}}}},{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{{\ensuremath{\mathsf{2}}}}}/{{{\ensuremath{\mathit{a}}}}},{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } & { \ensuremath{\,\rightarrowtriangle\,}}{\ensuremath{{{\ensuremath{\mathit{exit}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathit{a}}}}},{{{\ensuremath{\mathit{b}}}}}}}}}\ifempty{{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } { \label{spec : conj:4}{\tag{s : conj:4}}}\\ { \ensuremath{{{\ensuremath{\mathit{fail}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathit{b}}}}}'}}}}\ifempty{{{{\ensuremath{\mathsf{2}}}}}/{{{\ensuremath{\mathit{a}}}}},{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{{\ensuremath{\mathsf{2}}}}}/{{{\ensuremath{\mathit{a}}}}},{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } & { \ensuremath{\,\rightarrowtriangle\,}}{\ensuremath{{{\ensuremath{\mathit{redo}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathit{a}}}}}}}}}\ifempty{{{{\ensuremath{\mathsf{1}}}}}/{{{\ensuremath{\mathit{a}}}}},{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{{\ensuremath{\mathsf{1}}}}}/{{{\ensuremath{\mathit{a}}}}},{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } { \label{spec : conj:5}{\tag{s : conj:5}}}\\ { \ensuremath{{{\ensuremath{\mathit{redo}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathit{a}}}}},{{{\ensuremath{\mathit{b}}}}}}}}}\ifempty{{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } & { \ensuremath{\,\rightarrowtriangle\,}}{\ensuremath{{{\ensuremath{\mathit{redo}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathit{b}}}}}}}}}\ifempty{{{{\ensuremath{\mathsf{2}}}}}/{{{\ensuremath{\mathit{a}}}}},{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{{\ensuremath{\mathsf{2}}}}}/{{{\ensuremath{\mathit{a}}}}},{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } { \label{spec : conj:6}{\tag{s : conj:6 } } } \intertext{disjunction } { \ensuremath{{{\ensuremath{\mathit{call}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathit{a}}}}};{{{\ensuremath{\mathit{b}}}}}}}}}\ifempty{{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } & { \ensuremath{\,\rightarrowtriangle\,}}{\ensuremath{{{\ensuremath{\mathit{call}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathit{a}}}}}}}}}\ifempty{{{{\ensuremath{\mathsf{1}}}}}/{{{\ensuremath{\mathit{a}}}}};{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{{\ensuremath{\mathsf{1}}}}}/{{{\ensuremath{\mathit{a}}}}};{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } { \label{spec : disj:1}{\tag{s : disj:1}}}\\ { \ensuremath{{{\ensuremath{\mathit{fail}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathit{a}}}}}}}}}\ifempty{{{{\ensuremath{\mathsf{1}}}}}/{{{\ensuremath{\mathit{a}}}}};{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{{\ensuremath{\mathsf{1}}}}}/{{{\ensuremath{\mathit{a}}}}};{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } & { \ensuremath{\,\rightarrowtriangle\,}}{\ensuremath{{{\ensuremath{\mathit{call}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathit{b}}}}}}}}}\ifempty{{{{\ensuremath{\mathsf{2}}}}}/{{{\ensuremath{\mathit{a}}}}};{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{{\ensuremath{\mathsf{2}}}}}/{{{\ensuremath{\mathit{a}}}}};{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } { \label{spec : disj:2}{\tag{s : disj:2}}}\\ { \ensuremath{{{\ensuremath{\mathit{fail}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathit{b}}}}}}}}}\ifempty{{{{\ensuremath{\mathsf{2}}}}}/{{{\ensuremath{\mathit{a}}}}};{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{{\ensuremath{\mathsf{2}}}}}/{{{\ensuremath{\mathit{a}}}}};{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } & { \ensuremath{\,\rightarrowtriangle\,}}{\ensuremath{{{\ensuremath{\mathit{fail}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathit{a}}}}};{{{\ensuremath{\mathit{b}}}}}}}}}\ifempty{{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } { \label{spec : disj:3}{\tag{s : disj:3}}}\\ { \ensuremath{{{\ensuremath{\mathit{exit}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathit{a}}}}}}}}}\ifempty{{{{\ensuremath{\mathsf{1}}}}}/{{{\ensuremath{\mathit{a}}}}};{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{{\ensuremath{\mathsf{1}}}}}/{{{\ensuremath{\mathit{a}}}}};{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } & { \ensuremath{\,\rightarrowtriangle\,}}{\ensuremath{{{\ensuremath{\mathit{exit}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathit{a}}}}};{{{\ensuremath{\mathit{b}}}}}}}}}\ifempty{{{\ensuremath{\mathbb{u}}}}{{{\ensuremath{\mathit{or({{\ensuremath{\mathsf{{{{\ensuremath{\mathit{a}}}}}}}}},{{\ensuremath{\mathsf{({{{\ensuremath{\mathsf{1}}}}}/{{{\ensuremath{\mathit{a}}}}};{{{\ensuremath{\mathit{b}}}}})}}}})}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{{\ensuremath{\mathit{or({{\ensuremath{\mathsf{{{{\ensuremath{\mathit{a}}}}}}}}},{{\ensuremath{\mathsf{({{{\ensuremath{\mathsf{1}}}}}/{{{\ensuremath{\mathit{a}}}}};{{{\ensuremath{\mathit{b}}}}})}}}})}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } { \label{spec : disj:4}{\tag{s : disj:4}}}\\ { \ensuremath{{{\ensuremath{\mathit{exit}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathit{b}}}}}}}}}\ifempty{{{{\ensuremath{\mathsf{2}}}}}/{{{\ensuremath{\mathit{a}}}}};{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{{\ensuremath{\mathsf{2}}}}}/{{{\ensuremath{\mathit{a}}}}};{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } & { \ensuremath{\,\rightarrowtriangle\,}}{\ensuremath{{{\ensuremath{\mathit{exit}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathit{a}}}}};{{{\ensuremath{\mathit{b}}}}}}}}}\ifempty{{{\ensuremath{\mathbb{u}}}}{{{\ensuremath{\mathit{or({{\ensuremath{\mathsf{{{{\ensuremath{\mathit{b}}}}}}}}},{{\ensuremath{\mathsf{({{{\ensuremath{\mathsf{2}}}}}/{{{\ensuremath{\mathit{a}}}}};{{{\ensuremath{\mathit{b}}}}})}}}})}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{{\ensuremath{\mathit{or({{\ensuremath{\mathsf{{{{\ensuremath{\mathit{b}}}}}}}}},{{\ensuremath{\mathsf{({{{\ensuremath{\mathsf{2}}}}}/{{{\ensuremath{\mathit{a}}}}};{{{\ensuremath{\mathit{b}}}}})}}}})}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } { \label{spec : disj:5}{\tag{s : disj:5}}}\\ { \ensuremath{{{\ensuremath{\mathit{redo}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathit{a}}}}};{{{\ensuremath{\mathit{b}}}}}}}}}\ifempty{{{\ensuremath{\mathbb{u}}}}{{{\ensuremath{\mathit{or({{\ensuremath{\mathsf{{{{\ensuremath{\mathit{c}}}}}}}}},{{\ensuremath{\mathsf{({{{\ensuremath{\mathit{n}}}}}/{{{\ensuremath{\mathit{a}}}}};{{{\ensuremath{\mathit{b}}}}})}}}})}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{{\ensuremath{\mathit{or({{\ensuremath{\mathsf{{{{\ensuremath{\mathit{c}}}}}}}}},{{\ensuremath{\mathsf{({{{\ensuremath{\mathit{n}}}}}/{{{\ensuremath{\mathit{a}}}}};{{{\ensuremath{\mathit{b}}}}})}}}})}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } & { \ensuremath{\,\rightarrowtriangle\,}}{\ensuremath{{{\ensuremath{\mathit{redo}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathit{c}}}}}}}}}\ifempty{{{{\ensuremath{\mathit{n}}}}}/{{{\ensuremath{\mathit{a}}}}};{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{{\ensuremath{\mathit{n}}}}}/{{{\ensuremath{\mathit{a}}}}};{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}}}}}}\rangle}}}}{\label{spec : disj:6}{\tag{s : disj:6 } } } \intertext{true } { \ensuremath{{{\ensuremath{\mathit{call}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathsf{true}}}}}}}}}\ifempty{{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } & { \ensuremath{\,\rightarrowtriangle\,}}{\ensuremath{{{\ensuremath{\mathit{exit}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathsf{true}}}}}}}}}\ifempty{{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } { \label{spec : true:1}{\tag{s : true:1}}}\\ { \ensuremath{{{\ensuremath{\mathit{redo}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathsf{true}}}}}}}}}\ifempty{{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } & { \ensuremath{\,\rightarrowtriangle\,}}{\ensuremath{{{\ensuremath{\mathit{fail}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathsf{true}}}}}}}}}\ifempty{{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } { \label{spec : true:2}{\tag{s : true:2 } } } \intertext{fail } { \ensuremath{{{\ensuremath{\mathit{call}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathsf{fail}}}}}}}}}\ifempty{{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } & { \ensuremath{\,\rightarrowtriangle\,}}{\ensuremath{{{\ensuremath{\mathit{fail}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathsf{fail}}}}}}}}}\ifempty{{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } { \label{spec : fail}{\tag{s : fail } } } \intertext{explicit unification } { \ensuremath{{{\ensuremath{\mathit{call}}}}\mathinner{{\ensuremath{\mathsf{{\ensuremath{{{\ensuremath{\mathit{t}}}}_{{\ensuremath{\mathit{1}}}}}}\mathord={\ensuremath{{{\ensuremath{\mathit{t}}}}_{{\ensuremath{\mathit{2}}}}}}}}}}\ifempty{{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } & { \ensuremath{\,\rightarrowtriangle\,}}\begin{cases } { \ensuremath{{{\ensuremath{\mathit{exit}}}}\mathinner{{\ensuremath{\mathsf{{\ensuremath{{{\ensuremath{\mathit{t}}}}_{{\ensuremath{\mathit{1}}}}}}\mathord={\ensuremath{{{\ensuremath{\mathit{t}}}}_{{\ensuremath{\mathit{2}}}}}}}}}}\ifempty{{{\ensuremath{\mathbb{u}}}}{\ensuremath{\sigma}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{\ensuremath{\sigma}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } , & \text{if } { \ensuremath{{\rm{\text{mgu}}}{\ensuremath{\boldsymbol{(}}}{\ensuremath{{{\ensuremath{\mathit{t}}}}_{{\ensuremath{\mathit{1}}}}}}{\ensuremath{\boldsymbol{,}}}{\ensuremath{{{\ensuremath{\mathit{t}}}}_{{\ensuremath{\mathit{2}}}}}}{\ensuremath{\boldsymbol{)}}}}}={\ensuremath{\sigma}}\\ { \ensuremath{{{\ensuremath{\mathit{fail}}}}\mathinner{{\ensuremath{\mathsf{{\ensuremath{{{\ensuremath{\mathit{t}}}}_{{\ensuremath{\mathit{1}}}}}}\mathord={\ensuremath{{{\ensuremath{\mathit{t}}}}_{{\ensuremath{\mathit{2}}}}}}}}}}\ifempty{{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } , & \text{otherwise } \end{cases}{\label{spec : unif:1}{\tag{s : unif:1}}}\medskip \\ { \ensuremath{{{\ensuremath{\mathit{redo}}}}\mathinner{{\ensuremath{\mathsf{{\ensuremath{{{\ensuremath{\mathit{t}}}}_{{\ensuremath{\mathit{1}}}}}}\mathord={\ensuremath{{{\ensuremath{\mathit{t}}}}_{{\ensuremath{\mathit{2}}}}}}}}}}\ifempty{{{\ensuremath{\mathbb{u}}}}{\ensuremath{\sigma}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{\ensuremath{\sigma}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } & { \ensuremath{\,\rightarrowtriangle\,}}{\ensuremath{{{\ensuremath{\mathit{fail}}}}\mathinner{{\ensuremath{\mathsf{{\ensuremath{{{\ensuremath{\mathit{t}}}}_{{\ensuremath{\mathit{1}}}}}}\mathord={\ensuremath{{{\ensuremath{\mathit{t}}}}_{{\ensuremath{\mathit{2}}}}}}}}}}\ifempty{{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{\ensuremath{\mathbb{u}}}}}}}}}\rangle}}}}{\label{spec : unif:2}{\tag{s : unif:2 } } } \intertext{user - defined atomary goal { \ensuremath{{{\ensuremath{\mathit{g}}}}_{{\ensuremath{\mathit{a } } } } } } } { \ensuremath{{{\ensuremath{\mathit{call}}}}\mathinner{{\ensuremath{\mathsf{{\ensuremath{{{\ensuremath{\mathit{g}}}}_{{\ensuremath{\mathit{a}}}}}}}}}}\ifempty{{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } & { \ensuremath{\,\rightarrowtriangle\,}}\begin{cases } { \ensuremath{{{\ensuremath{\mathit{call}}}}\mathinner{{\ensuremath{\mathsf{{\ensuremath{{\ensuremath{\sigma}}{\ensuremath{\boldsymbol{(}}}{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\boldsymbol{)}}}}}}}}}\ifempty{{\ensuremath{{{\ensuremath{\mathit{g}}}}_{{\ensuremath{\mathit{a}}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{\ensuremath{{{\ensuremath{\mathit{g}}}}_{{\ensuremath{\mathit{a}}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } , & \text{if } { { { \ensuremath{\mathit{h}}}}}{\ensuremath{\mathrel{\mathord:\mathord-}}}{{{\ensuremath{\mathit{b}}}}}\text { is a fresh renaming of a } \\ & \hspace*{-2.7cm}\text{clause in { \ensuremath{\mit\pi } } , } \text{and } { \ensuremath{{\rm{\text{mgu}}}{\ensuremath{\boldsymbol{(}}}{\ensuremath{{{\ensuremath{\mathit{g}}}}_{{\ensuremath{\mathit{a}}}}}}{\ensuremath{\boldsymbol{,}}}{{{\ensuremath{\mathit{h}}}}}{\ensuremath{\boldsymbol{)}}}}}={\ensuremath{\sigma}},\text { and } { \ensuremath{{\ensuremath{\sigma}}{\ensuremath{\boldsymbol{(}}}{\ensuremath{{{\ensuremath{\mathit{g}}}}_{{\ensuremath{\mathit{a}}}}}}{\ensuremath{\boldsymbol{)}}}}}={\ensuremath{{{\ensuremath{\mathit{g}}}}_{{\ensuremath{\mathit{a}}}}}}\\ { \ensuremath{{{\ensuremath{\mathit{fail}}}}\mathinner{{\ensuremath{\mathsf{{\ensuremath{{{\ensuremath{\mathit{g}}}}_{{\ensuremath{\mathit{a}}}}}}}}}}\ifempty{{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } , & \text{otherwise } \end{cases}{\label{spec : atom:1}{\tag{s : atom:1}}}\medskip\\ { \ensuremath{{{\ensuremath{\mathit{exit}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathit{b}}}}}}}}}\ifempty{{\ensuremath{{{\ensuremath{\mathit{g}}}}_{{\ensuremath{\mathit{a}}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{\ensuremath{{{\ensuremath{\mathit{g}}}}_{{\ensuremath{\mathit{a}}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } & { \ensuremath{\,\rightarrowtriangle\,}}{\ensuremath{{{\ensuremath{\mathit{exit}}}}\mathinner{{\ensuremath{\mathsf{{\ensuremath{{{\ensuremath{\mathit{g}}}}_{{\ensuremath{\mathit{a}}}}}}}}}}\ifempty{{{\ensuremath{\mathbb{u}}}}{{{\ensuremath{\mathit{by({{\ensuremath{\mathsf{{{{\ensuremath{\mathit{b}}}}}}}}},{{\ensuremath{\mathsf{{\ensuremath{{{\ensuremath{\mathit{g}}}}_{{\ensuremath{\mathit{a}}}}}}}}}})}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{{\ensuremath{\mathit{by({{\ensuremath{\mathsf{{{{\ensuremath{\mathit{b}}}}}}}}},{{\ensuremath{\mathsf{{\ensuremath{{{\ensuremath{\mathit{g}}}}_{{\ensuremath{\mathit{a}}}}}}}}}})}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } { \label{spec : atom:2}{\tag{s : atom:2}}}\\ { \ensuremath{{{\ensuremath{\mathit{fail}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathit{b}}}}}}}}}\ifempty{{\ensuremath{{{\ensuremath{\mathit{g}}}}_{{\ensuremath{\mathit{a}}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{\ensuremath{{{\ensuremath{\mathit{g}}}}_{{\ensuremath{\mathit{a}}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } & { \ensuremath{\,\rightarrowtriangle\,}}{\ensuremath{{{\ensuremath{\mathit{fail}}}}\mathinner{{\ensuremath{\mathsf{{\ensuremath{{{\ensuremath{\mathit{g}}}}_{{\ensuremath{\mathit{a}}}}}}}}}}\ifempty{{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } { \label{spec : atom:3}{\tag{s : atom:3}}}\\ { \ensuremath{{{\ensuremath{\mathit{redo}}}}\mathinner{{\ensuremath{\mathsf{{\ensuremath{{{\ensuremath{\mathit{g}}}}_{{\ensuremath{\mathit{a}}}}}}}}}}\ifempty{{{\ensuremath{\mathbb{u}}}}{{{\ensuremath{\mathit{by({{\ensuremath{\mathsf{{{{\ensuremath{\mathit{b}}}}}}}}},{{\ensuremath{\mathsf{{\ensuremath{{{\ensuremath{\mathit{g}}}}_{{\ensuremath{\mathit{a}}}}}}'}}}})}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{{\ensuremath{\mathit{by({{\ensuremath{\mathsf{{{{\ensuremath{\mathit{b}}}}}}}}},{{\ensuremath{\mathsf{{\ensuremath{{{\ensuremath{\mathit{g}}}}_{{\ensuremath{\mathit{a}}}}}}'}}}})}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } & { \ensuremath{\,\rightarrowtriangle\,}}{\ensuremath{{{\ensuremath{\mathit{redo}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathit{b}}}}}}}}}\ifempty{{\ensuremath{{{\ensuremath{\mathit{g}}}}_{{\ensuremath{\mathit{a}}}}}}'{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{\ensuremath{{{\ensuremath{\mathit{g}}}}_{{\ensuremath{\mathit{a}}}}}}'{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } { \label{spec : atom:4}{\tag{s : atom:4 } } } \end{aligned}\ ] ] spec : about event : * _ current goal _ is a generalization of _ selected literal _ : rather than focusing upon single literals , we focus upon goals . * _ ancestor _ of a goal is defined in a disambiguating manner , via _ tags_. * the notion of _ environment _ is generalized , to contain following _ bets _ : 1 . variable bindings , 2 . choices taken ( or - branches ) , 3 . used predicate definitions . + environment is represented by one stack , storing each bet as soon as it is computed . for an event to represent the state of pure prolog execution , suffices here one environment and one ancestor stack . about transitions : * port transition relation is functional . the same holds for its converse , if restricted on _ legal events _ , i.e. events that can be reached from an _ initial event _ of the form @xmath20 . * this uniqueness of legal derivations enables _ forward and backward _ derivation steps , in the spirit of the byrd s article . * _ modularity _ of derivation : the execution of a goal can be abstracted like for example @xmath21 . notice the same a - stack . by _ atom _ or _ atomary goal _ we denote only user - defined predications . so @xmath12 , @xmath22 or @xmath23 shall not be considered atoms . the most general unifiers @xmath24shall be chosen to be idempotent , i.e.@xmath25 . the names @xmath26 or @xmath27 of should only suggest that the argument is related to @xmath28 or @xmath29 , but the actual retrieval is determined by the tags @xmath30 and @xmath31 , saying that respectively the first or the second conjunct are currently being tried . for example , the rule states that the call of @xmath32 leads to the call of @xmath28 with immediate ancestor @xmath33 . this kind of add - on mechanism is necessary to be able to correctly handle a query like @xmath34 where retrieval by unification would get stuck on the first conjunct . note the requirement @xmath35 in . since the clauses are in canonical form , unifying the head of a clause with a goal could do no more than rename the goal . since we do not need a renaming of the goal , we may fix the mgu to just operate on the clause . [ logupdate : more ] observe how and serve to implement the _ logical update view _ of lindholm and okeefe @xcite , saying that the definition of a predicate shall be fixed at the time of its call . this is further explained in the following remark . although we memorize the used predicate definition _ on exit _ , the definition will be unaffected by exit bindings , because _ bindings are applied lazily _ : instead of `` eagerly '' applying any bindings as they occur ( e.g. in @xmath23 , in resolution or in read ) , we chose to do this only in conjunction ( in rule ) and nowhere else . due to the rules and , the exit bindings shall not affect the predicate definition like e.g. @xmath36 . also , lazy bindings enable less ` jumpy ' trace . a jumpy trace can be illustrated by the following exit event ( assuming we applied bindings eagerly ) : @xmath37,b,[o|b])}}}}}}}}\ifempty{\{{{{\ensuremath{\mathsf{2}}}}}/([i|b]=[i|b]),append([],b , b){\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}\}{{\ensuremath{\mathbb{\sigma } } } } } { } { \ifempty{\{{{{\ensuremath{\mathsf{2}}}}}/([i|b]=[i|b]),append([],b , b){\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}\}}{}{{{,\ , { } } } { { \ensuremath{\mathsf{\{{{{\ensuremath{\mathsf{2}}}}}/([i|b]=[i|b]),append([],b , b){\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}\ } } } } } } \ifempty{{{\ensuremath{\mathbb{\sigma}}}}}{}{{{,\ , { } } } { { \ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma } } } } } } } } } } } } \ ] ] the problem consists in exiting the goal @xmath38,b , b)}$ ] via @xmath39,b,[o|b])}$ ] , the latter of course being no instance of the former . by means of lazy binding , we avoid the jumpiness , and at the same time make memoing definitions on exit possible . to ensure that the trace of a query execution shows the correct bindings , an event shall be printed only after the current substitution has been applied to it . a perhaps more important collateral advantage of lazy binding is that a successful derivation ( see ) can always be abstracted as follows : @xmath40 even if @xmath41 happened to get further instantiated in the course of this derivation . the instantiation will be reflected in the b - stackbut not in the goal itself . let @xmath1 be a program . _ port transition relation _ @xmath42wrt @xmath1is defined in . the converse relation shall be denoted by @xmath43 . if @xmath44 , we say that @xmath45 _ leads to _ @xmath46 . an event @xmath46 can be _ entered _ , if some event leads to it . an event @xmath46 can be _ left _ , if it leads to some event . the relation @xmath42 is functional , i.e. for each event @xmath46 there can be at most one event @xmath45 such that @xmath47 . the premisses of the transition rules are mutually disjunct , i.e. there are no critical pairs . [ ex : rel : converse ] the converse of the port transition relation is not functional , since there may be more than one event leading to the same event : @xmath48 we could have prevented the ambiguous situation above and made converse relation functional as well , by giving natural conditions on redo - transitions for atomary goal and unification . however , further down it will be shown that , for events that are _ legal _ , the converse relation is functional anyway . let @xmath1 be a program . let @xmath49 , @xmath46 be events . @xmath1-derivation of @xmath46 from @xmath49 _ , written as @xmath50 , is a path from @xmath49 to @xmath46 in the port transition relation wrt @xmath1 . we say that @xmath46 can be _ reached _ from @xmath49 . an _ initial event _ is any event of the form @xmath51 , where @xmath0 is a goal . the goal @xmath0 of an initial event is called a _ top - level goal _ , or a _ query_. let @xmath1 be a program . if there is a goal @xmath0such that @xmath52 is a @xmath1-derivation , then we say that @xmath50 is a _ legal @xmath1-derivation _ , @xmath46 is a _ legal @xmath1-event _ , and @xmath53 is a @xmath1-_execution _ of the query @xmath0 . a legal event @xmath46 is a _ final _ event wrt program @xmath1 , if there is no transition @xmath54 wrt @xmath1 . if @xmath55 is an event , and @xmath56 , then we say that @xmath57 is the _ parent _ of @xmath11 . function is defined as follows : @xmath58 and analogously for disjunction . let @xmath46 be an event with the port @xmath59 . if @xmath60 is one of @xmath61 , then @xmath46 is a _ push _ event . if @xmath60 is one of @xmath62 , then @xmath46 is a _ pop _ event . [ lem : finalevent ] if @xmath46 is a legal pop event , and its a - stack is not empty , then according to the rules ( see also appendix [ appendix : leave ] ) , the possibilities to leave an exit event are : @xmath63 these rules state that it is always possible to leave an exit event @xmath64 , save for the following two restrictions : the parent goal may not be @xmath12 , @xmath22 or a unification ; and if the parent goal @xmath65 is a disjunction , then there has to hold @xmath66 i.e. it is not possible to leave an event @xmath67 if @xmath68 ( and similarly for the second disjunct ) . the first restriction is void , since a parent can not be @xmath12 , @xmath22 or a unification anyway , according to the rules . it remains to show that the second restriction is also void , i.e. a legal exit event has necessarily the property . looking at the rules for entering an exit event , we note that the goal part of an exit event either comes from the a - stack , or is @xmath12 or @xmath23 . the latter two possibilities we may exclude , because @xmath69 can only be derived from @xmath70 , which can not be reached if @xmath71 . similarly for unification . so the goal part of a legal exit event must come from the a - stack . the elements of the a - stackoriginate from call / redo events , and they have the property . in conclusion , we can always leave a legal exit event with a nonempty a - stack . similarly for a fail event . [ lem : uniq ] if @xmath46 is a legal event , then @xmath46 can have only one legal predecessor , and only one successor . in case @xmath46 is non - initial , there is exactly one legal predecessor . in case @xmath46 is non - final , there is exactly one successor . the successor part follows from the functionality of @xmath42 . looking at the rules , we note that only two kinds of events may have more than one predecessor : @xmath72 and @xmath73 . let @xmath73 be a legal event . its predecessor may have been @xmath74 , on the condition that @xmath75 and @xmath76 have no mgu ( rule ) , or it could have been @xmath77 ( rule ) . in the latter case , @xmath77 must be a legal event , so the b - stack@xmath78 had to be derived . the only rule able to derive such a b - stackis , on the condition that the previous event was @xmath74 and @xmath79 . hence , there can be only one legal predecessor of @xmath73 , depending solely on @xmath75 and @xmath76 . by a similar argument we can prove that @xmath72 can have only one legal predecessor . this concludes the proof of functionality of the converse relation , if restricted to the set of legal events . as a notational convenience , all the events which are not final and do not lead to any further events by means of transitions with respect to the given program , are said to lead to the _ impossible event _ , written as @xmath80 . analogously for events that are not initial events and can not be entered . in particular , @xmath81 and @xmath82 with respect to any program . some impossible events are : @xmath83 , @xmath84 ( can not be entered , non - initial ) , and @xmath85 ( can not be left , non - final ) . [ lem : illegal ] if @xmath86 , then @xmath46 is not legal . if @xmath87 , then @xmath46 is not legal . let @xmath88 . if @xmath46 is legal , then , because of the uniqueness of the transition , @xmath45 has to be legal as well . [ lem : uptodate ] for a legal call event @xmath89 holds that @xmath90 , meaning that the substitutionsfrom the b - stack are _ already applied _ upon the goal to be called . in other words , the goal of any legal call event is up - to - date relative to the current substitution . notice that this property holds only for call events . concatenation of stacks we denote by @xmath91 . concatenating to both stacks of an event we denote by @xmath92 : if @xmath93 , then @xmath94 . let @xmath1 be a program . let @xmath95 be one of @xmath96 . if @xmath97 is a legal @xmath1-derivation , then for every a - stack@xmath98and for every b - stack@xmath99such that @xmath100 is a legal event , holds : @xmath101 is also a legal @xmath1-derivation . observe that our rules ( with the exception of ) refer only to the existence of the top element of some stack , never to the emptiness of a stack . since the top element of a stack @xmath102 can not change after appending another stack to @xmath102 , it is possible to emulate each of the original derivation steps using the ` new ' stacks . it remains to consider the rule , which applies the whole current substitution upon the second conjunct . first note that any variables in a legal derivation stem either from the top - level goal or are fresh . according to the , a call event is always up - to - date , i.e. the current substitution has already been applied to the goal . the most general unifiers may be chosen to be idempotent , so a multiple application of a substitution amounts to a single application . hence , if @xmath100 is a legal event , the substitution of @xmath99cannot affect any variables of the original derivation . uniqueness and modularity of legal port derivations allow us to succinctly define some traditional notions . [ def : success ] a goal @xmath11is said to terminate wrt program @xmath1 , if there is a @xmath1-derivation @xmath103 where @xmath95 is one of @xmath104 . in case of @xmath105 , the derivation is _ successful _ , otherwise it is _ failed_. in a failed derivation , @xmath106 . in a successful derivation @xmath107 is @xmath108 , restricted upon the variables of @xmath11 , called the _ computed answer substitution _ for @xmath11 . uniqueness of legal derivation steps enables _ forward and backward _ derivation steps , in the spirit of the byrd s article . push events ( call , redo ) are more amenable to forward steps , and pop events ( exit , fail ) are more amenable to backward steps . we illustrate this by a small example . if the events on the left - hand sides are legal , the following are legal derivations ( for appropriate @xmath109 , @xmath110 ) : @xmath111 the first statement claims : if @xmath112 is legal , then it was reached via @xmath113 . without inspecting @xmath99 , in general it is not known whether a disjunction succeeded via its first , or via its second member . but in this particular disjunction , the second member can not succeed : assume there are some @xmath114 , @xmath109 with @xmath115 . according to the rules : @xmath116 so according to , @xmath117 is not a legal event , which proves . similarly , the non - legal derivation proves . modularity of legal derivations enables _ abstracting the execution _ of a goal , like in the following example . assume that a goal @xmath118 succeeds , i.e. @xmath119 . then we have the following legal derivation : @xmath120 if @xmath118 fails , then we have : @xmath121 in this paper we give a simple mathematical definition s : ppof the 4-port model of pure prolog . some potential for formal verification of pure prolog has been outlined . there are two interesting directions for future work in this area : \(1 ) formal specification of the control flow of _ full standard prolog _ ( currently we have a prototype for this , within the 4-port model ) \(2 ) formal specification and proof of some non - trivial program properties , like adequacy and non - interference of a practical program transformation . concerning attempts to formally define the 4-port model , we are aware of only few previous works . one is a graph - based model of tobermann and beckstein @xcite , who formalize the graph traversal idea of byrd , defining the notion of a _ trace _ ( of a given query with respect to a given program ) , as a path in a trace graph . the ports are quite lucidly defined as hierarchical nodes of such a graph . however , even for a simple recursive program and a ground query , with a finite sld - tree , the corresponding trace graph is infinite , which limits its applicability . another model of byrd box is a continuation - based approach of jahier , ducass and ridoux @xcite . there is also a stack - based attempt in @xcite , but although it provides for some parametrizing , it suffers essentially the same problem as the continuation - based approach , and also the prototypical implementation of the tracer given in @xcite , taken as a specification of prolog execution : in these three attempts , a port is represented by some semantic action ( e.g. writing of a message ) , instead of a formal method . therefore it is not clear how to use any of these models to prove some port - related assertions . in contrast to the few specifications of the byrd box , there are many more general models of pure ( or even full ) prolog execution . due to space limitations we mention here only some models , directly relevant to s : pp , and for a more comprehensive discussion see e.g. @xcite . comparable to our work are the stack - based approaches . strk gives in @xcite , as a side issue , a simple operational semantics of pure logic programming . a state of execution is a stack of frame stacks , where each frame consists of a goal ( ancestor ) and an environment . in comparison , our state of execution consists of exactly one environment and one ancestor stack . the seminal paper of jones and mycroft @xcite was the first to present a stack - based model of execution , applicable to pure prolog with cut added . it uses a sequence of frames . in these stack - based approaches ( including our previous attempt @xcite ) , there is no _ modularity _ , i.e.it is not possible to abstract the execution of a subgoal . many thanks for helpful comments are due to anonymous referees . dedc96 lawrence byrd . understanding the control flow of prolog programs . in s. a. trnlund , editor , _ proc . of the 1980 logic programming workshop _ , pages 127138 , debrecen , hungary , 1980 . also as d. a. i. research paper no.151 . p. deransart , a. ed - dbali , and l. cervoni . . springer - verlag , 1996 . e. jahier , m. ducass , and o. ridoux . specifying byrd s box model with a continuation semantics . in _ proc . of the wlpe99 , las cruces , nm _ , volume 30 of _ entcs_. elsevier , 2000 . http://www.elsevier.nl/locate/entcs/volume30.html . n. d. jones and a. mycroft . stepwise development of operational and denotational semantics for prolog . in _ proc . of the 1st int . symposium on logic programming ( slp84 ) _ , pages 281288 , atlantic city , 1984 . m. kula and c. beierle . defining standard prolog in rewriting logic . in k. futatsugi , editor , _ proc . of the 3rd int . workshop on rewriting logic and its applications ( wrla 2000 ) , kanazawa _ , volume 36 of _ entcs_. elsevier , 2001 . http://www.elsevier.nl/locate/entcs/volume36.html . a. king and l. lu . a backward analysis for constraint logic programs . , 2(4):517547 , 2002 . m. kula . a rewriting prolog semantics . in m. leuschel , a. podelski , r. ramakrishnan c. and u. ultes - nitsche , editors , _ proc . of the cl 2000 workshop on verification and computational logic ( vcl 2000 ) , london _ , 2000 . t. lindgren . control flow analysis of prolog ( extended remix ) . technical report 112 , uppsala university , 1995 . http://www.csd.uu.se/papers/reports.html . t. lindholm and r. a. okeefe . efficient implementation of a defensible semantics for dynamic prolog code . in _ proc . of the 4th int . conference on logic programming ( iclp87 ) _ , pages 2139 , melbourne , 1987 . robert f. strk . the theoretical foundations of lptp ( a logic program theorem prover ) . , 36(3):241269 , 1998 . source distribution http://www.inf.ethz.ch/staerk/lptp.html . g. tobermann and c. beckstein . what s in a trace : the box model revisited . in _ proc . of the 1st int . workshop on automated and algorithmic debugging ( aadebug93 ) , linkping _ , volume 749 of _ lncs_. springer - verlag , 1993 . @xmath19 \intertext{leaving a call event } { \ensuremath{{{\ensuremath{\mathit{call}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathit{a}}}}},{{{\ensuremath{\mathit{b}}}}}}}}}\ifempty{{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } & { \ensuremath{\,\rightarrowtriangle\,}}{\ensuremath{{{\ensuremath{\mathit{call}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathit{a}}}}}}}}}\ifempty{{{{\ensuremath{\mathsf{1}}}}}/{{{\ensuremath{\mathit{a}}}}},{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{{\ensuremath{\mathsf{1}}}}}/{{{\ensuremath{\mathit{a}}}}},{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } { \tag{s : conj:1}}\\ { \ensuremath{{{\ensuremath{\mathit{call}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathit{a}}}}};{{{\ensuremath{\mathit{b}}}}}}}}}\ifempty{{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } & { \ensuremath{\,\rightarrowtriangle\,}}{\ensuremath{{{\ensuremath{\mathit{call}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathit{a}}}}}}}}}\ifempty{{{{\ensuremath{\mathsf{1}}}}}/{{{\ensuremath{\mathit{a}}}}};{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{{\ensuremath{\mathsf{1}}}}}/{{{\ensuremath{\mathit{a}}}}};{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } { \tag{s : disj:1}}\\ { \ensuremath{{{\ensuremath{\mathit{call}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathsf{true}}}}}}}}}\ifempty{{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } & { \ensuremath{\,\rightarrowtriangle\,}}{\ensuremath{{{\ensuremath{\mathit{exit}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathsf{true}}}}}}}}}\ifempty{{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } { \tag{s : true:1}}\\ { \ensuremath{{{\ensuremath{\mathit{call}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathsf{fail}}}}}}}}}\ifempty{{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } & { \ensuremath{\,\rightarrowtriangle\,}}{\ensuremath{{{\ensuremath{\mathit{fail}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathsf{fail}}}}}}}}}\ifempty{{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } { \tag{s : fail}}\\ { \ensuremath{{{\ensuremath{\mathit{call}}}}\mathinner{{\ensuremath{\mathsf{{\ensuremath{{{\ensuremath{\mathit{t}}}}_{{\ensuremath{\mathit{1}}}}}}\mathord={\ensuremath{{{\ensuremath{\mathit{t}}}}_{{\ensuremath{\mathit{2}}}}}}}}}}\ifempty{{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } & { \ensuremath{\,\rightarrowtriangle\,}}\begin{cases } { \ensuremath{{{\ensuremath{\mathit{exit}}}}\mathinner{{\ensuremath{\mathsf{{\ensuremath{{{\ensuremath{\mathit{t}}}}_{{\ensuremath{\mathit{1}}}}}}\mathord={\ensuremath{{{\ensuremath{\mathit{t}}}}_{{\ensuremath{\mathit{2}}}}}}}}}}\ifempty{{{\ensuremath{\mathbb{u}}}}{\ensuremath{\sigma}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{\ensuremath{\sigma}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } , & \text{if } { \ensuremath{{\rm{\text{mgu}}}{\ensuremath{\boldsymbol{(}}}{\ensuremath{{{\ensuremath{\mathit{t}}}}_{{\ensuremath{\mathit{1}}}}}}{\ensuremath{\boldsymbol{,}}}{\ensuremath{{{\ensuremath{\mathit{t}}}}_{{\ensuremath{\mathit{2}}}}}}{\ensuremath{\boldsymbol{)}}}}}={\ensuremath{\sigma}}\\ { \ensuremath{{{\ensuremath{\mathit{fail}}}}\mathinner{{\ensuremath{\mathsf{{\ensuremath{{{\ensuremath{\mathit{t}}}}_{{\ensuremath{\mathit{1}}}}}}\mathord={\ensuremath{{{\ensuremath{\mathit{t}}}}_{{\ensuremath{\mathit{2}}}}}}}}}}\ifempty{{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } , & \text{otherwise } \end{cases}{\tag{s : unif:1}}\medskip \\ { \ensuremath{{{\ensuremath{\mathit{call}}}}\mathinner{{\ensuremath{\mathsf{{\ensuremath{{{\ensuremath{\mathit{g}}}}_{{\ensuremath{\mathit{a}}}}}}}}}}\ifempty{{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } & { \ensuremath{\,\rightarrowtriangle\,}}\begin{cases } { \ensuremath{{{\ensuremath{\mathit{call}}}}\mathinner{{\ensuremath{\mathsf{{\ensuremath{{\ensuremath{\sigma}}{\ensuremath{\boldsymbol{(}}}{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\boldsymbol{)}}}}}}}}}\ifempty{{\ensuremath{{{\ensuremath{\mathit{g}}}}_{{\ensuremath{\mathit{a}}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{\ensuremath{{{\ensuremath{\mathit{g}}}}_{{\ensuremath{\mathit{a}}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } , & \text{if } { { { \ensuremath{\mathit{h}}}}}{\ensuremath{\mathrel{\mathord:\mathord-}}}{{{\ensuremath{\mathit{b}}}}}\text { is a fresh renaming of a } \\ & \hspace*{-2.7cm}\text{clause in { \ensuremath{\mit\pi } } , } \text{and } { \ensuremath{{\rm{\text{mgu}}}{\ensuremath{\boldsymbol{(}}}{\ensuremath{{{\ensuremath{\mathit{g}}}}_{{\ensuremath{\mathit{a}}}}}}{\ensuremath{\boldsymbol{,}}}{{{\ensuremath{\mathit{h}}}}}{\ensuremath{\boldsymbol{)}}}}}={\ensuremath{\sigma}},\text { and } { \ensuremath{{\ensuremath{\sigma}}{\ensuremath{\boldsymbol{(}}}{\ensuremath{{{\ensuremath{\mathit{g}}}}_{{\ensuremath{\mathit{a}}}}}}{\ensuremath{\boldsymbol{)}}}}}={\ensuremath{{{\ensuremath{\mathit{g}}}}_{{\ensuremath{\mathit{a}}}}}}\\ { \ensuremath{{{\ensuremath{\mathit{fail}}}}\mathinner{{\ensuremath{\mathsf{{\ensuremath{{{\ensuremath{\mathit{g}}}}_{{\ensuremath{\mathit{a}}}}}}}}}}\ifempty{{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } , & \text{otherwise } \end{cases}{\tag{s : atom:1}}\medskip\\ \intertext{leaving a redo event } { \ensuremath{{{\ensuremath{\mathit{redo}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathit{a}}}}},{{{\ensuremath{\mathit{b}}}}}}}}}\ifempty{{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } & { \ensuremath{\,\rightarrowtriangle\,}}{\ensuremath{{{\ensuremath{\mathit{redo}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathit{b}}}}}}}}}\ifempty{{{{\ensuremath{\mathsf{2}}}}}/{{{\ensuremath{\mathit{a}}}}},{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{{\ensuremath{\mathsf{2}}}}}/{{{\ensuremath{\mathit{a}}}}},{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } { \tag{s : conj:6}}\\ { \ensuremath{{{\ensuremath{\mathit{redo}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathit{a}}}}};{{{\ensuremath{\mathit{b}}}}}}}}}\ifempty{{{\ensuremath{\mathbb{u}}}}{{{\ensuremath{\mathit{or({{\ensuremath{\mathsf{{{{\ensuremath{\mathit{c}}}}}}}}},{{\ensuremath{\mathsf{({{{\ensuremath{\mathit{n}}}}}/{{{\ensuremath{\mathit{a}}}}};{{{\ensuremath{\mathit{b}}}}})}}}})}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{{\ensuremath{\mathit{or({{\ensuremath{\mathsf{{{{\ensuremath{\mathit{c}}}}}}}}},{{\ensuremath{\mathsf{({{{\ensuremath{\mathit{n}}}}}/{{{\ensuremath{\mathit{a}}}}};{{{\ensuremath{\mathit{b}}}}})}}}})}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } & { \ensuremath{\,\rightarrowtriangle\,}}{\ensuremath{{{\ensuremath{\mathit{redo}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathit{c}}}}}}}}}\ifempty{{{{\ensuremath{\mathit{n}}}}}/{{{\ensuremath{\mathit{a}}}}};{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{{\ensuremath{\mathit{n}}}}}/{{{\ensuremath{\mathit{a}}}}};{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}}}}}}\rangle}}}}{\tag{s : disj:6}}\\ { \ensuremath{{{\ensuremath{\mathit{redo}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathsf{true}}}}}}}}}\ifempty{{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } & { \ensuremath{\,\rightarrowtriangle\,}}{\ensuremath{{{\ensuremath{\mathit{fail}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathsf{true}}}}}}}}}\ifempty{{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } { \tag{s : true:2}}\\ { \ensuremath{{{\ensuremath{\mathit{redo}}}}\mathinner{{\ensuremath{\mathsf{{\ensuremath{{{\ensuremath{\mathit{t}}}}_{{\ensuremath{\mathit{1}}}}}}\mathord={\ensuremath{{{\ensuremath{\mathit{t}}}}_{{\ensuremath{\mathit{2}}}}}}}}}}\ifempty{{{\ensuremath{\mathbb{u}}}}{\ensuremath{\sigma}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{\ensuremath{\sigma}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } & { \ensuremath{\,\rightarrowtriangle\,}}{\ensuremath{{{\ensuremath{\mathit{fail}}}}\mathinner{{\ensuremath{\mathsf{{\ensuremath{{{\ensuremath{\mathit{t}}}}_{{\ensuremath{\mathit{1}}}}}}\mathord={\ensuremath{{{\ensuremath{\mathit{t}}}}_{{\ensuremath{\mathit{2}}}}}}}}}}\ifempty{{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{\ensuremath{\mathbb{u}}}}}}}}}\rangle}}}}{\tag{s : unif:2}}\\ { \ensuremath{{{\ensuremath{\mathit{redo}}}}\mathinner{{\ensuremath{\mathsf{{\ensuremath{{{\ensuremath{\mathit{g}}}}_{{\ensuremath{\mathit{a}}}}}}}}}}\ifempty{{{\ensuremath{\mathbb{u}}}}{{{\ensuremath{\mathit{by({{\ensuremath{\mathsf{{{{\ensuremath{\mathit{b}}}}}}}}},{{\ensuremath{\mathsf{{\ensuremath{{{\ensuremath{\mathit{g}}}}_{{\ensuremath{\mathit{a}}}}}}'}}}})}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{{\ensuremath{\mathit{by({{\ensuremath{\mathsf{{{{\ensuremath{\mathit{b}}}}}}}}},{{\ensuremath{\mathsf{{\ensuremath{{{\ensuremath{\mathit{g}}}}_{{\ensuremath{\mathit{a}}}}}}'}}}})}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } & { \ensuremath{\,\rightarrowtriangle\,}}{\ensuremath{{{\ensuremath{\mathit{redo}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathit{b}}}}}}}}}\ifempty{{\ensuremath{{{\ensuremath{\mathit{g}}}}_{{\ensuremath{\mathit{a}}}}}}'{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{\ensuremath{{{\ensuremath{\mathit{g}}}}_{{\ensuremath{\mathit{a}}}}}}'{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } { \tag{s : atom:4}}\\ \intertext{leaving an exit event } { \ensuremath{{{\ensuremath{\mathit{exit}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathit{a}}}}}'}}}}\ifempty{{{{\ensuremath{\mathsf{1}}}}}/{{{\ensuremath{\mathit{a}}}}},{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{{\ensuremath{\mathsf{1}}}}}/{{{\ensuremath{\mathit{a}}}}},{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } & { \ensuremath{\,\rightarrowtriangle\,}}{\ensuremath{{{\ensuremath{\mathit{call}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathit{b}}}}}''}}}}\ifempty{{{{\ensuremath{\mathsf{2}}}}}/{{{\ensuremath{\mathit{a}}}}},{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{{\ensuremath{\mathsf{2}}}}}/{{{\ensuremath{\mathit{a}}}}},{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } , \text { with } { { { \ensuremath{\mathit{b}}}}}''{\ensuremath{\mathrel{\joinrel{:=}}}}{{\ensuremath{{\ensuremath{{\rm{\text{substof}}}{\ensuremath{\boldsymbol{(}}}{{\ensuremath{\mathbb{\sigma}}}}{\ensuremath{\boldsymbol{)}}}}}{\ensuremath{\boldsymbol{(}}}{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\boldsymbol { ) } } } } } } { \tag{s : conj:2}}\\ { \ensuremath{{{\ensuremath{\mathit{exit}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathit{b}}}}}'}}}}\ifempty{{{{\ensuremath{\mathsf{2}}}}}/{{{\ensuremath{\mathit{a}}}}},{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{{\ensuremath{\mathsf{2}}}}}/{{{\ensuremath{\mathit{a}}}}},{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } & { \ensuremath{\,\rightarrowtriangle\,}}{\ensuremath{{{\ensuremath{\mathit{exit}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathit{a}}}}},{{{\ensuremath{\mathit{b}}}}}}}}}\ifempty{{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } { \tag{s : conj:4}}\\ { \ensuremath{{{\ensuremath{\mathit{exit}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathit{a}}}}}}}}}\ifempty{{{{\ensuremath{\mathsf{1}}}}}/{{{\ensuremath{\mathit{a}}}}};{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{{\ensuremath{\mathsf{1}}}}}/{{{\ensuremath{\mathit{a}}}}};{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } & { \ensuremath{\,\rightarrowtriangle\,}}{\ensuremath{{{\ensuremath{\mathit{exit}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathit{a}}}}};{{{\ensuremath{\mathit{b}}}}}}}}}\ifempty{{{\ensuremath{\mathbb{u}}}}{{{\ensuremath{\mathit{or({{\ensuremath{\mathsf{{{{\ensuremath{\mathit{a}}}}}}}}},{{\ensuremath{\mathsf{({{{\ensuremath{\mathsf{1}}}}}/{{{\ensuremath{\mathit{a}}}}};{{{\ensuremath{\mathit{b}}}}})}}}})}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{{\ensuremath{\mathit{or({{\ensuremath{\mathsf{{{{\ensuremath{\mathit{a}}}}}}}}},{{\ensuremath{\mathsf{({{{\ensuremath{\mathsf{1}}}}}/{{{\ensuremath{\mathit{a}}}}};{{{\ensuremath{\mathit{b}}}}})}}}})}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } { \tag{s : disj:4}}\\ { \ensuremath{{{\ensuremath{\mathit{exit}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathit{b}}}}}}}}}\ifempty{{{{\ensuremath{\mathsf{2}}}}}/{{{\ensuremath{\mathit{a}}}}};{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{{\ensuremath{\mathsf{2}}}}}/{{{\ensuremath{\mathit{a}}}}};{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } & { \ensuremath{\,\rightarrowtriangle\,}}{\ensuremath{{{\ensuremath{\mathit{exit}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathit{a}}}}};{{{\ensuremath{\mathit{b}}}}}}}}}\ifempty{{{\ensuremath{\mathbb{u}}}}{{{\ensuremath{\mathit{or({{\ensuremath{\mathsf{{{{\ensuremath{\mathit{b}}}}}}}}},{{\ensuremath{\mathsf{({{{\ensuremath{\mathsf{2}}}}}/{{{\ensuremath{\mathit{a}}}}};{{{\ensuremath{\mathit{b}}}}})}}}})}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{{\ensuremath{\mathit{or({{\ensuremath{\mathsf{{{{\ensuremath{\mathit{b}}}}}}}}},{{\ensuremath{\mathsf{({{{\ensuremath{\mathsf{2}}}}}/{{{\ensuremath{\mathit{a}}}}};{{{\ensuremath{\mathit{b}}}}})}}}})}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } { \tag{s : disj:5}}\\ { \ensuremath{{{\ensuremath{\mathit{exit}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathit{b}}}}}}}}}\ifempty{{\ensuremath{{{\ensuremath{\mathit{g}}}}_{{\ensuremath{\mathit{a}}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{\ensuremath{{{\ensuremath{\mathit{g}}}}_{{\ensuremath{\mathit{a}}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } & { \ensuremath{\,\rightarrowtriangle\,}}{\ensuremath{{{\ensuremath{\mathit{exit}}}}\mathinner{{\ensuremath{\mathsf{{\ensuremath{{{\ensuremath{\mathit{g}}}}_{{\ensuremath{\mathit{a}}}}}}}}}}\ifempty{{{\ensuremath{\mathbb{u}}}}{{{\ensuremath{\mathit{by({{\ensuremath{\mathsf{{{{\ensuremath{\mathit{b}}}}}}}}},{{\ensuremath{\mathsf{{\ensuremath{{{\ensuremath{\mathit{g}}}}_{{\ensuremath{\mathit{a}}}}}}}}}})}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{{\ensuremath{\mathit{by({{\ensuremath{\mathsf{{{{\ensuremath{\mathit{b}}}}}}}}},{{\ensuremath{\mathsf{{\ensuremath{{{\ensuremath{\mathit{g}}}}_{{\ensuremath{\mathit{a}}}}}}}}}})}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } { \tag{s : atom:2}}\\ \intertext{leaving a fail event } { \ensuremath{{{\ensuremath{\mathit{fail}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathit{a}}}}}'}}}}\ifempty{{{{\ensuremath{\mathsf{1}}}}}/{{{\ensuremath{\mathit{a}}}}},{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{{\ensuremath{\mathsf{1}}}}}/{{{\ensuremath{\mathit{a}}}}},{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } & { \ensuremath{\,\rightarrowtriangle\,}}{\ensuremath{{{\ensuremath{\mathit{fail}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathit{a}}}}},{{{\ensuremath{\mathit{b}}}}}}}}}\ifempty{{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } { \tag{s : conj:3}}\\ { \ensuremath{{{\ensuremath{\mathit{fail}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathit{b}}}}}'}}}}\ifempty{{{{\ensuremath{\mathsf{2}}}}}/{{{\ensuremath{\mathit{a}}}}},{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{{\ensuremath{\mathsf{2}}}}}/{{{\ensuremath{\mathit{a}}}}},{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } & { \ensuremath{\,\rightarrowtriangle\,}}{\ensuremath{{{\ensuremath{\mathit{redo}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathit{a}}}}}}}}}\ifempty{{{{\ensuremath{\mathsf{1}}}}}/{{{\ensuremath{\mathit{a}}}}},{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{{\ensuremath{\mathsf{1}}}}}/{{{\ensuremath{\mathit{a}}}}},{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } { \tag{s : conj:5}}\\ { \ensuremath{{{\ensuremath{\mathit{fail}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathit{a}}}}}}}}}\ifempty{{{{\ensuremath{\mathsf{1}}}}}/{{{\ensuremath{\mathit{a}}}}};{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{{\ensuremath{\mathsf{1}}}}}/{{{\ensuremath{\mathit{a}}}}};{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } & { \ensuremath{\,\rightarrowtriangle\,}}{\ensuremath{{{\ensuremath{\mathit{call}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathit{b}}}}}}}}}\ifempty{{{{\ensuremath{\mathsf{2}}}}}/{{{\ensuremath{\mathit{a}}}}};{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{{\ensuremath{\mathsf{2}}}}}/{{{\ensuremath{\mathit{a}}}}};{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } { \tag{s : disj:2}}\\ { \ensuremath{{{\ensuremath{\mathit{fail}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathit{b}}}}}}}}}\ifempty{{{{\ensuremath{\mathsf{2}}}}}/{{{\ensuremath{\mathit{a}}}}};{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{{\ensuremath{\mathsf{2}}}}}/{{{\ensuremath{\mathit{a}}}}};{{{\ensuremath{\mathit{b}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } & { \ensuremath{\,\rightarrowtriangle\,}}{\ensuremath{{{\ensuremath{\mathit{fail}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathit{a}}}}};{{{\ensuremath{\mathit{b}}}}}}}}}\ifempty{{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } { \tag{s : disj:3}}\\ { \ensuremath{{{\ensuremath{\mathit{fail}}}}\mathinner{{\ensuremath{\mathsf{{{{\ensuremath{\mathit{b}}}}}}}}}\ifempty{{\ensuremath{{{\ensuremath{\mathit{g}}}}_{{\ensuremath{\mathit{a}}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{\ensuremath{{{\ensuremath{\mathit{g}}}}_{{\ensuremath{\mathit{a}}}}}}{\ensuremath{\mathop{\bullet}}}{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } & { \ensuremath{\,\rightarrowtriangle\,}}{\ensuremath{{{\ensuremath{\mathit{fail}}}}\mathinner{{\ensuremath{\mathsf{{\ensuremath{{{\ensuremath{\mathit{g}}}}_{{\ensuremath{\mathit{a}}}}}}}}}}\ifempty{{{\ensuremath{\mathbb{u}}}}{{\ensuremath{\mathbb{\sigma}}}}}{}{{\langle\textstyle\frac{{{\ensuremath{\mathit{{{\ensuremath{\mathbb{\sigma}}}}}}}}}{{{\ensuremath{\mathsf{{{\ensuremath{\mathbb{u}}}}}}}}}\rangle } } } } { \tag{s : atom:3 } } \ ] ] spec : assume the following program @xmath1 : @xmath123 @xmath42@xmath124 @xmath42@xmath125 @xmath42@xmath126 @xmath42@xmath127 @xmath42@xmath128}}}{\ensuremath{\mathop{\bullet}}}{}{{{\ensuremath{\mathit{nil}}}}}\ } } { } { \ifempty{\{one(x , y){\ensuremath{\mathop{\bullet}}}{}1/one(x , y),two(x , y){\ensuremath{\mathop{\bullet}}}{}post(x , y){\ensuremath{\mathop{\bullet}}}{}1/post(x , y),fail{\ensuremath{\mathop{\bullet}}}{}{{{\ensuremath{\mathit{nil}}}}}\}}{}{{{,\ , { } } } \colorbox{svetlosivo}{{{\ensuremath{\mathsf{\{one(x , y){\ensuremath{\mathop{\bullet}}}{}1/one(x , y),two(x , y){\ensuremath{\mathop{\bullet}}}{}post(x , y){\ensuremath{\mathop{\bullet}}}{}1/post(x , y),fail{\ensuremath{\mathop{\bullet}}}{}{{{\ensuremath{\mathit{nil}}}}}\ } } } } } } } \ifempty{\{{\colorbox{sivo}{\bf\color{white}{\mathversion{bold}[{\ensuremath{x\mathord/1}}]}}}{\ensuremath{\mathop{\bullet}}}{}{{{\ensuremath{\mathit{nil}}}}}\}}{}{{{,\ , { } } } { { \ensuremath{\mathit{\{{\colorbox{sivo}{\bf\color{white}{\mathversion{bold}[{\ensuremath{x\mathord/1}}]}}}{\ensuremath{\mathop{\bullet}}}{}{{{\ensuremath{\mathit{nil}}}}}\ } } } } } } } $ ] @xmath42@xmath129}}}{\ensuremath{\mathop{\bullet}}}{}{{{\ensuremath{\mathit{nil}}}}}\ } } { } { \ifempty{\{1/one(x , y),two(x , y){\ensuremath{\mathop{\bullet}}}{}post(x , y){\ensuremath{\mathop{\bullet}}}{}1/post(x , y),fail{\ensuremath{\mathop{\bullet}}}{}{{{\ensuremath{\mathit{nil}}}}}\}}{}{{{,\ , { } } } \colorbox{svetlosivo}{{{\ensuremath{\mathsf{\{1/one(x , y),two(x , y){\ensuremath{\mathop{\bullet}}}{}post(x , y){\ensuremath{\mathop{\bullet}}}{}1/post(x , y),fail{\ensuremath{\mathop{\bullet}}}{}{{{\ensuremath{\mathit{nil}}}}}\ } } } } } } } \ifempty{\{{{{\ensuremath{\mathit{by({{\ensuremath{\mathsf{x{\ensuremath{\mathord=}}{}1}}}},{{\ensuremath{\mathsf{one(x , y)}}}})}}}}}{\ensuremath{\mathop{\bullet}}}{}{\colorbox{sivo}{\bf\color{white}{\mathversion{bold}[{\ensuremath{x\mathord/1}}]}}}{\ensuremath{\mathop{\bullet}}}{}{{{\ensuremath{\mathit{nil}}}}}\}}{}{{{,\ , { } } } { { \ensuremath{\mathit{\{{{{\ensuremath{\mathit{by({{\ensuremath{\mathsf{x{\ensuremath{\mathord=}}{}1}}}},{{\ensuremath{\mathsf{one(x , y)}}}})}}}}}{\ensuremath{\mathop{\bullet}}}{}{\colorbox{sivo}{\bf\color{white}{\mathversion{bold}[{\ensuremath{x\mathord/1}}]}}}{\ensuremath{\mathop{\bullet}}}{}{{{\ensuremath{\mathit{nil}}}}}\ } } } } } } } $ ] @xmath42@xmath130}}}{\ensuremath{\mathop{\bullet}}}{}{{{\ensuremath{\mathit{nil}}}}}\ } } { } { \ifempty{\{2/one(x , y),two(x , y){\ensuremath{\mathop{\bullet}}}{}post(x , y){\ensuremath{\mathop{\bullet}}}{}1/post(x , y),fail{\ensuremath{\mathop{\bullet}}}{}{{{\ensuremath{\mathit{nil}}}}}\}}{}{{{,\ , { } } } \colorbox{svetlosivo}{{{\ensuremath{\mathsf{\{2/one(x , y),two(x , y){\ensuremath{\mathop{\bullet}}}{}post(x , y){\ensuremath{\mathop{\bullet}}}{}1/post(x , y),fail{\ensuremath{\mathop{\bullet}}}{}{{{\ensuremath{\mathit{nil}}}}}\ } } } } } } } \ifempty{\{{{{\ensuremath{\mathit{by({{\ensuremath{\mathsf{x{\ensuremath{\mathord=}}{}1}}}},{{\ensuremath{\mathsf{one(x , y)}}}})}}}}}{\ensuremath{\mathop{\bullet}}}{}{\colorbox{sivo}{\bf\color{white}{\mathversion{bold}[{\ensuremath{x\mathord/1}}]}}}{\ensuremath{\mathop{\bullet}}}{}{{{\ensuremath{\mathit{nil}}}}}\}}{}{{{,\ , { } } } { { \ensuremath{\mathit{\{{{{\ensuremath{\mathit{by({{\ensuremath{\mathsf{x{\ensuremath{\mathord=}}{}1}}}},{{\ensuremath{\mathsf{one(x , y)}}}})}}}}}{\ensuremath{\mathop{\bullet}}}{}{\colorbox{sivo}{\bf\color{white}{\mathversion{bold}[{\ensuremath{x\mathord/1}}]}}}{\ensuremath{\mathop{\bullet}}}{}{{{\ensuremath{\mathit{nil}}}}}\ } } } } } } } $ ] @xmath42@xmath131}}}{\ensuremath{\mathop{\bullet}}}{}{{{\ensuremath{\mathit{nil}}}}}\ } } { } { \ifempty{\{two(1,y){\ensuremath{\mathop{\bullet}}}{}2/one(x , y),two(x , y){\ensuremath{\mathop{\bullet}}}{}post(x , y){\ensuremath{\mathop{\bullet}}}{}1/post(x , y),fail{\ensuremath{\mathop{\bullet}}}{}{{{\ensuremath{\mathit{nil}}}}}\}}{}{{{,\ , { } } } \colorbox{svetlosivo}{{{\ensuremath{\mathsf{\{two(1,y){\ensuremath{\mathop{\bullet}}}{}2/one(x , y),two(x , y){\ensuremath{\mathop{\bullet}}}{}post(x , y){\ensuremath{\mathop{\bullet}}}{}1/post(x , y),fail{\ensuremath{\mathop{\bullet}}}{}{{{\ensuremath{\mathit{nil}}}}}\ } } } } } } } \ifempty{\{{{{\ensuremath{\mathit{by({{\ensuremath{\mathsf{x{\ensuremath{\mathord=}}{}1}}}},{{\ensuremath{\mathsf{one(x , y)}}}})}}}}}{\ensuremath{\mathop{\bullet}}}{}{\colorbox{sivo}{\bf\color{white}{\mathversion{bold}[{\ensuremath{x\mathord/1}}]}}}{\ensuremath{\mathop{\bullet}}}{}{{{\ensuremath{\mathit{nil}}}}}\}}{}{{{,\ , { } } } { { \ensuremath{\mathit{\{{{{\ensuremath{\mathit{by({{\ensuremath{\mathsf{x{\ensuremath{\mathord=}}{}1}}}},{{\ensuremath{\mathsf{one(x , y)}}}})}}}}}{\ensuremath{\mathop{\bullet}}}{}{\colorbox{sivo}{\bf\color{white}{\mathversion{bold}[{\ensuremath{x\mathord/1}}]}}}{\ensuremath{\mathop{\bullet}}}{}{{{\ensuremath{\mathit{nil}}}}}\ } } } } } } } $ ] @xmath42@xmath132}}}{\ensuremath{\mathop{\bullet}}}{}{{{\ensuremath{\mathit{nil}}}}}\ } } { } { \ifempty{\{(1/(y{\ensuremath{\mathord=}}{}a);y{\ensuremath{\mathord=}}{}b){\ensuremath{\mathop{\bullet}}}{}two(1,y){\ensuremath{\mathop{\bullet}}}{}2/one(x , y),two(x , y){\ensuremath{\mathop{\bullet}}}{}post(x , y){\ensuremath{\mathop{\bullet}}}{}1/post(x , y),fail{\ensuremath{\mathop{\bullet}}}{}{{{\ensuremath{\mathit{nil}}}}}\}}{}{{{,\ , { } } } \colorbox{svetlosivo}{{{\ensuremath{\mathsf{\{(1/(y{\ensuremath{\mathord=}}{}a);y{\ensuremath{\mathord=}}{}b){\ensuremath{\mathop{\bullet}}}{}two(1,y){\ensuremath{\mathop{\bullet}}}{}2/one(x , y),two(x , y){\ensuremath{\mathop{\bullet}}}{}post(x , y){\ensuremath{\mathop{\bullet}}}{}1/post(x , y),fail{\ensuremath{\mathop{\bullet}}}{}{{{\ensuremath{\mathit{nil}}}}}\ } } } } } } } \ifempty{\{{{{\ensuremath{\mathit{by({{\ensuremath{\mathsf{x{\ensuremath{\mathord=}}{}1}}}},{{\ensuremath{\mathsf{one(x , y)}}}})}}}}}{\ensuremath{\mathop{\bullet}}}{}{\colorbox{sivo}{\bf\color{white}{\mathversion{bold}[{\ensuremath{x\mathord/1}}]}}}{\ensuremath{\mathop{\bullet}}}{}{{{\ensuremath{\mathit{nil}}}}}\}}{}{{{,\ , { } } } { { \ensuremath{\mathit{\{{{{\ensuremath{\mathit{by({{\ensuremath{\mathsf{x{\ensuremath{\mathord=}}{}1}}}},{{\ensuremath{\mathsf{one(x , y)}}}})}}}}}{\ensuremath{\mathop{\bullet}}}{}{\colorbox{sivo}{\bf\color{white}{\mathversion{bold}[{\ensuremath{x\mathord/1}}]}}}{\ensuremath{\mathop{\bullet}}}{}{{{\ensuremath{\mathit{nil}}}}}\ } } } } } } } $ ] @xmath42@xmath133}}}{\ensuremath{\mathop{\bullet}}}{}{{{\ensuremath{\mathit{by({{\ensuremath{\mathsf{x{\ensuremath{\mathord=}}{}1}}}},{{\ensuremath{\mathsf{one(x , y)}}}})}}}}}{\ensuremath{\mathop{\bullet}}}{}{\colorbox{sivo}{\bf\color{white}{\mathversion{bold}[{\ensuremath{x\mathord/1}}]}}}{\ensuremath{\mathop{\bullet}}}{}{{{\ensuremath{\mathit{nil}}}}}\ } } { } { \ifempty{\{(1/(y{\ensuremath{\mathord=}}{}a);y{\ensuremath{\mathord=}}{}b){\ensuremath{\mathop{\bullet}}}{}two(1,y){\ensuremath{\mathop{\bullet}}}{}2/one(x , y),two(x , y){\ensuremath{\mathop{\bullet}}}{}post(x , y){\ensuremath{\mathop{\bullet}}}{}1/post(x , y),fail{\ensuremath{\mathop{\bullet}}}{}{{{\ensuremath{\mathit{nil}}}}}\}}{}{{{,\ , { } } } \colorbox{svetlosivo}{{{\ensuremath{\mathsf{\{(1/(y{\ensuremath{\mathord=}}{}a);y{\ensuremath{\mathord=}}{}b){\ensuremath{\mathop{\bullet}}}{}two(1,y){\ensuremath{\mathop{\bullet}}}{}2/one(x , y),two(x , y){\ensuremath{\mathop{\bullet}}}{}post(x , y){\ensuremath{\mathop{\bullet}}}{}1/post(x , y),fail{\ensuremath{\mathop{\bullet}}}{}{{{\ensuremath{\mathit{nil}}}}}\ } } } } } } } \ifempty{\{{\colorbox{sivo}{\bf\color{white}{\mathversion{bold}[{\ensuremath{y\mathord / a}}]}}}{\ensuremath{\mathop{\bullet}}}{}{{{\ensuremath{\mathit{by({{\ensuremath{\mathsf{x{\ensuremath{\mathord=}}{}1}}}},{{\ensuremath{\mathsf{one(x , y)}}}})}}}}}{\ensuremath{\mathop{\bullet}}}{}{\colorbox{sivo}{\bf\color{white}{\mathversion{bold}[{\ensuremath{x\mathord/1}}]}}}{\ensuremath{\mathop{\bullet}}}{}{{{\ensuremath{\mathit{nil}}}}}\}}{}{{{,\ , { } } } { { \ensuremath{\mathit{\{{\colorbox{sivo}{\bf\color{white}{\mathversion{bold}[{\ensuremath{y\mathord / a}}]}}}{\ensuremath{\mathop{\bullet}}}{}{{{\ensuremath{\mathit{by({{\ensuremath{\mathsf{x{\ensuremath{\mathord=}}{}1}}}},{{\ensuremath{\mathsf{one(x , y)}}}})}}}}}{\ensuremath{\mathop{\bullet}}}{}{\colorbox{sivo}{\bf\color{white}{\mathversion{bold}[{\ensuremath{x\mathord/1}}]}}}{\ensuremath{\mathop{\bullet}}}{}{{{\ensuremath{\mathit{nil}}}}}\ } } } } } } } $ ] @xmath42@xmath134}}}{\ensuremath{\mathop{\bullet}}}{}{{{\ensuremath{\mathit{by({{\ensuremath{\mathsf{x{\ensuremath{\mathord=}}{}1}}}},{{\ensuremath{\mathsf{one(x , y)}}}})}}}}}{\ensuremath{\mathop{\bullet}}}{}{\colorbox{sivo}{\bf\color{white}{\mathversion{bold}[{\ensuremath{x\mathord/1}}]}}}{\ensuremath{\mathop{\bullet}}}{}{{{\ensuremath{\mathit{nil}}}}}\ } } { } { \ifempty{\{two(1,y){\ensuremath{\mathop{\bullet}}}{}2/one(x , y),two(x , y){\ensuremath{\mathop{\bullet}}}{}post(x , y){\ensuremath{\mathop{\bullet}}}{}1/post(x , y),fail{\ensuremath{\mathop{\bullet}}}{}{{{\ensuremath{\mathit{nil}}}}}\}}{}{{{,\ , { } } } \colorbox{svetlosivo}{{{\ensuremath{\mathsf{\{two(1,y){\ensuremath{\mathop{\bullet}}}{}2/one(x , y),two(x , y){\ensuremath{\mathop{\bullet}}}{}post(x , y){\ensuremath{\mathop{\bullet}}}{}1/post(x , y),fail{\ensuremath{\mathop{\bullet}}}{}{{{\ensuremath{\mathit{nil}}}}}\ } } } } } } } \ifempty{\{{{{\ensuremath{\mathit{or({{\ensuremath{\mathsf{y{\ensuremath{\mathord=}}{}a}}}},{{\ensuremath{\mathsf{(1/(y{\ensuremath{\mathord=}}{}a);y{\ensuremath{\mathord=}}{}b)}}}})}}}}}{\ensuremath{\mathop{\bullet}}}{}{\colorbox{sivo}{\bf\color{white}{\mathversion{bold}[{\ensuremath{y\mathord / a}}]}}}{\ensuremath{\mathop{\bullet}}}{}{{{\ensuremath{\mathit{by({{\ensuremath{\mathsf{x{\ensuremath{\mathord=}}{}1}}}},{{\ensuremath{\mathsf{one(x , y)}}}})}}}}}{\ensuremath{\mathop{\bullet}}}{}{\colorbox{sivo}{\bf\color{white}{\mathversion{bold}[{\ensuremath{x\mathord/1}}]}}}{\ensuremath{\mathop{\bullet}}}{}{{{\ensuremath{\mathit{nil}}}}}\}}{}{{{,\ , { } } } { { \ensuremath{\mathit{\{{{{\ensuremath{\mathit{or({{\ensuremath{\mathsf{y{\ensuremath{\mathord=}}{}a}}}},{{\ensuremath{\mathsf{(1/(y{\ensuremath{\mathord=}}{}a);y{\ensuremath{\mathord=}}{}b)}}}})}}}}}{\ensuremath{\mathop{\bullet}}}{}{\colorbox{sivo}{\bf\color{white}{\mathversion{bold}[{\ensuremath{y\mathord / a}}]}}}{\ensuremath{\mathop{\bullet}}}{}{{{\ensuremath{\mathit{by({{\ensuremath{\mathsf{x{\ensuremath{\mathord=}}{}1}}}},{{\ensuremath{\mathsf{one(x , y)}}}})}}}}}{\ensuremath{\mathop{\bullet}}}{}{\colorbox{sivo}{\bf\color{white}{\mathversion{bold}[{\ensuremath{x\mathord/1}}]}}}{\ensuremath{\mathop{\bullet}}}{}{{{\ensuremath{\mathit{nil}}}}}\ } } } } } } } $ ] @xmath42@xmath135}}}{\ensuremath{\mathop{\bullet}}}{}{{{\ensuremath{\mathit{by({{\ensuremath{\mathsf{x{\ensuremath{\mathord=}}{}1}}}},{{\ensuremath{\mathsf{one(x , y)}}}})}}}}}{\ensuremath{\mathop{\bullet}}}{}{\colorbox{sivo}{\bf\color{white}{\mathversion{bold}[{\ensuremath{x\mathord/1}}]}}}{\ensuremath{\mathop{\bullet}}}{}{{{\ensuremath{\mathit{nil}}}}}\ } } { } { \ifempty{\{2/one(x , y),two(x , y){\ensuremath{\mathop{\bullet}}}{}post(x , y){\ensuremath{\mathop{\bullet}}}{}1/post(x , y),fail{\ensuremath{\mathop{\bullet}}}{}{{{\ensuremath{\mathit{nil}}}}}\}}{}{{{,\ , { } } } \colorbox{svetlosivo}{{{\ensuremath{\mathsf{\{2/one(x , y),two(x , y){\ensuremath{\mathop{\bullet}}}{}post(x , y){\ensuremath{\mathop{\bullet}}}{}1/post(x , y),fail{\ensuremath{\mathop{\bullet}}}{}{{{\ensuremath{\mathit{nil}}}}}\ } } } } } } } \ifempty{\{{{{\ensuremath{\mathit{by({{\ensuremath{\mathsf{(y{\ensuremath{\mathord=}}{}a;y{\ensuremath{\mathord=}}{}b)}}}},{{\ensuremath{\mathsf{two(1,y)}}}})}}}}}{\ensuremath{\mathop{\bullet}}}{}{{{\ensuremath{\mathit{or({{\ensuremath{\mathsf{y{\ensuremath{\mathord=}}{}a}}}},{{\ensuremath{\mathsf{(1/(y{\ensuremath{\mathord=}}{}a);y{\ensuremath{\mathord=}}{}b)}}}})}}}}}{\ensuremath{\mathop{\bullet}}}{}{\colorbox{sivo}{\bf\color{white}{\mathversion{bold}[{\ensuremath{y\mathord / a}}]}}}{\ensuremath{\mathop{\bullet}}}{}{{{\ensuremath{\mathit{by({{\ensuremath{\mathsf{x{\ensuremath{\mathord=}}{}1}}}},{{\ensuremath{\mathsf{one(x , y)}}}})}}}}}{\ensuremath{\mathop{\bullet}}}{}{\colorbox{sivo}{\bf\color{white}{\mathversion{bold}[{\ensuremath{x\mathord/1}}]}}}{\ensuremath{\mathop{\bullet}}}{}{{{\ensuremath{\mathit{nil}}}}}\}}{}{{{,\ , { } } } { { \ensuremath{\mathit{\{{{{\ensuremath{\mathit{by({{\ensuremath{\mathsf{(y{\ensuremath{\mathord=}}{}a;y{\ensuremath{\mathord=}}{}b)}}}},{{\ensuremath{\mathsf{two(1,y)}}}})}}}}}{\ensuremath{\mathop{\bullet}}}{}{{{\ensuremath{\mathit{or({{\ensuremath{\mathsf{y{\ensuremath{\mathord=}}{}a}}}},{{\ensuremath{\mathsf{(1/(y{\ensuremath{\mathord=}}{}a);y{\ensuremath{\mathord=}}{}b)}}}})}}}}}{\ensuremath{\mathop{\bullet}}}{}{\colorbox{sivo}{\bf\color{white}{\mathversion{bold}[{\ensuremath{y\mathord / a}}]}}}{\ensuremath{\mathop{\bullet}}}{}{{{\ensuremath{\mathit{by({{\ensuremath{\mathsf{x{\ensuremath{\mathord=}}{}1}}}},{{\ensuremath{\mathsf{one(x , 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\ifempty{\{{\colorbox{sivo}{\bf\color{white}{\mathversion{bold}[{\ensuremath{x\mathord/1}}]}}}{\ensuremath{\mathop{\bullet}}}{}{{{\ensuremath{\mathit{nil}}}}}\}}{}{{{,\ , { } } } { { \ensuremath{\mathit{\{{\colorbox{sivo}{\bf\color{white}{\mathversion{bold}[{\ensuremath{x\mathord/1}}]}}}{\ensuremath{\mathop{\bullet}}}{}{{{\ensuremath{\mathit{nil}}}}}\ } } } } } } } $ ] @xmath42@xmath164 @xmath42@xmath165 @xmath42@xmath166 @xmath42@xmath167 @xmath42@xmath168

a simple mathematical definition of the 4-port model for pure prolog is given . the model combines the intuition of ports with a compact representation of execution state . forward and backward derivation steps are possible . the model satisfies a modularity claim , making it suitable for formal reasoning .

cs0310020

dissipationless n - body simulations of stellar systems indicate that scaling relations such as the so - called ` fundamental plane ' ( hereon , fp ) , that is , the systematic deviation from the expectations of the virial theorem applied to these systems , could be reproduced from the final products of hierarchical merging of galactic model progenitors @xcite . however , not all evolutionary conditions lead to fp - like relations : simple gravitational collapses do not . that is , objects resulted from mergers form a slightly non - homologous family ( and a fp - like relation ) , whereas collapses are homologous among themselves ( and show no deviation from the virial expectations ; see @xcite ) . at the same time , kandrup and collaborators @xcite argued on the existence of ` mesoscopic constraints ' of pure gravitational origin in systems relaxing towards virialization ( hereon , the ` kandrup effect ' ) . these constraints were inferred from the general preservation of the ` coarse - grained ' partitioning of the ranked energy distribution of particles , and seemed to regulate somehow the gravitational evolution of these galaxy models towards equilibrium . these constraints were also indirectly shown to be partially ` broken ' ( violated ) in mergers and fully operative in collapses @xcite . the effect of incomplete mixing of phase space in dissipationless gravitational collapses was known already since the decade of 80s ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? the surviving memory of initial conditions in the sense of an almost linear dependence of the final ( after the collapse ) energies with the initial energies ( in cosmological initial conditions ) was first demonstrated in @xcite . a more detailed investigation of this effect in n - body systems resulting from cosmological collapses is given in @xcite . such clues lead us to inquire whether the ` kandrup effect ' and the scaling relations of gravitational systems ( like the fp ) could be deeply related in some way . here we present a ` global map ' indicating where mesoscopic constraints could be mostly operative , in a full cosmological simulation . this paper is organized as follows . in section 2 , we study the ` kandrup effect ' in terms of dark matter halos . in section 3 , we investigate the behaviour of halos in terms of arnold s theorem on the ordering of charcteristic frequencies under the imposition of a linear constraint . in section 4 , we discuss our results . in the study of kandrup et al . , the distribution of the energy of the particles in systems resulting from collisions and merging of two model galaxies was analysed in detail . they have found that there is a ` coarse - grained ' sense in which the _ ordering _ of the mean energy of given collections of particles of the systems is strictly _ not violated _ through the gravitational evolution of the models towards equilibrium . the method consists of sorting the particles of a given initial model according to their energies . the models are partitioned into a few , ` mesoscopic ' ( around 5 to 10 ) bins of equal number of particles and for each of these bins , the mean energy is calculated . finally , the bins are ranked with the first one initially containing the most bound particles ( the most negative mean energy ) whereas the last bin contains the least bounded particles ( the least negative mean energy ) . the mean energies of these same collections of particles are then recalculated for the final model and compared with their initial values . from such an analysis , kandrup et al . found that the mean energy rank ordering of fixed collections of particles is preserved along the evolution . here analyse the ` kandrup effect ' in larger gravitationally - dominated structures , like clusters and superclusters of galaxies ( see also @xcite ) . to this end , we have analysed a @xmath0-cdm n - body simulation output of the virgo consortium . the analysis is identical to that of @xcite , but here the initial condition is the z=10 simulation box , and the final condition , the z=0 box ( the boxes have a @xmath1 mpc comoving size , where each particle has a mass of @xmath2 ) . signs of the ` kandrup effect ' were searched for the @xmath3 most massive halos found in the z=0 box , identified by the use of a simple ` friends - of - friends ' algorithm ( @xcite ) , setting periodic boundary conditions and a searching length of @xmath4 mpc . the energy of a particle considered in our work is the mechanical comoving one - particle energy . it was not calculated with respect to the local center of mass of the particular clumps , but with respect to the comoving reference frame ( that is , the frame which moves with the cosmological expansion of the simulation box ) . the comoving energy of a particle @xmath5 was calculated classically from : @xmath6 with comoving position @xmath7 and peculiar velocity @xmath8 . units used were mpc for length , gyr for time and @xmath9 for mass . the energy associated to the dynamics of expansion of the cosmological box does not enter into the above computations . at this point we remark that in the present simulation scenario ( @xmath0-cdm ) , the nonlinear collapse of sub - galactic mass halos are the first expected events after recombination . these small mass units will subsequently cluster together in a hierarchy of larger and larger objects ( bottom - up structure formation scenario ) . this is in contrast to top - down pictures where the formation of very massive objects comes first , as for instance , in the hot dark matter scenario . from the spherical top - hat collapse model , a reasonable estimate for the limit to the redshift at which a given halo becames virialized or formed ( @xmath10 ) is @xcite : @xmath11 the less massive halo analysed from the set of 31 objects has a mass of @xmath12 . assuming that its velocity dispersion is of order @xmath13 ( a typical figure for that mass scale ) and the last term of the expression above is of order @xmath14 , we find that @xmath10 for this halo is approximately @xmath15 . higher mass halos will have their @xmath10 s even smaller than in above case . for instance , the most massive halo in the simulation has @xmath16 . assuming @xmath17 , we find @xmath18 . since all the 31 halos analysed have masses within that range , their condition at @xmath19 ( the initial dump of the simulation ) reasonably represents a linear stage of their evolution . another way to see this point is that the particle mass of the simulation is @xmath20 . hence , such a particle is representing a large sub - galactic object , and considering its internal velocity dispersion of @xmath21 , then @xmath22 . of course , this is the mass resolution of the simulation , it is assumed to already represent a collapsed unit at @xmath19 . in other words , that gives an idea of the redshift formation limit for the smallest mass unit resolved in the simulation . hence , the 31 most massive halos at @xmath19 ( the first redshift dump available from the simulation ) are safely in their linear phases of evolution , so that the comparison of the ` kandrup effect ' seems quite satisfactory to be performed in reasonably equal grounds for all 31 halos . we have defined a ` violation ' parameter rate , @xmath23 , which measures _ the degree of rank ordering violation of the mean energy of @xmath24 fixed collections of particles _ , compared at @xmath19 and @xmath25 , so that if @xmath26 , we retrieve the ` kandrup effect ' ( no energy rank ordering violation ) , whereas @xmath27 means that all energy bins presented ordering violation . notice that the @xmath23 parameter penalizes " cases where there is a significant relative change in energy cell position ( in the sense of ordering ) . for example , if only the 10th energy cell crosses all the other @xmath28 cells , and all these @xmath28 cells keep their ordering unaltered , the @xmath23 parameter in this case is _ not _ assigned @xmath29 ( meaning that only one energy cell , out of @xmath24 , crosses some other cell - whatever how many places in ordering ) , but in fact our criteria assigns @xmath30 , meaning that _ all _ cells are relatively crossed in this case . so the @xmath23 parameter , although not a weighted parameter , measures in fact a relative crossing percentage . an illustration of the ` kandrup effect ' and its violation , measured by the @xmath23 parameter , is presented in figures 1a and 1b , for the 31 halos in order of increasing mass . we remark that one expects that the values of energies of the particles in a clump ( after the formation - collapse of it ) are spread in a wider range than the range of their initial energies . during the collapse and relaxation , a number of particles in the clump loose energy and they are trapped in a deeper potential well near the center of the clump , while other particles gain energy and they may even escape the clump after the collapse . this produces a wider range of energies at the final configuration . however , in the various panels of figs . 1a and 1b we see that the final energies are almost always distributed in a smaller range than the initial energies ( with only three or four exceptions ) . such a description is generally the correct expectation for energies evaluated at a fixed background . but in the present case , the energies are calculated with respect to the comoving frame . a qualitative understanding of the results of figures 1a and 1b is the following . first , consider two particles ( @xmath4 and @xmath31 ) comoving with the expansion . then , after a given time @xmath32 ( where @xmath33 is the initial time considered ) , their comoving coordinates are @xmath34 ; @xmath35 , and their potential energy ( with respect to the comoving frame ) is unchanged , although they might seem drifting apart from each other for an observer at a fixed reference frame . now consider that the two particles just detach from the expansion ( ` turn - around ' ) : even though with respect to a fixed frame their potential energy may look unchanged , it does get more negative with respect to the comoving frame ( because @xmath36 ; @xmath37 ) . subsequently , the particles get bound , their potential energy gets more negative in both fixed and comoving frames , but it is then clear that it gets even more negative in the latter case , because the background is expanding . the same reasoning is natural to escaping particles . also , notice that the velocity used is the peculiar velocity , which is the velocity substracted from the velocity of expansion . the overall result to the energy calculated this way is that the particles will present final energies almost always distributed in a smaller range than the initial energies . notice that figs . 1a and 1b refer to the initial and final _ mean _ energies of fixed collections of particles . the particles have been initially ranked according to their energies ( comoving ) , and binned to a fixed number of particles per ranked energy cell . the initial distribution of the energies within a given ` mesoscopic ' energy cell is relatively uniform . however , when the halos collapse or merge , this distribution tends to get skewed towards more negative energy values . some few particles do escape and carry energy , forming a tail in the distribution . the mean energy per cell gets more negative due to the effect explained above . also , in most cases of figs . 1a and 1b , the final energies are concentrated in lower values than the initial energies , but in one case particles seem to have gained energy as they form a clump . the method used to isolate the cosmological halos is friends - of - friends , which tends to extract overdensities in a given distribution of points . we expect that several such overdensities may be considered as virialized or quasi - virialized halos , but some ( few ) could in fact be artifacts , one of which may be the suspect case mentioned here . in figure 2 , @xmath23 is plotted in terms of the mass of the identified structures in the z=0 box . this figure shows that _ there is a sense in which larger and larger structures seem to evolve towards a preservation of the ` mesoscopic ' mean energies _ , whereas smaller and smaller structures tend to violate the ` kandrup effect ' . intermediate - sized structures present intermediate values of @xmath23 . from the above trend , we infer that the dynamics of the smaller structures is probably being dominated by merging processes , whereas larger structures seem to be ruled by the collapse mechanism ( intermediate - sized clusters could be collapsing structures accreting small mass systems ) . in order to address this question , we attempt to quantify whether it is mainly the inner radial bins of the halo which violate mean energy rank , or the outer ones . at this point , it is important to notice that in a spherical gravitational potential , @xmath38 , the orbit of a given particle is confined to a plane defined by its angular momentum vector , @xmath39 . the circular orbits are the ones which have the greatest @xmath40 by unit mass . for a given energy per unit mass , @xmath41 , among all possible @xmath40s , there will be a maximum @xmath42 , which corresponds to the circular orbit ( c.f . the curve @xmath42 divides the [ e , l ] plane into a forbidden ( above the curve ) and a populated ( below the curve ) region , were the particles of a given gravitational system will lie . ( deviations from spherical symmetry are recognized as the presence of a few particles above the @xmath42 curve in the forbidden region ) . it can be easily shown that the curve @xmath42 tends asimptotically to @xmath43 for very bound ( negative ) particle energies , and @xmath44 for @xmath45 . hence , particles with very negative energies ( boundest particles ) tend to have lower @xmath40 , and tend to be at the inner radii of the system ; on the other hand , particles less gravitationally bound tend to have a larger range of @xmath40s and mainly populate the outer radii of the system . hence it can be reasonalbly assumed that the most negative energy bins analysed are composed of particles mostly at the inner radii of the halos . having said this , we have analysed the behaviour of the 3 innermost and 3 outermost energy bins , as explained in the following . for each halo , we have assigned a value of 1 ( = yes ) whenever rank violation is found within each of these two sets of 3 bins ( separately ) , and assigned value 0 ( = no ) if rank is preserved . figure 3 illustrates the results found . it can be seen that the 3 outermost energy bins violate rank ordering in 19 out of the 31 halos , whereas for the 3 innermost bins the statistics is 29 out of 31 . so , whenever rank violation occurs , it does mainly at the most bounded energy bins , which means that , for the most part , the particles at the inner radii of the model are mainly affected , not the outer ones . at this point , two natural questions are : first , could the phenomenum observed in fig . [ fa4.5.mer ] be artificaly created solely because of the different number of particles involved in each halo ? that is , could it be that , depending on the number of particles ( namely , mass ) of the individual halo in question , the system could artifically be more fine - partitioned relatively to a system with more particles , and hence the former system could show up more violation of the ` kandrup effect ' than the latter ? in fact , kandrup et al . have analysed the problem of coarse - graining ( partitioning the energy space into 5 and 20 bins ) and found no significant dependence on coarse - graining . we adopt a cautionary position and identify the halos in which poor statistics may be a problem . we find that 7 out of the 31 evaluated halos have total number of particles less than 200 , which implies less than 20 particles per bin . figure 2 indicates the halos where the phenomenum ( namely , rank violation as mass decreases ) could be an artifact due to mass ( particle ) resolution . however , for the majority of halos , the number of particles per energy cell is reasonably larger than the number of coarse - grained cells adopted ( @xmath24 cells , in the present case ) . a second , related question is : ( i ) what is the coarse - graining level relevant to the observation of the ` kandrup effect ' , and ( ii ) why it happens ? concerning the latter part of the question , one of the possible explanations for the ` kandrup effect ' , given by kandrup et al . themselves , but not proven , is the existence of some _ constraint _ operative only at the level of _ collections _ of particles ranked by mean energy . we try to find some clues on this problem by attempting to answer the first part of the above question . one independent method which may prove interesting is to directly verify the validity of the theorem described by @xcite , concerning the behaviour of the characteristic frequencies of a dynamical system under the imposition of a linear constraint . arnold s theorem describes how the characteristic frequencies ( @xmath46 ) of a system with @xmath47 degrees of freedom are distributed relatively to the characteristic frequencies ( @xmath48 ) of a system obtained from the former under the imposition of a linear contraint , reducing the dimensionality of the given system to @xmath49 . the theorem goes as follows . let @xmath50 be the @xmath47 characteristic frequencies of the original system where no constraints are imposed , and @xmath51 the @xmath49 characteristic frequencies of the system with a constraint . then the theorem asserts that : @xmath52 let us define the characteristic frequency of a particle of the simulation as @xmath53 , where @xmath39 is the angular momentum of the particle relatively to the comoving center of mass ( cm ) of the system and @xmath54 the comoving distance of the given particle to the cm . we have taken the mean frequency values of particles grouped according to the same rank ordering method of kandrup et al . , and then proceeded as follows . we focused on the five halos that mostly violate the ` kandrup effect ' and the five ones mostly obeying it . we have partitioned these halos into @xmath28 orderly energy coarse grainings ( bins ) at @xmath25 . at @xmath19 , however , the @xmath28 corresponding bins were repartitioned into @xmath24 bins . this allowed us to directly compare the characteristic mean frequencies of each bin per halo at the two different redshifts . that is , from eq.(4 ) above , the primed values now refer to the mean frequencies of each of the @xmath28 bins analysed at @xmath25 ( per halo ) , and the unprimed values , to the @xmath24 bins at @xmath19 ( per halo ) . then , for each halo , we were able to analyse whether arnold s theorem ( eq . 4 ) would be violated or not , and at what level . we found that the percentages of halos matching arnold s theorem , bin by bin , for the halos mostly obeying the ` kandrup effect ' ( in order of decreasing mass ) , were : @xmath55 , @xmath55 , @xmath56 , @xmath57 , and @xmath58 . for the halos violating energy rank : @xmath59 , @xmath60 , @xmath57 , @xmath57 , and @xmath61 . hence , the most massive halos tend to obey arnold s theorem ( on the ordering of the characteristic frequencies ) more frequently than less massive halos . our general conclusion is drawn from a combination of previous and present clues on the phenomenum of relaxation . collapses are quite different from mergers , despite the fact that both are the end results of a dynamics based on fluctuations on the gravitational potential . these main differences are at the heart of the connection between the ` kandrup effect ' and the fp @xmath62 virial relations . from the current interpretation , merging of stellar systems occurs due to a transfer of the orbital energy to the particles within the systems through tidal interactions . this mechanism increases the internal energy of the systems at the expense of their orbital energy . if the orbital energy is less negative ( approaches @xmath63 ) , there is plenty of time for the particles to withdraw energy from the orbit of the pair of merging models . this process involves periodically , slowly evolving , large fluctuations on the potential , which takes a larger amount of time to stabilize . it was already evident that such slow of stabilization process could be responsable for producing non - homology among the simulated mergers @xcite . collapses , on the contrary , are generally much more ` violent ' than mergers , even those resulting from relatively ` warm ' initial conditions . the present analysis corroborates the idea that collapses and mergers behave differently from the point of view of their energy rank preservation / violation , considering that both mechanisms are dominant at different scaling ( mass ) regimes . indeed , we find that : * more massive halos tend to better preserve their mean energy rank ordering than less massive halos . * whenever rank violation occurs , it does mainly at the most bounded energy bins , which means that , for the most part , the particles at the inner radii of the model are mainly affected , not the outer ones . this is in agreement with the qualitative idea that the central region of the halo is where the gravitational potential fluctuates more strongly during the collapse , whereas in the outer regions of the halo the fluctuations are softer and dump rapidly . in the case of mergers , however , it is expected that the outer regions will be submitted to equally large fluctuations because of the anisotropy of the merging geometry as compared to a simple , isotropic collapse . indeed it is interesting to notice that rank violation of the 3 outermost bins does occur much more frequently for less massive halos , where merging is expected to occur more frequently as well . * the process of mesoscopic ` mixing ' in the energy space seems to be inefficient as compared with ` mixing ' in configuration and/or velocity space , an effect which is at odds with lynden - bell s theory of complete ` violent relaxation ' @xcite ; see discussion in kandrup et al . however , this effect seems to be more evident depending on the process of virialization ( collapses seem more prone to energy rank preservation ) . * we provide a general method to evaluate the coarse - graining level ( number of particles per energy cell ) relevant to reasonably measure signatures of the ` kandrup effect ' . we have observed that arnold s theorem , applied to the collections of particles of each energy bin , tends to be valid more frequently when the ` kandrup effect ' is operative , than when it is not . this may be related to the presence of a ` mesoscopic constraint ' operative at the level of particles , as observed by kandrup et al . alternatively , when arnold s theorem is shown not to be valid , the ` kandrup effect ' is expected to be violated . in this case , one could suspect of small particle number statistics or , otherwise , intrinsic ( real ) energy rank violation . we briefly assess how our results could be affected by a different choice of evaluation of energies . kandrup et al . analysed the merging of two galaxies and have computed the energy of a given particle considering two choices : kandrup et al . show that , if the initial center of mass velocity is relatively small , the tendency ( preservation of the ordering of the mean ranked energies of collections of particles ) is little affected ( methods 1 and 2 give similar results ) , although the effect does get weaker if the velocity of the cm is reasonably large . notice however that the simulations of kandrup et al . were performed on a fixed background . in the present situation the energies are all comoving quantities , the velocity of expansion of the cosmological background does not enter into the above computations , so it is our opinion that our choice of evalutation of energies is reasonable in the cosmological set . a posteriori , our results ( fig.1a and 1b ) do reproduce the ` kandrup effect ' . we also consider the question whether our results depend on the position of the particular clump inside the simulation box . a qualitative reasoning indicates that they must not , since we calculate the comoving energies with no reference to the relative location within the simulation box . also , the halos analysed in the paper are distributed at several locations within the simulation box , and the results are reasonably similar among them , irrespective of their position , depending mostly on the halo mass . we would like to draw attention to the relation of our results to those of the recent paper by @xcite . these authors have studied the velocity distribution function ( vdf ) of different gravitational systems ( from isotropic to highly non - isotropic structures ) , simulated from diverse n - body experiments , including cosmological ones , and found that the vdf has a universal profile . this may indicate that some fraction of the particles of a gravitational system exchange energy during the relaxation process only until the universal vdf has been reached . it would be highly interesting to understand the relation between energy rank preservation and the universal vdf . in summary , the scenario presented here , although admittedly schematic , does offer some preliminary important clues on the origin of the scaling relations of virialized systems . however , a rigorous understanding of the relaxation process still lacking @xcite , in order to reach a complete understanding of the phenomena outlined in this paper . the simulations in this paper were carried out by the virgo supercomputing consortium using computers based at computing centre of the max - planck society in garching and at the edinburgh parallel computing centre . the data are publicly available at www.mpa-garching.mpg.de/numcos . we thank dr . andr l. b. ribeiro for several discussions , and the anonymous referee for useful suggestions . thanks instituto nacional de pesquisas espaciais ( inpe ) , diviso de astrofsica ( das / cea ) , brazil , for using its facilities when most part of this paper was written . 99 arnold , v. i. , _ mathematical methods of classical mechanics _ , 2nd . edition , 1989 , springer - verlag , new york . bekki , k. , 1998 , apj , 496 , 713 . binney & tremaine , galactic dynamics , princeton university press . capelato , h. v. , de carvalho , r. r. & carlberg , r. g. , 1995 , apj , 451 , 525 . dantas , c. c. , ribeiro , a. l. b. , capelato , h. v. , and de carvalho , r. r. , 2000 , apj , 528 , l5 . dantas , c. c. , capelato , h. v. , de carvalho , r. r. , & ribeiro , a. l. b. , 2002 , a&a , 384 , 772 . dantas , c. c. , capelato , h. v. , ribeiro , a. l. b. & de carvalho , r. r. , 2003 , mnras , 340 , 398 . dantas , c. c. & ramos , f. m. , 2005 , apj , submitted . djorgovski , s. g. & davis , m. , apj , 1987 , 313 , 59 . djorgovski , s. g. , in proc . moriond astrophysics workshop , 1988 , starbursts and galaxy evolution , ed . t. x. thuan et al . ( gif sur yvette : editions frontires ) , 549 . djorgovski , s. g. , in cosmology and large - scale structure in the universe , asp conference series , vo . 24 , 1992 , reinaldo r. de carvalho , ( ed . ) , 19 . djogorvski , s. g. & santiago , b. x. , 1993 , in _ structure , dynamics and chemical evolution of elliptical galaxies _ , proc . eso / eipc workshop , ed . i. j. danziger , w. w. zeilinger , & k. kjar ( eso conf . and workshop proc . 45 ) ( garching : eso ) , 59 dressler , a. , lynden - bell , d. , burstein , d. , davies , r. l. , faber , s. m. , terlevich , r. j. , & wegner , g. , 1987 , apj , 313 , 42 . fof algorithm , http://www-hpcc.astro.washington.edu/tools/fof.html . hansen , s. h. et al . , astro - ph/0505420 hjorth , j. & madsen , j. , 1995 , apj 445 , 55 . kadrup , h. e. , mahon , m. e. , & smith , h. , 1993 , a&a 271 , 440 . longair , m. , galaxy formation , springer , page 366 . lynden - bell , d. , 1967 , mnras 136 101 . may & van albada , 1984 , mnras , 209 , 15 padmanabhan , t. , 1990 , physics reports , 188 , no.5 , 285 - 362 . pahre , m. , 1998 , phd thesis , california institute of technology . quinn & zurek , 1988 , apj , 331 , 1 van albada 1982 , mnras , 201 , 939 voglis , hiotelis & hoeflich , 1991 , a & a , 249 , 5 voglis , hiotelis & harsoula , 1994 , astrophysics and space science , 226 , 213 . zaroubi et al . , 1996 , apj , 457 , 50 white , s. , 1979 , mnras , 189 , 831 ) at @xmath25 ( final condition ) and @xmath19 ( initial condition ) , connected by line segments in order to illustrate the ordering of the mean energy of given collections of particles , from the most gravitationally bounded ones ( most negative energies ) to the less bounded ones ( less negative energies ) . the panels are ordered from the less massive halos ( top left panel ) to the most massive halos ( bottom right panel ) . the second panel ( from left to right ) at the third row is possibly an overdensity artifact ( not a virialized halo ) . [ kan],width=642,height=718 ] ) as a function of the halo mass in a z=0 @xmath0-cdm cosmological box . possible dynamical events are identified in this figure . the object marked with an ` x ' is possibly an overdensity artifact ( not a virialized halo ) . [ fa4.5.mer],width=604,height=453 ]

we present an analysis of the behaviour of the ` coarse - grained ' ( ` mesoscopic ' ) rank partitioning of the mean energy of collections of particles composing virialized dark matter halos in a @xmath0-cdm cosmological simulation . we find evidence that rank preservation depends on halo mass , in the sense that more massive halos show more rank preservation than less massive ones . we find that the most massive halos obey arnold s theorem ( on the ordering of the characteristic frequencies of the system ) more frequently than less massive halos . this method may be useful to evaluate the coarse - graining level ( minimum number of particles per energy cell ) necessary to reasonably measure signatures of ` mesoscopic ' rank orderings in a gravitational system .

astro-ph0508488

explosive nuclear burning in astrophysical environments produces unstable nuclei , which again can be targets for subsequent reactions . in addition , it involves a very large number of stable nuclei , which are not fully explored by experiments . thus , it is necessary to be able to predict reaction cross sections and thermonuclear rates with the aid of theoretical models . explosive burning in supernovae involves in general intermediate mass and heavy nuclei . due to a large nucleon number they have intrinsically a high density of excited states . a high level density in the compound nucleus at the appropriate excitation energy allows to make use of the statistical model approach for compound nuclear reactions ( e.g. @xcite ) , which averages over resonances . in this paper , we want to present new results obtained within this approach and outline in a clear way , where its application is valid . it is often colloquially termed that the statistical model is only applicable for intermediate and heavy nuclei . however , the only necessary condition for its application is a large number of resonances at the appropriate bombarding energies , so that the cross section can be described by an average over resonances . this can in specific cases be valid for light nuclei and on the other hand not be valid for intermediate mass nuclei near magic numbers . thus , another motivation of this investigation is to explore the nuclear chart for reactions with a sufficiently high level density , implying automatically that the nucleus can equilibrate in the classical compound nucleus picture . as the capture of an alpha particle leads usually to larger q - values than neutron or proton captures , the compound nucleus is created at a higher excitation energy . therefore , it is often even possible to apply the hauser - feshbach formalism for light nuclei in the case of alpha - captures . another advantage of alpha - captures is that the capture q - values vary very little with the n / z - ratio of a nucleus , for nuclei with z@xmath350 . for z@xmath450 , entering the regime of natural alpha - decay , very small alpha - capture q - values can be encountered for proton - rich nuclei . such nuclei on the other hand do not play a significant role in astrophysical environments , maybe with exception of the p - process . this means that in the case of alpha - captures the requirement of large level densities at the bombarding energy is equally well fulfilled at stability as for unstable nuclei . opposite to the behavior for alpha - induced reactions , the reaction q - values for proton or neutron captures vary strongly with the n / z - ratio , leading eventually to vanishing q - values at the proton or neutron drip line . for small q - values the compound nucleus is created at low excitation energies and also for intermediate nuclei the level density can be quite small . therefore , it is not advisable to apply the statistical model approach close to the proton or neutron drip lines for intermediate nuclei . for neutron captures close to the neutron drip line in r - process applications it might be still permissable for heavy and often deformed nuclei , which have a high level density already at very low excitation energies . in astrophysical applications usually different aspects are emphasized than in pure nuclear physics investigations . many of the latter in this long and well established field were focused on specific reactions , where all or most `` ingredients '' , like optical potentials for particle and alpha transmission coefficients , level densities , resonance energies and widths of giant resonances to be implementated in predicting e1 and m1 gamma - transitions , were deduced from experiments . this of course , as long as the statistical model prerequisites are met , will produce highly accurate cross sections . for the majority of nuclei in astrophysical applications such information is not available . the real challenge is thus not the well established statistical model , but rather to provide all these necessary ingredients in as reliable a way as possible , also for nuclei where none of such informations are available . in addition , these approaches should be on a similar level as e.g. mass models , where the investigation of hundreds or thousands of nuclei is possible with managable computational effort , which is not always the case for fully microscopic calculations . the statistical model approach has been employed in calculations of thermonuclear reaction rates for astrophysical purposes by many researchers @xcite , who in the beginning only made use of ground state properties . later , the importance of excited states of the target was pointed out @xcite . the compilations @xcite are presently the ones utilized in large scale applications in all subfields of nuclear astrophysics , when experimental information is unavailable . existing global optical potentials , mass models to predict q - values , deformations etc . , but also the ingredients to describe giant resonance properties have been quite successful in the past ( see e.g. the review by @xcite ) . the major remaining uncertainty in all existing calculations stems from the prediction of nuclear level densities , which in earlier calculations gave uncertainties even beyond a factor of 10 at the neutron separation energy @xcite , about a factor of 8 @xcite , and a factor of 5 even in the most recent calculations ( e.g. @xcite ; see fig.3.16 in @xcite ) . in nuclear reactions the transitions to lower lying states dominate due to the strong energy dependence . because the deviations are usually not as high yet at low excitation energies , the typical cross section uncertainties amounted to a smaller factor of 23 . we want to show in this paper , after a short description of the model and the required nuclear input , the implementation of a novel treatment of level density descriptions @xcite , where the level density parameter is energy dependent and shell effects vanish at high excitation energies . this is still a phenomenological approach , making use of a back - shifted fermi - gas model , rather than a combinatorial approach based on microscopic single - particle levels . but it is the first one leading to a reduction of the average cross section uncertainty to a factor of about 1.4 , i.e. an average deviation of about 40% from experiments , when only employing global predictions for all input parameters and no specific experimental knowledge . the degree of precision of the present approach will give astrophysical nucleosynthesis calculations a much higher predictive power . in order to give a guide for its application , we also provide a map of the nuclear chart which indicates where the statistical model requirements are fulfilled and its predictions therefore safe to use . a high level density in the compound nucleus permits to use averaged transmission coefficients @xmath5 , which do not reflect a resonance behavior , but rather describe absorption via an imaginary part in the ( optical ) nucleon - nucleus potential @xcite . this leads to the well known expression @xmath6 for the reaction @xmath7 from the target state @xmath8 to the exited state @xmath9 of the final nucleus , with a center of mass energy e@xmath10 and reduced mass @xmath11 . @xmath12 denotes the spin , @xmath13 the corresponding excitation energy in the compound nucleus and @xmath14 the parity of excited states . when these properties are used without subscripts they describe the compound nucleus , subscripts refer to states of the participating nuclei in the reaction @xmath7 and superscripts indicate the specific excited states . experiments measure @xmath15 , summed over all excited states of the final nucleus , with the target in the ground state . target states @xmath16 in an astrophysical plasma are thermally populated and the astrophysical cross section @xmath17 is given by @xmath18 the summation over @xmath19 replaces @xmath20 in eq . [ cslab ] by the total transmission coefficient @xmath21 here @xmath22 is the channel separation energy , and the summation over excited states above the highest experimentally known state @xmath23 is changed to an integration over the level density @xmath24 . the summation over target states @xmath16 in eq . [ csstar ] has to be generalized accordingly . in addition to the ingredients required for eq . [ cslab ] , like the transmission coefficients for particles and photons , width fluctuation corrections @xmath25 have to be employed . they define the correlation factors with which all partial channels for an incoming particle @xmath26 and outgoing particle @xmath27 , passing through the excited state @xmath28 , have to be multiplied . this takes into account that the decay of the state is not fully statistical , but some memory of the way of formation is retained and influences the available decay choices . the major effect is elastic scattering , the incoming particle can be immediately re - emitted before the nucleus equilibrates . once the particle is absorbed and not re - emitted in the very first ( pre - compound ) step , the equilibration is very likely . this corresponds to enhancing the elastic channel by a factor @xmath29 . in order to conserve the total cross section , the individual transmission coefficients in the outgoing channels have to be renormalized to @xmath30 . the total cross section is proportional to @xmath31 and , when summing over the elastic channel ( @xmath32 ) and all outgoing channels ( @xmath33 ) , one obtains the condition @xmath31=@xmath34 . we can ( almost ) solve for @xmath35 @xmath36 this requires an iterative solution for @xmath37 ( starting in the first iteration with @xmath31 and @xmath38 ) , which converges fast . the enhancement factor @xmath29 has to be known in order to apply eq . [ widthcorr ] . a general expression in closed form was derived @xcite , but is computationally expensive to use . a fit to results from monte carlo calculations gave @xcite @xmath39 for a general discussion of approximation methods see @xcite . [ widthcorr ] and [ newcorr ] redefine the transmission coefficients of eq . [ cslab ] in such a manner that the total width is redistributed by enhancing the elastic channel and weak channels over the dominant one . cross sections near threshold energies of new channel openings , where very different channel strengths exist , can only be described correctly , when taking width fluctuation corrections into account . of the thermonuclear rates presently available in the literature , only those by thielemann et al . @xcite include this effect , but their level density treatment still contains large uncertainties . the width fluctuation corrections of @xcite are only an approximation to the correct treatment . however , it was shown that they are quite adequate @xcite . the important ingredients of statistical model calculations as indicated in eqs . [ cslab ] through [ tot ] are the particle and gamma - transmission coefficients @xmath5 and the level density of excited states @xmath24 . therefore , the reliability of such calculations is determined by the accuracy with which these components can be evaluated ( often for unstable nuclei ) . in the following we want to discuss the methods utilized to estimate these quantities and recent improvements . the transition from an excited state in the compound nucleus @xmath28 to the state @xmath40 in nucleus @xmath41 via the emission of a particle @xmath26 is given by a summation over all quantum mechanically allowed partial waves @xmath42 here the angular momentum @xmath43 and the channel spin @xmath44 couple to @xmath45 . the transition energy in channel @xmath26 is @xmath46=@xmath47 . the individual particle transmission coefficients @xmath48 are calculated by solving the schrdinger equation with an optical potential for the particle - nucleus interaction . all early studies of thermonuclear reaction rates @xcite employed optical square well potentials and made use of the black nucleus approximation . we employ the optical potential for neutrons and protons given by @xcite , based on microscopic infinite nuclear matter calculations for a given density , applied with a local density approximation . it includes corrections of the imaginary part @xcite . the resulting s - wave neutron strength function @xmath49 is shown and discussed in @xcite , where several phenomenological optical potentials of the woods - saxon type and the equivalent square well potential used in earlier astrophysical applications are compared . the purely theoretical approach gives the best fit . it is also expected to have the most reliable extrapolation properties for unstable nuclei . a good overview on different approaches can be found in @xcite . deformed nuclei were treated in a very simplified way by using an effective spherical potential of equal volume , based on averaging the deformed potential over all possible angles between the incoming particle and the orientation of the deformed nucleus . in most earlier compilations alpha particles were also treated by square well optical potentials . we employ a phenomenological woods - saxon potential @xcite based on extensive data @xcite . for future use , for alpha particles and heavier projectiles , it is clear that the best results can probably be obtained with folding potentials ( e.g. @xcite ) . the gamma - transmission coefficients are treated as follows . the dominant gamma - transitions ( e1 and m1 ) have to be included in the calculation of the total photon width . the smaller , and therefore less important , m1 transitions have usually been treated with the simple single particle approach ( @xmath50 @xcite ) , as also discussed in @xcite . the e1 transitions are usually calculated on the basis of the lorentzian representation of the giant dipole resonance ( gdr ) . within this model , the e1 transmission coefficient for the transition emitting a photon of energy @xmath51 in a nucleus @xmath52 is given by @xmath53 here @xmath54 accounts for the neutron - proton exchange contribution @xcite and the summation over @xmath41 includes two terms which correspond to the split of the gdr in statically deformed nuclei , with oscillations along ( i=1 ) and perpendicular ( i=2 ) to the axis of rotational symmetry . many microscopic and macroscopic models have been devoted to the calculation of the gdr energies ( @xmath55 ) and widths ( @xmath56 ) . analytical fits as a function of a and z were also used @xcite . we make use of the ( hydrodynamic ) droplet model approach @xcite for @xmath57 , which gives an excellent fit to the gdr energies and can also predict the split of the resonance for deformed nuclei , when making use of the deformation , calculated within the droplet model . in that case , the two resonance energies are related to the mean value calculated by the relations @xcite @xmath58 , @xmath59 . @xmath60 is the ratio of the diameter along the nuclear symmetry axis to the diameter perpendicular to it , and can be obtained from the experimentally known deformation or mass model predictions . see @xcite for a detailed description of the approach utilized to calculate the gamma - transmission coefficients for the cross section determination shown in this work . while the method as such is well seasoned , considerable effort has been put into the improvement of the input for statistical hauser - feshbach models ( e.g. @xcite ) . however , the nuclear level density has given rise to the largest uncertainties in the description of nuclear reactions @xcite . for large scale astrophysical applications it is also necessary to not only find reliable methods for level density predictions , but also computationally feasible ones . such a model is the non - interacting fermi - gas model @xcite . most statistical model calculations use the back - shifted fermi - gas description @xcite . more sophisticated monte carlo shell model calculations @xcite , as well as combinatorial approaches ( see e.g. @xcite ) , have shown excellent agreement with this phenomenological approach and justified the application of the fermi - gas description at and above the neutron separation energy . here we want to apply an energy - dependent level density parameter @xmath0 together with microscopic corrections from nuclear mass models , which leads to improved fits in the mass range @xmath2 . mostly the back - shifted fermi - gas description , assuming an even distribution of odd and even parities ( however , see e.g. @xcite for doubts on the validity of this assumption at energies of astrophysical interest ) , is used @xcite : @xmath61 with @xmath62 the spin dependence @xmath63 is determined by the spin cut - off parameter @xmath64 . thus , the level density is dependent on only two parameters : the level density parameter @xmath0 and the backshift @xmath1 , which determines the energy of the first excited state . within this framework , the quality of level density predictions depends on the reliability of systematic estimates of @xmath0 and @xmath1 . the first compilation for a large number of nuclei was provided by @xcite . they found that the backshift @xmath1 is well reproduced by experimental pairing corrections . they also were the first to identify an empirical correlation with experimental shell corrections @xmath65 @xmath66 where @xmath65 is negative near closed shells . since then , a number of compilations have been published and also slightly different functional dependencies have been proposed ( for references , see e.g. @xcite ) , but they did not necessarily lead to better predictive power . improved agreement with experimental data was found @xcite by dividing the nuclei into three classes [ ( i ) those within three units of magic nucleon numbers , ( ii ) other spherical nuclei , ( iii ) deformed nuclei ] and fitting separate coefficients @xmath67 , @xmath68 for each class . in that case the mass formula in ref . @xcite was used . for the backshift @xmath1 the description @xmath69 was employed , deriving @xmath70 from the pairing correlation of a droplet model nuclear mass formula with the values @xmath71 with this treatment smaller deviations were found , compared to previous attempts @xcite . however , the number of parameters was considerably increased at the same time . the back - shifted fermi - gas approach diverges for @xmath72 ( i.e. @xmath73 , if @xmath1 is a positive backshift ) . in order to get the correct behavior at very low excitation energies , the fermi - gas description can be combined with the constant temperature formula ( @xcite ; @xcite and references therein ) @xmath74 the two formulations are matched by a tangential fit determining @xmath5 . an improved approach has to consider the energy dependence of the shell effects which are known to vanish at high excitation energies @xcite . although , for astrophysical purposes only energies close to the particle separation thresholds have to be considered , an energy dependence can lead to a considerable improvement of the global fit . this is especially true for strongly bound nuclei close to magic numbers . an excitation - energy dependent description was initially proposed by @xcite for the level density parameter @xmath0 : @xmath75\quad,\ ] ] where @xmath76 and @xmath77 the values of the free parameters @xmath78 , @xmath79 and @xmath80 are determined by fitting to experimental level density data . the shape of the function @xmath81 permits the two extremes : ( i ) for small excitation energies the original form of eq . [ aovera ] is retained with @xmath82 being replaced by @xmath83 , ( ii ) for high excitation energies @xmath0/@xmath84 approaches the continuum value obtained for infinite nuclear matter . previous attempts to find a global description of level densities used shell corrections @xmath85 derived from comparison of liquid - drop masses with experiment ( @xmath86 ) or the `` empirical '' shell corrections @xmath82 given by @xcite . a problem connected with the use of liquid - drop masses arises from the fact that there are different liquid - drop model parametrizations available in the literature which produce quite different values for @xmath85 @xcite . however , in addition the meaning of the correction parameter inserted into the level density formula ( eq . [ endepa ] ) has to be reconsidered . the fact that nuclei approach a spherical shape at high excitation energies has to be included . actually , the correction parameter @xmath87 should describe properties of a nucleus differing from the _ spherical _ macroscopic energy and include terms which are vanishing at higher excitation energies . the latter requirement is mimicked by the form of eq . [ endepa ] . therefore , the parameter should rather be identified with the so - called `` microscopic '' correction @xmath88 than with the shell correction . the mass of a nucleus with deformation @xmath89 can then be written as @xcite @xmath90 alternatively , one can write @xmath91 with @xmath92 being the shell - plus - pairing correction . the confusion about the term `` microscopic correction '' , which is sometimes used in an ambiguous way , is also pointed out in @xcite . thus , the above mentioned ambiguity follows from the inclusion of deformation - dependent effects into the macroscopic part of the mass formula . another important ingredient is the pairing gap @xmath93 , related to the backshift @xmath1 . instead of assuming constant pairing ( cf . @xcite ) or a fixed dependence on the mass number @xmath84 ( cf . eq . [ pairing ] ) , we determine the pairing gap @xmath93 from differences in the binding energies ( or mass differences , respectively ) of neighboring nuclei . thus , for the neutron pairing gap @xmath94 one obtains @xcite @xmath95\quad,\ ] ] where @xmath96 is the binding energy of the nucleus @xmath97 . similarly , the proton pairing gap @xmath98 can be calculated . at low energies , this description is again combined with the constant temperature formula ( eq . [ ctemp ] ) as described above . in our study we utilized the microscopic correction of a most recent mass formula @xcite , calculated with the finite range droplet model frdm ( using a folded yukawa shell model with lipkin - nogami pairing ) in order to determine the parameter @xmath83=@xmath88 . the backshift @xmath1 was calculated by setting @xmath99=1/2@xmath100 and using eq . [ pair ] . in order to obtain the parameters @xmath78 , @xmath79 , and @xmath80 , we performed a fit to experimental data on s - wave neutron resonance spacings of 272 nuclei at the neutron separation energy . the data were taken from a recent compilation @xcite . another recent investigation @xcite also attempted to fit level density parameters , but made use of a slightly different description of the energy dependence of @xmath0 and different pairing gaps . as a quantitative overall estimate of the agreement between calculation and experiment , one usually quotes the averaged ratio @xcite @xmath101^{1/2}\quad,\ ] ] with @xmath102 being the number of nuclei for which level densities @xmath24 are experimentally known . as best fit we obtain an averaged ratio @xmath103 with the parameter values @xmath104 , @xmath105 , @xmath106 . the ratios of experimental to predicted level densities ( i.e. theoretical to experimental level spacings ) for the nuclei considered are shown in fig . [ figrat ] . as can be seen , for the majority of nuclei the absolute deviation is less than a factor of 2 . this is a satisfactory improvement over the theoretical level densities previously used in astrophysical cross section calculations , where deviations of a factor 34 @xcite , or even in excess of a factor of 10 @xcite were found . such a direct comparison was rarely shown in earlier work . mostly the level density parameter @xmath0 , entering exponentially into the level density , was displayed . closely examining the nuclei with the largest deviations in our fit , we were not able to find any remaining correlation of the deviation with separation energy ( i.e. excitation energy ) or spin . although we quoted the value of the parameter @xmath79 above ( and will do so below ) as we left it as an open parameter in our fits , one can see that it is always small and can be set to zero without considerable increase in the obtained deviation . therefore , it is obvious that actually only two parameters are needed for the level density description . as an alternative to the frdm mass formula @xcite , in fig . [ hilffit ] we show the results when making use of the well - known mass formula by hilf et al . @xcite which turned out to be successful in predicting properties of nuclei at and close to stability . the parameter set @xmath107 , @xmath108 , @xmath109 yields an averaged ratio of @xmath110 . it can be seen from fig . [ hilffit ] that not only the average scatter is somewhat larger than with the frdm input , but also that this mass formula has problems in the higher mass regions . only an artificial alteration by about @xmath111 mev or more of the microscopic term in the deformed mass regions @xmath112 and @xmath113 can slightly improve the fit but the remaining scatter still leads to @xmath114 . the difference in the calculated level density from the frdm and the hilf mass formulae is plotted in fig . [ hcomp ] . the latter mass formula leads to a significantly higher level density ( by about a factor of 10 ) around the neutron magic number @xmath115 , whereas the level density remains lower ( by a factor of 0.07 ) close to the drip lines for @xmath116 . a fit comparable to the quality of the frdm approach can be obtained when employing a mass formula from an extended thomas - fermi plus strutinsky integral model ( etfsi ) @xcite . in order to extract a microscopic correction for this already microscopic calculation , we subtracted the frdm spherical macroscopic part @xmath117 ( see eq . [ emic ] ) from the etfsi mass and took this difference to be the etfsi microscopic correction . the pairing gaps were calculated as described above . this leads to a fit with @xmath118 , yielding the parameter values @xmath119 , @xmath120 and @xmath121 . however , although the fit is closer to the one obtained with frdm than the hilf result , the deviations for unstable nuclei are somewhat larger . the maximum deviation is a factor of about 38 for etfsi , as compared to a factor of 16 for the hilf approach . both formulae yield lower level densities than the frdm for nuclei close to the dripline with @xmath122 and higher level densities for neutron rich nuclei close to the magic shell at @xmath115 . the ratios of the level density from the etfsi approach to those of the frdm are shown in fig . [ etfsicomp ] . different combinations of masses and microscopic corrections from other models ( droplet model by myers and swiatecki @xcite , cameron - elkin mass formula and shell corrections @xcite ) were also tried but did not lead to better results . our fit to experimental level densities is also better than a recent analytical bcs approach @xcite which tried to implement level spacings from the etfsi model in a consistent fashion . to see the effect of the new level density description ( utilizing frdm input ) on the calculated cross sections , 30 kev neutron capture cross sections from experiment @xcite are compared to our calculations in fig . [ bakaplot ] . plotted are only nuclei for which the statistical model can be applied to calculate the cross section , using the criteria derived in the next section . an improvement in the overall deviation can be seen , compared to previous calculations @xcite . however , one systematic deviation can clearly be seen in the @xmath123 region . that `` peak '' is not caused by a deficiency in the general level density description but by the microscopic input . the frdm model overestimates the microscopic corrections close to the @xmath124 shell @xcite . we see that the uncertainty in level density translates into a similar uncertainty of the neutron capture cross sections which are used here as a representative example for applications to capture cross sections . although this does not seem to be a dramatic improvement for the experimental cross sections of stable nuclei over the previous approach @xcite , the purely empirical and also artificial division of nuclei into three classes of level density treatments could be avoided . the reason is that the excitation energy dependence was treated in the generalized ansatz of @xcite , ensuring the correct energy dependence which will also yield correct results when the adjustment is not done at the typical separation energy of 812 mev for stable nuclei but also for nuclei far from stability with smaller separation energies . the remaining uncertainty in the extrapolation is the reliability far off stability of the nuclear - structure model from which the microscopic corrections and pairing gaps ( and the masses ) are taken . however , recent investigations in astrophysics and nuclear physics have shown the robustness of the frdm approach @xcite . recently improved purely microscopic models have exhibited similar behavior towards the drip lines @xcite but there are no large scale calculations over the whole chart of nuclei available yet which include deformation . therefore , the frdm model used here is among the most reliable ones available at present . having a reliable level density description also permits to analyze when and where the statistical model approach is valid . generally speaking , in order to apply the model correctly a sufficiently large number of levels in the compound nucleus is needed in the relevant energy range which can act as doorway states to forming a compound nucleus . in the following this is discussed for neutron- , proton- and alpha - induced reactions with the aid of the level density approach presented above . this section is intended to be a guide to a meaningful and correct application of the statistical model . the nuclear reaction rate per particle pair at a given stellar temperature @xmath5 is determined by folding the reaction cross section with the maxwell - boltzmann ( mb ) velocity distribution of the projectiles @xcite @xmath125 two cases have to be considered : reactions between charged particles and with neutrons . the nuclear cross section for charged particles is strongely suppressed at low energies due to the coulomb barrier . for particles having energies less than the height of the coulomb barrier , the product of the penetration factor and the mb distribution function at a given temperature results in the so - called gamow peak , in which most of the reactions will take place @xcite . location and width of the gamow peak depend on the charges of projectile and target , and on the temperature of the interacting plasma . when introducing the astrophysical @xmath85 factor @xmath126 ( with @xmath60 being the sommerfeld parameter ) , one can easily see the two contributions of the velocity distribution and the penetrability in the integral : @xmath127 \quad,\ ] ] where the quantity @xmath128 arises from the barrier penetrability . taking the first derivative of the integrand yields the location of the gamov peak @xmath129 @xcite , @xmath130 with the charges @xmath131 , @xmath132 and the reduced mass @xmath16 of the involved nuclei at a temperature @xmath133 given in 10@xmath134 k. the effective width @xmath93 of the energy window can be derived as @xmath135 in the case of neutron - induced reactions the effective energy window has to be derived in a slightly different way . for s - wave neutrons ( @xmath136 ) the energy window is simply given by the location and width of the peak of the mb distribution function . for higher partial waves the penetrability of the centrifugal barrier shifts the effective energy @xmath129 to higher energies , similar to the gamov peak . for neutrons with energies less than the height of the centrifugal barrier this can be approximated by @xcite @xmath137 the energy @xmath129 will always be comparatively close to the neutron separation energy . using the above effective energy windows for charged and neutral particle reactions , a criterion for the applicability can be derived from the level density . for a reliable application of the statistical model a sufficient number of nuclear levels has to be within the energy window , thus contributing to the reaction rate . for narrow , isolated resonances , the cross sections ( and also the reaction rates ) can be represented by a sum over individual breit - wigner terms . at higher energies , with increasing level density , the sum over resonances may be approximated by an integral over @xmath13 @xcite . numerical test calculations were made in order to find the average number of levels per energy window which is sufficient to allow the substitution of the sum by an integral over the hf cross section . [ levcrit ] shows the dependence of the ratio between sum and integral @xcite on the number of levels in the energy window . to achieve 20% accuracy , about 10 levels are needed in the worst case ( non - overlapping , narrow resonances ) . usually , neutron s - wave resonances are comparatively broad and thus a smaller number of levels could be sufficient . however , applying the statistical model ( i.e. integrating over a level density instead of summing up over levels ) for a level density which is not sufficiently large , results in an overestimation of the actual cross section , as can be seen in fig . [ levcrit ] and was also shown in ref . therefore , in the following we will assume a conservative limit of 10 contributing resonances in the effective energy window for charged and neutral particle - induced reactions . fixing the required number of levels within the energy window of width @xmath93 , one can find the minimum temperature at which the above described condition is fulfilled . those temperatures ( above which the statistical model can be used ) are plotted in a logarithmic color scale in figs . [ neutcrit][alph ] . for neutron - induced reactions fig . [ neutcrit ] applies , fig . [ prot ] describes proton - induced reactions , and fig . [ alph ] alpha - induced reactions . plotted is always the minimum stellar temperature @xmath138 ( in 10@xmath139 k ) for the compound nucleus of the reaction . it should be noted that the derived temperatures will not change considerably even when changing the required level number within a factor of about two , because of the exponential dependence of the level density on the excitation energy . this permits to read directly from the plot whether the statistical model cross section can be `` trusted '' or whether single resonances or other processes ( e.g. direct reactions ) have also to be considered . ( however , this does not necessarily mean that the statistical cross section is always negligible in the latter cases , since the assumed condition is quite conservative ) . the above plots can give hints on when it is safe to use the statistical model approach and which nuclei have to be treated with special attention for specific temperatures . thus , information on which nuclei might be of special interest for an experimental investigation may also be extracted . in the first part of the paper we described the most recent approaches being used for the application of statistical model calculations in astrophysical applications . in the second part we focussed on the level density description which contained the largest error when using the properties described before . we were able to improve considerably the prediction of nuclear level densities by employing an energy - dependent description for the level density parameter @xmath0 and by properly including microscopic corrections and back - shifts . all nuclei can now be described with a single parameter set consisting of just three parameters . the globally averaged deviation of prediction from experiment of about 1.5 translates into a somewhat lower error in the final cross sections due to the dominance of transitions to states with low excitation energies . this will also make it worthwile to recalculate the cross sections and thermonuclear rates for many astrophysically important reactions in the intermediate and heavy mass region . finally , we also presented a `` map '' as a guide for the application of the statistical model for neutron- , proton- and alpha - induced reactions . [ neutcrit ] , [ prot ] , [ alph ] ( as well as figs . [ hcomp ] , [ etfsicomp ] ) as full size color plots can be obtained from the first author . the above plots can give hints on when it is safe to use the statistical model approach and which nuclei have to be treated with special attention at a given temperature . thus , information on which nuclei might be of special interest for an experimental investigation may also be extracted . it should be noted that we used very conservative assumptions in deriving the above criteria for the applicability of the statistical model . we thank f. kppeler and co - workers for making a preliminary version of their updated neutron - capture cross section compilation available to us . we also thank p. mller for discussions . this work was supported in part by the swiss nationalfonds . tr is acknowledging support by the austrian academy of sciences . w. hauser and h. feshbach , _ phys . a _ * 87 * , 366 ( 1952 ) . c. mahaux and h.a . weidenmller , _ ann . _ * 29 * , 1 ( 1979 ) . e. gadioli and p.e . hodgson , _ pre - 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the prediction of cross sections for nuclei far off stability is crucial in the field of nuclear astrophysics . in recent calculations the nuclear level density as an important ingredient to the statistical model ( hauser - feshbach ) has shown the highest uncertainties . we present a global parametrization of nuclear level densities within the back - shifted fermi - gas formalism . employment of an energy - dependent level density parameter @xmath0 , based on microscopic corrections from a recent frdm mass formula , and a backshift @xmath1 , based on pairing and shell corrections , leads to a highly improved fit of level densities at the neutron - separation energy in the mass range @xmath2 . the importance of using proper microscopic corrections from mass formulae is emphasized . the resulting level description is well suited for astrophysical applications . the level density can also provide clues to the applicability of the statistical model which is only correct for a high density of excited states . using the above description , one can derive a `` map '' for the applicability of the model to reactions of stable and unstable nuclei with neutral and charged particles .

astro-ph9706294

today , with a vast amount of publications being produced in every discipline of scientific research , it can be rather overwhelming to select a good quality work ; that is enriched with original ideas and relevant to scientific community . more often this type of publications are discovered through the citation mechanism . it is believed that an estimate measure for scientific credibility of a paper is the number of citations that it receives , though this should not be taken too literally since some publications may have gone unnoticed or have been forgotten about over time . knowledge of how many times their publications are cited can be seen as good feedback for the authors , which brings about an unspoken demand for the statistical analysis of citation data . one of the impressive empirical studies on citation distribution of scientific publications @xcite showed that the distribution is a power - law form with exponent @xmath0 . the power - law behaviour in this complex system is a consequence of highly cited papers being more likely to acquire further citations . this was identified as a _ preferential attachment _ process in @xcite . the citation distribution of scientific publications is well studied and there exist a number of network models @xcite to mimic its complex structure and empirical results @xcite to confirm predictions . however , they seem to concentrate on the total number of citations without giving information about the issuing publications . the scientific publications belonging to a particular research area do not restrict their references to that discipline only , they form bridges by comparing or confirming findings in other research fields . for instance most _ small world network models _ @xcite presented in statistical mechanics , reference a sociometry article @xcite which presents the studies of milgram on the small world problem . this is the type of process which we will investigate with a simple model that only considers two research areas and referencing within and across each other . the consideration of cross linking also makes the model applicable to _ the web of human sexual contacts _ @xcite , where the interactions between males and females can be thought of as two coupled growing networks . this paper is organized as follows : in the proceeding section the model is defined and analyzed with a rate equation approach @xcite . in the final section discussions and comparisons of findings with the existing data are presented . one can visualize the proposed model with the aid of fig . ( [ coupled ] ) that attempts to illustrate the growth mechanism . we build the model by the following considerations . initially , both networks @xmath1 and @xmath2 contains @xmath3 nodes with no cross - links between the nodes in the networks . at each time step two new nodes with no incoming links , one belonging to network @xmath1 and the other to @xmath2 , are introduced simultaneously . the new node joining to @xmath1 with @xmath4 outgoing links , attaches @xmath5 fraction of its links to pre - existing nodes in @xmath1 and @xmath6 fraction of them to pre - existing nodes in @xmath2 . the similar process takes place when a new node joins to @xmath2 , where the new node has @xmath7 outgoing links from which @xmath8 of them goes to nodes in @xmath2 and the complementary @xmath9 goes to @xmath1 . the attachments to nodes in either networks are preferential and the rate of acquiring a link depends on the number of connections and the initial attractiveness of the pre - existing nodes . we define @xmath10 as the average number of nodes with total @xmath11 number of connections that includes the incoming intra - links @xmath12 and the incoming cross - links @xmath13 in network @xmath1 at time @xmath14 . similarly , @xmath15 is the average number of nodes with @xmath16 connections at time @xmath14 in network @xmath2 . notice that the indices are discriminative and the order in which they are used is important , as they indicate the direction that the links are made . further more we also define @xmath17 and @xmath18 the average number of nodes with @xmath12 and @xmath19 incoming intra - links to @xmath1 and @xmath2 respectively . finally , we also have @xmath20 and @xmath21 to denote the average number of nodes in @xmath1 and @xmath2 with @xmath13 and @xmath22 incoming cross - links . to keep this paper less cumbersome we will only analyse the time evolution of network @xmath1 and apply our results to network @xmath2 . in addition to this , we only need to give the time evolution of @xmath23 , defined as the joint distribution of intra - links and cross - links . using this distribution we can find all other distributions that are mentioned earlier . the time evolution of @xmath23 can be described by a rate equation @xmath24\nonumber\\ & & + p_{ba}m_{b}[(k_{aa}+k_{ba}-1+a)n_{a}(k_{aa},k_{ba}-1,t)\nonumber\\ & & -(k_{aa}+k_{ba}+a)n_{a}(k_{aa},k_{ba},t)]\}+ \delta_{k_{aa}0}\delta_{k_{ba}0}.\end{aligned}\ ] ] the form of the eq . ( [ na ] ) seems very similar to the one used in @xcite . in that model the rate of creating links depends on the out - degree of the issuing nodes and the in - degree of the target nodes . here we are concerned with two different types of in - degrees namely intra- and cross - links of the nodes . on the right hand side of eq . ( [ na ] ) the terms in first square brackets represent the increase in the number of nodes with @xmath11 links when a node with @xmath25 intra - links acquires a new intra - link and if the node already has @xmath11 links this leads to reduction in the number . similarly , for the second square brackets where the number of nodes with @xmath11 links changes due to the incoming cross - links . the final term accounts for the continuous addition of new nodes with no incoming links , each new node could be thought of as the new publication in a particular research discipline . the normalization factor @xmath26 sum of all degrees is defined as @xmath27 we limit ourself to the case of preferential linear attachment rate@xcite @xmath28 shifted by @xmath29 , the initial attractiveness @xcite of nodes in @xmath1 , which ensures that there is a nonzero probability of any node acquiring a link . the nature of @xmath30 lets one to obtain , as @xmath31 @xmath32 where @xmath33 is the average total in - degree in network @xmath1 . ( [ mat ] ) implying that @xmath26 is linear in time . similarly , it is easy to show that @xmath34 is also linear function of time . we use these relations in eq . ( [ na ] ) to obtain the time independent recurrence relation @xmath35n_{a}(k_{aa},k_{ba})\nonumber\\ = p_{aa}m_{a}(k_{aa}+k_{ba}+a-1)n_{a}(k_{aa}-1,k_{ba})\nonumber\\ + p_{ba}m_{b}(k_{aa}+k_{ba}+a-1)n_{a}(k_{aa},k_{ba}-1)\nonumber\\ + ( a+<m_{a}>)\delta_{k_{aa}0}\delta_{k_{ba}0}.\end{aligned}\ ] ] the expression in eq . ( [ arec ] ) does not simplify however , it lets us to obtain the total in - degree distribution @xmath36 writing @xmath37 and since @xmath38 then @xmath39 satisfies @xmath40n_{a}(k_{a})=<m_{a}>(k_{a}+a-1)n_{a}(k_{a}-1 ) \nonumber\\ + ( a+<m_{a}>)\delta_{k_{a}0}.\end{aligned}\ ] ] solving eq . ( [ narec ] ) for @xmath41 yields , @xmath42 with @xmath43 as @xmath44 eq . ( [ nagamma ] ) gives the asymptotic behaviour of the total in - degree distribution in @xmath1 @xmath45 which is a power - law form with an exponent @xmath46 that only depends on the average total in - degree and the initial attractiveness of the nodes . similarly , we can write the total in - degree distribution in network @xmath2 for the asymptotic limit of @xmath47 as @xmath48 again , the exponent depends upon the initial attractiveness @xmath49 of nodes and the average total incoming links @xmath50 . we now move on to analyse @xmath17 , the distribution of the average number of nodes with @xmath12 intra - links in network @xmath1 . in citation network one can think of these links being issued from the same subject class as the receiving nodes and in the case of human sexual contact network , they represent the homosexual interactions . since @xmath51 which can also be written as @xmath52 , a linear function of time . then summing eq . ( [ arec ] ) over all possible values of @xmath13 @xmath53 we get @xmath54n_{aa}(k_{aa})+<m_{a } > \sum_{k_{ba}=0}^{\infty}k_{ba}n_{a}(k_{aa},k_{ba})\nonumber\\ = p_{aa}m_{a}(k_{aa}+a-1)n_{aa}(k_{aa}-1)+p_{aa}m_{a } \sum_{k_{ba}=0}^{\infty}k_{ba}n_{a}(k_{aa}-1,k_{ba})\nonumber\\ + p_{ba}m_{b}(k_{aa}+a)n_{aa}(k_{aa})+p_{ba}m_{b } \sum_{k_{ba}=0}^{\infty}(k_{ba}-1)n_{a}(k_{aa},k_{ba}-1)\nonumber\\ + ( a+<m_{a}>)\delta_{k_{aa}0}.\end{aligned}\ ] ] for large @xmath12 eq . ( [ aarec ] ) reduces to @xmath55n_{aa}(k_{aa})=\nonumber\\ p_{aa}m_{a}(k_{aa}+a-1)n_{aa}(k_{aa}-1)+(a+<m_{a}>)\delta_{k_{aa}0}.\end{aligned}\ ] ] iterating former relation for @xmath56 yields @xmath57 where @xmath58 in the asymptotic limit as @xmath59 eq . ( [ aagamma ] ) has a power - law form @xmath60 that depends upon both @xmath5 and the coupling parameter @xmath61 . similarly , the time independent recurrence relation for @xmath18 has the same form as eq . ( [ aarec ] ) with the only difference being the parameters . therefore we will simply give the power - law distribution @xmath62 where the other coupling parameter @xmath63 is revealed in the exponent . finally , the distribution of average number of nodes with incoming cross - links @xmath20 in @xmath1 can be found by summing over @xmath23 for all its intra - links @xmath64 as before @xmath65 is also linear in time . when the cross links @xmath13 are large enough , then from eq . ( [ arec ] ) we obtain @xmath66 where @xmath67 in the asymptotic limit as @xmath68 the distribution @xmath69 has a power - law form and similarly for the network @xmath2 as @xmath68 @xmath70 unlike the case in intra - links , here the exponents are inversely proportional to the coupled parameters @xmath61 and @xmath63 respectively . for the sake of simplicity , we set the number of outgoing links of the new nodes in either networks to be the same , i.e. @xmath71 . furthermore taking the rate of cross linking to be @xmath72 and the rate of intra linking @xmath73 , consequently we have @xmath74 , and @xmath75 as the coupling parameter . in the weak coupling case , the cross linking is negligibly small i.e. @xmath76 then the power - law exponent of the intra - link distribution is @xmath77 equal to total link distribution @xmath78 . this gives a solution obtained in@xcite and when @xmath79 we recover the exponent @xmath80 , the empirical findings in @xcite . thus , varying @xmath75 in @xmath83 yields any values of @xmath84 between @xmath85 and @xmath86 . on the contrary , the exponent of cross - link distribution @xmath87 decreases from @xmath86 to @xmath88 , as @xmath75 increases from @xmath89 to @xmath90 . taking @xmath79 gives @xmath91 supposing @xmath92 , which seems reasonable for consideration of citation networks , we find that @xmath93 and @xmath94 . the former result coincides with the distribution of connectivities for the electric power grid of southern california @xcite . where the system is small and the local interactions is of importance hence there seems to be some analogy to the intra - linking process . for the latter , as far as we are aware there is none empirical studies present in the published literature . now , consider the web of human sexual contacts@xcite . if we let @xmath1 to represent males and @xmath2 females that is @xmath95 and @xmath96 then @xmath97 are the power - law exponents of the degree distributions of the sexes . where @xmath98 and @xmath49 denote the male and female attractiveness respectively and usually @xmath99 is considered @xcite . by setting @xmath100 , @xmath101 and @xmath81 that is , cross links are predominant then as in @xcite we obtain @xmath102 for males and @xmath103 for females . the exponents @xmath104 and @xmath105 have been observed for the cumulative distributions in empirical study @xcite . the model we studied here seems to have the flexibility to represent variety of complex systems . we would like to thank the china scholarship council , epsrc for their financial support and geoff rodgers for useful discussions .

we introduce and solve a model which considers two coupled networks growing simultaneously . the dynamics of the networks is governed by the new arrival of network elements ( nodes ) making preferential attachments to pre - existing nodes in both networks . the model segregates the links in the networks as intra - links , cross - links and mix - links . the corresponding degree distributions of these links are found to be power - laws with exponents having coupled parameters for intra- and cross - links . in the weak coupling case the model reduces to a simple citation network . as for the strong coupling , it mimics the mechanism of _ the web of human sexual contacts_.

cond-mat0112052

agn have first been discovered in the radio and soon after searched in the optical band . consequently , they have been classified using their optical characteristics and mainly divided into two categories : type 1 ( agn1 ) and 2 ( agn2 ) according to the presence or not of broad emission lines in their optical spectra ( we will keep this definition of agn1 throughout this paper ) . before the advent of the last generation of hard x - ray telescopes , agn samples where predominantly based on agn1 selected either in the optical or , later on , in the soft x - rays by _ einstein _ and _ rosat_. in these bands the evolution of agn1 has been well measured ( see e.g. della ceca et al . 1992 ; boyle et al . 2000 ; miyaji , hasinger , & schmidt 2000 ) . on the contrary the production of samples of agn2 has been difficult at any wavelength and limited to few local surveys . the general picture was in favor of a model in which agn1 objects were associated to agn with low absorption in the hard x - rays while agn2 to obscured sources with large column densities and spectra strongly depressed in the soft x - rays , as expected in the unification models ( e.g. antonucci 1993 ) . in the last decade the advent of the _ asca _ and _ bepposax _ satellites has allowed for the first time the detection and identification of agn as the main counterparts of hard ( 2 - 10 kev ) x - ray sources down to fluxes @xmath14 erg @xmath3 s@xmath4 , more than 2 orders of magnitude fainter than _ heao1 _ ( wood et al . 1984 ) . these identifications accounted for about 30% of the 2 - 10 kev hard x - ray background ( ueda et al . 1998 ; fiore et al . recently the new generation of x - ray satellites such as _ chandra _ and _ xmm - newton _ , have reached fluxes 100 times fainter , identifying hundreds of sources and almost resolving the hard ( 2 - 10 kev ) x - ray background ( e.g. mushotzky et al . 2000 ; fiore et al . 2000 ; giacconi et al . 2001 ; hornschemeier et al . 2001 ; hasinger et al . 2001 ; tozzi et al . 2001 ; baldi et al . 2001 ) . thanks to their excellent angular resolution ( @xmath11 - 5@xmath15 ) , the first spectroscopic identifications projects have been able to observe faint ( i@xmath123 ) optical counterparts . at variance with the classical type-1/type-2 model in the optical , a significant number of the counterparts ( @xmath130% ) resulted to be apparently optical normal galaxies , with x - ray luminosities @xmath16@xmath10@xmath17 erg s@xmath4 typical of agn activity , and moreover part of the optical type 1 agns resulted to be absorbed in the hard x - rays ( see e.g. fiore et al . 2000 ; barger et al . 2001 ; tozzi et al . 2001 ; hornschemeier et al . 2001 ; comastri et al . 2002 ) . these observations have complicated the picture of the agn model . in this framework the computation of the density of agn has become an even more difficult task . in fact , it is not clear how to classify the sources and to take into account the selection biases introduced by the observation in the 2 - 10 kev range , where the absorption still play a relevant role . these recent deep surveys with _ chandra _ and _ xmm - newton _ have reached fluxes @xmath1 @xmath18 erg @xmath3 s@xmath4(2 - 10 kev ) in quite small areas ( less than 1 deg@xmath19 ) . as a consequence these surveys are not able to provide statistical significant samples at brighter fluxes ( @xmath20 erg @xmath3 s@xmath4 ; 5 - 10 kev ) where the density of sources is about 5/deg@xmath19 ( fiore et al . 2001 ) and tens of square degrees are to be covered . such data are necessary to provide large numbers of spectroscopic identified sources in a wide range of x - ray fluxes in order to cover as much as possible the @xmath21 plane and hence to derive their x - ray luminosity function ( lf ) . in this paper we report the results of the spectroscopic identifications of one of such brighter samples . the x - ray sources have been detected by the _ bepposax_-mecs instruments in the 5 - 10 kev band in the framework of the high energy llarge area survey ( hellas ) . preliminary results have been presented in fiore et al . ( 1999 ) and la franca et al . the whole survey and the catalogue is described by fiore et al . the data have been analyzed in the framework of the synthesis models for the x - ray background by comastri et al . ( 2001 ) , and the correlation with the soft x - rays has been investigated by vignali et al . ( 2001 ) . in section 2 we describe our x - ray and optical observations . in section 3 we present an analysis of the evolution of agn in the 2 - 10 kev band . because of the reasons previously described , the selection and definition of type 2/absorbed sources is still not clear , and thus we restricted our evolutionary studies to type 1 agn only . the results are discussed in section 4 . the spectroscopic follow up of the hellas sources has been carried out in a subsample enclosed in a region with @xmath22 , and outside @xmath23 and @xmath24 . in this region the number of sources is 118 out of a total of 147 . their flux distribution is shown in figure 1 and the sky coverage is shown in figure 2 and listed in table 1 . the _ bepposax _ x - ray positions have an uncertainty of about 1 - 1.5 arcmin , the larger at larger off - axis distances . we have thus searched for optical counterparts having r magnitude brighter than 21.0 in a circular region of 1 - 1.5 arcmin of radius around the hellas positions ( see below and section 3.1.1 for a discussion on the choice of this optical limit ) . in the case of large off - axis distances , the larger error - boxes ( 1.5@xmath25 ) have been used . 25 sources have been identified with cross - correlation with existing catalogues ( ned ) , and 49 have been investigated at the telescope . the total resulting sample of 74 sources has been built up in such a way that to a ) randomly sample the flux distribution of the parent catalogue of 118 sources ( see figure 1 ) , and b ) to avoid possible biases introduced by the cross - correlation with existing catalogues ( see appendix a for a detailed discussion ) . cl = 0.2 cm 1998 dec 28 : 3 nights & eso/3.6 m efosc2 1999 jun 8 : 3 nights & eso/3.6 m efosc2 2000 jan 3 : 3 nights & eso/3.6 m efosc2 2000 jul29 : 3 nights & eso/3.6 m efosc2 [ tab2 ] the follow - up observations have been carried out using the efosc2 at the 3.6 m eso telescope . previously published identifications ( fiore et al . 1999 ) have been carried out with the rc spectrograph ( rcsp ) at the kitt peak 4 m telescope , the fast spectrograph at the whipple 60@xmath15 telescope , and the hawaii 88@xmath15 telescope . long slit and multi object spectroscopy have been carried with resolution between 7 and 16 . the reduction process used standard midas facilities ( banse et al . the raw data were sky - subtracted and corrected for pixel - to - pixel variations by division with a suitably normalized exposure of the spectrum of an incandescent source ( flat - field ) . wavelength calibrations were carried out by comparison with exposures of he - ar , he , ar and ne lamps . relative flux calibration was carried out by observations of spectrophotometric standard stars ( oke 1990 ) . the campaign dates of the observations carried out at eso are listed in table 2 . we have divided the identified sources in agn1 , agn2 , galaxies with narrow emission lines showing moderate - to - high degree of excitation ( elg ) , bl lac objects , clusters of galaxies and stars . agn2 includes agn1.5 , agn1.8 and agn1.9 . no distinction has been made between radio - loud and radio quite objects . the radio properties of the hellas sources will be discussed in a forthcoming paper ( p. ciliegi , in preparation ) . the r - band magnitudes of these identifications have been pushed down to the limit of obtaining about less than 10@xmath26 probability of having a spurious identification , due to chance coincidences , for each of the hellas sources ( i.e. 90@xmath26 reliability for the whole sample ) . for agn1 we have chosen a limit of r=21 where the surface density is @xmath160 - 70/deg@xmath19 ( zitelli et al . tresse et al . ( 1996 ) found that 17@xmath26 of galaxies with @xmath27 host an emission line spectrum not typical of excitation from starburst ( but typical of agn2 activity ) , this percentage reduces to at minimum 8% if the maximum likely effect of absorption under the balmer lines is taken into account . they also found that the galaxies having forbidden emission lines without being agn2 are at least 36@xmath26 . these objects are mainly starburst galaxies and low ionization narrow emission line galaxies ( liner ) . we will call here all these sources simply emission line galaxies ( elg ) . we have assumed roughly constant these percentages and thus chosen a limit of r=19 for agn2 where the surface density of all galaxies is about 500/deg@xmath19 and consequently the expected density of agn2 is @xmath170 - 80/deg@xmath19 . for elg we have chosen a limit of r=17.5 where the surface density of all galaxies is about 160/deg@xmath19 , and therefore the expected density of elg is @xmath160/deg@xmath19 . the classification of the narrow emission line galaxies has been carried out using standard line ratio diagnostics ( tresse et al . some of the spectra showed h@xmath28 in emission and h@xmath29 having a ) very small emission , or b ) absent , or c ) small absorption . such spectra show however spectral features typical of strong - emission - line spectra such as [ oii]@xmath30 , [ oiii]@xmath31 , and [ sii]@xmath32 . in such cases the [ oiii]/h@xmath29 push the diagnostic ratio in the locus of agn2 , but stellar absorption at h@xmath29 could be significant affecting the classification . following tresse et al . ( 1996 ) , we have tested the classification under the assumption of a possible absorption of ew=2 over h@xmath29 . we have spectroscopically identified 36 out of the 49 hellas sources whose field has been investigated at the telescopes . for 13 sources we have found no optical counterparts brighter than the chosen magnitude limits ( see discussion above ) . the astrometric accuracy of the new generation x - ray telescopes ( _ chandra _ and _ xmm - newton _ ) would probably allow an unambiguous identification of the counterparts of these hellas sources , but in the framework of this work these fields have been declared _ empty_. these sources have harder x - ray spectra than the total sample . they have an average softness ratio of @xmath33 in comparison with the average value @xmath34 of the total sample of 118 sources , where s and h are the 1.3 - 4.5 kev the 4.5 - 10 kev counts , respectively ( see fiore et al . the absence of bright counterparts and the average harder x - ray spectrum favor the hypothesis that most of these sources preferably harbor absorbed agn . if these sources correspond to optical normal galaxies , such as those observed by _ chandra _ and _ xmm - newton _ ( see e.g. fiore et al . 2000 ; barger et al . 2001 ; tozzi et al . 2001 ; hornschemeier et al . 2001 ; comastri et al . 2002 ) , we would have not been able to properly identify them inside our error - boxes . however , our sample is statistically well defined , and from the number of empty fields we can directly derive an upper limit of @xmath118% ( 13/74 ) for the presence of these optical normal galaxies in our survey . the list of the observed hellas sources with their most probable spectroscopic identification is shown in table 3 . 80% of our identifications are inside an error - box of 60 arcsecs . 2 bright galaxies ( r@xmath115 - 16 ) are at a distance of about 110@xmath15 . the 13 empty fields are listed in table 4 . the optical spectra are shown in figure 3 ; as a reference , a list of the most typical emission lines for agn are over - plotted with the corresponding redshift . in table 5 the ew , fwhm , fwzi in the observed frame of the most relevant emission lines are listed . in total , 63@xmath26 ( 74/118 ) of our hellas subsample has been searched for spectroscopic identification . 61 have been identified : 37 agn1 , 9 agn2 , 5 elg , 6 clusters , 2 bl lac , 1 radio galaxy and 1 star . ccrrrllll = 0.2 cm 00 26 36.5 @xmath3519 44 14 & 00 26 36.1 @xmath3519 44 16 & 7 & 13 & 3.4 & 18.1 & agn2 & 0.238 + 00 27 09.9 @xmath3519 26 31 & 00 27 09.8 @xmath3519 26 14 & 18 & 6 & 1.8 & 17.7 & agn1 & 0.227@xmath36 + 00 45 49.6 @xmath3525 15 14 & 00 45 46.3 @xmath3525 15 50 & 59 & 24 & 3.3 & 17.5 & agn2 & 0.111 + 01 21 56.8 @xmath3558 44 05 & 01 21 56.9 @xmath3558 44 42 & 37 & 16 & 2.6 & 16.8 & agn2 & 0.118 + 01 34 33.3 @xmath3529 58 38 & 01 34 33.8 @xmath3529 58 16 & 24 & 6 & 0.9 & 18.0 & agn1 & 2.217 + 01 35 30.2 @xmath3529 51 22 & 01 35 32.7 @xmath3529 52 02 & 53 & 8 & 0.9 & 17.7 & agn1 & 1.344 + 01 40 08.9 @xmath3567 48 13 & 01 40 14.7 @xmath3567 48 54 & 53 & 8 & 2.8 & 12.4 & star & 0.000 + 02 42 01.8 @xmath3700 00 46 & 02 42 00.9 @xmath3700 00 22 & 29 & 10 & 1.5 & 18.6 & agn1 & 1.112@xmath38 + 03 15 45.0 @xmath3555 29 26 & 03 15 47.5 @xmath3555 29 04 & 31 & 15 & 2.7 & 17.9 & agn1 & 0.464@xmath38 + 03 17 32.4 @xmath3555 20 12 & 03 17 32.7 @xmath3555 20 26 & 14 & 21 & 4.1 & 17.5 & agn1 & 0.406@xmath38 + 03 33 09.6 @xmath3536 19 40 & 03 33 12.2 @xmath3536 19 48 & 33 & 15 & 4.0 & 17.5 & blac & 0.308@xmath38 + 03 34 07.4 @xmath3536 04 22 & 03 34 07.5 @xmath3536 04 04 & 19 & 5 & 1.9 & 20.1 & agn1 & 0.904 + 03 36 51.3 @xmath3536 15 57 & 03 36 54.0 @xmath3536 16 06 & 34 & 15 & 3.7 & 17.7 & agn1 & 1.537@xmath38 + 04 37 14.5 @xmath3547 30 58 & 04 37 11.8 @xmath3547 31 48 & 57 & 16 & 2.7 & 17.3 & agn1 & 0.142 + 04 38 47.9 @xmath3547 29 07 & 04 38 47.0 @xmath3547 28 02 & 67 & 20 & 4.7 & 20.5 & agn1 & 1.453 + 06 46 39.3 @xmath3544 15 35 & 06 46 37.6 @xmath3544 15 34 & 19 & 17 & 4.3 & 16.6 & agn1 & 0.153 + 07 21 29.6 @xmath3771 14 04 & 07 21 36.2 @xmath3771 13 24 & 52 & 8 & 0.8 & 17.7 & agn1 & 0.232@xmath38 + 07 41 40.3 @xmath3774 14 58 & 07 41 50.0 @xmath3774 14 48 & 41 & 23 & 30.7 & .... & clust . & 0.216@xmath38 + 07 43 09.1 @xmath3774 29 20 & 07 43 12.6 @xmath3774 29 36 & 22 & 7 & 6.0 & 16.4 & agn1 & 0.312@xmath38 + 08 37 37.2 @xmath3725 47 49 & 08 37 37.1 @xmath3725 47 52 & 4 & 12 & 2.6 & 16.9 & agn1 & 0.077 + 08 38 59.9 @xmath3726 08 14 & 08 38 59.2 @xmath3726 08 14 & 10 & 23 & 16.4 & 15.3 & elg & 0.048 + 10 32 15.8 @xmath3750 51 04 & 10 32 16.2 @xmath3750 51 20 & 18 & 10 & 3.1 & 15.9 & agn1 & 0.174 + 10 54 19.8 @xmath3757 25 09 & 10 54 21.2 @xmath3757 25 44 & 38 & 13 & 2.6 & 18.5 & agn2 & 0.205@xmath38 + 11 01 46.4 @xmath3772 26 11 & 11 01 48.8 @xmath3772 25 38 & 36 & 22 & 7.3 & 16.7 & agn1 & 1.460@xmath38 + 11 02 37.2 @xmath3772 46 38 & 11 02 36.8 @xmath3772 46 40 & 3 & 21 & 7.9 & 15.1 & agn1 & 0.089@xmath38 + 11 06 14.0 @xmath3772 43 16 & 11 06 16.1 @xmath3772 44 14 & 59 & 9 & 1.7 & 18.5 & agn1 & 0.680@xmath38 + 11 18 11.9 @xmath3740 28 33 & 11 18 13.8 @xmath3740 28 38 & 23 & 4 & 0.8 & 18.7 & agn1 & 0.387 + 11 18 46.2 @xmath3740 27 39 & 11 18 48.7 @xmath3740 26 48 & 59 & 5 & 1.4 & 18.5 & agn1 & 1.129 + 12 04 07.6 @xmath3728 08 31 & 12 04 04.0 @xmath3728 07 24 & 82 & 16 & 5.1 & .... & clust . & 0.167@xmath38 + 12 17 50.3 @xmath3730 07 08 & 12 17 52.1 @xmath3730 07 02 & 25 & 20 & 3.5 & 14.0 & bllac & 0.237@xmath38 + 12 18 55.0 @xmath3729 58 12 & 12 18 52.5 @xmath3729 59 02 & 59 & 13 & 2.0 & 18.6 & agn2 & 0.176 + 12 19 45.7 @xmath3747 20 43 & 12 19 52.3 @xmath3747 20 58 & 69 & 8 & 1.2 & 19.3 & agn1 & 0.654@xmath38 + 12 22 06.8 @xmath3775 26 17 & 12 22 07.0 @xmath3775 26 18 & 2 & 7 & 2.5 & .... & clust . & 0.240@xmath38 + 12 40 26.0 @xmath3505 13 20 & 12 40 27.8 @xmath3505 14 02 & 50 & 12 & 3.1 & 18.8 & agn1 & 0.300 + 12 40 29.6 @xmath3505 07 46 & 12 40 36.4 @xmath3505 07 52 & 102 & 17 & 1.9 & 15.2 & elg & 0.008 + 13 04 38.2 @xmath3510 15 47 & 13 04 35.6 @xmath3510 15 48 & 39 & 6 & 1.4 & 20.1 & agn1 & 2.386 + 13 05 32.3 @xmath3510 32 36 & 13 05 33.0 @xmath3510 33 20 & 45 & 22 & 19.3 & 14.9 & agn1 & 0.278@xmath38 + 13 36 34.3 @xmath3533 57 48 & 13 36 39.0 @xmath3533 57 58 & 60 & 22 & 3.2 & 10.5 & radiog . & 0.013@xmath38 + 13 42 59.3 @xmath3700 01 38 & 13 42 56.5 @xmath3700 00 58 & 59 & 21 & 3.2 & 18.7 & agn1 & 0.804@xmath38 + 13 48 20.8 @xmath3530 11 06 & 13 48 19.5 @xmath3530 11 56 & 52 & 14 & 2.2 & 15.3 & agn2 & 0.128 + 13 48 45.4 @xmath3530 29 37 & 13 48 44.7 @xmath3530 29 46 & 13 & 14 & 5.1 & 17.1 & agn1 & 0.330 + 13 50 09.4 @xmath3530 19 55 & 13 50 15.4 @xmath3530 20 10 & 80 & 11 & 5.1 & 16.5 & elg & 0.074 + 13 53 54.6 @xmath3718 20 33 & 13 53 54.4 @xmath3718 20 16 & 18 & 18 & 6.8 & 17.3 & agn1 & 0.217 + 14 17 12.5 @xmath3724 59 29 & 14 17 18.8 @xmath3724 59 30 & 86 & 13 & 0.7 & 19.5 & agn1 & 1.057@xmath38 + 14 18 31.1 @xmath3725 11 07 & 14 18 31.2 @xmath3725 10 50 & 17 & 9 & 6.1 & .... & clust . & 0.240@xmath38 + 15 19 39.9 @xmath3765 35 46 & 15 19 33.7 @xmath3765 35 58 & 41 & 14 & 9.4 & 14.4 & agn2 & 0.044 + 15 28 47.3 @xmath3719 39 10 & 15 28 47.7 @xmath3719 38 54 & 18 & 5 & 1.6 & 20.3 & agn1 & 0.657@xmath39 + 16 26 59.9 @xmath3755 28 21 & 16 26 59.0 @xmath3755 27 24 & 57 & 11 & 12.1 & .... & clust . & 0.130@xmath38 + 16 34 11.8 @xmath3759 45 29 & 16 34 18.5 @xmath3759 45 44 & 53 & 3 & 0.8 & 19.0 & agn2 & 0.341@xmath40 + 16 49 57.9 @xmath3704 53 33 & 16 50 00.0 @xmath3704 54 00 & 42 & 20 & 9.5 & .... & clust . & 0.154@xmath38 + 16 50 40.1 @xmath3704 37 17 & 16 50 42.7 @xmath3704 36 18 & 71 & 25 & 12.2 & 14.6 & agn2 & 0.031 + 16 52 38.0 @xmath3702 22 18 & 16 52 37.5 @xmath3702 22 06 & 14 & 5 & 0.7 & 20.7 & agn1 & 0.395 + 20 42 47.6 @xmath3510 38 31 & 20 42 53.0 @xmath3510 38 26 & 80 & 21 & 5.3 & 17.9 & agn1 & 0.363 + 20 44 34.8 @xmath3510 27 34 & 20 44 34.8 @xmath3510 28 08 & 35 & 15 & 2.0 & 17.7 & agn1 & 2.755 + 22 26 30.3 @xmath3721 11 57 & 22 26 31.5 @xmath3721 11 38 & 26 & 14 & 3.9 & 17.6 & agn1 & 0.260 + 23 15 36.4 @xmath3559 03 40 & 23 15 46.8 @xmath3559 03 14 & 85 & 8 & 2.0 & 11.2 & elg & 0.044@xmath38 + 23 19 22.1 @xmath3542 41 50 & 23 19 32.0 @xmath3542 42 28 & 116 & 21 & 5.7 & 16.5 & elg@xmath41 & 0.101 + 23 27 28.7 @xmath3708 49 30 & 23 27 28.7 @xmath3708 49 26 & 4 & 7 & 0.5 & 18.5 & agn1@xmath42 & 0.154 + 23 29 02.4 @xmath3708 34 39 & 23 29 05.8 @xmath3708 34 16 & 56 & 20 & 2.9 & 20.3 & agn1 & 0.953 + 23 31 55.6 @xmath3719 38 34 & 23 31 54.3 @xmath3719 38 36 & 19 & 17 & 3.7 & 18.8 & agn1 & 0.475 + 23 55 53.3 @xmath3728 36 06 & 23 55 54.3 @xmath3728 35 58 & 16 & 12 & 4.2 & 17.9 & agn1 & 0.731@xmath38 + [ tab3 ] cc = 0.2 cm 01 34 28.6 @xmath3530 06 04 & 1.33 + 01 34 49.6 @xmath3530 02 34 & 0.71 + 09 46 05.3 @xmath3514 02 59 & 2.82 + 09 46 32.8 @xmath3514 06 16 & 3.22 + 13 04 24.3 @xmath3510 23 53 & 1.28 + 13 04 45.1 @xmath3505 33 37 & 1.26 + 13 48 24.3 @xmath3530 25 47 & 3.15 + 14 11 58.7 @xmath3503 07 02 & 3.93 + 22 03 00.5 @xmath3532 04 18 & 2.83 + 22 31 49.6 @xmath3711 32 08 & 1.94 + 23 02 30.1 @xmath3708 37 06 & 2.67 + 23 02 36.2 @xmath3708 56 42 & 3.17 + 23 16 09.8 @xmath3559 11 24 & 1.32 + [ tab4 ] because of the difficulties previously described in quantifying all the selection biases in building up samples of type 2 ( and/or absorbed ) agn , we have decided to limit our evolutionary analysis to agn1 only . see appendix a for a discussion on the absence of substantial biases on our sample of agn1 . we have assumed h@xmath43=50 km / s / mpc and either the ( @xmath12,@xmath13)=(1.0,0.0 ) or the ( @xmath12,@xmath13)=(0.3,0.7 ) cosmologies . we have computed the 2 - 10 kev luminosity for agn1 assuming a typical spectral energy distribution as computed by pompilio , la franca , & matt ( 2000 ) , which assumes a power slope in energy with index @xmath44=0.9 ( @xmath45 ) and takes into account the reflection . this spectrum is roughly approximated by a single power @xmath44=0.6 in the range 2 - 50 kev ( i.e. up to @xmath46=4 ) . this is in agreement with the average slope of the spectra of agn1 from the hellas sample ( vignali et al . 2001 ) . as our spectroscopic identifications are limited to magnitudes brighter than @xmath47 , in order to estimate the density of agn1 , we have evaluated which is their completeness in our sample . we have assumed that all agn1 follow the same distribution of the ratio of the 5 kev to r - band optical luminosity log(l@xmath48/l@xmath49 ) without dependencies on luminosity and/or redshift . we have estimated the relationship between optical and hard x - ray luminosity for agn1 as follow . we selected a mixed sample of both 25 optical and 25 hard x - ray selected type 1 agn , mostly in the redshift range 0@xmath50@xmath46@xmath502 , from mineo et al . ( 2000 ) , george et al . ( 2000 ) and akiyama et al . the 25 optically selected agn1 have an average log(l@xmath48/l@xmath49 ) ratio of -4.78 with a standard deviation of 0.59 , while the 25 x - ray selected agn1 have an average log(l@xmath48/l@xmath49 ) ratio of -4.38 with a standard deviation of 0.58 . in figure 4 the histogram of the ratio of 5 kev and r - band optical luminosity of the total sample of 50 agn1 at @xmath46=0 is shown . we fitted the distribution with a gaussian having mean -4.58 and standard deviation 0.61 . we have estimated that with this distribution , a limit of @xmath47 corresponds to only 1 agn1 lost from our hellas survey , and consequently we can say that a limit of @xmath47 assure a high level of completeness for the identification of agn1 at the fluxes of our hellas survey . at variance , as already discussed , our hellas identification programme is biased against absorbed type 2 sources which probably harbor in the 13 empty fields found . although the incompleteness for agn1 in the hellas sample is quite low , during the computation of the lf of agn1 , we have estimated the correct area coverage for our survey by multiplying the area coverage corresponding to each point of the @xmath21 plane for : a ) the spectroscopic completeness of the hellas sample ( 74/118 ) , and b ) the fraction of found agn1 as a function of @xmath46 , as described before . in order to increase the statistical significance of our analysis , the hellas data have been combined to other hard x - ray samples of agn1 identified by grossan ( 1992 ) , boyle et al . ( 1998 ) , and akiyama et al . ( 2000 ) . the sample of grossan ( 1992 ) consists of 84 agn1 and 12 agn2 detected by _ predominantly at low redshift @xmath27 . the sample covers an area of 26919 deg@xmath19 down to a flux limit of @xmath51 erg @xmath3 s@xmath4 . the quoted fluxes at 5 kev have been converted to fluxes in the 2 - 10 kev assuming @xmath52 . the sample of boyle et al . ( 1998 ) consists of 12 agn1 and 6 agn2 detected by _ asca_. 2 - 10 kev counts have been converted to fluxes with a count - rate - to - flux conversion factor of @xmath53 erg @xmath3 s@xmath4/count , assuming @xmath52 ( georgantopoulos et al . , 1997 ) . the sky coverage has been taken from georgantopoulos et al . ( 1997 , see their figure 1 ) . the sample of akiyama et al . ( 2000 ) consists of 25 agn1 , 5 agn2 , 2 clusters , 1 star and 1 unidentified source . the sources have been detected by _ asca_. in order to convert the counts of the sky coverage presented in their table 1 , a conversion has been applied assuming that 1 count corresponds to @xmath54 erg @xmath3 s@xmath4 in the 2 - 10 kev band for an x - ray source with a slope @xmath44=0.6 . a total of 158 agn1 have been used in the redshift range 0@xmath50@xmath46@xmath503 . their luminosity - redshift distribution is shown in figure 5 . in this distribution a gap between the sample of grossan from _ hea01 _ and the other fainter samples from _ asca _ and _ bepposax _ is evident . this is because surveys at fluxes in the range @xmath55@xmath56 erg @xmath3 s@xmath4 in the 2 - 10 kev energy band where some hundreds of deg@xmath19 need to be covered are still missing . these surveys are in program with _ xmm - newton _ ( see e.g. barcons et al . 2001 ) . the best - fit parameters for the 2 - 10 kev lf and its cosmological evolution have been derived by minimization of the @xmath57 computed by comparison of the observed and expected number of agn1 in 6@xmath583 bins in the @xmath21 plane . the adopted luminosity / redshift grid was : 6 log@xmath16 bins , with @xmath59log@xmath16=1 in the range 42.2 - 48.2 , and 3 redshift bins ( 0.0 - 0.2 , 0.2 - 1.0 , 1.0 - 3.0 ) . table 6 summarizes the results for a number of different models . the errors quoted for the parameters are 68@xmath26 ( 1@xmath60 ) confidence intervals . they correspond to variations of @xmath61=1.0 , obtained perturbing each parameter in turn with respect to its best - fit value , and looking for a minimization with the remaining parameters free to float . the fitted lf have been tested with the 2d kolmogorov - smirnov ( 2dks ) test . two different functional forms have been used . following ceballos and barcons ( 1996 ) and boyle et al . ( 1998 ) , we used the pure luminosity evolution ( ple ) for the agn1 lf . we also used the luminosity dependent density evolution ( ldde ) model similar to the one fitted in the soft x - rays by miyaji et al . ( 2000 ) . the local ( @xmath46=0 ) agn1 lf used for the ple model has been represented by a two - power - law : @xmath62 where @xmath63 is expressed in units of @xmath64 . a standard power - law luminosity evolution model was used to parameterize the cosmological evolution of this lf : @xmath65 . in the case of the ldde model , as an analytical expression of the present day ( @xmath46=0 ) luminosity function , we used the smoothly connected two power - law form : @xmath66^{-1}.\ ] ] the description of the evolution is given by a factor @xmath67 such that : @xmath68 where @xmath69 } ) } & ( z \leq z_{\rm c } ; l_{\rm x}<l_{\rm a } ) \\ ( 1+z)^{p1 } & ( z \leq { z_{\rm c } } ; l_{\rm x}\ge l_{\rm a})\nonumber \\ e(l_{\rm x},{z_{\rm c } } ) \left[(1+z)/(1+{z_{\rm c } } ) \right]^{p2 } & ( z>{z_{\rm c } } ) . \\ \end{array } \right . \ ] ] lclllllllr = 0.2 cm 0 ) ple from boyle & 1.0,0.0 & 1.73 & 2.96 & 44.16 & 2.00 & ... & @xmath70 & @xmath71 & 24 + 1 ) ple & 1.0,0.0 & 1.83 & 3.00 & 44.17 & 2.12 & ... & @xmath72 & @xmath73 & 35 + 2 ) ple with @xmath74 & 1.0,0.0 & 1.87 & 3.03 & 44.17 & 2.52 & 1.39 & @xmath75 & @xmath76 & 34 + 3 ) ldde & 1.0,0.0 & 0.68 & 2.02 & 44.03 & 4.66 & 1.55@xmath40 & @xmath77 @xmath78 & @xmath79 & 40 + 4 ) ple & 0.3,0.7 & 1.93 & 2.97 & 44.24 & 2.22 & ... & @xmath80 & @xmath79 & 56 + 5 ) ple with @xmath74 & 0.3,0.7 & 1.94 & 2.97 & 44.24 & 2.26 & 2.38 & @xmath80 & @xmath81 & 54 + 6 ) ldde & 0.3,0.7 & 0.69 & 1.95 & 43.96 & 4.56 & 1.58@xmath40 & @xmath82 @xmath41 & @xmath83 & 60 + + 7 ) ldde softx & 1.0,0.0 & 0.62@xmath40 & 2.25@xmath40 & 44.13 & 5.40@xmath40 & 1.55@xmath40 & @xmath84 @xmath85 & @xmath86 & 48 + + 68@xmath26 confidence errors & & @xmath87 & @xmath88 & @xmath89 & @xmath90 & @xmath91 & 8@xmath26 + [ tab6 ] the parameters are defined as in miyaji et al . in particular , @xmath44 is proportional to the decrease of the density evolution of the faint objects . in this parameterization of the lf @xmath16 is expressed in units of @xmath92 . in both models the lf has been computed for luminosities brighter than logl@xmath93=42.2 . using this parameterization , for ( @xmath12,@xmath13)=(1.0,0.0 ) , a fit was done by keeping fixed the parameters @xmath94=@xmath95 ( the redshift at which the evolution stops ) , @xmath44=@xmath96 , and the ratio @xmath97 to the value found by miyaji et al . ( 2000 ) for agn1 only ( their ldde1 model ) in the soft x - rays , and leaving all the remaining parameters ( the two slopes @xmath98 and @xmath99 , the break luminosity @xmath100 , and the speed of the density evolution @xmath101 ) free to vary . in the case of an ( @xmath12,@xmath13)=(0.3,0.7 ) universe , as the fit for agn1 only from miyaji et al . ( 2000 ) was not available , their parameters from the fit for all agns ( agn1+agn2 ) have been used : @xmath94=@xmath102 , @xmath44=@xmath103 , @xmath104 . we started our computation of the lf in an ( @xmath12,@xmath105)=(1.0,0.0 ) cosmology . previous estimates of the shape and evolution of the lf of agn1 in the 2 - 10 kev range are from ceballos and barcons ( 1996 ) , and boyle et al . boyle et al . ( 1998 ) combined the local sample of 84 agn1 observed by _ heao1 _ from grossan ( 1992 ) with a fainter sample of 12 agn1 observed by _ asca _ ( see their distribution in the l@xmath106-@xmath46 plane in figure 4 ) . they found that the 2 - 10 kev agn x - ray lf is best represented by a two - power - law function evolving according to a pure luminosity evolution ( ple ) model : @xmath5@xmath6@xmath107 ( see model 0 in table 6 ) . in figure 6 the lf from only the 37 agn1 from hellas in three redshift intervals ( 0.0@xmath50@[email protected] , 0.2@xmath50@[email protected] , and 1.0@xmath50@[email protected] ) is shown . for the sake of comparison , the best - fit lf from boyle et al . ( 1998 ) is also shown . the data have been represented by correcting for evolution within the redshift bins as explained in la franca & cristiani ( 1997 ) . the error bars in the figures are based on poisson statistics at the 68@xmath26 confidence level . the data are in rough agreement with the previous estimate of boyle et al . ( 1998 ) , but show an excess of faint agn1 at redshift larger than @[email protected] . this feature is still present after combining our data with the other agn1 samples from grossan ( 1992 ) , boyle et al . ( 1998 ) , and akyiama et al . ( 2000 ) ( figure 7 ) . as already discussed , these samples all together collect 158 agn1 . these data have a @xmath57 probability of 0.04 to be drawn from a ple model such as that computed by boyle et al . ( 1998 ) ( model 0 in table 6 ) . we did not use the 2dks as this test is not appropriate in this case : it uses the cumulative distributions in the @xmath21 plane regardless of the normalization of the lf . our best - fit to the data with a ple model found a lf with similar slopes and break luminosity as that one of boyle et al . ( 1998 ) but with a slightly larger evolution ( @xmath108 instead of @xmath8=@xmath109 ) , and a significantly 20(@xmath110)@xmath26 larger normalization ( the normalization has a poisson uncertainty of 8@xmath26 . see model 1 in table 6 ) . our larger values of the evolution parameter @xmath8 and normalization @xmath111 are originated by the necessity of better fitting the observed higher density of faint agn1 at high redshift . the 2dks test gives a probability of 0.22 for this fit ( see table 6 ) . a even better probability of 0.31 is obtained if a stop in the evolution is applied at redshift @xmath74=@xmath112 ( model 2 in table 6 ) and a larger evolution ( @xmath8=@xmath113 ) is used . although our fits are already statistically adequate , our ple models do not fully describe an over - density of faint agn1 which is observed at high redshift . the data are not sufficiently faint to properly probe this part of the lf , but as this feature is similar to what observed in the soft x - rays ( miyaji et al . 2000 ) , we tried to obtain a even better fit of the data , by using the luminosity dependent density evolution ( ldde ) model similar to the one fitted in the soft x - rays , as described in the previous section . this model differentiates from the ple especially in the faint part of the lf , at luminosities lower than @xmath100 . in this part of the lf the density evolution decreases in proportion to the faintness of the objects . as our data just start to probe the part of the lf which is fainter than @xmath100 , we can not expect to find directly a fit to all the parameters of the ldde model to our data , and we limited our analysis to a check of the compatibility ( with limited changes ) of the ldde parameters found by miyaji et al . ( 2000 ) to our data . with this model we found the best - fit to our data ( model 3 : ldde , figure 8) by keeping fixed the parameters describing the stop in the evolution ( @xmath94=@xmath95 ) , and the dependence on luminosity of the density evolution ( @xmath44=@xmath96 ) to the value found by miyaji et al . ( 2000 ) for agn1 only , and leaving all the remaining parameters free to vary ( see section 3.2 ) . the 2dks test gives , for this model , a probability of 0.47 . if an ( @xmath12,@xmath13)=(0.3,0.7 ) cosmology is assumed , the ple models provide an ever better representation of the data in comparison with what found in the ( @xmath12,@xmath13)=(1.0,0.0 ) universe ( see figure 9 ) . the 2dks probability is 0.70 and 0.47 , with and without the introduction of the @xmath74 parameter , respectively . in this case even the simple ple model obtain a quite good fit of the data , and the introduction of the @xmath74 parameter is necessary to stop the evolution only at redshifts larger than 2.4 . however our data contain not enough agn1 at redshift larger than 2 in order to obtain an accurate measure of the @xmath74 parameter ( see figure 4 ) , therefore the errors in this parameter are quite large . also the ldde model obtain a satisfactory fit of the data ( see figure 10 ) . thanks to the identification of 61 sources of the hellas sample we have been able to double the number of hard x - ray agn1 available for statistical analysis at fluxes in the range @xmath114@xmath55 erg @xmath3 s@xmath4 . in total we can use 74 agn1 at these fluxes ( 37 from hellas ) , which combined with the local sample of grossan have allowed to show directly the shape of the lf of agn1 as function of redshift and measure its evolution . the ple models provide satisfactory fits of the data both in the ( @xmath12,@xmath13)=(1.0,0.0 ) and in the ( @xmath12,@xmath13)=(0.3,0.7 ) cosmologies . our estimate of the lf in the ( @xmath12,@xmath13)=(1.0,0.0 ) has a significantly larger normalization in comparison to the previous measure from boyle et al . ( 1998 ) . the data start to probe in the hard x - rays the faint part of the lf where the excess of density of agn1 has been observed in the soft x - rays , justifying the implementation of the ldde models . however , in both cosmologies , the statistic is not significant enough to distinguish between the ple and ldde models ( see table 6 ) . in fact , in the prediction of the differential counts of agn1 shown in figure 11 , the models differentiates at fluxes fainter than @xmath115 erg @xmath3 s@xmath4 , where the statistic is still poor . the new upcoming fainter surveys from _ chandra _ and _ xmm - newton _ will easily test which model is correct . in table 6 the percentages of the contribution to the 2 - 10 kev x - ray background are shown . the x - ray background has been computed integrating the lf up to @xmath116 for @xmath16@xmath11710@xmath118 @xmath119 . at variance with the results of 24% obtained from the evolution of the lf derived by boyle et al . ( 1998 ) , our models reproduces from @xmath135% up to @xmath160% of the xrb . we used @xmath120 erg @xmath3 s@xmath4 deg@xmath121 from chen , fabian and gendrau ( 1997 ) . the highest percentages would probably imply that part of the absorbed population necessary to reproduce the xrb is already included in the agn1 at high redshift . however , more detailed analysis of this issue are beyond the scope of this paper . the percentages would decreas if a value f @xmath122 erg @xmath3 s@xmath4 deg@xmath121 from vecchi et al . ( 1999 ) is assumed . it is interesting to notice that agn1 in the 2 - 10 kev range show an evolution up to @xmath123 which is fairly well compatible with what observed in the soft x - rays . in fact we found a good fit of the data in the ( @xmath12,@xmath13)=(1.0,0.0 ) universe if we assume exactly the same parameters found in the soft x - rays by miyaji et al . ( 2000 ) for agn1 , by only looking for a fit with the break luminosity @xmath100 ( model 7 in table 6 ) . miyaji et al . ( 2000 ) found log@xmath100=43.78 . the value @xmath100 found by our fit in the 2 - 10 kev band is log@xmath100=44.13 . this luminosity difference implies an x - ray spectrum for agn1 with slope @xmath44=0.6 , which is the same slope used in our computation of the lf . we note however that miyaji et al . ( 2000 ) used a slope @xmath44=1.0 in their computations of the soft x - ray lf . therefore , taken at a face value , the match between the soft and hard x - rays lfs implies that agn1 have a broad band concave spectrum getting steeper going toward lower energies . as already discussed in the previous section , the ( @xmath12,@xmath13)=(0.3,0.7 ) cosmology for agn1 alone has not been analyzed by miyaji et al . ( 2000 ) . the agreement of the evolution measured in the soft and hard x - rays is also shown in figure 12 , where the evolution of the density of agn1 brighter than logl@xmath93=44.8 is shown . the soft x - ray data from miyaji et al . ( 2000 ) have been over - plotted assuming a slope @xmath52 , which corresponds to a limit logl@xmath106(0.5 - 2 kev)@xmath11744.5 . the continuous line are the predictions of our ldde model for ( @xmath12,@xmath13)=(1.0,0.0 ) . based on observations collected at the european southern observatory , chile , eso n@xmath124 : 62.p-0783 , 63.o-0117(a ) , 64.o-0595(a ) , 65.o-0541(a ) . 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 . this research has been partially supported by asi contract ars-99 - 75 , murst grants cofin-98 - 02 - 32 , cofin-99 - 034 , cofin-00 - 02 - 36 and a 1999 cnaa grant . as already described , 25 sources have been identified by cross - correlation with existing catalogues . however , as the sample is not fully spectroscopic identified , the cross - correlation with the existing catalogues could alter the average characteristics ( fluxes , percentages of classes of counterparts , etc . ) of the subsample of identified sources . namely , the subsample could not be representative of the whole sample . let s explain this with an example . in an ideal sample of 50 agn1 and 50 agn2 a random identification of a subsample of 40 sources will identify about 50@xmath26 agn1 and 50@xmath26 agn2 according to sampling errors . but if a cross - correlation is first made with a catalogue of only agn1 ( let say 30 ) and later only 10 sources are randomly identified , the fraction of agn1 in the total identified subsample of 40 sources will be artificially increased ( they will be at minimum 75@xmath26 ) . our subsample is risking the same sort of bias , and we wish to quantify it . a kolmogorov - smirnov ( ks ) two - sample test gives a 9@xmath26 probability that the x - ray flux distribution of the 25 sources identified through the cross - correlation with catalogues belong to the same parent population of the total sample of 118 sources . although not significant , this low probability is due to the average slightly brighter fluxes of the cross - correlated subsample ( @xmath125 ) in comparison to the total sample ( @xmath126 ) . we have thus tried to populate the faintest bins during the observing runs at the telescope , and , indeed , the ks test gives a 66@xmath26 probability that the whole subsample of the 74 identified sources ( included the empty fields ) belong to the same parent population of the total sample of 118 sources ( see figure 1 ) . in this way , our observing runs at the telescope have recovered the possible alteration on the fraction of classes of sources which are identified . as the catalogues used for the cross correlation are mainly populated by agn1 ( we used ned ) , their fraction in our sample , as explained before , could be artificially increased . agn1 are 14 out of the 25 cross - correlated sources ( 56@xmath26 ) . the fraction of agn1 is , as expected ( but not significantly ) , lower for the sources observed at the telescope : 23 out of the 49 ( 47@xmath26 ) . the two values are not statistically distinguishable from the observed fraction 50@xmath26 of agn1 in the whole sample , which we thus consider representative of the whole hellas sample . la franca f. , fiore f. , vignali c. , comastri a. , pompilio f. , 2001 , in asp conf . 232 , the new era of wide field astronomy , ed . clowes , a.j . adamson & g.e . bromage ( san francisco : asp ) , 96 , preprint(astro - ph/00111008 ) = 0.2 cm -10.34 & 55.51 -11.85 & 55.51 -11.89 & 55.48 -11.93 & 55.39 -11.97 & 55.28 -12.00 & 55.05 -12.04 & 54.66 -12.08 & 54.04 -12.11 & 53.16 -12.15 & 51.97 -12.19 & 50.63 -12.22 & 49.04 -12.26 & 47.13 -12.30 & 44.88 -12.34 & 42.44 -12.37 & 39.93 -12.41 & 37.49 -12.45 & 34.88 -12.48 & 31.93 -12.52 & 28.83 -12.56 & 25.67 -12.59 & 22.75 -12.63 & 20.03 -12.67 & 17.53 -12.70 & 15.23 -12.74 & 13.16 -12.78 & 11.32 -12.82 & 9.73 -12.85 & 8.30 -12.89 & 6.96 -12.93 & 5.67 -12.96 & 4.56 -13.00 & 3.60 -13.04 & 2.88 -13.07 & 2.35 -13.11 & 1.96 -13.15 & 1.60 -13.19 & 1.27 -13.22 & 0.94 -13.26 & 0.68 -13.30 & 0.47 -13.33 & 0.33 -13.37 & 0.22 -13.41 & 0.14 -13.44 & 0.08 -13.48 & 0.05 -13.52 & 0.03 -13.56 & 0.01 -13.59 & 0.00 [ tab1 ] = 0.2 cm h002636@xmath35194416 & 0.238 & fwzi & ... & 62.1 & 38.1 & 62.1 & ... & ... + & & fwhm & ... & 19.7 & 15.4 & 18.5 & ... & ... + & & ew & ... & 82.4 & 14.6 & 133.1 & ... & ... + + h004546@xmath35251550 & 0.111 & fwzi & ... & ... & ... & 40.1 & 104.2 & 44.1 + & & fwhm & ... & ... & ... & 14.6 & 69.7 & 50.4 + & & ew & ... & -1.3 & -0.2 & 5.6 & 22.2 & 2.5 + + h012157@xmath35584442 & 0.118 & fwzi & ... & 44.1 & 38.1 & 64.1 & 120.2 & 48.1 + & & fwhm & ... & 19.7 & 13.1 & 16.8 & 45.0 & 80.0 + & & ew & ... & 30.4 & 2.9 & 21.4 & 53.5 & 20.8 + + h013434@xmath35295816 & 2.217 & fwzi & @xmath127452.7 & ... & ... & ... & ... & ... + & & fwhm & @xmath127123.6 & ... & ... & ... & ... & ... + & & ew & @xmath127156.5 & ... & ... & ... & ... & ... + + h013533@xmath35295202 & 1.344 & fwzi & 282.5 & ... & ... & ... & ... & ... + & & fwhm & 67.6 & ... & ... & ... & ... & ... + & & ew & 61.2 & ... & ... & ... & ... & ... + + h033408@xmath35360403 & 0.904 & fwzi & 412.6 & 40.1 & ... & ... & ... & ... + & & fwhm & 174.0 & 26.5 & ... & ... & ... & ... + & & ew & 120.5 & 6.1 & ... & ... & ... & ... + + h043712@xmath35473148 & 0.142 & fwzi & ... & 36.0 & 122.2 & 64.1 & 162.3 & 70.1 + & & fwhm & ... & 16.4 & 36.4 & 20.9 & 39.7 & 73.0 + & & ew & ... & 5.2 & 65.4 & 23.6 & 241.2 & 17.0 + + h043847@xmath35472802 & 1.453 & 372.6 & ... & ... & ... & ... & ... + & & 64.9 & ... & ... & ... & ... & ... + & & 61.4 & ... & ... & ... & ... & ... + + h064638@xmath35441534 & 0.153 & fwzi & ... & 60.1 & 128.2 & 60.1 & 122.2 & 46.1 + & & fwhm & ... & 18.3 & 47.2 & 21.3 & 36.1 & 29.0 + & & ew & ... & 7.9 & 61.0 & 78.9 & 168.4 & 10.0 + + h083737@xmath37254752 & 0.077 & fwzi & ... & 52.1 & 69.4 & 56.8 & 327.2 & 60.1 + & & fwhm & ... & 40.3 & 28.8 & 15.2 & 71.8 & 43.0 + & & ew & ... & 51.8 & 16.0 & 11.3 & 291.6 & 17.0 + + h083859@xmath37260814 & 0.048 & fwzi & ... & ... & 36.7 & 40.1 & 110.2 & 70.1 + & & fwhm & ... & ... & 11.7 & 23.2 & 42.1 & 32.2 + & & ew & ... & -8.9 & 1.9 & 4.1 & 46.0 & 6.7 + + h103216@xmath37505120 & 0.174 & fwzi & ... & 30.0 & 203.7 & 36.7 & 150.2 & 33.4 + & & fwhm & ... & 12.0 & 68.1 & 13.2 & 79.9 & 15.0 + & & ew & ... & 5.8 & 65.1 & 13.1 & 163.2 & 6.5 + + h111814@xmath37402838 & 0.387 & fwzi & 340.5 & 50.1 & ... & 40.1 & ... & ... + & & fwhm & 184.0 & 46.7 & ... & 10.3 & ... & ... + & & ew & 393.8 & 6.7 & -0.7 & 6.7 & ... & ... + + h111849@xmath37402648 & 1.129 & fwzi & @xmath127200.3 & ... & ... & ... & ... & ... + & & fwhm & @xmath12765.6 & ... & ... & ... & ... & ... + & & ew & @xmath12759.9 & ... & ... & ... & ... & ... + + h121853@xmath37295902 & 0.176 & fwzi & ... & 43.4 & 36.1 & 56.8 & 153.6 & 50.1 + & & fwhm & ... & 12.4 & 12.1 & 12.5 & 26.8 & 20.9 + & & ew & ... & 25.4 & 17.8 & 199.6 & 186.1 & 22.6 + + h124028@xmath35051402 & 0.300 & fwzi & ... & 80.1 & 166.9 & 63.4 & ... & ... + & & fwhm & ... & 25.6 & 61.3 & 23.9 & ... & ... + & & ew & ... & 36.1 & 90.8 & 67.1 & ... & ... + [ tab5 ] = 0.18 cm h124036@xmath35050752 & 0.008 & fwzi & ... & ... & 20.0 & -10.0 & 90.1 & 76.8 + & & fwhm & ... & ... & 13.9 & -7.0 & 34.0 & 26.0 + & & ew & ... & ... & 0.8 & -0.4 & 26.3 & 6.8 + + h130436@xmath35101549 & 2.386 & fwzi & @xmath36468.7 & ... & ... & ... & ... & ... + & & fwhm & @xmath127128.4 & ... & ... & ... & ... & ... + & & ew & @xmath127537.0 & ... & ... & ... & ... & ... + + h134820@xmath35301156 & 0.128 & fwzi & ... & 52.1 & ... & 56.1 & 112.2 & ... + & & fwhm & ... & 21.5 & ... & 22.0 & 53.9 & ... + & & ew & ... & 19.0 & -0.4 & 12.7 & 63.0 & ... + + h134845@xmath35302946 & 0.330 & fwzi & ... & 36.0 & 220.4 & 72.1 & ... & ... + & & fwhm & ... & 15.8 & 83.0 & 20.5 & ... & ... + & & ew & ... & 2.3 & 84.2 & 55.8 & ... & ... + + h135015@xmath35302010 & 0.074 & fwzi & ... & 60.1 & ... & 52.1 & 168.3 & 56.1 + & & fwhm & ... & 23.1 & ... & 24.5 & 64.4 & 31.7 + & & ew & ... & 100.5 & -0.6 & 16.0 & 48.8 & 8.0 + + h135354@xmath37182016 & 0.217 & fwzi & ... & 52.1 & 24.0 & 64.1 & 252.4 & 40.1 + & & fwhm & ... & 13.7 & 9.3 & 19.7 & 95.0 & 47.9 + & & ew & ... & 47.3 & 2.1 & 22.8 & 154.0 & 12.4 + + h151934@xmath37653558 & 0.044 & fwzi & ... & 36.0 & ... & 44.1 & 92.2 & 44.1 + & & fwhm & ... & 13.6 & ... & 9.7 & 9.0 & 14.5 + & & ew & ... & 43.6 & -1.3 & 22.6 & 26.4 & 6.1 + + h163419@xmath37594504 & 0.341 & fwzi & 144.2 & 48.1 & 72.1 & 56.1 & ... & ... + & & fwhm & 19.9 & 12.9 & 12.9 & 13.7 & ... & ... + & & ew & 33.1 & 19.7 & 30.8 & 267.2 & ... & ... + + h165043@xmath37043618 & 0.031 & fwzi & ... & 48.1 & 36.1 & 52.1 & 124.2 & 56.1 + & & fwhm & ... & 40.1 & 27.2 & 21.0 & 49.1 & 32.6 + & & ew & ... & 45.9 & 2.8 & 26.6 & 43.0 & 12.3 + + h165238@xmath37022206 & 0.395 & fwzi & 184.3 & 56.1 & 164.3 & 64.1 & ... & ... + & & fwhm & 70.0 & 24.1 & 129.5 & 21.5 & ... & ... + & & ew & 66.6 & 22.5 & 37.5 & 68.5 & ... & ... + + h204253@xmath35103826 & 0.363 & fwzi & ... & 72.1 & 184.3 & 76.1 & ... & ... + & & fwhm & ... & 22.5 & 48.5 & 22.3 & ... & ... + & & ew & ... & 15.4 & 73.0 & 201.8 & ... & ... + + h204435@xmath35102808 & 2.755 & fwzi & @xmath127248.4 & ... & ... & ... & ... & ... + & & fwhm & @xmath127101.9 & ... & ... & ... & ... & ... + & & ew & @xmath12756.6 & ... & ... & ... & ... & ... + + h222632@xmath37211138 & 0.260 & fwzi & ... & ... & 125.5 & 26.7 & ... & ... + & & fwhm & ... & ... & 41.6 & 7.9 & ... & ... + & & ew & ... & ... & 36.4 & 9.3 & ... & ... + + h231932@xmath35424228 & 0.101 & fwzi & ... & 68.1 & 40.1 & 56.1 & 172.3 & 64.1 + & & fwhm & ... & 23.1 & 26.3 & 21.8 & 46.9 & 38.4 + & & ew & ... & 20.4 & 5.0 & 24.2 & 77.6 & 7.5 + + h232729@xmath37084926 & 0.154 & fwzi & ... & 28.0 & 92.2 & 60.1 & 116.2 & 56.1 + & & fwhm & ... & 15.9 & 80.3 & 18.7 & 34.2 & 37.5 + & & ew & ... & 13.0 & 16.8 & 19.1 & 60.9 & 16.2 + + h232906@xmath37083416 & 0.953 & fwzi & 368.6 & ... & ... & ... & ... & ... + & & fwhm & 76.1 & ... & ... & ... & ... & ... + & & ew & 133.6 & ... & ... & ... & ... & ... + + h233154@xmath37193836 & 0.475 & fwzi & 344.5 & ... & ... & ... & ... & ... + & & fwhm & 75.0 & ... & ... & ... & ... & ... + & & ew & 93.5 & ... & ... & ... & ... & ... + [ tab5 ]

we present optical spectroscopic identifications of hard x - ray ( 5 - 10 kev ) selected sources belonging to the hellas sample obtained with _ bepposax _ down to a 5 - 10 kev flux limit of @xmath0@xmath1@xmath2 erg @xmath3 s@xmath4 . the sample consists of 118 sources . 25 sources have been identified trough correlations with catalogues of known sources . 49 have been searched for spectroscopic identification at the telescope . 13 fields resulted empty down to r=21 . 37 sources have been identified with type 1 agn and 9 with type 2 agn . the remaining are : 5 narrow emission line galaxies , 6 clusters , 2 bl lac , 1 radio galaxy and 1 star . combining these objects with other hard x - ray selected agns from _ asca _ and _ heao1 _ , we find that the local luminosity function of type 1 agn in the 2 - 10 kev band is fairly well represented by a double - power - law - function . there is evidence for significant cosmological evolution according to a pure luminosity evolution ( ple ) model @xmath5@xmath6@xmath7 , with @xmath8=2.12 and @xmath8=2.22 ( @xmath9@xmath10@xmath11 ) in a ( @xmath12,@xmath13)=(1.0,0.0 ) and in a ( @xmath12,@xmath13)=(0.3,0.7 ) cosmology , respectively . the data show an excess of faint high redshift type 1 agn which is well modeled by a luminosity dependent density evolution ( ldde ) , similarly to what observed in the soft x - rays . however , in both cosmologies , the statistic is not significant enough to distinguish between the ple and ldde models . the fitted models imply a contribution of agn1 to the 2 - 10 kev x - ray background from 35% up to 60% .

astro-ph0112455

the theoretical description of hadron production at large transverse momentum ( @xmath2 ) in either hadronic or nuclear collisions at high energies is traditionally framed in a two - step process that involves first a hard scattering of partons , followed by the fragmentation of the scattered parton to the detected hadron @xcite . the first part is calculable in perturbation qcd , while the second part makes use of fragmentation functions that are determined phenomenologically . such a production mechanism has recently been found to be inadequate for the production of particles at intermediate @xmath2 in heavy - ion collisions @xcite . instead of fragmentation it is the recombination of partons that is shown to be the more appropriate hadronization process , especially when the soft partons are involved . although at extremely high @xmath2 fragmentation is still dominant , it is desirable to have a universal description that can be applied to any @xmath2 , based on the same hadronization scheme . to achieve that goal it is necessary that the fragmentation process can be treated as the result of recombination of shower partons in a jet . the purpose of this paper is to take that first step , namely : to introduce the notion of shower partons and to determine their distributions in order to represent the phenomenological fragmentation functions in terms of recombination . the subject matter of this work is primarily of interest only to high - energy nuclear collisions because hadronization in such processes is always in the environment of soft partons . semi - hard shower partons initiated by a hard parton can either recombine among themselves or recombine with soft partons in the environment . in the former case the fragmentation function is reproduced , and nothing new is achieved . it is in the latter case that a very new component emerges in heavy - ion collisions , one that has escaped theoretical attention thus far . it should be an important hadronization process in the intermediate @xmath2 region . our main objective here is to quantify the properties of shower partons and to illustrate the importance of their recombination with thermal partons . the actual application of the shower parton distributions ( spd ) developed here to heavy - ion collisions will be considered elsewhere @xcite . the concept of shower partrons is not new , since attempts have been made to generate such partons in pqcd processes as far as is permitted by the validity of the procedure . two notable examples of such attempts are the work of marchesini and webber @xcite and geiger @xcite . however , since pqcd can not be used down to the hadronization scale , the branching or cascading processes terminate at the formation of color - singlet pre - hadronic clusters , which can not easily be related to our shower partons and their hadronization . we shall discuss in more detail at the end of secs . iii and iv the similarities and differences in the various approaches . the fragmentation of a parton to a hadron is not a process that can be calculated in pqcd , although the @xmath1 evolution of the fragmentation function ( ff ) is calculable . the ff s are usually parameterized by fitting the data from @xmath3 annihilations @xcite as well as from @xmath4 and @xmath5 collisions @xcite . although the qcd processes of generating a parton shower by gluon radiation and pair creation can not be tracked by perturbative methods down to low virtuality , we can determine the spd s phenomenologically in much the same way that the ff s themselves are , except that we fit the ff s , whereas the ff s are determined by fitting the data . an important difference is that both the shower partons and their distributions are defined in the context of the recombination model , which is the key link between the shower partons ( inside the black box called ff ) and the observed hadron ( outside the black box ) . in the recombination model the generic formula for a hadronization process is @xcite @xmath6 where @xmath7 is the joint distribution of a quark @xmath8 at momentum fraction @xmath9 and an antiquark @xmath10 at @xmath11 , and @xmath12 is the recombination function ( rf ) for the formation of a meson at @xmath13 . we have written the lhs of eq . ( [ 1 ] ) as @xmath14 , the invariant ff , but the rhs would have the same form if the equation were written for the inclusive distribution , @xmath15 , of a meson produced in a collisional process . in the former case of fragmentation , @xmath16 refers to the shower partons initiated by a hard parton . in the latter case of inclusive production , @xmath16 refers to the @xmath8 and @xmath10 that are produced by the collision and are to recombine in forming the detected meson . the equations for the two cases are similar because the physics of recombination is the same . in either case the major task is in the determination of the distribution @xmath16 . we now focus on the fragmentation problem and regard eq . ( [ 1 ] ) as the basis of the recombination model for fragmentation . the lhs is the ff , known from the parameterization that fits the data . the rhs has the rf that is known from previous studies of the recombination model @xcite and will be specified in the next section . thus it is possible to determine the properties of @xmath16 from eq.([1 ] ) . to facilitate that determination we shall assume that @xmath16 is factorizable except for kinematic constraints , i.e. , in schematic form we write it as @xmath17 where @xmath18 denotes the distribution of shower parton @xmath8 with momentum fraction @xmath9 in a shower initiated by a hard parton @xmath19 . the exact form with proper kinematic constraints will be described in detail in the next section . here we remark on the general implications of eqs . ( [ 1 ] ) and ( [ 2 ] ) . the important point to emphasize is that we are introducing the notion of shower partons and their momentum distributions @xmath20 . the significance of the spd is not to be found in problems that involve only the collisions of leptons and hadrons , for which the fragmentation of partons is known to be an adequate approach , and the recombination of shower partons merely reproduces what is already known . the knowledge about the spd becomes crucial when the shower partons recombine with other partons that are not in the jet but are in the ambient environment . we shall illustrate this important point later . it should be recognized that the spd that we shall determine through the use of eqs . ( [ 1 ] ) and ( [ 2 ] ) depends on the specific form of @xmath12 , which in turn depends on the wave function of the meson produced . it would be inconsistent to use our @xmath21 given below in conjunction with some approximation of the rf that differs significantly from our @xmath22 . the recombination of two shower partons must recover the ff from which the spd s are obtained . finally , we remark that @xmath21 should in principle depend on @xmath1 at which the @xmath23 is used for its determination , since @xmath1 evolution affects both . it is outside the scope of this paper to treat the @xmath1 dependence of @xmath21 . our aim here is to show how @xmath21 can be determined phenomenologically , and how it can be applied , when @xmath1 is fixed . the same method can be used to determine @xmath21 at other values of @xmath1 . in practice , the @xmath1 dependence of @xmath21 is not as important as the inclusion of the role of the shower partons in the first place at any reasonably approximate @xmath1 in heavy - ion collisions where hard partons are produced in a range of transverse momentum . in order to solve eqs . ( [ 1 ] ) and ( [ 2 ] ) for @xmath21 , we first point out that there are various @xmath24 functions corresponding to various fragmentation processes . we shall select five of them , from which we can determine five spd s . three of them form a closed set that involves no strange quarks or mesons . let us start with those three . consider the light quarks @xmath25 , @xmath26 , @xmath27 , @xmath28 , and gluon @xmath29 . they can all fragment into pions . to reduce them to three essential ff s , we consider the three basic types @xmath30 , @xmath31 and @xmath32 , that correspond to valence , sea and gluon fragmentation , respectively . if the fragmenting quark has the same flavor as that of a valence quark in @xmath33 , then the valence part of the fragmentation is described by @xmath30 , e.g. , @xmath34 , @xmath35 , @xmath36 , the subscript @xmath37 referring to the valence component . all other cases of quark fragmentation are described by @xmath31 , e.g. , @xmath38 , @xmath39 , @xmath40 . if the initiating parton is a gluon , then we have @xmath32 for any state of @xmath33 . those ff s are given by ref.@xcite in parametric form . we shall use them even though they are older than the more recent ones @xcite , which do not give the @xmath30 and @xmath31 explicitly . our emphasis here is not on accuracy , but on the feasibility of extracting the spd s from the ff s of the type discussed above . we shall determine @xmath21 from the bkk parameterization @xcite with @xmath1 fixed at 100 gev@xmath41 and demonstrate that the use of shower partons is important in heavy - ion collisions . for the spd s we use the notation @xmath42 and @xmath43 for valence and sea - quark distributions , respectively , in a shower initiated by a quark or antiquark , and @xmath44 for any light quark distribution in a gluon - initiated shower . that is , for example , @xmath42 @xmath45 , @xmath46 , @xmath47 . it should be recognized that @xmath43 also describes the sea quarks of the same flavor , such as @xmath48 , so that the overall distribution of shower quark that has the same flavor as the initiating quark ( e.g. @xmath49 ) is given by @xmath50 it is evident from the above discussion that there is a closed relationship that is independent of other unknowns . it follows from eq . ( [ 1 ] ) when restricted to sea - quark fragmentation : @xmath51 the sea - spd @xmath52 can be determined from this equation alone . in eq.([4 ] ) we have exhibited the argument of the second @xmath43 function that reflects the momentum constraint , i.e. , if one shower parton has momentum fraction @xmath9 , then the momentum fraction of the other recombining shower parton can not exceed @xmath53 , and can only be a fraction of the balance @xmath54 . symmetrization of @xmath9 and @xmath11 is automatic by virtue of the invariance of @xmath55 under the exchange of @xmath9 and @xmath11 . after @xmath52 is determined from eq . ( [ 4 ] ) , we next can obtain @xmath42 from @xmath56 where the curly brackets define the symmetrization of the leading parton momentum @xmath57 \quad . \label{6}\end{aligned}\ ] ] finally , we have the closed equation for the gluon - initiated shower @xmath58 in this non - strange sector we have 3 spd s ( @xmath43 , @xmath42 and @xmath44 ) to be determined from the 3 phenomenological ff s ( @xmath31 , @xmath30 and @xmath59 ) . in extending the consideration to the strange sector , we must make use of @xmath43 and @xmath44 determined in the above set and two new ff s @xmath60 and @xmath61 that describe the fragmentation of a non - strange quark and gluon , respectively , to a kaon . that is , we have @xmath62 @xmath63 where @xmath64 and @xmath65 are two additional spd s specifying the strange quark distributions in showers initiated by non - strange and gluon partons , respectively . @xmath66 is the rf for kaon . to complete the description of the integral equations , we now specify the rf s . they depend on the square of the wave functions of the mesons , @xmath33 and @xmath67 , whose structures in momentum space have been quantified in the valon model @xcite . unlike the case of the proton , whose structure is well studied by deep inelastic scattering so that the valon distribution can be obtained from the parton distribution functions @xcite , the rf for the pion relies on the parton distribution of the pion probed by drell - yan process @xcite . the derivation of the rf s for both @xmath33 and @xmath67 is given in @xcite ; they are @xmath68 @xmath69 the @xmath70 functions guarantee the momentum conservation of the recombining quarks and antiquarks , which are dressed and become the valons of the produced hadrons . since the recombination process involves the quarks and antiquarks , one may question the fate of the gluons . this problem has been treated in the formulation of the recombination model @xcite , where gluons are converted to quark - antiquark pairs in the sea before hadronization . that is , the sea is saturated by the conversion to carry all the momentum , save the valence parton . such a procedure has been shown to give the correct normalization of the inclusive cross section of hadronic collisions @xcite . in the present problem of parton fragmentation we implement the recombination process in the same framework , although gluon conversion is done only implicitly . what is explicit is that the gluon degree of freedom is not included in the list of shower partons . it means that in the equations for @xmath30 , @xmath31 , and @xmath32 ( and likewise in the strange sector ) only @xmath42 , @xmath43 and @xmath44 appear ; they are the spd s of quarks and antiquarks that are to recombine . those quarks and antiquarks must include the converted sea , since they are responsible for reproducing the ff s through eqs . ( [ 4 ] ) , ( [ 5 ] ) and ( [ 7 ] ) without gluons . thus the shower partons whose momentum distributions we calculate are defined by those equations that have no gluon component for recombination , and would not be the same as what one would conceptually get ( if it were possible ) in a pqcd calculation that inevitably has both quarks and gluons . it should be noted that our procedure of converting gluons to @xmath71 pairs is essentially the same as what is done in @xcite , whose branching processes terminate at the threshold of the non - perturbative regime . in that approach nearby quarks and antiquarks that are the products of the conversion from different gluons form color - singlet clusters of various invariant masses that subsequently decay ( or fragment as in strings ) sequentially through resonances to the lowest lying hadron states @xcite . similar but not identical approach is taken in @xcite , where gluons are not directly converted to @xmath71 pairs , but are either absorbed or annihilated by @xmath72 born - diagram processes . we now proceed to solve the integral equations for the five ff s , which are known from ref . those equations relate them to the five unknown spd s : @xmath42 , @xmath43 , @xmath44 , @xmath64 and @xmath65 . if those equations were algebraic , we obviously could solve them for the unknowns . being integral equations , they can nevertheless be `` solved '' by a fitting procedure that should not be regarded as being unsatisfactory for lack of mathematical rigor , since the ff s themselves are obtained by fitting the experimental data in some similar manner . indeed , the ff s in the next - to - leading order are given in parameterized forms @xcite @xmath73 where the parameters for @xmath74 gev@xmath41 are given in table i for @xmath75 and @xmath76 . .parameters in eq . ( [ 12 ] ) for @xmath77gev@xmath41 . [ cols="<,^,^,^,^",options="header " , ] it is evident from fig . 1 that all the fits are very good , except in the low @xmath13 region of @xmath78 . in the latter case we are constrained by the condition @xmath79 that is imposed by the requirement that there can be only one valence quark in the shower partons . however , the fit for @xmath80 is excellent , and that is the important region for the determination of @xmath81 . in application to @xmath82 , say , the @xmath25 quark in the shower must have both valence and sea quarks so the shower distribution for the @xmath25 quark is always the sum : @xmath83 . since @xmath52 is large at small @xmath84 , and is accurately determined , the net result for @xmath85 should be quite satisfactory . it is remarkable how well the ff s in fig . 1 are reproduced in the recombination model . the corresponding spd s that make possible the good fit are shown in fig . they have very reasonable properties , namely : ( a ) valence quark is harder ( b ) sea quarks are softer , ( c ) gluon jet has higher density of shower partons , and ( d ) the density of produced @xmath86 quarks is lower than that of the light quarks . for valence+sea quark ( solid line ) , sea quark ( dash - dot line ) and thermal partons ( dashed line).,scaledwidth=45.0% ] it is appropriate at this point to relate our approach to those of marchesini - webber @xcite and geiger @xcite , which are serious attempts to incorporate the qcd dynamics in their description of the branching and collision processes . the former is done in the momentum space only , whereas the latter is formulated in space - time as well as in momentum space . the parton cascade model of geiger is a very ambitious program that treats a large variety of processes ranging from @xmath3 annihilation @xcite to deep inelastic scattering @xcite to hadronic and nuclear collisions @xcite . the evolution of partons is tracked by use of relativistic transport equations with gain and loss terms . cluster formation takes into account the invariant distance between near - neighbor partons . cluster decay makes use of the hagedorn spectrum and the particle data table . because of the complexity of the problems both qcd models are implemented by monte carlo codes the predictive power of the models is exhibited as numerical outputs that can not easily be adapted for comparison with our results on the spd s . our approach makes no attempt to treat the qcd dynamics ; however , the spd s obtained are guaranteed to reproduce the ff s on the one hand , and are conveniently parameterized for use in other context that goes beyond fragmentation , as we shall show in the next section . from the way the color - singlet clusters are treated in the qcd models , it is clear that our shower partons do not correspond to the partons of those models at the end of their evolution processes , except in the special case when the cluster consists of only one particle . in our approach the non - perturbative part of how the shower partons dress themselves and recombine to form hadrons with the proper momentum - fraction distributions is contained in the rf s . such shower partons that are ready to hadronize are sufficiently far from other shower partons as to be independent from them . in general , they can not be identified with the @xmath8 and @xmath87 that form the color - singlet clusters in the qcd models , but are more closely related to the constituents of the final hadrons , as in the case of quarkonium formation @xcite . the distribution of those constituents in a hard - parton shower can not be displayed in the qcd models , but are determined by us by solving eqs . ( [ 4 ] ) and ( [ 9 ] ) as we have stated in the introduction , the purpose of determining the spd s is for their application to problems where the ff s are insufficient to describe the physics involved . we consider in this section two such problems as illustrations of the usefulness of the spd s . the first is when a hard parton is produced in the environment of thermal partons , as in heavy - ion collisions . the second is the determination of two - pion distribution in a jet . let us suppose that a @xmath25 quark is produced at @xmath88 gev / c in a background of thermal partons whose invariant @xmath89 distribution is @xmath90 let the parameters @xmath91 and @xmath92 be chosen to correspond to a typical situation in au+au collisions at @xmath93 gev @xcite @xmath94 the high-@xmath89 @xmath25 quark generates a shower of partons with various flavors . consider specifically @xmath25 and @xmath28 in that shower . the valence quark distribution is given by @xmath95 , while the @xmath28 sea - quark distribution ( including the ones converted from the gluons ) is given by @xmath96 . in fig . 3 we plot @xmath97 for ( a ) @xmath25 quark ( valence and sea ) in solid line , ( b ) @xmath28 sea antiquark in dash - dot line , and ( c ) @xmath28 thermal antiquark in dashed line . they correspond to @xmath98 ( invariant distributions @xmath99 , @xmath43 , and @xmath100 , respectively ) , in which @xmath101 and @xmath96 are evaluated at @xmath102 , with @xmath103 gev / c . note that the thermal distribution is higher than the shower parton distributions for @xmath104 gev / c . that makes a crucial difference in the recombination of those partons . such a thermal distribution is absent in @xmath105 collisions , whose soft partons are at least two orders of magnitude lower . in @xmath3 annihilation there are , of course , no soft partons at all . in @xmath2 arising from thermal - shower recombination ( solid line ) and shower - shower recombination , i.e. fragmentation ( dash - dot line).,scaledwidth=45.0% ] we now calculate the production of @xmath106 from the assemblage of @xmath25 and @xmath28 partons . the thermal - shower ( @xmath107 ) recombination gives rise to @xmath108 where eq . ( [ 10 ] ) has been used in an equation such as eq.([1 ] ) for @xmath109 , but expressed for @xmath110 . using eqs . ( [ 3 ] ) , ( [ 15 ] ) and the parametrizations given in table ii , the integral in eq . ( [ 17 ] ) can readily be evaluated . the result is shown by the solid line in fig . 4 . it is to be compared with the @xmath2 distribution from the fragmentation of the @xmath25 quark to @xmath106 , which is @xmath111 since this is just retracing the path in which we obtained @xmath67 and @xmath43 from the @xmath112 function in the first place , eq . ( [ 18 ] ) can more directly be identified with @xmath113 . \label{19}\end{aligned}\ ] ] the result is shown by the dash - dot line in fig . evidently , the contribution from the thermal - shower recombination is much more important than that from fragmentation in the range of @xmath2 shown . despite the fact that @xmath114 is lower than @xmath115 for @xmath116 gev / c , its dominance at @xmath117 gev / c is enough to result in the @xmath118 recombination to dominate over the @xmath119 recombination for all @xmath120 gev / c . this example demonstrates the necessity of knowing the spd s in a jet , since @xmath101 is used in eq . ( [ 17 ] ) . if @xmath121 recombination is the only important contribution as in @xmath105 collisions , then fragmentation as in eq . ( [ 19 ] ) is all that is needed , and the search for spd s plays no crucial role . in realistic problems the hard - parton momentum @xmath122 has to be integrated over the weight of the jet cross section . however , for our illustrative purpose here , that is beside the point . our next example is the study of the dihadron distribution in a jet . we need only carry out the investigation here for a jet in vacuum , since the replacement of a shower parton by a thermal parton for a jet in a medium is trivial , having seen how that is done in the replacement of eq . ( [ 18 ] ) by ( [ 17 ] ) in the case of the single - particle distribution . consider the joint distribution of two @xmath106 in a jet initiated by a hard @xmath25 quark . as we shall work in the momentum fraction variables , the value of the momentum of the initiating @xmath25 quark is irrelevant , except that it should be high . let @xmath123 and @xmath124 denote the momentum fractions of the two @xmath106 , and @xmath125 denotes that of the @xmath19th parton , @xmath126 . then , since only one @xmath25 quark can be valence , the other three quarks being in the sea , we have one @xmath67 , three @xmath43 , and two @xmath22 functions . combinatorial complications arise when we impose the condition that @xmath127 for @xmath128 . there are two methods to keep the accounting of the different orderings of the four @xmath125 . _ method 1 . _ let one ordering be @xmath129 there are 4 ! ways to rearrange the four @xmath125 in all orders . however , they are to be convoluted with @xmath130 , which is symmetric in @xmath131 , and similarly with @xmath132 . thus there are @xmath133 independent terms . since @xmath67 can appear at any one of the four positions in eq . ( [ 20 ] ) , we have altogether 24 terms . thus we have @xmath134 r_{\pi}(x_1,x_2,x_1 ) r_{\pi}(x_3,x_4,x_2 ) , \label{21}\end{aligned}\ ] ] where @xmath135 symbolizes the permutation of all @xmath125 and summing over all four positions of @xmath67 , but eliminating redundant terms that are symmetric under the interchanges of @xmath131 and @xmath136 . _ method 2 . _ let us fix the ordering in eq . ( [ 20 ] ) but permute the contributing @xmath125 to @xmath123 and @xmath124 . there are six arrangements of @xmath125 and @xmath137 in @xmath138 , while counting in @xmath139 and @xmath140 is unnecessary . let us denote the summation over them by @xmath141 . thus we have @xmath142 \left [ { 1 \over 6}\sum_q r_{\pi}(x_i , x_j , x_1 ) r_{\pi}(x_{i'},x_{j'},x_2)\right ] \label{22}\end{aligned}\ ] ] where @xmath143 denotes summing over the four positions of @xmath67 . equation ( [ 22 ] ) is equivalent to ( [ 21 ] ) . it should be noted that not all terms in these equations can be expressed in the form factorizable ff s . one example that can is @xmath144 r_{\pi}(x_1,x_2,x_1)\nonumber\\ & & \times l\left ( { x_3 \over 1-x_1-x_2 } \right)l\left ( { x_4 \over 1-x_1-x_2-x_3 } \right)r_{\pi}(x_3,x_4,x_2)\nonumber\\ & & = d^{\pi^+}_u(x_1 ) d^{\pi^+}_s\left(x_2/(1-x_1)\right ) .\label{23}\end{aligned}\ ] ] because of the presence of terms that can not be written in factorizable form , the two - particle distribution can not be adequately represented by the ff s only . correlated distribution in a @xmath25-quark initiated jet.,scaledwidth=45.0% ] using the spd s obtained in the previous section , we get the results shown in fig . 5 , which exhibits the @xmath124 distribution for four fixed values of @xmath123 . this type of correlation in parton fragmentation has never been calculated before . although the shapes of the @xmath124 distributions look similar in the log scale in fig . 5 , there is significant attenuation as @xmath145 for each value of @xmath123 . thus the effective slope becomes steeper for larger @xmath123 . recent experiments at rhic have begun to measure the distribution of particles associated with triggers restricted to a small interval . the extension of our calculation here to such problems will need the input of jet cross sections for all hard partons in heavy - ion collisions and the participation of thermal partons in the recombination . here we only demonstrate the utility of the spd s in the study of dihadron correlation . we have described the fragmentation process in the framework of recombination . the shower parton distributions obtained are shown to be useful in problems where the knowledge of the fragmentation functions alone is not sufficient to provide answers to questions concerning the interaction between a jet and its surrounding medium or between particles within a jet . such questions arise mainly in nuclear collisions at high energies . in our view the basic hadronization process is recombination , even for fragmentation in vacuum . since the recombination process can only be formulated in the framework of a model , the shower parton distributions obtained are indeed model dependent . that is a price that must be paid for the study of hadrons produced at intermediate @xmath2 where the interaction between soft and semi - hard partons can not be ignored , and where perturbative qcd is not reliable . once recombination is adopted for treating hadronization in that @xmath2 range , the extension to higher @xmath2 can remain in the recombination framework , since the fragmentation process is recovered by the recombination of two shower partons . for hadron production in heavy - ion collisions at super high energies , such as at lhc , then the high density of hard partons produced will require the consideration of recombination of hard partons from overlapping jets . thus it is sensible to remain in the recombination mode for all @xmath2 . we have shown in this paper how the spd s can be determined from the ff s . although we have determined the spd s at only one value of @xmath1 for the ff s , it is clear that the same procedure can be followed for other value of @xmath1 . the formal description of how the @xmath1 dependences of the ff s can be transferred to the @xmath1 dependences of the spd s is a problem that is worth dedicated attention . while the numerical accuracy of the spd s obtained here can still be improved , especially at lower @xmath1 , for the purpose of phenomenological applications the availability of the parametrizations given in table ii is far more important than not taking into account at all the shower partons and their interactions with the medium in the environment . the @xmath1 evolution of the spd s may have to undergo a long process of investigatory evolution of its own just as what has happened to the ff s . that can proceed in parallel to the rich phenomenology that can now be pursued in the application of the role of shower partons to heavy - ion collisions . we are grateful to s. kretzer for a helpful communication . this work was supported , in part , by the u. s. department of energy under grant no . de - fg03 - 96er40972 and by the ministry of education of china under grant no .

we develop the notion of shower partons and determine their distributions in jets in the framework of the recombination model . the shower parton distributions obtained render a good fit of the fragmentation functions . we then illustrate the usefulness of the distributions in a problem where a jet is produced in the environment of thermal partons as in heavy - ion collisions . the recombination of shower and thermal partons is shown to be more important than fragmentation . application to the study of two - particle correlation in a jet is also carried out . @xmath0 2@xmath1

hep-ph0312271

a qn is the explosive transition of a massive neutron star ( ns ) to a quark star ( qs ; the compact remnant ) . it ejects the outermost layers of the ns as the relativistic qn ejecta with kinetic energy exceeding excess @xmath2 erg . the interaction of this ejecta with its surroundings leads to unique phenomena and has important implications to astrophysics . when occurring in binaries , quark - novae ( qne ) have the potential to transform our view of binary evolution and has serious implications to both high - energy astrophysics and cosmology . after a description of the qn and its energetics in section 2 , we briefly review two cases of qne in binaries . the first case is a qn - ia ( section 3 ) which is a qn going off in a short period binary consisting of ( the exploding ) ns and a white dwarf ( wd ) which is the mass reservoir . the extremely dense relativistic qn ejecta impacts ( shocks , compresses and heats ) the wd and triggers the thermonuclear run - away of a traditional type ia . along side the type ia , the spinning - down qs provides an additional power source which tampers " with the energy budget . in the second case , we show that a qn occurring in a massive binary can account for the exotic " light - cuves of double - humped hydrogen poor slsne ( section 4 ) . we summarize in section 5 . we define @xmath3 as the critical mass for a non - rotating ns to undergo quark deconfinement in its core . the presence of enough strange quarks in the deconfined core of the ns then triggers the conversion of hadrons ( i.e. matter made of _ up _ and _ down _ quarks ) to the conjectured more stable _ ( uds ) _ matter ( i.e. matter made of free _ up _ , _ down _ and _ strange _ quarks)@xcite . in a qn@xcite , the _ ( ud)-to-(uds ) _ conversion front propagates toward the surface of the ns while harnessing neutrino@xcite , photon@xcite and gravitational energy@xcite possibly yielding a detonative regime . micro - physics driven hydrodynamical simulations of this conversion process seem to indicate that a detonation may indeed occur@xcite and when coupled with gravitational collapse may lead to a universal mechanism for the ejection of the ns outermost layers ( @xmath4 of qn ejecta ) with a universal kinetic energy , @xmath5 , of a few times @xmath2 erg ( i.e. with an associated lorentz factor exceeding @xmath6)@xcite . thus the kinetic energy released in a qn exceeds that of a supernova by at least an order of magnitude . the neutron - rich qn ejecta provides a favorable site for nucleosynthesis@xcite . when this ejecta ( expanding radially outward from the parent ns ) collides with the preceding sn ejecta , it re - energizes and re - brightens the sn yielding a superluminous sn@xcite . this double - detonation generates a superluminous double - peaked light - curve if the time - delay between the sn and the qn exceeds a few days . we account for the luminosity@xcite , the photometric / spectroscopic signatures@xcite as well as introduce nuclear / spallation signatures resulting from the interaction of the ultra - relativistic qn ejecta with the sn shell and circumstellar material@xcite . for shorter time - delays of less than a day , the qn kinetic energy is lost to pdv work but the collision between the qn ejecta with the sn ejecta yields unique nuclear signatures which may explain existing observations@xcite . the qs shows features reminiscent of soft gamma repeaters @xcite while the explosion energetics and variability are reminiscent of gamma - ray bursts @xcite . when occurring in binaries , the more complex interactions with the companion result in even more interesting features . we review the key signatures and main implications to astrophysics in this paper . we first discuss what happens when a ns in a close binary with a wd companion explodes as a qn . in this scenario , roche - lobe overflow disrupts the wd which produces a carbon - oxygen ( co ) torus surrounding the ns @xcite . alternatively , the ns may fully merge with the wd so that the ns now is in the core of the wd when the qn occurs . the qn will be triggered following sufficient mass accretion . some of the relativistic qn ejecta will impact ( shock , heat and compress ) the disrupted wd inducing a runaway nuclear burning of the co in an event we termed a qn - ia since it is `` type ia''-like explosion . a crucial difference here however is the qs which provides extra power through magnetic braking spin - down and consequently a qn - ia ( which spectrum resembles a type - ia sn ) is powered by a combination of @xmath1ni decay and the spin - down luminosity of the qs . this has drastic consequences for cosmological models if qne - ia contaminate the sample of type ia sne used as distance indicators in cosmology as discussed below . the spin - down contribution yields a _ red - shift - dependent phillips - like relation _ ( ( * ? ? ? * figure 1 ) shows the correlation between peak absolute magnitude and light - curve shape ) which means that they can confuse ( i.e. are _ not _ rejected by the ) light - curve fitters used for cosmology ( ( * ? ? ? * figure 4 ) ) . the rate of qne - ia may be a significant fraction of the observed type ia sne and may be dominant at higher redshift@xcite . this is especially egregious given that the qn - ia light - curve varies with redshift . to estimate the effect of contamination , we analyzed hundreds of synthetic qne - ia light - curves using the salt2 light - curve fitting software@xcite to find the difference ( @xmath7 ) between the actual distance modulus and the fitted distance modulus as a function of redshift , @xmath8 . most of the simulated qne - ia were best fitted@xcite with : @xmath9 for @xmath10 there is a strong correlation since @xmath11 , but at @xmath12 the correlation is much weaker @xmath13 . we conclude that if qne - ia represent an important fraction of the sne used in the work which estimates the accelerating expansion of the universe@xcite , this may have drastically altered the statistics and conclusions of those studies @xcite . it is thus vital to differentiate between qne - ia and standard sne - ia . applying our correction above to the union2.1 data@xcite we obtain the true distance moduli of the observed sne - ia ( if they are indeed qne - ia ) as shown by red crosses in ( * ? ? * figure 6 ) . this demonstrates how the sne - ia distance moduli , when corrected , lay very close to the @xmath14 , @xmath15 curve . a qn - ia may have already been observed in sn 2014j @xcite . sn 2014j s @xmath16 mass was estimated to be @xmath17 @xcite , based on @xmath18 decay lines . however , based on the peak luminosity , @xmath19 is expected . in the qn - ia model , the discrepancy can be accounted for by the qs spin - down power . perhaps the best prospect for observationally distinguishing a qn - ia from a sn - ia is the detection of the gravitational wave signal produced during the explosive transition of the ns to a qs . if the qn is asymmetric , it should emit a gravitational wave signal that could be observable by advanced ligo@xcite . this would be followed by another signal from the exploding wd ; here the time delay between the qn and the exploding wd is the time it takes the qn ejecta to reach the disrupted wd plus the burning time of the wd . other types of type ia sne ( i.e. single or double degenerate scenarios ) would lack such a dual signal . another strong observational signature of a qn - ia is high - energy emission ( specifically x - ray signatures@xcite ) from the qs which would be impossible in standard sne - ia channels . the qs , being an aligned rotator , would be radio quiet @xcite . these signature may be observed in the near future in sn 2014j . ) by the qn . the second hump is from the bh - accretion phase after the qs merges with the co core . the circles show u - band observations of the slsn asassn-15lh@xcite.,width=288 ] qne are likely to occur in binaries experiencing ce phases where the ns can accrete enough mass to reach @xmath3 . in particular , in massive binaries experiencing two ce phases , the ns would have access to two mass reservoirs@xcite . in this picture , the ns would have evolved earlier from its more massive progenitor and is companion to the second star during its ce phase . when the system enters the ce phase ( in which the hydrogen envelope is ejected ) , it leaves behind a close binary consisting of the ns and the giant star s he core . at this point the he - core has a mass of a few @xmath20 and an orbital separation of @xmath21 ( ( * ? ? ? * figure 1 ) ) . during this phase , the ns accretes up to @xmath22 and relies on the second ce to grow in mass . the he core then expands causing a second he - rich ce phase . sufficient mass is accreted onto the ns during this second ce and the ns reaches @xmath3 and undergoes a qn explosion inside the expanded hydrogen - poor envelope . the qn ejecta shocks and unbinds the ce providing a bright , short - lived hump matching those observed in double - humped slsne@xcite . following the qn , the remaining system consists of a qs and the co core ( of mass @xmath23 ) of the he star . orbital decay lead to a merger a few days to a few weeks following the qn event . the qs then rapidly accretes from the co core leading to the collapse of the qs into a black hole ( bh ) . the remainder of the co core is subsequently accreted by the bh . the accretion luminosity powers the long lasting main hump of the double - humped slsn . figure 1 shows our fit to the recently discovered slsn asassn-15lh@xcite . we fit it with a ce mass and radius of @xmath24 , respectively . the bh - accretion parameters are @xmath25 erg s@xmath26 for the initial accretion luminosity with an injection power in time @xmath27 . the time delay between the qn event and the onset of bh - accretion is 20 days . these parameters are similar to those we used to fit a number of hydrogen - poor slsne@xcite ( ( * ? ? ? * table 2 ) and ( * ? ? ? * figure 2 ) ) . our model can also fit double - peaked slsne showing late - time emission ( e.g. iptf13ehe and lsq14bdq ) which we modelled as the collision between the he - rich ce ( ejected by the qn ) and the hydrogen - rich ( i.e. first ) ce ejected during the firts ce phase ( ( * ? ? ? * figure 1 ) ) . the available accretion energy @xmath28 ( @xmath29 is the bh - accretion efficiency ) is enough to account for the extreme radiation released during the long - lasting hump in slsne . the qn is key to our model since besides accounting for the first peak , it also ejects the second ce at speeds of a few @xmath30 km s@xmath26 which ensures a very efficient harnessing of the bh - accretion input power by the very large envelope a few days to a few weeks following the qn event . the qn deposits its momentum and energy impulsively in the ce which makes our model fundamentally different from those involving spin - down power where the energy is deposited gradually . qne should be common in binaries where accretion onto the ns from a companion ( i.e. the disrupted wd in lmxbs ) or during a ce phase ( i.e. during massive binaries evolution ) can drive the ns above the critical mass , @xmath3 , triggering the qn . the ability of the qn model in binaries to fit slsne in general ( see http://www.quarknova.ca/lcgallery.html ) and in particular a number of double - humped slsne ia , suggests that qne may be an important component of massive binary evolution and may even be responsible for ce ejection . the qn - ia model has two main interesting features : first , the detonation of the wd in the qn - ia scenario is explained by standard shock physics governing the interaction of the qn ejecta and the wd . secondly , the qn - ia provides an elegant explanation for the correlation between peak magnitude and light - curve shape through the contribution of spin - down energy to the light - curve . our model can be tested by further work including simulations of qne in binary evolution . our model relies on the feasibility of the qn explosion which requires sophisticated simulations of the burning of a ns to a qs which are being pursued . preliminary simulations with consistent treatment of nuclear and neutrino reactions , particle diffusion and hydrodynamics show instabilities which could lead to a detonation@xcite . we also propose that a core - collapse " qn could result from the collapse of the quark matter core@xcite which provides another avenue for the explosion . the research of ro , dl and nk is funded by the natural sciences and engineering research council of canada . j.e.s is funded by the university of florida theoretical astrophysics fellowship . 90 a. r. bodmer , _ phrvd _ , * 4 * , 1601 , ( 1971 ) e. witten , _ phrvd _ , * 30 * , 272 , ( 1984 ) r. ouyed , j. dey and m. dey , _ a&a _ , * 390 * , 39 , ( 2002 ) p. kernen , r. , ouyed and p. jaikumar , _ apj _ , * 618 * , 485 ( 2005 ) c. vogt , r. , rapp and r. ouyed , _ nuclear physics a _ , * 735 * , 543 ( 2004 ) b. niebergal , r. ouyed and p. jaikumar , phrvc , * 82 * , 062801 ( 2010 ) r. ouyed , b. niebergal and p. jaikumar , explosive combustion of a neutron star into a quark star : the non - premixed scenario " in proceedings of compact stars in the qcd phase diagram ( csqcdiii ) , eds . l. paulucci , j. e. horvath , m. chiapparini , and r. negreiros , http://www.slac.stanford.edu/econf/c121212/ [ arxiv:1304.8048 ] ( 2013 ) r. ouyed , and d. leahy , d. , _ apj _ , * 696 * , 562 ( 2009 ) p. jaikumar , b. meyer , b. s. otsuki , and r. ouyed , _ * 471 * , 227 ( 2007 ) m. kostka , n. koning , z. shand , et al . , _ a&a _ , * 568 * , a97 ( 2014 ) d. leahy and r. ouyed , _ mnras _ , * 387 * , 1193 , ( 2008 ) r. ouyed , m. kostka , n. koning , et al . 2012 , _ mnras _ , 423 , 1652 ( 2012 ) m. kostka , n. , koning , d. leahy , et al . , _ revmexastron.astrop._ , * 50 * , 167 ( 2014 ) r. ouyed , d. , leahy , a. , ouyed and p. jaikumar , _ phrvlett . _ , * 107 * , 151103 ( 2011 ) r. ouyed , d. leahy and n. koning , _ raa _ , * 15 * , 483 ( 2015 ) r. ouyed , d. leahy and b. niebergal , _ a&a _ * 473 * , 357 , ( 2007a ) r. ouyed , d. leahy and b. niebergal , _ a&a _ * 475 * , 63 , ( 2007b ) j. e. staff , r. ouyed and m. bagchi , m. 2007 , _ apj _ , * 667 * , 340 ( 2007 ) r. ouyed and j. e. staff , _ raa _ , * 13 * , 435 , ( 2013 ) r. ouyed , n. koning , d. leahy , et al . , _ raa _ , * 14 * , 497 , ( 2014 ) j. guy et al . , _ a&a _ , * 466 * , 11 , ( 2007 ) s. perlmutter , et al . _ apj _ , * 517 * , 565 , ( 1999 ) a. g. riess , et al . , _ aj _ , * 116 * , 1009 , ( 1998 ) n. suzuki , d. , rubin , c. , lidman , et al . _ apj _ , * 746 * , 85 ( 2012 ) r. ouyed , d. leahy , n. koning and j. e. staff , _ apj _ , * 801 * , 64 , ( 2015 ) e. churazov et al . , _ nature _ , * 512 * , 406 , ( 2014 ) j. e. staff , p. jaikumar , v. chan and r. ouyed , _ apj _ , bf 751 , 24 , ( 2012 ) r. ouyed , b. niebergal , w. dobler and d. leahy , _ apj _ , * 653 * 558 ( 2006 ) r. ouyed , d. leahy and n. koning , _ mnras _ , * 454 * , 2353 , ( 2015 ) r. ouyed , d. leahy and n. koning , _ apj _ , in press ( 2016 ) , http://arxiv.org/abs/1510.06135 s. dong , b. j. shappee , j. l. prieto , et al . science magazine , _ 351 _ , issue 6270 , * 257 * ( 2016 )

the explosive transition of a massive neutron star to a quark star ( the quark - nova ; qn ) releases in excess of @xmath0 erg in kinetic energy which can drastically impact the surrounding environment of the qn . a qn is triggered when a neutron star gains enough mass to reach the critical value for quark deconfinement to happen in the core . in binaries , a neutron star has access to mass reservoirs ( e.g. accretion from a companion or from a common envelope ; ce ) . we explain observed light - curves of hydrogen - poor superluminous supernovae ( slsne ia ) in the context of a qn occurring in the second ce phase of a massive binary . in particular this model gives good fits to light - curves of slsne with double - humped light - curves . our model suggests the qn as a mechanism for ce ejection and that they be taken into account during binary evolution . in a short period binary with a white dwarf companion , the neutron star can quickly grow in mass and experience a qn event . part of the qn ejecta collides with the white dwarf ; shocking , compressing ; and heating it to driving a thermonuclear runaway producing a sn ia impostor ( a qn - ia ) . unlike `` normal '' type ia supernovae where no compact remnant is formed , a qn - ia produces a quark star undergoing rapid spin - down providing additional power along with the @xmath1ni decay energy . type ia sne are used as standard candles and contamination of this data by qne - ia can infer an incorrect cosmology .

1601.04235

most fundamental physical stellar parameters of field white dwarfs , such as effective temperature , surface gravity , and magnetic field strength can directly be measured with high precision from spectroscopic observations . assuming a mass - radius relation , both mass and radius may be inferred independently of the distance . determining these properties also for the accreting white dwarfs in cataclysmic variables ( cvs ) is a relatively new research field , essential not only for testing stellar ( binary ) evolution theory , but for understanding the physics of accretion in this whole class of binaries . the last years saw a rapid growth of identified polars , cvs containing a synchronously rotating magnetic white dwarf . despite the large number of know systems ( @xmath6 ) rather little is known about the temperatures of the accreting white dwarfs in these systems . the main reasons for this scarcity are twofold . ( a ) in the easily accessible optical wavelength band , the white dwarf photospheric emission is often diluted by cyclotron radiation from the accretion column below the stand - off shock , by emission from the secondary star , and by light from the accretion stream . even when the accretion switches off almost totally and the white dwarf becomes a significant source of the optical flux ( e.g. schwope et al . 1993 ) , the complex structure of the zeeman splitted balmer lines and remnant cyclotron emission complicate a reliable temperature determination . ( b ) at ultraviolet wavelengths the white dwarf entirely dominates the emission of the system during the low state and may be a significant source even during the high state . however , the faintness of most polars requires time - consuming space based observations ( e.g. stockman et al . iue observations of rxj1313.2@xmath03259 ( henceforth rxj1313 ) were carried out in march , 1996 . one swp ( 1150@xmath01980 ) and one lwp ( 1950@xmath03200 ) low resolution spectrum were obtained on march 2 and march 6 , respectively ( table1 ) . the lwp image was taken prior to the failure of gyro#5 , read - out of the image had to await that control over the spacecraft was re - established . both observations were taken through the large aperture , resulting in a spectral resolution of @xmath7 . because of the faintness of rxj1313 , the exposure time of the swp spectrum was chosen roughly equal to the orbital period . the spectra have been processed through the iuenewsips pipeline , yielding flux and wavelength calibrated spectra . the swp spectrum is shown in fig[f - swp ] . it is a blue continuum with a flux decline below @xmath8 . due to the long exposure time , the spectrum is strongly affected by cosmic ray hits . some emission of civ@xmath91550 and heii@xmath91640 may be present in the spectrum of rxj1313 , but from the present data no secure detection of line emission can be claimed . the absence / weakness of emission lines strongly indicates that the iue observations were taken during a period of very low accretion activity . the broad flux turnover below @xmath8 is reminiscent of the photospheric absorption observed during low states , e.g. in amher @xcite or dpleo @xcite . our first approach was , thus , to fit the swp data with non - magnetic pure hydrogen white dwarf model spectra @xcite . however , none of the models could satisfyingly describe the observed spectrum . while the continuum requires a rather low temperature , @xmath10k , the steep slope in the narrow core of the absorption ( @xmath11 ) is in disagreement with the very broad line of such low - temperature models . lrr imageno . & start ( ut ) & exp . time ( sec ) + swp56879l & 02mar199608:01:49 & 13800 + lwp32069l & 06mar199618:20:31 & 2100 + the analysis of low - state ultraviolet spectroscopy of other polars taught us that the white dwarfs often have a non - uniform temperature distribution over their surface @xcite , possibly due to heating by low - level accretion @xcite . we , therefore , fitted the iue data of rxj1313 with a two - temperature model , using again our non - magnetic pure hydrogen model spectra and leaving four free parameters ; the temperatures and scaling factors of both components . the best fit is achieved by a white dwarf with a `` base '' temperature of @xmath1k and a `` spot '' temperature of @xmath2k ( fig.[f - swp ] ) . for a distance @xmath12pc , as derived by thomas et al . @xcite , the white dwarf radius resulting from the scaling factors is @xmath13 cm . assuming the hamada - salpeter ( 1961 ) mass - radius relation for a carbon core , the corresponding mass is @xmath14 , which is consistent with the mass derived by thomas et al . @xcite . because the iue / swp observation represents the orbital mean of the ultraviolet emission of rxj1313 , the spot size can not be directly estimated . assuming that the ultraviolet - bright spot shows a similar variation as the x - ray spot observed with rosat @xcite , we estimate a fractional area @xmath3 . for a somewhat larger spot , the temperature would be correspondingly lower . fig.[f - overall ] shows the iue swp and lwp spectra along with an average optical low state spectrum , as well as the two - component model . the flux of the lwp spectrum is somewhat lower than predicted by the model , which could be due either to heavy underexposure ( table1 ) or to the fact that the lwp spectrum covers only @xmath15 of the orbital period , possibly resulting in a lower spot - contribution than in the orbital - averaged swp spectrum , or both . the agreement of the model spectra with observed optical flux is reasonably good , especially when considering that only the 12251900 range was used for the fit and that the ultraviolet and optical spectra were taken at different epochs . the summed spectrum of the white dwarf model and a red dwarf matching the red end of the rxj1313 spectrum has @xmath16 , which is in agreement with the observed low - state magnitude of the system @xcite . during the low state , the optical and ultraviolet emission of rxj1313 is , hence , dominated by its two stellar components . for completeness , we mention that an additional possible source of absorption is the interstellar gas . we computed the interstellar profile for the absorption column derived from the x - ray data , @xmath17 @xcite . the width of this line is smaller than the geocoronal emission in the swp spectrum . thus , interstellar absorption can not explain the narrow `` core '' observed in the iue spectrum . a major uncertainty in the computation of realistic hydrogen line profiles in magnetic atmospheres is the treatment of the stark broadening in the presence of a magnetic field . the stark broadening of the individual zeeman components is smaller than that of the entire transition in the non - magnetic case , but no detailed calculations are available . this uncertainty can be taken into account by treating the amount of the stark broadening as a free parameter in the model atmosphere calculation and calibrating it with observations @xcite . for , this approach is , however , difficult . on one hand , there are only few single magnetic white dwarfs for which good ultraviolet spectroscopy has been obtained . on the other hand , the three zeeman components of become visible as individual absorption features only for fields @xmath18 mg . for lower field strengths the profile is still dominated by the stark effect and the zeeman shifts introduce only an additional broadening which is , again , difficult to quantify . an additional problem in the computation of synthetic profiles arises for low - temperature atmospheres ( @xmath19k ) . in ultraviolet observations of non - magnetic white dwarfs in this temperature range , quasi - molecular absorption of @xmath20 and @xmath21 produces strong absorption features at @xmath8 and @xmath22 , respectively , which are overlayed on the red wing of @xcite . calculations of these transitions in the presence of a strong magnetic field have not yet been approached . we have retrieved the iue spectra available for all magnetic white dwarfs listed by jordan @xcite , and find that in at best two of them the @xmath20 feature can be identified ( bpm25114 , @xmath23 mg and kuv23162@xmath01230 , @xmath24 mg ) . also , none of the accreting magnetic white dwarfs in polars with @xmath19k observed in the ultraviolet display noticeable @xmath20 absorption @xcite . from fig.[f - swp ] it is apparent that also the spectrum of rxj1313 is devoid of noticeable absorption at 1400 and 1600 . in summary , observations indicate that the @xmath20 and @xmath21 quasi - molecular absorption lines may be weaker in a strongly magnetic atmosphere than in a non - magnetic one . assuming a magnetic field strength of @xmath25 mg for rxj1313 , as derived by thomas et al . @xcite from the cyclotron emission , the expected shift of the @xmath26 components is @xmath27 , causing the centre of the @xmath28 component to coincide with the steepest slope of the profile . while the zeeman effect may broaden the observed profile , the reduced stark broadening will cause an opposite effect . we estimate that the use of non - magnetic model spectra in the analysis of the profile may cause a temperature error of a about @xmath29k . we conclude that the theoretical uncertainties in the stark broadening do presently not warrant the use of magnetic model spectra . the narrow core in the broad absorption observed in rxj1313 can not be produced by magnetic effects supporting our interpretation of a rather cool white dwarf with a localized hot region . it is well established that the white dwarfs in cvs tend to be hotter than single white dwarfs . this observational result suggests that accretional heating takes place in addition to the secular core cooling of the white dwarfs in cvs ( e.g. sion 1991,1999 ) . furthermore , the white dwarfs in cvs below the period gap are on average cooler than those in cvs above the gap ( gnsicke 1997,1998 ; sion 1991,1999 ) . a combination of two effects is thought to be responsible for this difference : ( i ) the average age of the short - period cvs below the period gap is about an order of magnitude larger than that of cvs above the gap @xcite and core cooling of their white dwarfs has progressed correspondingly ; ( ii ) the average accretion rate in short - period cvs is about an order of magnitude lower than in long - period cvs @xcite , resulting in reduced accretional heating . warner @xcite shows admittedly only for a small sample of cvs that the expected correlation between accretion rate and white dwarf temperature does , in fact , exist . rxj1313 is the polar with the fourth - longest period . it is , therefore , expected to be rather young , to experience a comparatively high time - averaged accretion rate , and to have a correspondingly hot white dwarf . contrary to these expectations , however , it harbours the coldest white dwarf of all the cvs above the period gap . in fact , the temperature of the white dwarf in rxj1313 is comparable to the average white dwarf temperature in short - period cvs . we suggest two possible scenarii which can explain the atypically low white dwarf temperature . \(a ) rxj1313 has only recently developed from a detached pre - cataclysmic binary into the semi - detached state . mass transfer is in the process of turning on and substantial heating of the white dwarf has not yet taken place . in this case , the observed effective temperature of the white dwarf allows to estimate a lower limit on the cooling age and , thereby , on the time elapsed since the system emerged from the common envelope . the time scale for the turn - on of the mass transfer is @xmath30yrs , which is short compared to the @xmath31yrs that a cv spends above the gap @xcite . the probability of finding a cv in this stage of its evolution is rather small , but non - zero . \(b ) rxj1313 is a `` normal '' long - period cv , but has more recently experienced a low accretion rate for a sufficiently long time interval ( @xmath5yrs ) which allowed its white dwarf to cool down to its current temperature . long - term ( @xmath32yrs ) fluctuations of the accretion rate about the secular mean predicted from angular momentum loss by magnetic braking @xcite are consistent with the large range of observed accretion rates at a given orbital period @xcite . two possible explanation for these fluctuations have been suggested . ( b1 ) a limit - cycle in the secondary s radius driven by irradiation from the hot primary ( king et al . 1995,1996 ) which causes a corresponding variation in the mass transfer rate . ( b2 ) cvs possibly enter a prolonged phase of low ( or zero ) @xmath33 following a classical nova eruption , referred to as hibernation @xcite . rxj1313 may be such a hibernating cv . however , the low temperature of the white dwarf in rxj1313 argues against a very recent nova explosion . in v1500cyg , the white dwarf cooled from the nuclear burning regime , i.e. several @xmath34k in 1975 to @xmath35k in 1992 @xcite . on the theoretical side , prialnik ( 1986 ) shows in a simulation of a 1.25 classical nova that the white dwarf reaches its minimum temperature @xmath36yrs after the nova explosion . two observational tests could help to decide whether rxj1313 is a rather `` fresh '' post - nova with its secondary only marginally filling its roche - lobe : ( 1 ) a nova eruption may break synchronization , as observed in v1500cyg @xcite , causing the orbit to widen and the secondary to retreat from its roche lobe . the resynchronization of the white dwarf spin with the orbital period occurs in v1500cyg apparently on a time scale of a few hundred years @xcite . more accurate ephemerides of rxj1313 than presently available ( thomas et al . 1999 ) are necessary to test for a small remnant asynchronism of the white dwarf spin . ( 2 ) the nova eruption may contaminate the binary system with material processed in the thermonuclear event , resulting in deviations from the typical populationi abundances found in most cvs @xcite . anomalous ultraviolet emission line ratios similar to those observed in post - novae have been found in the asynchronous polar bycam @xcite . the white dwarf in bycam has @xmath37k @xcite , which is in rough in agreement with the temperature expected a few 1000 years after a nova explosion . bycam contains also a slightly asynchronously rotating white dwarf , which leaves the question of the expected time scale for resynchronization somewhat unsettled . high - state ultraviolet observations of the cno lines in rxj1313 do not exist so far and will serve to test the post - nova hypothesis . if no evidence for a nova event should be found , rxj1313 is either very young as a cv or experiences a prolonged low state in some kind of mass transfer cycle . before we discuss the temperature of the white dwarf photosphere , = 15000k , we comment on the `` warm '' ultraviolet - bright spot with @xmath38k . rxj1313 is yet another polar in which the white dwarf appears to have a non - uniform temperature distribution . other examples are amher @xcite , dpleo @xcite , or qstel @xcite . these hot spots are best explained by the localized irradiation of the photosphere with cyclotron and x - ray photons from the accretion funnel which is continuously fed at a low rate . in the case of rxj1313 , we estimate the luminosity of the ultraviolet - bright spot to be @xmath39 , corresponding to an accretion rate of @xmath40 , which is consistent with the low - state accretion rate , @xmath41 , derived by thomas et al . @xcite . we now discuss accretion heating of the white dwarf in rxj1313 . as shown by giannone & weigert @xcite and by sion @xcite this is an inherently time - dependent process . accretion compresses the outer non - degenerate layers of the white dwarf which heat up approximately adiabatically if the accretion rate is high . the core suffers some compression , too , which heats primarily the non - degenerate ions . for intermittent accretion , the thermal inertia of the deep heating produces a time delay which causes an enhanced luninosity long after accretion stopped . for very low accretion rates , @xmath42 , on the other hand , prolonged accretion may lead to a quasi - stationary state in which the energy loss balances compressional heating @xcite and the temperature profile of the envelope remains stationary . the simplest way to view compressional heating is to consider the energy released when the accreted mass is added . since the envelope mass is small and represents a practically constant fraction of the white dwarf mass , @xmath33 increases the mass of the degenerate core of mass @xmath43 , radius , and temperature . the energy released per unit time is g@xmath33(1/ 1/ ) of which some fraction feeds the initial degeneracy of the electrons reaching . apart from a factor of order unity , this energy equals the work performed by compression , @xmath44)/dt with @xmath45 the pressure and @xmath46 the density , integrated over the envelope . note that this energy release is different from that freed at the surface , which equals g@xmath33/ in an am her star , and represents the additional energy released by compression of the envelope of the white dwarf between radii and . if compression is adiabatic , the work performed is used to increase the internal energy of the gas , as prescribed by the first law of thermodynamics . in the isothermal case , the released energy would completely appear as radiative loss . we consider here the case of slow compression and assume that the energy released by accretion at a rate @xmath33 equals the increment in luminosity @xmath47 where g is the gravitational constant , and we estimate that @xmath48 is between 0.5 and 1.0 . core heating is a minor effect and adds only @xmath49% to @xmath50 . hence , the compressional energy is primarily released in the envelope , at an approximately constant rate per radius interval . in equilibrium , accretion at a rate @xmath33 can maintain an effective temperature of the white dwarf defined by , even if the white dwarf had cooled to a substantially lower temperature before the onset of accretion . in their discovery paper , thomas et al . ( 1999 ) derive a mass of the white dwarf in rxj1313 of @xmath51 and a secondary mass of @xmath52 , with uncertainties of about 0.10 . there was some concern about the mass ratio which should be @xmath53/@xmath54 for stable mass transfer . for definiteness , we assume here @xmath55 . thomas et al . ( 1999 ) also derived a mass accretion rate which is very low compared to other long period cvs . they observed the system over seven years and found that it hovers most of the time at low accretion luminosities corresponding to @xmath42 . only twice was the system found in an intermediate state with an accretion rate of @xmath56 , during the rosat all - sky - survey and in a subsequent optical follow - up observation in february 1993 . it was never observed at an accretion rate of @xmath57 , the typical value of cvs with = 45 h @xcite . to be sure , the derived accretion rates depend ( i ) on the adopted white dwarf mass and ( ii ) on the soft x - ray temperature and the bolometric fluxes of the quasi - blackbody source , and the quoted rates are probably uncertain by a factor of @xmath58 . a white dwarf of 0.5 has a core radius = @xmath59 cm and a radius @xmath60 cm at = 15000k . for these parameters , we find an equilibrium temperature from compressional heating alone of @xmath61 k , where @xmath62 k is the accretion rate in units of @xmath63 . the observed temperature of 15000k can be maintained by an accretion rate of @xmath64 for @xmath65 , which is within the observed range of mass transfer rates . since the internal energy source of the white dwarf will contribute to the observed luminosity , the actual accretion rate required to maintain the photosphere at 15000 k may be somewhat lower . alternatively , the efficiency @xmath48 of converting the compressional energy release into the observed luminosity may be lower . within the uncertainties , however , it is also possible that the present temperature is almost entirely due to compressional heating and that the white dwarf had cooled to a temperature substantially below 15000 k prior to the onset of mass transfer . in any case , the cooling age of @xmath66yrs to reach 15000 k @xcite is a lower limit to the actual pre - cv age of the white dwarf . since the kelvin - helmholtz time scale of the envelope is roughly of the order of @xmath67yrs , the low temperature of the white dwarf requires @xmath33 to have been low for a comparable length of time . if rxj1313 is a cv in the process of turning on mass transfer we would expect that the accretion rate in rxj1313 would ultimately reach @xmath57 at which time the white dwarf has been compressionally heated to @xmath68k , the temperature typically observed in cvs with @xmath69h @xcite . we conclude that the low temperature of the white dwarf in rxj1313 is consistent with compressional heating by mass accretion at a rate substantially lower than the expected for a long period cv . the system has not passed through a phase of high accretion rate within at least the last @xmath67yrs which is the approximate kelvin - helmholtz time scale for the envelope . there are three possible previous histories of rxj1313 : ( a ) it is a young cv in the process of turning on the mass transfer ; ( b1 ) it is in a long - lasting phase of low accretion within an irradiation - driven limit cycle ; ( b2 ) it has passed through a nova outburst shutting off mass transfer for a prolonged period . we can not presently distinguish between cases ( a ) and ( b1 ) , while observational tests of ( b2 ) have been suggested above . we thank klaus reinsch for the optical spectrum of rxj1313 and for useful comments on the manuscript , and stefan jordan for discussions on magnetic model atmospheres . this research was supported by the dlr under grant 50or96098 and 50or99036 .

we present low - state iue spectroscopy of the rosat - discovered polar rxj1313.2@xmath03259 . the swp spectrum displays a broad absorption profile , which can be fitted with a two - temperature model of a white dwarf of @xmath1k with a hot spot of @xmath2k which covers @xmath3 of the white dwarf surface . the white dwarf temperature is atypically low for the long orbital period ( 4.18h ) of rxj1313.2@xmath03259 . this low temperature implies either that the system is a young cv in the process of switching on mass transfer or that it is an older cv found in a prolonged state of low accretion rate , much below that predicted by standard evolution theory . in the first case , we can put a lower limit on the life time as pre - cv of @xmath4yrs . in the second case , the good agreement of the white dwarf temperature with that expected from compressional heating suggests that the system has experienced the current low accretion rate for an extended period @xmath5yrs . a possible explanation for the low accretion rate is that rxj1313.2@xmath03259 is a hibernating post nova and observational tests are suggested .

astro-ph9910456

the recent discovery of huge quantities of dust ( @xmath5 ) in very high redshifted galaxies and quasars ( isaak et al . 2002 ; bertoldi et al . 2003 ) suggests that dust was produced efficiently in the first generation of supernovae ( sne ) . theoretical studies ( kozasa et al . 1991 ; todini & ferrara 2001 , hereafter tf ; nozawa etal . 2003 , n03 ) predicted the formation of a significant quantity of dust ( @xmath6 @xmath7 ) in the ejecta of type ii sne , and the predicted dust mass is believed to be sufficient to account for the quantity of dust observed at high redshifts ( maiolino et al . 2006 ; meikle et al . recently , a model of dust evolution in high redshift galaxies ( dwek et al . 2007 ) indicates that at least 1 @xmath7 of dust per sn is necessary for reproducing the observed dust mass in one hyperluminous quasar at @xmath8 . observationally , the presence of freshly formed dust has been confirmed in a few core - collapsed sne , such as sn1987a , which clearly have showed several signs of dust formation in the ejecta ( see mccray 1993 for details ) . the highest dust mass obtained so far for sn 1987a is @xmath9 @xmath7 @xcite . spitzer _ and _ hst _ observations ( sugerman et al . 2006 ) showed that up to 0.02 @xmath7 of dust formed in the ejecta of sn2003gd with the progenitor mass of 612 @xmath7 , and the authors concluded that sne are major dust factories . however , from the detailed analysis of the late time mid infrared observations , meikle et al . ( 2007 ) found that the mass of freshly formed dust in the same sn is only @xmath10 , and failed to confirm the presence 0.02 @xmath7 dust in the ejecta . the aforementioned results show that the derived dust mass is model - dependent , and that the amount of dust that really condenses in the ejecta of core - collapsed sne is unknown . cassiopeia a ( cas a ) is the only galactic supernova remnant ( snr ) that exhibits clear evidence of dust formed in ejecta ( lagage et al . 1996 ; arendt et al . 1999 , hereafter adm ) . the amount of dust that forms in the ejecta of young snr is still controversial . previous observations inferred only @xmath11 of dust at temperatures between 90 and 350 k ( adm ; douvion et al . 2001 , hereafter d01 ) ; this estimate is 2 to 3 orders of magnitude too little to explain the dust observed in the early universe . recent submillimeter observations of casa and kepler with scuba @xcite revealed the presence of large amounts of cold dust ( @xmath12 at 1520 k ) missed by previous iras / iso observations . on the other hand , highly elongated conductive needles with mass of only 10@xmath13 to 10@xmath14 @xmath7 could also explain a high sub - mm flux of cas a , when including grain destruction by sputtering ( dwek 2004 ) , though the physicality of such needles is doubtful @xcite . while @xcite showed that much of the 160@xmath0 m emission observed with multiband imaging photometer for _ spitzer _ ( mips ) is foreground material , suggesting there is no cold dust in cas a , @xcite used co emission towards the remnant to show that up to about a solar mass of dust could still be associated with the ejecta , not with the foreground material . these controversial scenarios of dust mass highlight the importance of correctly identifying the features and masses of dust freshly formed in cas a. the galactic young snr cas a allows us to study in detail the distribution and the compositions of the dust relative to the ejecta and forward shock with infrared spectrograph onboard the _ spitzer space telescope_. cas a is one of the youngest galactic snrs with an age of 335 yr attributed to a sn explosion in ad 1671 . the progenitor of cas a is believed to be a wolf - rayet star with high nitrogen abundance @xcite and to have a mass of 15 - 25 m@xmath15 @xcite or 29 - 30 m@xmath15 @xcite . the predicted dust mass formed in sne depends on the progenitor mass ; for a progenitor mass of 15 to 30 m@xmath15 , the predicted dust mass is from 0.3 to 1.1 m@xmath15 ( no3 ) and from 0.08 to 1.0 m@xmath15 ( tf ) , respectively . in this paper , we present _ spitzer _ infrared spectrograph ( irs ) mapping observations of cas a , and identify three distinct classes of dust associated with the ejecta and discuss dust formation and composition with an estimate of the total mass of freshly formed dust . we performed _ spitzer _ irs mapping observations covering nearly the entire extent of cas a on 2005 january 13 with a total exposure time of 11.3 hr . the short low ( sl : 5 - 15 @xmath0 m ) and long low ( ll : 15 - 40 @xmath0 m ) irs mapping involved @xmath1616@xmath17 and @xmath18 pointings , producing spectra every 5@xmath1 and 10@xmath1 , respectively . the spectra were processed with the s12 version of the irs pipeline using the cubism package ( kennicutt et al . 2003 ; smith et al . 2007 ) , whereby backgrounds were subtracted and an extended emission correction was applied . the spectral resolving power of the irs sl and ll modules ranges from 62 to 124 . the irs spectra of cas a show bright ejecta emission lines from ar , ne , s , si , o , and fe and various continuum shapes as indicated by the representative spectra in figure 1 . the most common continuum shape exhibits a large bump peaking at 21 @xmath0 m as shown by spectrum `` a '' in figure [ sixspec ] . this `` 21 @xmath0m - peak '' dust is often accompanied by the silicate emission feature at 9.8 @xmath0 m which corresponds to the stretching mode . a second class of continuum shapes exhibits a rather sharp rise up to 21 @xmath0 m and then stays flat thereafter . this `` weak-21 @xmath0 m dust '' is often associated with relatively strong ne lines ( in comparison with ar lines ) and is indicated by spectrum `` b '' in figure [ sixspec ] . the third type of dust continuum is characterized by a smooth and featureless , gently rising spectrum with strong [ ] + [ ] and [ ] emission lines as shown by spectra `` c '' and `` d '' in figure [ sixspec ] . the spectrum `` d '' shows double line structures that may be due to doppler - resolved lines of [ ] at 26 @xmath0 m and [ ] at 35 @xmath0 m . note that the `` featureless '' dust ( spectrum `` d '' in fig . [ sixspec ] ) is a class of dust , separate from the interstellar / circumstellar dust ( spectrum `` e '' in fig . [ sixspec ] ) heated by the forward shock . the interstellar / circumstellar dust spectrum in cas a has no associated gas line emission . the `` broad '' continuum ( see figure 7b of ennis et al . 2006 ) is a combination of the spectra `` c '' and `` e '' . the spectrum `` c '' has contamination from the shock heated dust in projection , and for simplicity it is excluded in estimating the masses of the freshly formed dust ( see 5 ) . the `` featureless '' dust lacks the gentle peak around 26 @xmath0 m and also lacks the interstellar silicate - emission feature between 9 @xmath0 m and 11 @xmath0 m observed in the spectra from the forward shock region . most importantly , the `` featureless '' dust accompanies relatively strong si and s ejecta lines and mostly from the interior of the remnant ( blue region in fig . [ images]f ) . we generated a map of the 21 @xmath0m - peak dust by summing the emission over 19 - 23 @xmath0 m after subtracting a baseline between 18 - 19 @xmath0 m and 23 - 24 @xmath0 m . the line - free dust map ( fig . [ images]a ) resembles the [ ar ii ] and [ o iv]+[fe ii ] ejecta - line maps , as shown in figures [ images]b and [ images]c , and we also find that the [ ne ii ] map is very similar to the [ ar ii ] map . the [ ] map shows a remarkable similarity to the 21 @xmath0m - peak dust map ( fig . [ images]a and [ images]b ) , thereby confirming this dust is freshly formed in the ejecta . maps of [ ] ( fig . [ images]d ) and [ o iv]+[fe ii ] ( fig . [ images]c ) shows significant emission at the center revealing ejecta that have not yet been overrun by the reverse shock ( unshocked ejecta ) . there is also [ ] and [ o iv]+[fe ii ] emission at the bright ring indicating that some of the si and o+fe ejecta have recently encountered the reverse shock . while the bright o+fe emission outlines the same bright ring structure as the [ ar ii ] and 21 @xmath0m - peak dust maps , the bright part of the si shell shows a different morphology from the other ejecta maps . we can characterize the spectra of our three dust classes by using the flux ratios between 17 @xmath0 m and 21 @xmath0 m and between 21 @xmath0 m and 24 @xmath0 m . although the spectra in cas a show continuous changes in continuum shape from strong 21 @xmath0 m peak to weak 21 @xmath0 m peak and to featureless , we can locate regions where each of the three classes dominates . figure [ images]f shows the spatial distribution of our three dust classes where red , green , and blue indicate 21 @xmath0m - peak dust , weak-21 @xmath0 m dust , and featureless dust , respectively . the flux ratios used to identify the three dust classes are as follows where @xmath19 is the flux density in the extracted spectrum at wavelength @xmath20 ( @xmath0 m ) : \1 ) 21 @xmath0m - peak dust : we use the ratio @xmath21 , where @xmath22 is the dispersion in @xmath23 over the remnant , which is equivalent to @xmath24 . the regions with 21 @xmath0m - peak dust coincide with the brightest ejecta . \2 ) weak-21 @xmath0 m dust : we use the ratio @xmath25 , which is equivalent to @xmath26 . the regions showing the weak-21 @xmath0 m continuum shape mostly coincide with faint ejecta emission , but not always . \3 ) featureless dust map : we use the ratio @xmath27 , which is equivalent to @xmath28 . this ratio also picks out circumstellar dust heated by the forward shock , so we used several methods to exclude and mitigate contamination from circumstellar dust emission . first , using x - ray and radio maps , we excluded the forward shock regions at the edge of the radio plateau @xcite . second , there are highly structured `` continuum - dominated '' x - ray filaments across the face of the remnant which are similar to the exterior forward shock filaments and may be projected forward shock emission @xcite . for our analysis , we excluded regions where there were infrared counterparts to the projected forward shock filaments . third , for simplicity we excluded regions with gently rising spectra identified by curve `` c '' ( the spectra which continues to rise to longer wavelengths ) in figure [ sixspec ] . this type of spectrum is mainly found on the eastern side of cas a where there is an h@xmath29 region , the northeast jet , and other exterior optical ejecta @xcite making it difficult to determine if the continuum emission is due to ejecta dust or circumstellar dust . however , note that some portion of the continuum in the spectra , `` c '' , is freshly formed dust . we finally excluded regions where there was a noticeable correlation to optical quasi - stationary flocculi @xcite which are dense circumstellar knots from the progenitor wind . the featureless dust emission appears primarily across the center of the remnant , as shown in figure [ images]d ( blue ) . the featureless dust is accompanied by relatively strong [ ] and [ ] and [ + ] lines , as shown by the spectrum `` d '' of figure [ sixspec ] . the [ ] + [ ] line map ( fig . [ images]c ) shows significant emission at the center as well as at the bright ring of the reverse shocked material . the [ ] line map shows different morphology than other line maps and the 21 @xmath0m - peak dust map ; depicting center - filled emission with a partial shell , as shown in figure [ images ] . this poses the following important question : why is the si map more center - filled than the ar map ? the answer is unclear because si and ar are both expected at similar depths in the nucleosynthetic layer ( e.g. woosley , heger & weaver 2002 ) . the relatively faint infrared emission of si and s at the reverse shock may imply relatively less si and s in the reverse shock . we suspect it is because the si and s have condensed to solid form such as mg protosilicate , mgsio@xmath3 , mg@xmath2sio@xmath30 and fes . in contrast , ar remains always in the gas and does not condense to dust , so it should be infrared or x - ray emitting gas . alternate explanation is that the ionization in the interior is due to photoionization from the x - ray shell ( see hamilton & fesen 1988 ) ; in this case , the lack of interior relative to might be due to its much higher ionization potential ( 16 ev compared to 8 ev ) . theoretical models of nucleosynthesis , accounting for heating , photoionization , and column density of each element would be helpful for understanding the distribution of nucleosynthetic elements . the si and s emission detected at the interior , is most likely unshocked ejecta where the revere shock has not yet overtaken the ejecta . the radial profile of unshocked ejecta is centrally peaked at the time of explosion , as shown by @xcite . the radial profile of unshocked fe ejecta is also expected to be center - filled for @xmath161000 yr old type ia snr of sn 1006 @xcite . the morphology of the featureless dust resembles that of unshocked ejecta , supporting the conclusion that the featureless dust is also freshly formed dust . the spectrum in figure [ sixspec ] ( curve `` d '' ) , shows the resolved two lines at 26 @xmath0 m and at 35 @xmath0 m . the two respective lines at @xmath1626 @xmath0 m may be resolved lines of [ ] and [ ] , and at @xmath1635 @xmath0 m [ ] and [ ] ( as expected that the unshocked ejecta near the explosion center have a low velocity ) ; alternatively , they could be highly doppler - shifted lines ( in this case the two lines at 26 @xmath0 m are both [ ] , and the two lines at 35 @xmath0 m are both [ ] ) . the newly revealed unshocked ejecta deserves extensive studies ; preliminary doppler - shifted maps were presented in @xcite and the detailed analysis of velocities and abundances of unshocked and shocked ejecta will be presented in future papers @xcite . we performed spectral fitting to the irs continua using our example regions in figure [ sixspec ] . included in the fitting are mips 24 @xmath0 m and 70 @xmath0 m fluxes @xcite , and the contribution of synchrotron emission ( figs . [ 21umpeakspec ] and [ weak21umspec ] ) , estimated from the radio fluxes @xcite and infrared array camera ( irac ) 3.6 @xmath0 m fluxes @xcite . we measured synchrotron radiation components for each position using radio maps and assuming the spectral index @xmath29=-0.71 @xcite where log s @xmath31 @xmath29 log @xmath32 . because the full - width - half - maximum of 24 @xmath0 m is smaller than the irs extracted region , the surface brightnesses for 24 @xmath0 m were measured using a 15@xmath1 box , the same size as the area used for the extracted irs ll spectra . we also made color corrections to each mips 24 @xmath0 m data point based on each irs spectrum and band - filter shape ; the correction was as high as 25% for some positions . while the uncertainty of calibration errors in irac is 3 - 4% , that of mips 24 @xmath0 m is better than 10% . the mips 70 @xmath0 m image @xcite , shown in figure [ images]e , clearly resolves cas a from background emission , unlike the 160 @xmath0 m image @xcite . most of the bright 70 @xmath0 m emission appears at the bright ring and corresponds to the 21 @xmath0 m dust map and the shocked ejecta , particularly [ ] , indicating that the 70 @xmath0 m emission is primarily from freshly formed dust in the ejecta . the 70 @xmath0 m emission also appears at the interior as shown in figure [ images]e . we measured the brightness for 70 @xmath0 m within a circle of radius 20@xmath1 for each position , accounting for the point - spread function ( note that when the emission is uniform , the aperture size does not affect the surface brightness ) . we estimated the uncertainties of the 70 @xmath0 m fluxes to be as large as 30% . the largest uncertainty comes from background variation due to cirrus structures based on our selection of two background areas , 5@xmath33 to the northwest and south of the cas a. the dust continuum is fit with the planck function ( b@xmath34 ) multiplied by the absorption efficiency ( @xmath35 ) for various dust compositions , varying the amplitude and temperature of each component . to determine the dust composition , we consider not only the grain species predicted by the model of dust formation in sne ( tf , n03 ) , but also mg protosilicates ( adm ) and feo @xcite as possible contributors to the 21 @xmath0 m feature . the optical constants of the grain species used in the calculation are the same as those of @xcite , except for amorphous si @xcite , amorphous sio@xmath2 @xcite , amorphous al@xmath2o@xmath3 @xcite , feo @xcite , and we apply mie theory @xcite to calculate the absorption efficiencies , q@xmath36 , assuming the grains are spheres of radii @xmath37 @xmath0 m . we fit both amorphous and crystalline grains for each composition , but it turned out that the fit results in cas a ( see 3 ) favor amorphous over crystalline grains . thus , default grain composition indicates amorphous , hereafter . for mg protosilicate , the absorption coefficients are evaluated from the mass absorption coefficients tabulated in @xcite , and we assume that the absorption coefficient varies as @xmath38 for @xmath39 40 @xmath0 m , typical for silicates . we fit the flux density for each spectral type using scale factors @xmath40 for each grain type @xmath41 , such that f@xmath42 = @xmath43 . note that the calculated values of q@xmath36/@xmath44 are independent of the grain size as long as 2@xmath45 @xmath461 where @xmath47 is the complex refractive index . thus the derived scale factor c@xmath48 as well as the estimated dust mass ( see 4 ) are independent of the radius of the dust . the dust compositions of the best fits are summarized in table 1 . the strong 21 @xmath0m - peak dust is best fit by mg proto - silicate , amorphous sio@xmath2 and feo grains ( with temperatures of 60 - 120 k ) as shown in figure [ 21umpeakspec ] . these provide a good match to the 21 @xmath0 m feature . adm suggested that the 21 @xmath0 m feature is best fit by mg proto - silicate while d01 suggested it is best fit by sio@xmath2 instead . we found , however , that sio@xmath2 produced a 21 @xmath0 m feature that was too sharp . we also fit the observations using mg@xmath2sio@xmath30 , which exhibits a feature around 20 @xmath0 m and the overall variation of absorption coefficients of mg@xmath2sio@xmath30 with wavelength might be similar to that of mg protosilicate @xcite . however , with mg@xmath2sio@xmath30 , the fit is not as good as that of mg protosilicate , not only at the 21-@xmath0 m peak , but also at shorter ( 10 - 20 @xmath0 m ) and longer ( 70 @xmath0 m ) wavelengths . thus , we use mg protosilicate and sio@xmath2 as silicates to fit the 21 @xmath0m - peak dust feature . the fit with mg protosilicate , sio@xmath2 and feo is improved by adding aluminum oxide ( al@xmath2o@xmath3 , 83 k ) and fes ( 150 k ) , where al@xmath2o@xmath3 improved the overall continuum shape between 10 - 70 @xmath0 m and fes improved the continuum between 30 - 40 @xmath0 m ( underneath the lines of si , s and fe ) , as shown figure [ 21umpeakspec ] . the silicate composition is responsible for the 21 @xmath0 m peak , suggesting that the dust forms around the inner - oxygen and s - si layers and is consistent with ar being one of the oxygen burning products . we also include amorphous mgsio@xmath3 ( 480 k ) and sio@xmath2 ( 300 k ) to account for the emission feature around the 9.8 @xmath0 m . the composition of the low temperature ( 40 - 90 k ) dust component necessary for reproducing 70 @xmath0 m is rather unclear . either al@xmath2o@xmath3 ( 80 k ) ( model a in table 1 ) or fe ( 100 k ) ( model b in table 1 and figure [ modelbspec ] ) can fit equally well , as listed in table 1 . we could use carbon instead of al@xmath2o@xmath3 or fe , but the line and dust compositions suggest the emission is from inner o , s - si layers , where carbon dust is not expected . there are still residuals in the fit from the feature peaking at 21 @xmath0 m ( 20 - 23 @xmath0 m ) , and an unknown dust feature at 11 - 12.5 @xmath0 m ( it is not a part of typical pah feature ) , as shown in figure [ 21umpeakspec ] . the former may be due to non - spherical grains or different sizes of grains . the weak 21 @xmath0 m continuum is fit by feo and mg@xmath2sio@xmath30 or mg protosilicate ( models c and d in table 1 ) since the curvature of the continuum changes at 20 - 21 @xmath0 m as shown in figure [ weak21umspec ] . to fit the rest of the spectrum , we use glassy carbon dust and al@xmath2o@xmath3 grains . the glassy carbon grains ( 220 k ) can account for the smooth curvature in the continuum between 8 - 14 @xmath0 m . carbon dust ( 80 k ) and al@xmath2o@xmath3 ( 100 k ) contribute to the continuum between 15 - 25 @xmath0 m . we could use fe dust instead , but we suspect carbon dust because of the presence of relatively strong ne line emission with the weak 21 @xmath0 m dust class . ne , mg , and al are all carbon burning products . we can not fit the spectrum replacing carbon by al@xmath2o@xmath3 with a single or two temperatures because @xmath49 of al@xmath2o@xmath3 has a shallow bump around 27 @xmath0 m , thus the fit requires three temperature components of al@xmath3o@xmath2 or a combination of two temperature components of al@xmath3o@xmath2 and a temperature component of carbon . the continuum between 33 - 40 @xmath0 m ( underneath the lines of si , s and fe ) can be optimally fit by fes grains . the 70 @xmath0 m image shown in figure [ images]e shows interior emission similar to the unshocked ejecta but that may also be due to projected circumstellar dust at the forward shock . in order to fit the featureless spectrum out to 70 @xmath0 m , we must first correct for possible projected circumstellar dust emission . the exterior forward shock emission is most evident in the northern and northwestern shell . taking the typical brightness in the nw shell ( @xmath1620 mjy sr@xmath50 ) , and assuming the forward shock is a shell with 12% radial thickness , the projected brightness is less than 4 - 10% of the interior emission ( @xmath1640 mjy sr@xmath50 after background subtraction ) . we assume that the remaining wide - spread interior 70 @xmath0 m emission is from relatively cold , unshocked ejecta . using the `` corrected '' 70 @xmath0 m flux , the featureless spectra are equally reproduced by three models ( models e , f , and g ) in table 1 and figures [ modelespec ] and [ flessspec ] . all fits include mgsio@xmath3 , feo and si , and either aluminum oxide , fe , or a combination of the two are required at long wavelength . carbon dust can also produce featureless spectra at low temperature but we exclude this composition because of the lack of ne ( produced from carbon burning ) . aluminum oxide and fe dust are far more likely to be associated with the unshocked ejecta because they result from o - burning and si - burning , respectively and the unshocked ejecta exhibit si , s , and o+fe line emission . however , one of the key challenges in sn ejecta dust is to understand featureless dust such as fe , c , and aluminum oxide , and to link it to the associated nucleosynthetic products . we estimated the amount of freshly formed dust in cas a based on our dust model fit to each of the representative 21 @xmath0m - peak , weak-21 @xmath0 m , and featureless spectra ( fig . [ sixspec ] ) . the dust mass of @xmath41-grain type is given by : @xmath51 where @xmath52 is the flux from @xmath41-grain species , @xmath53 is the distance , @xmath54 is the planck function , @xmath55 is the bulk density , and @xmath44 is the dust particle size . by employing the scale factor @xmath40 and the dust temperature @xmath56 derived from the spectral fit , the total dust mass is given by @xmath57 , where _ @xmath58 _ is the solid angle of the source . the total mass of the 21 @xmath0m - peak dust is then determined by summing the flux of all the pixels in the 21 @xmath0m - peak dust region ( red region in fig . [ images]f ) and assuming each pixel in this region has the same dust composition as the spectrum in fig . [ 21umpeakspec ] . we took the same steps for the weak-21 @xmath0 m dust and the featureless dust . the estimated total masses for each type of dust using a distance of 3.4 kpc ( reed et al . 1995 ) are listed in table 1 . using the least massive composition in table 1 for each of the three dust classes yields a total mass of m@xmath4 ( the sum of masses from models a , d , and f ) . using the most - massive composition for each of the three dust classes yields a total mass of m@xmath4 ( the sum of masses from models b , c , and e ) . the primary uncertainty in the total dust mass between and m@xmath4 is due to the selection of the dust composition , in particular for the featureless dust . we also extracted a global spectrum of cas a , but excluding most of the exterior forward shock regions . the spectrum is well fit with the combination of our three types of dust ( including all compositions from models a - g ) , as shown in figure [ totalspec ] . we used the dust composition of models a - g as a guideline in fitting the global spectrum , because the dust features ( which were noticeable in representative spectra ) were smeared out . our goal in fitting the global spectrum is to confirm consistency between the mass derived from global spectrum and that derived from representative spectra described above . the total estimated mass from the global spectrum fit is @xmath160.028 m@xmath4 , being consistent to the mass determined from the individual fits to each dust class . the respective dust mass for each grain composition is listed in table 3 . the masses of mgsio@xmath3 , sio@xmath2 , fes and si are more than a factor of ten to hundred smaller than the predictions ; the predictions ( n03 and tf ) also have the dust features at 9 @xmath0 m for mgsio@xmath3 , 21 @xmath0 m for sio@xmath2 , and 30 - 40 @xmath0 m for fes stronger than the observed spectra if the dust mass is increased . the carbon mass is also a factor of 10 lower than the predictions . we were not able to fit the data with as much carbon dust mass as expected , even if we use the maximum carbon contribution allowed from the spectral fits . we find an estimated total freshly - formed dust mass of - m@xmath4 is required to produce the mid - infrared continuum up to 70 @xmath0 m . the dust mass we derive is orders of magnitude higher than the two previous infrared estimates of 3.5@xmath5910@xmath14 m@xmath4 and 7.7@xmath59 10@xmath60 m@xmath4 , which are derived by extrapolation from 1.6@xmath5910@xmath13 m@xmath4 ( d01 ) and 2.8@xmath59 10@xmath61 m@xmath4 ( adm ) for selected knots , respectively . one of the primary reasons for our higher mass estimate is that we include fluxes up to 70 @xmath0 m while the fits in d01 and adm accounted for dust emission only up to 30 and @xmath1640 @xmath0 m , respectively . the cold dust ( 40 - 150 k ) has much more mass than the warmer ( @xmath62150 k ) dust . in addition , our irs mapping over nearly the entire extent of cas a with higher spatial and spectral resolutions provides more accurate measurements , while d01 and adm covered only a portion of the remnant . in addition , adm use only mg protosilicate dust ; the absorption coefficient for mg protosilicate is a few times larger than those of other compositions . our dust mass estimate is also at least one order of magnitude higher than the estimate of 3@xmath59 10@xmath14 m@xmath4 by @xcite . they fitted msx and _ spitzer _ mips data with mg protosilicate . note that they used only one composition . they derived a freshly synthesized dust mass of 3@xmath5910@xmath14 m@xmath4 at a temperature of 79 - 82 k and a smaller dust mass of 5@xmath5910@xmath61 m@xmath4 at a higher temperature of 226 - 268 k , and they explained that the mass estimate depends on the chosen dust temperature . as adm mentioned , the absorption coefficient for mg protosilicate is a few times larger than those of other compositions . therefore , even including the long - wavelength data , the estimated mass was small since only mg protosilicate was modeled . with the photometry in @xcite , one could easily fit the data with only mg protosilicate and would not need additional grain compositions . however , with the accurate irs data , many dust features and the detailed continnum shape could not be fit solely with the mg protosilicate . note that the continuum shapes of weak 21 @xmath0 m dust and featureless " dust are very different from the shape of protosilicate absorption coefficient . therefore , it was necessary to include many other compositions in order to reproduce the observed irs spectra . it should be noted here that , in contrast with the previous works , we introduced si and fe bearing materials such as si , fe , fes and feo . we explain why we included such dust in our model fitting as follows . firstly , we included si and fe dust because these elements are significant outputs of nucleosynthesis ; indeed @xcite show that si and fe are primary products in the innermost layers of the ejecta . secondly , we observed strong si and fe lines in the infrared and x - ray spectra ; strong si lines were detected in the _ spitzer _ spectra , as shown in figure [ sixspec ] ( also see d01 ) , and the fe line detection at 17.9 @xmath0 m is also shown in figure [ weak21umspec ] . ( the fe maps at 17.9 @xmath0 m and at 1.64 @xmath0 m were presented in @xcite and @xcite , respectively . ) si and fe lines from ejecta are also bright in x - ray emission @xcite . thirdly , dust such as si , fe , feo and fes is predicted to form in the ejecta of population iii supernovae ( n03 ) . tf and n03 predict fe@xmath3o@xmath30 instead of feo in the uniformly mixed ejecta where the elemental composition is oxygen - rich , but the kind of iron - bearing grains in oxygen - rich layers of the ejecta is still uncertain , partly because the surface energy of iron is very sensitive to the concentration of impurities such as o and s ( as was discussed by @xcite ) , and partly because the chemical reactions at the condensation of fe - bearing dust is not well understood . depending on the elemental composition and the physical conditions in the ejecta , it is possible that fe , feo and/or fes form in the oxygen - rich layers of galactic sne . the observations of cas a favor feo dust over fe@xmath3o@xmath30 , in order to match the spectral shape of the 21 @xmath0m - peak dust and the weak-21 @xmath0 m dust . this aspect should be explored theoretically in comparison with the observations in the future . our total mass estimate is also about one order of magnitude higher than the estimate of 6.9@xmath5910@xmath14 m@xmath4 by @xcite , who used iras fluxes ( possibly confused by background cirrus ) and assumed a silicate type dust as stellar or supernova condensates being present in supernova cavity and heated up by the reverse shock . our estimated mass is much less than 1 m@xmath4 , which @xcite suggested may still be associated with the ejecta , after accounting for results of high - resolution co observations . our estimated mass of to m@xmath15 is only derived for wavelengths up to 70 @xmath0 m , so it is still possible that the total freshly - formed dust mass in cas a is higher than our estimate because there may be colder dust present . future longer - wavelength observations with herschel , scuba-2 and alma are required to determine if this is the case . also note that we did not include any mass from fast moving knots projected into the same positions as the forward shock , such as in the northeast and southwest jets , and the eastern portions of the snr outside the 21 @xmath0m - peak dust region ( see fig . [ images]e ) , because such dust could not be cleanly separated from the interstellar / circumstellar dust . we can use our dust mass estimate in conjunction with the models of n03 and tf to understand the dust observed in the early universe . if the progenitor of cas a was 15 m@xmath4 , our estimated dust mass ( - m@xmath15 ) is 718% of the 0.3 m@xmath15 predicted by the models . if the progenitor mass was 30 m@xmath4 , then the dust mass is 25% of the 1.1 m@xmath15 predicted by the models . one reason our dust mass is lower than predicted by the models is that we can not evaluate the mass of very cold dust residing in the remnant from the observered spectra up to 70 @xmath0 m as described above , unless the predicted mass is overestimated . another reason is that when and how much dust in the remnant is swept up by the reverse shock is highly dependent on the thickness of the hydrogen envelope at the time of explosion and that the evolution and destruction of dust grains formed in sne strongly depend not only on their initial sizes but also the density of ambient interstellar medium ( nozawa et al . dust formation occurs within a few hundred days after the sn explosion ( kozasa et al . 1989 ; tf ; n03 ) . without a thick hydrogen envelope , given an age for cas a of @xmath16300 years , a significant component of dust may have already been destroyed if dust grains formed in the ejecta were populated by very small - sized grains ; otherwise , it is possible that some grain types may be larger , which would increase the inferred mass . we observed most of the dust compositions predicted by sn type ii models , and the global ejecta composition is consistent with the unmixed - case n03 model than mixed - case model ; however , note that different morphologies of ar and si maps imply that some degree of mixing has occurred . our estimated dust mass with _ spitzer _ data is one order of magnitude smaller than the predicted models of dust formation in sne ejecta by n03 and tf , but one to two orders of magnitude higher than the previous estimations . we now compare the dust mass in high - redshift galaxies with the observed dust mass of cas a based on the chemical evolution model of morgan & edmunds ( 2001 ) . by a redshift of 4 , sne have been injecting dust in galaxies for over 2 billion years and there is enough dust from sne to explain the lower limit on the dust masses ( @xmath167@xmath5910@xmath63 m@xmath4 ) inferred in submm galaxies and distant quasars @xcite . it should be noted with the dust mass per sn implied by our results for cas a alone , the interpretation of dust injection from sne is limited , because the amount of dust built up over time is strongly dependent on the initial mass function , stellar evolution models and star formation rates @xcite , and destruction rates in supernova are believed to be important at timescales greater than a few billion years . additional infrared / submm observations of other young supernova remnants and supernovae are crucial to measure physical processes of dust formation in sne including the dust size distribution , composition and dependence on nucleosynthetic products and environment , and to understand the dust in the early universe in terms of dust injection from sne . we presented _ spitzer _ irs mapping covering nearly the entire extent of cas a and examined if sne are primary dust formation sites that can be used to explain the high quantity of dust observed in the early universe . the irs spectra of cas a show a few dust features such as an unique 21 @xmath0 m peak in the continuum from mg protosilicate , sio@xmath2 , and feo . we observed most of the dust compositions predicted by sn type ii dust models . however , the dust features in cas a favour mg protosilicate rather than mg@xmath2sio@xmath30 , and feo rather than fe@xmath3o@xmath30 . the composition infers that the ejecta are unmixed . our total estimated dust mass with _ spitzer _ observations ranging from 5.5 - 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peak ( a ) & a & * mg protosilicate * , * mgsio@xmath3 * , sio@xmath2 , feo , fes , si , _ al@xmath2o@xmath3 _ & ar & inner - o , s - si & 0.0030 + 21@xmath0m - peak & b & * mg protosilicate * , * mgsio@xmath3 * , feo , sio@xmath2 , feo , fes , si , _ fe _ & ar & inner - o , s - si & 0.0120 + weak-21@xmath0 m ( b ) & c & * c - glass * , * feo * , al@xmath2o@xmath3 , si , _ mg@xmath2sio@xmath30 _ & ne , si , ar ( s , o+fe ) & c - burning & 0.0180 + weak-21@xmath0 m & d & * c - glass * , * feo * , al@xmath2o@xmath3 , si , fes , _ mg protosilicate _ & ne , si , ar ( s , o+fe ) & c - burning & 0.0157 + featureless ( d ) & e & * mgsio@xmath3 * , * si * , fes , _ fe , mg@xmath2sio@xmath30 _ & si , s , ( o+fe ) & o , al burning ( fe - si - s ) & 0.0245 + featureless & f & * mgsio@xmath3 * , * si * , fes , _ fe , al@xmath2o@xmath3 _ & si , s , ( o+fe ) & o , al burning ( fe - si - s ) & 0.0171 + featureless & g & * mgsio@xmath3 * , * si * , fes , _ al@xmath2o@xmath3 , mg@xmath2sio@xmath30 _ & si , s , ( o+fe ) & o , al burning ( fe - si - s ) & 0.0009 + lllllllllllllll [ dustmasstab ] al@xmath2o@xmath3 & 6.66e-05 ( 083 ) & 0.00e+00 ( 000 ) & 5.13e-05 ( 105 ) & 1.03e-04 ( 100 ) & 0.00e+00 ( 000 ) & 8.13e-04 ( 050 ) & 6.50e-04 ( 060 ) & + c glass & 0.00e+00 ( 000 ) & 0.00e+00 ( 000 ) & 2.08e-03 ( 80/180 ) & 1.07e-03 ( 80/220 ) & 0.00e+00 ( 000 ) & 0.00e+00 ( 000 ) & 0.00e+00 ( 000 ) & + mgsio@xmath3 & 1.19e-08 ( 480 ) & 1.19e-08 ( 480 ) & 0.00e+00 ( 000 ) & 0.00e+00 ( 000 ) & 2.55e-05 ( 110 ) & 3.19e-05 ( 110 ) & 2.55e-05 ( 110 ) & + mg@xmath2sio@xmath30 & 0.00e+00 ( 000 ) & 0.00e+00 ( 000 ) & 7.89e-05 ( 120 ) & 0.00e+00 ( 000 ) & 1.72e-06 ( 130 ) & 0.00e+00 ( 000 ) & 3.00e-06 ( 130 ) & + mg protosilicate & 5.00e-05 ( 120 ) & 4.67e-05 ( 120 ) & 0.00e+00 ( 000 ) & 3.77e-05 ( 120 ) & 0.00e+00 ( 000 ) & 0.00e+00 ( 000 ) & 0.00e+00 ( 000 ) & + sio@xmath2 & 2.23e-03 ( 060/300 ) & 1.40e-03 ( 065/300 ) & 0.00e+00 ( 000 ) & 0.00e+00 ( 000 ) & 0.00e+00 ( 000 ) & 0.00e+00 ( 000 ) & 0.00e+00 ( 000 ) & + si & 4.34e-04 ( 096 ) & 4.34e-04 ( 100 ) & 1.63e-03 ( 090 ) & 8.17e-03 ( 080 ) & 9.32e-04 ( 090 ) & 1.24e-04 ( 120 ) & 6.21e-05 ( 120 ) & + fe & 0.00e+00 ( 075 ) & 9.82e-03 ( 110 ) & 0.00e+00 ( 000 ) & 0.00e+00 ( 000 ) & 2.16e-02 ( 95/135 ) & 1.36e-02 ( 100/150 ) & 0.00e+00 ( 000 ) & + feo & 1.13e-04 ( 105 ) & 2.11e-04 ( 095 ) & 1.39e-02 ( 060 ) & 5.97e-03 ( 065 ) & 0.00e+00 ( 000 ) & 0.00e+00 ( 000 ) & 0.00e+00 ( 000 ) & + fes & 1.20e-04 ( 150 ) & 2.11e-04 ( 150 ) & 0.00e+00 ( 000 ) & 3.40e-04 ( 120 ) & 1.94e-03 ( 055 ) & 2.59e-03 ( 055 ) & 1.29e-04 ( 100 ) & + llllllllll [ dustmasstab ] al@xmath2o@xmath3 & 2.40e-04 @xmath16 9.00e-03 & 8.20e-04 & 5.1300e-05 & 1.22e-04 ( 105 ) + carbon & 7.00e-02 @xmath16 3.00e-01 & 1.07e-03 & 2.0767e-03 & 2.04e-03 ( 070/265 ) + mgsio@xmath3 & 2.00e-03 @xmath16 7.00e-3 & 2.55e-05 & 2.5500e-05 & 1.65e-04 ( 110 ) + mg@xmath2sio@xmath30 & 3.70e-02 @xmath16 4.40e-1 & 3.00e-06 & 8.0620e-05 & 3.21e-05 ( 120 ) + mg protosilicate & none & 8.77e-05 & 4.6710e-05 & 6.70e-05 ( 110 ) + sio@xmath2 & 2.50e-02 @xmath16 1.400e-01 & 2.23e-03 & 1.3964e-03 & 1.35e-03 ( 065 ) + si & 7.00e-02 @xmath16 3.00e-01 & 8.66e-03 & 2.9989e-03 & 4.42e-03 ( 080 ) + fe & 2.00e-02 @xmath16 4.00e-02 & 0.00e+00 & 3.1459e-02 & 1.03e-02 ( 090 ) + feo & none & 6.08e-03 & 1.4136e-02 & 6.23e-03 ( 070 ) + fes & 4.00e-02 @xmath16 1.10e-01 & 5.90e-04 & 2.1501e-03 & 2.90e-03 ( 090 ) +

we performed _ spitzer _ infrared spectrograph mapping observations covering nearly the entire extent of the cassiopeia a supernova remnant ( snr ) , producing mid - infrared ( 5.5 - 35 @xmath0 m ) spectra every 5@xmath1 - 10@xmath1 . gas lines of ar , ne , o , si , s and fe , and dust continua were strong for most positions . we identify three distinct ejecta dust populations based on their continuum shapes . the dominant dust continuum shape exhibits a strong peak at 21 @xmath0 m . a line - free map of 21 @xmath0m - peak dust made from the 19 - 23 @xmath0 m range closely resembles the [ ar ii ] , [ o iv ] , and [ ne ii ] ejecta - line maps implying that dust is freshly formed in the ejecta . spectral fitting implies the presence of sio@xmath2 , mg protosilicates , and feo grains in these regions . the second dust type exhibits a rising continuum up to 21 @xmath0 m and then flattens thereafter . this `` weak 21 @xmath0 m '' dust is likely composed of al@xmath2o@xmath3 and c grains . the third dust continuum shape is featureless with a gently rising spectrum and is likely composed of mgsio@xmath3 and either al@xmath2o@xmath3 or fe grains . using the least massive composition for each of the three dust classes yields a total mass of m@xmath4 . using the most - massive composition yields a total mass of m@xmath4 . the primary uncertainty in the total dust mass stems from the selection of the dust composition necessary for fitting the featureless dust as well as 70 @xmath0 m flux . the freshly formed dust mass derived from cas a is sufficient from sne to explain the lower limit on the dust masses in high redshift galaxies .

0709.2880

the impossibility of superluminal communication through the use of quantum entanglement has already been vividly discussed in the past , see for example @xcite . recently this topic has re - entered the stage of present research in the context of quantum cloning : the no - signalling constraint has been used to derive upper bounds for the fidelity of cloning transformations @xcite . as the connection between approximate cloning and no - signalling is still widely debated , we aim at clarifying in this paper the quantum mechanical principles that forbid superluminal communication , and at answering the question whether they are the same principles that set limits to quantum cloning . our scenario throughout the paper for the attempt to transmit information with superluminal speed is the well - known entanglement - based communication scheme@xcite . the idea is the following : two space - like separated parties , say alice and bob , share an entangled state of a pair of two - dimensional quantum systems ( qubits ) , for example the singlet state @xmath0 . alice encodes a bit of information by choosing between two possible orthogonal measurement bases for her qubit and performing the corresponding measurement . by the reduction postulate , the qubit at bob s side collapses into a pure state depending on the result of the measurement performed by alice . if a perfect cloning machine were available , bob could now generate an infinite number of copies of his state , and therefore would be able to determine his state with perfect accuracy , thus knowing what basis alice decided to use . in this way , transfer of information between alice and bob would be possible . in particular , if they are space - like separated , information could be transmitted with superluminal speed . the same transfer of information could evidently also be obtained if it were possible to determine the state of a single quantum system with perfect accuracy , which is also impossible @xcite . one might ask the question whether approximate cloning allows superluminal communication @xcite : with imperfect cloning bob can produce a number of imperfect copies , and thus get some information about his state . but this information is never enough to learn alice s direction of measurement . this has been shown in ref . @xcite for a specific example . more generally , as we will show in this paper , the reason is that _ no _ local linear transformation can lead to transmission of information through entanglement , but any cloning operation consistent with quantum mechanics has to be linear . the fact that non - locality of quantum entanglement can not be used for superluminal communication , has been phrased as `` peaceful coexistence '' @xcite between quantum mechanics and relativity , a much - cited expression . here we emphasize that this consistency is not a coincidence , but a simple consequence of linearity and completeness of quantum mechanics . our arguments go beyond previous work @xcite , as we consider the most general evolution on alice s and bob s side in the form of local maps . recently , this consistency has been exploited in order to devise new methods to derive bounds or constraints for quantum mechanical transformations @xcite . however , in this paper we will show that the principles underlying the impossibility of 1 ) superluminal signalling and 2 ) quantum cloning beyond the optimal bound allowed by quantum mechanics @xcite , are not the same . in particular , the impossibility of information transfer by means of quantum entanglement is due only to linearity and preservation of trace of local operations . in this section we want to show how the impossibility of superluminal communication arises by assuming only completeness and linearity of local maps on density operators . we consider the most general scenario where alice and bob share a global quantum state @xmath1 of two particles and are allowed to perform any local map , which we denote here with @xmath2 and @xmath3 , respectively . the local map can be any local transformation , including a measurement averaged over all possible outcomes ( which , in fact , can not be known by the communication partner ) . alice can choose among different local maps in order to encode the message `` @xmath4 '' that she wishes to transmit , namely she encodes it by performing the transformation @xmath5 on her particle . bob can perform a local transformation @xmath6 on his particle ( e.g. cloning ) and then a local measurement @xmath7 to decode the message ( @xmath8 is a povm @xcite ) . the impossibility of superluminal communication in the particular case where bob performs only a measurement has been demonstrated in ref . @xcite . here we follow a more general approach , discussing the roles of `` completeness '' and linearity of any local map involved . by `` completeness '' of a map we mean that the trace is preserved under its action , namely @xmath9\equiv { \mbox{tr}}[\rho_a]\ ] ] for any @xmath10 @xcite . linearity of the map on trace - class operators of the form @xmath11 , allows to extend the completeness condition to the whole hilbert space , namely @xmath12\equiv { \mbox{tr}}[\rho_{ab}]\;,\ ] ] and analogously for the partial trace @xmath13\equiv { \mbox{tr}}_a[\rho_{ab } ] \label{part}\;,\ ] ] on bob s side , only linearity without completeness is needed for the local map , leading to the equality @xmath14= { \mbox{$\cal b$}}\,{\mbox{tr}}_a[{\mbox{$\cal a$}}\otimes{\mbox{$1 \hspace{-1.0 mm } { \bf l}$}}(\rho_{ab})]\;.\label{gcomp}\ ] ] as we will show in the following , the above equations are the fundamental ingredients and the only requirements for local maps to prove the impossibility of superluminal communication . we will now compute the conditional probability @xmath15 that bob records the result @xmath16 when the message @xmath4 was encoded by alice : @xmath17\;.\ ] ] by exploiting eqs . ( [ gcomp ] ) and ( [ part ] ) we have @xmath18 ) ] \nonumber \\&= & { \mbox{tr}}_b[\pi_r\,{\mbox{$\cal b$}}\ , ( { \mbox{tr}}_a[\rho_{ab}])]\equiv p(r ) \;.\label{gcomp2}\end{aligned}\ ] ] the conditional probability is therefore independent of the local operation @xmath19 that alice performed on her particle , and therefore the amount of transmitted information vanishes . note that the speed of transmission does not enter in any way , i.e. _ any _ transmission of information is forbidden @xcite , in particular superluminal transmission . we want to stress that this result holds for all possible linear local operations that alice and bob can perform , and also for any joint state @xmath1 . in particular , it holds for any kind of linear cloning transformation performed at bob s side ( notice that ideal cloning is a non - linear map ) . notice also that any operation that is physically realizable in standard quantum mechanics ( completely positive map ) is linear and complete , and therefore it does not allow superluminal communication . we also emphasize here that the `` peaceful coexistence '' between quantum mechanics and relativity is automatically guaranteed by the linearity and completeness of any quantum mechanical process . actually , as shown in the diagram [ maps ] , the set of local quantum mechanical maps is just a subset of the local maps that do not allow superluminal communication . .5truecm = .8 .5truecm in the next section we will show how superluminal communication could be achieved if one would give up the linearity requirement for the local maps , by discussing some explicit examples . our examples are based on the scenario where alice and bob share an entangled state of two qubits and alice performs a projection measurement with her basis oriented along the direction @xmath20 . the final state of bob , who does not know the result of the measurement , is given by @xmath21 where @xmath22 denote the probabilities that alice finds her qubit oriented as @xmath23 , and @xmath24 are the corresponding final density operators at bob s side after he performed his local transformation . notice that the evolved state of bob , as in the following examples , can be a joint state of a composite system with more than one qubit . if the information is encoded in the choice of two possible different orientations @xmath25 and @xmath26 of the measurement basis , the impossibility of superluminal communication corresponds to the condition @xmath27 for all choices of @xmath28 and @xmath29 . in the following section we give some explicit examples of local maps on bob s side . notice that we will intentionally leave the ground of quantum mechanics ( an explicit example of a superluminal communication scheme based on the use of non - linear evolutions is also given in ref . @xcite ) . \(1 ) _ example of a linear , non - positive @xmath30 cloning transformation which does not allow superluminal communication : _ the evolved state at bob s side after his transformation is a state of two qubits given by @xmath31 \label{rhoout1}\end{aligned}\ ] ] where @xmath32 is the bloch vector which is cloned and @xmath33 is the shrinking factor . the above map is non - positive for @xmath34 @xcite . this is the case , for instance , for @xmath35 and @xmath36 . such a transformation violates the upper bound of the @xmath30 universal quantum cloner @xcite but , as this is a linear transformation , eq . ( [ gcomp2 ] ) holds . therefore the cloning is `` better '' than the optimal one , and the no - signalling condition ( [ nsc ] ) is still fulfilled . this means that we can go beyond the laws of quantum mechanics ( complete positivity ) without necessarily creating the possibility of superluminal communication . \(2 ) _ example of non - linear , positive or non - positive @xmath30 cloning transformation which does allow superluminal communication : _ consider bob s transformation @xmath37\ , \label{rhoout2}\end{aligned}\ ] ] where @xmath38 denotes a function of the component @xmath39 of the bloch vector , which is such that this map acts non - linearly on a convex combination of density matrices . for odd functions , namely @xmath40 one does not violate the no - signalling condition for a maximally entangled state because taking @xmath41 it follows that @xmath42 does not depend on @xmath43 , whereas for even non - constant functions one does . however , for odd functions the no - signalling condition is in general violated for partially entangled pure states , i.e. @xmath44 in eq . ( [ final ] ) . it is interesting to see that in this non - physical case superluminal communication is achieved when sharing less than maximal entanglement . depending on the value of the parameter @xmath45 this map can be positive or non - positive . examples of non - positive maps can for instance be found by violating the condition @xmath46 ( compare with previous example ) . \(3 ) _ example of a non - linear , positive @xmath47 cloning transformation which does allow superluminal communication : _ consider @xmath48 where @xmath49 is orthogonal to @xmath50 . the no - signalling condition ( [ nsc ] ) for two different choices of basis @xmath51 and @xmath52 with equiprobable outcomes is violated because @xmath53 which holds for any value @xmath54 . it is then possible to devise a measurement procedure that distinguishes between the left and right hand side of eq . ( [ dist ] ) , thus allowing to transmit information faster than light . in order to illustrate this we give an explicit example with @xmath55 . let us denote the right hand side of equation ( [ nonlinear ] ) as @xmath56 . we choose @xmath57 and @xmath58 and a povm measurement on the clones given by the operators @xmath59 and @xmath60 , which are the projectors over the subspaces spanned by @xmath61 and @xmath62 , respectively . with this measurement the probabilities for outcome 0 and 1 depend on alice s choice of measurement basis . we denote as @xmath63 the probability that bob finds outcome 0 , if alice measured in the basis @xmath64 , and arrive at @xmath65=0 \;,\nonumber \\ p(1|\psi)&=&1-p(0|\psi)=1 \;.\end{aligned}\ ] ] analogously , for the other choice of alice s basis one has @xmath66=\frac{1}{2 } \;,\nonumber \\ p(1|\phi)&=&1-p(0|\phi)=\frac{1}{2 } \;.\end{aligned}\ ] ] therefore , we can distinguish between the two different choices of bases . note that , when giving up the constraint of linearity , one could send signals superluminally even for fidelities smaller than those of optimal quantum cloning . similar arguments hold for the transformation @xmath67 we have shown that the `` peaceful coexistence '' between quantum mechanics and relativity is automatically guaranteed by the linearity and completeness ( i.e. trace - preserving property ) of any quantum mechanical process : hence , any approximate optimal quantum cloning , as a particular case of a linear trace - preserving map , can not lead to signalling . for the sake of illustration , in figure [ maps ] we summarize the set of local maps . this set is divided into linear and non - linear maps . any linear trace - preserving map forbids superluminal signalling . reversely , the no - signalling condition implies only linearity , as shown in refs.@xcite and @xcite . the positive maps contain the linear maps allowed by quantum mechanics ( qm ) , namely the completely positive trace - preserving maps . both trace - preservation and positivity crucial for quantum mechanics are not implied by the no - signalling constraint . in particular , positivity seems to be unrelated with no - signalling . hence , there is room for maps that go beyond quantum mechanics , but still preserve the constraint of no - superluminal signalling , and example 1 ) above shows that this is the case . from what we have seen we can conclude that any bound on a cloning fidelity can not be derived from the no - signalling constraint alone , but only in connection with other quantum mechanical principles : example 3 ) shows how the cloning fidelity is unrelated to the no - signalling condition . quantum mechanics as a complete theory , however , naturally guarantees no - signalling , and obviously gives the correct known upper bounds on quantum cloning . we thank c. fuchs , g. c. ghirardi , l. hardy and a. peres for fruitful discussions . db acknowledges support by the esf programme qit , and from deutsche forschungsgemeinschaft under sfb 407 and schwerpunkt qiv . the theoretical quantum optics group of pavia acknowledges the european network equip and cofinanziamento 1999 `` quantum information transmission and processing : quantum teleportation and error correction '' for partial support . 99 g. ghirardi , a. rimini , and t. weber , lett . nuovo cimento * 27 * , 293 ( 1980 ) . n. herbert , found . * 12 * , 1171 ( 1982 ) . w. k. wootters , w. h. zurek , nature * 299 * , 802 ( 1982 ) . g. ghirardi , r. grassi , a. rimini and t. weber , europhys . lett . * 6 * , 95 ( 1988 ) . h. scherer and p. busch , phys . a * 47 * , 1647 ( 1993 ) . g. svetlichny , found . * 28 * , 131 ( 1998 ) . a. peres , phys . a * 61 * , 022117 ( 2000 ) . n. gisin , phys . a * 242 * , 1 ( 1998 ) . l. hardy and d.d . song , phys . a * 259 * , 331 ( 1999 ) . s. ghosh , g. car and a. roy , phys . a * 261 * , 17 ( 1999 ) . pati , quant - 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preserving since generally alice s map is a measurement , and summing over all possible outcomes ( i.e. for the completeness of the measurement ) leads to a linear trace - preserving map ( non linear state reduction maps are always linear on average ) . a. peres , private communication . n. gisin , helv . phys . acta * 62 * , 363 ( 1989 ) . n. gisin , phys . lett . a * 143 * , 1 ( 1990 ) . v. buek and m. hillery , phys . a * 54 * , 1844 ( 1996 ) .

we show that non - locality of quantum mechanics can not lead to superluminal transmission of information , even if most general local operations are allowed , as long as they are linear and trace preserving . in particular , any quantum mechanical approximate cloning transformation does not allow signalling . on the other hand , the no - signalling constraint on its own is not sufficient to prevent a transformation from surpassing the known cloning bounds . we illustrate these concepts on the basis of some examples .

quant-ph0010070

one of the most important motivations of these series of conferences is to promote vigorous interaction between statisticians and astronomers . the organizers merit our admiration for bringing together such a stellar cast of colleagues from both fields . in this third edition , one of the central subjects is cosmology , and in particular , statistical analysis of the large - scale structure in the universe . there is a reason for that the rapid increase of the amount and quality of the available observational data on the galaxy distribution ( also on clusters of galaxies and quasars ) and on the temperature fluctuations of the microwave background radiation . these are the two fossils of the early universe on which cosmology , a science driven by observations , relies . here we will focus on one of them the galaxy distribution . first we briefly review the redshift surveys , how they are built and how to extract statistically analyzable samples from them , considering selection effects and biases . most of the statistical analysis of the galaxy distribution are based on second order methods ( correlation functions and power spectra ) . we comment them , providing the connection between statistics and estimators used in cosmology and in spatial statistics . special attention is devoted to the analysis of clustering in fourier space , with new techniques for estimating the power spectrum , which are becoming increasingly popular in cosmology . we show also the results of applying these second - order methods to recent galaxy redshift surveys . fractal analysis has become very popular as a consequence of the scale - invariance of the galaxy distribution at small scales , reflected in the power - law shape of the two - point correlation function . we discuss here some of these methods and the results of their application to the observations , supporting a gradual transition from a small - scale fractal regime to large - scale homogeneity . the concept of lacunarity is illustrated with some detail . we end by briefly reviewing some of the alternative measures of point statistics and structure functions applied thus far to the galaxy distribution : void probability functions , counts - in - cells , nearest neighbor distances , genus , and minkowski functionals . cosmological datasets differ in several respects from those usually studied in spatial statistics . the point sets in cosmology ( galaxy and cluster surveys ) bear the imprint of the observational methods used to obtain them . the main difference is the systematically variable intensity ( mean density ) of cosmological surveys . these surveys are usually magnitude - limited , meaning that all objects , which are brighter than a pre - determined limit , are observed in a selected region of the sky . this limit is mainly determined by the telescope and other instruments used for the program . apparent magnitude , used to describe the limit , is a logarithmic measure of the observed radiation flux . it is usually assumed that galaxies at all distances have the same ( universal ) luminosity distribution function . this assumption has been tested and found to be in satisfying accordance with observations . as the observed flux from a galaxy is inversely proportional to the square of its distance , we can see at larger distances only a bright fraction of all galaxies . this leads directly to the mean density of galaxies that depends on their distance from us @xmath0 . one can also select a distance limit , find the minimum luminosity of a galaxy , which can yet be seen at that distance , and ignore all galaxies that are less luminous . such samples are called volume - limited . they are used for some special studies ( typically for counts - in - cells ) , but the loss of hard - earned information is enormous . the number of galaxies in volume - limited samples is several times smaller than in the parent magnitude - limited samples . this will also increase the shot ( discreteness ) noise . in addition to the radial selection function @xmath1 , galaxy samples also are frequently subject to angular selection . this is due to our position in the galaxy we are located in a dusty plane of the galaxy , and the window in which we see the universe , also is dusty . this dust absorbs part of galaxies light , and makes the real brightness limit of a survey dependent on the amount of dust in a particular line - of - sight . this effect has been described by a @xmath2 law ( @xmath3 is the galactic latitude ) ; in reality the dust absorption in the galaxy is rather inhomogeneous . there are good maps of the amount of galactic dust in the sky , the latest maps have been obtained using the cobe and iras satellite data @xcite . edge problems , which usually affect estimators in spatial statistics , also are different for cosmological samples . the decrease of the mean density towards the sample borders alleviates these problems . of course , if we select a volume - limited sample , we select also all these troubles ( and larger shot noise ) . from the other side , edge effects are made more prominent by the usual observing strategies , when surveys are conducted in well - defined regions in the sky . thus , edge problems are only partly alleviated ; maybe it will pay to taper our samples at the side borders , too ? some of the cosmological surveys have naturally soft borders . these are the all - sky surveys ; the best known is the iras infrared survey , dust is almost transparent in infrared light . the corresponding redshift survey is the pscz survey , which covers about 85% of the sky @xcite . a special follow - up survey is in progress to fill in the remaining galactic zone - of - avoidance region , and meanwhile numerical methods have been developed to interpolate the structures seen in the survey into the gap @xcite . another peculiarity of galaxy surveys is that we can measure exactly only the direction to the galaxy ( its position in the sky ) , but not its distance . we measure the radial velocity @xmath4 ( or redshift @xmath5 , @xmath6 is the velocity of light ) of a galaxy , which is a sum of the hubble expansion , proportional to the distance @xmath7 , and the dynamical velocity @xmath8 of the galaxy , @xmath9 . thus we are differentiating between redshift space , if the distances simply are determined as @xmath10 , and real space . the real space positions of galaxies could be calculated if we exactly knew the peculiar velocities of galaxies ; we do not . the velocity distortions can be severe ; well - known features of redshift space are fingers - of - god , elongated structures that are caused by a large radial velocity dispersion in massive clusters of galaxies . the velocity distortions expand a cluster in redshift space in the radial direction five - ten times . for large - scale structures the situation is different , redshift distortions compress them . this is due to the continuing gravitational growth of structures . these differences can best be seen by comparing the results of numerical simulations , where we know also the real - space situation , in redshift space and in real space . the last specific feature of the cosmology datasets is their size . up to recent years most of the datasets have been rather small , of the order of @xmath11 objects ; exceptions exist , but these are recent . such a small number of points gives a very sparse coverage of three - dimensional survey volumes , and shot noise has been a severe problem . this situation is about to change , swinging to the other extreme ; the membership of new redshift surveys already is measured in terms of @xmath12 ( 160,000 for the 2df survey , quarter of a million planned ) and million - galaxy surveys are on their way ( the sloan survey ) . more information about these surveys can be found in their web pages : _ http:/-2pt / www.mso.anu.edu.au/2dfgrs/ _ for the 2df survey and _ http:/-2pt / www.sdss.org/ _ for the sloan survey . this huge amount of data will force us to change the statistical methods we use . nevertheless , the deepest surveys ( e.g. , distant galaxy cluster surveys ) will always be sparse , so discovering small signals from shot - noise dominated data will remain a necessary art . there are several related quantities that are second - order characteristics used to quantify clustering of the galaxy distribution in real or redshift space . the most popular one in cosmology is the two - point correlation function , @xmath13 . the infinitesimal interpretation of this quantity reads as follows : @xmath14dv_1 dv_2\ ] ] is the joint probability that in each one of the two infinitesimal volumes @xmath15 and @xmath16 , with separation vector @xmath17 , lies a galaxy . here @xmath18 is the mean number density ( intensity ) . assuming that the galaxy distribution is a homogeneous ( invariant under translations ) and isotropic ( invariant under rotations ) point process , this probability depends only on @xmath19 . in spatial statistics , other functions related with @xmath20 are commonly used : @xmath21 where @xmath22 is the second - order intensity function , @xmath23 is the pair correlation function , also called the radial distribution function or structure function , and @xmath24 is the conditional density proposed by . different estimators of @xmath20 have been proposed so far in the literature , both in cosmology and in spatial statistics . the main differences are in correction for edge effects . comparison of their performance can be found in several papers @xcite . there is clear evidence that @xmath20 is well described by a power - law at scales @xmath25 mpc where @xmath26 is the hubble constant in units of 100 km s@xmath27mpc@xmath28 : @xmath29 with @xmath30 and @xmath31 mpc . this scaling behavior is one of the reasons that have lead some astronomers to describe the galaxy distribution as fractal . a power - law fit for @xmath32 permits to define the correlation dimension @xmath33 . the extent of the fractal regime is still a matter of debate in cosmology , but it seems clear that the available data on redshift surveys indicate a gradual transition to homogeneity for scales larger than 1520 @xmath34 mpc @xcite . moreover , in a fractal point distribution , the correlation length @xmath35 increases with the radius of the sample because the mean density decreases @xcite . this simple prediction of the fractal interpretation is not supported by the data , instead @xmath35 remains constant for volume - limited samples with increasing depth @xcite . several versions of the volume integral of the correlation function are also frequently used in the analysis of galaxy clustering . the most extended one in spatial statistics is the so - called ripley @xmath36-function @xmath37 although in cosmology it is more frequent to use an expression which provides directly the average number of neighbors an arbitrarily chosen galaxy has within a distance @xmath0 , @xmath38 or the average conditional density @xmath39 again a whole collection of estimators are used to properly evaluate these quantities . pietronero and coworkers recommend to use only minus estimators to avoid any assumption regarding the homogeneity of the process . in these estimators , averages of the number of neighbors within a given distance are taken only considering as centers these galaxies whose distances to the border are larger than @xmath0 . however , caution has to be exercised with this procedure , because at large scales only a small number of centers remain , and thus the variance of the estimator increases . integral quantities are less noisy than the corresponding differential expressions , but obviously they do contain less information on the clustering process due the fact that values of @xmath40 and @xmath41 for two different scales @xmath42 and @xmath43 are more strongly correlated than values of @xmath44 and @xmath45 . scaling of @xmath46 provides a smoother estimation of the correlation dimension . if scaling is detected for partition sums defined by the moments of order @xmath47 of the number of neighbors @xmath48 the exponents @xmath49 are the so - called generalized or multifractal dimensions @xcite . note that for @xmath50 , @xmath51 is an estimator of @xmath52 and therefore @xmath49 for @xmath50 is simply the correlation dimension . if different kinds of cosmic objects are identified as peaks of the continuous matter density field at different thresholds , we can study the correlation dimension associated to each kind of object . the multiscaling approach @xcite associated to the multifractal formalism provides a unified framework to analyze this variation . it has been shown @xcite that the value of @xmath33 corresponding to rich galaxy clusters ( high peaks of the density field ) is smaller than the value corresponding to galaxies ( within the same scale range ) as prescribed in the multiscaling approach . finally we want to consider the role of lacunarity in the description of the galaxy clustering @xcite . in fig . [ lacun ] , we show the space distribution of galaxies within one slice of the las campanas redshift survey , together with a fractal pattern generated by means of a rayleigh - lvy flight @xcite . both have the same mass - radius dimension , defined as the exponent of the power - law that fits the variation of mass within concentric spheres centered at the observer position . @xmath53 the best fitted value for both point distributions is @xmath54 as shown in the left bottom panel of fig . [ lacun ] . the different appearance of both point distributions is a consequence of the different degree of lacunarity . have proposed to quantify this effect by measuring the variability of the prefactor @xmath55 in eq . [ mrr ] , @xmath56 the result of applying this lacunarity measure is shown in the right bottom panel of fig . [ lacun ] . the visual differences between the point distributions are now well reflected in this curve . @xmath57 curves in the lower left panel , but the lacunarity curves ( in the lower right panel ) differ considerably . the solid lines describe the galaxy distribution , dotted lines the model results . from . ] the current statistical model for the main cosmological fields ( density , velocity , gravitational potential ) is the gaussian random field . this field is determined either by its correlation function or by its spectral density , and one of the main goals of spatial statistics in cosmology is to estimate those two functions . in recent years the power spectrum has attracted more attention than the correlation function . there are at least two reasons for that the power spectrum is more intuitive physically , separating processes on different scales , and the model predictions are made in terms of power spectra . statistically , the advantage is that the power spectrum amplitudes for different wavenumbers are statistically orthogonal : @xmath58 here @xmath59 is the fourier amplitude of the overdensity field @xmath60 at a wavenumber @xmath61 , @xmath62 is the matter density , a star denotes complex conjugation , @xmath63 denotes expectation values over realizations of the random field , and @xmath64 is the three - dimensional dirac delta function . the power spectrum @xmath65 is the fourier transform of the correlation function @xmath66 of the field . estimation of power spectra from observations is a rather difficult task . up to now the problem has been in the scarcity of data ; in the near future there will be the opposite problem of managing huge data sets . the development of statistical techniques here has been motivated largely by the analysis of cmb power spectra , where better data were obtained first , and has been parallel to that recently . the observed samples can be modeled by an inhomogeneous point process ( a gaussian cox process ) of number density @xmath67 : @xmath68 where @xmath64 is the dirac delta - function . as galaxy samples frequently have systematic density trends caused by selection effects , we have to write the estimator of the density contrast in a sample as @xmath69 where @xmath70 is the selection function expressed in the number density of objects . the estimator for a fourier amplitude ( for a finite set of frequencies @xmath71 ) is @xmath72 where @xmath73 is a weight function that can be selected at will . the raw estimator for the spectrum is @xmath74 and its expectation value @xmath75 where @xmath76 is the window function that also depends on the geometry of the sample volume . symbolically , we can get the estimate of the power spectra @xmath77 by inverting the integral equation @xmath78 where @xmath79 denotes convolution , @xmath80 is the raw estimate of power , and @xmath81 is the ( constant ) shot noise term . in general , we have to deconvolve the noise - corrected raw power to get the estimate of the power spectrum . this introduces correlations in the estimated amplitudes , so these are not statistically orthogonal any more . a sample of a characteristic spatial size @xmath82 creates a window function of width of @xmath83 , correlating estimates of spectra at that wavenumber interval . as the cosmological spectra are usually assumed to be isotropic , the standard method to estimate the spectrum involves an additional step of averaging the estimates @xmath84 over a spherical shell @xmath85 $ ] of thickness @xmath86 in wavenumber space . the minimum - variance requirement gives the fkp @xcite weight function : @xmath87 and the variance is @xmath88 where @xmath89 is the number of coherence volumes in the shell . the number of independent volumes is twice as small ( the density field is real ) . the coherence volume is @xmath90 . as the data sets get large , straight application of direct methods ( especially the error analysis ) becomes difficult . there are different recipes that have been developed with the future data sets in mind . a good review of these methods is given in . the deeper the galaxy sample , the smaller the coherence volume , the larger the spectral resolution and the larger the wavenumber interval where the power spectrum can be estimated . the deepest redshift surveys presently available are the pscz galaxy redshift survey ( 15411 redshifts up to about @xmath91mpc , see ) , the abell / aco rich galaxy cluster survey , 637 redshifts up to about 300@xmath92mpc @xcite ) , and the ongoing 2df galaxy redshift survey ( 141400 redshifts up to @xmath93mpc @xcite ) . the estimates of power spectra for the two latter samples have been obtained by the direct method @xcite . [ 2dfpower ] shows the power spectrum for the 2df survey . the covariance matrix of the power spectrum estimates in fig . [ 2dfpower ] was found from simulations of a matching gaussian cox process in the sample volume . the main new feature in the spectra , obtained for the new deep samples , is the emergence of details ( wiggles ) in the power spectrum . while sometime ago the main problem was to estimate the mean behaviour of the spectrum and to find its maximum , now the data enables us to see and study the details of the spectrum . these details have been interpreted as traces of acoustic oscillations in the post - recombination power spectrum . similar oscillations are predicted for the cosmic microwave background radiation fluctuation spectrum . the cmb wiggles match the theory rather well , but the galaxy wiggles do not , yet . the probability that a randomly placed sphere of radius @xmath0 contains exactly @xmath81 galaxies is denoted by @xmath94 . in particular , for @xmath95 , @xmath96 is the so - called void probability function , related with the empty space function or contact distribution function @xmath97 , more frequently used in the field of spatial statistics , by @xmath98 . the moments of the counts - in - cells probabilities can be related both with the multifractal analysis @xcite and with the higher order @xmath99-point correlation functions @xcite . in spatial statistics , different quantities based on distances to nearest neighbors have been introduced to describe the statistical properties of point processes . @xmath100 is the distribution function of the distance @xmath0 of a given point to its nearest neighbor . it is interesting to note that @xmath97 is just the distribution function of the distance @xmath0 from an arbitrarily chosen point in @xmath101 not being an event of the point process to a point of the point process ( a galaxy in the sample in our case ) . the quotient @xmath102 introduced by is a powerful tool to analyze point patterns and has discriminative power to compare the results of @xmath81-body models for structure formation with the real distribution of galaxies @xcite . one very popular tool for analysis of the galaxy distribution is the genus of the isodensity surfaces . to define this quantity , the point process is smoothed to obtain a continuous density field , the intensity function , by means of a kernel estimator for a given bandwidth . then we consider the fraction of the volume @xmath103 which encompasses those regions having density exceeding a given threshold @xmath104 . the boundary of these regions specifies an isodensity surface . the genus @xmath105 of a surface @xmath106 is basically the number of holes minus the number of isolated regions plus 1 . the genus curve shows the variation of @xmath105 with @xmath103 or @xmath104 for a given window radius of the kernel function . an analytical expression for this curve is known for gaussian density fields . it seems that the empirical curve calculated from the galaxy catalogs can be reasonably well fitted to a gaussian genus curve @xcite for window radii varying within a large range of scales . a very elegant generalization of the previous analysis to a larger family of morphological characteristics of the point processes is provided by the minkowski functionals . these scalar quantities are useful to study the shape and connectivity of a union of convex bodies . they are well known in spatial statistics and have been introduced in cosmology by . on a clustered point process , minkowski functionals are calculated by generalizing the boolean grain model into the so - called germ - grain model . this coverage process consists in considering the sets @xmath107 for the diagnostic parameter @xmath0 , where @xmath108 represents the galaxy positions and @xmath109 is a ball of radius @xmath0 centered at point @xmath110 . minkowski functionals are applied to sets @xmath111 when @xmath0 varies . in @xmath101 there are four functionals : the volume @xmath112 , the surface area @xmath113 , the integral mean curvature @xmath114 , and the euler - poincar characteristic @xmath115 , related with the genus of the boundary of @xmath111 by @xmath116 . application of minkowski functionals to the galaxy cluster distribution can be found in . these quantities have been used also as efficient shape finders by . this work was supported by the spanish mcyt project aya2000 - 2045 and by the estonian science foundation under grant 2882 . enn saar is grateful for the invited professor position funded by the vicerrectorado de investigacin de la universitat de valncia . peacock j a , cole s , norberg p , baugh c m , bland - hawthorn j , bridges t , cannon r d , colless m , collins c , couch w , dalton g , deeley k , propris r d , driver s p , efstathiou g , ellis r s , frenk c s , glazebrook k , jackson c , lahav o , lewis i , lumsden s , maddox s , percival w j , peterson b a , price i , sutherland w taylor k 2001 _ nature _ * 410 * , 169173 . percival w j , baugh c m , bland - hawthorn j , bridges t , cannon r , cole s , colless m , collins c , couch w , dalton g , propris r d , driver s p , efstathiou g , ellis r s , frenk c s , glazebrook k , jackson c , lahav o , lewis i , lumsden s , maddox s , moody s , norberg p , peacock j a , peterson b a , sutherland w taylor k 2001 . astro - ph/0105252 , submitted to mon . not . soc . saunders w ballinger b e 2000 _ in _ r. c kraan - korteweg , p. a henning h andernach , eds , ` the hidden universe , asp conference series ' astronomical society of the pacific , san francisco . astro - ph/0005606 , in press . saunders w , sutherland w j , maddox s j , keeble o , oliver s j , rowan - robinson m , mcmahon r g , efstathiou g p , tadros h , white s d m , frenk c s , carramiana a hawkins m r s 2000 _ mon . not . soc . _ * 317 * , 5564 . schmoldt i m , saar v , saha p , branchini e , efstathiou g p , frenk c s , keeble o , maddox s , mcmahon r , oliver s , rowan - robinson m , saunders w , sutherland w j , tadros h white s d m 1999 _ astron . j. _ * 118 * , 11461160 .

in this introductory talk we will establish connections between the statistical analysis of galaxy clustering in cosmology and recent work in mainstream spatial statistics . the lecture will review the methods of spatial statistics used by both sets of scholars , having in mind the cross - fertilizing purpose of the meeting series . special topics will be : description of the galaxy samples , selection effects and biases , correlation functions , nearest neighbor distances , void probability functions , fourier analysis , and structure statistics .

astro-ph0203251

the identification of the higgs boson(s ) is one of the main goals of the large hadron collider ( lhc ) being built at cern . there are expectations that there exists a ` light ' higgs boson with mass @xmath3 gev . in this mass range , its detection at the lhc will be challenging . there is no obvious perfect detection process , but rather a range of possibilities , none of which is compelling on its own . some of the processes are listed in table 1 , together with the predicted event rates for the integrated luminosity of 30 fb@xmath4 expected over the first two or three year period of lhc running . we see that , _ either _ large signals are accompanied by a huge background , _ or _ the processes have comparable signal and background rates for which the number of higgs events is rather small . here we wish to draw particular attention to process ( c ) , which is often disregarded ; that is the exclusive signal @xmath5 , where the + sign indicates the presence of a rapidity gap . it is possible to install proton taggers so that the ` missing mass ' can be measured to an accuracy @xmath6 gev @xcite . then the exclusive process will allow the mass of the higgs to be measured in two independent ways . first the tagged protons give @xmath7 and second , via the @xmath1 decay , we have @xmath8 , although now the resolution is much poorer with @xmath9 gev . the existence of matching peaks , centered about @xmath10 , is a unique feature of the exclusive diffractive higgs signal . besides its obvious value in identifying the higgs , the mass equality also plays a key role in reducing background contributions . another advantage of the exclusive process @xmath11 , with @xmath1 , is that the leading order @xmath12 background subprocess is suppressed by a @xmath13 selection rule @xcite . [ cols="<,^,^,^,^ " , ] the radiation associated with the @xmath14 hard subprocess is not the only way to populate and to destroy the rapidity gaps . there is also the possibility of soft rescattering in which particles from the underlying event populate the gaps . the probability , @xmath15 , that the gaps survive the soft rescattering was calculated using a two - channel eikonal model , which incorporates high mass diffraction @xcite . including this factor , and the nlo @xmath16 factor , the cross section is predicted to be @xcite @xmath17 for the production of a standard model higgs boson of mass 120 gev at the lhc ) at the tevatron , 0.2 fb , is too low to provide a viable signal . ] . it is estimated that there may be a factor two uncertainty in this prediction @xcite . the event rate in entry ( c ) of table 1 includes a factor 0.6 for the efficiency associated with proton tagging , 0.6 for @xmath18 and @xmath19 tagging , 0.5 for the @xmath20 jet polar angle cut , @xmath21 , ( necessary to reduce the @xmath2 qcd background ) and 0.67 for the @xmath1 branching fraction @xcite . hence the original @xmath22 events is reduced to an observable signal of 11 events , as shown in table 1 . the advantage of the @xmath23 signal is that there exists a @xmath13 selection rule , which requires the leading order @xmath24 background subprocess to vanish in the limit of massless quarks and forward outgoing protons limit , the two born - level diagrams ( figs . 2(a , b ) _ without _ the emission of the gluon ) cancel each other . ] . however , in practice , lo background contributions remain . the prolific @xmath25 subprocess may mimic @xmath2 production since we may misidentify the outgoing gluons as @xmath18 and @xmath19 jets . assuming the expected 1% probability of misidentification , and applying @xmath21 jet cut , gives a background - to - signal ratio @xmath26 . secondly , there is an admixture of @xmath27 production , arising from non - forward going protons which gives @xmath28 . thirdly , for a massive quark there is a contribution to the @xmath13 cross section of order @xmath29 , leading to @xmath26 , where @xmath30 is the transverse energy of the @xmath18 and @xmath19 jets . next , we have the possibility of nlo @xmath31 background contributions . of course , the extra gluon may be observed experimentally and these background events eliminated . however , there are exceptions . the extra gluon may go unobserved in the direction of a forward proton . this background may be effectively eliminated by requiring the equality @xmath32 . then we may have soft gluon emission . first , we note that emission from an outgoing @xmath18 or @xmath19 is not a problem , since we retain the cancellation between the crossed and uncrossed graphs . emission from the virtual @xmath18 line is suppressed by at least a factor of @xmath33 ( in the amplitude ) , where @xmath34 and @xmath35 are the energies of the outgoing soft gluon and an outgoing @xmath18 quark in the @xmath24 centre - of - mass frame . the potential danger is gluon emission from an incoming gluon , see fig . 2 . the first two diagrams no longer cancel , as the @xmath2 system is in a colour - octet state . however , the third diagram has precisely the colour and spin structure to restore the cancellation . thus soft gluon emissions from the initial colour - singlet @xmath36 state factorize and , due to the overriding @xmath13 selection rule , qcd @xmath2 production is still suppressed . the remaining danger is large angle hard gluon emission which is collinear with either the @xmath18 or @xmath19 jet , and therefore unobservable . if the cone angle needed to separate the @xmath37 jet from the @xmath18 ( or @xmath19 ) jet is @xmath38 then the expected background from unresolved three jet events leads to @xmath39 . the nnlo @xmath40 background contributions are found to be negligible ( after requiring @xmath41 ) , as are soft pomeron - pomeron fusion contributions to the background ( and to the signal ) @xcite . so , in total , double - diffractive higgs production has a signal - to - background ratio of about three , after including the @xmath16 factors . identifying a ` light ' higgs will be a considerable experimental challenge . all detection processes should be considered . from table 1 we see that valuable information can be obtained from weak boson fusion , where the higgs and the accompanying jets are produced at high @xmath42 . for example , process ( d ) is based on the @xmath43 decay for which the background is small @xcite , whereas process ( f ) exploits rapidity gaps so that the larger @xmath1 signal may be isolated @xcite , provided the pile - up problems can be overcome @xcite . here we have drawn attention to the exclusive @xmath11 signal , process ( c ) . the process has the advantage that the signal exceeds the background . the favourable signal - to - background ratio is offset by a low event rate , caused by the necessity to preserve the rapidity gaps so as to ensure an exclusive signal . nevertheless , entry ( c ) of table 1 shows that the signal has reasonable significance in comparison to the standard @xmath44 and @xmath45 search modes . moreover , the advantage of the matching higgs peaks , @xmath46 , can not be overemphasized . spectrum , see process ( a ) of table 1 . ] we thank albert de roeck , risto orava and andrei shuvaev for valuable discussions , and the eu , pparc and the leverhulme trust for support . xx d. zeppenfeld et al . , phys . rev . * d62 * ( 2000 ) 013009 . v. drollinger , t. mller and d. denegri , cms note , hep - ph/0111312 ; + j. goldstein et al . * 86 * ( 2001 ) 1694 . a. de roeck , v.a . khoze , a.d . martin , r. orava and m.g . ryskin , durham report , hep - ph/0207042 . khoze , a.d . martin and m.g . ryskin , eur . j. * c23 * ( 2002 ) 311 . d. zeppenfeld , hep - ph/0203123 ; n.kauer , t. plehn , d. rainwater and d. zeppenfeld , phys . * b503 * ( 2001 ) 113 . khoze , a.d . martin and m.g . ryskin , eur . j. * c21 * ( 2001 ) 99 . z. bern , l. dixon and c. schmidt , hep - ph/0206194 . khoze , a.d . martin and m.g . ryskin , eur . j. * c14 * ( 2000 ) 525 . khoze , a.d . martin and m.g . ryskin , eur . j. * c19 * ( 2001 ) 477 . khoze , a.d . martin and m.g . ryskin , eur . j. * c18 * ( 2000 ) 167 .

we show that exclusive double - diffractive higgs production , @xmath0 , followed by the @xmath1 decay , could play an important role in identifying a ` light ' higgs boson at the lhc , provided that the forward outgoing protons are tagged . we predict the cross sections for the signal and for all possible @xmath2 backgrounds . ippp/02/41 + dcpt/02/82 + 3 july 2002 * forward proton tagging as a way to identify a light higgs boson at the lhc * a.d . martin , v.a . khoze and m.g . ryskin institute for particle physics phenomenology , + university of durham , dh1 3le , uk

hep-ph0207062

we are interested in the following nonconvex semidefinite programming problem : @xmath1 where @xmath2 is convex , @xmath3 is a nonempty , closed convex set in @xmath4 and @xmath5 ( @xmath6 ) are nonconvex matrix - valued mappings and smooth . the notation @xmath7 means that @xmath8 is a symmetric negative semidefinite matrix . optimization problems involving matrix - valued mapping inequality constraints have large number of applications in static output feedback controller design and topology optimization , see , e.g. @xcite . especially , optimization problems with bilinear matrix inequality ( bmi ) constraints have been known to be nonconvex and np - hard @xcite . many attempts have been done to solve these problems by employing convex semidefinite programming ( in particular , optimization with linear matrix inequality ( lmi ) constraints ) techniques @xcite . the methods developed in those papers are based on augmented lagrangian functions , generalized sequential semidefinite programming and alternating directions . recently , we proposed a new method based on convex - concave decomposition of the bmi constraints and linearization technique @xcite . the method exploits the convex substructure of the problems . it was shown that this method can be applied to solve many problems arising in static output feedback control including spectral abscissa , @xmath9 , @xmath10 and mixed @xmath11 synthesis problems . in this paper , we follow the same line of the work in @xcite to develop a new local optimization method for solving the nonconvex semidefinite programming problem . the main idea is to approximate the feasible set of the nonconvex problem by a sequence of inner positive semidefinite convex approximation sets . this method can be considered as a generalization of the ones in @xcite . 0.1 cm _ contribution . _ the contribution of this paper can be summarized as follows : * we generalize the inner convex approximation method in @xcite from scalar optimization to nonlinear semidefinite programming . moreover , the algorithm is modified by using a _ regularization technique _ to ensure strict descent . the advantages of this algorithm are that it is _ very simple to implement _ by employing available standard semidefinite programming software tools and _ no globalization strategy _ such as a line - search procedure is needed . * we prove the convergence of the algorithm to a stationary point under mild conditions . * we provide two particular ways to form an overestimate for bilinear matrix - valued mappings and then show many applications in static output feedback . 0.1 cm _ outline . _ the next section recalls some definitions , notation and properties of matrix operators and defines an inner convex approximation of a bmi constraint . section [ sec : alg_and_conv ] proposes the main algorithm and investigates its convergence properties . section [ sec : app ] shows the applications in static output feedback control and numerical tests . some concluding remarks are given in the last section . in this section , after given an overview on concepts and definitions related to matrix operators , we provide a definition of inner positive semidefinite convex approximation of a nonconvex set . let @xmath12 be the set of symmetric matrices of size @xmath13 , @xmath14 , and resp . , @xmath15 be the set of symmetric positive semidefinite , resp . , positive definite matrices . for given matrices @xmath16 and @xmath17 in @xmath12 , the relation @xmath18 ( resp . , @xmath19 ) means that @xmath20 ( resp . , @xmath21 ) and @xmath22 ( resp . , @xmath23 ) is @xmath24 ( resp . , @xmath25 ) . the quantity @xmath26 is an inner product of two matrices @xmath16 and @xmath17 defined on @xmath12 , where @xmath27 is the trace of matrix @xmath28 . for a given symmetric matrix @xmath16 , @xmath29 denotes the smallest eigenvalue of @xmath16 . [ de : psd_convex]@xcite a matrix - valued mapping @xmath30 is said to be positive semidefinite convex ( _ psd - convex _ ) on a convex subset @xmath31 if for all @xmath32 $ ] and @xmath33 , one has @xmath34 if holds for @xmath35 instead of @xmath36 for @xmath37 then @xmath38 is said to be _ strictly psd - convex _ on @xmath39 . in the opposite case , @xmath38 is said to be _ psd - nonconvex_. alternatively , if we replace @xmath36 in by @xmath40 then @xmath38 is said to be psd - concave on @xmath39 . it is obvious that any convex function @xmath2 is psd - convex with @xmath41 . a function @xmath42 is said to be _ strongly convex _ with parameter @xmath43 if @xmath44 is convex . the notation @xmath45 denotes the subdifferential of a convex function @xmath46 . for a given convex set @xmath39 , @xmath47 if @xmath48 and @xmath49 if @xmath50 denotes the normal cone of @xmath39 at @xmath51 . the derivative of a matrix - valued mapping @xmath38 at @xmath51 is a linear mapping @xmath52 from @xmath4 to @xmath53 which is defined by @xmath54 for a given convex set @xmath55 , the matrix - valued mapping @xmath56 is said to be differentiable on a subset @xmath16 if its derivative @xmath57 exists at every @xmath58 . the definitions of the second order derivatives of matrix - valued mappings can be found , e.g. , in @xcite . let @xmath59 be a linear mapping defined as @xmath60 , where @xmath61 for @xmath62 . the adjoint operator of @xmath8 , @xmath63 , is defined as @xmath64 for any @xmath65 . finally , for simplicity of discussion , throughout this paper , we assume that all the functions and matrix - valued mappings are _ twice differentiable _ on their domain . let us first describe the idea of the inner convex approximation for the scalar case . let @xmath42 be a continuous nonconvex function . a convex function @xmath66 depending on a parameter @xmath67 is called a convex overestimate of @xmath68 w.r.t . the parameterization @xmath69 if @xmath70 and @xmath71 for all @xmath72 . let us consider two examples . 0.1 cm _ example 1 . _ let @xmath46 be a continuously differentiable function and its gradient @xmath73 is lipschitz continuous with a lipschitz constant @xmath74 , i.e. @xmath75 for all @xmath76 . then , it is well - known that @xmath77 . therefore , for any @xmath78 we have @xmath79 with @xmath80 . moreover , @xmath81 for any @xmath51 . we conclude that @xmath82 is a convex overestimate of @xmath46 w.r.t the parameterization @xmath83 . now , if we fix @xmath84 and find a point @xmath85 such that @xmath86 then @xmath87 . consequently if the set @xmath88 is nonempty , we can find a point @xmath85 such that @xmath86 . the convex set @xmath89 is called an inner convex approximation of @xmath90 . 0.1 cm _ example 2 . _ @xcite we consider the function @xmath91 in @xmath92 . the function @xmath93 is a convex overestimate of @xmath46 w.r.t . the parameterization @xmath94 provided that @xmath95 . this example shows that the mapping @xmath96 is not always identity . let us generalize the convex overestimate concept to matrix - valued mappings . [ def : over_relaxation ] let us consider a psd - nonconvex matrix mapping @xmath97 . a psd - convex matrix mapping @xmath98 is said to be a psd - convex overestimate of @xmath38 w.r.t . the parameterization @xmath69 if @xmath99 and @xmath100 for all @xmath76 and @xmath101 in @xmath102 . let us provide two important examples that satisfy definition [ def : over_relaxation ] . _ example 3 . _ let @xmath103 be a bilinear form with @xmath104 , @xmath105 and @xmath106 arbitrarily , where @xmath16 and @xmath17 are two @xmath107 matrices . we consider the parametric quadratic form : @xmath108 one can show that @xmath109 is a psd - convex overestimate of @xmath110 w.r.t . the parameterization @xmath111 . indeed , it is obvious that @xmath112 . we only prove the second condition in definition [ def : over_relaxation ] . we consider the expression @xmath113 . by rearranging this expression , we can easily show that @xmath114 . now , since @xmath104 , by @xcite , we can write : @xmath115 note that @xmath116 . therefore , we have @xmath117 for all @xmath118 and @xmath119 . _ example 4 . _ let us consider a psd - noncovex matrix - valued mapping @xmath120 , where @xmath121 and @xmath122 are two psd - convex matrix - valued mappings @xcite . now , let @xmath122 be differentiable and @xmath123 be the linearization of @xmath122 at @xmath124 . we define @xmath125 . it is not difficult to show that @xmath126 is a psd - convex overestimate of @xmath127 w.r.t . the parametrization @xmath128 . [ re : nonunique_of_bmi_app ] _ example 3 _ shows that the `` lipschitz coefficient '' of the approximating function is @xmath129 . moreover , as indicated by _ examples _ 3 and 4 , the psd - convex overestimate of a bilinear form is not unique . in practice , it is important to find appropriate psd - convex overestimates for bilinear forms to make the algorithm perform efficiently . note that the psd - convex overestimate @xmath130 of @xmath131 in _ example 3 _ may be less conservative than the convex - concave decomposition in @xcite since all the terms in @xmath130 are related to @xmath132 and @xmath133 rather than @xmath16 and @xmath17 . let us recall the nonconvex semidefinite programming problem . we denote by @xmath134 the feasible set of and @xmath135 the relative interior of @xmath136 , where @xmath137 is the relative interior of @xmath3 . first , we need the following fundamental assumption . [ as : a1 ] the set of interior points @xmath138 of @xmath136 is nonempty . then , we can write the generalized kkt system of as follows : @xmath139 any point @xmath140 with @xmath141 is called a _ kkt point _ of , where @xmath142 is called a _ stationary point _ and @xmath143 is called the corresponding lagrange multiplier . the main step of the algorithm is to solve a convex semidefinite programming problem formed at the iteration @xmath144 by using inner psd - convex approximations . this problem is defined as follows : @xmath145 here , @xmath146 is given and the second term in the objective function is referred to as a regularization term ; @xmath147 is the parameterization of the convex overestimate @xmath148 of @xmath149 . let us define by @xmath150 the solution mapping of [ eq : convx_subprob ] depending on the parameters @xmath151 . note that the problem [ eq : convx_subprob ] is convex , @xmath152 is multivalued and convex . the feasible set of [ eq : convx_subprob ] is written as : @xmath153 the algorithm for solving starts from an initial point @xmath154 and generates a sequence @xmath155 by solving a sequence of convex semidefinite programming subproblems [ eq : convx_subprob ] approximated at @xmath156 . more precisely , it is presented in detail as follows . [ alg : a1 ] * initialization . * determine an initial point @xmath157 . compute @xmath158 for @xmath6 . choose a regularization matrix @xmath159 . set @xmath160 . * iteration @xmath161 ( @xmath162 ) * perform the following steps : * _ step 1 . _ for given @xmath156 , if a given criterion is satisfied then terminate . * _ solve the convex semidefinite program [ eq : convx_subprob ] to obtain a solution @xmath163 and the corresponding lagrange multiplier @xmath164 . * _ update @xmath165 , the regularization matrix @xmath166 ( if necessary ) . increase @xmath161 by @xmath167 and go back to step 1 . * * the core step of algorithm [ alg : a1 ] is step 2 where a general convex semidefinite program needs to be solved . in practice , this can be done by either implementing a particular method that exploits problem structures or relying on standard semidefinite programming software tools . note that the regularization matrix @xmath168 can be fixed at @xmath169 , where @xmath43 is sufficiently small and @xmath170 is the identity matrix . since algorithm [ alg : a1 ] generates a feasible sequence @xmath155 to the original problem and this sequence is strictly descent w.r.t . the objective function @xmath46 , _ no globalization strategy _ such as line - search or trust - region is needed . we first show some properties of the feasible set @xmath171 defined by . for notational simplicity , we use the notation @xmath172 . [ le : feasible_set ] let @xmath173 be a sequence generated by algorithm [ alg : a1 ] . then : * @xmath174 the feasible set @xmath175 for all @xmath176 . * @xmath177 it is a feasible sequence , i.e. @xmath178 . * @xmath179 @xmath180 . * @xmath181 for any @xmath182 , it holds that : @xmath183 where @xmath184 is the strong convexity parameter of @xmath46 . for a given @xmath156 , we have @xmath185 and @xmath186 for @xmath6 . thus if @xmath187 then @xmath188 , the statement a ) holds . consequently , the sequence @xmath189 is feasible to which is indeed the statement b ) . since @xmath163 is a solution of [ eq : convx_subprob ] , it shows that @xmath190 . now , we have to show it belongs to @xmath191 . indeed , since @xmath192 by definition [ def : over_relaxation ] for all @xmath6 , we conclude @xmath193 . the statement c ) is proved . finally , we prove d ) . since @xmath163 is the optimal solution of [ eq : convx_subprob ] , we have @xmath194 for all @xmath187 . however , we have @xmath195 due to c ) . by substituting @xmath196 in the previous inequality we obtain the estimate d ) . now , we denote by @xmath197 the lower level set of the objective function . let us assume that @xmath198 is continuously differentiable in @xmath199 for any @xmath67 . we say that the _ robinson qualification _ condition for [ eq : convx_subprob ] holds at @xmath124 if @xmath200 for @xmath6 . in order to prove the convergence of algorithm [ alg : a1 ] , we require the following assumption . [ as : a2 ] the set of kkt points of is nonempty . for a given @xmath67 , the matrix - valued mappings @xmath198 are continuously differentiable on @xmath199 . the convex problem [ eq : convx_subprob ] is solvable and the robinson qualification condition holds at its solutions . we note that if algorithm 1 is terminated at the iteration @xmath161 such that @xmath201 then @xmath156 is a stationary point of . [ th : convergence ] suppose that assumptions a.[as : a1 ] and a.[as : a2 ] are satisfied . suppose further that the lower level set @xmath199 is bounded . let @xmath202 be an infinite sequence generated by algorithm [ alg : a1 ] starting from @xmath157 . assume that @xmath203 . then if either @xmath46 is strongly convex or @xmath204 for @xmath182 then every accumulation point @xmath205 of @xmath206 is a kkt point of . moreover , if the set of the kkt points of is finite then the whole sequence @xmath207 converges to a kkt point of . first , we show that the solution mapping @xmath150 is _ closed_. indeed , by assumption a.[as : a2 ] , [ eq : convx_subprob ] is feasible . moreover , it is strongly convex . hence , @xmath208 , which is obviously closed . the remaining conclusions of the theorem can be proved similarly as ( * ? ? ? * theorem 3.2 . ) by using zangwill s convergence theorem @xcite of which we omit the details here . [ rm : conclusions ] note that the assumptions used in the proof of the closedness of the solution mapping @xmath209 in theorem [ th : convergence ] are weaker than the ones used in ( * ? ? ? * theorem 3.2 . ) . in this section , we present some applications of algorithm [ alg : a1 ] for solving several classes of optimization problems arising in static output feedback controller design . typically , these problems are related to the following linear , time - invariant ( lti ) system of the form : @xmath210 where @xmath211 is the state vector , @xmath212 is the performance input , @xmath213 is the input vector , @xmath214 is the performance output , @xmath215 is the physical output vector , @xmath216 is state matrix , @xmath217 is input matrix and @xmath218 is the output matrix . by using a static feedback controller of the form @xmath219 with @xmath220 , we can write the closed - loop system as follows : @xmath221 the stabilization , @xmath9 , @xmath222 optimization and other control problems of the lti system can be formulated as an optimization problem with bmi constraints . we only use the psd - convex overestimate of a bilinear form in _ example 3 _ to show that algorithm [ alg : a1 ] can be applied to solving many problems ins static state / output feedback controller design such as : * sparse linear static output feedback controller design ; * spectral abscissa and pseudospectral abscissa optimization ; * @xmath223 optimization ; * @xmath224 optimization ; * and mixed @xmath225 synthesis . these problems possess at least one bmi constraint of the from @xmath226 , where @xmath227 , where @xmath118 and @xmath28 are matrix variables and @xmath228 is a affine operator of matrix variable @xmath28 . by means of _ example 3 _ , we can approximate the bilinear term @xmath229 by its psd - convex overestimate . then using schur s complement to transform the constraint @xmath230 of the subproblem [ eq : convx_subprob ] into an lmi constraint @xcite . note that algorithm [ alg : a1 ] requires an interior starting point @xmath231 . in this work , we apply the procedures proposed in @xcite to find such a point . now , we summary the whole procedure applying to solve the optimization problems with bmi constraints as follows : [ scheme : a1 ] + _ step 1 . _ find a psd - convex overestimate @xmath232 of @xmath233 w.r.t . the parameterization @xmath234 for @xmath235 ( see _ example 1 _ ) . + _ step 2 . _ find a starting point @xmath157 ( see @xcite ) . + _ step 3 . _ for a given @xmath156 , form the convex semidefinite programming problem [ eq : convx_subprob ] and reformulate it as an optimization with lmi constraints . + _ step 4 . _ apply algorithm [ alg : a1 ] with an sdp solver to solve the given problem . now , we test algorithm [ alg : a1 ] for three problems via numerical examples by using the data from the comp@xmath236ib library @xcite . all the implementations are done in matlab 7.8.0 ( r2009a ) running on a laptop intel(r ) core(tm)i7 q740 1.73ghz and 4 gb ram . we use the yalmip package @xcite as a modeling language and sedumi 1.1 as a sdp solver @xcite to solve the lmi optimization problems arising in algorithm [ alg : a1 ] at the initial phase ( phase 1 ) and the subproblem [ eq : convx_subprob ] . the code is available at http://www.kuleuven.be/optec/software/bmisolver . we also compare the performance of algorithm [ alg : a1 ] and the convex - concave decomposition method ( ccdm ) proposed in @xcite in the first example , i.e. the spectral abscissa optimization problem . in the second example , we compare the @xmath10-norm computed by algorithm [ alg : a1 ] and the one provided by hifoo @xcite and penbmi @xcite . the last example is the mixed @xmath237 synthesis optimization problem which we compare between two values of the @xmath238-norm level . we consider an optimization problem with bmi constraint by optimizing the spectral abscissa of the closed - loop system @xmath239 as @xcite : @xmath240 here , matrices @xmath216 , @xmath217 and @xmath218 are given . matrices @xmath241 and @xmath220 and the scalar @xmath242 are considered as variables . if the optimal value of is strictly positive then the closed - loop feedback controller @xmath219 stabilizes the linear system @xmath243 . by introducing an intermediate variable @xmath244 , the bmi constraint in the second line of can be written @xmath245 . now , by applying scheme [ scheme : a1 ] one can solve the problem by exploiting the sedumi sdp solver @xcite . in order to obtain a strictly descent direction , we regularize the subproblem [ eq : convx_subprob ] by adding quadratic terms : @xmath246 , where @xmath247 . algorithm [ alg : a1 ] is terminated if one of the following conditions is satisfied : * the subproblem [ eq : convx_subprob ] encounters a numerical problem ; * @xmath248 ; * the maximum number of iterations , @xmath249 , is reached ; * or the objective function of is not significantly improved after two successive iterations , i.e. @xmath250 for some @xmath251 and @xmath252 , where @xmath253 . we test algorithm [ alg : a1 ] for several problems in comp@xmath236ib and compare our results with the ones reported by the _ convex - concave decomposition method _ ( ccdm ) in @xcite . -0.45 cm .computational results for in comp@xmath254ib [ cols= " < , > , > , > , > , > , > , > , > , > " , ] here , @xmath225 are the @xmath223 and @xmath224 norms of the closed - loop systems for the static output feedback controller , respectively . with @xmath255 , the computational results show that algorithm [ alg : a1 ] satisfies the condition @xmath256 for all the test problems . the problems ac11 and ac12 encounter a numerical problems that algorithm [ alg : a1 ] can not solve . while , with @xmath257 , there are @xmath258 problems reported infeasible , which are denoted by `` - '' . the @xmath224-constraint of three problems ac11 and nn8 is active with respect to @xmath257 . we have proposed a new iterative procedure to solve a class of nonconvex semidefinite programming problems . the key idea is to locally approximate the nonconvex feasible set of the problem by an inner convex set . the convergence of the algorithm to a stationary point is investigated under standard assumptions . we limit our applications to optimization problems with bmi constraints and provide a particular way to compute the inner psd - convex approximation of a bmi constraint . many applications in static output feedback controller design have been shown and two numerical examples have been presented . note that this method can be extended to solve more general nonconvex sdp problems where we can manage to find an inner psd - convex approximation of the feasible set . this is also our future research direction .

in this work , we propose a new local optimization method to solve a class of nonconvex semidefinite programming ( sdp ) problems . the basic idea is to approximate the feasible set of the nonconvex sdp problem by inner positive semidefinite convex approximations via a parameterization technique . this leads to an iterative procedure to search a local optimum of the nonconvex problem . the convergence of the algorithm is analyzed under mild assumptions . applications in static output feedback control are benchmarked and numerical tests are implemented based on the data from the compl@xmath0ib library .

1202.5488

the dynamics of fluids can exhibit hysteresis . for example , a flag - like object shows bistability between flapping and nonflapping states @xcite . hysteresis also exists in vortex shedding dynamics behind rigid objects , such as a vibrating cylinder @xcite , a multiple cylinder arrangement @xcite , a long cylinder in a three - dimensional flow @xcite , and a rod in a soap film @xcite . in these experiments , the transitions between laminar flow and vortex shedding states occur in a hysteretic manner as a function of the reynolds number . it is known that the taylor couette flow also exhibits hysteresis @xcite . in superfluids , hysteresis has been observed in rotating toroidal systems @xcite . in this paper , we consider the transition between a laminar flow state and a quantized vortex shedding state around an obstacle moving in a bose einstein condensate ( bec ) . in a superfluid , the velocity field around an obstacle is irrotational below the critical velocity . when the velocity of the obstacle exceeds the critical velocity , quantized vortices are created and released behind the obstacle , as observed in a trapped bec stirred by an optical potential @xcite . the critical velocity for vortex creation and the dynamics of quantized vortex shedding in superfluids have been studied theoretically by many researchers @xcite . the purpose of the present paper is to show that superfluids undergo hysteretic changes between stationary laminar flow and periodic shedding of quantized vortices . consider an obstacle with gradually increasing velocity ; on reaching the critical velocity @xmath0 , periodic vortex shedding starts . now consider an obstacle with gradually decreasing velocity from above @xmath0 ; the vortex shedding stops at a velocity @xmath1 . we show that there is a bistability between these flow patterns , i.e. , @xmath2 . although hysteretic vortex shedding under a moving potential was reported in ref . @xcite , the mechanism has not been studied in detail . in the present paper , we show that the hysteretic behaviors are due to the fact that released vortices enhance the flow velocity around the obstacle and induce subsequent vortex creation . we show that the hysteretic behavior is observed for a circular obstacle moving in a two - dimensional ( 2d ) superfluid and a spherical obstacle moving in a three - dimensional ( 3d ) superfluid . this paper is organized as follows . section [ s : formulation ] formulates the problem and describes the numerical method . the hysteretic dynamics are studied for a 2d system in sec . [ s:2d ] and for a 3d system in sec . [ s:3d ] . conclusions are given in sec . [ s : conc ] . we study the dynamics of a bec at zero temperature using mean - field theory . the system is described by the gross pitaevskii ( gp ) equation , @xmath3 where @xmath4 is the macroscopic wave function , @xmath5 is the atomic mass , @xmath6 is an external potential , and @xmath7 is the @xmath8-wave scattering length . we consider situations in which a localized potential @xmath9 moves at a velocity @xmath10 , i.e. , the potential @xmath11 has a form , @xmath12 we transform eq . ( [ gp ] ) into the frame of reference of the moving potential @xmath11 by substituting the unitary transformation @xmath13 \psi(\bm{r } , t)\ ] ] into eq . ( [ gp ] ) , which yields @xmath14 in the following , the velocity vector is taken as @xmath15 where @xmath16 is the unit vector in the @xmath17 direction . we consider an infinite system , in which the atomic density @xmath18 far from the moving potential is constant @xmath19 . for the density @xmath19 , the healing length @xmath20 and the sound velocity @xmath21 are defined as @xmath22 which determine the characteristic time scale , @xmath23 the chemical potential for the density @xmath19 is given by @xmath24 normalizing eq . ( [ gp2 ] ) by the quantities in eqs . ( [ xi])([mu ] ) , we obtain @xmath25 where @xmath26 , @xmath27 , @xmath28 , @xmath29 , and @xmath30 are dimensionless quantities . the independent parameters in eq . ( [ gpn ] ) are only @xmath31 and @xmath32 . we numerically solve eq . ( [ gpn ] ) using the pseudo - spectral method @xcite . the initial state is the stationary state of eq . ( [ gpn ] ) for a velocity @xmath33 below the critical velocity @xmath0 for vortex nucleation , which is prepared by the imaginary - time propagation method @xcite . the initial state is a stationary laminar flow and contains no vortices . to break the exact numerical symmetry , a small random noise is added to each mesh of the initial state . the real - time propagation of eq . ( [ gpn ] ) is then calculated with a change in the velocity @xmath33 or the potential @xmath9 to trigger the vortex creation . the size of the space is taken to be large enough and the periodic boundary condition imposed by the pseudo - spectral method does not affect the dynamics around the potential . first , we consider a 2d space . typically , the size of the numerical space is taken to be @xmath34 in @xmath17 and @xmath35 in @xmath36 , and is divided into a @xmath37 mesh . the obstacle potential is given by @xmath38 where @xmath39 is the radius of the circular potential . numerically , a value that is significantly larger than the chemical potential is used for @xmath40 in eq . ( [ u ] ) . the following results are qualitatively the same as those for a gaussian potential in place of the rigid circular potential in eq . ( [ u ] ) . ( left panels ) and phase @xmath41 ( right panels ) profiles for @xmath42 and @xmath43 . to trigger the vortex shedding , the additional potential given by eq . ( [ uadd ] ) is applied during @xmath44 . the arrows in the phase profiles indicate the directions in which the quantized vortices are rotating . the size of each panel is @xmath45 . see the supplemental material for a movie of the dynamics . , width=302 ] figure [ f : dynamics ] shows the time evolution of the density @xmath18 and phase @xmath41 profiles . the initial state is the stationary state for the velocity @xmath46 and radius @xmath47 , as shown in fig . [ f : dynamics](a ) . this stationary laminar flow state is stable . to trigger the vortex shedding , we apply an additional potential , @xmath48 / \xi^2}\ ] ] during @xmath44 , in addition to the circular potential in eq . ( [ u ] ) . this additional potential perturbs the edge of the circular potential , at which quantized vortex creation is induced , as shown in fig . [ f : dynamics](b ) . subsequently , quantized vortices are periodically created one after the other @xcite , as shown in figs . [ f : dynamics](c ) and [ f : dynamics](d ) , even after the perturbation potential is removed at @xmath49 and the velocity @xmath46 is smaller than the critical velocity @xmath0 . this result indicates that there are at least two stable flow patterns for the same parameters : a stationary laminar flow and periodic vortex shedding . at @xmath50 for the same parameters as those in fig . [ f : dynamics ] . the dashed square is magnified in the inset , where the vertical lines indicate @xmath51 , 275 , and 320 . ( b)(e ) density @xmath18 profiles ( left panels ) and velocity @xmath52 profiles ( right panels ) . the crosses in ( b ) indicate the positions @xmath53 at which the velocities are plotted in ( a ) . the arrows in the density profiles indicate the directions in which the quantized vortices are rotating . the size of each panel is @xmath54 . , width=302 ] the velocity field of the atomic flow has the form , @xmath55 figure [ f : velocity](a ) shows the time evolution of the velocities @xmath56 at @xmath53 . these positions are indicated by the crosses in fig . [ f : velocity](b ) . for the stationary flow ( @xmath57 ) , the velocities are @xmath58 . the fluctuations around @xmath21 are due to the small numerical noises added to the initial state . at @xmath59 , the additional potential given by eq . ( [ uadd ] ) is applied and a clockwise vortex is released from near the position @xmath60 . as a consequence , @xmath61 suddenly decreases . it can also be seen in fig . [ f : velocity](c ) that the released vortex decreases the velocity field in the vicinity of its creation . the clockwise vortex shedding then induces counterclockwise vortex creation , as shown in fig . [ f : velocity](d ) . immediately after that ( @xmath62-@xmath63 ) , @xmath64 increases rapidly , which is followed by a sudden decrease due to the shedding of another counterclockwise vortex , as shown in fig . [ f : velocity](d ) . this periodic vortex shedding is repeated indefinitely . the dynamics shown in fig . [ f : velocity ] implies that the release of a vortex induces the creation of a subsequent vortex , i.e. , periodic vortex shedding is taking place . at @xmath65 for @xmath47 . the velocity is increased as @xmath66 ( red , solid line ) or decreased as @xmath67 ( blue , dashed line ) . the insets show the density profiles at @xmath68 and @xmath69 for the increase in @xmath70 , and @xmath71 and @xmath72 for the decrease in @xmath70 . , width=302 ] to show the hysteresis clearly , we gradually increase and decrease the velocity @xmath70 around the critical velocity . figure [ f : hysteresis ] shows the time evolution of the flow velocity @xmath56 at @xmath65 . when the velocity @xmath70 is gradually increased , the vortex shedding starts at the critical velocity @xmath73 . on the other hand , when @xmath70 is decreased from above @xmath0 , the periodic vortex shedding continues for @xmath74 , eventually stopping at the lower critical velocity @xmath75 . the fluctuation in @xmath56 for @xmath76 is due to the remnant disturbing waves . ) and ( [ vf2 ] ) , respectively . ( b ) @xmath52 in eq . ( [ vf1 ] ) for @xmath77 and @xmath78 . ( d ) @xmath52 in eq . ( [ vf2 ] ) for @xmath79 , @xmath80 , @xmath81 , and @xmath82 . the size of each panel in ( b ) and ( d ) is @xmath54 . , width=302 ] the velocity field around a circular obstacle can be analyzed using the point - vortex model for an inviscid incompressible fluid . the situation in fig . [ f : velocity](c ) is modeled as in fig . [ f : analytic](a ) , where a clockwise vortex is located at @xmath83 and the circle of radius @xmath39 contains a counterclockwise vortex . the complex velocity field in which the normal component @xmath84 vanishes at @xmath85 is given by @xmath86 where @xmath87 , @xmath88 , @xmath89 , and @xmath90 @xcite . the first term on the right - hand side of eq . ( [ vf1 ] ) approaches a uniform flow @xmath91 at infinity @xmath92 and the second term represents a flow generated by the vortices located at @xmath93 and the origin . the flow velocity at @xmath94 is @xmath95 and @xmath96 , which indicates that the flow velocity at @xmath97 is enhanced by the vortices . thus , once a vortex is released from @xmath98 , the next vortex is created at @xmath97 , which results in the dynamics shown in fig . [ f : velocity](d ) . the velocity field in eq . ( [ vf1 ] ) for @xmath99 and @xmath78 is shown in fig . [ f : analytic](b ) , which is very similar to fig . [ f : velocity](c ) . the situation in fig . [ f : velocity](d ) is modeled by fig . [ f : analytic](c ) , for which the velocity field is given by @xmath100 where @xmath101 and @xmath102 . the flow velocity at @xmath103 is @xmath104.\ ] ] when @xmath105 is positive , the first term in the square bracket of eq . ( [ vx2 ] ) , i.e. , the vortex at @xmath106 , enhances the flow velocity . the vortex released from @xmath97 therefore induces the creation of the subsequent vortex at @xmath97 . the velocity field in eq . ( [ vf2 ] ) for @xmath107 , @xmath78 , @xmath80 , and @xmath82 is shown in fig . [ f : analytic](d ) , which well reproduces fig . [ f : velocity](d ) . thus , vortices shed behind an obstacle induce the creation of an additional vortex , resulting in periodic vortex shedding , and ultimately hysteretic mechanism that allows this behavior to continue below the critical velocity @xmath0 . and the velocity @xmath33 . the vortex shedding always occurs in the `` vortex shedding '' region , no vortices are created in the `` no vortex '' region , and hysteresis appears in the `` bistability '' region . the boundaries between the bistability and vortex - shedding regions and the bistability and no - vortex regions are @xmath0 and @xmath1 , respectively . , width=302 ] figure [ f : diagram ] shows the radius @xmath39 and the velocity @xmath33 dependence of the flow patterns . the vortex shedding always occurs in the `` vortex shedding '' region and vortices are never created in the `` no vortex '' region . the `` bistability '' region lies between these two regions , in which a stationary laminar flow is stable but periodic vortex shedding is kept once it starts . for @xmath108 , the bistability region disappears , probably because @xmath109 in eq . ( [ vx2 ] ) is small and hence the enhancement of the successive vortex creation is less effective . although the bistability region may also exist for @xmath110 , it is difficult to determine the precise value of @xmath1 numerically , since the vortex shedding dynamics are aperiodic for large @xmath39 , and are dependent on infinitesimal numerical noises . ) moves in the @xmath111 direction at a velocity @xmath112 ; the dynamics are given in the frame of reference of the potential . an additional potential given by eq . ( [ add3d ] ) is applied during @xmath113 . the size of the cuboidal frame is @xmath114 . see the supplemental material for a movie of the dynamics . , width=302 ] next we examine a 3d system . we use a gaussian potential for the moving obstacle as @xmath115 a spherical rigid potential analogous to that in eq . ( [ u ] ) gives similar results . we prepare the initial state of a stationary laminar flow with @xmath116 , which is below the critical velocity @xmath117 for vortex creation . in order to trigger the vortex shedding , an additional potential @xmath118 / \xi^2}\ ] ] is applied during @xmath119 , where @xmath120 . this additional potential is located at the edge of the potential @xmath9 in eq . ( [ gaussian ] ) , triggering the generation of a quantized vortex ring , as shown in fig . [ f:3d](b ) . subsequent vortex creation is induced and periodic vortex shedding begins , as shown in figs . [ f:3d](c ) and [ f:3d](d ) , respectively . this result indicates that the bistability between the stationary laminar flow and periodic vortex shedding also exists in a 3d system . we note that the vortex rings in a 3d system are topologically different from the vortex pairs in a 2d system . a vortex - antivortex pair in a 2d system corresponds to a vortex ring in a 3d system , since they propagate without changing their shapes . by contrast , a vortex - vortex pair in a 2d system , such as that seen in fig . [ f : velocity](e ) , has no counterpart in a 3d system , since two vortices rotate around one another for a vortex - vortex pair . such rotation would tangle vortex rings in a 3d system . it is interesting that both 2d and 3d systems exhibit bistability despite the topological difference . we investigated the dynamics of a bec with a moving obstacle potential , and found bistability between stationary laminar flow and periodic vortex shedding . when the velocity of the obstacle is gradually increased , quantized vortex shedding starts at the critical velocity @xmath0 . on the other hand , when the velocity is gradually decreased from above @xmath0 , the vortex shedding stops at a velocity @xmath1 . we found that @xmath121 for an appropriately sized obstacle potential ( fig . [ f : hysteresis ] ) . for a velocity @xmath122 , a stationary laminar flow is stable , but periodic vortex shedding is maintained once it starts ( figs . [ f : dynamics ] and [ f : velocity ] ) . such hysteretic behavior originates from the fact that the vortices released behind the obstacle enhance the velocity field around the obstacle , inducing subsequent vortex generation ( fig . [ f : analytic ] ) . the bistability between the stationary laminar flow and periodic vortex shedding exists not only in 2d systems but also in 3d systems ( fig . [ f:3d ] ) . we thank t. kishimoto for fruitful discussions . this work was supported by a grant - in - aid for scientific research ( no.26400414 ) and a grant - in - aid for scientific research on innovative areas `` fluctuation & structure '' ( no . 25103007 ) from the ministry of education , culture , sports , science and technology of japan .

it is shown using numerical simulations that flow patterns around an obstacle potential moving in a superfluid exhibit hysteresis . in a certain velocity region , there is a bistability between stationary laminar flow and periodic vortex shedding . the bistability exists in two and three dimensional systems .

1405.4577

a proper choice of order parameters is the most important perspective for building a sound model for liquid crystals . the order parameters should be simple and explanatory in terms of mathematics and physics , while efficient in computations . from the viewpoint of theoretical analysis , it is desirable to adopt order parameters that are as simple as possible in order to capture essential phenomena of liquid crystals , with four phases the most important : isotropic(@xmath0 ) , nematic(@xmath1 ) , smectic - a(@xmath2 ) , and smectic - c(@xmath3 ) phases . as for computational aspects , the order parameters should be discretely representable with reasonable dimensions while keeping the energy functional well - posed , concise , and efficient . the classical models for static liquid crystals can be classified into three scales : molecular models , tensor models , and vector models . each of these models have merits and limitations in respect to mathematics , physics and computations . the molecular models for liquid crystals are based on microscopic statistical physics . by means of the cluster expansion , onsager @xcite pioneered the molecular field theory , in which the order parameter is the density function for the position and orientation of molecules . onsager s molecular model is established on sound physical principles , and contains no adjustable parameters . however , the molecular model is not clearly related to macroscopic properties , and the high dimension of the order parameter imposes considerable obstacles in computations . based on onsager s molecular theory , the mcmillan model @xcite and a list of molecular models @xcite @xcite @xcite parameterized the density function by some scalar order parameters to model the smectic phases . by lowering the dimensions , these model are very efficient in computations . however , the order parameters of these models are spatially invariant , so that they can not model detailed physical phenomena with confined geometry and spatial variance . on the other hand , a vector model for liquid crystals was phenomenologically proposed by oseen @xcite , where the order parameter is a vector field @xmath4 representing the director . as a development of landau - ginzburg theory and oseen - frank theory , chen and lubensky @xcite introduced the density as another order parameter to characterize @xmath1-@xmath2-@xmath3 phase transitions . the chen - lubensky model is famous for its conciseness . the internal coefficients can be measured through experiments . however , this model presumes liquid crystals to be uniaxial , and the director @xmath5 is singular at defect points , which renders it difficult to characterize some small scale phenomena , such as defects and interfaces . overcoming the drawbacks of the molecular models and the vector models , the well - known landau - de gennes model @xcite was proposed with the energy functional @xmath6&=\int_\omega\left ( \frac{a(t - t^*)}{2}{\mathrm{tr}}(q^2)-\frac{b}{3}{\mathrm{tr}}(q^3)+\frac{c}{4}({\mathrm{tr}}(q^2))^2\right ) { \mathrm{d}}\tx\\ & + \int_\omega\left ( l_1 \vert \nabla q \vert^2+l_2 \partial_j q_{ik}\partial_k q_{ij}+l_3 \partial_j q_{ij } \partial_kq_{ik}+l_4 q_{lk } \partial_k q_{ij } \partial_l q_{ij}\right ) { \mathrm{d}}\tx , \end{aligned}\ ] ] where @xmath7 , @xmath8 , @xmath9 , and @xmath10 are constants . the order parameter is a @xmath11 symmetric traceless tensor field @xmath12 , which is the second moment of molecules orientation at every point . the q - tensor is a desirable order parameter providing information on both the preferred molecular orientation and the degree of orientational order at every given point , while capturing essential physical properties . various phenomena were studied with the landau - de gennes model , e.g. phase transitions in confined geometries @xcite @xcite , wetting phenomena @xcite @xcite , surface - induced bulk alignment @xcite @xcite , and defects and disclinations @xcite @xcite . the order parameter and the energy functional of the landau - de gennes model are also simple enough to perform rigorous mathematical analysis . ball and majumdar proved that the energy of landau - de gennes model is unbounded from below as @xmath13 , and proposed to modify the entropy term from a polynomial into a thermotropic one in order to avoid the unboundedness @xcite . furthermore , ball and zarnescu proved that for simply - connected domains and in sobolev space @xmath14 with corresponding boundary conditions , the landau - de gennes theory and the oseen - frank theory coincide @xcite . various generalizations of the landau - de gennes model @xcite @xcite @xcite @xcite were proposed to include smectic phases . the work of pajak and osipov @xcite is a generalization of the mcmillan model @xcite and the landau - de gennes model , which starts from the self - consistent field theory and adopts the one mode approximation to parameterize the density function . the details of this model are provided in section [ section : gmm ] . this model is efficient in computations , but the order parameters are spatially invariant , failing to characterize some physical phenomena with confined geometry and spatial variance . the works of mukherjee @xcite and biscari _ @xcite are also generalizations of the landau - de gennes theory . the q - tensor is coupled with the complex smectic order parameter , and the spatial inhomogeneity of the order parameters enables the model to characterize smectic phases . however , it is noteworthy that these models did not explain how the energy functionals were derived in details . in a recent paper by han _ @xcite , a systematic way of modeling static liquid crystals with uniaxial molecules was proposed . to be more precise , starting from onsager s molecular theory , a new q - tensor model was presented incorporating the bingham closure and a taylor expansion with truncation at low order moments . the coefficients in the new q - tensor model were approximated in terms of the microscopic shape factor @xmath15 by assuming the interaction potential to be the volume exclusion potential of rigid rod - like molecules . here , @xmath16 is the diameter of the semisphere at the ends of the rods , and @xmath17 is the length of the rods . in modeling the nematic phase , the oseen - frank model and the ericksen model were derived using the new q - tensor model by assuming a constant density . three elastic constants @xmath18 , @xmath19 , and @xmath20 , measuring the strains on liquid crystals in deformation , were calculated analytically . in modeling the smectic phase , under the uniaxial assumption , some preliminary numerical results regarding @xmath0-@xmath1-@xmath2 phase transitions were presented . the energy functional of the new q - tensor model in a modified version reads @xmath21=f_{bulk } + f_{elastic , 2}+ f_{elastic , 4},\\ \end{aligned}\ ] ] where @xmath22 { \mathrm{d}}\tx,\\ \nonumber\beta f_{elastic , 2}= & \frac{1}{2}\int_\omega\big[-g_1 \vert \nabla c\vert^2 + g_2 \vert \nabla(cq)\vert^2+g_3\partial_i(cq_{ik})\partial_j(cq_{jk } ) -g_4 \partial_i(cq_{ij})\partial_j(c)\\ \label{elastic2}&+ g_5\vert \nabla(cq_4)\vert^2+g_6\partial_i(cq_{4iklm } ) \partial_j(cq_{4jklm})+g_7 \partial_i(cq_{4ijkl})\partial_j(cq_{kl})\big ] { \mathrm{d}}\tx,\\ \nonumber\beta f_{elastic , 4}=&\frac{1}{2}\int_{\omega}\big[h_1\vert \nabla^2 c\vert^2 + h_2 \vert \nabla^2 ( cq)\vert^2 \\ \label{elastic4}&+ h_3\partial_{ij}(cq_{ij})\partial_{kl}(cq_{kl } ) + h_4 \partial_{ik}(cq_{ip})\partial_{jk}(cq_{jp } ) \big ] { \mathrm{d}}\tx.\end{aligned}\ ] ] here , @xmath23 is the thermodynamic beta . @xmath24 is the density . q - tensor @xmath12 is the traceless second moment of the orientational variable @xmath25 , and @xmath26 is the traceless fourth moment of the orientational variable @xmath25 . @xmath27 is the bingham distribution parameters , and @xmath28 is the normalizing constant for the bingham distribution . the bulk energy @xmath29 contains the entropy and the quadratic terms of the order parameters . the second order elastic energy @xmath30 contains the derivative terms of the order parameters ; the fourth order elastic energy @xmath31 contains the second order derivative terms of the order parameters . note that here the fourth order elastic energy @xmath31 is truncated : only the positive definite terms are preserved in order to ensure the lower boundedness of the energy functional . the detailed descriptions of the energy functional are presented in the main body of this paper . in this paper , regarding the new q - tensor model as a phenomenological model , we focus on its density variation effects , and show its effectiveness in terms of mathematics , physics , and computations . the introduction of the density @xmath24 as another order parameter exhibits many benefits . it empowers the model to characterize the smectic phases , and to characterize the density variations in small scale phenomena such as defects and interfaces . since a few fourier modes are enough to characterize the profiles of the order parameters , the model can be solved with low computational costs . numerical experiments are performed to study @xmath0-@xmath1-@xmath2-@xmath3 phase transitions , and the results are compared with experimental results . the @xmath0-@xmath1 interface problem is also studied in the new q - tensor model setup , where the numerical results are compared with previous theoretical results and experimental results . in addition , we will elaborate on the strong connections of the new q - tensor model with the classical models of liquid crystals in three scales . derived from onsager s molecular theory , the new q - tensor model can generate all the other classical models with some assumptions and approximations . figure [ relations ] shows the classical models in three scales and their relations . this paper is organized as follows . in section [ section : prop ] , we demonstrate the mathematical properties of the new q - tensor model and the numerical methods to compute the energy functional . in section [ section : phy ] , the numerical results regarding @xmath0-@xmath1-@xmath2-@xmath3 phase transitions and @xmath0-@xmath1 interface are presented . in section [ section : compare ] , we compare the q - tensor model with the classical models . we give several concluding remarks in section [ section : sum ] . some detailed calculations involved in the paper are provided in the appendix . in this paper , all vectors will be expressed by boldface letters . the juxtaposition of a pair of vectors @xmath32 denotes the dyadic product of @xmath25 and @xmath5 . doubly contracted tensor products are represented by a colon . the einstein summation convention for tensors is used . in this section , we review the new q - tensor model previously derived from onsager s molecular theory in the paper @xcite , and discuss the numerical methods . we will demonstrate the merits of density as the order parameter in terms of concise mathematics , efficient computations , and explanatory physics . the new q - tensor model gives the free energy of liquid crystals as equation ( [ energy0 ] ) , with the definitions of notations as follows . consider the rod - like liquid crystal molecules in domain @xmath33 . the molecules take positional coordinate @xmath34 , and orientational coordinate @xmath35 . the positional coordinate @xmath36 is dimensionless , which is the ratio of the physical position @xmath37 and the length of the molecules @xmath17 . we introduce the density @xmath24 and the q - tensor @xmath12 as order parameters . the density @xmath24 is dimensionless , representing the number of molecules in volume @xmath38 . the tensor @xmath12 is the second moment of @xmath25 with respect to @xmath39 , @xmath40 where @xmath39 is the probability density function of molecules with respect to orientation @xmath25 at given position @xmath41 . this definition requires the q - tensor to be a symmetric , traceless @xmath11 matrix whose eigenvalues @xmath42 are constrained by the inequalities @xmath43 let @xmath44 be the fourth moment of @xmath25 in terms of @xmath39 , which is defined as @xmath45 where @xmath46 denotes the symmetrization of the tensor with respect to all permutations of indices . various closure models were proposed to represent the relationship between @xmath44 and @xmath47 @xcite . among various closure models , we choose the bingham closure model , which has the following good properties . * the maximizer @xmath48 , maximizing the entropy @xmath49 subject to the q - tensor @xmath12 conforming ( [ defq ] ) and ( [ qinequ ] ) , follows the bingham distribution @xcite . * the bingham closure automatically ensures ( [ qinequ ] ) , which preserve the physical meanings of the q - tensor . * the dynamic q - tensor model , derived from doi s kinetic theory using the bingham closure , obeys the energy dissipation law @xcite . * some numerical results show that the bingham closure gives the best approximation to doi s kinetic theory in simulating complex flows of liquid - crystalline polymers @xcite the bingham distribution of @xmath50 reads @xmath51 where @xmath52 and @xmath53 is any traceless symmetric matrix in @xmath54 . under this assumption , it has been proven that given any @xmath12 which conforms ( [ qinequ ] ) , @xmath53 and @xmath50 can be uniquely determined @xcite . therefore , the dimension of the probability density function @xmath50 reduces to five , the same as that of @xmath12 and @xmath53 . as a generalization of the landau - de gennes theory , the introduction of the density @xmath24 as another order parameter enables the model to characterize smectic phases and density variations in physical phenomena such as defects and interfaces . the major phases of rod - like molecule liquid crystals can be characterized by @xmath24 and @xmath12 as * @xmath0 phase : @xmath55 , @xmath56 . * @xmath1 phase : @xmath55 , @xmath57 . * @xmath2 phase : @xmath58 and @xmath47 are one - dimensional and periodic . the director @xmath5 , defined as the principal eigenvector of q - tensor , is parallel to the layer normal @xmath59 . * @xmath3 phase : @xmath58 and @xmath47 are one - dimensional and periodic . the director @xmath5 and the layer normal @xmath59 yield angle @xmath60 , which represents the average of tilt angles between the rod - like molecules and the layer normal . biaxiality is an important property in smectic phases , and in other physical phenomena such as defects and interfaces . in this paper , biaxiality is given by a common mathematical definition @xmath61 this definition requires the biaxiality @xmath62 within the interval @xmath63 $ ] . if the q - tensor is uniaxial , @xmath62 will be zero ; if the q - tensor shows strong biaxiality , @xmath62 will tend to one . other notations in the energy functional ( [ energy0 ] ) are as follows . @xmath23 is the thermodynamic beta . the @xmath64 dimensionless coefficients @xmath65 , @xmath66 , and @xmath67 are determined in terms of the microscopic molecule s interaction potential , and are functions of temperature . we can infer their approximate range of values by assuming the interaction potential to be the volume exclusion potential of rigid rod - like molecules @xcite . to ensure that the energy is bounded from below , it is necessary that @xmath68 and @xmath69 be positive . otherwise , assuming the order parameters to be a series of highly oscillating functions , it is easy to show that the energy functional tends to negative infinity . we consider the one - dimensional q - tensor model . we note here that the one - dimensional case is significant and sufficiently representative , since @xmath0 , @xmath1 , @xmath2 , and @xmath3 phases can all be represented in one dimension . for periodic boundary conditions , the spectral method is efficient and accurate enough to compute the energy functional , by representing the order parameters with a few fourier modes . as another method , we can discretize the order parameters in nodal space and use finite difference method to calculate the derivatives . consider the reduced energy functional on the interval @xmath70 $ ] , which takes the form @xmath71&= \int_0^h c(x)(\ln c(x ) + b_q : q-\ln z(x ) ) { \mathrm{d}}x\\ & + \frac{1}{2 } \int_0^h \big[a_{1}c^2 -a_{2 } ( cq_{ij})^2-a_{3 } ( cq_{4ijkl})^2\big ] { \mathrm{d}}x \\ & + \frac{1}{2}\int_0^h \big [ -g_1 ( \frac{{\mathrm{d}}}{{\mathrm{d}}x } c)^2 + g_2 ( \frac{{\mathrm{d}}}{{\mathrm{d}}x}(cq_{ij}))^2+g_3(\frac{{\mathrm{d}}}{{\mathrm{d}}x}(cq_{1k}))^2 -g_4 \frac{{\mathrm{d}}}{{\mathrm{d}}x}(cq_{11})\frac{{\mathrm{d}}}{{\mathrm{d}}x}(c)\\ & \quad\quad+g_5 ( \frac{{\mathrm{d}}}{{\mathrm{d}}x}(cq_{4ijkl}))^2 + g_6(\frac{{\mathrm{d}}}{{\mathrm{d}}x}(cq_{41klm}))^2+g_7 \frac{{\mathrm{d}}}{{\mathrm{d}}x}(cq_{411kl})\frac{{\mathrm{d}}}{{\mathrm{d}}x}(cq_{kl } ) \big ] { \mathrm{d}}x\\ & + \frac{1}{2}\int_0^h \big [ h_1(\frac{{\mathrm{d}}^2}{{\mathrm{d}}x^2}(c))^2+h_2(\frac{{\mathrm{d}}^2}{{\mathrm{d}}x^2}(cq_{pq}))^2+h_3(\frac{{\mathrm{d}}^2}{{\mathrm{d}}x^2}(cq_{11}))^2 + h_4(\frac{{\mathrm{d}}^2}{{\mathrm{d}}x^2}(cq_{1p}))^2 \big ] { \mathrm{d}}x . \end{aligned}\ ] ] we consider how to numerically represent the order parameters . in the following , we will show that @xmath72 can be represented by five independent scalar variables @xmath73 , @xmath74 , @xmath75 , @xmath76 , and @xmath77 . the bingham assumption implies that @xmath27 , @xmath47 , and @xmath44 are mutually determined and can be diagonalized simultaneously . let the eigenvalue decomposition of @xmath27 be @xmath78 , where @xmath79 is a traceless diagonal matrix and @xmath80 is an orthogonal matrix . using a change of variables in the integral ( [ defq ] ) and ( [ defq4 ] ) , @xmath47 and @xmath44 can be represented as @xmath81 here , @xmath82 and @xmath83 are the diagonal forms of @xmath47 and @xmath44 respectively , which can be computed using @xmath79 as @xmath84 we use @xmath85 and @xmath86 to represent the traceless diagonal matrix @xmath79 , @xmath87 and use euler angles @xmath60 , @xmath88 , and @xmath89 to represent the orthogonal transformation matrix @xmath80 , @xmath90 here , the first angle @xmath60 is the angle between the principal eigenvector of @xmath47 and the x - axis . above all , the q - tensor @xmath72 can be represented by @xmath73 , @xmath74 , @xmath75 , @xmath76 , and @xmath77 . in the meanwhile , @xmath91 , @xmath92 , and @xmath93 can also be computed using these five variables . one way to discretize the order parameters is to expand them by fourier modes @xmath94 where @xmath95 . the dimension of variables is @xmath96 . the derivatives of @xmath97 and @xmath98 are calculated using fast fourier transform ( fft ) . in the following section , we use this method in computing the phase transition problem ; empirically , @xmath99 is enough to characterize the profiles of the order parameters . as another way to represent the order parameters , we can discretize the interval @xmath70 $ ] with @xmath100 nodes . the dimension of variables is @xmath101 . the derivatives of the order parameters are computed using the finite difference method . in the following section , we use this method in computing the @xmath0-@xmath1 interface problem , and @xmath102 is accurate enough empirically . the minimum of the energy functional can be found by standard methods . we use numerical differentiation to calculate the gradient of the objective function , and use quasi - newton methods , such as the bfgs method to solve the optimization problem . the computations for both of the two problems discussed in the following section converge in about 100 steps with absolute precision of @xmath103 . the overall computational cost is as moderate as the cost of the landau - de gennes model . it is worth pointing out that the q - tensor model may have some local minima . in the simulation , we try different random initial values to ensure that we find the global minimum . in practice , there are no other local minimum points near the global minimum point . in the paper @xcite where the new q - tensor model was first introduced , the coefficients @xmath65 , @xmath66 , and @xmath67 were expressed in terms of the molecule s shape factor @xmath15 , under the assumption that the interaction is the hard core potential of the rod - like molecules without attraction effects , with @xmath16 the diameter of semi - sphere at two ends of the rod and @xmath17 the length of the rod . if we take account of the attraction effects between molecules , these coefficients would be functions of temperature , and alter within a few order of magnitudes as the temperature changes @xcite . in this paper , we regard the new q - tensor model as a phenomenological model . we set these coefficients near the values deduced using the hard core potential . fix @xmath104 the coefficients @xmath65 and @xmath66 are derived in terms of microscopic shape factor @xmath105 . we set @xmath106 to ensure the lower boundedness , and set @xmath107 and @xmath108 since they are order of magnitude smaller . two remaining coefficients @xmath109 and @xmath110 are not fixed . empirically , they are more important to determine the phases of liquid crystals . we will set these two coefficients near @xmath111 and @xmath112 within an order of magnitude . note that we can not relate the coefficients @xmath65 , @xmath66 , @xmath67 with any specific liquid crystal materials so far , and the relationship is still under investigation . we consider the phase transition problem and the @xmath0-@xmath1 interface problem . the phase transition problem is to find the most stable phase of liquid crystals given the average density , to find the characterizations of the various phases , and to study the transition of phases from one to another as the average density increases . in some suitable environments , @xmath0 and @xmath1 phases will coexist and take up different regions in the liquid crystal materials . the @xmath0-@xmath1 interface problem is to investigate the interfacial region between the @xmath0 and @xmath1 phases . the mathematical formulation of these two problems will be given below . density variations are essential in both problems . in the phase transition problem , the density variation serves to lower the second order elastic energy @xmath30 , which renders the smectic phase stable . in the @xmath0-@xmath1 interface problem , given the chemical potential and the grand potential density , the density at @xmath0 and @xmath1 phases are different , which renders a natural variation of density at the interface . first , we consider the phase transition problem . since @xmath0 and @xmath1 phases are spatially homogeneous , and @xmath2 and @xmath3 phases are layered , we presume the solution to be periodic to reduce the computational costs . therefore , we consider the domain @xmath113 $ ] , and assume that the order parameters enjoy the periodic boundary conditions . given the average density @xmath114 , we need to minimize the average free energy @xmath115 , where @xmath116 is defined as equation ( [ equation reduced ] ) . we have the optimization problem @xmath117 spectral methods are used to calculate the free energy . the period @xmath118 , which is also a variable that needs to be optimized , represents the layer thickness in the smectic phases . figure [ illustration smectic ] illustrates the microscopic configuration of molecules of the smectic phases . the following phase transitions are found given various sets of @xmath109 and @xmath110 . * @xmath119 , @xmath120 . + @xmath121 . * @xmath122 , @xmath123 . + @xmath124 . * @xmath125 , @xmath126 . + @xmath127 . and the layer thickness @xmath118 are their important characterizations . ] -@xmath3 transition is second order . ] -@xmath3 transition is second order . ] the one - dimensional phase diagrams of @xmath0-@xmath1-@xmath2-@xmath3 transitions are presented using @xmath128 coordinates in figure [ fig : energy vs density ] , and using @xmath129 coordinates in figure [ fig : angle vs density ] . as the average density @xmath114 increases , the system exhibits @xmath0 , @xmath1 , @xmath2 , and @xmath3 phases sequentially . as shown in figure [ fig : energy vs density ] , @xmath0-@xmath1 and @xmath1-@xmath2 phase transitions are first order . @xmath0 and @xmath1 phases coexist as @xmath114 is within @xmath130 ; @xmath1 and @xmath2 phases coexist as @xmath114 is within @xmath131 . the free energies of both phases in these regions are local minima , which are confirmed by numerically evaluating the hessian matrix at the minimum points . it is noteworthy that the @xmath1-@xmath2 phase transition is of weak first order : at both of the local minimum points , the smallest eigenvalue of the hessian is near 0 , which indicates that the energy barrier between these two stable phases is rather small . from the physical point of view , the average density @xmath132 states that on average there are @xmath133 liquid crystal molecules with shape factor @xmath134 in a cube with volume @xmath38 , and the molecules take up roughly @xmath135 percent the volume of the entire space . as shown in figure [ fig : angle vs density ] , the @xmath2-@xmath3 phase transition is of second order . after @xmath114 exceeds @xmath136 , the tilt angle of director increases from @xmath137 to @xmath138 degrees continuously . in one layer , the centers of molecules align in order , but not exactly in a plane , which renders the layer thickness larger than the length of molecules . the optimized layer thickness for both @xmath2 and @xmath3 phases is roughly @xmath139 , i.e. @xmath140 times the length of the molecule , and it varies slightly within the order @xmath141 as the average density increases . the layer thickness is quite sensitive to the coefficients @xmath66 and @xmath67 . if @xmath110 decreases from @xmath142 to @xmath143 , the layer thickness will decrease from @xmath139 to @xmath144 . this relation between the coefficients and the layer thickness is explained using the vector model , with its details discussed in section [ section : cl ] . figure [ separ ] shows the components of the free energy as the average density increases , which exhibits the order of transitions more clearly . the definitions of these components are given as ( [ bulk ] ) , ( [ elastic2 ] ) , and ( [ elastic4 ] ) , and the entropy is the first part of the bulk energy @xmath29 . the entropy decreases at the beginning . it has positive leaps at @xmath0-@xmath1 and @xmath1-@xmath2 critical points , and finally increases gradually . the bulk energy increases gradually with a leap at @xmath1-@xmath2 critical point . the elastic energy @xmath30 and @xmath145 are zero in @xmath0 and @xmath1 phases . the second order elastic energy @xmath30 drops and continuously decreases after reaching @xmath1-@xmath2 critical point , while the fourth order elastic energy @xmath31 soars up and continuously increases after @xmath1-@xmath2 point , with a smaller magnitude than @xmath30 . all these components at @xmath2-@xmath3 critical point are continuous . the total energy increases steadily without significant leaps . the second order elastic energy @xmath30 contains two parts : derivative terms in the density @xmath146 and the oseen - frank energy which contains the derivative terms of director @xmath5 . this decomposition is further discussed in section [ section : cl ] . in smectic phases , the oseen - frank energy is zero for there is no distortion of the director . the derivative terms in the density @xmath146 in @xmath30 with negative coefficients are the reason for the density - varied phases to have a lower free energy . the second order derivative terms in the density in @xmath31 with positive coefficients are the reason for the total energy to be stable . -axis in the direction normal to the layer . ( a ) the relative density @xmath147 , defined as @xmath148 . ( b ) the eigenvalues of q - tensor , written as @xmath149 , where @xmath150 is perpendicular to @xmath5 ; @xmath151 is the principal nematic order parameter , and @xmath152 characterizes biaxial effects . ( c ) the biaxiality , defined as @xmath153 . ] -axis in the direction normal to the layer . ( a ) the relative density @xmath147 , defined as @xmath148 . ( b ) the eigenvalues of q - tensor , written as @xmath149 , where @xmath150 is perpendicular to @xmath5 ; @xmath151 is the principal nematic order parameter , and @xmath152 characterizes biaxial effects . ( c ) the biaxiality , defined as @xmath153 . ] -axis in the direction normal to the layer . ( a ) the relative density @xmath147 , defined as @xmath148 . ( b ) the eigenvalues of q - tensor , written as @xmath149 , where @xmath150 is perpendicular to @xmath5 ; @xmath151 is the principal nematic order parameter , and @xmath152 characterizes biaxial effects . ( c ) the biaxiality , defined as @xmath153 . ] the representative use of the order parameters and the biaxiality characterizing the smectic phase is shown in figure [ fig : s ] . the relative density @xmath154 is defined as @xmath148 . the q - tensor can be written as @xmath149 , where @xmath150 is perpendicular to the director @xmath5 ; @xmath155 is the principal nematic order parameter , and @xmath156 characterizes biaxial effects . the biaxiality @xmath157 is defined as equation ( [ biaxiality ] ) . in the smectic phase , the relative density @xmath154 and order parameter @xmath155 fluctuate within one layer . the maximum point of @xmath146 and @xmath155 are identical , where the centers of molecules concentrate and the molecules are more likely to point to the preferred direction . in the @xmath2 phase , the biaxiality @xmath157 vanishes , which supports that the uniaxial approximation often adopted in modeling @xmath2 . in @xmath3 phase , @xmath157 vanishes when the relative density @xmath154 is large . at the region between two layers where the relative density @xmath154 is small , the biaxiality @xmath157 is large . this phenomena can be explained intuitively : the group of molecules at the vicinity region between two layers have more freedom of orientation and are less symmetric than the group of molecules at the center of one layer . the @xmath0-@xmath1-@xmath2 transitions and @xmath0-@xmath1-@xmath3 transitions are presented given other sets of coefficients . the one - dimensional phase diagrams are presented using @xmath128 coordinates in figure [ fig : ina ] characterizing @xmath0-@xmath1-@xmath2 transitions , and in figure [ fig : inc ] characterizing @xmath0-@xmath1-@xmath3 transitions . in the @xmath2 phase , the layer thickness is roughly @xmath158 . in @xmath3 phase , the layer thickness is roughly @xmath159 , and the tilt angle is @xmath160 degree . the @xmath1-@xmath3 transition is also first order . -@xmath1-@xmath3 transitions . the remaining coefficients are @xmath125 and @xmath126 . ] -@xmath1-@xmath3 transitions . the remaining coefficients are @xmath125 and @xmath126 . ] according to liquid crystal experiments , the @xmath0-@xmath1 transition and the @xmath1-@xmath2 transition could be either first or second order , depending on the liquid crystal materials . the @xmath0-@xmath1 transition is most commonly believed to be of first order @xcite , but some experiments of mbba [ ( 4-butyl - phenyl)-(4-methoxy - benzylidene)-amine ] reported a second order transitions @xcite . the study of 8cb-10cb mixtures @xcite confirmed the prediction made by halperin _ that fluctuation change the second order @xmath1-@xmath2 transition into a weak first order one . the study of 4-n - alkoxybenzylidene-4-phenylazoaniline @xcite drew the conclusion that decreasing the length of the alkyl end chain drives the first order @xmath1-@xmath2 transition to be second order . according to monte - carlo simulation based on molecular theory , assuming the molecule s shape to be spherocylindrical , mcgrother _ et al . _ @xcite , bolhuis _ et al . _ @xcite , and lolson _ et al . _ @xcite showed that the @xmath1-@xmath2 transition is a first order one , but it tends to become continuous as the shape factor @xmath161 . in comparison , our simulation results predict the @xmath0-@xmath1 transition is of first order ; the energy barrier between the @xmath1 and @xmath2 phases is rather small , which indicates that the @xmath1-@xmath2 transition is of weak first order . for the @xmath2-@xmath3 transition , the study of 8s5 [ 4-n - pentyl - phenylthiol-4-n - octyloxybenzoate ] @xcite reported a second order transition , and the tilt angle increases from @xmath137 degree to @xmath162 degree continuously . the study of c7 [ 4-(3-methyl-2-chloropentanoyloxy)-4-heptyloxybiphenyl ] @xcite reported a first order @xmath2-@xmath3 transition , and the tilt angle ranges from 0 to 34 degree . in our simulations , the @xmath2-@xmath3 transition is of second order , and the tilt angle ranges from 0 to 23 degree . for the smectic layer thickness , most experiments indicated that the layer thickness of the smectic phase is around the molecule s length @xmath17 . a study of 8cb [ 4-n - octyl-4-cyanobiphenyl ] and 8ocb [ 4-n - octyloxy-4-cyanobiphenyl ] @xcite showed that when molecules form dimers , one smectic layer would contain two molecule layer overlapped a bit , such that the layer thickness of the smectic phase is larger than @xmath17 but within @xmath163 . in our simulations , the predicted layer thickness is larger than most experiments results . however , our predicted layer thickness is in a reasonable range , which is larger than the length @xmath17 of the molecules but within @xmath163 varying with different coefficients . consider the liquid crystals in the entire space . @xmath0 and @xmath1 are at two opposite ends of the space , sharing the same chemical potential and the same grand potential density . assume the problem is one dimensional , and the order parameters vary in one dimension and are homogeneous in the other two dimensions . given the angle @xmath60 between the interface normal and the director of the nematic phase , the configuration of the interface is determined . the microscopic configuration of molecules of @xmath0-@xmath1 interface is illustrated as figure [ snapshots ] . -@xmath1 interface , for the three different anchoring conditions . the configuration with tilt angle @xmath164 is the most stable . ] consider the region @xmath165 $ ] . we need to minimize the grand potential @xmath166=&\int_{-h}^h c ( x)(\ln c ( x ) + b_q : q-\ln z ( x ) ) { \mathrm{d}}x\\ + & \frac{1}{2}\big \ { \int_{-h}^h \big[a_{1}c^2 -a_{2 } ( cq_{ij})^2-a_{3 } ( cq_{4ijkl})^2\big ] { \mathrm{d}}x \\ + & \int_{-h}^h \big [ -g_1 ( \frac{{\mathrm{d}}}{{\mathrm{d}}x } c)^2 + g_2 ( \frac{{\mathrm{d}}}{{\mathrm{d}}x}(cq_{ij}))^2+g_3(\frac{{\mathrm{d}}}{{\mathrm{d}}x}(cq_{1k}))^2 -g_4 \frac{{\mathrm{d}}}{{\mathrm{d}}x}(cq_{11})\frac{{\mathrm{d}}}{{\mathrm{d}}x}(c)\\ & \quad\quad+g_5 ( \frac{{\mathrm{d}}}{{\mathrm{d}}x}(cq_{4ijkl}))^2 + g_6(\frac{{\mathrm{d}}}{{\mathrm{d}}x}(cq_{41klm}))^2+g_7 \frac{{\mathrm{d}}}{{\mathrm{d}}x}(cq_{411kl})\frac{{\mathrm{d}}}{{\mathrm{d}}x}(cq_{kl } ) \big ] { \mathrm{d}}x\\ + & \int_{-h}^h \big [ h_1(\frac{{\mathrm{d}}^2}{{\mathrm{d}}x^2}(c))^2+h_2(\frac{{\mathrm{d}}^2}{{\mathrm{d}}x^2}(cq_{pq}))^2+h_3(\frac{{\mathrm{d}}^2}{{\mathrm{d}}x^2}(cq_{11}))^2 + h_4(\frac{{\mathrm{d}}^2}{{\mathrm{d}}x^2}(cq_{1p}))^2 \big ] { \mathrm{d}}x \big \}\\ - & \mu \int_{-h}^h c(x ) { \mathrm{d}}x , \end{aligned}\ ] ] under proper boundary conditions . the chemical potential is chosen to meet the coexistence condition , @xmath167 . the order parameters for the stable isotropic phase are @xmath168 and @xmath56 ; the order parameters for the stable nematic phase are @xmath169 and @xmath170 with @xmath171 . we anchor the angle @xmath60 between the director and the interface normal to be constant all over the space , and perform simulations with a series of @xmath60 . thus , the boundary conditions are @xmath172 where @xmath173 , @xmath174 , @xmath175 , @xmath176 , @xmath177 . the length @xmath118 is chosen large enough , which is @xmath178 in our computations . the coefficients @xmath65 , @xmath66 , and @xmath67 are given in equations ( [ coefficients ] ) with @xmath179 , @xmath180 . note that the coefficients @xmath66 here are all deduced by assuming the interaction potential to be the volume exclusion potential of rigid rod - like molecules . the results are summarized as follows . ; b ) the principal nematic order parameter @xmath155 ; c ) the biaxial effect @xmath156 . the definitions of @xmath155 and @xmath156 are given by denoting q - tensor as @xmath181 , where @xmath150 is perpendicular to @xmath182 . the centers of these profiles are shifted by @xmath183 in order to clearly display them for different tilt angles . ] ; b ) the principal nematic order parameter @xmath155 ; c ) the biaxial effect @xmath156 . the definitions of @xmath155 and @xmath156 are given by denoting q - tensor as @xmath181 , where @xmath150 is perpendicular to @xmath182 . the centers of these profiles are shifted by @xmath183 in order to clearly display them for different tilt angles . ] ; b ) the principal nematic order parameter @xmath155 ; c ) the biaxial effect @xmath156 . the definitions of @xmath155 and @xmath156 are given by denoting q - tensor as @xmath181 , where @xmath150 is perpendicular to @xmath182 . the centers of these profiles are shifted by @xmath183 in order to clearly display them for different tilt angles . ] figure [ curve1 ] ( a ) shows the profile of the density @xmath146 . the density profile for tilt angle @xmath60 greater than @xmath184 is a monotonically increasing function ; nevertheless , for the tilt angle smaller than @xmath184 , it displays a shallow dip near the isotropic side of the interface , and a small oscillation near the nematic side of the interface . the shallow dip near the isotropic side was captured by chen and noolandi @xcite in numerical simulation of onsager s molecular theory , in which it was explained by the competition between the entropy and the excluded - volume interaction . the oscillating behavior was noticed by allen @xcite in monte carlo simulations ; however , in monte carlo simulations , the oscillating behavior emerged for all cases of the tilt angle , and was explained as boundary effects . in the new q - tensor model , the oscillating behavior can be explained by the negative coefficients of the derivative terms of density @xmath146 : with oscillation , the elastic energy @xmath30 can be lower . on the contrary , the fourth order elastic energy @xmath31 serves to stabilize the energy at the interface . the interfacial width of the density is roughly @xmath185 . it is at its narrowest at @xmath164 . figure [ curve1 ] ( b ) shows the principal nematic order parameter @xmath155 . unlike the density profile , the profile of @xmath155 is always a monotonic function . the centers of the profiles for @xmath146 and @xmath155 are different . there is a `` phase shift '' for @xmath155 to the isotropic side with roughly @xmath186 compared to the density profile . unlike the density profile , the interfacial width of @xmath155 remains identical at @xmath185 as tilt angle @xmath60 increases . the interfacial width is not sensitive to the coefficients @xmath67 . no matter how we alter the coefficients @xmath67 with an order of magnitude , the interfacial width is always roughly @xmath185 ; nevertheless , as @xmath67 increases , the oscillation near the nematic side is mitigated but still exists . figure [ curve1 ] ( c ) shows the biaxiality parameter @xmath156 . the biaxial effect only appears significantly in the interfacial region . it is opposite between the isotropic side and the nematic side , and is stronger at the isotropic side . the biaxiality is the strongest for @xmath164 . there is no biaxiality for the @xmath187 case because of the rotational symmetry . in general , the biaxial effect for @xmath0-@xmath1 interface is quite weak . ] figure [ surface ] shows the dependence of the isotropic - nematic surface energy on the angle @xmath60 . the surface energy is defined as the difference between the grand potential of the interface and that of the isotropic phase . the surface energy is a monotonic function of the tilt angle and has a minimum at @xmath188 with value @xmath189 . therefore , the interface prefers to have a tilt angle of @xmath190 , which is the configuration as figure [ snapshots ] ( c ) . in another word , if we do not assume the tilt angle @xmath60 constant , but just anchor it at the boundary , it would tend to @xmath190 at the interface , which was the setup and situation in the study by monte carlo simulations @xcite . there are several previous studies of the @xmath0-@xmath1 interface problem investigated using onsager s molecular theory with volume exclusion potential for rod - like molecules @xcite @xcite @xcite . these simulations gave the interfacial width roughly at @xmath163 , which agree with the new q - tensor model . chen and noolandi @xcite predicted that the surface energy attains minimum at @xmath177 with value @xmath191 . koch and harlen @xcite used a generalization of onsager s approach and did a more comprehensive investigation . they predicted that the surface energy attains minimum at @xmath177 with value @xmath192 . our new q - tensor model gives it a lower value @xmath189 . since the new q - tensor model is a phenomenological model , and we alter the coefficients in our models , the value of the surface energy is only comparable in orders . to find an accurate value of the surface energy , it is better to use onsager s molecular theory . the experimental study of mbba @xcite showed that the surface energy attains minimum at the tilt angle @xmath177 , which is consistent with our calculations . however , the study of ncb @xcite showed that the minimum of surface energy is attained at a tilt angle close to @xmath193 . in the meanwhile , the experimental values of the interfacial width for the ncb are of the order of @xmath194 ( with molecule s length about @xmath195 ) , in which the ratio is much larger than the ratio in our work and numerical simulations with hard rod molecules @xcite @xcite @xcite . the numerical simulation of the @xmath0-@xmath1 interface problem in modeling specific materials is still under investigation . in the paper @xcite , the oseen - frank model and the ericksen model were derived using the new q - tensor model by assuming a constant density . in this section , we show the effectiveness of the new q - tensor model by comparing it with other classical models in various scales aiming to model density variations . de gennes pioneered the q - tensor , which provides information on both the preferred molecular orientation and the degree of orientational order , to describe nematics phases @xcite . later on , by exploiting an analogy between smectic liquid crystals and superconductors , de gennes again proposed an independent , complex order parameter that allows for the description of nematic - smectic phase transitions @xcite . successive models generalized the landau - de gennes model and coupled the nematic order tensor with the complex smectic order parameter . in the work of biscari _ et al . _ @xcite , the free energy contained the polynomial terms and the first order derivative terms . in the work of mukherjee @xcite , the free energy contained the polynomial terms , the first order and the second order derivative terms of these order parameters . these models successfully modeled the @xmath0 , @xmath1 , @xmath2 , and @xmath3 phases with computational efficiency . however , it is noteworthy that the complex order parameter is not suitable for numerical simulations of liquid crystals in confined geometries , such as disclinations and defects . the work of pevnyi _ et . @xcite addressed the above problem and replaced the complex order parameter with the density @xmath24 . the other order parameter @xmath5 entered the free energy through the second order tensor @xmath196 , so we regard this model as a tensor model . this model gave reasonable numerical results to model the @xmath2 phase in confined geometries with defects . the free energy contains the polynomial terms of the density , and the second order derivative terms of the order parameters , which can be viewed as a reduced case of the new q tensor model . the new q - tensor model are more generalized comparing to the above models . for the new q - tensor model , the entropy term is in the thermotropic form defined by ball and majumdar @xcite , but no longer has the polynomial terms as in the landau - de gennes model . such an entropy term ensures the boundedness of the eigenvalues of the q - tensor , preserving its physical meaning . if we assume @xmath24 is constant , @xmath197 would degenerate to the energy functional of the landau - de gennes model , which is able to characterize the stable nematic phase @xcite . the introduction of @xmath26 is necessary , since otherwise the elastic constant @xmath18 would equal to @xmath20 , which is not always true . moreover , if we need to model density variation phenomena in which @xmath24 is not constant , @xmath197 would not be bounded from below , for the reason that the derivative terms of @xmath24 in @xmath30 are negative . it is necessary to consider at least the second order derivative terms @xmath31 to stabilize the energy functional . based on onsager s molecular theory , assuming a two particle interaction potential and introducing an orientational order parameter , maier and saupe proposed a molecular field theory to characterize the @xmath0-@xmath1 transition @xcite . adding a positional order parameter , mcmillan extended the theory to include the @xmath2 phase @xcite . successive improvements were made to account for more complicated potentials , such as the work of mederos _ et al . _ @xcite which incorporated the hard core interaction , and the work of gorkunov _ et al . _ @xcite which considered the fragments attraction and repulsion interaction . most recently , pajak and osipov @xcite proposed a generalization of the mcmillan model including @xmath198-@xmath199-@xmath200-@xmath201 phases . in the smectic phase , let @xmath202 be the unit layer normal , @xmath203 be the unit normal of the tilt plane , and @xmath204 be their unit orthogonal complement . the layer thickness is presumed to be @xmath205 . the order parameters of the generalized mcmillan model are @xmath206 , @xmath207 , and @xmath208 , defined in terms of the density function of molecules @xmath209 as @xmath210 in the @xmath211 framework , these order parameters have components decomposition @xmath212 the free energy of the generalized mcmillan model can be expressed as @xmath213-k_b t c_0 \ln z,\ ] ] which represents the sum of the entropy and the quadratic terms of @xmath206 , @xmath214 , @xmath215 , @xmath216 , @xmath217 , @xmath218 , and @xmath219 . here , @xmath114 is the average density , @xmath220 is the maier - saupe constant which determines the @xmath0-@xmath1 transition temperature , @xmath221 are constants , and @xmath222 is the partition function which can be represented by the order parameters . the generalized mcmillan model can be directly derived from the new q - tensor model if we assume @xmath223 where @xmath24 and @xmath12 are the order parameters of the new q - tensor model , and @xmath206 , @xmath224 , and @xmath225 are the order parameters of the generalized mcmillan model . intuitively , the order parameters of the generalized mcmillan model are the fourier coefficients of the order parameters of the new q - tensor model , and the generalized mcmillan model can be interpreted as the one mode approximation of the new q - tensor model . the generalized mcmillan models are mechanistic models based on self - consistent field theory . the coefficients are linked with microscopic quantities , i.e. the parameters in interaction potentials . another advantage of the generalized mcmillan model is its elegance and simplicity , where the energy functional can be expressed by several scalar parameters . however , the order parameters of the generalized mcmillan model are averaged over the entire space , failing to characterize some small scale phenomena where the domain has confined geometry and spatial variance , for example , phase transitions in confined geometries , wetting phenomena , surface - induced bulk alignment , defects and disclinations , and interfaces . the chen - lubensky model @xcite is a vector model based on the landau - ginzburg theory , characterizing the @xmath1-@xmath2-@xmath3 transitions . the order parameters are the density @xmath24 and the director @xmath4 . the corresponding free energy consists of two parts , @xmath226 where @xmath227 contains terms up to second order derivatives of the density @xmath24 , @xmath228 ^ 2 -c_{1 } ( \ttn \cdot \nabla c)^2+\frac{c_{1}^2}{4d_{1}}c^2 \\ & \qquad+c_{2}(\delta_{ij}-n_in_j ) \nabla _ i c \nabla_j c + d_2 [ ( \delta_{ij}-n_in_j)\nabla_i \nabla_j c]^2\big ) { \mathrm{d}}\tx , \end{aligned}\ ] ] and @xmath229 is the oseen - frank energy for distortions in terms of the nematic director @xmath4 , @xmath230 ^ 2 + k_3 \vert \ttn \times ( \nabla \times \ttn ) \vert ^2\big ) { \mathrm{d}}\tx . \end{aligned}\ ] ] the coefficients @xmath231 , @xmath232 , @xmath233 , @xmath234 , @xmath235 , and @xmath236 are determined through experiments . these coefficients are intuitive : @xmath236 are elastic constants ; @xmath231 and @xmath233 determine the horizontal period ; @xmath232 and @xmath234 determine the perpendicular period . in the paper @xcite where the new q - tensor model was first introduced , assuming the q - tensor uniaxial and @xmath24 and @xmath237 constant , the oseen - frank energy @xmath229 was deduced and the elastic constants @xmath18 , @xmath19 , @xmath20 were expressed analytically . in the following paragraphs , making alternative assumptions that q - tensor is uniaxial and constant , we will deduce a vector model similar to the free energy @xmath227 . assume q - tensor is uniaxial and spatially homogeneous , @xmath238 where @xmath239 , @xmath240 , and @xmath5 are constant . under this assumption , the energy functional can be formulated as @xmath241 = & \int_\omega c \ln ( c ) { \mathrm{d}}\tx + \frac{1}{2}\int_\omega \biggl(\hat a_1 c^2 -\hat g_1 \vert \nabla c \vert^2 - \hat g_2 ( \ttn \cdot \nabla c)^2 + \hat h_1 ( \nabla^2 c : \ttn\ttn)^2\\ & + \hat h_2 ( \nabla^2 c:\ttn\ttn)\triangle c + \hat h_3 ( \triangle c ) ^2 + \hat h_4\vert \ttn \cdot \nabla^2 c \vert^2 + \hat h_5 \vert \nabla^2 c\vert^2 \biggl ) { \mathrm{d}}\tx,\\ \end{aligned}\ ] ] where @xmath242 , @xmath243 , and @xmath244 are functions of @xmath65 , @xmath66 , @xmath67 , @xmath239 , and @xmath240 , linear with respect to @xmath65 , @xmath66 , @xmath67 , and quadratic with respect to @xmath239 and @xmath240 . the expressions of @xmath242 , @xmath243 , @xmath244 in terms of @xmath65 , @xmath66 , @xmath67 , @xmath239 , @xmath240 are listed in the appendix . the deduced vector model is able to characterize @xmath0-@xmath1-@xmath2-@xmath3 phase transitions . let the average density be @xmath245 , the entropy term is almost linear to @xmath114 , and the interaction terms are quadratic to @xmath114 . as @xmath114 is small , the entropy term dominates , and the system exhibits isotropic phases . as @xmath114 increases , the interaction terms dominate , and the system transits to nematic and smectic phases . two quantities are important in smectic phases : layer thickness and tilt angle . to derive these quantities , assuming a given @xmath239 , we plug a trial function @xmath246 into the energy functional , where @xmath247 , @xmath248 , and @xmath249 . the free energy becomes @xmath250\right\}{\mathrm{d}}\tx , \end{aligned}\ ] ] where @xmath251 is a quadratic function of @xmath252 and @xmath253 . minimizing the free energy over @xmath254 , @xmath255 , and @xmath256 , we obtain the minimizer @xmath257 , @xmath258 , and @xmath259 , with relations @xmath260 these two equations can be solved iteratively , and @xmath259 is easy to compute but can not be expressed explicitly . the system will exhibit * @xmath2 phase , if @xmath261 , @xmath262 , and @xmath263 . let @xmath264 $ ] . the layer thickness is @xmath265 . * @xmath3 phase , if @xmath266 , @xmath267 , and @xmath263 . the layer thickness is @xmath268 , and the tilt angle is @xmath258 . * no smectic phase , otherwise . the analytical derivation above can explain the relations between the coefficients @xmath65 , @xmath66 , @xmath67 , and the tilt angle and layer thickness . it can also help to set the coefficients in the new q - tensor model . in comparison with the deduced vector model , the chen - lubensky model lacks entropy terms which leads to its failure in characterizing the isotropic phase . in the chen - lubensky model , the oseen - frank energy and the derivative terms of @xmath24 are added up directly . on the contrary , in the new q - tensor model , the oseen - frank energy and the derivative terms of @xmath24 are deduced based on different assumptions as discussed previously . if no assumptions are made , there will be cross terms incorporating the gradient of @xmath5 and the gradient of @xmath24 . furthermore , there are two extra terms in the deduced vector model , @xmath269 and @xmath270 . these two terms may lead to different phenomena in the subspace perpendicular to the director . except for these differences , the deduced vector model agrees with the chen - lubensky model . in this work , we have investigated the new q - tensor theory for liquid crystals focusing on density variations . phenomenologically , the @xmath0-@xmath1-@xmath2-@xmath3 phase transitions are predicted , and the @xmath0-@xmath1 interface is investigated . we have also drawn comparisons of the proposed model with the classical models , and strong connections are found . we have shown that all these classical models can be derived from the new q - tensor model . in comparison , the high dimension of the order parameter sets considerable obstacles in computations for the onsager s molecular model . the order parameter @xmath4 adopted in chen - lubensky model fails to explain the degree of orientational order . landau - de gennes model is famous for its tractability in mathematical analysis and computations . however , generalizations of q - tensor model are needed to include smectic phase . the generalized mcmillan model is a good approximation of the onsager s theory , and efficient in computations . the weakness is in the ultimate energy functional which fails to reflect detailed physical understanding . in addition , the generalized landau - de gennes models did not explain how their energy functionals were derived in details . on the other hand , the new q - tensor model can be derived from onsager s molecular theory . the introduction of density enables us to characterize smectic phases and the density variations in physical phenomena . the new q - tensor model captures much of the essential physics while remaining mathematically tractable and efficient computationally . compared to all classical models , the new q - tensor model is a good trade - off . one notes that the coefficients @xmath65 , @xmath66 , @xmath67 in the new q - tensor model can not be related with any specific liquid crystal materials so far . the physical relevance of the coefficients is still under investigation . the new q - tensor model studies the phase transitions as a function of the average density . it is difficult to consider temperature - driven phase transitions , for which one needs to include the attraction interaction . this will be studied in future work . defects are essential physical phenomena for liquid crystals . the effects of density variations in defects characterized by the new q - tensor model will also be studied in the future . the authors wish to thank dr . jiequn han and dr . wei wang for their help in discussions and advices for writings . pingwen zhang is partly supported by nsf of china under grant 11421101 and 11421110001 . the relations between @xmath242 , @xmath243 , @xmath244 ( the coefficients in vector model with energy functional ( [ vector model ] ) ) and @xmath65 , @xmath66 , @xmath67 ( the coefficients in the new q - tensor model with energy functional ( [ energy0 ] ) ) are as follows . @xmath239 and @xmath240 are the principal nematic order parameters defined as ( [ s2s4 ] ) . , _ molecular order and surface tension for the nematic - isotropic interface of mbba , deduced from light reflectivity and light scattering measurements _ , molecular crystals and liquid crystals , 22 ( 1973 ) , pp . 317 - 331 .

in this article , we study the new q - tensor model previously derived from onsager s molecular theory by han _ et al . _ [ _ arch . rational mech . anal . _ , 215.3 ( 2014 ) , pp . 741 - 809 ] for static liquid crystal modeling . taking density and q - tensor as order parameters , the new q - tensor model not only characterizes important phases while capturing density variation effects , but also remains computationally tractable and efficient . we report the results of two numerical applications of the model , namely the isotropic nematic smectic - a smectic - c phase transitions and the isotropic nematic interface problem , in which density variations are indispensable . meanwhile , we show the connections of the new q - tensor model with classical models including generalized landau - de gennes models , generalized mcmillan models , and the chen - lubensky model . the new q - tensor model is the pivot and an appropriate trade - off between the classical models in three scales . liquid crystals , q - tensor model , density variations , smectic phase , phase transition , isotropic - nematic interface

1410.1612

radio galaxies ( rgs ) represent the largest single objects in the universe . powered by an active galactic nucleus ( agn ) jets emerge from the central engine , which very likely is a super - massive black hole accreting matter surrounding it . there is a huge range of linear extent of the rgs : from less than @xmath0 pc gigahertz - peaked spectrum ( gps ) , through @xmath0 @xmath1 pc compact steep spectrum ( css ) , @xmath1 @xmath2 pc normal - size sources , up to greater than 1 mpc giant radio galaxies ( grg ) . the three largest grgs , recognized up to now , are shown in fig . 1 . although giant - size radio sources are very rare among rgs , from many years they have been of a special interest for several reasons . their very large angular size on the sky give an excellent opportunity for the study of radio source physics . they are also very useful to study the density and evolution of the intergalactic and intracluster environment . one of the key issues of the current research is attempt to trace an individual evolution of rgs . is there a single evolutionary scheme governing the linear extent of radio sources , or do small and large sources evolve in a different way ? to answer this question , in a number of papers , both theoretical and observational , attempts were undertaken to recognize factors which may differentiate giants from normal - size sources . it seems that there is no a single factor responsible for the size of classical radio sources ; the large size of grgs probably results from a combination of different factors like : age of a source , jet power , density of environment , etc . still very limited number of well studied grgs is a reason of that uncertainty . therefore the phenomenon of grg is still open for a further research . during the iau symposium no . 199 ( december 1999 ) machalski & jamrozy ( 2002 ) presented an evidence that only a very small fraction of expected faint grgs of fanaroff - riley ( 1974 ) type ii ( frii ) was detected at that time . in order to find those missed giant sources we inspected the radio maps available from the large radio surveys : nvss ( condon et al . , 1998 ) and the first part of first ( becker et al . , 1995 ) . the maps of these surveys , made with two different angular resolution ( 45@xmath3 and 5@xmath3 , respectively ) at the same observing frequency of 1.4 ghz , allowed ( i ) an effective removal of confusing sources , ( ii ) a reliable determination of morphological type of the giant candidate , and ( iii ) a detection of the compact radio core necessary for the proper identification of the source with its parent optical object . as the result we selected a sample of 36 grg candidates ( cf . machalski et al . , 2001 ) . in order to identify their host galaxy , to determine its distance and other physical properties , we have carried out several radio and optical observations of the sample sources . in particular , we already made optical spectroscopy and got redshift for 17 out of 36 galaxies ( spectroscopic redshifts of 5 sample galaxies were available prior our research ) . out of 22 galaxies , 19 host giant radio sources . in the meantime , similar efforts have been undertaken by schoenmakers et al . ( 2001 ) and lara et al . owing to the above studies , the statistics of giant radio galaxies is enlarged . the numbers of frii - type grgs , expected from our population analysis ( machalski & jamrozy 2002 ) , are recalled in table 1 and compared with the observed numbers . the observed numbers denoted by an asterisk refer to the data available in 1999 , while other are from the beginning of the year 2003 . lccc & @xmath4 mjy & @xmath5 jy & @xmath6 jy + observed & 64/11@xmath7 & 31/26@xmath7 & 11/9@xmath7 + expected & 350 & 45.7 & 8.8 + obs / expected & 18%/3%@xmath7 & 68%/57%@xmath7 & 122%/100%@xmath7 + two examples of grgs from our sample are shown in fig . 2 . j1343 + 3758 with the linear size of 3.14 mpc has appeared to be the third largest source in the universe ( machalski & jamrozy 2000 ) , while j1604 + 3438 represents a very rare type of agn a so - called double - double rg ( cf . schoenmakers et al . 2000 ) which shows two pairs of lobes likely originating from an old and a new cycle of activity . low - resolution optical spectra of host galaxies of these two giant radio sources are shown in fig . 3 . some of the above data are used to constrain the existing analytical models of a dynamical evolution of frii - type radio sources ( machalski et al . 2003 ; chyy et al . our investigations of the new giant radio sources are in progress . however , we would like to extend them on grgs on the southern sky . there are several scientific reasons for such a project , and the main of them are : * all of the recent systematic search for new giants ( lara et al . 2001 , machalski et al . 2001 , schoenmakers et al . 2001 ) were performed on the northern sky . furthermore , only about 17% of the presently known grgs have negative declinations , and gross of them are high flux density ( @xmath80.5 jy ) nearby objects . therefore , one can expect a large number of undetected yet grgs on the southern hemisphere very useful for a further enlargement of their still unsatisfactory statistics . * the development of astronomical high - technology facilities i.e. the existing and planned large optical telescopes on the south ( vlt , salt ) is very rapid . therefore , it should be easy to attain the redshift of new grg hosting galaxies which is the crucial observational parameter for determination of all physical parameters of the radio sources like their distance , projected linear size , volume of their lobes or cocoon , luminosity , etc . the above needs low - resolution spectroscopic observations of usually faint optical counterparts ( which in many cases have very low apparent magnitudes @xmath9 ) in a reasonably short time . * there is a high probability that the planned powerful radio interferometer the square kilometer area ( ska ) will be located in the south ( there are two southern site candidates for its possible location , i.e. australia or south africa ) . it would give the opportunity to detect and study low surface - brightness grgs with a very high angular resolution and sensitivity in the future . this , in turn , would allow to recognize a very last stage of the dynamical evolution of classical double radio galaxies and learn a typical lifetime and end of that physical process . becker , r.h . , white , r.l . , & helfand , d.j . 1995 , , 450 , 559 chyy , k. , jamrozy , m. , kleinman , s.j . , krzesiski , j. , machalski , j. , mcmillan , nitta , a. , serafimovich , n. , & zola , s. 2003 , baltic astronomy , ( proceedings of the jenam 2003 , astro - ph/0310606 ) condon , j.j . , cotton , w.d . , greisen , e.w . , yin , q.f . , perley , r.a . , taylor , g.b . , & broderick , j.j . 1998 , , 115 , 1693 fanaroff , b.l . , & riley , j.m . 1974 , mnras , 167 , 31 lara , l. , cotton , w.d . , feretti , l. , giovannini , g. , marcaide , j.m . , marquez , i. , & venturi , t. 2001 , , 370 , 409 machalski , j. , chyy , k. , & jamrozy , m. 2003 , submitted for mnras , ( astro - ph/0210546 ) machalski , j. , & jamrozy , m. 2002 , in iau symposium 199 , the universe at low radio frequencies , ed . a. p. rao , g. swarup , & gopal - krishna ( san francisco : asp ) , 203 machalski , j. , jamrozy , m. , & zola , s. 2001 , , 371 , 445 machalski , j. , & jamrozy , m. 2000 , , 363 , l17 schoenmakers , a.p . , de bruyn , a.g . , rttgering , h.j.a . , & van der laan , h. 2001 , , 374 , 861 schoenmakers , a.p . , de bruyn , a.g . , rttgering , h.j.a . , van der laan , h. , & kaiser , c.r . 2000 , , 315 , 371

an extensive search for distant giant radio galaxies on the southern hemisphere is justified . we emphasize the crucial role of optical spectroscopy in determination of their basic physical parameters , i.e. the distance , projected linear size , volume of their lobes or cocoon , luminosity , etc . , and argue that salt will be the best instrument for such a task . # 1_#1 _ # 1_#1 _ = # 1 1.25 in .125 in .25 in

astro-ph0401182

the unexpected behavior of some fusion - evaporation cross sections at energies far below the coulomb barrier , has challenged the theoretical models to explain the , so called , fusion hinderance phenomena in true coupled - channels calculations ( ccc ) for reactions such as @xmath4ni+@xmath4ni , @xmath1ni+@xmath1ni , and @xmath1ni+@xmath2mo @xcite . the ccc could , however , be sensitive to the so far unobserved , hence not - included , high - lying states . misicu and esbensen @xcite were the first who succeeded in describing the above said three reactions in terms of a density - dependent m3y interaction , modified by adding a repulsive core potential @xcite . the repulsive core changes the shape of the inner part of the potential in terms of a thicker barrier ( reduced curvature @xmath5 ) and shallower pocket . here , deformations are included up to hexadecapole ( @xmath6 ) and the orientation degrees of freedom is integrated over all the allowed values in the same plane . the dynamical cluster - decay model ( dcm ) of preformed clusters by gupta and collaborators @xcite is found recently @xcite to have barrier modification effects as the inbuilt property , where barrier lowering " at sub - barrier energies arise in a natural way in its fitting of the only parameter of model , the neck - length parameter . the difference of actually - calculated barrier from the actually - used barrier height , corresponding to the neck - length parameter for best - fitted fusion - evaporation cross section , gives the barrier lowering " in dcm , whose values are found to increase as the incident energy decreases to sub - barrier energies . calculations are based on @xmath7 deformations and orientation @xmath8-dependent nuclear proximity potential of blocki _ et al . _ @xcite . very recently , the wong formula @xcite is also extended by gupta and collaborators @xcite to include its angular momentum @xmath0-summation explicitly , which is also shown to contain the barrier modification effects due to the @xmath0-dependent barriers . however , for the @xmath3ni - based fusion - evaporation cross sections @xcite , a further modification of barriers is found essential for below - barrier energies , which is implemented empirically either by lowering the barrier " or narrowing the barrier curvature " by a fixed amount for all @xmath0 s in the potential calculated by using the proximity potential of blocki _ et al . _ @xcite , but with multipole deformations @xmath6 and @xmath8-integrated for co - planer nuclei . apparently , the depth of the potential pocket plays no role , in both the dcm and @xmath0-summed wong formula ( the two models are same for capture reactions ) . in this contribution , we use within the @xmath0-summed wong model , the nuclear proximity potential obtained recently @xcite for the skyrme nucleus - nucleus interaction in the semiclassical etf approach . using sedf , the universal function of proximity potential is obtained as a sum of the parametrized spin - orbit - density - independent and the spin - orbit - density - dependent universal functions ( uf s ) , with different parameters of uf s obtained for different skyrme forces @xcite . this method has the advantage of introducing the barrier modifications at sub - barrier energies , if needed , by either ( i ) modifying the fermi density parameters ( the half - density radii and/ or surface thicknesses , for exact " sedf calculations @xcite ) , ( ii ) the constants of the parametrized uf s @xcite or ( iii ) change the skyrme force itself since a different skyrme force would give different barrier characteristics ( height and curvature ) . this later possibility is exploited here in this paper . it is possible that some skyrme force would fit the data for one reaction , but not for another reaction and hence requiring barrier modification " or another skyrme force . section 2 gives briefly the semiclassical etf method using sedf , including details of approximations used for adding densities . section 3 discusses the @xmath0-summed wong formula @xcite . our calculations are given in section 4 , and a brief summary of results in section 5 . the sedf in semiclassical etf method provides a convenient way for calculating the interaction potential between two nuclei . in the hamiltnian density , the kinetic energy density @xmath9 as well as the spin - orbit density @xmath10 are functions of the nucleon density @xmath11 , @xmath12 . for the composite system , the densities can be added in either adiabatic or sudden approximation , but we are interested in sudden densites since the different terms of skyrme hamiltonian density are then found to constitute the nuclear proximity potential @xcite . the sudden densities are defined with or without exchange effects ( due to anti - symmetrization ) , and the one without exchange effects is also refered to as frozen density @xcite . in etf method , the lowest order @xmath9 is the thomas fermi ( tf ) kinetic energy density @xmath13 , which already contains a large part of the exchange effects , and that the higher order terms include exchange effects in full . here we limit @xmath14 and @xmath15 to second order terms for reasons of being enough for numerical convergence @xcite . the nucleus - nucleus interaction potential in sedf , based on semiclassical etf model , is @xmath16 , \label{eq:1}\ ] ] where the skyrme hamiltonian density @xmath17 \nonumber\\ & & + \frac{1}{12}t_3\rho^\alpha\left[(1+\frac{1}{2}x_3)\rho^2-(x_3+\frac{1}{2 } ) ( \rho_n^2+\rho_p^2)\right ] + \frac{1}{4}\left[t_1(1+\frac{1}{2}x_1)+t_2(1+\frac{1}{2}x_2)\right ] \rho\tau \nonumber\\ & & -\frac{1}{4}\left[t_1(x_1+\frac{1}{2})-t_2(x_2+\frac{1}{2})\right ] ( \rho_n\tau_n+\rho_p\tau_p ) + \frac{1}{16}\left[3t_1(1+\frac{1}{2}x_1)-t_2(1+\frac{1}{2}x_2)\right ] ( \vec{\nabla}\rho)^2 \nonumber\\ & & -\frac{1}{16}\left[3t_1(x_1+\frac{1}{2})+t_2(x_2+\frac{1}{2})\right ] \left[(\vec{\nabla}\rho_n)^2+(\vec{\nabla}\rho_p)^2\right ] \nonumber\\ & & -\frac{1}{2}w_0\left[\rho\vec{\nabla}\cdot\vec{j}+\rho_n\vec{\nabla}\cdot\vec{j_n } + \rho_p\vec{\nabla}\cdot\vec{j_p}\right ] . % & = & h(\rho)+h(\vec{j } ) . \label{eq:2}\end{aligned}\ ] ] here , @xmath18 , @xmath19 , @xmath20 are the nuclear , kinetic energy and spin - orbit densities , respectively . m is the nucleon mass . @xmath21 , @xmath22 ( @xmath23=0,1,2,3 ) , @xmath24 and @xmath25 are the skyrme force parameters , fitted by different authors to ground state properties of various nuclei ( see , e.g. , @xcite ) . of the available forces , we use the old , well known siii and sv forces . coulomb effects are added directly . recently , agrawal _ et . _ @xcite modified the hamiltonian density ( [ eq:2 ] ) on two accounts , and obtained a new force gski : ( i ) the third term in ( [ eq:2 ] ) is replaced as @xmath26 , \label{eq:3}\ ] ] and ( ii ) a new term due to tensor coupling with spin and gradient is added as @xmath27 we have also used this new gski force , with additional six , two each of @xmath28 , @xmath29 and @xmath24 , constants . the kinetic energy density in etf method , up to second order @xcite , for @xmath30=@xmath31 or @xmath32 is @xmath33 with @xmath34 as the effective mass form factor , @xmath35 note that both @xmath36 and @xmath34 are each functions of @xmath11 and/ or @xmath37 only . the spin @xmath10 is a purely quantal property , and hence has no contribution in the lowest ( tf ) order . however , at the etf level , the second order contribution gives @xmath38 note , @xmath39 is also a function of @xmath11 and/ or @xmath37 alone . next , for the proximity potential we introduce the slab approximation of semi - infinite nuclear matter with surfaces parallel to @xmath40 plane , moving in @xmath41-direction , and separated by distance @xmath42 having minimum value @xmath43 . then , following blocki _ _ @xcite and gupta _ et al . _ @xcite , the interaction potential @xmath44 between two nuclei separated by @xmath45 , is given as @xmath46\right\}dz \nonumber\\ & = & 4\pi\bar{r}\gamma b \phi(d ) . \label{eq:8}\end{aligned}\ ] ] where @xmath47 is the mean curvature radius , and @xmath48 is the interaction energy per unit area between the flat slabs giving the universal function @xmath49 in terms of a dimensionless variable @xmath50 , with sufrace width @xmath51=0.99 fm . the nuclear surface energy constant @xmath52 $ ] mev @xmath53 . @xmath49 can be calculated exactly " or parametrized in terms of exponential and/or polynomial functions @xcite . for axially deformed and oriented nuclei , @xmath47 is given in terms of the radii of curvature @xmath54 and @xmath55 in the principal planes of curvature of each of the two nuclei ( @xmath23=1,2 ) at the points of closest approach ( defining @xmath43 ) , by @xmath56sin^2\phi + \left [ { { { 1}\over { r_{11}r_{22}}}+{{1}\over { r_{21}r_{12}}}}\right ] cos^2\phi . \label{eq : 9}\ ] ] here , @xmath57 is the azimuthal angle between the principal planes of curvature of two nuclei ( @xmath57=@xmath58 for co - planar nuclei ) . the four principal radii of curvature are given in terms of radii @xmath59 and their first and second order derivatives @xmath60 and @xmath61 _ w.r.t . _ @xmath62 , where @xmath63 , \label{eq:10}\ ] ] with @xmath64 as the spherical or half - density nuclear radius , @xmath65=2,3,4 ... , as multipole deformations , and @xmath66 as an angle between radius vector @xmath59 and symmetry axis , measured clockwise from symmetry axis . for the estimation of @xmath43 , we refer to @xcite for @xmath57=0 and to @xcite for @xmath670 . for nuclear denesity @xmath68 of each nucleus ( @xmath23=1,2 ) , we use the temperature t - dependent , two - parameter fermi density ( fd ) distribution , which for the slab approximation is given by @xmath69^{-1 } -\infty\leq z\leq \infty \label{eq:11}\ ] ] with @xmath70 , and central density @xmath71^{-1}$ ] . then , since @xmath72 , following our earlier work @xcite , for nuclen density we define @xmath73 with half density radii @xmath64 and surface thickness parameters @xmath74 in eq . ( [ eq:11 ] ) at @xmath75=0 , obtained by fitting the experimental data to respective polynomials in nuclear mass region @xmath76=4 - 238 , as @xmath77 @xmath78 the t - dependence in the above formulas are then introduced as in ref . @xcite , @xmath79,\qquad a_{0i}(t)=a_{0i}(t=0)[1 + 0.01t^2 ] . \label{eq:15}\ ] ] also , the surface width @xmath51 is made t - dependent @xcite , @xmath80 , where t is related to the incoming center - of - mass energy @xmath81 or the compound nucleus excitation energy @xmath82 via the entrance channel @xmath83-value , as @xmath84 . next , for the composite system , @xmath85 , and the @xmath86 and @xmath87 are added as per prescrption used . for sudden approximation ( with exchange effects ) , @xmath88 and for the frozen approximation ( equivalently , sudden without exchange effects ) , @xmath89 with @xmath90 , @xmath91 , and @xmath92 . in the following , we consider only frozen densities ( sudden without exchange ) , since @xmath0-summed wong formula within sudden approximation does not fit the @xmath1ni - based reactions data for use of siii force @xcite . finally , adding the coulomb and centrifugal interactions to the nuclear interaction potential @xmath44 , we get the total interaction potential for deformed and oriented nuclei @xcite , as @xmath93 with non - sticking moment - of - inertia @xmath94 ( = @xmath95 ) for @xmath96 . eq.([eq:18 ] ) gives the barrier height @xmath97 , position @xmath98 , and the curvature @xmath99 for each @xmath0 , to be used in extended @xmath0-summed wong s formula @xcite , discussed in the following section . according to wong @xcite , in terms of @xmath0 partial waves , the fusion cross - section for two deformed and oriented nuclei colliding with @xmath81 is @xmath100 with @xmath101 as the reduced mass . here , @xmath102 is the transmission coefficient for each @xmath0 which describes the penetration of barrier @xmath103 . using hill - wheeler @xcite approximation , the penetrability @xmath102 , in terms of its barrier height @xmath104 and curvature @xmath105 , is @xmath106^{-1 } , \label{eq:20}\ ] ] with @xmath99 evaluated at the barrier position @xmath107 corresponding to @xmath97 . note , the @xmath0-dependent potentials are required here , given by eq.([eq:18 ] ) . carrying out the @xmath0-summation in eq.([eq:19 ] ) empirically for a best fit to measured cross - section @xcite , and on integrating over the angles @xmath8 and @xmath57 , we get the fusion cross - section @xmath108 . we have made our calculations for the @xmath1ni+@xmath2mo reaction , using siii , sv and gski forces , with frozen densities . 1(a ) shows for one @xmath81 and fixed ( @xmath8 , @xmath57 ) , a comparison of interaction potentials for the three forces , illustrating their barrier height and position to be force - dependent . this is an interesting property , which we use to fit the fusion - evaporation cross - section in fig . interesting enough , the data @xcite for the above said @xmath1ni+@xmath2mo reaction fit the @xmath8-integrated ( @xmath57=0@xmath109 ) @xmath0-summed wong formula for only the new force gski . the other forces ( siii and sv ) would apparently need additional barrier modification effects to be added empirically @xcite . from the deduced @xmath110-values , presented in fig . 1(c ) , we notice that @xmath110 as a function of @xmath81 vary smoothly only for gski force , achieving zero value at sub - barrier energies and a tendency to saturate at an above - barrier energy . for the other forces ( siii and sv ) , non - fitting the data , the @xmath110 varies with @xmath81 erratically , in particular at sub - barrier energies , but could also be smoothed by adding futher barrier lowering " or barrier narrowing " emprically @xcite . concluding , the @xmath0-summed wong expression , using the barriers calculated in frozen - density approximation in semiclassical extended thomas fermi method , based on skyrme energy density formalism , describes the fusion - evaporation cross - section data for @xmath1ni+@xmath2mo reaction nicely with the new syrme force gski only . the variation of deduced @xmath110 with @xmath81 is found smooth . other skyrme forces ( siii and sv ) demand additional barrier modifications at sub - barrier energies for this reaction . however , the same calculation when applied @xcite to @xmath3ni+@xmath3ni data result in a similar good fit , with a smooth dependence of @xmath110 on @xmath81 , for only the skyrme force siii . thus , barrier - modification or no barrier - modification in @xmath0-summed wong expression depends on the choice of skyrme force in semiclassical etf method for frozen densities . ( a ) interaction potentials for one @xmath81 and fixed ( @xmath8 , @xmath57 ) , ( b ) fusion - evaporation cross - section as a function of @xmath81 for @xmath1ni+@xmath2mo reaction , calculated by using @xmath0-summed wong formula integrated over @xmath8 ( @xmath57=0@xmath109 ) and compared with experimental data @xcite , and ( c ) deduced @xmath110 values _ vs. _ @xmath81 , for the skyrme forces siii , sv and gski , using frozen densities . , title="fig : " ] ( a ) interaction potentials for one @xmath81 and fixed ( @xmath8 , @xmath57 ) , ( b ) fusion - evaporation cross - section as a function of @xmath81 for @xmath1ni+@xmath2mo reaction , calculated by using @xmath0-summed wong formula integrated over @xmath8 ( @xmath57=0@xmath109 ) and compared with experimental data @xcite , and ( c ) deduced @xmath110 values _ vs. _ @xmath81 , for the skyrme forces siii , sv and gski , using frozen densities . , title="fig : " ] ( a ) interaction potentials for one @xmath81 and fixed ( @xmath8 , @xmath57 ) , ( b ) fusion - evaporation cross - section as a function of @xmath81 for @xmath1ni+@xmath2mo reaction , calculated by using @xmath0-summed wong formula integrated over @xmath8 ( @xmath57=0@xmath109 ) and compared with experimental data @xcite , and ( c ) deduced @xmath110 values _ vs. _ @xmath81 , for the skyrme forces siii , sv and gski , using frozen densities . , title="fig : " ] the financial support from the department of sc . & tech . ( dst ) , govt . of india , and council of sc . & industial research ( csir ) , new delhi is gratefully acknowledged . 99 jiang c l _ et al . _ 2004 _ phys . lett . _ * 93 * 012701 ; jiang c l _ et al . _ 2005 _ phys . rev . c. _ * 71 * 044613 misicu s and esbensen h 2006 _ phys . lett . _ * 96 * 112701 misicu s and greiner w 2004 _ phys . c _ * 69 * 054601 see , e.g. , the review : gupta r k , arun s k , kumar r , and niyti 2008 _ int . rev . phys . ( irephy ) _ * 2 * 369 gupta r k 2010 _ lecture notes in physics , clusters in nuclei " _ , ed beck c , vol . i , ( springer verlag ) p 223 arun s k , kumar r , and gupta r k 2009 _ j. phys . g : nucl . part . phys . _ * 36 * 085105 gupta r k , arun s k , kumar r and bansal m 2009 _ nucl . phys . a _ * 834 * 176c blocki j , randrup j , swiatecki w j , and tsang c f 1977 _ ann . _ * 105 * 427 wong c y 1973 _ phys . lett . _ * 31 * , 766 kumar r , bansal m , arun s k , and gupta r k 2009 _ phys . c _ * 80 * 034618 bansal m and gupta r k 2010 _ phys . c _ , to be published . gupta r k , singh d , kumar r and greiner w 2009 _ j. phys . g : nucl . part . phys . _ * 36 * 075104 gupta r k , singh d , and greiner w 2007 _ phys . c _ * 75 * 024603 chattopadhyay p and gupta r k 1984 _ phys . c _ * 30 * 1191 li g -q 1991 _ j. phys . g : nucl . part . * 17 * 1 bartel j and bencheikh k 2002 _ eur . j. a _ * 14 * 179 brack m , guet c , and hakansson h -b 1985 _ phys . _ * 123 * 275 friedrich j and reinhardt p -g 1986 _ phys . rev . c _ * 33 * 335 agrawal b k , dhiman s k , and kumar r 2006 _ phys . rev . c. _ * 73 * 034319 gupta r k , singh n , and manhas m 2004 _ phys . rev . c _ * 70 * 034608 manhas m and gupta r k 2005 _ phys . c _ * 72 * 024606 shlomo s and natowitz j b 1991 _ phys . c _ * 44 * 2878 royer g and mignen j 1992 _ j. phys . g : nucl . part . phys . _ * 18 * 1781 kumar r and gupta r k , _ proc . int . symp . on nucl . _ 2009 * 54 * , ( brns - dae , govt . of india ) p 288 gupta r k _ et al . _ 2005 _ j. phys . g : nucl . part . phys . _ * 31 * 631 hill d l and wheeler j a 1959 _ phys . rev . _ * 89 * 1102 ; thomas t d 1959 _ phys . rev . _ * 116 * 703 .

we obtain the nuclear proximity potential by using semiclassical extended thomas fermi ( etf ) approach in skyrme energy density formalism ( sedf ) , and use it in the extended @xmath0-summed wong formula under frozen density approximation . this method has the advantage of allowing the use of different skyrme forces , giving different barriers . thus , for a given reaction , we could choose a skyrme force with proper barrier characteristics , not - requiring extra barrier lowering " or barrier narrowing " for a best fit to data . for the @xmath1ni+@xmath2mo reaction , the @xmath0-summed wong formula , with effects of deformations and orientations of nuclei included , fits the fusion - evaporation cross section data exactly for the force gski , requiring additional barrier modifications for forces siii and sv . however , the same for other similar reactions , like @xmath3ni+@xmath3ni , fits the data best for siii force . hence , the barrier modification effects in @xmath0-summed wong expression depends on the choice of skyrme force in extended etf method .

1010.3321

in 1983 thouless @xcite proposed a simple pumping mechanism to produce , even in the absence of an external bias , a quantized electron current through a quantum conductor by an appropriate time - dependent variation of the system parameters . experimental realizations of quantum pumps using quantum dots ( qds ) were already reported in the early 90 s @xcite . more recently , due to the technological advances in nano - lithography and control , such experiments have risen to a much higher sophistication level , making it possible to pump electron @xcite and spin @xcite currents through open nanoscale conductors , as well as through single and double qds @xcite . early theoretical investigations where devoted to the adiabatic pumping regime within the single - particle approximation @xcite . this is well justified for experiments with open qds , where interaction effects are believed to be weak @xcite and the typical pumping parameters are slow with respect the characteristic transport time - scales , such as the electron dwell time @xmath0 . this time - scale separation enormously simplifies the analysis of the two - time evolution of the system . within the adiabatic regime , inelastic and dissipation @xcite effects of currents generated by quantum pumps were analyzed . furthermore , issues like counting statistics @xcite , memory effects @xcite , and generalizations of charge pumping to adiabatic quantum spin pumps were also proposed and studied @xcite . non - adiabatic pumping has been theoretically investigated within the single - particle picture , either by using keldysh non - equilibrium green s functions ( negf ) with an optimal parametrization of the carrier operators inspired by bosonization studies @xcite , or by a flouquet analysis of the @xmath1-matrix obtained from the scattering approach @xcite . while the first approach renders complicated integro - differential equations for the green s functions associated to the transport , the second one gives a set of coupled equations for the flouquet operator . it is worth to stress that , in both cases the single - particle picture is crucial to make the solution possible and it is well established that both methods are equivalent @xcite . several works have provided a quite satisfactory description of quantum pumping for weakly interacting systems . in contrast , the picture is not so clear for situations where interaction effects are important . different approximation schemes have been proposed to deal with pumping in the presence of interactions and to address charging effects , which are not accounted for in a mean - field approximation . typically , two limiting regimes have been studied , namely , the one of small pumping frequencies @xmath2 , such that @xmath3 ( adiabatic limit ) @xcite and the one of very high frequencies , @xmath4 ( sudden or diabatic limit ) @xcite . nonadiabatic pumping is mainly studied as a side effect of photon - assisted tunneling @xcite , where @xmath4 . unfortunately , it is quite cumbersome to calculate corrections to these limit cases . for instance , the analysis of higher - order corrections to the adiabatic approximation for the current gives neither simple nor insightful expressions @xcite . in addition to the theoretical interest , a comprehensive approach bridging the limits of @xmath4 and @xmath5 has also a strong experimental motivation : most current experimental realizations of quantum pumping deal with qds in the coulomb blockade regime and @xmath6 . this regime was recently approached ( from below ) by means of a diagrammatic real - time transport theory with a summation to all orders in @xmath2 @xcite . however , the derivation implied the weak tunnel coupling limit , whereas experiments @xcite typically rely on tunnel coupling variations which include both weak and strong coupling . to address the above mentioned issues and to account for the different time scales involved it is natural to use a propagation method in the _ time domain _ @xcite . in this work we express the current operator in terms of density matrices in the heisenberg representation . we obtain the pumped current by truncating the resulting equations - of - motion for the many - body problem . the time - dependence is treated exactly by means of an auxiliary - mode expansion @xcite . this approach provides a quite amenable path to circumvent the usual difficulties of dealing with two - time green s functions @xcite . moreover , it has been successfully applied to systems coupled to bosonic reservoirs @xcite and to the description of time - dependent electron - transport using generalized quantum master equations for the reduced density matrix @xcite . since the auxiliary - mode expansion is well controlled @xcite , the accuracy of our method is determined solely by the level of approximation used to treat the many - body problem . the formalism we put forward is illustrated by the study of the charge pumped through a qd in the coulomb - blockade regime by varying its resonance energy and couplings to the leads . the external drive is parametrized by a single pulse , whose duration and amplitude can be arbitrarily varied . by doing so , the formalism is capable to reproduce all known results of the adiabatic limit and to explore transient effects beyond this simple limit . the paper is organized as follows . in sec . [ sec : model ] we present the resonant - level model , as well the theoretical framework employed in our analysis . in sec . [ sec : prop ] we introduce the general propagation scheme , suitable to calculate the pumping current at the adiabatic regime and beyond it . next , in sec . [ sec : app ] , we discuss few applications of the method . finally , in sec . [ sec : conclusion ] we present our conclusions . the standard model to address electron transport through qds is the anderson interacting single - resonance model coupled to two reservoirs , one acting as a source and the other as a drain . despite its simplicity , the model provides a good description for coulomb - blockade qds and for qds at the kondo regime , where the electrons are strongly correlated . in this paper we address the coulomb - blockade regime , for qds whose typical line width @xmath7 is much smaller than the qd mean level spacing @xmath8 , justifying the use of the anderson single - resonance model . in addition , in the coulomb blockade regime @xmath7 is much smaller than the resonance charging energy @xmath9 . the total hamiltonian is given by the usual threefold decomposition into a quantum dot hamiltonian @xmath10 , a hamiltonian @xmath11 representing the leads , and a coupling term @xmath12 , namely [ eq : hamiltonian ] @xmath13 the qd is modeled by a single level of energy @xmath14 , which can be occupied by spin - up and spin - down electrons , which interact through a contact interaction of strength @xmath9 . the qd hamiltonian reads @xmath15 where @xmath16 , @xmath17 and @xmath18 stand for electron number , creation and annihilation operator for the respective spin state @xmath19 in the dot . the two reservoirs , labeled as @xmath20 ( left ) and @xmath21 ( right ) , are populated by non - interacting electrons , whose hamiltonian reads @xmath22 where @xmath23 and @xmath24 stand for the electron creation and annihilation operators for the @xmath25-reservoir state @xmath26 , respectively . the reservoir single - particle energies have the general form @xmath27 with the @xmath28 accounting for a time - dependent bias . the stationary current due to a time - dependent bias was already addressed several years ago @xcite . for pumping , we take @xmath29 , as usual . finally , the coupling hamiltonian is given by @xmath30 with @xmath31 denoting the coupling matrix element between the qd and the reservoir @xmath25 . we are interested in the electronic current from reservoir @xmath25 to the qd state @xmath32 , which can be obtained from current operator @xmath33\,.\end{aligned}\ ] ] here and in the following we use units where the elementary charge @xmath34 and the reduced planck constant @xmath35 , unless otherwise indicated . to calculate @xmath36 we use the following equations of motion , which are obtained from the hamiltonian [ eqs . ] by means of the heisenberg equation , [ eq : eomtheisenberg ] @xmath37\;.\label{eq : eomtheisenberg3}\end{aligned}\ ] ] analogous equations hold for @xmath38 and @xmath39 . in the spirit of the scheme introduced by caroli and co - workers @xcite , we assume an initially uncorrelated density operator of the combined system , _ i.e. _ , we set @xmath40 for @xmath41 . further , we apply the so - called wide - band limit @xcite , where the square of the tunneling element @xmath42 is inversely proportional to the density of states @xmath43 at energy @xmath44 . by means of the lead green function @xcite @xmath45\ ] ] we can define the decay rate [ eq : gammawbl ] @xmath46 which becomes local in time in the wide - band limit , namely @xmath47 in the following we replace the sum in eq . by the expression involving the @xmath8-function in eq . . the equation of motion for the reservoir operators @xmath48 , eq . , is now readily integrated , yielding [ eq : eomtresop ] @xmath49 where we have used the lead green functions , eq . , and introduced @xmath50 equations are used to rewrite eq . as @xmath51 { \hat{c}}_{s}(t ) \notag\\ & + \sum\limits_{\alpha k } t^*_{\alpha k}(t ) { \hat{b}}_{\alpha k s}(t)\ , . \label{eq : ceom}\end{aligned}\ ] ] here the wide - band limit , eq . , is employed to obtain the decay term , proportional to @xmath52 . similarly , we can rewrite eq . as @xmath53 - { \mathrm{i}}\gamma(t ) { \hat{n}}_{\bar{s}}(t)\;.\notag\end{aligned}\ ] ] here again the time integral of @xmath54 is reduced to a decay width due to the wide - band limit . the expression for the time - dependent current is given by the expectation value of the current operator @xmath55 defined in eq . . as will become clear later on , it is useful to write this expectation value as @xmath56 with the _ current matrices of the first order _ @xmath57 these current matrices are an essential ingredient of our propagation scheme , which is based on finding equations of motion for @xmath58 . such equations have been derived starting from a negf formalism for non - interacting electrons @xcite . exactly as for the operator equations above we can use @xmath59 from eq . and employ the wide - band limit for the current matrices defined in eq . . this leads to the following decomposition [ eq : negfdefpi1 ] @xmath60 having derived all relevant equations of motion for the operators we can specify the respective equations for the two contributions @xmath61 and @xmath62 . the term @xmath61 is the simplest and is basically given by the equation of motion for @xmath63 , cf . the corresponding equation for the occupation @xmath64 reads @xmath65 the above relation can be viewed as the charge conservation equation for the qd . the rate by which the charge in the qd changes is equal to the total electronic currents . the first term at the _ r.h.s . _ of the equation can be interpreted as the current flowing into the qd , whereas the second term gives the current flowing out . since we do not consider a spin - dependent driving or spin - polarized initial states it is @xmath66 . this relation is not explicitly used in the derivation , but is employed as a consistency check throughout the analysis . the evaluation of @xmath62 requires the solutions for both , the lead operator @xmath48 and the dot operator @xmath18 . using those , we write @xmath67 here we have introduced the abbreviation @xmath68\ ] ] and used that @xmath69 with @xmath70 the fermi function describing the equilibrium occupation of lead @xmath25 . the last term in eq . uses the _ auxiliary current matrices of the second order _ @xmath71 which will be subject to further approximations in the following . before we turn to the approximations , we would like to briefly discuss the physical meaning of @xmath72 . the equation of motion for the two - electron density matrix @xmath73 reads @xmath74 which follows from eq . . the two - electron density matrix may be interpreted as the occupation of one quantum - dot level under the condition that the other one is occupied . the rate of change of this conditional occupation is consequently given by tunneling into and out of the respective dot state under the same condition . the latter process is described by the first term on the _ r.h.s . _ of eq . . the former process is governed by the auxiliary current matrices @xmath72 , which can be rewritten in the suggestive form @xmath75 consequently , the current matrices @xmath72 describe the _ conditional current _ from reservoir @xmath25 into the quantum - dot level with spin @xmath32 . the simplest approximation to @xmath72 consists in using the following factorization @xmath76 inserting this expression into eq . , results in the following equation of motion @xmath77 { \pi''}_{\alpha k s}(t ) \notag\\ & + t^*_{\alpha k}(t ) f_{\alpha k}.\end{aligned}\ ] ] this result is equivalent to the hartree - fock approximation applied to the anderson model standard two - electron green function @xcite . as any mean field approach , it does not lead to a double resonance green function , which is required to properly account for charging effects . hence , as it is well known , a good description of the coulomb - blockade regime requires going beyond this level of truncation in the equations of motion . instead of factorizing @xmath72 directly , we proceed by deriving its equation of motion . by means of eqs . we get @xmath78 { \phi}_{\alpha k s}(t ) + \sum\limits_{\alpha',k ' } t^*_{\alpha ' k'}(t ) { \left\langle { \hat{b}}^\dagger_{\alpha k s}(t ) { \hat{b}}_{\alpha ' k ' s}(t ) { \hat{n}}_{\bar{s}}(t ) \right\rangle}\notag\\ & + \sum\limits_{\alpha',k ' } \left [ t_{\alpha ' k'}(t ) { \left\langle { \hat{b}}^\dagger_{\alpha k s}(t ) { \hat{c}}_{s}(t ) { \hat{b}}^\dagger_{\alpha ' k ' \bar{s}}(t ) { \hat{c}}_{\bar{s}}(t ) \right\rangle } - t^*_{\alpha ' k'}(t ) { \left\langle { \hat{b}}^\dagger_{\alpha k s}(t ) { \hat{c}}_{s}(t ) c^\dagger_{\bar{s}}(t ) { \hat{b}}_{\alpha ' k ' \bar{s}}(t ) \right\rangle } \right ] . \label{eq : phieom}\end{aligned}\ ] ] note that the term proportional to @xmath9 has only four operators in the expectation values because of @xmath79 . the approximation consists in neglecting matrix elements involving opposite spins , which renders the following factorizations @xmath80 this approximation for the density matrices is equivalent to the truncation scheme employed in the negf approach used for the study of coulomb blockade regime ( high - temperature limit of the anderson model ) @xcite . as a result of the factorization , we obtain the following compact equation of motion for the approximated second - order current matrices @xmath81 { \tilde{\phi}}_{\alpha k s}(t ) \notag\\ & + t^*_{\alpha k}(t)\ , f_{\alpha k}\ , n_{\bar{s}}(t)\;. \label{eq : pieomhubbard}\end{aligned}\ ] ] the equations of motion for @xmath82 [ eq . ] , @xmath83 [ eq . with @xmath72 replaced by @xmath84 and for @xmath85 [ eq . ] form a closed set of equations , which can be solved by means of an auxiliary - mode expansion discussed below . the general idea of the auxiliary - mode expansion consists in making use of a contour integration and the residue theorem to perform the energy integration , for instance , in eq . . to this end the fermi function is expanded in a sum over simple poles ( or auxiliary modes ) and the respective integrals are given as finite sums , cf . appendix [ sec : appexp ] . the transition to auxiliary modes ( denoted by the index @xmath86 ) is facilitated by the following set of rules [ eq : rules ] @xmath87 which are derived in appendix [ sec : appexp ] . the first rule replaces the reservoir energy @xmath44 by the ( complex ) pole @xmath88 of the expansion , cf . the second rule replaces the fermi function by the respective weight , which is the same for all auxiliary modes . finally , the third rule provides the actual expansion for the current matrices . applying these rules , the current matrices become @xmath89 the equation of motion for the auxiliary matrix @xmath90 is obtained from eq . . one arrives at @xmath91 { \pi''}_{\alpha s p } ( t ) \nonumber\\ & & + \frac{1}{\beta}t_{\alpha}(t ) + u \:{\phi}_{\alpha s p } ( t ) \ , . \label{eq : auxeomtpi}\end{aligned}\ ] ] the equations of motion for the auxiliary matrices @xmath92 are quite similar to those of eq . , namely , @xmath93 { \tilde{\phi}}_{\alpha s p } ( t ) \nonumber\\ & + \frac{1}{\beta}t_{\alpha}(t)\ , n_{\bar{s}}(t)\;. \label{eq : auxeomtphi}\end{aligned}\ ] ] the solution of the above equations still requires a complete description of the population dynamics given by @xmath82 . the latter can be directly obtained from eq . in terms of the current matrices @xmath94 this concludes the derivation of the auxiliary mode propagation scheme . the set of equations to , with initial conditions @xmath95 , @xmath96 , and @xmath97 , can be solved numerically using standard algorithms . before the desired time dependence of the parameters @xmath98 and @xmath99 sets in , the system has to be propagated until a steady state is reached . in this way , transient effects arising from the choice of the initial state are avoided . for convenience we derive in appendix [ sec : appstat ] the expressions for the stationary occupations , which may also be used as initial values for @xmath100 . in this section we present two applications of the formalism developed above . as shown below , one of the interesting features of non - adiabatic pumping is an increasing delay in the current response to the external drive with growing driving speed . hence , in distinction to the adiabatic limit , the current caused by a train of pulses can show interesting transient effects , whenever the pulse period is shorter than the system response time . to better understand non - adiabatic driving effects , we focus our analysis on single pulses and vary the speed by which their shape is changed . it is worth stressing that our propagation method does not possess restrictions on the time dependence of the system driving parameters . in other words , the external time - dependent drive can be just a single pulse or a train of pulses , it can also be either fast or slow as compared with the system internal time scales . let us begin by discussing the current generated by a single gaussian voltage pulse changing the resonance energy as @xmath101\;.\end{aligned}\ ] ] here @xmath102 sets the pumping time - scale . we take @xmath103 to be time - independent and equal for both leads , @xmath104 . since thereby @xmath105 , we will consider only @xmath106 in the following . figure [ fig : single]a shows the time dependence of the resonance energy according to eq . . the two bottom panels show the instantaneous current @xmath106 as a function of time for both the non - interacting ( @xmath107 ) and the interacting ( @xmath108 ) case . in the limit of large @xmath102 , we use as a check for our results an analytical expression for the pumped current @xmath106 , obtained for @xmath107 within the adiabatic approximation @xcite . for different pulse lengths @xmath102 and @xmath107 and @xmath109 . parameters used : @xmath110 , @xmath111 , @xmath112 , @xmath113 and @xmath114 ( number of auxiliary modes ) . dots denote the adiabatic limit @xcite . ] here , due to the l / r symmetry , there is no net charge flowing through the qd . at any given time both leads pump the same amount of charge in or out . in the driving scheme defined by eq . , the qd is initially nearly empty . at @xmath115 , the resonance energy favors an almost full occupation . for very slow pumping , large @xmath116 , the current @xmath106 depends only on the resonance energy @xmath14 : as the resonance dives into the fermi sea , the qd is loaded with charge and the process is reversed as @xmath14 starts increasing . this is no longer true when the drive is faster and @xmath116 decreases : now one observes a retardation effect , namely , the @xmath106 depends not only on the resonance position , but also on driving speed . for fast driving one needs to integrate @xmath106 over times much longer than @xmath102 to observe a vanishing net charge per pulse . the pumped currents @xmath117 characterize the time - dependent electron response to the external drive . however , in most applications one is only interested in the charge pumped per cycle @xmath118 or per pulse @xmath119 . in the latter case , @xmath119 is given as time integral over the current which we write in a symmetric way @xmath120\;.\ ] ] one of the beautiful lessons learned form the investigation of adiabatic pumping , establishes a proportionality relation between @xmath119 and the area swapped by the time - dependent driving forces in parameter space @xcite . in other words , the total charge flowing through a qd per cycle ( or per pulse ) in a _ single - parameter adiabatic pump _ vanishes . due to the constraints of single - parameter pumps , in most applications at least two parameters are used @xcite . on the other hand , by using a single - gate modulation one can realize a constrained two - parameter pump @xcite , which implies that the time - dependence of the parameters is ultimately coupled due to the modulation of only a single gate voltage . in the following we will investigate the implications of this scenario for non - adiabatic pumping . ( upper row ) and the decay widths @xmath121 ( lower row , blue / full line ) and @xmath122 ( lower row , red / broken line ) for three different cases : a ) @xmath123 , b ) @xmath124 , and c ) @xmath125 . the dotted lines indicate the chemical potential in the reservoirs and the shaded area shows the times when the resonance energy is below the chemical potential . ] specifically , let us consider voltage pulses of the form @xmath126 \;.\ ] ] here @xmath102 measures the characteristic pulse time , whereas @xmath8 governs the time the pulse sets in . the numerical factor @xmath127 ensures that @xmath102 is the full width at half maximum of the pulse , which simplifies the following discussion . by tuning the delay one can conveniently switch between a single parameter ( @xmath128 ) and a two parameter setup ( @xmath129 ) . further , the time - dependence of the resonance energy and the coupling strengths ( decay widths ) are chosen as @xmath130\;,\\ \gamma_{\rm l}(t ) & = \frac{\gamma_0}{2 } \left [ 1 + s(t , \delta_{\rm l } ) \right]\;.\end{aligned}\ ] ] this choice takes into account that the coupling strengths depend exponentially on the gate voltage @xcite . the constraint is imposed by setting @xmath131 and the specific value of @xmath132 . for this driving parameterization , the resonance and the decay widths are @xmath133 and @xmath134 , respectively , for both asymptotic limits of @xmath135 . in the following , the parameters are taken as @xmath136 , @xmath137 , @xmath138 , and interaction energy either @xmath107 or @xmath109 . in fig . [ fig : appscheme3 ] the time dependence of @xmath139 and @xmath140 is illustrated for three cases @xmath124 and @xmath141 . as mentioned above , in each case the coupling to the right reservoir @xmath122 follows the time dependence of the resonance energy . when the latter attains its minimal value at @xmath115 , which brings the energy well below the chemical potential of the reservoirs , the coupling to the right reservoir is _ minimal_. on the other hand , the behavior of the coupling to the left reservoir can be influenced by the value of @xmath132 . for @xmath123 the _ maximum _ of @xmath121 comes before @xmath115 , while for @xmath142 it is attained after @xmath115 . in the case @xmath124 the coupling to the left reservoir is maximal simultaneously with @xmath122 being minimal at @xmath115 . in the following the response to these drivings will be investigated . vs pulse - shift @xmath132 in the long - pulse limit ( upper panel ) and at @xmath143 ( lower panel ) . the dashed lines indicate half of the non - interacting result . ] knowing the time dependence shown in fig . [ fig : appscheme3 ] one can readily predict the behavior of @xmath119 in the adiabatic limit . in this case , electron flow occurs when the resonance energy matches the chemical potential of the reservoirs . in our pulse scheme , @xmath14 equals the chemical potential at @xmath144 and @xmath145 corresponding to the onset of charging and de - charging of the qd , respectively . further , the direction of the net current is determined by the difference of the couplings to the reservoirs at these very times . for example , for @xmath123 one finds @xmath146 while charging and @xmath147 while de - charging . consequently , the net current is directed from left to right and @xmath119 is expected to be _ positive_. for @xmath142 the situation is opposite and @xmath119 should be _ finally , for @xmath124 the couplings are equal at both instants of time and the net current is vanishing . these expectations are confirmed by our results for the adiabatic regime , @xmath148 , and different values of @xmath149 , which are shown in fig . [ fig : appmonoqpphi]a . as already mentioned , one observes @xmath150 for @xmath124 ( monoparametric pumping ) . as @xmath151 begins to increase , @xmath152 increases as well . in this scenario , when the resonance energy matches the chemical potential , electrons load the dot from the left ( or right ) and later they are unloaded to the right ( or left , depending on the sign on @xmath8 ) . for larger values of @xmath151 , the left reservoir participates less in the loading or unloading of the qd and the charge per pulse vanishes accordingly . for interaction strengths @xmath153 the double occupation of the qd is suppressed and consequently , in the adiabatic regime , @xmath119 is half the value of @xmath119 for the non - interacting case . the numerical results indicate that within the hubbard i approximation , @xmath108 does not introduce new time scales to the problem for @xmath148 , and its major effect is to correct the spin degeneracy factor in the equations for the @xmath107 case . none of the aforementioned features are observed in the non - adiabatic pumping regime . figure [ fig : appmonoqpphi]b shows , for example , that , for short pulses there is no simple relation between @xmath119 for @xmath107 and for @xmath108 . moreover , compared to the adiabatic regime the charge per pulse can be substantially larger in this regime . unfortunately , the behavior of @xmath119 in this regime is not as easily predicted in general , since the evolution of the parameters @xmath154 , @xmath121 , @xmath155 after the onset of loading and unloading has to be taken into account . this is because in the non - adiabatic regime the qd charging and de - charging is delayed with respect to the external system changes , as it was shown in sec . [ sec : comparison ] . taking , for example , the case @xmath124 one finds from fig . [ fig : appscheme3]b , that @xmath146 while the resonance energy is below the chemical potential and charging occurs . during the de - charging , when @xmath156 , one finds @xmath157 . consequently , the current is expected to flow mainly from left to right , which leads to a positive charge per cycle . this is confirmed by the results shown in fig . [ fig : appmonoqpphi]b . the quantitative behavior depends on the precise magnitude of the delay , which is determined by the pulse length . however , from the analysis presented above and for sufficiently short pulses one concludes that @xmath119 has to be positive independent of @xmath132 . the interesting implications of this result will be discussed at the end of this section . finally , in fig . [ fig : appmonoqp ] we summarize and corroborate the discussion of the non - adiabatic pumping . it shows the charge pumped due the pulse as a function of pulse length @xmath116 in the non - interacting ( @xmath107 ) and the coulomb blockade regime ( @xmath158 ) . in the latter case @xmath159 for all pulse lengths . as discussed above , the amount of pumped charge @xmath119 depends very strongly on the value of @xmath116 . in the limit of large pulse lengths , @xmath119 approaches the respective adiabatic value , while for @xmath160 the charge per pulse vanishes . moreover , one finds that @xmath119 is indeed positive for small pulse lengths . this has the intriguing consequence that the charge per pulse can change its sign sweeping from short to long pulses . this is shown in fig . [ fig : appmonoqp]b for @xmath125 , where @xmath119 is negative in the adiabatic regime . a more general and quantitative analysis of this effect is certainly desirable , but beyond the scope of this article . it may lead , however , to interesting new applications . it is also worth to mention , that by changing the pumping parameters it is possible to optimize the charge pumped per pulse and in particular to find situations where @xmath161 , which may be very interesting for metrology purposes @xcite . we presented a new method to analyze non - adiabatic charge pumping through single - level quantum dots that takes into account coulomb interactions . the method is based on calculating the time evolution for single - electron density matrices . the many - body aspects of the problem are approximated by truncating the equations of motion one order beyond mean field . the novelty is the way the time evolution is treated : by means of an auxiliary - mode expansion , we obtain a propagation scheme that allows for dealing with arbitrary driving parameters , fast and slow . the method presented in this paper can be applied to a wide range of coupling parameters @xmath162 , provided one avoids the kondo regime . hence , we are not restricted to the weak coupling limit where @xmath119 , the charge pumped per pulse , is rather small . the presented results for single - pulses are also valid for pulse trains , provided the time between the pulses is sufficiently long . one can expect to find qualitatively new and interesting effects by decreasing the time lag . the propagation scheme allows , in principle , for studying transient effects . in addition , by propagating over a periodic sequence of pulses it constitutes a complementary approach to the more familiar periodic driving . in this regard , our propagation scheme has the potential to be a valuable tool and provide deeper insights into non - adiabatic quantum pumps . this work is supported in part by cnpq ( brazil ) . here we motivate the rules given in sect . [ sec : prop ] . to begin with we introduce correlation functions , which can be approximated by finite sums . then we write the current matrices in terms of these finite sums . as we will show later , in the present case we have to consider the following reservoir correlation function @xmath163 \right\ } \;,\label{eq : corrfuncint}\end{aligned}\ ] ] where the line - width function @xmath164 is defined as usual @xcite @xmath165 in the second line we have used the wide - band limit . in order to perform the energy integration in eq . we expand the fermi function @xmath166 as a finite sum over simple poles @xmath167 with @xmath168 and @xmath169 . instead of using the matsubara expansion @xcite , with poles @xmath170 , we use a partial fraction decomposition of the fermi function @xcite , which converges much faster than the standard matsubara expansion . in this case the poles @xmath171 are given by the eigenvalues @xmath172 of a @xmath173 matrix @xcite . the poles are arranged such that all poles @xmath174 ( @xmath175 ) are in the upper ( lower ) complex plane . as in the matsubara expansion all poles have the same weight . employing the expansion given by eq . , one can evaluate the energy integrals by contour integration in the upper or lower complex plane depending on the sign of @xmath176 . thereby , the integral in eq . becomes a ( finite ) sum of the residues . for @xmath177 one gets [ eq : corrfuncexp ] @xmath178 with the auxiliary modes for reservoir @xmath25 given by @xmath179+x_p/\beta.\ ] ] here , @xmath180 is the chemical potential and @xmath181 is due to the time - dependent single - particle energies @xmath182 of the reservoir hamiltonian [ eq.([eq : reshamilop ] ) ] . the set of equations and can be formally solved . in order to write down these solutions we define the following functions # 1^#1 @xmath183 } \;,\\ g^u_s ( t , t ' ) & \equiv \exp { - { \mathrm{i}}\int^t_{t ' } dt '' \left[\varepsilon_s(t '' ) + u - { \mathrm{i}}\frac{\gamma(t'')}{2}\right ] } \;.\end{aligned}\ ] ] with these definitions the formal solution of eq . reads @xmath184\,,\end{aligned}\ ] ] where we have assumed @xmath185 , corresponding to our choice of an initially uncorrelated density matrix ( see sec . [ sec : setup ] ) . an analogous equation holds for @xmath85 , again with @xmath186 . we can combine these two expressions to get for the second part of the current matrix @xmath187 where we have used the definition of the correlation function @xmath188 given by eq . . finally , by means of the expansion of the correlation functions we obtain an expansion of the current matrices @xmath189 which resembles the last rule of eqs using the explicit expression for @xmath90 and taking the time derivative one can easily verify the first two rules given by eqs . . similarly , one also obtains an expression for @xmath190 , which reads @xmath191 the time derivative of this expression is given by eq . . if neither the couplings @xmath42 ( and thus @xmath7 ) nor the levels @xmath192 or @xmath44 depend on time the level occupations @xmath193 and the currents @xmath194 converge to stationary values . these values can be obtained by setting all time derivatives in the respective equations of motion to zero . in order to simplify the notation we characterize the stationary values by omitting the time argument . within the hartree - fock approximation [ sec . [ sec : hfapp ] ] we get from eq . @xmath196 plugging this into eq . , changing the @xmath197 summation into an integral over @xmath198 and using the definition , we get for the wide - band limit [ eq . ] @xmath199 equation is a non - linear equation for @xmath193 and has to be solved numerically . we obtain the stationary conditional current @xmath200 for the hubbard i approximation [ sec . [ sec : hiapp ] ] from eq . as @xmath201 this expression can be used for the stationary @xmath62 in eq . @xmath202 } \end{aligned}\ ] ] we use eq . and the definition and finally get for the occupation the following integral @xmath203\\ a'(\varepsilon ) & \equiv \frac{1}{(\varepsilon{-}\varepsilon_{s})^{2}+\left(\frac{\gamma}{2}\right)^{2 } } \\ a''(\varepsilon ) & \equiv a'(\varepsilon ) \frac{u\left[4(\varepsilon{-}\varepsilon_{s})-u\right ] } { ( \varepsilon{-}\varepsilon_{s}{-}u)^{2}+\left(\frac{3\gamma}{2}\right)^{2 } } \,.\end{aligned}\ ] ] this time the equation is linear in @xmath204 and can be solved explicitly . in the limits @xmath205 and @xmath206 it is @xmath207 and @xmath208 , respectively . the former limit corresponds to non - interacting electrons and eq . gives the correct expression for the occupation @xcite . the latter case describes the situation with very strong interactions .

we study non - adiabatic charge pumping through single - level quantum dots taking into account coulomb interactions . we show how a truncated set of equations of motion can be propagated in time by means of an auxiliary - mode expansion . this formalism is capable of treating the time - dependent electronic transport for arbitrary driving parameters . we verify that the proposed method describes very precisely the well - known limit of adiabatic pumping through quantum dots without coulomb interactions . as an example we discuss pumping driven by short voltage pulses for various interaction strengths . such finite pulses are particular suited to investigate transient non - adiabatic effects , which may be also important for periodic drivings , where they are much more difficult to reveal .

1110.5298

the collective dynamics of complex distributed systems often can be usefully described in terms of a superposition of rate processes or frequencies which determine the changes in macroscopically measurable variables as energy flows through the system ; that is , a dynamical model expressed as a system of coupled ordinary differential equations in a few averaged state variables or mode coefficients and several , independently tunable , parameters that represent physical properties or external controls . this type of reduced ( or low - order or low - dimensional ) modelling averages over space , mode spectrum structure , single - particle dynamics and other details , but the payoff lies in its amenity to sophisticated analytic theory and methods that enable us to track important qualitative features in the collective dynamics , such as singularities , bifurcations , and stability changes , broadly over the parameter space . motivated by the need for improved guidance and control of the ( mostly bad ) behaviour of fusion plasmas in magnetic containers , i elaborate in this work a case study in bifurcation and stability analysis in which reduced dynamical system modelling yields new global and predictive information about gradient driven turbulence flow energetics that is complementary to direct numerical simulation and can guide experimental design . reduced dynamical models are powerful tools for describing and analysing complex systems such as turbulent plasmas and fluids , primarily because they are supported by well - developed mathematics that gives qualitative and global insight , such as singularity , bifurcation , stability , and symmetry theory . in principle one can map analytically the bifurcation structure of the entire phase and parameter space of a reduced dynamical system , but this feat is not possible for an infinite - dimensional system , or partial differential equations , and not practicable for systems of high order . the usefulness of such models seems to be no coincidence , too : in turbulent systems generally , which in detail are both complex and complicated , the dynamics seems to take place in a low - dimensional subspace @xcite . it seems paradoxical that enthusiasm for low - dimensional modelling and qualitative analysis of fluid and plasma systems has paced the ever larger direct numerical simulations of their flow fields . this is an exemplar of how the simplexity and complicity @xcite juxtaposition can work well : these methods affirm each other , for both have common ground in the universal conservation equations for fluid flow ( as well as separate bases in mathematics and computational science ) . developments in one feed developments in the other . reduced dynamical models can give insights into the physics and dynamics of a system in a way that is complementary to brute - force numerical simulations of the detailed , spatially distributed models from which they are derived . in practice this complementarity means that low - order models ( which capture few or no spatial modes ) can be used to channel information gleaned from the generic , qualitative structure of the parameter space attractors , critical points of onset , stability properties , and so on to numerical simulations ( which contain all spatial modes but , on their own , bring little physical understanding ) , giving them purpose and meaning . in turn the fluid simulations serve as virtual experiments to validate the low - order approach . it is reasonable , therefore , to assert that improved low - dimensional dynamical models for plasmas and fluids could provide numerical experimenters with new and interesting challenges that will continue to push the limits of computational science and technology . fusion plasmas in magnetic containers , such as those in tokamak or stellarator experiments , are strongly driven nonequilibrium systems in which the kinetic energy of small - scale turbulent fluctuations can drive the formation of large - scale coherent structures such as shear and zonal flows . this inherent tendency to self - organize is a striking characteristic of flows where lagrangian fluid elements see a predominantly two - dimensional velocity field , and is a consequence of the inverse energy cascade @xcite . the distinctive properties of quasi two - dimensional fluid motion are the basis of natural phenomena such as zonal and coherent structuring of planetary flows , but are generally under - exploited in technology . in plasmas the most potentially useful effect of two - dimensional fluid motion is suppression of high wavenumber turbulence that generates cross - field transport fluxes and degrades confinement @xcite . suppression of turbulent transport can manifest temporally as a spontaneous and more - or - less abrupt enhancement of sheared poloidal or zonal flows and concomitant damping of density fluctuations , and spatially as the rapid development of a localized transport barrier or steep density gradient . the phenomenon is often called low- to high - confinement ( l h ) transitions and has been the subject of intensive experimental , _ in numero , _ and theoretical and modelling investigations since the 1980s . the large and lively primary literature on reduced dynamical models for confinement transitions and associated oscillations in plasmas represents a sort of consensus on the philosophy behind qualitative analysis , if not on the details of the models themselves . what motivates this approach is the predictive power that a unified , low - order description of the macroscopic dynamics would have in the management of confinement states . since it is widely acknowledged that control of turbulent transport is crucial to the success of the world - wide fusion energy program @xcite it is important to develop predictive models for efficient management of access to , and sustainment of , high confinement rgimes . for example , if one plans to maintain a high confinement state at a relatively low power input by exploiting the hysteresis in the transition it would be useful , not to mention cheaper , to know in advance which parameters control the shape and extent of hysteresis , or whether it can exist at all in the operating space of a particular system , or whether a transition will be oscillatory . however , it has been shown @xcite that many of the models in the literature are structurally flawed . they often contain pathological or persistent degenerate ( higher order ) singularities . an associated issue is that of overdetermination , where near a persistent degenerate singularity there may be more defining equations than variables . consequently much of the discussion in the literature concerning confinement transitions is qualitatively wrong . * such models can not possibly have predictive power*. the heart of the matter lies in the mapping between the bifurcation structure and stability properties of a dynamical model and the physics of the process it is supposed to represent : if we probe this relationship we find that degenerate singularities ought to correspond to some essential physics ( such as fulfilling a symmetry - breaking imperative , or the onset of hysteresis ) , or they are pathological . in the first case we can usually unfold the singularity in a physically meaningful way ; in the other case we know that something is amiss and we should revise our assumptions . degenerate singularities are good because they provide opportunities to improve a model and its predictive capabilities , but bad when they are not recognized as such . [ [ section ] ] the literature on confinement transitions has two basic strands : ( 1 ) transitions are an internal , quasi two - dimensional flow , phenomenon and occur spontaneously when the rate of upscale transfer of kinetic energy from turbulence to shear and zonal flows exceeds the nonlinear dissipation rate @xcite ; ( 2 ) transitions are due to nonambipolar ion orbit losses near the plasma edge , the resulting electric field providing a torque which drives the poloidal shear flow nonlinearly @xcite . these two different views of the physics behind confinement transitions are smoothly reconciled for the first time in this work . a systematic methodology for characterizing the equilibria of dynamical systems involves finding and classifying high - order singularities then perturbing around them to explore and map the bifurcation landscape @xcite . broadly , this paper is about applying singularity theory as a diagnostic tool while an impasto picture of confinement transition dynamics is compounded . the bare - bones model is presented in section [ two ] in section [ three ] the global consequences of local symmetry - breaking are explored , leading to the discovery of an organizing centre and trapped degenerate singularities . this leads in to section [ four ] where i unfold a trapped singularity smoothly by introducing another layer that models the neglected physics of downscale energy transfer . section [ five ] follows the qualitative changes to the bifurcation and stability structure that are due to potential energy dissipative losses . in section [ six ] the unified model is presented , in which is included a direct channel between gradient potential energy and shear flow kinetic energy . the results and conclusions are summarized in section [ seven ] in the edge region of a plasma confinement experiment such as a tokamak or stellarator potential energy is stored in a steep pressure gradient which is fed by a power source near the centre . gradient potential energy @xmath0 is converted to turbulent kinetic energy @xmath1 , which is drawn off into stable shear flows , with kinetic energy @xmath2 , and dissipation channels . the energetics of this simplest picture of confinement transition dynamics are schematized in fig . [ fig1](a ) . ( nomenclature for the quantities here and in the rest of the paper is defined in table [ tab1 ] . ) energy transfer diagrams for the gradient - driven plasma turbulence shear flow system . annotated arrows denote rate processes ; curly arrows indicate dissipative channels , straight arrows indicate inputs and transfer channels between the energy - containing subsystems . see text for explanations of each subfigure . ] a skeleton dynamical system for this overall process can be written down directly from fig . [ fig1](a ) by inspection : @xmath3 the power input @xmath4 is assumed constant and the energy transfer and dissipation rates generally may be functions of the energy variables . a more physics - based derivation of this system was outlined in ball ( 2002)ball:2002 , in which averaged energy integrals were taken of momentum and pressure convection equations in slab geometry , using an electrostatic approximation to eliminate the dynamics of the magnetic field energy . equations [ e1 ] are fleshed out by substituting specific rate - laws for the general rate expressions on the right hand sides : [ e2 ] @xmath5 where @xmath6 . the rate expressions in eqs [ e2 ] were derived in sugama and horton ( 1995)sugama:1995 and ball ( 2002)ball:2002 from semi - empirical arguments or given as ansatzes . ( rate - laws for bulk dynamical processes are not usually derivable purely from theory , and ultimately must be tested against experimental evidence . ) the rest of this paper is concerned with the character of the equilibria of eqs [ e2 ] and modifications and extensions to this system . we shall study the type , multiplicity , and stability of attractors , interrogate degenerate or pathological singularities where they appear , and classify and map the bifurcation structure of the system . in doing this we shall attempt to answer questions such as : are eqs [ e2 ] or modified versions a good that is , predictive model of the system ? does the model adequately reflect the known phenomenology of confinement transitions in fusion plasmas ? what is the relationship between the bifurcation properties of the model and the physics of confinement transitions ? the equilibrium solutions of eqs [ e2 ] are shown in the bifurcation diagrams of fig . [ fig2 ] , where the shear flow @xmath7 is chosen as the state variable and the power input @xmath4 is chosen as the principal bifurcation or control parameter . ( in these and subsequent bifurcation diagrams stable equilibria are indicated by solid lines and unstable equilibria are indicated by dashed lines . ) several bifurcation or singular points are evident . the four points in ( a ) annotated by asterisks , where the stability of solutions changes , are hopf bifurcations to limit cycles , which are discussed in section [ three - two ] . on the line @xmath8 the singularity * p * is found to satisfy the defining and non - degeneracy conditions for a pitchfork , @xmath9 where @xmath10 is the bifurcation equation derived from the zeros of eqs [ e2 ] , @xmath11 represents the chosen state variable , @xmath12 represents the chosen control or principal bifurcation parameter , and the subscripts denote partial derivatives . in the qualitatively different bifurcation diagrams ( a ) and ( b ) the dissipative parameter @xmath13 is relaxed either side of the critical value given in ( c ) , where the perfect , twice - degenerate pitchfork is represented . thus for @xmath14 ( a poorly dissipative system ) the turning points in ( a ) appear and the system may also show oscillatory behaviour . the dynamics are less interesting for @xmath15 ( a highly dissipative system ) as in ( b ) because the turning points , and perhaps also the hopf bifurcations , can not occur . however , * p * is persistent through variations in @xmath13 or any other parameter in eqs [ e2 ] . ( this fact was not recognized in some previous models for confinement transitions , where such points were wrongly claimed to represent second - order phase transitions . ) typically the pitchfork is associated with a fragile symmetry in the dynamics of the modelled physical system . the symmetry in this case is obvious from fig . [ fig2 ] : in principle the shear flow can be in either direction equally . in real life ( or _ in numero _ ) , experiments are always subject to perturbations that determine a preferred direction for the shear flow , and the pitchfork is inevitably dissolved . in this case the perturbation is an effective force or torque from any asymmetric shear - inducing mechanism , such as friction with neutrals in the plasma or external sources , and acts as a shear flow driving rate . assuming this rate to be small and independent of the variables over the characteristic timescales for the other rate processes in the system , we may revise the shear flow evolution eq . [ e2c ] as @xmath16 where the symmetry - breaking term @xmath17 models the shear flow drive . the corresponding energy transfer schematic is shown in fig . [ fig1](b ) . the pitchfork * p * in fig . [ fig2 ] ( c ) can now be obtained exactly by applying the conditions ( [ e3 ] ) to the zeros of eqs [ e2a ] , [ e2b ] , and [ e4 ] , with [ e2d ] , and with @xmath18 and @xmath19 : @xmath20 the other singularity * t * on @xmath8 satisfies the defining and non - degeneracy conditions for a transcritical bifurcation , @xmath21 it is once - degenerate and also requires the symmetry - breaking parameter for exact definition . a bifurcation diagram where * p * is fully unfolded , that is , for @xmath22 and @xmath23 , is shown in fig . this diagram is rich with information that speaks of the known and predicted dynamics of the system and of ways in which the model can be improved further , and which can not be inferred or detected from the degenerate bifurcation diagrams of fig . it is worthwhile to step through fig . [ fig3 ] in detail , with the energy schema fig . [ fig1 ] ( b ) at hand . let us begin on the stable branch at @xmath24 . here the pressure gradient is being charged up , the gradient potential energy is feeding the turbulence , and the shear flow is small but positive because the sign of the perturbation @xmath17 is positive . as the power input @xmath4 is increased quasistatically the shear flow begins to grow but at the turning point , where solutions become unstable , there is a discontinuous transition to the upper stable branch in @xmath7 and @xmath0 , ( a ) and ( b ) , and to the lower stable branch in @xmath1 , ( c ) . at the given value of @xmath13 the hysteresis is evident : if we backtrack a little the back transition takes place at a lower value of @xmath4 . continuing along the upper stable branch in @xmath7 we encounter another switch in stability ; this time at a hopf bifurcation to stable period one limit cycles . ( in this and other diagrams the amplitude envelopes of limit cycle branches are marked by large solid dots . ) from the amplitude envelope we see that the oscillations grow as power is fed to the system then are extinguished rather abruptly , at a second hopf bifurcation where the solutions regain stability . the shear flow decreases toward zero as the pressure - dependent anomalous viscosity , the second term in eq . [ e2d ] , takes over to dissipate the energy at high power input . the system may also be evolved to an equilibrium on the antisymmetric , @xmath25 branch in fig . [ fig3 ] ( a ) , by choosing initial conditions appropriately or a large enough kick . however , if the power input then falls below the turning point at @xmath26 we see an interesting phenomenon : the shear flow spontaneously reverses direction . the transient would nominally take the system toward the nearest stable attractor , the lower @xmath27 branch , but since it would then be sitting very close to the lower @xmath27 turning point small stochastic fluctuations could easily induce the transition to the higher @xmath27 branch . see the inset zoom - in over this region in ( a ) . here is an example of a feature that is unusual in bifurcation landscapes , a domain over which there is fivefold multiplicity comprising three stable and two unstable equilibria . two more examples of threefold stable domains will be shown in section [ five ] the same equilibria depicted using @xmath0 and @xmath1 as dynamical variables in ( b ) and ( c ) are annotated to indicate whether they correspond to the @xmath27 or @xmath25 domain . for clarity the amplitude envelopes of the limit cycle solutions are omitted from ( b ) and ( c ) . in the remainder of this paper i concentrate on the @xmath27 branches and ignore the @xmath25 domain . now we approach the very heart of the model , the organizing centre ; strangely enough via the branch of _ unstable _ solutions that is just evident in fig . [ fig3 ] ( a ) and ( b ) in the top left - hand corner and ( c ) in the lower left hand corner . the effects of symmetry - breaking are more far - reaching than merely providing a local universal unfolding of the pitchfork , for this branch of equilibria was trapped as a singularity at @xmath28 for @xmath29 . the organizing centre itself , described as a metamorphosis in ball ( 2002)ball:2002 , can be encountered by varying @xmath17 . the sequence in fig . [ fig4 ] tells the story visually . the `` new '' unstable branch develops a hopf bifurcation at the turning point . ( strictly speaking , this is a degenerate hopf bifurcation , called dze , where a pair of complex conjugate eigenvalues have zero real and imaginary components . ) as @xmath17 is tuned up to 0.08 ( a ) a segment of stable solutions becomes apparent as the hopf bifurcation moves away from the turning point ; the associated small branch of limit cycles can also just be seen . at @xmath30 , the metamorphosis , the `` new '' and `` old '' branches exchange arms , ( b ) . the metamorphosis satisfies the conditions ( [ trans ] ) and is therefore an unusual , non - symmetric , transcritical bifurcation . it signals a profound change in the _ type _ of dynamics that the system is capable of . for @xmath31 a transition must still occur at the lower limit point , but there is no classical hysteresis , ( c ) and ( d ) . in fact classical hysteresis is ( locally ) forbidden by the non - degeneracy condition @xmath32 in eqs [ trans ] . various scenarios are possible in this rgime , including a completely non - hysteretic transition , a forward transition to a stable steady state and a back transition from a large period limit cycle , or forward and back transitions occurring to and from a limit cycle . the symmetry - broken model comprises eqs [ e2a ] , [ e2b ] , and [ e4 ] , with [ e2d ] . the bifurcation structure , some of which is depicted in figs [ fig3 ] and [ fig4 ] , predicts various behaviours : * shear flow suppression of turbulence ; * smooth , hysteretic , non - hysteretic , and oscillatory transitions ; * spontaneous and kicked reversals in direction of shear flow ; * saturation then decrease of the shear flow with power input due to pressure - dependent anomalous viscosity ; * a metamorphosis of the dynamics through a transcritical bifurcation . a critical appraisal of experimental evidence that supports the qualitative structure of this model is given in ball ( 2004)ball:2004a . with the exception of the last item all of the above dynamics have been observed in magnetically contained fusion plasma systems . the model would therefore seem to be a `` good '' and `` complete '' one , in the sense of being free of pathological or persistent degenerate singularities and reflecting observed behaviours . however , there are several outstanding issues that suggest the model is still incomplete . one issue arises as a gremlin in the bifurcation structure that makes an unphysical prediction , another comes from a thermal diffusivity term that was regarded as negligible in previous work on this model . a third issue arises from the two strands in literature on the physics of confinement transitions : the model as it stands does not describe confinement transitions due to a nonlinear electric field drive . the first issue of incompleteness concerns a pathology in the bifurcation structure of the model , implying infinite growth of shear flow as the power input _ falls_. before we pinpoint the culprit singularity , it is illuminating to evince the physical or unphysical situation through a study of the role of the thermal capacitance parameter @xmath33 , which regulates the contribution of the pressure gradient dynamics , eq . [ e2a ] , to the oscillatory dynamics of the system . conveniently , we can use @xmath33 as a second parameter to examine the stability of steady - state solutions around the hopf bifurcations in fig . [ fig4 ] without quantitative change . the machinery for this study consists of the real - time equations [ e2a ] , [ e2b ] , and [ e4 ] recast in `` stretched time '' @xmath34 , [ e6 ] @xmath35 and the two - parameter locus of hopf bifurcations in the real - time system shown in fig . we consider two cases . the curve is the locus of the hopf bifurcations in fig . [ fig4](c ) over variations in @xmath33 . ] 1 . the high - capacitance rgime : + the maximum on the curve in fig . [ fig5 ] marks a degenerate hopf bifurcation , a dze point . here the two hopf bifurcations at lower @xmath4 in fig . [ fig4](c ) merge and are extinguished as @xmath33 increases . ( this merger through a dze has obviously occurred in fig . [ fig4](d ) , where @xmath17 is varied rather than @xmath33 . ) the surviving upper hopf bifurcation moves to higher @xmath4 as @xmath36 increases further . + in this high - capacitance rgime the dynamics becomes quasi one - dimensional on the stretched timescale . to see this formally , define @xmath37 and multiply the stretched - time equations [ e6b ] and [ e6c ] through by @xmath38 . in the limit @xmath39 the kinetic energy variables are slaved to the pressure gradient ( or potential energy ) dynamics . switching back to real time and multiplying eq . [ e2a ] through by @xmath38 , for @xmath40 @xmath41 . the kinetic energy subsystems see the potential energy as a constant , `` infinite source '' . it is conjectured that the surviving upper hopf bifurcation moves toward @xmath42 and for @xmath43 the dynamics becomes largely oscillatory in real time , with energy simply sloshing back and forth between the turbulence and the shear flow . 2 . the low - capacitance rgime : + the minimum on the curve in fig . [ fig5 ] marks another degenerate hopf bifurcation , also occurring at a dze point . here the two hopf bifurcations at higher @xmath4 in fig . [ fig4](c ) merge and are extinguished as @xmath33 decreases . the surviving hopf bifurcation moves to lower @xmath4 and higher @xmath7 as @xmath36 decreases further . this scenario is illustrated in fig . [ trap4a ] , where the steady - state curve and limit cycle envelope are roughly sketched in a decreasing @xmath33 sequence . + as @xmath33 is decreased further than the minimum in fig . [ fig5 ] the remaining hopf bifurcation slides up the steady - state curve , which becomes stable toward unrealistically high @xmath7 and low @xmath4 . ] + in this low - capacitance rgime the dynamics also becomes quasi one - dimensional , and as @xmath44 the conjectured fate of the surviving hopf bifurcation is a double zero eigenvalue trap at @xmath45 . to see why this can be expected , consider again the stretched - time system , eqs [ e6 ] . for @xmath46 we have @xmath47 and @xmath48 . on the stretched timescale the potential energy subsystem sees the kinetic energy subsystems as nearly constant , and @xmath49 . reverting to real time , as @xmath50 we have @xmath51 ; the potential energy is reciprocally slaved to the kinetic energy dynamics . + the anomaly in this low - capacitance picture is that , as the power input @xmath4 ebbs , the shear flow can grow quite unrealistically . with diminishing @xmath33 the hopf bifurcation moves upward along the curve , the branch of limit cycles shrinks , and the conjugate pair of pure imaginary eigenvalues approaches zero . it would seem , therefore , that some important physics is still missing from the model . what is not shown in figs [ fig3 ] and [ fig4 ] ( because a log scale is used for illustrative purposes ) is a highly degenerate branch of equilibria that exists at @xmath52 where @xmath53 and @xmath54 ; it is shown in fig . [ fig7](a ) . for @xmath55 there is a trapped degenerate turning point , annotated as s4 , where the `` new '' branch crosses the @xmath52 branch . the key to its release ( or unfolding ) lies in recognizing that kinetic energy in large - scale structures inevitably feeds the growth of turbulence at smaller scales , as well as vice versa @xcite . in a flow where lagrangian fluid elements locally experience a velocity field that is predominantly two - dimensional there will be a strong tendency to upscale energy transfer ( or inverse energy cascade , see kraichnan and montgomery ( 1980)kraichnan:1980 ) , but the net rate of energy transfer to high wavenumber ( or kolmogorov cascade , see ball ( 2004)ball:2004a ) is not negligible . what amounts to an ultraviolet catastrophe in the physics when energy transfer to high wavenumber is neglected maps to a trapped degenerate singularity in the mathematical structure of the model . the trapped singularity s4 may be unfolded smoothly by including a simple , conservative , back - transfer rate between the shear flow and turbulent subsystems : [ e7 ] @xmath56 the model now consists of eqs [ e7 ] and [ e2a ] , with [ e2d ] , and the corresponding energy transfer schematic is fig . [ fig1](c ) . the back- transfer rate coefficient @xmath57 need not be identified with any particular animal in the zoo of plasma and fluid instabilities , such as the kelvin - helmholtz instability ; it is simply a lumped dimensionless parameter that expresses the inevitability of energy transfer to high wavenumber . the manner and consequences of release of the turning point s4 can be appreciated from fig . [ fig7](b ) , from which we learn a salutary lesson : unphysical equilibria and singularities should not be ignored . the unfolding of s4 creates a maximum in the shear flow , and ( apparently ) a _ fourth _ hopf bifurcation is released from a trap at infinity . at the given values of the other parameters this unfolding of s4 has the effect of forming a finite - area isola of steady - state solutions , but it is important to visualize this ( or , indeed , any other ) bifurcation diagram as a slice of a three - dimensional surface of steady states , where the third coordinate is another parameter . ( isolas of steady - state solutions were first reported in the chemical engineering literature , where nonlinear dynamical models typically include a thermal or chemical autocatalytic reaction rate @xcite . ) in fig . [ fig8 ] we see two slices of this surface , prepared in order to demonstrate that the metamorphosis identified in section [ 3.3 ] is preserved through the unfolding of s4 . here the other turning points are labelled s1 , s2 , and s3 . walking through fig . [ fig8 ] we make the forward transition at s1 and progress along this branch through the onset of an a limit cycle rgime , as in fig . [ fig3 ] . for obvious reasons we now designate this segment as the _ intermediate _ shear flow branch , and the isola or peninsula as the _ high _ shear flow branch . in ( a ) a back - transition occurs at s2 . the system can only reach a stable attractor on the isola by a transient , either a non - quasistatic jump in a second parameter or an evolution from initial conditions within the appropriate basin of attraction . in ( b ) as we make our quasistatic way along the intermediate branch with diminishing @xmath4 the shear flow begins to grow , then passes through a second oscillatory domain before reaching a maximum and dropping steeply ; the back transition in this case occurs at s4 . in the model so far the only outlet channel for the potential energy is conversion to turbulent kinetic energy , given by the conservative transfer rate @xmath58 . however , in a driven dissipative system such as a plasma other conduits for gradient potential energy may be significant . the cross - field thermal diffusivity , a neoclassical transport quantity @xcite is often assumed to be negligible in the strongly - driven turbulent milieu of a tokamak plasma @xcite , but here eq . [ e2 ] is modified to include explicitly a linear `` infinite sink '' thermal energy dissipation rate : @xmath59 following thyagaraja et al . ( 1999)thyagaraja:1999 @xmath60 is taken as as a lumped dimensionless parameter and the rate term @xmath61 as representing all non - turbulent or residual losses such as neoclassical and radiative losses . the model now consists of eqs [ e7 ] and [ e8 ] , with [ e2d ] and the corresponding energy schematic is fig . [ fig1](d ) . this simple dissipative term has profound effects on the bifurcation structure of the model , and again the best way to appreciate them is through a guided walking tour of the bifurcation diagrams . in fig . [ fig9 ] the series of bifurcation diagrams has been computed for increasing values of @xmath60 and a connected slice of the steady state surface ( i.e. , using a set of values of the other parameters for which the metamorphosis has already occurred ) . a qualitative change is immediately apparent , which has far - reaching consequences : for @xmath62 the two new turning points s5 and s6 appear , born from a local cusp singularity that was trapped at @xmath63 . overall , from ( a ) to ( e ) we see that s1 does not shift significantly but that the peninsula becomes more tilted and shifts to higher @xmath4 , but let us begin the walk at s1 in ( b ) . here , as in fig . [ fig8 ] , the transition occurs to an intermediate shear flow state and further increments of @xmath4 take the system through an oscillatory rgime . but the effect of decreasing @xmath4 is radically different : at s6 a discontinuous transition occurs to a high shear flow state on the stable segment of the peninsula . from this point we may step forward through the shear flow maximum and fall back to the intermediate branch at s5 . we see that over the range of @xmath4 between s5 and s6 the system has five steady states , comprising three stable interleaved with two unstable steady states . as in fig . [ fig8](b ) a back transition at low @xmath4 occurs at s4 . the tristable rgime in ( b ) has disappeared in ( c ) in a surprisingly mundane way : not through a singularity but merely by a shift of the peninsula toward higher @xmath4 . but this shift induces a _ different _ tristable rgime through the creation of s7 and s8 at another local cusp singularity . in ( d ) s4 and s7 have been annihilated at yet another local cusp singularity . it is interesting and quite amusing to puzzle over the 2-parameter projection of these turning points s1 , s8 , s7 , and s4 followed over @xmath60 , it is given in fig . [ fig9c-2par ] . the origins of the three local cusps can be read off the diagram , keeping in mind that the crossovers are a _ trompe de loeil _ : they are nonlocal . the turning points s1 , s8 , s7 , and s4 are followed over @xmath60 . @xmath64 , @xmath65 , @xmath66 , @xmath67 , @xmath68 , @xmath69 , @xmath70 , @xmath71 . ] at s5 in fig . [ fig9](c ) , ( d ) , and ( e ) the system transits to a limit cycle , rather than to a stable intermediate steady state . shown in fig . [ fig10 ] are the bifurcation diagrams in @xmath1 and @xmath0 corresponding to fig . [ fig9](e ) . the pressure gradient jumps at s1 because the power input exceeds the distribution rates , and oscillatory dynamics between the energy subsystems sets in abruptly at s5 . the turbulence is enormously suppressed due to uptake of energy by the shear flow , but rises again dramatically with this hard onset of oscillations . the early theoretical work on confinement transitions attempted to explain edge l h transitions exclusively in terms of the electric field driving torque created by nonambipolar ion orbit losses @xcite , with no coupling to the internal dynamics of energy transfers from the potential energy reservoir in the pressure gradient . the electric field is bistable , hence the transition to a high shear flow , or high confinement , rgime is discontinuous and hysteretic . although there are many supporting experiments @xcite , this exposition of the physics behind confinement transitions is incomplete because it can not explain shear flow suppression of turbulence a well - known characteristic of l h transitions . here this `` electric field bifurcation '' physics is treated as a piece of a more holistic physical picture and a simple model for the rate of shear flow generation due to this physics is used to create a unified dynamical model for confinement transitions . following the earlier authors this rate is given as @xmath72 $ ] , which simply says that the rate at which ions are preferentially lost , and hence flow is generated , is proportional to a collision frequency @xmath73 times the fraction of those collisions that result in ions with sufficient energy to escape . the form of the energy factor assumes an ion distribution that is approximately maxwellian and @xmath74 , analogous to an activation energy , is proportional to the square of the critical escape velocity . in this form of the rate expression i have explicitly included the temperature - dependence of @xmath75 , through @xmath0 , which couples it to the rest of the system . if @xmath76 is high the rate is highly temperature ( pressure gradient ) sensitive . ( for heuristic purposes constant density is assumed , constants and numerical factors are normalized to 1 , and the relatively weak temperature dependence of the collision rate @xmath73 is ignored . ) for convenience the equations for the unified model are gathered together : [ e9 ] @xmath77 \label{e8c}\\ & \mu(p , n ) = bp^{-3/2 } + a pn . \tag{\ref{e2d}}\end{aligned}\ ] ] the corresponding energy schematic is fig . [ fig1](e ) where it is seen that @xmath78 is a competing potential energy conversion channel , that can dominate the dynamics when the critical escape velocity @xmath76 is low or the pressure is high . this is exactly what we see in the bifurcation diagrams , fig . [ fig11 ] . overall , the effect of this contribution to shear flow generation from the ion orbit loss torque is to elongate and flatten the high shear flow peninsula . the hopf bifurcations that are starred in ( a ) , where the contribution is relatively small , have disappeared in ( b ) at a dze singularity . what this means is that as @xmath79 begins to take over * there is no longer a practicably accessible intermediate branch * , as can be seen in ( c ) where the intermediate branch is unstable until the remaining hopf bifurcation is encountered at extremely high @xmath4 . locally , in the transition region , as @xmath80 becomes significant the bifurcation diagram begins to look more like the simple s - shaped , cubic normal form schematics with classical hysteresis presented by earlier authors . however , this * unified * model accounts for shear flow suppression of the turbulence ( d ) , whereas theirs could not . the generation of stable shear flows in plasmas , and the associated confinement transitions and oscillatory behaviour in tokamaks and stellarators , is regulated by reynolds stress decorrelation of gradient - driven turbulence and/or by an induced bistable radial electric field . * these two mechanisms are smoothly unified by the first smooth road through the singularity and bifurcation structure of a reduced dynamical model for this system*. the model is constructed self - consistently , beginning from simple rate - laws derived from the basic pathways for energy transfer from pressure gradient to shear flows . it is iteratively strengthened by finding the singularities and allowing them to `` speak for themselves '' , then matching up appropriate physics to the unfoldings of the singularities . the smooth road from turbulence driven to electric field driven shear flows crosses interesting territory : * hysteresis is possible in both rgimes and is governed by different physics . * a metamorphosis of the dynamics is encountered , near which hysteretic transitions are forbidden . the metamorphosis is a robust organizing centre of codimension 1 , even though there are singularities of higher codimension in the system . * oscillatory and tristable domains are encountered . * to travel the smooth road several obstacles are successively negotiated in physically meaningful ways : a pitchfork is dissolved , simultaneously releasing a branch of solutions from a singular trap at infinity , a singularity is released from a trap at zero power input , and another is released from a trap at zero thermal diffusivity . in particular , these results suggest strategies for controlling access to high confinement states and manipulating oscillatory behaviour in fusion experiments . more generally i have shown that low - dimensional models have a useful role to play in the study of one of the most formidable of complex systems , a strongly driven turbulent plasma . having survived such a trial - by - ordeal , the methodology is expected to continue to develop as a valuable tool for taming this and other complex systems . .glossary of nomenclature [ cols= " < , < " , ] this work is supported by the australian research council . i thank the referees for helpful comments that have resulted in a better paper , and for their positive endorsements . diamond , p. h. , shapiro , v. , shevchenko , v. , kim , y. b. , rosenbluth , m. n. , carreras , b. a. , sidikman , k. , lynch , v. e. , garcia , l. , terry , p. w. , and sagdeev , r. z. ( 1992 ) . self - regulated shear flow turbulence in confined plasmas : basic concepts and potential applications to the l@xmath81h transition . , 2:97113 . fujisawa , a. , iguchi , h. , minami , t. , yoshimura , y. , tanaka , k. , itoh , k. , sanuki , h. , lee , s. , kojima , m. , itoh , s .- i . , yokoyama , m. , kado , s. , okamura , s. , akiyama , r. , ida , k. , isobe , m. , and s. nishimura , m. ( 2000 ) . experimental study of the bifurcation nature of the electrostatic potential of a toroidal helical plasma . , 7(10):41524183 .

a case study in bifurcation and stability analysis is presented , in which reduced dynamical system modelling yields substantial new global and predictive information about the behaviour of a complex system . the first smooth pathway , free of pathological and persistent degenerate singularities , is surveyed through the parameter space of a nonlinear dynamical model for a gradient - driven , turbulence shear flow energetics in magnetized fusion plasmas . along the route various obstacles and features are identified and treated appropriately . an organizing centre of low codimension is shown to be robust , several trapped singularities are found and released , and domains of hysteresis , threefold stable equilibria , and limit cycles are mapped . characterization of this rich dynamical landscape achieves unification of previous disparate models for plasma confinement transitions , supplies valuable intelligence on the big issue of shear flow suppression of turbulence , and suggests targeted experimental design , control and optimization strategies .

physics0410160

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

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

nlin0303010

the top quark is the heaviest known elementary particle , with a mass very close to the electroweak symmetry - breaking scale . as such , the top could be sensitive to physics beyond the standard model ( sm ) @xcite . new particles decaying to @xmath1 pairs can be scalar or vector , color - singlet or color - octet ; a scalar resonance is predicted in two - higgs - doublets models @xcite ; vector particles appear as massive @xmath4-like bosons in extended gauge theories @xcite , or as kaluza - klein states of the gluon and @xmath4 boson @xcite , or as colorons @xcite . searches for a color - singlet particle decaying to a @xmath1 pair have been performed by cdf and d0 collaborations in run i @xcite and run ii @xcite . in this letter we describe a search for a massive color - octet vector particle @xmath5 , which we call generically a `` massive gluon '' . we assume @xmath5 has a much stronger coupling to the top quark than to the lighter quarks , @xmath6 @xcite . we also assume the massive - massless gluon coupling is negligible . a feynman diagram for @xmath1 production via massive - gluon is shown in fig . [ fig : feynmann ] . the coupling strength of massive gluons to light and top quarks are @xmath7 and @xmath8 , respectively , where @xmath9 is the coupling constant of the sm strong interaction . in @xmath1 production , there are three observable parameters : the product of massive gluon coupling strength @xmath10 , mass @xmath11 , and width @xmath12 . in this analysis we consider only the possibility of novel strong - sector production ; we assume that the weak decay of the top quark follows the sm . since the color and the current structures of @xmath5 and sm gluon ( @xmath13 ) are identical , interference between processes through massive gluon and massless gluon will produce a @xmath1 invariant mass distribution with an enhanced signal that has a characteristic form @xcite as shown later in fig . [ fig : ttinvmass2 ] . if the coupling of @xmath5 to quarks is assumed to be parity - conserving , the production matrix element can be written as @xmath14 , @xmath15 , and @xmath16 are the @xmath1 invariant mass squared , the velocity of the top quark , and the angle between the top quark and the incident quark in the @xmath1 center of mass system , respectively . the propagator factors of gluons , massive gluons , and interference are @xmath17 we search for @xmath1 pairs produced by massive and massless gluons , where interference between these diagrams ( denoted by @xmath18 ) is considered , by examining the invariant mass spectrum of observed @xmath1 candidate events . production via a massive gluon in the `` lepton+jets '' decay channel.,width=192 ] the search is based on data collected with the cdf ii detector between march 2002 and may 2007 at the fermilab tevatron @xmath19 collider , corresponding to an integrated luminosity of about 1.9 fb@xmath0 . the cdf ii detector is a general purpose detector which is azimuthally and forward - backward symmetric . the detector consists of a charged particle tracking system composed of silicon microstrip detectors and a gas drift chamber inside a 1.4 t magnetic field , surrounded by electromagnetic and hadronic calorimeters and enclosed by muon detectors . the details of the detector are described elsewhere @xcite . the cross section for standard model @xmath1 production in @xmath19 collisions at @xmath20 tev is dominated by @xmath21 annihilation ( @xmath22 ) . the remaining @xmath23 is attributed to gluon - gluon fusion @xcite . standard model top quarks decay almost exclusively to @xmath24 . the search presented here focuses on the @xmath1 event topology wherein one @xmath25 boson decays hadronically while the other decays to an electron or muon and its corresponding neutrino . both @xmath2 quarks and the two decay quarks of the second @xmath25 boson appear as jets in the detector . accordingly , @xmath1 candidates in this `` lepton + jets '' channel are characterized by a single lepton , missing transverse energy @xmath26 @xcite , due to the undetected neutrino , and four jets . we use lepton triggers that require an electron or muon with @xmath27 gev/@xmath28 . events included in our analysis must first contain an isolated electron ( muon ) with @xmath29 gev ( @xmath30 gev@xmath31 ) in the central detector region with @xmath32 . electron and muon identification methods are described in ref . we remove events which have multiple leptons , cosmic ray muons , electrons from photon - conversion , or tracks , such that its momenta added to the primary lepton momenta gives an invariant mass equal to the @xmath4 mass . the position of the primary vertex ( along the beam ) is required to be within 60 cm of the center of the nominal beam intersection and consistent with the reconstructed @xmath33 position of the high-@xmath34 lepton . events must also feature at least 20 gev of @xmath26 attributable to the presence of a high-@xmath34 neutrino , as well as exactly four jets with @xmath35 and @xmath29 gev ( jet @xmath36 corrections are described in ref . jets are clustered with a cone - based algorithm , with a cone size @xmath37 . we reduce non-@xmath1 backgrounds by requiring at least one jet identified by the displaced secondary vertex `` @xmath2-tagging '' algorithm @xcite as being consistent with the decay of a long - lived @xmath2 hadron . the background to the @xmath1 signal is mostly from @xmath25+jets events with a falsely - reconstructed secondary vertex ( mistags ) or from @xmath25+jets events where one or more jets are due to heavy - flavor quarks . smaller contributions are from qcd multi - jet production , in which either the @xmath25 signature is faked when jets contain semi - leptonic @xmath2-hadron decays or when jets are mis - reconstructed and appear as electrons and missing @xmath36 , single top quark production , diboson ( @xmath38 , @xmath39 , @xmath40 ) production , and @xmath4 bosons produced in association with multiple jets . the methods used to estimate the backgrounds are detailed in ref . the @xmath41 process is taken as a background for this search , which is estimated by assuming @xmath42 gluon fusion fraction from ref . the estimated backgrounds in the sample are summarized in table [ tab : backgroundcomposition ] . the diboson and @xmath41 background processes are modeled with pythia @xcite , @xmath25+jets and @xmath4+jets processes with alpgen @xcite , and qcd with data . .background composition to the @xmath43 process and the expected numbers of events . systematic uncertainties coming from the background estimation method are listed . electroweak includes single top quark , diboson production , and @xmath4 bosons + jets productions . a luminosity of 1.9 fb@xmath0 is assumed . [ cols="<,^",options="header " , ] [ tab : lowerupperlimits ] we thank the fermilab staff and the technical staffs of the participating institutions for their vital contributions . this work was supported by the u.s . department of energy and national science foundation ; the italian istituto nazionale di fisica nucleare ; the ministry of education , culture , sports , science and technology of japan ; the natural sciences and engineering research council of canada ; the national science council of the republic of china ; the swiss national science foundation ; the a.p . sloan foundation ; the bundesministerium fr bildung und forschung , germany ; the world class university program , the national research foundation of korea ; the science and technology facilities council and the royal society , uk ; the institut national de physique nucleaire et physique des particules / cnrs ; the russian foundation for basic research ; the ministerio de ciencia e innovacin , and programa consolider - ingenio 2010 , spain ; the slovak r&d agency ; and the academy of finland . 99 c.t . hill and s.j . parke , phys . rev . d 49 ( 1994 ) 4454 . hill , phys . b 345 ( 1995 ) 483 . k. j. f. gaemers and f. hoogeveen , phys . b 146 ( 1984 ) 347 . d. dicus , a. stange and s. willenbrock , phys . b 333 ( 1994 ) 126 . a. leike , phys . ( 1999 ) 143 . b. lillie , l. randall , l .- t . wang , j. high energy phys . 09 ( 2007 ) 074 . t. g. rizzo , phys . d 61 ( 2000 ) 055005 . t. affolder _ ( cdf collaboration ) , phys . 85 ( 2000 ) 2062 . v. m. abazov _ ( d0 collaboration ) , phys . 92 ( 2004 ) 221801 . t. aaltonen _ ( cdf collaboration ) , phys . ( 2008 ) 231801 . t. aaltonen _ ( cdf collaboration ) , phys . d 77 ( 2008 ) 051102 . t. m. abazov _ ( d0 collaboration ) , phys . b 668 ( 2008 ) 98 . d. acosta _ et al . _ ( cdf collaboration ) , phys . rev . d 71 ( 2005 ) 032001 . m. cacciari _ et al . _ , j. high energy phys . 0404 ( 2004 ) 68 . we use a coordinate system defined about the proton beam direction , which is taken as the @xmath33 axis ; the @xmath44 axis lies in the horizontal plane . then @xmath16 is the usual polar angle and @xmath45 is the azimuthal angle . we define the pseudorapidity @xmath46 of a particle s three - momentum as @xmath47 . the transverse energy and momentum are defined as @xmath48 and @xmath49 where @xmath50 is the energy measured by the calorimeter and @xmath51 is the momentum measured in the tracking system . the missing transverse energy is defined as @xmath52 where @xmath53 is a unit vector in the transverse plane that points from the event vertex to the azimuth of the @xmath54 calorimeter tower . a. abulencia _ ( cdf collaboration ) , j. phys . g : nucl . part . 34 ( 2007 ) 2457 . a. bhatti _ et al . _ , methods , a 566 ( 2006 ) 375 . t. sjstrand _ et al . _ , computer physics commun . 135 ( 2001 ) 238 . et al . _ , j. high energy phys . 0307 ( 2003 ) 001 . k. kondo , j. phys . 57 ( 1988 ) 4126 ; k. kondo , hep - ex/0508035 . a. abulencia _ ( cdf collaboration ) , phys . d 73 ( 2006 ) 092002 . j. pumplin _ et al . _ , j. high energy phys . 0207 ( 2002 ) 012 . a. d. martin _ et al . _ , b 356 ( 1995 ) 89 . g. marchesini and b.r . webber , nucl . b 310 ( 1988 ) 461 ; g. marchesini _ 67 ( 1992 ) 465 . s. frixione _ et al . _ , j. high energy phys . 06 ( 2002 ) 029 . a. abulencia _ et al . _ , phys . rev . d 73 ( 2006 ) 032003 .

we present the result of a search for a massive color - octet vector particle , ( e.g. a massive gluon ) decaying to a pair of top quarks in proton - antiproton collisions with a center - of - mass energy of 1.96 tev . this search is based on 1.9 fb@xmath0 of data collected using the cdf detector during run ii of the tevatron at fermilab . we study @xmath1 events in the lepton+jets channel with at least one @xmath2-tagged jet . a massive gluon is characterized by its mass , decay width , and the strength of its coupling to quarks . these parameters are determined according to the observed invariant mass distribution of top quark pairs . we set limits on the massive gluon coupling strength for masses between 400 and 800 gev@xmath3 and width - to - mass ratios between 0.05 and 0.50 . the coupling strength of the hypothetical massive gluon to quarks is consistent with zero within the explored parameter space . massive gluon , top quark 13.85.rm , 14.65.ha , 14.80.-j

0911.3112

the evidence for the existence of dark matter in various astrophysical systems has been gathering over the past three decades . it is now well - recognized that the presence of dark matter is required in order to explain the observations of galaxies and other astrophysical systems on larger scales . the clearest support for the existence of dark matter comes from the now well - known observation of nearly flat rotation curves or constant rotation velocity in the outer parts of galaxies @xcite . surprisingly the rotation velocity is observed to remain nearly constant till the last point at which it can be measured . the simple principle of rotational equilibrium then tells one that the amount of dark to visible mass must increase at larger radii . thus the existence of the dark matter is deduced from its dynamical effect on the visible matter , namely the stars and the interstellar gas in galaxies . the presence of dark matter in the elliptical galaxies is more problematic to ascertain since these do not contain much interstellar hydrogen gas which could be used as a tracer of their dynamics , and also because these galaxies are not rotationally supported . these galaxies are instead supported by pressure or random motion of stars ( see binney @xcite for details of physical properties of the spiral and elliptical galaxies ) . as a result , the total mass can not be deduced using the rotation curve for elliptical galaxies . instead , here the motions of planetary nebulae which arise from old , evolved stars , as well as lensing , have been used to trace the dark matter @xcite . the fraction of dark matter at four effective radii is still uncertain with values ranging from 20% to 60% given in the literature , for the extensively studied elliptical galaxy ngc 3379 @xcite . historically the first evidence for the unseen or dark matter was found in _ clusters _ of galaxies . assuming the cluster to be in a virial equilibrium , the total or the virial mass can be deduced from the observed kinematics . zwicky @xcite noted that there is a discrepancy of a factor of @xmath2 10 between the observed mass in clusters of galaxies and the virial mass deduced from the kinematics . in other words , the random motions are too large for the cluster to be bound and a substantial amount of dark matter ( @xmath2 10 times the visible matter in galaxies ) is needed for the clusters of galaxies to remain bound . this discrepancy remained a puzzle for over four decades , and was only realized to be a part of the general trend after the galactic - scale dark matter was discovered in the late 1970 s . on the much larger cosmological scale , there has been some evidence for non - baryonic dark matter from theoretical estimates of primordial elements during big bang nucleosynthesis and measurements of them , particularly , primordial deuterium . accurate measurements of the cosmic microwave background radiation ( cmbr ) could as well give information about the total dark matter relic density of the universe . the satellite based cobe experiment was one of the first experiments to provide accurate mapping " of the cmbr@xcite . the recent high precision determination of the cosmological parameters using type i supernova data @xcite as well as precise measurements of the cosmic background radiation by the wmap collaboration @xcite has pinpointed the total relic dark matter density in the early universe with an accuracy of a few percent . accordingly , dark matter forms almost 26% of all the matter density of the universe , with visible matter about 4% and the dark energy roughly about 70% of the total energy density . this goes under the name of @xmath3cdm model with @xmath3 standing for dark energy and denoted by the einstein s constant , and cdm standing for cold dark matter @xcite . numerical simulations for the currently popular scenario of galaxy formation , based on the @xmath4cdm model , predicts a universal profile for the dark matter in halos of spherical galaxies @xcite . while this model was initially successful , over the years many discrepancies between the predictions from it and the observations have been pointed out . the strongest one has been the ` cusp - core ' issue of the central mass distribution . while navarro _ _ @xcite predict a cuspy central mass distribution , the observations of rotation curves of central regions of galaxies , especially the low surface brightness galaxies , when modeled show a flat or cored density distribution @xcite . a significantly different alternative to the dark matter , which can be used to explain the rotation curves of the galaxies and clusters was proposed early on by milgrom . he claimed that @xcite for low accelerations , newtonian law has to be modified by addition of a small repulsive term . this idea is known as mond or the modified newtonian dynamics . while initially this idea was not taken seriously by the majority of astrophysics community , it has gained more acceptance in the recent years . for example some of the standard features seen in galaxies such as the frequency of bars can be better explained under the mond paradigm , see tiret _ _ @xcite . for a summary of the predictions and comparisons of these two alternatives ( dark matter and mond ) , see combes _ et al . _ @xcite . so far the most direct empirical proof for the existence of dark matter , and hence the evidence against mond , comes from the study of the so - called bullet cluster@xcite . this is a pair of galaxies undergoing a supersonic collision at a redshift of @xmath5 . the main visible baryonic component in clusters is hot , x - ray emitting gas . in a supersonic collision , this hot gas would collide and be left at the center of mass of the colliding system while the stars will just pass through since they occupy a small volume . in the bullet cluster , the gravitational potential as traced by the weak - lensing shows peaks that are separated from the central region traced by the hot gas . in mond , these two would be expected to coincide , since the gravitational potential would trace the dominant visible component namely the hot gas , while if there is dark matter it would be expected to peak at the location of the stellar component in the galaxies . the latter case is what has been observed as can been seen in fig.1 of @xcite . for the rest of the article , we will not consider the mond explanation , but instead take the view point that the flat rotation curves of galaxies and clusters at large radii as an evidence for the existence of dark matter . furthermore , we believe that the dark matter explanation is much simpler and more natural compared to the mond explanation . despite the fact that the existence of dark matter has been postulated for over three decades , there is still no consensus of what its constituents are . this has been summarized well in many review articles . refs @xcite are couple of examples that span from the early to recent times on this topic . over the years , both astrophysicists as well as particle physicists have speculated on the nature of dark matter . the baryonic dark matter in the form of low - mass stars , binary stars , or jupiter - like massive planets were ruled out early on ( see @xcite for a summary ) . from the amount of dark matter required to explain the flat rotation curves , it can be shown that the number densities required of these possible constituents would be large , and hence it would be hard to hide these massive objects . because , if present in these forms , they should have been detected either from their absorption or from their emission signals . it has also been proposed that the galactic dark matter could be in the form of dense , cold molecular clumps @xcite , though this has not yet been detected . this alternative can not be expected to explain the dark matter necessary to fit " the observations of clusters , or indeed the elliptical galaxies since the latter have very little interstellar gas . there is also a more interesting possibility of the dark matter being essentially of baryonic nature , but due to the dynamics of the qcd phase transition in the early universe which left behind a form of _ cold _ quark - gluon - plasma , the baryon number content of the dark matter is hidden from us . this idea was first proposed by witten in 1984 @xcite , who called these quantities as quark nuggets . an upper limit on the total number of baryons in a quark nugget is determined by the baryon to photon ratio in the early universe ( see for example @xcite ) . taking in to consideration these constraints , it is possible to fit the observed relic density with a mass ( density ) distribution of the quark nuggets @xcite . for the observational possibilities of such quark nuggets , see for example , ref.@xcite . from a more fundamental point of view , it is not clear what kind of elementary particle could form dark matter . the standard model of particle physics describes all matter to be made up of quarks and leptons of which neutrinos are the _ only _ ones which can play the role of dark matter as they are electrically neutral . however with the present indications from various neutrino oscillation experiments putting the standard model neutrino masses in the range @xmath6 ev @xcite they will not form significant amount of dark matter . there could however , be non - standard sterile neutrinos with masses of the order of kev - mev which could form _ _ warm__dark matter ( for reviews , see refs . cold dark matter ( cdm ) , on the other hand , is favored over the _ warm _ dark matter by the hierarchical clustering observed in numerical simulations for large scale structure formation , see for example ref.@xcite . recent analysis including x - ray flux observations from coma cluster and andromeda galaxy have shown that the room for sterile neutrino warm dark matter is highly constrained @xcite . however , if one does not insist that the total relic dark matter density is due to sterile neutrinos then , it is still possible that they form a sub - dominant _ warm _ component of the total dark matter @xcite relic density . the standard model thus , needs to be extended to incorporate a dark matter candidate . the simplest extensions would be to just include a new particle which is a singlet under the sm gauge group ( _ i.e. _ , does not carry the standard model interactions ) . further , we might have to impose an additional symmetry under which the dark matter particle transforms non - trivially to keep it stable or at least sufficiently long lived with a life time typically larger than the age of the universe . some of the simplest models would just involve adding additional light ( @xmath2 gev ) scalar particles to the sm and with an additional @xmath7 symmetry ( see for example , boehm _ et al . _ similar extensions of sm can be constructed with fermions too @xcite . an interesting aspect of these set of models is that they can be tested at existing @xmath8 colliders like for the example , the one at present at frascati , italy @xcite . a heavier set of dark matter candidates can be achived by extending the higgs sector by adding additional higgs scalar doublets . these go by the name of _ inert _ higgs models @xcite . in this extension , there is a _ additional _ neutral higgs boson which does not have sm gauge interactions ( hence inert ) , which can be a dark matter candidate . with the inclusion of this extra inert higgs doublet , the sm particle spectrum has some added features , like it can evade the up to 1.5 tev while preserving the perturbativity of higgs couplings up to high scales and further it is consistent with the electroweak precision tests @xcite . on the other hand , there exist extensions of the standard model ( generally labeled beyond standard model ( bsm ) physics ) which have been constructed to address a completely different problem called the hierarchy problem . the hierarchy problem addresses the lack the symmetry for the mass of the higgs boson in the standard model and the consequences of this in the light of the large difference of energy scales between the weak interaction scale ( @xmath9 gev ) and the quantum gravity or grand unification scale ( @xmath10 gev ) . such a huge difference in the energy scales could destabilize the higgs mass due to quantum corrections . to protect the higgs mass from these dangerous radiative corrections , new theories such as supersymmetry , large extra dimensions and little higgs have been proposed . it turns out that most of these bsm physics models contain a particle which can be the dark matter . a few examples of these theories and the corresponding candidates for dark matter are as follows . ( i ) axions are pseudo - scalar particles which appear in theories with peccei - quinn symmetry @xcite proposed as solution to the _ strong cp _ problem of the standard model . they also appear in superstring theories which are theories of quantum gravity . the present limits on axions are @xcite extremely strong from astrophysical data . in spite of this , there is still room for axions to form a significant part of the dark matter relic density . \(ii ) supersymmetric theories @xcite which incorporate fermion - boson interchange symmetry are proposed as extensions of standard model to protect the higgs mass from large radiative corrections . the dark matter candidate is the lightest supersymmetric particle ( lsp ) which is stable or sufficiently long lived as mentioned before - parity . if this symmetry is exact , the particle is stable . if is broken very mildly , the lsp could be sufficiently long lived , close to the age of the universe . ] . depending on how supersymmetry is broken @xcite , there are several possible dark matter candidates in these models . in some models , the lightest supersymmetric particle and hence the dark matter candidate is a neutralino . the neutralino is a linear combination of super - partners of @xmath11 as well as the neutral higgs bosons . the other possible candidates are the super - partner of the graviton , called the gravitino and the super - partners of the axinos , the scalar saxion and the fermionic axino . these particles also can explain the observed relic density @xcite . \(iii ) other classic extensions of the standard model either based on additional space dimensions or larger symmetries also have dark matter candidates . in both versions of the extra dimensional models , _ i.e. , _ the arkani - hamed , dimopoulos , dvali ( add ) @xcite and randall - sundrum ( rs ) @xcite , models , the lightest kaluza - klein particle can be considered as the dark matter candidate @xcite . similarly , in the little - higgs models where the higgs boson is a pseudo - goldstone boson of a much larger symmetry , a symmetry called t - parity @xcite assures us a stable and neutral particle which can form the dark matter . very heavy neutrinos with masses of @xmath12 can also naturally appear within some classes of randall - sundrum and little higgs models . under suitable conditions , these neutrinos can act like cold dark matter . ( for a recent study , please see @xcite ) . in addition to these particles , more exotic candidates like simpzillas @xcite and wimpzillas @xcite with masses close to the gut scale ( @xmath13 gev ) have also been proposed in the literature . indirect searches like icecube @xcite ( discussed below ) already have strong constraints on simpzillas . if the dark matter candidate is indeed a new particle and it has interactions other than gravitational interactions , then the most probable interactions it could have are the weak interactions . this weakly interacting particle , dubbed as wimp ( weakly interacting massive particle ) could interact with ordinary matter and leave traces of its nature . there are two ways in which the wimp could be detected ( a ) direct detection : here one looks for the interaction of the wimp on a target , the target being typically nuclei in a scintillator . it is expected that the wimps present all over the galaxy scatter off the target nuclei once in a while . measuring the recoil of the nuclei in these rarely occurring events would give us information about the properties of the wimp . the scattering cross section would depend on whether it was elastic or inelastic and is a function of the spin of the wimp . there are more than 20 experiments located all over the world , which are currently looking for wimp through this technique . some of them are dama , cdms , cresst , cuoricino , drift _ etc . _ ( b ) indirect detection : when wimps cluster together in the galatic halo , they can annihilate with themselves giving rise to electron - positron pairs , gamma rays , proton - anti - proton pairs , neutrinos _ etc . _ the flux of such radiation is directly proportional to the annihilation rate and the the wimp matter density . observation of this radiation could lead to information about the mass and the cross section strength of the wimps . currently , there are several experiments which are looking for this radiation ( i ) magic , hess , cangaroo , fermi / glast , egret _ etc . _ look for the gamma ray photons . ( ii ) heat , caprice , bess , pamela , ams can observe anti - protons and positron flux . ( iii)very highly energetic neutrinos / cosmic rays @xmath14 a few tev to multi - tev can be observed by large detectors like amanda , antares , icecube _ etc . _ ( for a more detailed discussion see @xcite ) . over the years , there have been indications of presence of the dark matter through both direct and indirect experiments . the most popular of these signals are integral and dama results ( for a nice discussion on these topics please see , @xcite ) . integral ( international gamma -ray astrophysics laboratory ) is a satellite based experiment looking for gamma rays in outer space . in 2003 , it has observed a very bright emission of the 511 kev photons from the galactic bulge @xcite at the centre . the 511 kev line is special as it is dominated by @xmath8 annihilations via the positronium . the observed rate of ( 3 - 15 ) @xmath15 positrons / sec in the inner galaxy was much larger than the expected rate from pair creation via cosmic ray interactions with the interstellar medium in the galactic bulge by orders of magnitude . further , the signal is approximately spherically symmetric with very little positrons from galactic bulge contributing to the signal @xcite . several explanations have been put forward to explain this excess . astrophysical entities like hypernovae , gamma ray burts and x - ray binaries have been proposed as the likely objects contributing to this excess . on the other hand , this signal can also be attributed to the presence of dark matter which could annihilate itself giving rise to electron - positron pairs . to explain the integral signal in terms of dark matter , extensions of standard model involving light @xmath16 particles and light gauge bosons ( @xmath17 ) are ideally suited . these models which have been already reviewed in the previous section , can be probed directly at the existing and future @xmath8 colliders and hence could be tested . until further confirmation from either future astrophysical experiments or through ground based colliders comes about , the integral remains an ` anomaly ' as of now . while the integral is an indirect detection experiment , the dama ( dark matter ) is a direct detection experiment located in the gran sasso mountains of italy . the target material consists of highly radio pure nai crystal scintillators ; the scintillating light from wimp - nucleon scattering and recoil is measured . the experiment looks for an annual modulation of the signal as the earth revolves around the sun @xcite . such modulation of the signal is due to the gravitational effects of the sun as well as rotatory motion of the earth . dama and its upgraded version dama / libra have collected data for seven annual cycles and four annual cycles respectively @xcite . ] . together they have reported an annual modulation at 8.2@xmath18 confidence level . if confirmed , the dama results would be the first direct experimental evidence for the existence of wimp dark matter particle . however , the dama results became controversial as this positive signal has not been confirmed by other experiments like xenon and cdms , which have all reported null results in the spin independent wimp - nucleon scattering signal region . the xenon 10 detector also at gran sasso laboratories uses a xenon target while measuring simultaneously the scintillation and ionization produced by the scattering of the dark matter particle . the simultaneous measurement reduces the background significantly down to @xmath19 . with a fiducial mass of 5.4 kg , they set an upper limit of wimp - nucleon spin independent cross section to be 8.8 @xmath20 for a wimp mass of 100 gev@xcite . an upgraded version xenon 100 has roughly double the fiducial mass has started taking data from oct 2009 . in the first results , they present null results , with upper limits of about @xmath21 for 55 gev wimps @xcite . these results severely constraint interpretation of the dama results in terms of an elastic spin independent wimp - nucleon scattering . the cdms ( cryogenic dark matter search ) experiment has 19 germanium detectors located in the underground soudan mine , usa . it is maintained at temperatures @xmath22 ( milli - kelvin ) . nuclear recoils can be seen " by measuring the ionisation energy in the detector . efficient separation between electron recoils and nuclear recoils is possible by employing various techniques like signal timing and measuring the ratios of the ionization energies . similar to xenon , this experiment @xcite too reported null results in the signal region and puts an upper limit @xmath23 on the wimp - nucleon cross - section for a wimp mass of around 60 gev . the cogent ( cryogenic germanium neutrino technology ) collaboration runs another recent experiment which uses ultra low noise germanium detectors . it is also located in the soudan man , usa . the experiment has one of the lowest backgrounds below 3 kevee ( kev electron equivalent ( ee ) ionisation energy ) . it could further go down to 0.4 kevee , the electron noise threshold . the first initial runs have again reported null results @xcite consistent with the observed background . at this point , the experiment did not have the sensitivity to confirm / rule out the dama results . however , later runs have shown some excess events over the expected background in the low energy regions @xcite . while , the collaboration could not find a suitable explanation for this excess ( as of now ) there is a possibility of these _ excess _ events having their origins in a very light wimp dark matter particle . however , care should be taken before proceeding with this interpretation as the cogent collaboration does not distinguish between electron recoils and nucleon recoils@xcite . in the light of these experimental results , the dama results are hard to explain . one of the ways out to make the dama results consistent with other experiments is to include an effect called channelling " which could be present only in the nai crystals which dama uses . however , even the inclusion of this effect does not improve the situation significantly . to summarize , the situation is as follows for various interpretations of the wimp - nucleon cross section . for esi ( elastic spin independent ) interpretation , the dama regions are excluded by both cdms as well as xenon 10 . this is irrespective of whether one considers the channeling effect or not . it is also hard to reconcile dama results with cogent in this case . for elastic spin dependent ( esd ) interpretation , the dama and cogent results though consistent with each other are in conflict with other experiments . for an interpretation in terms of wimp - proton scattering , the results are in conflict with several experiments like simple , picasso etc . on the other hand , an interpretation in terms of wimp - neutron scattering is ruled out by xenon and cdms data . for the inelastic dark matter interpretations , spin -independent cross section with a medium mass ( @xmath24 ) wimp is disfavored by cresst as well as cdms data . for a low mass ( close to 10 gev ) wimp , with the help of channeling in the nai crystals , it is possible to explain the dama results , in terms of spin- independent inelastic dark matter - nucleon scattering . however , the relevant parameters ( dark matter mass and mass splittings ) should be fine tuned and further , the wimp velocity distribution in the galaxy should be close to the escape velocity . inelastic spin dependent interpretation of the dama results is a possibility ( because it can change relative signals at different experiments @xcite ) which does not have significant constraints from other experiments . however , it has been shown@xcite that inelastic dark matter either with spin dependent or spin independent interpretation of the dama results is difficult to reconcile with the cogent results , unless one introduces substantial exponential background in the cogent data . the focus of the present topical review is a set of new experimental results which have appeared over the past year . in terms of the discussion in the previous section , these experiments follow indirect " methods to detect dark matter . the data from these experiments seems to be pointing to either discovery " of the dark matter or some yet non - understood new astrophysics being operative within the vicinity of our galaxy . the four main experiments which have led to this excitement are ( i ) pamela@xcite ( ii ) atic@xcite ( iii ) hess@xcite and ( iv ) fermi@xcite . all of these experiments involve international collaborations spanning several nations . while pamela and fermi are satellite based experiments , atic is a balloon borne experiment and hess is a ground based telescope . all these experiments contain significant improvements in technology over previous generation experiments of similar type . the h.e.s.s experiment has a factor @xmath25 improvement in @xmath26-ray flux sensitivity over previous experiments largely due to its superior rejection of the hadronic background . similarly , atic is the next generation balloon based experiment equipped to have higher resolution as well as larger statistics . similar statements also hold for the satellite based experiments , pamela and fermi . it should be noted that the satellite based experiments have some inherent advantages over the balloon based ones . firstly , they have enhanced data taking period , unlike the balloon based ones which can take data only for small periods . and furthermore , these experiments also do not have problems with the residual atmosphere on the top of the instrument which plagues the balloon based experiments . . the figures of pamela and atic are reproduced from their original papers cited above . [ pamela],title="fig:",scaledwidth=40.0% ] . the figures of pamela and atic are reproduced from their original papers cited above . [ pamela],title="fig:",scaledwidth=50.0% ] the satellite - based _ payload for antimatter matter exploration and light - nuclei astrophysics _ or pamela collects cosmic ray protons , anti - protons , electrons , positrons and also light nuclei like helium and anti - helium . one of the main strengths of pamela is that it could distinguish between electrons and anti - electrons , protons and anti - protons and measure their energies accurately . the sensitivity of the experiment in the positron channel is up to approximately 300 gev and in the anti - proton channel up to approximately 200 gev . since it was launched in june 2006 , it was placed in an elliptical orbit at an altitude ranging between @xmath27 km with an inclination of 70.0 . about 500 days of data was analyzed and recently presented . the present data is from 1.5 gev to 100 gev has been published in the journal nature @xcite . in this paper , pamela reported an excess of positron flux compared to earlier experiments . in the left panel of the fig . [ pamela ] , we see pamela results along with the other existing results . the y - axis is given by @xmath28 , which @xmath29 represents the flux of the corresponding particle . according to the analysis presented by pamela , the results of pamela are consistent with the earlier experiments up to 20 gev , taking into consideration the solar modulations between the times of pamela and previous experiments . particles with energies up to 20 gev are strongly effected by solar wind activity which varies with the solar cycle . on the other hand , pamela has data from 10 gev to 100 gev , which sees an increase in the positron flux ( fig . [ pamela ] ) . the only other experimental data in this energy regime ( up to 40 gev ) are the ams and heat , which while having large errors are consistent with the excess seen by pamela . in the low energy regime most other experiments are in accordance with each other but have large error bars . , scaledwidth=80.0% ] cosmic ray positrons at these energies are expected to be from secondary sources _ i.e. _ as result of interactions of primary cosmic rays ( mainly protons and electrons ) with interstellar medium . the flux of this secondary sources can be estimated by numerical simulations . there are several numerical codes available to compute the secondary flux , the most popular publicly available codes being galprop @xcite and crpropa @xcite . these codes compute the effects of interactions and energy loses during cosmic ray propagation within galactic medium taking also in to account the galactic magnetic fields . galprop solves the differential equations of motion either using a 2d grid or a 3d grid while crpropa does the same using a 1d or 3d grids . while galprop contains a detailed exponential model of the galactic magnetic fields , crpropa implements only extragalactic turbulent magnetic fields . in particular crpropa is not optimised for convoluted galactic magnetic fields . for this reason , galprop is best suited for solving diffusion equations involving low energy ( gev - tev ) cosmic rays in galactic magnetic fields . the main input parameters of the galprop code are the primary cosmic ray injection spectra , the spatial distribution of cosmic ray sources , the size of the propagation region , the spatial and momentum diffusion coefficients and their dependencies on particle rigidity . these inputs are mostly fixed by observations , like the interstellar gas distribution is based on observations of neutral atomic and molecular gas , ionized gas ; cross sections and energy fitting functions are build from nuclear data sheets ( based on las almos nuclear compilation of nuclear crosssections and modern nuclear codes ) and other phenomenological estimates . interstellar radiation fields and galactic magnetic fields are based on various models present in literature . the uncertainties in these inputs would constitute the main uncertainties in the flux computation from galprop . recently , a new code called crt which emphasizes more on the minimization of the computation time was introduced . here most of the input parameters are user defined @xcite . finally , using the popular monte carlo routine geant @xcite one can construct cosmic ray propagation code as has been done by @xcite . on the other hand , dark matter relic density calculators like darksusy @xcite also compute cosmic ray propagation in the galaxies required for indirect searches of dark matter . it is further interfaced with galprop . in summary , galprop is most suited for the present purposes _ i.e , _ understanding of pamela and atic data which is mostly in the gev - tev range . it has been shown the results from these experiments do not vary much if one instead chooses to use a geant simulation . in fact , most of the experimental collaborations use galprop for their predictions of secondary cosmic ray spectrum . in the left panel of fig . [ pamela ] , the expectations based on galprop are given as a solid line running across the figure . from the figure it is obvious that pamela results show that the positron fraction increases with energy compared to what galprop expects . the excess in the positron fraction as measured by pamela with respect to galprop indicates that this could be a result due to new primary sources rather than secondary sources this new primary source could be either dark matter decay / annihilation or a nearby astrophysical object like a pulsar . before going to the details of the interpretations , let us summarize the results from atic and fermi also . _ advanced thin ionization calorimeter _ or in short atic is a balloon - borne experiment to measure energy spectrum of individual cosmic ray elements within the region of gev up to almost a tev ( thousand gev ) with high precision . as mentioned , this experiment was designed to be a high - resolution and high statistics experiment in this energy regime compared to the earlier ones . atic measures all the components of the cosmic rays such as electrons , protons ( and their anti - particles ) with high energy resolution , while distinguishing well between electrons and protons . atic ( right panel in fig . [ pamela ] ) presented its primary cosmic ray electron ( @xmath1 + ps . ) spectrum between the energies 3 gev to about 2.5 tev @xmath30 . thus it is normalized by a factor @xmath31 . ] . the results show that the spectrum while agreeing with the galprop expectations up to 100 gev , show a sharp increase above 100 gev . the total flux increases till about 600 gev where it peaks and then sharply falls till about 800 gev . thus , atic sees an excess of the primary cosmic ray ( @xmath1 + ps . ) spectrum between the energy range @xmath32 gev . the rest of the spectrum is consistent with the expectations within the errors . what is interesting about such peaks in the spectrum is that , if they are confirmed they could point towards a breit - wigner resonance in dark matter annihilation cross section with a life time as given by its width . as we will discuss in the next section , this possibility is severely constrained by the data from the fermi experiment . another ground based experiment sensitive to cosmic rays within this energy range is h.e.s.s which can measure gamma rays from few hundred gev to few tev . this large reflecting array telescope operating from namibia has presented data ( shown in figure [ fermi ] ) from @xmath33 gev to about 5 tev . it could confirm neither the peaking like behavior at 600 gev nor the sharp cut - off at 800 gev of the atic data . the atic results can be made consistent with those of hess . this would require a @xmath34 overall normalisation of the hess data . such a normalisation is well within the uncertainty of the energy resolution of hess . however notice that hess data does not have a sharp fall about and after 800 gev . the large area telescope ( lat ) is one of the main components of the fermi gamma ray space telescope , which was launched in june 2008 . due to its high resolution and high statistical capabilities , it has been one of the most anticipated experiments in the recent times . fermi can measure gamma rays between 20 mev and 300 gev with high accuracy and primary cosmic ray electron ( @xmath1 + ps . ) flux between 20 gev and 1 tev . the energy resolution averaged over the lat acceptance is 11% fwhm ( full - width - at - half - maximum ) for 20 - 100 gev , increasing to 13% fwhm for 150 - 200 gev . the photon angular resolution is less than 0.1 over the energy range of interest ( 68% containment ) . the fermi - lat collaboration has recently published its six month data on the primary cosmic ray electron flux . more than 4 million electron events above 20 gev were selected in survey ( sky scanning ) mode from 4 august 2008 to 31 january 2009 . the systematic error on the absolute energy of the lat was determined to be @xmath35 for 20 - 300 gev . please see table i for more details on the errors in @xcite . in fig . [ fermi ] we reproduce the result produced by the fermi collaboration . they find that the primary cosmic ray electron spectrum more or less goes along the expected lines up to 100 gev ( its slightly below the expected flux between 10 and 50 gev ) , however above 100 gev , there is strong signal for an excess of the flux ranging up to 1 tev . the fermi data thus confirms the excess in the electron spectrum which was seen by atic , the excess however has a much flatter profile with respect to the peak seen by atic . thus , atic could in principle signify a resonance in the spectrum , whereas fermi can not . however , in comparing both the spectra from the figures presented above , one should keep in mind that the fermi excess is in the total electron spectrum ( @xmath36 ) whereas the atic data is presented in terms of positron excess only . if the excess in fermi is caused by the excess only through excess positrons , one should expect that the fermi spectra to also have similar peak like behavior at 600 gev . from fig.([fermi ] ) , where both fermi and atic data are presented , we see that the atic data points are far above that of fermi s . lets now summarise the experimental observations @xcite which would require an interpretation : * the excess in the flux of positron fraction @xmath37 measured by pamela up to 100 gev . * the lack of excess in the anti - proton fraction measured by pamela up to 100 gev . * the excess in the total flux @xmath38 in the spectrum above 100 gev seen by fermi , hess _ etc . _ while below 100 gev , the measurements have been consistent with galprop expectations . * the absence of peaking like behavior as seen by atic , which indicates a long lived particle , in the total electron spectrum measured by fermi . two main interpretations have been put forward : ( a ) a nearby astrophysical source which has a mechanism to accelerate particles to high energies and ( b ) a dark matter particle which decays or annihilates leading to excess of electron and positron flux . which of the interpretations is valid will be known within the coming years with enhanced data from both pamela and fermi . let us now turn to both the interpretations : pulsars and supernova shocks have been proposed as likely astrophysical local sources of energetic particles that could explain the observed excess of the positron fraction @xcite . in the high magnetic fields present in the pulsar magnetosphere , electrons can be accelerated and induce an electromagnetic cascade through the emission of curvature radiation . this can lead to a production of high energy photons above the threshold for pair production ; and on combining with the number density of pulsars in the galaxy , the resulting emission can explain the observed positron excess @xcite . the energy of the positrons tell us about the site of their origin and their propagation history @xcite . the cosmic ray positrons above 1 tev could be primary and arise due to a source like a young plusar within a distance of 100 pc @xcite . this would also naturally explain the observed anisotropy , as argued for two of the nearest pulsars , namely @xmath39 and the @xmath40 @xcite . on a similar note , diffusive shocks as in a supernova remnant hardens the spectrum , hence this process can explain the observed positron excess above 10 gev as seen from pamela @xcite . another possible astrophysical source that has been proposed is the pion production during acceleration of hadronic cosmic rays in the local sources @xcite . it has been argued @xcite that the measurement of secondary nuclei produced by cosmic ray spallation can confirm whether this process or pulsars are more important as the production mechanism . it has been show that the present data from atic - ii supports the hadronic model and can account for the entire positron excess observed . if the excess observed by pamela , hess and fermi is not due to some yet not fully - understood astrophysics but is a signature of the dark matter , then there are two main processes through which such an excess can occur : 1 . the annihilation of dark matter particles into standard model ( sm ) particles and 2 . the decay of the dark matter particle into sm particles . interpretation in terms of annihilating dark matter , however , leads to conflicts with cosmology . the observed excesses in the pamela / fermi data would set a limit on the product of annihilation cross section and the velocity of the dark matter particle in the galaxy ( for a known dark matter density profile ) . annihilation of the dark matter particles also happens in the early universe with the same cross section but at much larger velocities for particles ( about 1000 times the particle velocities in galaxies ) . the resultant relic density is not compatible with observations . the factor @xmath41 difference in the velocities should some how be compensated in the cross sections . this can be compensated by considering boost " factors for the particles in the galaxy which can enhance the cross section by several orders of magnitude . the boost factors essentially emanate from assuming local substructures for the dark matter particles , like clumps of dark matter and are typically free parameters of the model ( see however , @xcite ) . another mechanism which goes by the name sommerfeld mechanism can also enhance the annihilation cross sections . for very heavy dark matter ( with masses much greater than the relevant gauge boson masses ) trapped in the galactic potential , non - perturbative effects could push the annihilation cross - sections to much larger values . for su(2 ) charged dark matter , the masses of dark matter particles should be @xmath42 @xcite . the sommerfeld mechanism is more general and applicable to other ( new ) interactions also@xcite . another way of avoiding conflict with cosmology would be to consider non - thermal production of dark matter in the early universe . before the release of fermi data , the annihilating dark matter model with a very heavy dark matter @xmath43tev was much in favour to explain the resonance peak " of the atic and the excess in pamela data . post fermi , whose data does not have sharp raise and fall associated with a resonance , the annihilating dark matter interpretation has been rendered incompatible . however , considering possible variations in the local astro physical background profile due to presence of local cosmic ray accelerator , it has been shown that it is still possible to explain the observed excess , along with fermi data with annihilating dark matter . the typical mass of the dark matter particle could lie even within sub - tev region @xcite and as low as @xmath44 gev @xcite . some more detailed analysis can be found in @xcite . several existing bsm physics models of annihilating dark matter become highly constrained or ruled out if one requires to explain pamela / atic and fermi data . the popular supersymmetric dm candidate neutralino with its annihilating partners such as chargino , stop , stau _ etc . _ , can explain the cosmological relic density but not the excess observed by pamela / atic . novel models involving a new _ dark force _ , with a gauge boson having mass of about 1 gev @xcite , which predominantly decays to _ leptons _ , together with the so - called sommerfeld enhancement seem to fit the data well . the above class of models , which are extensions of standard model with an additional @xmath7 gauge group , caught the imagination of the theorists @xcite . a similar supersymmetric version of this mechanism where the neutralinos in the mssm can annihilate to a scalar particle , which can then decay the observed excess in the cosmic ray data @xcite . models involving type ii seesaw mechanism @xcite have also been considered recently where neutrino mass generation is linked with the positron excess . in addition to the above it has been shown that extra dimensional models with kk gravitions can also produce the excess @xcite . models with nambu - goldstone bosons as dark matter have been studied in @xcite . in the case of decaying dark matter , the relic density constraint of the early universe is not applicable , however , the lifetime of the dark matter particle ( typically of a mass of @xmath45 tev ) should be much much larger than ( @xmath46 times ) the age of the universe@xcite . such a particle can fit the data well . a crucial difference in this picture with respect to the annihilation picture is that the decay rate is directly proportional to the density of the dark matter ( @xmath47 ) , whereas the annihilation rate is proportional to its square , ( @xmath48 ) . the most promising candidates in the decaying dark matter seem to be a fermion ( scalar ) particle decaying in to @xmath49 _ etc . _ ( @xmath50 _ etc . _ ) @xcite . in terms of the bsm physics , supersymmetric models with a heavy gravitino and small r - parity violation have been proposed as candidates for decaying dark matter @xcite . a heavy neutralino with @xmath51-parity violation can also play a similar role @xcite stated above . a recent more general model independent analysis has shown that , assuming the galprop background , gravitino decays can not simultaneously explain both pamela and fermi excess . however , the presence of additional astrophysical sources can change the situation @xcite . independent of the gravitino model , it has been pointed out that , the decays of the dark matter particle could be new signals for unification where the dark matter candidate decays through dimension six operators suppressed by two powers of gut scale @xcite . finally , there has also been some discussion about the possibilities of dark matter consisting of not one particle but two particles , of which one is the decaying partner . this goes under the name of and analysis of this scenario has been presented by @xcite . we have so far mentioned just a sample of the theoretical ideas proposed in the literature . several other equally interesting and exciting ideas have been put forward , which have not been presented to avoid the article becoming too expansive . an interesting aspect about the present situation is that , future data from pamela and fermi could distinguish whether the astrophysical interpretation _ i.e. _ in terms of pulsars or the particle physics interpretation in terms of dark matter is valid @xcite . pamela is sensitive to up to 300 gev in its positron fraction and this together with the measurement of the total electron spectrum can strongly effect the dark matter interpretations . fermi with its improved statistics , can on the other hand look for anisotropies within its data @xcite which can exist if the pulsars are the origin of this excess . further measurements of the anti - deuteron could possibly gives us a hint why there is no excess in the anti - proton channel @xcite . similarly neutrino physics experiments could give us valuable information on the possible models@xcite . finally , the large hadron collider could also give strong hints on the nature of dark matter through direct production @xcite . as we have been preparing this note , there has been news from one the experiments called cdms - ii ( cryogenic dark matter search experiment)@xcite . as mentioned before this experiment conducts direct searches for wimp dark matter by looking at collisions of wimps on super - cooled nuclear target material . the present and final analysis of this experiment have shown two events in the signal region , with the probability of observing two or more background events in that region being close to @xmath52 . thus , while these results are positive and encouraging , they are not conclusive . however these results already set a stringent upper bound on the wimp - nucleus cross section for a wimp mass of around 70 gev . the exclusion plots in the parameter space of wimp cross section and wimp mass are presented in the paper @xcite . the interpretations of this positive signal are quite different compared to the signal of pamela and fermi . while pamela and fermi as we have seen would require severe modifications for the existing beyond standard model ( bsm ) models of dark matter , cdms results if confirmed would prefer the existing bsm dark matter candidates like neutralino of the supersymmetry . there are ways of making both pamela / fermi and cdms - ii consistent through dark matter interpretations , however , we will not discuss it further here . finally , it has been shown that it is possible to make cdms - ii results consistent with dama annual modulation results by assuming a spin - dependent inelastic scattering of wimp on nuclei @xcite . in the present note , we have tried to convey exciting developments which have been happening recently within the interface of astrophysics and particle physics , especially on the one of the most intriguing subjects of our time , namely , the _ dark matter_. though it has been proposed about sixty years ago , so far we have not have any conclusive evidence of its existence other than through gravitational interactions , or we do not of its fundamental composition . experimental searches which have been going on for decades have not bore fruit in answering either of these questions . for these reasons , the present indications from pamela and fermi have presented us with a unique opportunity of unraveling at least some of mystery surrounding the dark matter . these experimental results , if they hold and get confirmed as due to dark matter , would strongly modify the way dark matter was perceived in the scientific community . as a closing point , let us note that there are several new experiments being planned to explore the dark matter either directly or indirectly and thus some information about the nature of the dark matter might just around the corner . we thank pamela collaboration , atic collaboration and fermi - 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it is well known that the dark matter dominates the dynamics of galaxies and clusters of galaxies . its constituents remain a mystery despite an assiduous search for them over the past three decades . recent results from the satellite - based pamela experiment detect an excess in the positron fraction at energies between @xmath0 gev in the secondary cosmic ray spectrum . other experiments namely atic , hess and fermi show an excess in the total electron ( ps . + @xmath1 ) spectrum for energies greater 100 gev . these excesses in the positron fraction as well as the electron spectrum could arise in local astrophysical processes like pulsars , or can be attributed to the annihilation of the dark matter particles . the second possibility gives clues to the possible candidates for the dark matter in galaxies and other astrophysical systems . in this article , we give a report of these exciting developments . = 22.8 cm .6 true cm

0909.1182

differential geometry has proven to be highly valuable in extracting the geometric meaning of continuum vector theories . of particular interest has been the dirac - khler formulation of fermionic field theory @xcite , which uses the antisymmetry inherent in the product between differential forms to describe the clifford algebra . in order to regularize calculations , we are required to introduce a discrete differential geometry scheme and it would be ideal if this had the same properties as the continuum and the correct continuum limit . however , defining such a scheme has proven to be very challenging . the difficulties are usually exhibited by the hodge star , which maps a form to its complement in the space , and the wedge product between forms . in a discretization , we would like the latter to allow the product rule to be satisfied and we would like both to be local . several discrete schemes have been proposed that address these difficulties with varying success . becher and joos @xcite used a lattice to define operators with many desirable properties , but that were non - local . to resolve the non - locality , they introduced translation operators . kanamori and kawamoto @xcite also used a lattice and introduced a specific non - commutativity between the fields and discrete forms . this allowed the product rule to be fulfilled , but they found that it became necessary to introduce a second orientation of form in order for their action to remain hermitian . instead of a lattice , balachandran _ et al _ @xcite used a quantized phase space to regularize their system , leading to a fuzzy coordinate space @xcite . in this paper , we shall build upon a proposal by adams @xcite in which he introduces two parallel lattices to maintain the locality of the hodge star and uses a finite element scheme to capture the properties of the wedge product . this proposal describes a local , discrete differential geometry for an arbitrary topological space and its formal aspects have been thoroughly studied by de beauc , samik sen , siddartha sen and czech @xcite . however , here we want to focus on its application to the dirac - khler formulation . in lattice quantum chromodynamics ( lattice qcd ) calculations , it is common to see the staggered fermion formulation used to describe fermions @xcite . this formulation addresses the problem of fermion doubling @xcite by reducing the number of degenerate fermions to @xmath0 in @xmath1 dimensional space - time . it is frequently used with the quarter - root trick @xcite to provide a description of a single fermion on the lattice , although this approach has attracted some controversy @xcite . the continuous dirac - khler formulation is regarded as providing the continuum limit for the staggered fermion formulation and so a discrete dirac - khler formulation with continuum properties can potentially offer great insight into how to develop non - degenerate , doubler - free fermionic field theories for the lattice . in this paper , we show how the two lattices of adams proposal can be used to describe chiral symmetry in the associated dirac - khler formulation . we also see how the idea of using more than one lattice can be extended to describe an exact flavour projection . we find that this necessitates the introduction of two new structures of lattice and a new operator . finally , we evaluate the path integral for this formulation , considering the effects of chiral and flavour projection . this builds on our previous work @xcite . our starting point is the _ complex _ , which is the space on which we define the discrete differential geometry . it comprises the points of the lattice , together with the links , faces , volumes and hyper - volumes between the points . each point , link , face , volume or hyper - volume is an example of a _ simplex _ and each simplex has an accompanying cochain . we denote a cochain by the vertices of its corresponding simplex . for example , we write the cochain for the simplex between vertices @xmath2 , @xmath3 , @xmath4 and @xmath5 from fig . [ twod ] as @xmath6 $ ] . each cochain is essentially a functional that acts upon a differential form of the same dimension as its simplex to give unity . for example , @xmath6 $ ] is defined such that @xmath7 the cochains act as the discrete differential forms of the theory and a general field is a linear combination of cochains . on the square @xmath8 , we write a general field as @xmath9)[a]+\tilde{\phi}([b])[b]+\tilde{\phi}([c])[c]+\tilde{\phi}([d])[d ] \\ & & + \tilde{\phi}([ab])[ab]+\tilde{\phi}([dc])[dc]+\tilde{\phi}([da])[da ] + \tilde{\phi}([cb])[cb ] \\ & & + \tilde{\phi}([abcd])[abcd ] \ . \end{array}\ ] ] to define the wedge product between cochains , we must first introduce the whitney map , which maps from the complex to the continuum , and the de rham map , which maps the other way . the whitney map , @xmath10 , replaces a cochain with a differential form of the same dimension as its accompanying simplex and introduces functions to interpolate in the regions between simplexes . for example , taking @xmath8 to be a unit square with origin @xmath2 , we introduce the interpolation functions @xmath11 where @xmath12 is the coordinate vector and this allows us to write @xmath13)[a]+\tilde{\phi}([b])[b]+\tilde{\phi}([c])[c]+\tilde{\phi}([d])[d]\right ) = \tilde{\phi}([a])\mu_1(x)\mu_2(x ) \\ \hspace{0.4 cm } + \tilde{\phi}([b])(1-\mu_1(x))\mu_2(x ) + \tilde{\phi}([c])(1-\mu_1(x))(1-\mu_2(x ) ) \\ \hspace{0.4 cm } + \tilde{\phi}([d])\mu_1(x)(1-\mu_2(x ) ) \\ w\left(\tilde{\phi}([da])[da ] + \tilde{\phi}([cb])[cb]+\tilde{\phi}([dc])[dc]+\tilde{\phi}([ab])[ab]\right ) = \\ \hspace{0.4 cm } \tilde{\phi}([da])\mu_1(x ) dx^2 + \tilde{\phi}([cb])(1-\mu_1(x))dx^2 + \tilde{\phi}([dc])(1-\mu_2(x))dx^1 \\ \hspace{0.4 cm } + \tilde{\phi}([ab])\mu_2(x ) dx^1\\ w\left(\tilde{\phi}([abcd])[abcd]\right ) = \tilde{\phi}([abcd])dx^1\wedge dx^2 . \end{array}\ ] ] the de rham map , @xmath14 , discretizes a field by integrating over the simplexes whose dimension match that of the accompanying differential form . @xmath14 also introduces a cochain of the appropriate dimension . thus , @xmath15 ) & = & \phi(x)|_{x = a } & \tilde{\phi}([b ] ) & = & \phi(x)|_{x = b}\\ \tilde{\phi}([c ] ) & = & \phi(x)|_{x = c } & \tilde{\phi}([d ] ) & = & \phi(x)|_{x = d}\\ \tilde{\phi}([dc ] ) & = & \int_{dc } \phi(x ) dx^1 & \tilde{\phi}([ab ] ) & = & \int_{ab } \phi(x ) dx^1\\ \tilde{\phi}([da ] ) & = & \int_{da } \phi(x ) dx^2 & \tilde{\phi}([cb ] ) & = & \int_{cb } \phi(x ) dx^2\\ \tilde{\phi}([abcd ] ) & = & \int_{abcd } \phi(x ) dx^1\wedge dx^2 \end{array}\ ] ] and @xmath16 & = & \tilde{\phi}([a])[a ] + \tilde{\phi}([b])[b ] + \tilde{\phi}([c])[c ] + \tilde{\phi}([d])[d ] \\ r\left[\phi(x,1)dx^1\right ] & = & \tilde{\phi}([dc])[dc]+\tilde{\phi}([ab])[ab ] \\ r\left[\phi(x,2)dx^2\right ] & = & \tilde{\phi}([da])[da]+\tilde{\phi}([cb])[cb ] \\ r\left[\phi(x,12)dx^1\wedge dx^2\right ] & = & \tilde{\phi}([abcd])[abcd ] \ . \end{array}\ ] ] the wedge product between two discrete fields , @xmath17 , now takes the form @xmath18 where @xmath19 is the wedge product of the continuum . we can take advantage of @xmath10 and @xmath14 to define the discrete exterior derivative as @xmath20 where @xmath21 is the exterior derivative from the continuum , @xmath22 . in the continuum , the hodge star is defined to be @xmath23 where we have written @xmath24 as shorthand for the product of forms @xmath25 and @xmath26 is the complement of @xmath27 in the space . @xmath28 is the levi - civita tensor . the square of the hodge star has the property @xmath29 where @xmath30 is the dimension of the form and @xmath1 the dimension of the space . to define the hodge star discretely requires the introduction of a second complex , known as the _ dual _ , in the same space as the first . the dual ( shown in fig . [ origcomp ] for two dimensions ) is aligned with the original complex so that the mid - points of complementary simplexes coincide . the hodge star is defined so that it maps a cochain from one complex to a cochain from the other complex with an aligned simplex . this gives the square of the operator , acting on a general cochain @xmath31 $ ] , the following local form : @xmath32 = ( -1)^{g(n - g)}[g]$ ] , where @xmath33 is dimension of the simplex and @xmath1 the dimension of the space . with the hodge star in place , the adjoint derivative can be defined as @xmath34 = ( -1)^{ng+n+1}*d * [ g]\ ] ] and the laplacian can be written @xmath35 , where @xmath36 is the dirac - khler operator . to define the inner product , we must introduce the barycentric subdivided complex @xcite . the vertices of this complex are formed from the midpoints of the simplexes on either the original or dual complex ( the result is the same , whichever we choose ) . these vertices are used in the construction of a new set of simplexes and a new whitney map , denoted @xmath37 , which interprets a cochain from either complex as a cochain on the barycentric subdivided complex and maps it to a product of differential forms and interpolating functions defined from the simplexes of the barycentric subdivided complex . the barycentric subdivided complex belonging to fig . [ origcomp ] is shown in fig . [ bary ] . this allows the inner product of two discrete fields , @xmath38 and @xmath39 , to be defined as @xmath40 in the dirac - khler basis , the clifford algebra is implemented with the clifford product , @xmath41 , acting on differential forms , @xmath42 formally , @xmath41 is defined to be @xmath43 where @xmath44 . for a one - form , acting upon a general field @xmath45 where @xmath46 is the contraction operator @xmath47 and @xmath48 is the linear combination of forms @xmath49 the correspondence between the dirac spinor , @xmath50 , and the dirac - khler field , @xmath48 , is established with @xmath51 where @xmath52 is defined to be @xmath53 in euclidean space - time . @xmath54 is shorthand for the product of the matrices @xmath55 , where @xmath56 take the form @xmath57 and @xmath58 are the pauli matrices . the matrix @xmath52 is pivotal to this correspondence because it has the properties @xmath59 which mean that @xmath60 and @xmath61 the components @xmath62 and @xmath63 are explicitly related by @xmath64 on the complex , we find that the fields do not exhibit the properties of eqs . ( [ corr1 ] ) and ( [ corr2 ] ) exactly . if this were the case , we would find that , referring to fig . [ twod ] , @xmath65+[ab]\right ) & \tilde{\wedge } & \left(\tilde{\phi}([ad])[ad]+\tilde{\phi}([cb])[cb]\right ) \\ & = & \int_{abcd}dx^1 dx^2 tr\left(\gamma_2\psi(x)\right)[abcd ] \ . \end{array}\ ] ] however , in the right hand side of this expression , the integration is over a domain of different dimension to that of the combination of @xmath66-matrices . using the definition of the de rham map and eq . ( [ equiv ] ) , we can see that no such field exists in the discretization , so , instead , the left hand side evaluates to @xmath67+[ab]\right ) & \tilde{\wedge } & \left(\tilde{\phi}([ad])[ad]+\tilde{\phi}([cb])[cb]\right ) \\ & = & \frac{1}{2}\left(\tilde{\phi}([da])+\tilde{\phi}([cb])\right)[abcd ] \ , \end{array}\ ] ] which is a first order approximation to the right hand side of eq . ( [ wedge ] ) . in the continuum , the columns of the four by four matrix , @xmath50 , each correspond to a separate flavour of field which can be isolated using the flavour projection @xmath68 , where @xmath69 and @xmath70 however , because the properties of eqs . ( [ corr1 ] ) and ( [ corr2 ] ) are only approximately captured on the complex , we can not use the discrete counterparts to @xmath71 to facilitate flavour projection and @xmath72 to generate exact chiral symmetry . however , we can take advantage of the relationship between the dual and the original complexes to implement an exact chiral symmetry , as we shall demonstrate in the next section . in subsequent sections , we introduce additional complexes to implement exact flavour projection . it was shown by rabin that , in the continuum , the chiral symmetry of dirac - khler fields is related to the hodge star @xcite . using equation ( [ equiv ] ) , we can show that the substitution @xmath73 is equivalent to the transformation @xmath74 where @xmath75 and @xmath76 are operators defined to be @xmath77 on the complex , the discrete fields are obtained from the continuous fields . as such , there is the potential to use @xmath78 to describe chiral symmetry . however , we can not use the formulation as it stands , because the fields associated with the simplexes from each complex are initially discretized by integrating over different domains . for example , referring to fig . [ origcomp ] , @xmath79)$ ] is obtained by sampling @xmath80 at @xmath2 and @xmath81)$ ] is obtained by integrating @xmath82 over @xmath83 . in this case , @xmath84 is not equivalent to @xmath85 because the domains do not agree . to attain this equivalence , we must modify the domain of integration used to initially discretize the fields on one of the complexes . whilst the choice is arbitrary , we will chose to modify the fields on the dual . we introduce a new de rham map , @xmath86 , that is identical to @xmath14 on the original complex , but that uses domains of integration on the dual that match simplices from the original complex . it is defined so that fields on the dual are discretized using domains of integration defined by the simplexes from the original complex related to the simplexes of the dual by their accompanying cochains and @xmath78 . formally , @xmath86 is defined to be @xmath87 & = & \sum_{h}\int_{h}\phi(x , h ) dx^h & \hspace{0.5cm}\mbox { on the original complex } \\ r_0[\phi(x , h)dx^h ] & = & \sum_{h}\int_{\bar{*}h}\phi(x , h ) dx^h & \hspace{0.5cm}\mbox { on the dual complex } . \end{array}\ ] ] here , @xmath27 is a simplex of the same space - time dimension as @xmath24 and @xmath88 maps a simplex from the dual to its counterpart on the original complex obtained as the simplex associated with the cochain @xmath89 $ ] . we continue to define the wedge product , clifford product , exterior derivative and adjoint derivative using @xmath14 . only for the initial discretization of the fields do we propose to use @xmath86 . with the fields discretized in this manner , we can generate an exact chiral symmetry with the operator @xmath90 acting on @xmath38 which has the property @xmath91 to implement chiral projection , we deconstruct @xmath78 into @xmath92 , which is the hodge star mapping cochains from the original complex to the dual , and @xmath93 which is the hodge star mapping cochains from the dual complex to the original . this allows us to write chiral projection as @xmath94 if we write @xmath95 , where @xmath96 and @xmath97 are the discrete fields on the original and dual complexes , respectively , then @xmath98 will project the right handed degrees of freedom of @xmath99 onto the original complex and the left handed degrees of freedom onto the dual . the dirac - khler field enjoys global @xmath100 flavour symmetry in the continuum . however , as can be seen from eq . ( [ dxproj ] ) , flavour projection only requires the subgroups generated by @xmath101 and @xmath102 . as we have seen , on the complex , the nive implementation of these symmetries is only approximate . however , just as we were able to use the dual and original complexes to describe an exact chiral symmetry , we can use the dual and original complexes to describe the @xmath103 symmetry needed for flavour projection . we write @xmath104 as the product of two operators @xmath105 and @xmath106 and @xmath107 as the product of the two projections @xmath108 and @xmath109 , such that @xmath110 where @xmath111 in the continuum , we can show that @xmath112 and we can use this to write @xmath106 as @xmath113 unfortunately , describing the symmetry generated by @xmath102 is more involved . when we apply @xmath114 to a general form , the form we obtain is complementary in all four dimensions to the original and this allows us to describe this process in terms of @xmath78 . if we consider applying @xmath115 to a general form , the form we obtain is complementary in the @xmath116 subspace , but equal to the original form in the @xmath117 subspace . by analogy , we need an operator that maps a form to its complement in the @xmath116 subspace , but not in the @xmath117 subspace . to this end we introduce @xmath118 , which , in the continuum , we define to be @xmath119 here , we have introduced @xmath120 as the components of @xmath27 belonging to the @xmath116 subspace and @xmath121 as the complementary operator in the @xmath116 subspace . @xmath122 is equal to @xmath27 in the @xmath117 subspace and complementary to @xmath27 in the @xmath116 subspace and @xmath123 is the complement of @xmath120 in the @xmath116 subspace . the square of @xmath118 has the property @xmath124 which is comparable to eq . ( [ starsq ] ) . with @xmath118 , we can show that @xmath125 where @xmath126 and @xmath127 is the number of components of @xmath27 in the @xmath116 subspace . to describe @xmath118 in the discretization , we are required to introduce a new complex , just as adams was required to introduce the dual to describe @xmath78 . the new complex should align with the original complex so that simplexes that are complementary in the @xmath116 subspace , but not the @xmath117 subspace , share midpoints . we christen the new complex the @xmath128 ( @xmath129-complement ) complex . in order to help visualize the alignment of the @xmath128 complex , in fig . [ 12c ] we show its analogue in two dimensions . here , the simplexes with coincident midpoints are complementary to each other in the @xmath130 direction , but not the @xmath131 direction and we would define the analogue of @xmath118 so that it maps between the following pairs of cochains from @xmath8 and @xmath83:- @xmath132 & \leftrightarrow & [ ab ] \\ { [ d ] } & \leftrightarrow & [ dc ] \\ { [ c ] } & \leftrightarrow & [ dc ] \\ { [ b ] } & \leftrightarrow & [ ab ] \\ { [ da ] } & \leftrightarrow & [ abcd ] \\ { [ cb ] } & \leftrightarrow & [ abcd ] \ . \end{array}\ ] ] it is worth mentioning that , in a two dimensional theory , we would not need to introduce this complex to isolate the flavours because the origianl and dual complexes would be sufficient . it is simply an analogue to the @xmath128 complex . in order to complete the description of flavour projection , we must go further and define a fourth complex . the term proportional to @xmath133 in eq . ( [ dxproj ] ) maps a form to its complement in the @xmath116 subspace , before mapping the resulting form to its complement in all four dimensions . the end result is a form complementary to the original in the @xmath117 subspace , but not in the @xmath116 subspace . in the continuum , this can be described as a combination of @xmath118 and @xmath78 , but to capture this map discretely requires us to introduce a fourth complex . the fourth complex is the dual of the @xmath128 complex ( or the @xmath129-complement of the dual , it can be viewed either way ) and we christen it the @xmath134 ( @xmath129 complement s dual ) complex . the two dimensional analogue of the @xmath134 complex , together with the original , dual and the analogue of the @xmath128 complex , is shown in fig . [ 12cd ] . to enable flavour projection between these four complexes , we must ensure that the fields are initially discretized using compatible domains of integration . to this end , we extend the definition of @xmath86 so that the fields on all four complexes are initially discretized using domains of integration taken from the original complex . @xmath86 becomes @xmath87 & = & \sum_h \int_{h } \phi(x , h ) dx^h & \hspace{0.5cm}\mbox { on the original complex } \\ r_0[\phi(x , h)dx^h ] & = & \sum_h \int_{\bar{*}h } \phi(x , h ) dx^h & \hspace{0.5cm}\mbox { on the dual complex } \\ r_0[\phi(x , h)dx^h ] & = & \sum_h \int_{\bar{\spadesuit}h } \phi(x , h)dx^h & \hspace{0.5cm}\mbox { on the $ 12c$ complex and } \\ r_0[\phi(x , h)dx^h ] & = & \sum_h \int_{\bar{*}\bar{\spadesuit}h } \phi(x , h)dx^h & \hspace{0.5cm}\mbox { on the $ 12cd$ complex } , \end{array}\ ] ] where @xmath135 is defined as before and @xmath136 maps a simplex to its complement in the @xmath116 subspace , but not the @xmath117 subspace . formally , @xmath137 generates the simplex associated with the cochain generated by @xmath138 $ ] . we must also extend the definition of @xmath106 from eq . ( [ projbeta ] ) to include the maps between the @xmath128 and @xmath134 complexes . if we rewrite @xmath99 as @xmath139 where @xmath140 and @xmath141 are the contributions from the @xmath128 and @xmath134 complexes , respectively , @xmath142 now takes the form @xmath143 where @xmath144 maps from the @xmath134 complex to the @xmath128 complex and @xmath145 maps the other way . to describe @xmath105 , we similarly deconstruct @xmath118 into @xmath146 which maps from the @xmath128 complex to the original , @xmath147 , which maps the other way , @xmath148 , which maps from the @xmath134 complex to the dual and @xmath149 which maps the other way . this allows us to write @xmath105 as @xmath150 and flavour projection can now be written as @xmath151 . one of the properties of the geometric discretization is that the dirac - khler operator maps the degrees of freedom from each complex in the same way . for example , referring to @xmath8 and @xmath83 from fig . [ origcomp ] , @xmath36 maps @xmath152)[abcd]$ ] to @xmath153)[da]$ ] and this is matched by the behaviour of @xmath36 on the dual which maps @xmath154)[c]$ ] to @xmath155)[dc]$ ] . consequently , the cancellation properties of @xmath152)$ ] and @xmath154)$ ] will be equally valid before and after the application of @xmath36 , so we have @xmath156\hat{\tilde{\phi}}=0 \ .\ ] ] to illustrate @xmath104 , we shall consider the effect of @xmath157 on @xmath99 . in particular , we will consider @xmath157 in stages , so first we apply @xmath158 to @xmath99 . this leaves the degrees of freedom belonging to the first and third columns of @xmath159 on the ordinary and dual complexes and the degrees of freedom belonging to the second and fourth columns of @xmath159 on the @xmath128 and @xmath134 complexes . if we now apply @xmath160 to this system , it will project between the original and dual complexes to leave the degrees of freedom belonging to the first column of @xmath159 on the original complex and the degrees of freedom belonging to the third column of @xmath159 on the dual complex . between the @xmath128 and @xmath134 complexes , @xmath160 will leave the degrees of freedom belonging to the second column of @xmath159 on the @xmath128 complex and the degrees of freedom belonging to the fourth column of @xmath159 on the @xmath134 complex . in the previous two sections , we showed that it was possible to implement exact chiral symmetry using the original and dual complexes and flavour projection using the original , dual , @xmath128 and @xmath134 complexes . however , because both projections use the original and the dual complexes in different ways , we can not implement chiral and flavour projection simultaneously using the formulation as it stands . to illustrate this point , we consider @xmath161 , which we write as @xmath162 . for the four complexes , @xmath163 becomes @xmath164 @xmath98 leaves the degrees of freedom belonging to the upper components of @xmath159 on the original and @xmath128 complexes and the degrees of freedom belonging to the lower components of @xmath159 on the dual and @xmath134 complexes . to this , we apply @xmath158 , which projects between the original and @xmath128 complexes to leave the degrees of freedom belonging to the upper components of the first and third columns of @xmath159 on the original complex and the degrees of freedom belonging to the upper components of the second and fourth columns of @xmath159 on the @xmath128 complex . @xmath158 also projects between the dual and @xmath134 complexes to leave the degrees of freedom belonging to the lower components of the first and third columns on the dual complex and the degrees of freedom belonging to the lower components of the second and fourth columns of @xmath159 on the @xmath134 complex . if we now consider applying @xmath160 to this system , we see that @xmath160 projects between the original and dual complexes to leave the degrees of freedom belonging to both the upper and lower components of the first and third columns on both complexes . it also projects between the @xmath128 and @xmath134 complexes to leave the degrees of freedom belonging to both the upper and lower components of the second and fourth columns on both complexes . because @xmath106 and @xmath163 both map between the original and dual complexes and the @xmath128 and @xmath134 complexes in different ways , we can not use these definitions for @xmath104 and @xmath163 to isolate non - degenerate , chiral dirac - khler fermions however , we can overcome this difficulty , by introducing a second set of complexes that are duplicates of the existing four . by introducing a second set , we can redefine the chiral projection , so that it maps between complexes from different sets , whilst continuing to define the flavour projection so that it maps between complexes from the same set . this arrangement allows us to use @xmath163 to place only the degrees of freedom belonging to the right handed components of @xmath159 on one set of complexes and the degrees of freedom belonging to the left handed components of @xmath159 on the other . the flavour projection within each set will now no longer mix different chiralities of field . to formally define this system , we label the first set of complexes @xmath2 and the duplicate set @xmath3 . we write the hodge star operator in the form @xmath165 , where @xmath166 labels the sets from which and to which the hodge star maps and @xmath167 label the complexes from which and to which it maps , respectively . because @xmath118 only maps between complexes of one set , it is unnecessary to modify its notation . the chiral projection operator now takes the form @xmath168 and we write @xmath169 . @xmath170 places the degrees of freedom belonging to the right handed components of @xmath159 on the complexes of set @xmath2 and the degrees of freedom belonging to the left handed components of @xmath159 on the complexes of set @xmath3 . the flavour projection operator is now @xmath171 , where @xmath172 as before , we describe the application of @xmath173 to @xmath98 in stages . @xmath174 leaves the degrees of freedom belonging to the upper components of the first and third columns of @xmath159 on the original and dual complexes of set @xmath2 and the lower components of the first and third columns on the original and dual complexes of set @xmath3 . it also leaves the degrees of freedom belonging to the upper components of the second and fourth columns of @xmath159 on the @xmath128 and @xmath134 complexes of set @xmath2 and the lower components of the second and fourth columns of @xmath159 on the @xmath128 and @xmath134 complexes of set @xmath3 . applying @xmath160 to this system will leave the degrees of freedom belonging to the upper components of the first column of @xmath159 on the original complex of set @xmath2 and the lower components of the first column of @xmath159 on the original complex of set @xmath3 . it will also leave the degrees of freedom belonging to the upper components of the third column of @xmath159 on the dual complex of set @xmath2 and the lower components of the third column of @xmath159 on the dual complex of set @xmath3 . similarly , it will leave the degrees of freedom belonging to the upper components of the second column of @xmath159 on the @xmath128 complex of set @xmath2 and the lower components of the second column of @xmath159 on the @xmath128 complex of set @xmath3 . lastly , it will also leave the degrees of freedom belonging to the upper components of the fourth column of @xmath159 on the @xmath134 complex of set @xmath2 and the lower components of the fourth column of @xmath159 on the @xmath134 complex of set @xmath3 . using these definitions , @xmath104 and @xmath163 can now be used to isolate a non - degenerate , chiral dirac - khler fermion on each complex . the flavour and chiral projections have important consequences for the path integral . if we initially consider only flavour projection , the action must include contributions from all four complexes . @xmath175 where , on each complex , we have @xmath176 @xmath159 contributes different degrees of freedom to the fields on each complex whose cochains are related by @xmath118 and @xmath78 , so we can write the path integral as the product of four path integrals @xmath177[d\tilde{\phi}_i]e^{-s_i(\tilde{\bar{\phi}}_i,\tilde{\phi}_i)}\right)^{l_0 } , \ ] ] where @xmath178 is an as yet undefined constant . because each complex lies in a separate space , we can evaluate each contribution separately and , on each complex , we have the standard result @xmath179[d\tilde{\phi}_i]\left(e^{-<\tilde{\bar{\phi}}_i , ( d-\delta)\tilde{\phi}_i>}\right)^{l_0 } = det\left[(d-\delta)\right]^{l_0 } , \ ] ] where @xmath36 is defined as a matrix operator . consequently , the full path integral takes the form @xmath180^{4l_0}\ ] ] and for this to have the correct continuum limit , we require that @xmath181 . interestingly , this means that the form of the path integral on each complex , prior to flavour projection , is @xmath179[d\tilde{\phi}_i]\left(e^{-<\tilde{\bar{\phi}}_i , ( d-\delta)\tilde{\phi}_i>}\right)^{\frac{1}{4 } } , \ ] ] which is exactly equivalent to that used in the quarter - root trick of lattice qcd @xcite . applying flavour projection to this system changes the relationship between @xmath38 and @xmath159 so that @xmath182 and @xmath183 each describe the degrees of freedom belonging to one column of @xmath184 and @xmath159 , respectively . this applies to the measures of integration as well as the fields in the action . one consequence of this projection is that several fields on each complex come to share the same structure in terms of @xmath185 , @xmath56 and @xmath159 . for example , after flavour projection , referring to fig . [ twod ] , @xmath186)=\psi(x)_1^{(1)}|_{x = a } , \hspace{1 cm } \tilde{\phi}_o([abcd])=-\int_{abcd}dx^1\wedge dx^2 \psi(x)_1^{(1 ) } \ .\ ] ] however , because these fields are integrated over different domains , they are independent and the path integral is evaluated in the same way to give @xmath187 . if we now extend the path integral so that the action permits chiral projection , we must include the contribution from both sets of complexes . the action becomes @xmath188 and , prior to flavour projection , we can evaluate the path integral separately on each complex to obtain @xmath189^{8l_0 } , \ ] ] which requires us to set @xmath190 , in order to obtain the correct continuum limit . applying both chiral and flavour projection to this system , as in the previous section , changes the relationship between @xmath38 and @xmath159 so that the path integral comes to be a product of four , single flavour , right - handed path integrals and four , single flavour , left - handed path integrals . as in the previous case , fields sharing a similar structure of @xmath185 , @xmath56 and @xmath159 are independent because their domains of integration differ . however , because the components of @xmath99 on each complex can be unambiguously described as left or right handed , the chiral projection sends half the components to zero . consequently , the effective dimension of @xmath36 is halved on each complex . here we have shown that it is possible to describe exact chiral symmetry for dirac - khler fermions using the two complexes of the geometric discretization . we have extended this idea to describe exact flavour projection and we have shown that this necessitated the introduction of two new structures of complex as well as a new operator . to describe simultaneous chiral and flavour projection , we introduced a duplicate set of complexes and we were required to carefully define chiral projection so that it operates between sets and flavour projection so that it operates within sets . this allowed us to project a single flavour of chiral field onto each complex . we have observed that evaluating the path integral on each of the four complexes , prior to flavour projection and without the provision for simultaneous chiral projection , leads to a form equivalent to that used in the quarter - root trick of lattice qcd . 999 e. kahler , der innere differentialkalkul , _ rend . mat . * 21 * ( 1962 ) , 425 p becher and h joos , the dirac - kahler equation and fermions on the lattice , _ z. phys . c _ * 15 * ( 1982 ) , 343 p becher and h joos , on the geometric lattice approximation to a realistic model of qcd , _ lett . nuovo cim . _ * 38 * ( 1983 ) , 293 i kanamori and n kawamoto , dirac - 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it is shown that an exact chiral symmetry can be described for dirac - khler fermions using the two complexes of the geometric discretization . this principle is extended to describe exact flavour projection and it is shown that this necessitates the introduction of a new operator and two new structures of complex . to describe simultaneous chiral and flavour projection , eight complexes are needed in all and it is shown that projection leaves a single flavour of chiral field on each .

0706.4385

in the last few years a new phenomenon has attracted attention of the community of soft condensed matter physicists appearance of attraction between like charged macromolecules in solutions containing multivalent ions . the problem is particularly fascinating because it contradicts our well established intuition that like charged entities should repel @xcite . the fundamental point , however , is that the electrolyte solutions are intrinsically complex systems for which many body interactions play a fundamental role . the attraction between like charged macromolecules is important for many biological systems . one particularly striking example is provided by the condensation of dna by multivalent ions such as @xmath1 , @xmath2 and various polyamines @xcite . this condensation provides an answer to the long standing puzzle of how a highly charged macromolecule , such as the dna , can be confined to a small volume of viral head or nuclear zone in procaryotic cell . evidently , the multivalent ions serve as a glue which keeps the otherwise repelling like - charged monomers in close proximity @xcite . in eukaryotic cells , the cytosol is traversed by a network of microtubules and microfilaments rigid chains of highly charged protein ( f - actin ) which in spite of large negative charge agglomerate to form filaments of cytoskeleton @xcite . the actin fibers are also an important part of the muscle tissue , providing a rail track for the motion of molecular motor myosin . although the nature of attraction between like charged macromolecules is still not fully understood , it seems clear that the attractive force is mediated by the multivalent counterions @xcite . a strong electrostatic attraction between the polyions and the oppositely charged multivalent counterions produces a sheath of counterions around each macromolecule . the condensed counterions can become highly correlated resulting in an overall attraction . it is important to note that the complex formed by a polyion and its associated counterions does not need to be neutral for the attraction to arise . under some conditions the correlation induced attraction can overcome the monopolar repulsion coming from the net charge of the complexes . recently a simple model was presented to account for the attraction between two lines of charges @xcite . each line had @xmath3 discrete uniformly spaced monomers of charge @xmath4 , and @xmath5 condensed counterions of charge @xmath6 free to move along the rod . the net charge of such a polyion - counterion complex is @xmath7 . nevertheless , it was found that if @xmath8 and @xmath9 , at sufficiently short distances , the two like - charged rods would attract @xcite . it was argued that the attraction resulted from the correlations between the condensed counterions and reached maximum at zero temperature . if @xmath10 the force was always found to be repulsive . clearly , a one dimensional line of charge is a dramatic oversimplification of the physical reality . if we are interested in studying the correlation induced forces between real macromolecules their finite radius must be taken into account @xcite . thus , a much more realistic model of a polyion is a cylinder with a uniformly charged backbone @xcite or with an intrinsic charge pattern @xcite as , e.g. , the helix structure of dna molecule . furthermore , the condensed counterions do not move along the line , but on the surface of the cylinder . unfortunately , these extended models are much harder to study analytically . in this paper we explore the effects of finite polyion diameter on the electrostatic interactions between the two polyions using monte carlo simulations . we find that the finite diameter and the associated angular degrees of freedom of condensed counterions significantly modify the nature of attraction . thus , although there is still a minimum charge which must be neutralized by the counterions in order for the attraction to appear , this fraction is no longer equal to @xmath11 as was the case for the line of charge model . we find that the critical fraction depends on the valence of counterions and is less than @xmath11 for @xmath9 . for monovalent counterions no attraction is found . the crystalline structure of the condensed counterions , as first suggested by simulations of gronbech - jensen _ et al . _ @xcite and refs . @xcite , is also not very obvious . in particular we find very similar distributions of condensed counterions in the regime of attractive and repulsive interactions . the structure of this paper is as follows . the model and the method of calculation are described in section [ model ] . in section [ results ] , we present the results of the simulations . the conclusions are summarized in section [ summary ] . the dna model considered here is an extension of the one proposed earlier by arenzon , stilck and levin @xcite . a similar model has been recently discussed by solis and olvera de la cruz @xcite . the polyions are treated as parallel rigid cylinders of radius @xmath12 and @xmath3 ionized groups , each of charge @xmath4 , uniformly spaced with separation @xmath13 along the principle axis , fig . [ modelfig ] . besides the fixed monomers , each polyion has @xmath5 condensed counterions with valence @xmath14 and charge @xmath6 , which are constrained to move on the surface of the cylinder . to locate a condensed counterion it is necessary to provide its longitudinal position , @xmath15 ( @xmath16 ) , and the transversal angle , @xmath17 ( @xmath18 ) . to simplify the calculations , the angular and the longitudinal degrees of freedom are discretized , see fig . [ modelfig ] . the surface of the cylinder is subdivided into @xmath3 parallel rings with a charged monomer at the center of each ring . each ring has @xmath19 sites available to the condensed counterions , see figs . [ modelfig ] and [ rings ] . the hardcore repulsion between the particles requires that a site is occupied by at most one condensed counterion . the two polyions are parallel , with the intermolecular space treated as a uniform medium of dielectric constant @xmath20 . = 0.4 we introduce occupation variables @xmath21 for the two polyions , so that @xmath22 and @xmath23 . thus , @xmath24 if the @xmath25th site of the @xmath26th polyion is occupied by a particle of valence @xmath27 ( negative core charge or counterion of valence @xmath14 , respectively ) , otherwise @xmath28 . note that the core charge is always `` occupied '' , while the counterions are free to move between the @xmath29 ring sites of each polyion . the hamiltonian for the interaction between the two polyions is , @xmath30 with the @xmath31 . all lengths are measured in units of @xmath13 ( for dna , @xmath32 ) . the dimensionless quantity , @xmath33 , is the manning parameter , which for dna is @xmath34 . the partition function is obtained by tracing over all the possible values of @xmath21 consistent with the constraint of fixed number of counterions per polyion , @xmath35 clearly , this is a very crude model of the interaction between two macromolecules in a polyelectrolyte solution . the molecular nature of the solvent is ignored . also the number of condensed counterions is fixed instead of being dependent on the separation between the particles . nevertheless , we believe that this simple model can provide some useful insights for the mechanism of attraction in real polyelectrolyte solutions . = 0.35 we are interested in statistical averages of observables such as the energy and the force between the two polyions . furthermore , to understand the nature of the interaction between the two macromolecules it is essential to study the correlations between the condensed counterions on the two polyions . the force is obtained from the partition function eq . ( [ partition ] ) , @xmath36 . from symmetry , only the @xmath37-component is different from zero . for finite macromolecules the symmetry between the two polyions can not be broken @xcite . hence it is impossible to produce a true crystalline order in a finite system at non - zero temperature . since within our simplified model the two polyions have exactly the same number of condensed counterions , the average angular counterion distribution @xmath38 must be symmetric with respect to the mid - plane @xmath39 , see fig . [ modelfig ] . the angle @xmath40 labels the site @xmath25 on polyion @xmath26 , see fig . [ rings ] . thus , @xmath41 denotes the occupation variable for the site 3 on the ring @xmath15 located on polyion 2 , with an angle of @xmath42 . indeed , fig . [ z20n7_ocup ] shows that the density profiles are completely symmetric ( up to fluctuations ) . in spite of this symmetry it is possible for the counterions on the two polyions to become highly correlated clearly , the strength of these correlations will depend on the product @xmath43 and the separation between the two macromolecules . considering fig . [ rings ] , it is evident that if the site @xmath44 on the first polyion is occupied , the likelihood of occupation of the site @xmath45 on the second polyion will be reduced . to explore the nature of electrostatic correlations , we define a counterion - hole correlation function between the adjacent rings on the two polyions , @xmath46 \rangle \nonumber\\ & & - \langle n_i^1(z,\theta^1_i)\rangle \ : \langle \left[1-n_j^2(z,\theta^2_j)\right ] \rangle \:.\end{aligned}\ ] ] here @xmath47 denotes the ensemble average . this function should be non - zero when sites on the two polyions are correlated , that is if one is occupied by a condensed counterion there is an increased probability of the second being empty . to calculate the force between the two polyions , we have performed a standard monte - carlo ( mc ) simulation with the usual metropolis algorithm @xcite . first , one counterion on polyion 1 is randomly chosen and displaced to a vacant position on the _ same _ polyion . this move is accepted or rejected according to the standard detailed balance criterion @xcite . we do not permit exchange of particles between the polyions . next , the same is done for polyion 2 . in one monte - carlo step ( mcs ) all @xmath48 condensed counterions on the two polyions are permitted to attempt a move . the long - ranged nature of the coulomb interaction requires evaluation of all the pair interactions in eq . ( [ hamiltonian ] ) at every mcs . due to the limited computational power available to us , we have confined our attention to relatively small systems with @xmath49 and @xmath50 . we have checked , however , that for @xmath50 the force has already reached the continuum limit and did not vary further with increase of @xmath19 . also we note that the `` thermodynamic limit '' is reached reasonably quickly , so that there is a good collapse of data already for @xmath51 , see fig . [ z20_force ] . 2000 mcs served to equilibrate the system , after which 500 samples were used to calculate the basic observables , namely , the mean force and energy . to obtain the correlation functions , 5000 samples were used with 5000 mcs for equilibration . the simulations were performed for @xmath34 and @xmath52 , relevant for dna . for monovalent counterions the simulation results indicate that the force is purely repulsive . this is in complete agreement with the experiments @xcite , which do not find any indication of dna condensation for monovalent counterions . = 0.35 for divalent counterions the force between the two complexes can becomes negative , indicating appearance of an effective attraction , fig . [ z20_force ] . the range of attraction is larger than was found for the one dimensional line of charge model , ref . @xcite . = 0.35 within the manning theory @xcite @xmath53 of the dna s charge is neutralized by the divalent counterions . however , there are indications that even a larger fraction of dna s charge can become neutralized by the multivalent ions if the counterion correlations are taken into account @xcite . in this case the interaction is purely attractive , with the range of about @xmath54 or @xmath55 ( @xmath0 surface - to - surface ) , fig . [ z20_force ] . = 0.4 = 0.4 a minimum number of condensed counterions is necessary for attraction to appear . in fig . [ d_0 ] we present the surface - to - surface separation , @xmath56 , below which the force between the two complexes becomes negative ( attractive ) , as a function of the number of multivalent counterions . for the case of dna with divalent counterions @xmath57 , the attraction appears only if @xmath58 of the core charge is neutralized . for @xmath59 this fraction decrease to @xmath60 . furthermore , decrease in the value of the manning parameter , @xmath61 , increases the minimum number of condensed counterions necessary for the attraction to appear . this is fully consistent with the fact that the attraction is mediated by the correlations between the condensed counterions . since raise in temperature tends to disorganize the system , the state of highest correlation between the condensed counterions corresponds to @xmath62 or @xmath63 . the surface - to - surface distance at which the attraction first appears tends to zero as the number of condensed counterions is diminished . we find @xmath64 , where the average counterion concentration is @xmath65 and the critical fraction @xmath66 depends on the valence of condensed counterions @xmath14 . from fig . [ d_0 ] it is evident that @xmath67 . this should be contrasted with the line of charge model ref . @xcite , for which @xmath68 . = 0.35 in fig . [ snaps ] we show two snapshots of the characteristic equilibrium configurations for ( a ) @xmath69 and ( b ) @xmath70 . looking at this figures it is difficult to see something that would distinguishes between them , both appear about the same . there is no obvious crystallization or transversal polarization suggested in previous studies @xcite . yet , the case ( a ) corresponds to the repulsive , while the case ( b ) corresponds to the attractive interaction between the polyions . to further explore this point , in figs . [ d32.8 ] and [ d16.65 ] we present the site - site correlation function , eq . ( [ correl ] ) , for macromolecules with @xmath71 and @xmath72 . for @xmath69 the surface - to - surface distance between the two polyions is sufficiently large for their condensed counterions to be practically uncorrelated , fig . [ d32.8 ] . on the other hand , for @xmath73 strong correlations between the condensed counterions are evident , fig . [ d16.65 ] . the fig . [ d16.65 ] shows that the sites two and three on the first polyion are strongly correlated with the sites seven and eight on the second polyion , respectively . it is these correlations between the adjacent sites on the two polyions which are responsible to the appearance of attraction between the two macromolecules when they are approximated , fig . [ z20_force ] . we have presented a simple model for polyion - polyion attraction inside a polyelectrolyte solution . it is clear from our calculations that the attraction results from the correlations between the condensed counterions and reaches maximum for @xmath62 . the thermal fluctuations tend to diminish the correlations , decreasing the amplitude of the attractive force . consistent with the experimental evidence , the attraction exists only in the presence of multivalent counterions . our simulations demonstrate that a critical number of condensed counterions is necessary for the appearance of attraction . the fraction of bare charge that must be neutralized for the attraction to arise depends on the valence of counterions . the larger the valence , the smaller the fraction of the bare polyion charger that must be neutralized for the attraction to appear . this result should be contrasted with the line of charge model @xcite for which the critical fraction was found to be equal to @xmath11 , independent of the counterion charge . we thank j. j. arenzon for helpful comments on simulations . this work was supported by cnpq conselho nacional de desenvolvimento cientfico e tecnolgico and finep financiadora de estudos e projetos , brazil .

a simple model is presented for the appearance of attraction between two like charged polyions inside a polyelectrolyte solution . the polyions are modeled as rigid cylinders in a continuum dielectric solvent . the strong electrostatic interaction between the polyions and the counterions results in counterion condensation . if the two polyions are sufficiently close to each other their layers of condensed counterions can become correlated resulting in attraction between the macromolecules . to explore the counterion induced attraction we calculate the correlation functions for the condensed counterions . it is found that the correlations are of very short range . for the parameters specific to the double stranded dna , the correlations and the attraction appear only when the surface - to - surface separation is less than @xmath0 . 2

cond-mat0011308

pulsars lose their spin energy via relativistic pulsar winds ( pws ) of charged particles . the pw shocks in the ambient medium and forms a pulsar wind nebula ( pwn ) whose synchrotron radiation can be observed in a very broad energy range , from the radio to tev @xmath8-rays ( see kaspi et al . 2006 , gaensler & slane 2006 , and kargaltsev & pavlov 2008 [ kp08 hereafter ] for recent reviews ) . the shocked pw is confined between the termination shock ( ts ) and contact discontinuity ( cd ) surface that separates the shocked pw from the shocked ambient medium between the cd and the forward shock ( fs ) . the shapes of the ts , cd , and fs depend on the wind outflow geometry and the ratio of the pulsar s speed to the sound speed in the ambient medium ( the mach number ) , @xmath9 . in particular , if the pulsar moves with a supersonic speed , @xmath10 , and the preshock pw is isotropic , then the ts , cd , and fs acquire bow - like shapes ahead of the pulsar , with the ts apex ( `` head '' ) at a distance @xmath11 , where @xmath12 is the ram pressure , @xmath13 the density of the ambient medium ( e.g. , bucciantini et al . 2005 ; hereafter b+05 ) . the shocked pw forms a tail behind the pulsar , with a flow speed significantly exceeding the pulsar s speed ( romanova et al . 2005 ; b+05 ) . among @xmath1460 pwne detected by _ , about 20 pwne show such bowshock - tail morphologies ( kp08 ) . such tails have been observed , for instance , behind the pulsars j17472958 ( gaensler et al . 2004 ) , j15095850 ( kargaltsev et al . 2008 ) , b0355 + 54 ( mcgowan et al . 2006 ) , and b1929 + 10 ( misanovic et al . 2008 ) , with very different spindown ages , @xmath15 , 160 , 620 , and 3100 kyr , respectively . we should note , however , that the detailed shape of the detected bowshock - tail pwne is often different from the idealized models , especially in the immediate vicinity of the pulsar , possibly because of anisotropies of the pulsar outflows . for instance , by analogy with a few bright , well - resolved pwne around young pulsars moving with subsonic velocities ( such as the crab pwn ; weisskopf et al . 2000 ) , one can expect that the pulsar outflows consist of equatorial and axial components , with respect to the spin axis , which are responsible for the `` tori '' and `` jets '' observed in these torus - jet pwne ( kp08 ) . one of the most peculiar pwne has been detected around the famous geminga pulsar ( psr j0633 + 1746 ) . geminga was discovered as a @xmath8-ray source @xmath8195 + 5 , with the _ sas-2 _ satellite ( e.g. , thompson et al . the period of geminga , @xmath16 ms , was discovered by halpern & holt ( 1992 ) in x - ray observations with the _ rntgen satellit _ ( _ rosat _ ) , and the period derivative , @xmath17 s s@xmath6 , was first measured by bertsch et al . ( 1992 ) in @xmath8-rays with the _ compton gamma ray observatory _ ( _ cgro _ ) . the period and its derivative correspond to the spindown age @xmath18 kyr and spindown power @xmath19 erg s@xmath6 . the geminga pulsar has also been detected in the optical ( halpern & tytler 1988 ; bignami et al . 1988 ) , near - ir ( koptsevich et al . 2001 ) , and uv ( kargaltsev et al . the distance to geminga , @xmath20 pc , was estimated from its annual parallax measured in observations with the _ hubble space telescope _ ( faherty et al . 2007 ) . its proper motion , @xmath21 mas / yr , corresponds to the transverse velocity , @xmath22 km s@xmath6 [ where @xmath23 . as this velocity considerably exceeds the typical sound speed in the interstellar medium ( ism ) , @xmath2430 km s@xmath6 , one should expect geminga to be accompanied by a bowshock - tail pwn , with @xmath25 cm , which corresponds to @xmath26 , where @xmath27 is the angle between the pulsar s velocity and the line of sight , and @xmath28 . _ xmm - newton _ observations of geminga in 2002 april , reported by caraveo et al . ( 2003 ; hereafter c+03 ) , revealed two @xmath29 long tails behind the pulsar , approximately symmetric with respect to the sky projection of the pulsar s trajectory ( see fig . 1 ) , with a luminosity of @xmath30 erg s@xmath6 in the 0.35 kev band . c+03 suggested that these tails are associated with a bowshock generated by the pulsar s motion , and , using the one - zone bowshock model by wilkin ( 1996 ) , predicted that the head of the bowshock , @xmath31@xmath32 ahead of the pulsar , is hidden in the bright wings of the pulsar point spread function ( psf ) in the _ xmm - newton _ image . the geminga field was observed in 2004 ( sanwal et al . 2004 ; pavlov et al . 2006 [ hereafter p+06 ] ) with the _ chandra _ advanced ccd imaging spectrometer ( acis ) , whose resolution , @xmath33 , is much better than that of the _ xmm - newton _ detectors . the most interesting finding of that observation was the detection of an axial tail behind the pulsar aligned with the direction of the pulsar s proper motion ( p+06 ; de luca et al . 2006 ; see fig . 2 , top ) . the axial tail , with a luminosity @xmath34 erg s@xmath6 , was seen up to @xmath35 from the pulsar , almost up to the boundary of the field of view ( fov ) . p+06 suggested that the axial tail may be a jet emanating from the pulsar magnetosphere . in addition to the axial tail , a faint arc - like structure was detected @xmath36@xmath37 ahead of the pulsar ( but no emission at @xmath0@xmath38 , contrary to the c+03 prediction ) , and a @xmath39 enhancement , apparently connecting the arc with one of the outer tails ( south of the axial tail ) , was noticed ( p+06 ) . no emission was detected from the other ( northern ) outer tail in that short , @xmath40 ks , exposure . to image the whole extent of the geminga pwn and study its tails in more detail , we observed this field with _ chandra _ acis in 2007 , with a longer exposure and a larger fov . in this paper , we report the results of this observation and compare them with the previous findings . we describe the data analysis and the observational results in 2 , and discuss the implications of these results in 3 . the geminga field was observed with _ chandra _ acis on 2007 august 27 for 78.12 ks ( obsid 7592 ) . the observation was taken in timed exposure ( te ) mode , with the frame time of 3.24 s. after removing 20 s of high background and correcting for the detector dead time , the scientific exposure time ( live time ) is 77,077 s. to maximize the signal - to - noise ratio ( @xmath41 ) for the very faint pwn emission , we imaged the field onto the front - illuminated i3 chip , which has a lower background than the commonly used ( and slightly more sensitive ) back - illuminated s3 chip . we used the very faint telemetry format to provide a better screening of background events . as putting the target at the acis - i aimpoint ( near the corner of the i3 chip ) could result in chip gaps crossing the pwn image , we moved the focus to the middle of node 2 on the i3 chip ( sim - z @xmath42 ) and applied the @xmath43 offset to put the pulsar at @xmath44 from the chip boundaries . to obtain a deeper pwn image and examine a possible pwn variability , we also used the previous _ chandra _ observation of geminga carried out on 2004 february 7 ( obsid 4674 ; 18,793 s scientific exposure ) . the details of that observation have been described by p+06 . here we only mention that the observation was taken in faint telemetry format , and the geminga pulsar and its pwn were imaged on 1/8 subarray of the s3 chip ( @xmath45 fov ) , which reduced the pileup in the pulsar image but did not allow us to image the whole pwn . we have used the chandra interactive analysis of observations ( ciao ) software ( ver . 4.0 ; caldb ver . 3.4.0 ) for the acis data analysis , starting from the level 1 event files . we have applied the standard grade filtering and used the energy range 0.38 kev to minimize the background contribution . we have also applied the exposure map correction , but found that the effects of nonuniform exposure and nonuniform ccd response in the pwn region are small ( except for the boundary of the 1/8 subarray used in the observation of 2004 ) . to confront the high - resolution _ chandra _ data with the _ xmm - newton _ results , we also used the data obtained with the mos1 and mos2 detectors of the european photon imaging camera ( epic ) on board _ xmm - newton_. in addition to the observation of 2002 april 45 ( obsid 011117010 ; 77.97 ks scientific exposure , after removing the periods of high background ) reported by c+03 and p+06 , we also used the data sets obtained in observations 0201350101 of 2004 march 13 ( 16.23 ks scientific exposure ) , 031159100 of 2006 march 17 ( 4.46 ks ) , 0400260201 , 2006 october 2 ( 19.70 ks ) , and 0400260301 of 2007 march 11 ( 23.83 ks ) . the total effective exposure of the five observations is 142.18 ks . all the observations were taken with medium filter in full frame mode , providing a @xmath46 diameter fov . the data reduction was performed with the scientific analysis system ( sas ) package ( ver . good events with patterns 012 and energies within the 0.38 kev range were selected for the data analysis . the _ chandra _ acis data of 2007 provide the high - resolution image of the entire geminga pwn for the first time ( see the top panel of fig . 2 and fig . 3 ) . below we will describe the observed properties of the pwn elements , and compare them with the results of the 2004 acis observation . to calculate the net source counts @xmath47 in the area @xmath48 of a pwn element , we use the formula @xmath49 , where @xmath50 is the total number of counts detected from the area @xmath48 , and @xmath51 is the number of background counts detected from the area @xmath52 . then the @xmath53 error of @xmath47 and the signal - to - noise ratio are given by @xmath54^{1/2}$ ] and @xmath55 , respectively . for the analysis of the 2004 data , we use a source - free rectangular background region ( @xmath56 arcsec@xmath57 , @xmath58 counts in the 0.38 kev band ) to the north of the geminga pulsar . for the 2007 data , we use the background measured from a source - free rectangular region , with the area @xmath59 arcsec@xmath57 , in the northeast portion of the acis - i3 chip ( unless stated otherwise ) . this region contains @xmath60 counts in the 0.38 kev band , which corresponds to the background brightness of @xmath61 counts arcsec@xmath62 s@xmath6 , a factor of 2 lower than in the _ chandra _ observation of 2004 . the values of @xmath47 and @xmath41 for the pwn elements are given in table 1 . for the spectral analysis , we have used the xspec package ( ver . 12.4.0 ) and fit the spectra with the absorbed power - law ( pl ) model ( wabs*powerlaw ) , with the fixed hydrogen column density @xmath63 cm@xmath62 ( halpern & wang 1997 ; de luca et al . as the number of counts in the pwn is small , we use the maximum likelihood method ( c - statistic ) for spectral fitting . table 1 provides the values of the photon index @xmath65 and the flux @xmath66 of the pwn elements . the brightest feature of the geminga pwn in the _ chandra _ data of 2004 is the axial tail ( a - tail hereafter ) , seen up to at least @xmath35 from the pulsar in the direction opposite to the pulsar s proper motion ( see p+06 and fig . 2 , middle ) . in the image from the 2007 observation , we see the a - tail up to at least @xmath2 ( @xmath67 pc ) from the pulsar ( fig . 2 , top ) , with @xmath68 source counts within the region of 706 arcsec@xmath57 area shown by the solid lines in figure 3 . the pl fit of its spectrum ( see table 1 and fig . 4 ) gives the photon index @xmath69 and the 0.38 kev luminosity @xmath70 erg s@xmath6 ( assuming an isotropic emission at @xmath71 pc ) , versus @xmath72 and @xmath73 erg s@xmath6 in the 2004 data , as measured in the 118 arcsec@xmath57 area rectangle that contains @xmath74 counts ( shown in the middle panel of fig . 2 ) . the a - tail looks patchy in both the 2004 and 2007 observations , with some `` blobs '' standing out . the blobs , labeled a , b , and c in figure 2 are at the distances of about @xmath75 , @xmath0 , and @xmath76 from the pulsar , respectively ( blob a is seen in the 2004 image , while blob b and blob c are seen in the 2007 image ) . they contain @xmath77 , @xmath78 , and @xmath79 source counts , respectively , within the @xmath80-radius circles around their centers . the analysis of the brightness distribution along the a - tail shows that the blobs are significant at @xmath81 levels ( i.e. , they are not just statistical fluctuations of the brightness distributions ) . for instance , the number of counts in the @xmath82 radius circle around the center of blob b ( 15 counts ) exceeds the average number of counts per the same 12.6 arcsec@xmath57 area in the a - tail ( @xmath83 counts ) at the @xmath84 level . the nonuniform surface brightness distribution along the a - tail , and the difference of these distributions in the 2004 and 2007 images are shown in figure 5 . the brightest in the 2007 data is blob c at the apparent end of the tail , centered at @xmath85 , @xmath86 ( j2000 ) . because of the small number of counts , we can not firmly determine whether the blob corresponds to a point source or an extended one . interestingly , the end portion of the tail looks attached to this blob , while the tail looks detached from the pulsar in both the 2004 and 2007 images . therefore , one could even speculate that the tail might belong not to geminga but to some unrelated field object ( e.g. , it might be a jet of an active galactic nucleus [ agn ] , accidentally oriented toward geminga in the sky projection ) . to check whether the blobs are indeed associated with the tail or they may be background sources , we examined the optical / nir catalogs . we found no optical counterparts to blob a and blob b , but we found an object at @xmath87 , @xmath88 ( j2000 ) , about @xmath89 from the center of blob c ( the coordinates are from the usno - b1.0 catalog [ monet et al . 2003 ] , with the quoted mean uncertainties of @xmath90 and @xmath91 in @xmath92 and @xmath93 , respectively ) . based on the magnitudes and colors ( e.g. , @xmath94 [ gsc2.3 catalog ; lasker et al . 2008 ] , @xmath95 , @xmath96 , @xmath97 [ 2mass catalog ; cutri et al . 2003 ] ) , this object could be a background k star . such a star could contribute to the x - ray emission of blob c. the observed x - ray flux in the @xmath80 radius aperture is @xmath98 erg cm@xmath62 s@xmath6 . according to maccacaro et al . ( 1988 ) , the x - ray / optical flux ratio , @xmath99 , corresponds to a k or m star , and it excludes an agn [ for which @xmath100 ) as the source of the x - ray and optical emission ( hence the tail is not an agn jet ) . thus , we can not rule out the possibility that a k star , accidentally projected onto the a - tail , is at least partly responsible for the brightened end of the tail in the 2007 image . if we exclude blob c , the tail s luminosity decreases by a factor of 1.5 , but the spectral slope remains virtually the same ( see table 1 and fig . 4 ) . this suggests that the star s contribution does not dominate in the blob c emission , but , because of the small number of counts and large statistical errors , we can not firmly conclude on the nature of blob c. in the observation of 2004 , blob c was imaged onto an underexposed part of the fov ( because of the dither ) , between the dashed lines in the middle panel of figure 2 . taking into account the shorter effective exposure of that observation ( but the higher sensitivity of the s3 chip ) , we expect @xmath101 counts to be detected in the @xmath82 radius circle around the position of the blob c centroid ; however , there are no counts within that circle . this may suggest some variability of the source , but the statistical significance of this difference is marginal ( e.g. , the probability of detecting zero counts when 4.2 counts are expected is 0.0145 , which corresponds to a @xmath102 significance ) . based on the acis count rate of blob c in 2007 , one could expect to detect about 90 counts in the @xmath103 radius aperture in the 142 ks mos1+mos2 exposure , but we found only @xmath104 counts in the mos data . furthermore , we note that the blob c position is projected onto the wings of the pulsar psf in the _ xmm - newton _ images ( see fig . 1 ) , whose contribution to the number of extracted counts is difficult to evaluate because of the `` spiky '' shape of the psf . anyway , the number of counts expected for blob c in the _ xmm - newton _ data significantly exceeds the measured one , suggesting variability of blob c. the a - tail images ( fig . 2 ) look appreciably different in the 2004 and 2007 data , in both the overall flux and the surface brightness distribution ( see fig . 5 ) . for instance , the flux , @xmath105 erg cm@xmath62 s@xmath6 , in the 118 arcsec@xmath106 area of the tail in the 2004 data is a factor of 6 higher than the flux from the same area in the 2007 data ( the difference between the count rates is significant at the @xmath107 level , with account for a factor of 1.6 higher sensitivity of the s3 chip compared to that of the i3 chip , for @xmath108 and @xmath109 cm@xmath62 ) . the different positions of the blobs in the 2004 and 2007 images suggest that the blobs are moving along the a - tail ( perhaps similar to the blobs in the vela pulsar jet ; pavlov et al . one might even speculate that , for instance , blob b is , in fact , blob a that had moved @xmath110 ( @xmath111 cm ) in 3.5 yr between the observations ( which would correspond to the transverse velocity of @xmath112 km s@xmath6 ) . however , as blob b could also form independently after the disappearance of blob a , this will remain a speculation until the characteristic blob lifetime is estimated in a series of monitoring observations . in the large - scale _ chandra _ image of 2007 ( fig . 3 ) , one can see a possible faint extension of the a - tail ( within the dashed polygon in fig . this faint portion has a factor of 1.5 higher observed flux ( but a factor of 2.4 lower average brightness ) than the bright portion ( see table 1 ) . however , its statistical significance is only @xmath113 , and it is not seen in the deep _ xmm - newton _ image ( see fig . 1 ) . therefore , this `` faint portion '' most likely represents a string of background fluctuations accidentally aligned in the a - tail direction . the two `` outer tails '' of the geminga pwn , seen up to @xmath114 from the pulsar , were originally detected in the _ xmm - newton _ observations of 2002 april ( c+03 ) . adding four shorter observations of 20042007 , which increases the total exposure by a factor of 1.8 , shows qualitatively the same picture ( fig . 1 ) . the two tails are approximately symmetric with respect to the pulsar s trajectory in the sky , forming a horseshoe - like structure . the southern and northern tails ( we will call them the s - tail and n - tail , for brevity ) are seen up to @xmath115 and @xmath116 from the pulsar , respectively , in the summed mos1+mos2 image . their typical width , @xmath117@xmath38 , is comparable to the _ xmm - newton_angular resolution . the tails in the pulsar vicinity ( within @xmath118 ) are immersed in the bright pulsar s image . the spectrum of the combined emission from the two tails , extracted from the @xmath119 elliptical regions shown in figure 1 ( about 560 source counts ) , can be described by a pl model with @xmath120 ( @xmath121 for 50 degrees of freedom [ d.o.f ] , for fixed @xmath122 cm@xmath62 ) . the unabsorbed flux and luminosity of the two tails , are @xmath123 erg cm@xmath62 s@xmath6 and @xmath124 erg s@xmath6 , in the 0.38 kev band . both the total flux and the spectral slope are consistent with those obtained by c+03 . the contribution of the s - tail into the total energy flux is about 76% . the average specific intensities in the s - tail and n - tail elliptical regions are @xmath125 and @xmath126 erg cm@xmath62 s@xmath6 arcsec@xmath62 , respectively . fitting the spectra of the s - tail and n - tail separately , we obtained @xmath127 and @xmath128 , respectively . the apparently large difference between the spectral slopes , @xmath129 , is not statistically significant . the spectra can also be fitted by the models for emission of an optically thin thermal plasma ( e.g , @xmath130 kev for the fit of the s - tail + n - tail spectrum with the mekal model ; @xmath131 for 50 d.o.f . ) . thanks to its high angular resolution , _ chandra _ observations make it possible to image the tails in the pulsar vicinity and resolve the tail structure . in the short observation of 2004 the n - tail was not detected , while the initial portion of the s - tail was detected with about @xmath39 significance ( p+06 ) . the entire extent of the tails could not be seen because the 1/8 subarray was used . in the _ chandra _ data of 2007 we have detected both outer tails . for the analysis , we divide each of the tails into two parts . the bright initial parts ( up to @xmath1 and @xmath132 from the pulsar , for the s - tail and n - tail , respectively ) are delineated by solid polygons in figure 3 , while the longer faint parts ( up to 42 and 36 from the pulsar , respectively ) are shown by dashed polygons . ( we note that the bright portion of the n - tail and a substantial part of the bright portion of the s - tail are hidden behind the pulsar image in the _ xmm - newton _ data . ) the statistical significance of the faint parts is marginal in the _ chandra _ data ( see table 1 ) , but their reality is supported by the _ xmm - newton _ data ( see fig . 6 , where the _ xmm - newton _ brightness contours are overlaid on the _ chandra _ image ) . the end parts of the s - tail in the _ chandra _ and _ xmm - newton _ images are slightly shifted with respect to each other , which might suggest possible variability of the outer tails . the _ chandra _ image resolves the tails in the transverse dimension , showing @xmath133 widths , but the image is not deep enough to infer the brightness distribution across the tails . also , the tails do not show sharp - cut outer boundaries , perhaps because of the same reason . figure 6 also shows the locations of four optical - nir sources that are projected onto the tails . we have already discussed one of them ( # 1 in fig . 6 , whose position coincides with blob c in the a - tail ) . the other three sources ( numbered 2 , 3 , and 4 in fig . 6 , with v magnitudes of 17.6 , 15.3 , and 13.4 , respectively , from the nomad catalog ; zacharias et al . 2004 ) are projected onto the s - tail . source 3 and source 4 are likely an f star and a k star , respectively , based on their optical - nir colors , while the magnitude errors for the fainter source 2 are too large to determine its nature . the @xmath134 radius circles around the positions of the sources 2 , 3 , and 4 contain 7 , 1 , and 11 counts , respectively . even if the x - ray emission at these locations is due to the optical sources , the total number of detected source counts from all the three sources ( assuming they are pointlike ) is only @xmath135 , while the number of counts from the entire s - tail is @xmath136 ( table 1 ) . therefore , their contribution to the observed x - ray emission from the entire s - tail is negligible . the pl fits of the outer tails spectra ( see table 1 and fig . 4 ) indicate that the spectral slope of the s - tail does not differ significantly from that of the a - tail ( e.g. , @xmath137 fits both spectra within the @xmath53 uncertainties ) , while the n - tail is apparently harder , in contradiction to the result found from the _ xmm - newton _ data . although the observed number of counts from the n - tail is a factor of 1.8 lower than that from the s - tail , their luminosities are comparable . table 1 also suggests that both the n - tail and s - tail spectra soften from the bright parts toward the extended faint portions , but the statistical significance of the softening is low ( e.g. , @xmath138 for the s - tail ) . the total luminosity of the two tails is @xmath139 erg s@xmath6 , in the 0.38 kev band , assuming isotropic emission . this value exceeds the estimate derived above from the _ xmm - newton _ data by a factor of about 3 , but that estimate was obtained for a fraction of outer tails , which did not include the bright part of the n - tail and included only a small portion of the bright part of the s - tail . the average specific intensities in the bright parts are about @xmath140 and @xmath141 erg cm@xmath62 s@xmath6 arcsec@xmath62 , for the s - tail and n - tail , respectively . similar to the a - tail , the bright and faint portions of the s - tail and the faint portion of the n - tail look patchy . to quantify the statistical significance of the patchiness , we have compared the net counts from brighter and fainter regions of equal size for each of these components . for the bright s - tail , faint s - tail , and faint n - tail , we found significancies of @xmath142 , @xmath143 , and @xmath144 , respectively . therefore , the patchiness is not ruled out , but a deeper observation is required to prove it firmly . a similar analysis of the _ xmm - newton _ data does not show a statistically significant patchiness , because of large noise and poor angular resolution . to examine variability of the outer tails , we have compared the count rates in the area of 462 arcsec@xmath57 of the bright part of the n - tail ( @xmath145 and @xmath146 net counts in the observations of 2007 and 2004 , respectively ) . accounting for the factor of 1.2 higher sensitivity of the s3 chip compared to the i3 chip ( for the spectral parameters derived from the 2007 observation ) , the difference between the count rates is significant at the @xmath147 level . on the other hand , the difference of the count rates in the 985 arcsec@xmath57 area of the bright part of the s - tail , which was detected in both the 2004 and 2007 _ chandra _ observations ( @xmath148 and @xmath149 net counts , respectively ) , is statistically insignificant . we have also looked for variability of the outer tails in the _ xmm - newton _ data , but found no statistically significant differences between the separate observations because of the strong noise . as it is natural to assume that the `` outer tails '' represent the sky projection of limb - brightened boundaries of a shell , one can expect some x - ray emission from the region between the outer tails , in addition to the a - tail . to look for this emission , we have inspected two inter - tail regions of combined area @xmath150 arcsec@xmath57 that exclude the entire ( bright plus faint ) a - tail and contain @xmath151 counts . using the background extracted from three source - free rectangles around the pwn ( @xmath152 counts in the combined area @xmath153 arcsec@xmath57 ) , we found @xmath154 net counts from the source " . adding the alleged faint portion of the a - tail , which gives @xmath155 counts in the area @xmath156 arcsec@xmath57 , we found @xmath157 net counts . thus , we conclude that there is no detectable emission from the region between the outer tails , and the `` faint portion '' of the a - tail is likely an illusion ( in agreement with our conclusion at the end of 2.2.1 ) . using the approach outlined by weisskopf et al . ( 2007 ) , we find the @xmath39 upper limit of 94 counts in the area @xmath156 arcsec@xmath57 ( 99% and 90% upper limits are 78 and 45 respectively ) . the corresponding @xmath39 upper limit on the surface brightness , @xmath158 counts arcsec@xmath62 , is lower than the average surface brightness of the outer tails ( e.g. , by factors of @xmath159 and @xmath160 for the `` entire '' s - tail and n - tail , respectively ) . the _ xmm - newton _ image also does not show detectable inter - tail emission . using the same approach , we found the @xmath39 upper limit of 37 counts in the @xmath161 box between the outer tails shown in figure 1 ( 99% and 90% upper limits are 32 and 22 counts , respectively ) . the corresponding @xmath39 upper limit on surface brightness , @xmath162 counts arcsec@xmath62 , is lower by factors of @xmath163 and @xmath164 than the surface brightnesses of the s - tail and n - tail , respectively . an arc - like diffuse emission region , about @xmath36@xmath37 ahead of the pulsar , was reported by p+06 from the 2004 _ chandra _ data . we have analyzed the data inside a polygon ( area @xmath165 arcsec@xmath57 ) and found @xmath166 source counts . our spectral analysis provides a photon index @xmath167 and the observed flux @xmath168 erg cm@xmath62 s@xmath6 , in the 0.38 kev band , consistent with the p+06 estimates . based on these results , the expected number of arc counts in the _ chandra _ observation of 2007 is @xmath169 . however , although some diffuse emission is seen at that site in the 2007 data , there are only @xmath170 source counts in the corresponding polygon ( at the same distance from the pulsar , which has moved @xmath171 in the sky in the 3.5 years ) , and the shape of the count distribution does not resemble an arc ( see the upper panel of fig . 2 ) . the spectral slope , @xmath172 , is consistent with that of the 2004 arc , but the observed flux , @xmath173 erg cm@xmath62 s@xmath6 , is a factor of 5 lower . this suggests that the emission ahead of the pulsar is variable , but the significance of this variability is not very high ( e.g. , @xmath174 in the difference between the expected and observed counts ) . the alleged arc can not be seen in the _ xmm - newton _ images because it is hidden in the pulsar psf . however , these images show a `` streak '' ahead of the pulsar , in the direction of the proper motion , which is best seen in the summed image ( bottom panels of fig one might speculate that this streak is a geminga pwn element ( e.g. , a forward jet ) . to check this hypothesis , we extracted 760 source counts from the @xmath175 rectangle that includes the streak ( shown in the bottom panels of fig . 1 ) and fit the spectrum with various models . we found that the very soft streak spectrum does not fit the pl model ( @xmath176 for 44 d.o.f . ) , but it fits the two - component pl+blackbody model ( @xmath177 , @xmath178 kev , @xmath179 for 42 d.o.f . ) that is consistent with the pulsar s spectrum ( e.g. , kargaltsev et al . therefore , we conclude that the streak is not related to the pwn , but it is an artificial spike - like feature in the mos psf caused by the `` spiders '' holding the x - ray telescopes ( see the xmm - newton users handbook , sec . 3.2.1.1 ) this conclusion is also supported by a lack of any excess above the background at the corresponding area in the _ chandra _ images . interestingly , the summed 2004 + 2007 _ chandra _ image in the bottom panel of figure 2 shows a hint of a short , @xmath180@xmath103 , jetlike structure ahead of the pulsar , with its end seemingly connecting to the n - tail . the number of counts is , however , too small to conclude whether this structure is an accidentally aligned superposition of events from the two images or there is indeed a forward jet , perhaps bent in the north - northwest direction into the n - tail , which we can not see in the separate images because of the scarce statistics . a deeper observation is needed to understand the true nature of this and other apparent structures in the immediate vicinity of the pulsar . the _ chandra _ observation of 2007 has confirmed the existence of three tail - like features in the geminga pwn image , with comparable luminosities , and allowed us to study the pwn in more detail ( e.g. , to detect the a - tail at larger distances from the pulsar and investigate the pwn in the immediate vicinity of the pulsar ) . moreover , it has provided first evidence of variability of the pwn elements , in particular , of the a - tail and the emission in the pulsar vicinity . the observed structure of the geminga pwn looks very unusual . although the overall appearance of the pwn , particularly the alignment of the tails with the pulsar s proper motion , leaves no doubts that the pwn structure is caused by the supersonic motion of the pulsar in the ism , none of the other @xmath1420 bowshock - tail pwne detected by _ chandra _ ( see kp08 ) show three distinct tails . to interpret the observed structure in terms of the pwn models , we should first understand the intrinsic three - dimensional morphology of the x - ray pwn , which is by no means obvious . at the first glance , the most natural interpretation of the pwn elements is that the `` outer tails '' , together with the `` arc '' that apparently connects the tails ahead of the pulsar , delineate the limb - brightened boundary of the sky projection of an optically thin shell , shaped approximately as a paraboloid of revolution , while the nearly straight axial tail represents a collimated outflow in the direction opposite to that of the pulsar s motion . on the other hand , one can not exclude the possibility that the outer tails are , in fact , hose - like structures , such as jets confined by their own magnetic fields and bent by the head wind of ism matter . moreover , one could even speculate that the axial `` tail '' is doppler - boosted emission from a narrow region of a shell formed by material flowing with relativistic speeds . below we will discuss these possibilities in more detail . the patchy outer tails of @xmath0@xmath38 width are seen up to @xmath181 ( @xmath182 pc ) from the pulsar . their x - ray luminosity , @xmath183 erg s@xmath6 in the 0.38 kev band , is a fraction of @xmath184 of the pulsar s spin - down power @xmath185 , lower than the typical @xmath186@xmath187 for younger pwne ( kp08 ) . the outer tails luminosity is a factor of @xmath188 lower than the pulsar s magnetospheric luminosity , while the pwn luminosity is usually higher than the magnetospheric luminosity for younger pulsars ( kargaltsev et al . 2007 ) . the spectrum of the outer tails can be described by the absorbed pl model with @xmath189 , which is apparently harder than the typical pwn spectra . the explanation of the outer tails properties depends on the topology of the pw outflow . let us assume that the outer tails are limb - brightened shell boundaries and explore the consequences of this assumption . first of all , the very fact that the shell boundaries are much brighter than the rest of the shell image ( see 2.2.2 ) implies a nonrelativistic speed of the bulk outflow along the shell . if it were relativistic , then , due to the doppler boosting , the brightest parts of the shell image would be not the boundaries but they would correspond to the smallest angles between the flow velocity and the line of sight . this inference restricts the number of possible interpretations of the shell . for instance , the shell can not be interpreted as synchrotron emission from the shocked pw immediately outside the ( bullet - like ) ts surface not only because the cylindrical radius of the shell is too large ( see p+06 ) , but also because the shocked pw is expected to flow with nearly relativistic speed , @xmath190@xmath191 , along the lateral ts boundary ( b+05 ) . as the brightness is proportional to @xmath192^{-\gamma-2}$ ] , where @xmath193 is the angle between the flow direction and the line of sight , the flow toward the observer would be a factor of @xmath194 brighter than the flow in the perpendicular direction , for the photon index @xmath195 . one might consider the possibility that the shell emission comes from the shocked ism material heated up to x - ray temperatures . in this case , we would associate the outer tails with the fs , and the emission mechanism with the thermal emission of an optically thin plasma rather than synchrotron emission from relativistic electrons . from the junction conditions at the shock front , the expected temperature of the shocked ism material at the head of the bowshock is @xmath196 kev [ where @xmath197 is the chemical weight , @xmath198 , and adiabatic index @xmath199 is assumed for the ism gas ] , and it should be even lower behind the pulsar because of the obliqueness of the shock . since the fit of the outer tails spectrum with the mekal model gives @xmath20018 kev ( see 2.2.2 ) , the expected temperature of the shocked ism gas looks too low to explain the hard spectrum of the tails emission unless @xmath201 km s@xmath6 , which would imply that the pulsar moves at a small inclination angle @xmath27 with respect to the line of sight , @xmath202 . however , since such a speed is higher than those observed for other pulsars , and the pwn appearance can hardly be reconciled with such small inclination angles , we can discard this interpretation . it seems more reasonable to assume that the shell is formed by the shocked pw flowing immediately inside the cd surface , where the magnetic field is compressed ( b+05 ) and the synchrotron radiation is enhanced . however , such an interpretation contradicts the available pwn models , which predict a nearly relativistic flow speed in the outer layers of the synchrotron emitting pwn ( hence dim boundaries and bright central part ) because of the doppler boosting ( see above ) . indeed , the simulated pwn images for @xmath203 ( see fig . 4 in b+05 ) show the brightest synchrotron emission from the bowshock head region , while throughout the entire pwn the brightness decreases from the axis toward the cd , in contrast to the observed images . ( this is partly caused by the assumption that the pwn magnetic field is purely toroidal , which reduces the synchrotron intensity at the pwn boundaries , where the magnetic field is parallel to the line of sight . however , even if the magnetic field is completely disordered , the brightness does not grow from the axis toward the boundaries ; see fig . 5 in b+05 . ) therefore , this interpretation of the outer tails implies that there is a mechanism that decelerates the flow . the deceleration can be provided by the shear ( kelvin - helmholtz ) instabilities at the cd , which can lead to advection of clumps of the heavier shocked ism material into the shocked pw and slow down the latter ( e.g. , b+05 ) . the diffuse appearance and the patchiness of the tails in the high - resolution _ chandra_images ( see 2.2.2 and figs . 2 and 3 ) are consistent with this hypothesis . it could be verified observationally if the speed of the nonuniformities in the outer tails , which should form in the process of mass loading , is measured in a series of deep observations . another apparent problem with the interpretation of the outer tails and the arc as traces of the cd surface is the discrepancy between the observed and predicted ratios of the cd s cylindrcal radius , @xmath204 , to the distance @xmath205 of the cd head from the pulsar . for instance , the b+05 model predicts @xmath206 while the observed ratio is @xmath207 ( if we interpret the `` arc '' ahead of the pulsar as the head of the cd surface . ) this means that either the inclination angle is small , @xmath208@xmath209 , or some assumptions of the b+05 model are violated . since the observed pwn shape can hardly be reconciled with such small inclination angles ( see fig . 3 in c+03 ) , we suggest that the discrepancy is caused by the assumption that the unshocked pw is isotropic . indeed , if the wind is predominantly equatorial ( i.e. , concentrated around the plane perpendicular to the pulsar s spin axis ) , and the spin axis is aligned with the direction of pulsar motion ( as observed in a number of young pulsars ) , then the lower wind ram pressure ahead of the pulsar should result in a smaller distance between the pulsar and the ts ( and cd ) apex . the only models of pwne with anisotropic wind outflow we are aware of have been presented by vigelius et al . these models consider only nonrelativistic flows and , more importantly , assume zero magnetic field , but they should satisfactorily describe the pwn morphology for small values of the pw magnetization parameter ( defined as the ratio of the poynting flux to the kinetic energy flux ) . although vigelius et al . did not directly consider the case of an equatorial outflow perpendicular to the pulsar velocity , some of the considered cases ( e.g. , fig . 6 in that paper ) qualitatively confirm our explanation . for completeness , we should also mention the interpretation suggested by c+03 , that the outer tails represent the synchrotron radiation of the pw in the interstellar magnetic field `` compressed in the bowshock '' by a factor of 4 ( for an adiabatic shock with @xmath210 and a large mach number ) , up to @xmath211 g . as c+03 assume a one - zone shock model , which apparently does not describe realistic pwne , this interpretation may not be directly applicable . one may speculate , however , that ultrarelativistic electrons from the high - energy tail of the electron energy distribution could leak from the shocked wind region ( incide the cd ) into the shocked ism region and generate synchrotron radiation in the interstellar magnetic field amplified at the fs . we should note , however , that only the magnetic field component parallel to the shock surface is amplified by this mechanism , and the amplification becomes insignificant behind the pulsar because of the shock obliqueness . therefore , we believe that there is no need to invoke this complicated hypothesis as long as the more straightforward explanation ( synchrotron radiation from the region of the cd surface ) seems viable . finally , we should explain the fact that the outer tails are not truly symmetric with respect to the trajectory of geminga on the sky , neither in shape ( especially close to the pulsar ) nor , particularly , in brightness ( the s - tail is considerably brighter ) . we could tentatively ascribe this asymmetry to nonuniform conditions ( density and/or temperature ) in the ambient medium . the nonuniformity is supported by the very large array ( vla ) and effelsberg radio telescope hi ( 21 cm line ) observations of the geminga field ( giacani et al . these observations have shown the pulsar and its x - ray pwn to be in a local minimum of the hi emission , surrounded by an open hi shell ( an incomplete ring with an average radius of @xmath212 ) that envelopes the southern part of the x - ray pwn , with the internal border of the shell close to the s - tail . the lack of neutral hydrogen in the vicinity of the pulsar can be explained by the ionization caused by the pulsar s uv and soft x - ray emission . the openness of the shell ( no hi emission northwest of the x - ray pwn ) might imply a higher temperature ( and perhaps a lower density ) of the ism in that direction . one may speculate that geminga is crossing a cold ism cloud and is now approaching the cloud s northwest boundary . it remains to be understood , however , how the relative brightness of the s - tail is connected with the alleged lower temperature and higher density in that region . we should also mention that , based on the hi radio results , one could expect an h@xmath213 pwn south of the s - tail , associated with the fs . however , c+03 report the nondetection of `` organized diffuse h@xmath213 emission from the x - ray structure surrounding geminga '' in a 5 hour observation with the vlt - antu telescope . their figure 2 shows an apparent filament at the outer border of the s - tail , but it is not immediately clear whether or not this feature is related to the pwn . to conclude , if the outer tails represent the limb - brightened boundaries of the sky projection of a shell , this shell is most likely the synchrotron radiation from the region of interaction of the shocked pw and shocked ism material , where the wind flow is decelerated to nonrelativistic velocities by the shear instability , which implies mass loading . the shape of the shell is somewhat different from the shape of the cd surface in the available numerical pwn models , perhaps because the models do not include the mass loading and proper anisotropy of the unshocked pw . the hypothesis that the outer `` tails '' represent the boundary of a shell is not the only possible explanation . in particular , as no emission is seen between the outer tails except for the a - tail ( see 2.2.2 ) , we can not exclude the possibility that the outer tails are in fact some collimated flows emanating from the pulsar magnetosphere , such as two jets aligned with the pulsar s spin axis near the pulsar and bent by the ram pressure at larger distances . this interpretation implies a large angle @xmath214 between the spin axis and pulsar s velocity ( @xmath215@xmath216 , as follows from figs . 2 and 3 ) and a sufficiently large angle @xmath217 between the spin axis and the line of sight supports the outer gap interpretation of geminga s @xmath8-ray emission ( romani & watters 2010 ) . ] . _ chandra _ observations have shown that jets emanating along the spin axes are ubiquitous among young pwne ( see , e.g. , weisskopf et al . 2000 ; pavlov et al . 2003 ) , and the spin axis is often approximately aligned with the pulsar velocity direction ( ng & romani 2007 ) . the mechanisms of jet formation and collimation are currently not certain . in the scenario discussed by benford ( 1984 ) , a fraction of electrons created in the vacuum gaps above the magnetic poles and accelerated along the open magnetic field lines is deflected toward the spin axis and forms a beam collimated by its own toroidal magnetic field . another mechanism of axial outflow formation has been discussed by komissarov & lyubarsky ( 2004 ) , who assume that the outflow is originally equatorial and show that the magnetic hoop stress can stop the outflow in the surface layers of the equatorial disk and redirect it into magnetically confined polar jets . if the pulsar were not moving with respect to the ambient medium , the jet matter would keep flowing along the spin axis until the jet is destroyed by the interaction with the medium . the ram pressure exerted onto the jets of a moving pulsar can bend the jets in the direction opposite to that of the pulsar s motion , so that the jets are seen as two tails behind the moving pulsar . this scenario allows one to explain the observed asymmetry of the geminga s outer tails ( see 2.2.2 ) . the asymmetry can be associated with the large ( but different from @xmath218 ) angle @xmath214 between the spin axis and the pulsar s velocity . at such an orientation the angles between the ram pressure direction and the jet matter velocity directions are acute and obtuse for the southeastern and northwestern jets , respectively , which means that bending the northwestern jet is more difficult . an additional reason for the asymmetry might be a deviation of the pulsar s magnetic field geometry from an ideal centered dipole , which would lead to different structures of the magnetic field at the two poles and different properties of the two jets . different brightness of the jets , especially in the pulsar vicinity , might be caused by doppler boosting ( if the angle @xmath217 between the spin axis and line of sight is different from @xmath218 ) , but it is hard to estimate the doppler factor and to infer the angles with the current noisy data . as the bent polar outflow interpretation of the outer tails requires a large value of the angle @xmath214 , while @xmath214 is apparently small for most pulsars , this interpretation implies that the outer tails are a rare phenomenon , in agreement with pwn observations that have not shown such tails in other pwne . it , however , remains to be understood why @xmath214 is so different for geminga . a theoretical study of the expected distribution of this angle using the physics of the neutron star birth is required to confirm this explanation . the straight , patchy a - tail is seen up to @xmath2 ( @xmath3 pc ) from the pulsar . its surface brightness is not only nonuniform but also variable , as we see from the comparison of the 2007 and 2004 data . assuming a nearly isotropic emission ( which , rigorously speaking , implies a nonrelativistic flow ) , the a - tail luminosity , @xmath219@xmath220 erg s@xmath6 in the 0.38 kev band , is @xmath221@xmath222 of geminga s spin - down power @xmath223 . the a - tail luminosity is @xmath2240.06 of the nonthermal ( magnetospheric ) luminosity of the geminga pulsar in the same energy range and is a factor of 24 lower than the total luminosity of the outer tails . there are three conceivable explanations of the a - tail : a jet emanating from the pulsar magnetosphere in the direction opposite to the pulsar velocity , a tail part of the bowshock - tail pwn created by the supesonic motion of the pulsar , and a doppler - boosted image of a shell into which a fraction of the relativistic pw is directed . we will discuss these interpretations below , taking into account their connection with the above - discussed interpretations of the outer tails . the interpretation of the a - tail as a pulsar jet , suggested by p+06 , is consistent with only one of the above - discussed interpretations of the outer tails , namely , the hypothesis that the outer tails represent a boundary of a shell ( e.g. , an equatorial outflow bent by the ram pressure ) . as pulsar jets emerge along the pulsar s spin axis , and the a - tail is aligned with the pulsar s trajectory in the sky , the jet interpretation of the a - tail implies that the spin axis is likely aligned with the pulsar s velocity . this suggests that the `` natal kick '' of the geminga pulsar was directed along the spin axis , which has important implications for the mechanisms of supernova explosion and neutron star formation ( e.g. , ng & romani 2007 ) . the lack of a clear ( counter)jet ahead of the pulsar ( see , however , the note at the end of 2.2.3 ) could be explained by doppler boosting ( the approaching jet is brighter than the receding counterjet , assuming the jet material flows with nearly relativistic velocities ) . alternatively , the counterjet can be partially or fully destroyed by the ism ram pressure , or the outflows in the opposite directions may be intrinsically different . as described in 2.2.1 , the surface brightness is distributed nonuniformly along the a - tail , with some `` blobs '' seen at different positions in the images of 2004 and 2007 , and a `` gap '' between the pulsar and the beginning of the a - tail . this means that there are regions of the enhanced magnetic field and/or higher density along the a - tail , which might be caused by discrete ejections from the magnetosphere , or they could be manifestations of some instabilities ( e.g. , the sausage instability , as discussed by pavlov et al . 2003 for the outer jet of the vela pwn ) or internal shocks in the jet flow . particularly interesting is the brightest blob c seen at the apparent end of the a - tail in the image of 2007 ( see fig . 2 ) . although a background k star could contribute to the blob c emission ( see 2.2.1 ) , our analysis suggests that the star s contribution is not dominant and blob c could be associated with the jet s termination shock . to understand the nature of the blobs , it would be important to study their evolution in a series of deep observations , which , in particular , would help estimate the flow speed in the jet . using the observed diameter of the alleged jet and the estimate for the energy injection rate , p+06 estimate the jet s magnetic field : @xmath225 . for such a magnetic field the expected jet length is @xmath226 pc , where @xmath227 is the bulk flow velocity in the jet , and @xmath228 is the synchrotron cooling time . the jet length estimated from the initial bright portion of the axial tail , @xmath229 pc , is much smaller than this value unless @xmath227 is much smaller than @xmath230 and/or @xmath231 is small , which seems unlikely . to explain this contradiction , p+06 speculate that the jet becomes uncollimated or destroyed well before it radiates its entire internal energy . to check such speculations and test the pulsar jet interpretation , deeper observations are required . a tail - like structure similar to the observed a - tail could form behind the supersonically moving pulsar due to the collimation of the shocked pw by the ram pressure ( b+05 ; romanova et al . 2005 ) . for instance , if we assume that the outer tails and the possible arc ahead of the pulsar delineate the cd surface ( see 3.1.1 ) , then the a - tail might be the shocked pw immediately outside ( and perhaps behind ) the bullet - like ts . p+06 have shown that this interpretation is not quantitatively consistent with the available simulations of bowshock - tail pwne , but those simulations do not take into account the intrinsic anisotropy of the pw . in addition , it would be difficult to explain the presence of the blobs and the variability of the a - tail in the framework of this interpretation . therefore , we consider this interpretation unlikely . if the outer tails are bent polar outflows ( 3.1.2 ) , then the ram - pressure confined shocked pw would be the only possible explanation for the a - tail . a more detailed interpretation of the a - tail would depend on the pw model . for instance , if the pw were intrinsically isotropic , then the a - tail might be interpreted as originating from the shocked pw sheath " immediately outside the ts , and the observed width of the tail would imply the distances @xmath232 and @xmath233 of the ts and cd heads from the pulsar . therefore , we would expect a bright arc ( brighter than any part of the axial tail ) @xmath234@xmath235 ahead of the pulsar . no such a bright arc is seen in the images , but this does not necessarily rule out the ts origin of the a - tail because the ts head could be closer to the pulsar and hidden within the pulsar image if the pw is anisotropic ( p+06 ) . in this interpretation , however , we have to assume that no emission is seen from the cd surface region , which looks somewhat unnatural . in the bent polar outflow interpretation of the outer tails , one could also assume that the a - tail tail is associated with the cd - confined cylindrical region behind the ( unresolved ) ts . in this case , for an isotropic pw , we would expect a bright arc @xmath236 ahead of the pulsar , which can easily be hidden within the pulsar image . in the framework of this interpretation , the non - uniformity and variability of the a - tail could be explained by shear instabilities at the cd surface ( cf . 3.1.1 ) , which could also decelerate the flow , so that the blobs velocity would be lower than that in the jet interpretation of the a - tail . we should note , however , that in such interpretations the polar outflows are more luminous than the ram - pressure confined tail ( perhaps an equatorial outflow ) , which has not been observed for any other pulsar . as we have mentioned above , the image of a shell formed by material outflowing with relativistic speeds may be strongly affected by doppler boosting , which brightens those parts of the shell where the angle @xmath193 between the bulk flow velocity and the line of sight is the smallest . for instance , a conical shell , in which the material flows from the cone vertex at the pulsar position , would look like a straight strip corresponding to the minimum @xmath193 . if the material flows from the head of a paraboloid - like shell ( e.g. , the cd surface ) , the observer would see a shorter strip , detached from the paraboloid head in general case . in principle , one could imagine that the a - tail is such a projection of the shell formed by the shocked pw that flows out with relativistic velocities between the ts and cd , while the outer tails are bent polar outflows ( jets ) . in this interpretation , the true transverse radius of the shell would be larger than the observed width of the a - tail . the blobs in the a - tail might be some local instabilities in the relativistic flow , which would likely move with relativistic bulk - flow velocities . therefore , it would be important to examine the blob motion in future observations . the new _ chandra _ and _ xmm - newton _ observations of the geminga pwn have confirmed that it has three tail - like components , unlike any other detected pwn . the new observations have allowed us to image the tails at larger distances from the pulsar and establish their patchy structure . comparing the new and previous _ observations , we have found indications of pwn variability , especially in the axial tail and the emission ahead of the pulsar . in particular , we found up to three blobs in the axial tail , at different positions in 2004 and 2007 . similar to other x - ray pwne , the geminga pwn is due to synchrotron radiation of shocked pw comprised of relativistic particles . based on the new and old observations , we have proposed several competing interpretations of the pwn structure . very likely , the outer tails delineate a limb - brightened boundary of a shell - like region of interaction of the shocked pw and shocked ism , while the axial tail is a pulsar jet along the spin axis aligned with the pulsar s trajectory . such an interpretation implies a nonrelativistic speed of the bulk outflow along the shell , possibly decelerated by the shear instability and mass loading . alternatively , the outer tailis could be polar outflows from the pulsar magnetosphere ( e.g. , pulsar jets along the spin axis ) , bent by the ism ram pressure , in which case the axial tail could be a shocked pw ( e.g. , an equatorial outflow ) collimated by the ism ram pressure exerted on the supersonically moving pwn . to discriminate between various interpretation of the observed pwn , a series of carefully designed _ chandra _ observations is required . in particular , such observations should allow one to measure the speeds of the bulk flows in the tails , which would distinguish fast jets from ram - pressure - confined pulsar winds slowed down by the interaction with the ambient ism . also , such observations should be deep enough to establish the true morphology of the emission in the immediate vicinity of the pulsar . for instance , if a deeper observation convincingly shows that there is an arc ahead of the pulsar connecting the two outer tails , then the bending axial outflows scenario will be ruled out . if , however , we see two straight tails originating from the pulsar in a direction inclined to the pulsar s velocity direction , then the tails can be interpreted as bent jets . in addition , the detailed modeling of anisotropic magnetic pw from a high - speed pulsar will also be extremely useful to properly interpret the observational data . we thank andrew melatos for useful discussions of the pwn modeling . support for this work was provided by the national aeronautics and space administration through _ chandra _ award number go7 - 8053a issued by the _ chandra _ x - ray observatory center , which is operated by the smithsonian astrophysical observatory for and on behalf of the national aeronautics space administration under contract nas8 - 03060 . the work by ggp was also partially supported by nasa grant nnx09ac84 g . kargaltsev , o. , & pavlov , g. g. 2008 , in aip conf . 983 , 40 years of pulsars : millisecond pulsars , magnetars , and more , ed.c . bassa , a. cumming , v. m. kaspi , & z. wang ( melville , ny : aip ) , 171 ( kp08 ) lrccccc a - tail ( 2007 ; bright ) & 705.6 & @xmath237 & @xmath238 & @xmath239 & @xmath240 & @xmath241 + a - tail ( 2007 ; bright , w / o blob c ) & 677.1 & @xmath242 & 5.3 & @xmath243 & @xmath244 & @xmath245 + a - tail ( 2007 ; faint ) & 2404.2 & @xmath246 & 1.7 & @xmath247 & @xmath248 & @xmath249 + a - tail ( 2004 ) & 118.1 & @xmath250 & 6.4 & @xmath251 & @xmath252 & @xmath253 + n - tail ( 2007 ; bright ) & 462.3 & @xmath254 & 4.5 & @xmath255 & @xmath256 & @xmath257 + n - tail ( 2007 ; faint ) & 4011.9 & @xmath258 & 2.2 & @xmath259 & @xmath260 & @xmath261 + n - tail ( 2007 ; entire ) & 4474.2 & @xmath262 & 3.4 & @xmath263 & @xmath264 & @xmath265 + s - tail ( 2007 ; bright ) & 2258.0 & @xmath266 & 7.1 & @xmath267 & @xmath268 & @xmath269 + s - tail ( 2007 ; faint ) & 3740.7 & @xmath270 & 1.3 & @xmath271 & @xmath272 & @xmath273 + s - tail ( 2007 ; entire ) & 5998.7 & @xmath274 & 5.0 & @xmath275 & @xmath276 & @xmath277 + s - tail ( 2004 ) & 985.3 & @xmath148 & 3.7 & @xmath278 & @xmath279 & @xmath280 + arc ( 2007 ) & 86.6 & @xmath170 & 3.4 & @xmath281 & @xmath282 & @xmath283 + arc ( 2004 ) & 86.6 & @xmath284 & 4.5 & @xmath285 & @xmath286 & @xmath287 + ) of the geminga pulsar and its pwn in the 0.58 kev band . the top , middle and bottom panels correspond to the observation of 20042007 ( 64 ks total scientific exposure ) , 2002 ( 78 ks ) , and 20022007 ( 142 ks ) , respectively . the images in the left panels are binned in @xmath288 pixels , while the images in the right panels are additionally smoothed with a @xmath289 fwhm gaussian . the ellipses ( @xmath119 ) show the regions for which the spectra and fluxes were measured , while the @xmath290 rectangle between the ellipses was used for estimating the upper limit on the surface brightness between the outer tails ( see 2.2.2 ) . the @xmath291 rectangle ahead of the pulsar was used to measure the spectrum of the `` streak '' ( see 2.2.3 ) . the @xmath36 radius circle in the bottom left panel is centered at the position of blob c found in the _ chandra _ observation of 2007 ( see fig . 2 and 2.2.1 ) . the source north - northwest of the pulsar is an 11-th magnitude k star ( c+03 ) . , title="fig:",width=257 ] ) of the geminga pulsar and its pwn in the 0.58 kev band . the top , middle and bottom panels correspond to the observation of 20042007 ( 64 ks total scientific exposure ) , 2002 ( 78 ks ) , and 20022007 ( 142 ks ) , respectively . the images in the left panels are binned in @xmath288 pixels , while the images in the right panels are additionally smoothed with a @xmath289 fwhm gaussian . the ellipses ( @xmath119 ) show the regions for which the spectra and fluxes were measured , while the @xmath290 rectangle between the ellipses was used for estimating the upper limit on the surface brightness between the outer tails ( see 2.2.2 ) . the @xmath291 rectangle ahead of the pulsar was used to measure the spectrum of the `` streak '' ( see 2.2.3 ) . the @xmath36 radius circle in the bottom left panel is centered at the position of blob c found in the _ chandra _ observation of 2007 ( see fig . 2 and 2.2.1 ) . the source north - northwest of the pulsar is an 11-th magnitude k star ( c+03 ) . , title="fig:",width=257 ] + ) of the geminga pulsar and its pwn in the 0.58 kev band . the top , middle and bottom panels correspond to the observation of 20042007 ( 64 ks total scientific exposure ) , 2002 ( 78 ks ) , and 20022007 ( 142 ks ) , respectively . the images in the left panels are binned in @xmath288 pixels , while the images in the right panels are additionally smoothed with a @xmath289 fwhm gaussian . the ellipses ( @xmath119 ) show the regions for which the spectra and fluxes were measured , while the @xmath290 rectangle between the ellipses was used for estimating the upper limit on the surface brightness between the outer tails ( see 2.2.2 ) . the @xmath291 rectangle ahead of the pulsar was used to measure the spectrum of the `` streak '' ( see 2.2.3 ) . the @xmath36 radius circle in the bottom left panel is centered at the position of blob c found in the _ chandra _ observation of 2007 ( see fig . 2 and 2.2.1 ) . the source north - northwest of the pulsar is an 11-th magnitude k star ( c+03 ) . , title="fig:",width=257 ] ) of the geminga pulsar and its pwn in the 0.58 kev band . the top , middle and bottom panels correspond to the observation of 20042007 ( 64 ks total scientific exposure ) , 2002 ( 78 ks ) , and 20022007 ( 142 ks ) , respectively . the images in the left panels are binned in @xmath288 pixels , while the images in the right panels are additionally smoothed with a @xmath289 fwhm gaussian . the ellipses ( @xmath119 ) show the regions for which the spectra and fluxes were measured , while the @xmath290 rectangle between the ellipses was used for estimating the upper limit on the surface brightness between the outer tails ( see 2.2.2 ) . the @xmath291 rectangle ahead of the pulsar was used to measure the spectrum of the `` streak '' ( see 2.2.3 ) . the @xmath36 radius circle in the bottom left panel is centered at the position of blob c found in the _ chandra _ observation of 2007 ( see fig . 2 and 2.2.1 ) . the source north - northwest of the pulsar is an 11-th magnitude k star ( c+03 ) . , title="fig:",width=257 ] + ) of the geminga pulsar and its pwn in the 0.58 kev band . the top , middle and bottom panels correspond to the observation of 20042007 ( 64 ks total scientific exposure ) , 2002 ( 78 ks ) , and 20022007 ( 142 ks ) , respectively . the images in the left panels are binned in @xmath288 pixels , while the images in the right panels are additionally smoothed with a @xmath289 fwhm gaussian . the ellipses ( @xmath119 ) show the regions for which the spectra and fluxes were measured , while the @xmath290 rectangle between the ellipses was used for estimating the upper limit on the surface brightness between the outer tails ( see 2.2.2 ) . the @xmath291 rectangle ahead of the pulsar was used to measure the spectrum of the `` streak '' ( see 2.2.3 ) . the @xmath36 radius circle in the bottom left panel is centered at the position of blob c found in the _ chandra _ observation of 2007 ( see fig . 2 and 2.2.1 ) . the source north - northwest of the pulsar is an 11-th magnitude k star ( c+03 ) . , title="fig:",width=257 ] ) of the geminga pulsar and its pwn in the 0.58 kev band . the top , middle and bottom panels correspond to the observation of 20042007 ( 64 ks total scientific exposure ) , 2002 ( 78 ks ) , and 20022007 ( 142 ks ) , respectively . the images in the left panels are binned in @xmath288 pixels , while the images in the right panels are additionally smoothed with a @xmath289 fwhm gaussian . the ellipses ( @xmath119 ) show the regions for which the spectra and fluxes were measured , while the @xmath290 rectangle between the ellipses was used for estimating the upper limit on the surface brightness between the outer tails ( see 2.2.2 ) . the @xmath291 rectangle ahead of the pulsar was used to measure the spectrum of the `` streak '' ( see 2.2.3 ) . the @xmath36 radius circle in the bottom left panel is centered at the position of blob c found in the _ chandra _ observation of 2007 ( see fig . 2 and 2.2.1 ) . the source north - northwest of the pulsar is an 11-th magnitude k star ( c+03 ) . , title="fig:",width=257 ] images of the geminga pulsar and its pwn from the _ chandra _ observations of 2007 and 2004 ( top and middle panels , respectively ) , and the combined image ( bottom ) , in the 0.38 kev range . the original data were binned in @xmath292 pixels and smoothed with a @xmath82 fwhm gaussian . the arrows show the pulsar s proper motion , whereas the dashed lines in the middle and bottom panels indicate the sky region for which the exposure was reduced by the telescope dithering ( from 18.8 to 0 ks , in the 2004 observation ) . three @xmath80-radius circles mark `` blobs '' in the a - tail ( see 2.2.1 ) . the label x marks a field star ( see p+06 ) . , title="fig:",height=294 ] + images of the geminga pulsar and its pwn from the _ chandra _ observations of 2007 and 2004 ( top and middle panels , respectively ) , and the combined image ( bottom ) , in the 0.38 kev range . the original data were binned in @xmath292 pixels and smoothed with a @xmath82 fwhm gaussian . the arrows show the pulsar s proper motion , whereas the dashed lines in the middle and bottom panels indicate the sky region for which the exposure was reduced by the telescope dithering ( from 18.8 to 0 ks , in the 2004 observation ) . three @xmath80-radius circles mark `` blobs '' in the a - tail ( see 2.2.1 ) . the label x marks a field star ( see p+06 ) . , title="fig:",height=294 ] + images of the geminga pulsar and its pwn from the _ chandra _ observations of 2007 and 2004 ( top and middle panels , respectively ) , and the combined image ( bottom ) , in the 0.38 kev range . the original data were binned in @xmath292 pixels and smoothed with a @xmath82 fwhm gaussian . the arrows show the pulsar s proper motion , whereas the dashed lines in the middle and bottom panels indicate the sky region for which the exposure was reduced by the telescope dithering ( from 18.8 to 0 ks , in the 2004 observation ) . three @xmath80-radius circles mark `` blobs '' in the a - tail ( see 2.2.1 ) . the label x marks a field star ( see p+06 ) . , title="fig:",height=294 ] image of the geminga pwn in the _ chandra _ data of 2007 . the solid and dashed contours show the regions used for extracting the spectra of brighter and fainter parts of the pwn tails , respectively . , width=451 ] ) , shown in the inset . the dashed ( black ) and solid ( red ) points ( with @xmath294 errors ) are for the 2004 and 2007 data , respectively . the count rates and their errors for the 2007 data are multiplied by a factor of 1.6 to account for the different sensitivities of the i3 and s3 chips . , width=480 ] image of the geminga pulsar and its pwn from the _ chandra _ data of 2007 binned in @xmath292 pixels and smoothed with an @xmath295 fwhm gaussian , with overlaid brightness contours from the _ xmm - newton _ image shown in fig . 1 . the green crosses with numbers indicate the positions of optical stars projected onto the pwn elements . , width=480 ]

previous observations of the middle - aged pulsar geminga with _ xmm - newton _ and _ chandra _ have shown an unusual pulsar wind nebula ( pwn ) , with a @xmath0 long central ( axial ) tail directed opposite to the pulsar s proper motion and two @xmath1 long , bent lateral ( outer ) tails . here we report on a deeper _ chandra _ observation ( 78 ks exposure ) and a few additional _ xmm - newton _ observations of the geminga pwn . the new _ chandra _ observation has shown that the axial tail , which includes up to three brighter blobs , extends at least @xmath2 ( i.e. , @xmath3 pc ) from the pulsar ( @xmath4 is the distance scaled to 250 pc ) . it also allowed us to image the patchy outer tails and the emission in the immediate vicinity of the pulsar with high resolution . the pwn luminosity , @xmath5 erg s@xmath6 , is lower than the pulsar s magnetospheric luminosity by a factor of 10 . the spectra of the pwn elements are rather hard ( photon index @xmath7 ) . comparing the two _ chandra _ images , we found evidence of pwn variability , including possible motion of the blobs along the axial tail . the x - ray pwn is the synchrotron radiation from relativistic particles of the pulsar wind ; its morphology is connected with the supersonic motion of geminga . we speculate that the outer tails are either ( 1 ) a sky projection of the limb - brightened boundary of a shell formed in the region of contact discontinuity , where the wind bulk flow is decelerated by shear instability , or ( 2 ) polar outflows from the pulsar bent by the ram pressure from the ism . in the former case , the axial tail may be a jet emanating along the pulsar s spin axis , perhaps aligned with the direction of motion . in the latter case , the axial tail may be the shocked pulsar wind collimated by the ram pressure .

1002.1109

the need for the efficient use of the scarce spectrum in wireless applications has led to significant interest in the analysis of cognitive radio systems . one possible scheme for the operation of the cognitive radio network is to allow the secondary users to transmit concurrently on the same frequency band with the primary users as long as the resulting interference power at the primary receivers is kept below the interference temperature limit @xcite . note that interference to the primary users is caused due to the broadcast nature of wireless transmissions , which allows the signals to be received by all users within the communication range . note further that this broadcast nature also makes wireless communications vulnerable to eavesdropping . the problem of secure transmission in the presence of an eavesdropper was first studied from an information - theoretic perspective in @xcite where wyner considered a wiretap channel model . in @xcite , the secrecy capacity is defined as the maximum achievable rate from the transmitter to the legitimate receiver , which can be attained while keeping the eavesdropper completely ignorant of the transmitted messages . later , wyner s result was extended to the gaussian channel in @xcite . recently , motivated by the importance of security in wireless applications , information - theoretic security has been investigated in fading multi - antenna and multiuser channels . for instance , cooperative relaying under secrecy constraints was studied in @xcite@xcite . in @xcite , for amplify and forwad relaying scheme , not having analytical solutions for the optimal beamforming design under both total and individual power constraints , an iterative algorithm is proposed to numerically obtain the optimal beamforming structure and maximize the secrecy rates . although cognitive radio networks are also susceptible to eavesdropping , the combination of cognitive radio channels and information - theoretic security has received little attention . very recently , pei _ et al . _ in @xcite studied secure communication over multiple input , single output ( miso ) cognitive radio channels . in this work , finding the secrecy - capacity - achieving transmit covariance matrix under joint transmit and interference power constraints is formulated as a quasiconvex optimization problem . in this paper , we investigate the collaborative relay beamforming under secrecy constraints in the cognitive radio network . we first characterize the secrecy rate of the amplify - and - forward ( af ) cognitive relay channel . then , we formulate the beamforming optimization as a quasiconvex optimization problem which can be solved through convex semidefinite programming ( sdp ) . furthermore , we propose two sub - optimal null space beamforming schemes to reduce the computational complexity . we consider a cognitive relay channel with a secondary user source @xmath0 , a primary user @xmath1 , a secondary user destination @xmath2 , an eavesdropper @xmath3 , and @xmath4 relays @xmath5 , as depicted in figure [ fig : channel ] . we assume that there is no direct link between @xmath0 and @xmath2 , @xmath0 and @xmath1 , and @xmath0 and @xmath3 . we also assume that relays work synchronously to perform beamforming by multiplying the signals to be transmitted with complex weights @xmath6 . we denote the channel fading coefficient between @xmath0 and @xmath7 by @xmath8 , the fading coefficient between @xmath7 and @xmath2 by @xmath9 , @xmath7 and @xmath1 by @xmath10 and the fading coefficient between @xmath7 and @xmath3 by @xmath11 . in this model , the source @xmath0 tries to transmit confidential messages to @xmath2 with the help of the relays on the same band as the primary user s while keeping the interference on the primary user below some predefined interference temperature limit and keeping the eavesdropper @xmath3 ignorant of the information . it s obvious that our channel is a two - hop relay network . in the first hop , the source @xmath0 transmits @xmath12 to relays with power @xmath13=p_s$ ] . the received signal at the @xmath14 relay @xmath7 is given by @xmath15 where @xmath16 is the background noise that has a gaussian distribution with zero mean and variance of @xmath17 . in the af scenario , the received signal at @xmath7 is directly multiplied by @xmath18 without decoding , and forwarded to @xmath2 . the relay output can be written as @xmath19 the scaling factor , @xmath20 is used to ensure @xmath21=|w_m|^2 $ ] . there are two kinds of power constraints for relays . first one is a total relay power constraint in the following form : @xmath22 where @xmath23^t$ ] and @xmath24 is the maximum total power . @xmath25 and @xmath26 denote the transpose and conjugate transpose , respectively , of a matrix or vector . in a multiuser network such as the relay system we study in this paper , it is practically more relevant to consider individual power constraints as wireless nodes generally operate under such limitations . motivated by this , we can impose @xmath27 or equivalently @xmath28 where @xmath29 denotes the element - wise norm - square operation and @xmath30 is a column vector that contains the components @xmath31 . @xmath32 is the maximum power for the @xmath14 relay node . the received signals at the destination @xmath2 and eavesdropper @xmath3 are the superposition of the messages sent by the relays . these received signals are expressed , respectively , as @xmath33 where @xmath34 and @xmath35 are the gaussian background noise components with zero mean and variance @xmath36 , at @xmath2 and @xmath3 , respectively . it is easy to compute the received snr at @xmath2 and @xmath3 as @xmath37 where @xmath38 denotes the mutual information . the interference at the primary user is latexmath:[\ ] ] where superscript @xmath43 denotes conjugate operation . then , the received snr at the destination and eavesdropper , and the interference on primary user can be written , respectively , as @xmath44 with these notations , we can write the objective function of the optimization problem ( i.e. , the term inside the logarithm in ( [ srate ] ) ) as @xmath45 if we denote @xmath46 , @xmath47 , define @xmath48 , and employ the semidefinite relaxation approach , we can express the beamforming optimization problem as @xmath49 the optimization problem here is similar to that in @xcite . the only difference is that we have an additional constraint due to the interference limitation . thus , we can use the same optimization framework . the optimal beamforming solution that maximizes the secrecy rate in the cognitive relay channel can be obtained by using semidefinite programming with a two dimensional search for both total and individual power constraints . for simulation , one can use the well - developed interior point method based package sedumi @xcite , which produces a feasibility certificate if the problem is feasible , and its popular interface yalmip @xcite . it is important to note that we should have the optimal @xmath50 to be of rank - one to determine the beamforming vector . while proving analytically the existence of a rank - one solution for the above optimization problem seems to be a difficult task , we would like to emphasize that the solutions are rank - one in our simulations . thus , our numerical result are tight . also , even in the case we encounter a solution with rank higher than one , the gaussian randomization technique is practically proven to be effective in finding a feasible , rank - one approximate solution of the original problem . details can be found in @xcite . obtaining the optimal solution requires significant computation . to simplify the analysis , we propose suboptimal null space beamforming techniques in this section . we choose @xmath51 to lie in the null space of @xmath52 . with this assumption , we eliminate @xmath3 s capability of eavesdropping on @xmath2 . mathematically , this is equivalent to @xmath53 , which means @xmath51 is in the null space of @xmath54 . we can write @xmath55 , where @xmath56 denotes the projection matrix onto the null space of @xmath54 . specifically , the columns of @xmath56 are orthonormal vectors which form the basis of the null space of @xmath54 . in our case , @xmath56 is an @xmath57 matrix . the total power constraint becomes @xmath58 . the individual power constraint becomes @xmath59 under the above null space beamforming assumption , @xmath60 is zero . hence , we only need to maximize @xmath61 to get the highest achievable secrecy rate . @xmath61 is now expressed as @xmath62 the interference on the primary user can be written as @xmath63 defining @xmath64 , we can express the optimization problem as @xmath65 this problem can be easily solved by semidefinite programming with bisection search @xcite . in this section , we choose @xmath51 to lie in the null space of @xmath52 and @xmath66 . mathematically , this is equivalent to requiring @xmath67 , and @xmath68 . we can write @xmath69 , where @xmath70 denotes the projection matrix onto the null space of @xmath54 and @xmath71 . specifically , the columns of @xmath70 are orthonormal vectors which form the basis of the null space . in our case , @xmath70 is an @xmath72 matrix . the total power constraint becomes @xmath73 . the individual power constraint becomes @xmath74 . with this beamforming strategy , we again have @xmath75 . moreover , the interference on the primary user is now reduced to @xmath76 which is the sum of the forwarded additive noise components present at the relays . now , the optimization problem becomes @xmath77 again , this problem can be solved through semidefinite programming . with the following assumptions , we can also obtain a closed - form characterization of the beamforming structure . since the interference experienced by the primary user consists of the forwarded noise components , we can assume that the interference constraint @xmath78 is inactive unless @xmath41 is very small . with this assumption , we can drop this constraint . if we further assume that the relays operate under the total power constraint expressed as @xmath79 , we can get the following closed - form solution : @xmath80 where @xmath81 is the largest generalized eigenvalue of the matrix pair @xmath82 . and positive definite matrix @xmath83 , @xmath84 is referred to as a generalized eigenvalue eigenvector pair of @xmath82 if @xmath84 satisfy @xmath85 @xcite . ] hence , the maximum secrecy rate is achieved by the beamforming vector @xmath86 where @xmath87 is the eigenvector that corresponds to @xmath88 and @xmath89 is chosen to ensure @xmath90 . the discussion in section [ sec : op ] can be easily extended to the case of more than one primary user in the network . each primary user will introduce an interference constraint @xmath91 which can be straightforwardly included into ( [ optimal ] ) . the beamforming optimization is still a semidefinite programming problem . on the other hand , the results in section [ sec : op ] can not be easily extended to the multiple - eavesdropper scenario . in this case , the secrecy rate for af relaying is @xmath92 , where the maximization is over the rates achieved over the links between the relays and different eavesdroppers . hence , we have to consider the eavesdropper with the strongest channel . in this scenario , the objective function can not be expressed in the form given in ( [ srate ] ) and the optimization framework provided in section [ sec : op ] does not directly apply to the multi - eavesdropper model . however , the null space beamforming schemes discussed in section [ sec : null ] can be extended to the case of multiple primary users and eavesdroppers under the condition that the number of relay nodes is greater than the number of eavesdroppers or the total number of eavesdroppers and primary users depending on which null space beamforming is used . the reason for this condition is to make sure the projection matrix @xmath93 exists . note that the null space of @xmath94 channels in general has the dimension @xmath95 where @xmath4 is the number of relays . we assume that @xmath96 , @xmath97 are complex , circularly symmetric gaussian random variables with zero mean and variances @xmath98 , @xmath99 , @xmath100 and @xmath101 respectively . in this section , each figure is plotted for fixed realizations of the gaussian channel coefficients . hence , the secrecy rates in the plots are instantaneous secrecy rates . in fig . [ fig:1 ] , we plot the optimal secrecy rates for the amplify - and - forward collaborative relay beamforming system under both individual and total power constraints . we also provide , for comparison , the secrecy rates attained by using the suboptimal beamforming schemes . the fixed parameters are @xmath102 , @xmath103 , and @xmath104 . since af secrecy rates depend on both the source and relay powers , the rate curves are plotted as a function of @xmath105 . we assume that the relays have equal powers in the case in which individual power constraints are imposed , i.e. , @xmath106 . it is immediately seen from the figure that the suboptimal null space beamforming achievable rates under both total and individual power constraints are very close to the corresponding optimal ones . especially , they are nearly identical in the high snr regime , which suggests that null space beamforming is optimal at high snrs . thus , null space beamforming schemes are good alternatives as they are obtained with much less computational burden . moreover , we interestingly observe that imposing individual relay power constraints leads to small losses in the secrecy rates . in fig . [ fig:11 ] , we change the parameters to @xmath107 , @xmath108 and @xmath104 . in this case , channels between the relays and the eavesdropper and between the relays and the primary - user are on average stronger than the channels between the relays and the destination . we note that beamforming schemes can still attain good performance and we observe similar trends as before . in fig . [ fig:2 ] , we plot the optimal secrecy rate and the secrecy rates of the two suboptimal null space beamforming schemes ( under both total and individual power constraints ) as a function of the interference temperature limit @xmath41 . we assume that @xmath109 . it is observed that the secrecy rate achieved by beamforming in the null space of both the eavesdropper s and primary user s channels ( bnep ) is almost insensitive to different interference temperature limits when @xmath110 since it always forces the signal interference to be zero regardless of the value of @xmath41 . it is further observed that beamforming in the null space of the eavesdropper s channel ( bne ) always achieves near optimal performance regardless the value of @xmath41 under both total and individual power constraints . in this paper , collaborative relay beamforming in cognitive radio networks is studied under secrecy constraints . optimal beamforming designs that maximize secrecy rates are investigated under both total and individual relay power constraints . we have formulated the problem as a semidefinite programming problem and provided an optimization framework . in addition , we have proposed two sub - optimal null space beamforming schemes to simplify the computation . finally , we have provided numerical results to illustrate the performances of different beamforming schemes . a. wyner `` the wire - tap channel , '' _ bell . syst tech . j _ , vol.54 , no.8 , pp.1355 - 1387 , jan 1975 . i. csiszar and j. korner `` broadcast channels with confidential messages , '' _ ieee trans . inform . theory _ , vol.it-24 , no.3 , pp.339 - 348 , may 1978 . v. nassab , s. shahbazpanahi , a. grami , and z .- q . luo , `` distributed beamforming for relay networks based on second order statistics of the channel state information , '' _ ieee trans . on signal proc . 56 , no 9 , pp . 4306 - 4316 , g. zheng , k. k. wong , a. paulraj , and b. ottersten , `` robust collaborative - relay beamforming , '' _ ieee trans . on signal proc . 57 , no . 8 , aug . 2009 z - q luo , wing - kin ma , a.m .- c . so , yinyu ye , shuzhong zhang `` semidefinite relaxation of quadratic optimization problems '' _ ieee signal proc . magn . 3 , may 2010 j. lofberg , `` yalmip : a toolbox for modeling and optimization in matlab , '' _ proc . the cacsd conf . _ , taipei , taiwan , 2004 . s. boyd and l. vandenberghe , convex optimization . cambridge , u.k . : cambridge univ . press , 2004 .

in this paper , a cognitive relay channel is considered , and amplify - and - forward ( af ) relay beamforming designs in the presence of an eavesdropper and a primary user are studied . our objective is to optimize the performance of the cognitive relay beamforming system while limiting the interference in the direction of the primary receiver and keeping the transmitted signal secret from the eavesdropper . we show that under both total and individual power constraints , the problem becomes a quasiconvex optimization problem which can be solved by interior point methods . we also propose two sub - optimal null space beamforming schemes which are obtained in a more computationally efficient way . _ index terms : _ amplify - and - forward relaying , cognitive radio , physical - layer security , relay beamforming .

1009.6197

sigmoidal input - output response modules are very well - conserved in cell signaling networks that might be used to implement binary responses , a key element in cellular decision processes . additionally , sigmoidal modules might be part of more complex structures , where they can provide the nonlinearities which are needed in a broad spectrum of biological processes [ 1,2 ] , such as multistability [ 3,4 ] , adaptation [ 5 ] , and oscillations [ 6 ] . there are several molecular mechanisms that are able to produce sigmoidal responses such as inhibition by a titration process [ 7,8 ] , zero - order ultrasensitivity in covalent cycles [ 9,10 ] , and multistep activation processes - like multisite phosphorylation [ 11 - 13 ] or ligand binding to multimeric receptors [ 14 ] . sigmoidal curves are characterized by a sharp transition from low to high output following a slight change of the input . the steepness of this transition is called ultrasensitivity [ 15 ] . in general , the following operational definition of the hill coefficient may be used to calculate the overall ultrasensitivity of sigmoidal modules : @xmath0 where ec10 and ec90 are the signal values needed to produce an output of 10% and 90% of the maximal response . the hill coefficient @xmath1 quantifies the steepness of a function relative to the hyperbolic response function which is defined as not ultrasensitive and has @xmath2 ( i.e. an 81-fold increase in the input signal is required to change the output level from 10% to 90% of its maximal value ) . functions with @xmath3 need a smaller input fold increase to produce such output change , and are thus called ultrasensitive functions . global sensitivity measures such the one described by eq . 1 do not fully characterize s - shaped curves , y(x ) , because they average out local characteristics of the analyzed response functions . instead , these local features are well captured by the logarithmic gain or response coefficient measure [ 16 ] defined as : equation 2 provides local ultrasensitivity estimates given by the local polynomial order of the response function . @xmath4 equation 2 provides local ultrasensitivity estimates given by the local polynomial order of the response function . mitogen activated protein ( map ) kinase cascades are a well - conserved motif . they can be found in a broad variety of cell fate decision systems involving processes such as proliferation , differentiation , survival , development , stress response and apoptosis [ 17 ] . they are composed of a chain of three kinases which sequentially activate one another , through single or multiple phosphorylation events . a thoughtful experimental and mathematical study of this kind of systems was performed by ferrell and collaborators , who analyzed the steady - state response of a mapk cascade that operates during the maturation process in xenopus oocytes [ 18 ] . they developed a biochemical model to study the ultrasensitivity displayed along the cascade levels and reported that the combination of the different ultrasensitive layers in a multilayer structure produced an enhancement of the overall system s global ultrasensitivity [ 18 ] . in the same line , brown et al . [ 19 ] showed that if the dose - response curve , f(x ) , of a cascade could be described as the mathematical composition of functions , fisi , that described the behavior of each layer in isolation ( i.e , @xmath5 then the local ultrasensitivity of the different layers combines multiplicatively : @xmath6 . in connection with this result , ferrell showed for hill - type modules of the form @xmath7 where the parameter ec50 corresponds to the value of input that elicits half - maximal output , and nh is the hill coefficient ) , that the overall cascade global ultrasensitivity had to be less than or equal to the product of the global ultrasensitivity estimations of each cascade s layer , i.e @xmath8 [ 20 ] . hill functions of the form given by eq . 3 are normally used as empirical approximations of sigmoidal dose - response curves , even in the absence of any mechanistic foundation [ 2 ] . however , it is worth noting that for different and more specific sigmoidal transfer functions , qualitatively different results could have been obtained . in particular , a supra - multiplicative behavior ( the ultrasensitivity of the combination of layers is higher than the product of individual ultrasensitivities ) might be observed for left - ultrasensitive response functions , i.e. functions that are steeper to the left of the ec50 than to the right [ 21 ] ) . in this case , the boost in the ultrasensitivity emerges from a better exploitation of the ultrasensitivity `` stored '' in the asymmetries of the dose - response functional form ( see [ 21 ] for details ) . as modules are embedded in bigger networks , constraints in the range of inputs that each module would receive ( as well as in the range of outputs that the network would be able to transmit ) could arise . we formalized this idea in a recent publication introducing the notion of dynamic range constraint of a module s dose - response function . the later concept is a feature inherently linked to the coupling of modules in a multilayer structure , and resulted a significant element to explain the overall ultrasensitivity displayed by a cascade [ 21 ] . besides dynamic range constraint effects sequestration - i.e. , the reduction in free active enzyme due to its accumulation in complex with its substrate- is another relevant process inherent to cascading that could reduce the cascade s ultrasensitivity [ 22 - 24 ] . moreover , sequestration may alter the qualitative features of any well - characterized module when integrated with upstream and downstream components , thereby limiting the validity of module - based descriptions [ 25 - 27 ] . all these considerations expose the relevance of studying the behavior of modular processing units embedded in their physiological context . although there has been significant progress in the understanding of kinase cascades , how the combination of layers affects the cascade s ultrasensitivity remains an open - ended question for the general case . sequestration and dynamic range constraints not only contribute with their individual complexity , but also usually occur together , thus making it more difficult to identify their individual effective contribution to the system s overall ultrasensitivity . in the present work , we have developed a method to describe the overall ultrasensitivity of a molecular cascade in terms of the effective contribution of each module . in addition , said method allows us to disentangle the effects of sequestration and dynamic range constraints . we used our approach to analyze a recently presented synthetic mapk cascade experimentally engineered by oshaughnessy et al . [ 28 ] . using a synthetic biology approach oshaughnessy et al . [ 28 ] constructed an isolated mammalian mapk cascade ( a raf - mek - erk system ) in yeast and analyzed its information processing capabilities under different rather well - controlled environmental conditions . they made use of a mechanistic mathematical description to account for their experimental observations . their model was very similar in spirit to huang - ferrell s with two important differences : a ) no phosphatases were included , and b ) the creation and degradation of all species was explicitly taken into account . interestingly , they reported that the multilayer structure of the analyzed cascades can accumulate ultrasensitivity supra - multiplicatively , and suggested that cascading itself and not any other process ( such as multi - step phosphorylation , or zero - order ultrasensitivity ) was at the origin of the observed ultrasensitivity . they called this mechanism , de - novo ultrasensitivity generation . as we found the proposed mechanism a rather appealing and unexpected way of ultrasensitivity generation , we wanted to further characterize it within our analysis framework . in particular , we reasoned that the methodology and concepts introduced in the present contribution were particularly well - suited to understand the mechanisms laying behind the ultrasensitivity behavior displayed by oshaughnessy cascade model . the paper was organized as follows . first , we presented a formal connection between local and global descriptors of a module s ultrasensitivity for the case of a cascade system composed of @xmath9 units . we then introduced the notion of hill input s working range in order to analyze the contribution to the overall system s ultrasensitivity of a module embedded in a cascade . next , we presented the oshaughnessy cascade analysis , in order to show the insights that might be gained using the introduced concepts and analysis methodologies . we concluded by presenting a summarizing discussion after which conclusions were drawn . the concept of ultrasensitivity describes a module s ability to amplify small changes in input values into larger changes in output values . it is customary to quantify and characterize the extent of the amplification both globally , using the hill coefficient @xmath1 defined in equation 1 , and locally , using the response coefficient , r(i ) , as a function of the module s input signal i ( equation 2 ) , we found a simple relationship between both descriptions considering the logarithmic amplification coefficient @xmath10 , defined as : @xmath11 @xmath10 describes ( in a logarithmic scale ) the change produced in the output when the input varies from a to b values . for instance , @xmath12 for an hyperbolic function evaluated between the inputs that resulted in 90% and 10% of the maximal output . in this case , the two considered input levels delimited the input range that should be considered for the estimation of the respective hill coefficient @xmath1 . , we called this input interval : the hill input s working range ( see fig 1a - b ) . ) , and the `` hill working range '' is the input range relevant for the calculation of the system s @xmath13 . in hill functions , inputs values much smaller than its ec50 produce local sensitivities around its hill coefficient.schematic response function diagrams for the composition of two hill type ultrasensitive modules ( d - e ) . the @xmath14 and @xmath15 are the input values that take the `` i '' module when the last module ( the second one ) reach the % 10 and % 90 of it maximal output ( @xmath16 ) . when @xmath17(d ) @xmath16 equals the maximum output of module 2 in isolation , thus , @xmath18 and @xmath19 match the ec10 and ec90 of module 2 in isolation . also the hill working range of module 1 is located in the input region below ec501 . on the other hand , when @xmath20 ( e ) @xmath16 is less than the maximum output of module 2 in isolation , thus , @xmath18 and @xmath19 differ from the ec10 and ec90 of module 2 in isolation . in this case the modules-1 hill working range tends to be centered in values higher than its ec50 , this will depends on modules-2 ultrasensitivity ( see supplementary materials [ sm1]),scaledwidth=100.0% ] [ fig1 ] taking into account eq . ( 4 ) , the parameter @xmath1 could be rewritten as follows , @xmath21 consequently , the hill coefficient could be interpreted as the ratio of the logarithmic amplification coefficients of the function of interest and an hyperbolic function , evaluated in the corresponding hill input s working range . it is worth noting that the logarithmic amplification coefficient that appeared in equation 5 equaled the slope of the line that passed through the points @xmath22 and @xmath23 in a log - log scale . thus , it was equal to the average response coefficient calculated over the interval @xmath24 $ ] in logarithmic scale ( see fig 1c ) . therefore , @xmath25 where @xmath26 denoted the mean value of the variable x over the range [ a , b ] . this last equation explicitly linked the local and global ultrasensitivity descriptions . in particular , it can be appreciated that the module s hill coefficient was related to the average response coefficient over the module s hill input s working range , in units of a reference hyperbolic curve . we generalized the last result to cast the overall global ultrasensitivity level of a multitier cascade in terms of logarithmic amplification coefficients . we proceeded by first considering two coupled ultrasensitive modules , disregarding effects of sequestration of molecular components between layers . in this case , the expression for the system s dose - response curve , @xmath27 , results from the mathematical composition of the functions , @xmath28 , which describe the input / output relationship of isolated modules @xmath29 : @xmath30 using eq . 5 , the system s hill coefficient @xmath1 resulted : @xmath31 where @xmath14 and @xmath15 delimited the hill input s working range of the composite system , i.e. the input values for the i - layer so that the last layer ( corresponding to i=2 in this case ) reached the 10% and 90% of it maximal output level , respectively ( see fig 1d-1e ) . it followed from equation 8 that the system s hill coefficient nh could be written as the product of two factors , @xmath32 and @xmath33 , which characterized local average sensitivities over the relevant input region for each layer : @xmath34 $ ] , with @xmath29 ( see fig 1d - e ) . the factor @xmath33 in equation 8 was formally equivalent to the hill coefficient of layer-2 but , importantly , now it was calculated using layer-1 hill input s working range limits , x101 and x901 , instead of the hill working range limits of layer-1 in isolation , @xmath35 and @xmath36 . on the other hand , @xmath32 was the amplification factor @xmath37 that logarithmically scaled the range @xmath38 $ ] into @xmath39 $ ] . in this context , we dubbed the @xmath40 coefficient : effective ultrasensitivity coefficient of layer - i , as it was associated to the effective contribution of layer - i to the system s overall ultrasensitivity . for the more general case of a cascade of @xmath9 modules we found that : @xmath41 this last equation , which hold exactly , showed a very general result . for the general case , the overall @xmath1 of a cascade could be understood as a multiplicative combination of the @xmath40 of each module . the connection between the global and local ultrasensitivity descriptors , provided by equation 9 , proved to be a useful tool to analyze ultrasensitivity in cascades , as it allowed assessing the effective contribution of each module to the system s overall ultrasensitivity . according to this equation , hill input s working range designated regions of inputs over which the mean local - ultrasensitivity value was calculated for each cascade level in order to set the system s @xmath1 . it was thus a significant parameter to characterize the overall ultrasensitivity of multilayer structures . in figures 1d-1e we illustrated , for the case of two composed hill functions , to what extent the actual location of these relevant intervals depended on the way in which the cascade layers were coupled . the ratio between @xmath42 and @xmath43 played a key role at this respect ( see fig 1 ) . we showed how this parameter sets the hill working ranges for the case of modules presenting different dose - response curves in sup mat . importantly this analysis highlighted the impact of the detailed functional form of a module s response curve on the overall system s ultrasensitivity in cascade architectures . local sensitivity features of the involved transfer functions were of the uttermost importance in this kind of setting and could be at the core of non - trivial phenomenology . for example a dose - response module with larger local ultrasensitivities than its overall global value might contribute with more ultrasensitivity to the system than the function s own @xmath1 ( see sup.mat . [ sm1 ] ) . in order to disentangle the different factors contributing to a cascade overall ultrasensitivity , we simultaneously considered three approximations of the system under study ( see fig 2 ) . in a first step ( fig 2a ) we numerically computed the transfer function , @xmath44 , of each module in isolation and calculated the respective hill coefficients @xmath45 . ) can be obtain by the mathematical composition of each module s transfer functions @xmath46 acting in isolation ( b ) . when the sequestration effect is taken into account , the layers embedded in the map kinase cascade may have a different dose - response curve from the isolated case ( c).,scaledwidth=100.0% ] [ fig2 ] we then studied the mathematical composition , @xmath47 , of the isolated response functions ( fig 2b ) : @xmath48 @xmath47 represented the transfer function of the kinase cascade when sequestration effects were completely neglected . following equation 9 effective ultrasensitivities , @xmath49 could be estimated and compared against the global ultrasensitivity that each module displayed in isolation , . thus , this second step aimed to specifically analyze to what extent the existence of _ hill s input working ranges _ impinged on ultrasensitivity features display by cascade arrangement of layers ( fig 2b ) . finally , in a third step , response functions were obtained considering a mechanistic model of the cascade . effective ultrasensitivity coefficients , @xmath50 , could then be estimated for each module in order to asses for putative sequestration effects that could take place in the system ( fig 2c ) . a sketch of the oshaughnessy et al . mapk cascade is shown in fig 3a . in our analysis , we defined the output of a module and the input to the next one as the total active form of a species , including complexes with the next layer s substrates . however , we excluded complexes formed by same layer components ( such as a complex between the phosphorylated kinase and its phosphatase ) , since these species are `` internal '' to each module . by doing this , we are able to consistently identify layers with modules ( the same input / output definition was used by ventura _ et al _ [ 25 ] ) . [ fig3 ] in order to study the contribution of each layer to the system s ultrasensitivity , we proceeded to numerically compute the transfer function of each module in isolation and then calculate their respective @xmath51 ( column 2 in table 1 ) . .[tab : table1]*3-step analysis of oshaughnessy _ et al . _ dual - step cascade . * + column 1 : hill coefficients of active species dose - response curves ( respect to estradiol ) . column 2 : hill coefficients of the modules in . column 3 : effective ultrasensitivities of the system given by the composition of the modules in isolation , which is represented by @xmath47 ( equation [ 6 ] ) where no sequestration effects are present . column 4 : effective ultrasensitivities in the original cascade . [ cols="<,<,<,<,<",options="header " , ] we verified that the hill function is not contributing as much ultrasensitivity as the original system s raf - mek module . the reason is that even the dose - response of active mek and the hill function appear to be similar , there are strong dissimilarities in their local ultrasensitivity behavior ( see fig 5b ) . this is particularly true for low input values , where the hill s input working range is located . in this region , the active mek curve presents local ultrasensitivity values larger than the hill function counterpart , thus the replacement by a hill function produce a reduction in the hill coefficient in this way , despite the high - quality of the fitting adjustment residual standard error=2.6 ) , the hill function approximation introduced non - trivial alterations in the system s ultrasensitivity as a technical glitch . there have been early efforts to interpret cascade system - level ultrasensitivity out of the sigmoidal character of their constituent modular components . usually they have focused either on a local or a global characterization of ultrasensitivity features . for instance , brown et al . [ 19 ] had shown that the local system s ultrasensitivity in cascades equals the product of the local ultrasensitivity of each layer . in turn , from a global ultrasensitivity perspective , ferrell [ 20 ] pointed out that in the composition of two hill functions , the hill coefficient results equal or less than the product of the hill coefficient of both curves ( @xmath52 ) . in this contribution we have found a mathematical expression ( equation 6 ) that linked both , the local and global ultrasensitivity descriptors in a fairly simple way . moreover we could provide a generalized result to handle the case of a linear arrangement of an arbitrary number of such modules ( equation 9 ) . noticeably , within the proposed analysis framework , we could decompose the overall global ultrasensitivity in terms of a product of single layer effective ultrasensitivities . these new parameters were calculated as local - ultrasensitivity values averaged over meaningful working ranges ( dubbed _ hill s input working ranges _ ) , and permitted to assess the effective contribution of each module to the system s overall ultrasensitivity . of course , the reason why we could state an exact general equation for a system - level feature in terms of individual modular information was that in fact system - level information was used in the definition of the _ hill s input working ranges _ that entered equation 9 . the specific coupling between ultrasensitive curves , set the corresponding _ hill s input working ranges _ , thus determining the effective contribution of each module to the cascade s ultrasensitivity . this process , which we called _ hill s input working range setting _ , has already been noticed by several authors [ 20 , 29,30 , 23 ] , but as far as we know this was the first time that a mathematical framework , like the one we present here , has been proposed for it . the value of the obtained expression ( equation 9 ) resides in the fact that not only it captured previous results , like ferrell s inequality , but also that it threw light about the mechanisms involved in the ultrasensitivity generation . for instance , the existence of supramultiplicative behavior in signaling cascades have been reported by several authors [ 23 , 28 ] but in many cases the ultimate origin of supramultiplicativity remained elusive . our framework naturally suggested a general scenario where supramultiplicative behavior could take place . this could occur when , for a given module , the corresponding _ s input working range _ was located in an input region with local ultrasensitivities higher than the global ultrasensitivity of the respective dose - response curve . in order to study how multiple ultrasensitive modules combined to produce an enhancement of the system s ultrasensitivity , we have developed an analysis methodology that allowed us to quantify the effective contribution of each module to the cascade s ultrasensitivity and to determine the impact of sequestration effects in the system s ultrasensitivity . this method was particularly well suited to study the ultrasensitivity in map kinase cascades . we used our methodology to revisit oshaughnessy et al . tunable synthetic mapk system [ 28 ] in which they claim to have found a new source of ultrasensitivity called : _ de novo _ ultrasensitivity generation . they explained this new effect in terms of the presence of intermediate elements in the kinase - cascade architecture . we started analyzing the mapk cascade . we found that sequestration was not affecting the system s ultrasensitivity and that the overall sub - multiplicative behavior was only due to a re - setting of the hill input s working range for the first and second levels of the cascade . then , to investigate the origin of the claimed _ de novo _ ultrasensitivity generation mechanism , we applied our framework on the single - step phosphorylation cascade . we found that the system s ultrasensitivity in the single - step cascade came only from the contribution of the last module , which behaved as a goldbeter - koshland unit with kinases working in saturation and phosphatases in a linear regime . therefore the ultrasensitivity in its single - step cascade was not generated by the cascading itself , but by the third layer , which itself was actually ultrasensitive . finally we analyzed the auxiliary model considered by oshaughnessy et al . in which the raf and mek layers were replaced by a hill function that is coupled to the erk layer . in this case , , even the original estradiol - mek input - output response curve could be fairly well fitted and global ultrasensitivity features were rather well captured , the replacement by a hill function produce a strong decrease in the systems ultrasensitivity . we found that the functional form of the hill function failed to reproduce original local ultrasensitivity features that were in fact the ones that , due to the particular hill working range setting acting in this case , were responsible for the overall systems ultrasensitivity behavior . the analyzed case was particularly relevant , as provided an illustrative example that warned against possible technical glitches that could arise as a consequence of the inclusion of approximating functions in mapk models . the study of signal transmission and information processing inside the cell has been , and still is , an active field of research . in particular , the analysis of cascades of sigmoidal modules has received a lot of attention as they are well - conserved motifs that can be found in many cell fate decision systems . in the present contribution we focused on the analysis of the ultrasensitive character of this kind of molecular systems . we presented a mathematical link between global and local characterizations of the ultrasensitive nature of a sigmoidal unit and generalized this result to handle the case of a linear arrangement of such modules . in this way , the overall system ultrasensitivity could define in terms of the effective contribution of each cascade tier . based on our finding , we proposed a methodological procedure to analyzed cascade modular systems , in particular mapk cascades . we used our methodology to revisit oshaughnessy et al . tunable synthetic mapk system [ 28 ] . in which they claim to find a new source of ultrasensitivity called : _ de novo _ ultrasensitivity generation , which they explained in terms of the presence of intermediate elements in the kinase - cascade architecture . with our framework we found that the ultrasensitivity did not come from a cascading effect but from a ` hidden ' first - order ultrasensitivity process in the one of the cascade s layer . from a general perspective , our framework serves to understand the origin of ultrasensitivity in multilayer structures , which could be a powerful tool in the designing of synthetic systems . in particular , in ultrasensitive module designing , our method can be used to guide the tuning of both the module itself and the coupling with the system , in order to set the working range in the region of maximal local ultrasensitivity . \1 . ferrell je , ha sh . ultrasensitivity part iii : cascades , bistable switches , and oscillators . trends in biochemical sciences . 2014 dec 31;39(12):612 - 8 . zhang q , sudin bhattacharya and melvin e. andersen ultrasensitive response motifs : basic amplifiers in molecular signalling networks open biol . 2013 3 130031 + 3 . angeli d , ferrell j e and sontag e d detection of multistability , bifurcations , and hysteresis in a large class of biological positive - feedback systems . pnas 2004 101 ( 7 ) 1822 - 7 + 4 . ferrell j e and xiong w bistability in cell signaling : how to make continuous processes discontinuous , and reversible processes irreversible chaos 2001 11 ( 1 ) 227 - 36 + 5 . srividhya j , li y , and pomerening jr open cascades as simple solutions to providing ultrasensitivity and adaptation in cellular signaling phys biol . 2011 8(4):046005 6 . kholodenko b n negative feedback and ultrasen- sitivity can bring about oscillations in the mitogen - acti- vated protein kinase cascades . eur j biochem 2000 267 15831588 . buchler n e and louis m. , molecular titration and ultrasensitivity in regulatory networks . 2008384 1106 - 19 . buchler n e and cross f r. , protein sequestration generates a flexible ultrasensitive response in a genetic network . mol syst biol . 20095 272 + 9 . goldbeter a and koshland d e an amplified sensitivity arising from covalent modification in biological systems pnas 1981 78 11 6840 - 6844 + 10 . ferrell j e and ha s h ultrasensitivity part ii : multisite phosphorylation , stoichiometric inhibitors , and positive feedback trends in biochemical sciences 2014 39 ( 11 ) : 556569 + 11 . ferrell j e tripping the switch fantastic : how a protein kinase cascade can convert graded inputs into switch - 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y f and ferrell j e ultrasensitivity in the mitogen - activated protein kinase cascade proc . natl . 1996 93 10078 - 10083 + 19 . brown gc , hoek j b , kholodenko b n , why do protein kinase cascades have more than one level ? , trends biochem sci . 1997 22 ( 8):288 . + 20 . ferrell j e how responses get more switch - like as you move down a protein kinase cascade trends biochem sci . 1997 22 ( 8):288 - 9 . altszyler e , ventura a , colman - lerner a , chernomoretz a. impact of upstream and downstream constraints on a signaling module s ultrasensitivity . physical biology . 2014 oct 14;11(6):066003 . bluthgen n , bruggeman f j , legewie s , herzel h , westerhoff h v and kholodenko b n effects of sequestration on signal transduction cascades . febs journal 2006 273 895 - 906 + 23 . e and slepchenko b m on sensitivity amplification in intracellular signaling cascades phys . biol . 2008 5 036004 - 12 + 24 . wang g , zhang m. tunable ultrasensitivity : functional decoupling and biological insights . scientific reports . 2016 6 20345 + 25 . ventura a c , sepulchre j a and merajver s d a hidden feedback in signaling cascades is revealed , plos comput biol . 2008 4(3):e1000041 + 26 . del vecchio d , ninfa a and sontag e modular cell biology : retroactivity and insulation . mol sys biol 2008 4 161 p161 . ventura a c , jiang p , van wassenhove l , del vecchio d , merajver s d , and ninfa a j signaling properties of a covalent modification cycle are altered by a downstream target pnas 2010 107 ( 22 ) 10032 - 10037 + 28 . oshaughnessy e c , palani s , collins j j , sarkar c a tunable signal processing in synthetic map kinase cascades cell 2011 7 144(1):119 - 31 + 29 . bluthgen n and herzel h map - kinase - cascade : switch , amplifier or feedback controller ? 2nd workshop on computation of biochemical pathways and genetic networks - berlin : logos - verlag 2001 55 - 62 + 30 . bluthgen n and herzel h how robust are switches in intracellular signaling cascades journal of theoretical biology 2003 225 293 - 300 + the _ hill s input working range _ delimits the region of inputs over which the mean local - ultrasensitivity value is calculated ( equation 9 ) . it is thus a significant parameter to get insights about the overall ultrasensitivity of multilayer structures . in what follows , we show that the actual location of this relevant interval depends on the way in which cascade layers are coupled . let s start by considering two coupled ultrasensitive modules . two different regimes could be identified depending whether the upstream module s maximum output was or was nt large enough to fully activate the downstream unit : a. in the first case i.e. when @xmath20 ( see fig 1d ) , @xmath18 and @xmath19 are equal to the @xmath53 and @xmath54 levels respectively . therefore , when coupled to module-1 , the hill input s working range of module-2 would not differ from the isolated case , and would equal the hill coefficient of this module acting in isolation : i.e @xmath55 . in addition , it can be seen that the hill input s working range of module-1 tends to be located at the low input - values region for increasing levels of the ratio @xmath56 . in this region the response coefficient of the hill functions achieve the highest values , with @xmath57 ( see fig 1c ) , thus , when calculating the average logarithmic gain , @xmath58 , we would obtain @xmath59 . finally , following equation 8 we get @xmath60 . it can be seen that the cascade behaves multiplicatively in this regime , which is consistent with ferrell s results [ 1 ] b. when the upstream module s maximal output is not enough to fully activate the downstream module , i.e. @xmath62 we will have different behaviors depending on module-2 ultrasensitivity : first let s see what happens in a case in which module-2 dose - response has @xmath63 , thus displaying a linear behavior at low input values ( see fig 6 ) . [ figs1 ] the linearity produces that @xmath64 and @xmath65 ( @xmath18 and @xmath19 of the linear curve ) match the % 10 and % 90 of @xmath66 , thus @xmath67 and @xmath68 coincide with @xmath35 and @xmath36 centering the hill working range around @xmath69 . furthermore , as a result of applying equation 4 , the system s behavior lies on module-1 and shows a multiplicative behavior , given the linearity of module-2 . @xmath70 on the other hand , it can be seen that if @xmath71 , then module-2 dose response has a power - law behavior at low input values ( see fig 6 ) . in this case , the non - linearity produces a shift in modules-2 working range toward higher values , which centers modules-1 hill working range in input values higher than @xmath69 . furthermore , given that @xmath58 decreases with @xmath72 , the modules-1 working range shift produces @xmath73 , then , @xmath74 finally we get that if @xmath20 and @xmath75 , the system shows a submultiplicative behavior ( consistent with ferrell s results [ 20 ] ) , which arises from a setting of modules-1 working range in a region with low local ultrasensitivity . although we show that the submultiplicativity occurs in the limit of @xmath20 , the same argument is still valid for @xmath62 . of course , the ultimate consequences in the coupling of two ultrasensitive modules will depend on the particular mathematical details of the transfer functions under consideration . in this way , a completely qualitatively different behavior could be found for a system composed of two modules characterized by golbeter - koshland , gk , response functions [ 2 ] . gk functions appear in the mathematical characterization of covalent modification cycles ( such as phosphorylation - dephosphorylation ) , ubiquitous in cell signaling , operating in saturation . for cases where the phosphatases , but not the kinases , work in saturation , gk functions present input regions with response coefficients higher than their overall @xmath1 ( see fig 2a - c ) . their _ hill input s working range _ are thus located in the region of greatest local ultrasensitivity , these functions are able to contribute with more effective ultrasensitivity than their global ultrasensitivity . therefore , cascades involving gk functions may exhibit supra - multiplicative behavior . for this kind of systems , fig 2d shows that , under regime ( a ) ( when the modules-2 ec50 is much lower than the gk maximal output level , o1 max ) the modules-1 hill input s working range is set in its linear regime ( @xmath76 ) , and the gk function does not contribute to the overall system s ultrasensitivity . on the other hand , fig 2e shows that the @xmath42 to @xmath43 relation can be tuned in order to set modules-1 hill input s working range in its most ultrasensitive region , producing an effective ultrasensitivity contribution , @xmath33 , even larger than the ultrasensitivity of the gk curve in isolation ( i.e. @xmath77 ) , resulting in supra - multiplicative behavior @xmath78 . our analysis highlights the impact of the detailed functional form of a module s response curve on the overall system s ultrasensitivity in cascade architectures . local sensitivity features of the involved transfer functions are of the uttermost importance in this kind of setting and could be at the core of non - trivial phenomenology . ferrell j e how responses get more switch - like as you move down a protein kinase cascade trends biochem sci . 1997 22 ( 8):288 - 9 . + 2 . goldbeter a and koshland d e an amplified sensitivity arising from covalent modification in biological systems pnas 1981 78 11 6840 - 6844 in fig 7a can be appreciated that sequestration effects were actually negligible for the mapkk and mapk layers , given that the input - output relation of the composition of isolated functions ( non - seq ) and embedded modules ( seq ) coincided . only for the mapkkk layer , sequestration effects produced a shift between both curves . noticeably , the corresponding hill working ranges changed accordingly , and the resulting overall ultrasensitivity did not get affected at all . hence we could finally conclude that in this particular system , even sequestration effects existed , the overall sub - multiplicative behavior was only due to a resetting of the _ hill input s working range _ for the first and second levels of the cascade . and @xmath15 of each layer of the original cascade , while the red solid vertical lines show the @xmath14 and @xmath15 of each layers in the system given by the composition of the modules in isolation ( @xmath47 , see eq [ 6 ] ) . it worth noting that the response curves that each module sustain in the non - sequestration scenario ( panels b - c ) will coincide with the isolated curves , with the exception that are limited in the spanned input region.,scaledwidth=100.0% ] [ figs2 ] the goldbeter - koshland function [ 1 ] is defined as @xmath79 in order to center the g - k function , we multiply the independent variable for a scale factor @xmath80 , @xmath81 , where @xmath80 was set in order to make the ec50 of g - k function coincides with the ec50 of erkpp curve

ultrasensitive response motifs , which are capable of converting graded stimulus in binary responses , are very well - conserved in signal transduction networks . although it has been shown that a cascade arrangement of multiple ultrasensitive modules can produce an enhancement of the system s ultrasensitivity , how the combination of layers affects the cascade s ultrasensitivity remains an open - ended question for the general case . here we have developed a methodology that allowed us to quantify the effective contribution of each module to the overall cascade s ultrasensitivity and to determine the impact of sequestration effects in the overall system s ultrasensitivity . the proposed analysis framework provided a natural link between global and local ultrasensitivity descriptors and was particularly well - suited to study the ultrasensitivity in map kinase cascades . we used our methodology to revisit oshaughnessy et al . tunable synthetic mapk cascade , in which they claim to have found a new source of ultrasensitivity : ultrasensitivity generated de novo , which arises due to cascade structure itself . in this respect , we showed that the system s ultrasensitivity in its single - step cascade did not come from a cascading effect but from a ` hidden ' first - order ultrasensitivity process in one of the cascade s layer . our analysis also highlighted the impact of the detailed functional form of a module s response curve on the overall system s ultrasensitivity in cascade architectures . local sensitivity features of the involved transfer functions were found to be of the uttermost importance in this kind of setting and could be at the core of non - trivial phenomenology associated to ultrasensitive motifs .

1608.08007

after carrying out a fourier transform and a multipole decomposition , the radial and time parts of the retarded green function for linear fields on a schwarzschild black hole can be written as @xmath15 where @xmath16 , @xmath17 is the multipole number , @xmath18 , @xmath19 and @xmath20 is the wronskian of the two functions @xmath21 and @xmath22 . these functions are linearly independent solutions of the radial ode @xmath23\right\ } \psi_{\ell}=0 \ ] ] where @xmath24 and @xmath25 . the solutions are uniquely determined when @xmath26 by the boundary conditions : @xmath27 as @xmath28 and @xmath29 as @xmath30 . the behaviour of the radial potential at infinity leads to a branch cut in the radial solution @xmath22 @xcite . the contour of integration in eq.([eq : green ] ) can be deformed in the complex-@xmath3 plane @xcite yielding a contribution from a high - frequency arc , a series over the residues ( the qnms ) and a contribution from the branch cut along the nia : @xmath31 where the bcms are @xmath32 with @xmath33 where @xmath34 $ ] is the discontinuity of @xmath22 across the branch cut . we present here methods for the analytic calculation of the bcms . we calculate @xmath21 using the jaff series , eq.39 @xcite . the coefficients of this series , which we denote by @xmath35 , satisfy a 3-term recurrence relation . we calculate @xmath22 using the series in eq.73 @xcite , which is in terms of the confluent hypergeometric @xmath36-function and the coefficients @xmath35 . this series has seldom been used and one must be aware that , in order for @xmath22 to satisfy the correct boundary condition , we must set @xmath37 , which itself has a branch cut . to find an expression for @xmath38 _ on _ the nia we exploit this series by combining it with the known behavior of the @xmath36-function across its branch cut : @xmath39 where we are taking the principal branch both for @xmath40 and for the @xmath36-function . in order to check the convergence of this series , we require the behaviour for large-@xmath41 of the coefficients @xmath35 . using the _ birkhoff series _ as in app.b @xcite , we find the leading order @xmath42 ( we have calculated up to four orders higher in @xcite ) as @xmath43 . we note that this behaviour corrects leaver s eq.46 @xcite in the power ` @xmath44 ' instead of ` @xmath45 ' . the integral test then shows that the series ( [ eq : leaver - liu series for deltagt ] ) converges for any @xmath46 . although convergent , the usefulness of ( [ eq : leaver - liu series for deltagt ] ) at small-@xmath47 is limited since convergence becomes slower as @xmath47 approaches 0 while , for large-@xmath47 , @xmath38 grows and oscillates for fixed @xmath48 and @xmath49 . therefore we complement our analytic method with asymptotic results for small and large @xmath8 . the small-@xmath50 asymptotics are based on an extension of the mst formalism @xcite . we start with the ansatz @xmath51 imposing eq.([eq : radial ode ] ) yields a 3-term recurrence relation for @xmath52 and requiring convergence as @xmath53 yields an equation for @xmath54 , that may readily be solved perturbatively in @xmath3 from starting values @xmath55 and @xmath56 . likewise for the coefficients @xmath52 , taking @xmath57 we obtain @xmath58 \\ & a_{2}^{\mu}= -\frac{(\ell+1-s)^2(\ell+2-s)^2}{4(\ell+1)(2\ell+1)(2\ell+3)^2}\omega^2+o\left(\omega^3\right ) \ ] ] while @xmath59 and @xmath60 are given by the corresponding terms with @xmath61 . ( apparent possible singularities in these coefficients are removable . ) the @xmath62 term in eq.([eq : f small - nu ] ) corresponds to page s eq.a.9 @xcite . to obtain higher order aymptotics we employ the barnes integral representation of the hypergeometric functions @xcite which involves a contour in the complex @xmath63-plane from @xmath64 to @xmath65 threading between the poles of @xmath66 , @xmath67 and @xmath68 . as @xmath69 double poles arise at the non - negative integers from 0 to @xmath70 , however we may move the contour to the right of all these ambient double poles picking up polynomials in @xmath48 with coefficients readily expanded in powers of @xmath3 , leaving a regular contour which admits immediate expansion in powers of @xmath8 . by the method of mst we can also construct @xmath22 and hence determine @xmath71 and @xmath72 . for compactness , we only give the following small-@xmath8 expressions for the case @xmath10 ( cases @xmath73 and @xmath12 are presented in @xcite ) , @xmath74 \nonumber\\ & -\frac { ( -1)^{\ell } \pi } { 2^{2 \ell-1 } } \left(\frac{(2\ell+1)\ell!}{((2 \ell+1)!!)^2}\right)^2 \nu^{2\ell+3 } \left[\frac{4(15\ell^2 + 15\ell-11)}{(2 \ell-1)(2\ell+1 ) ( 2 \ell+3 ) } \left(\ln ( 2 \nu ) + h_{\ell}-4 h_{2 \ell}+ \gamma_e \right)\right . \\ & \left . -4 \left(-8 h_{\ell}{}^2 + 8 h_{\ell}+3 h_\ell^{(2)}+ 2h_{\infty}^{(2 ) } \right)+ \frac{512\ell^6 + 2016 \ell^5 + 1616 \ell^4 - 1472 \ell^3- 1128 \ell^2 + 722 \ell-59}{(2 \ell-1)^2 ( 2 \ell+1)^2 ( 2 \ell+3)^2}\right ] + o(\nu^{2\ell+3 } ) \nonumber\end{aligned}\ ] ] where @xmath75 is the @xmath17-th harmonic number of order @xmath48 . we note that the @xmath76 term at second - to - leading order originates both in @xmath71 and in @xmath72 . in fact , both functions possess a @xmath76 already at next - to - leading order for small-@xmath47 , but they cancel each other out in @xmath77 . similarly , the coefficient of a potential term in @xmath77 of order @xmath78 is actually zero . let us now investigate the branch cut contribution to the black hole response to an initial perturbation given by the field @xmath79 and its time derivative @xmath80 at @xmath81 : @xmath82 \nonumber\end{aligned}\ ] ] we obtain the asymptotics of the response for late times @xmath83 using eqs.([eq : deltag in terms of deltag ] ) and ( [ eq : f small - nu])([q / w^2 s=0 gral l ] ) . we note the following features . the orders @xmath84 and @xmath85 in the bcms @xmath38 yield tail terms behaving like @xmath86 and @xmath87 , respectively . we have thus generalized leaver s eq.56 @xcite to finite values of @xmath48 . furthermore , eq.56 @xcite is an expression containing the leading orders from @xmath79 and from @xmath80 . however , the next - to - leading order from @xmath80 will be of the same order as the leading - order from @xmath79 . in our approach above we consistently give a series in small-@xmath47 , thus obtaining the correct next - to - leading order term for large-@xmath88 in the power - law tail . importantly , we also obtain the following two orders in the perturbation response : @xmath89 and @xmath90 . we note the interesting @xmath89 behaviour , which is due to the @xmath91 term in eq.([q / w^2 s=0 gral l ] ) . to the best of our knowledge , this is the first time in the literature that any of the above features has been obtained . the logarithmic behaviour is not completely surprising given the calculations in @xcite . however , one may be led to a wrong logarithmic behaviour @xcite if the calculations are not performed in detail . in order to exemplify our results , we give the explicit asymptotic behaviour in the case @xmath10 and @xmath92 and initial data @xmath93 and @xmath94 , with @xmath95 . the perturbation response due to the branch cut at @xmath96 at late times is given by @xmath97t^{-7 } + o\left(t^{-7}\right ) . \nonumber\end{aligned}\ ] ] fig . [ fig : perturbation response ] shows that these asymptotics are in excellent agreement with a numerical solution of the wave equation . to the gaussian described above eq.([eq : pert_asymp ] ) compared to the late - time asymptotics . solid - red : numerical solution ; dashed - black : eq.([eq : pert_asymp ] ) ; lower curves : numerical solution minus the first ( green ) , first 2 ( blue ) and first 4 ( cyan ) terms in eq.([eq : pert_asymp ] ) . ] at large-@xmath47 , we obtain the asymptotics @xmath98},&\!\!\ ! s=0,2\\ \dfrac{-\sqrt{\pi}i\lambda\sin(2\pi{\nu})}{{\nu}^{3/2}},&\!\!\!s=1 \end{cases } \\ & { f_{\ell}}(r ,- i\nu ) \sim \begin{cases } ( -1)^{s/2 } e^{\nu r_*}+\sin(2\pi{\nu})e^{-\nu r _ * } , & s=0,2\\ \dfrac{\sqrt{\pi}\lambda}{2{\nu}^{1/2}\sin(2\pi{\nu})}e^{\nu r_*}+e^{-\nu r_*},&s=1 \end{cases } \nonumber\end{aligned}\ ] ] these asymptotics show a divergence in @xmath99 when @xmath100 . they also lead to a divergence in the perturbation response at fixed @xmath101 and @xmath48 for a non - compact gaussian as initial data . both types of divergences are expected to cancel out with the other contributions to the green function . we have thus provided a complete account of the bcms for all frequencies along the nia ; the behaviour is illustrated in fig.[fig : deltag s = l=2 and s=0,l=1 ] . as a function of @xmath47 for @xmath102 and @xmath103 . ( a ) using eq.([eq : leaver - liu series for deltagt ] ) ; dashed - green : @xmath104 , continuous - blue : @xmath10 , @xmath92 , dot - dashed - orange : @xmath105 . note the interesting behaviour near the algebraically - special frequency @xcite at @xmath106 for @xmath107 . ( b ) @xmath10 , @xmath92 for small @xmath47 ; continuous - blue using eq.([eq : leaver - liu series for deltagt ] ) ; dashed - red using eq.([q / w^2 s=0 gral l ] ) to @xmath108 see @xcite . ( c ) @xmath10 , @xmath92 for large @xmath47 ; continuous - blue using eq.([eq : leaver - liu series for deltagt ] ) ; dashed - red using the asymptotics of eq.([eq : large_nu ] ) . , width=309 ] [ cols="^,^ " , ] we present here an analysis for large-@xmath1 of the electromagnetic qnms . we may find solutions of eq.([eq : radial ode ] ) valid for fixed @xmath109 as expansion in powers of @xmath110 as @xmath111 , @xmath112 , starting with the two independent solutions : @xmath113 and @xmath114 . we may express any higher order solution in terms of the @xmath115-order green function as @xmath116\frac{\psi_i^{({k}-1)}(u)}{\sqrt{\nu } } \right . \nonumber \\ & \left.- ( 2u)^{1/2}\left[4d^2 -2d - \lambda\right]\frac{\psi_i^{({k}-2)}(u)}{{\nu } } \right\}\end{aligned}\ ] ] where @xmath117 . from this expression , it follows that @xmath118 \nonumber \end{aligned}\ ] ] where @xmath119 . in addition , for @xmath120 , @xmath121 and @xmath122 $ ] are both real . it follows that along @xmath123 , up to power law corrections , @xmath124 equating asymptotic expansions at @xmath125 yields @xmath126 , @xmath127 and also serves to determine @xmath128 ( except when @xmath129 for @xmath130 , which do not contribute to the qnm condition ) . by matching the @xmath134 to wkb solutions along @xmath120 and @xmath135 we are able to find large-@xmath8 asymptotics for @xmath22 . also , we may use the exact monodromy condition , @xmath136 , to obtain large-@xmath47 asymptotics for @xmath21 . the asymptotic qnm condition ( @xmath137 ) in the 4th quadrant then becomes @xmath138 it is straightforward to find the qnm frequencies to _ arbitrary _ order in @xmath1 in terms of the @xmath132 by systematically solving eq.([eq : qnm cond ] ) . explicitly , using the values in eq.([eq : alpha values ] ) , we have @xmath139}{96n^{5/2 } } + o\!\left({n^{-3}}\right ) \nonumber\end{aligned}\ ] ] it is remarkable that the terms in the expansion show the behaviour @xmath140 to all orders . in fig . [ fig : qnm numeric closed form ] we compare these asymptotics with the numerical data in @xcite . in @xcite we apply the method used to obtain eq.([eq : qnm s=1 ] ) to the cases @xmath10 and @xmath12 and we obtain the corresponding qnm frequencies up to order @xmath141 and have agreement with @xcite . we are thankful to sam dolan and , particularly , to barry wardell for helpful discussions . thanks luis lehner and the perimeter institute for theoretical physics for hospitality and financial support . m. c. is supported by a ircset - marie curie international mobility fellowship in science , engineering and technology . a.o . acknowledges support from science foundation ireland under grant no 10/rfp / phy2847 . 37ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12 & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty @noop * * , ( ) link:\doibase 10.1103/physrevlett.74.2414 [ * * , ( ) ] , @noop * * , ( ) link:\doibase 10.1088/0264 - 9381/26/16/163001 [ * * , ( ) ] , @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1103/physrevd.78.044006 [ * * , ( ) ] , link:\doibase 10.1103/physrevd.62.024027 [ * * , ( ) ] , @noop * * , ( ) , @noop * * , ( ) link:\doibase 10.1016/0370 - 2693(95)01148-j [ * * , ( ) ] , link:\doibase 10.1103/physrevlett.81.4293 [ * * , ( ) ] , link:\doibase 10.1103/physrevlett.90.081301 [ * * , ( ) ] , link:\doibase 10.1103/physrevlett.100.141301 [ * * , ( ) ] , link:\doibase 10.1103/physrevd.84.084031 [ * * , ( ) ] , @noop * * , ( ) @noop ( ) , link:\doibase 10.1063/1.1626805 [ * * , ( ) ] , @noop * * , ( ) , link:\doibase 10.1016/j.physletb.2007.04.068 [ * * , ( ) ] link:\doibase 10.1103/physrevd.69.044004 [ * * , ( ) ] , link:\doibase 10.1103/physrevd.62.064009 [ * * , ( ) ] , link:\doibase 10.1103/physrevd.5.2419 [ * * , ( ) ] link:\doibase 10.1103/physrevd.5.2439 [ * * , ( ) ] @noop ( ) , @noop * * , ( ) , link:\doibase 10.1088/0264 - 9381/28/9/094021 [ * * , ( ) ] , link:\doibase 10.1038/nphys1907 [ * * , ( ) ] , @noop @noop * * , ( ) , @noop * * , ( ) link:\doibase 10.1103/physrevd.52.2118 [ * * , ( ) ] , @noop _ _ ( , ) @noop * * , ( ) @noop * * , ( ) , @noop _ _ ( , , ) @noop * * , ( ) , @noop _ _ ( , ) @noop

linear field perturbations of a black hole are described by the green function of the wave equation that they obey . after fourier decomposing the green function , its two natural contributions are given by poles ( quasinormal modes ) and a largely unexplored branch cut in the complex - frequency plane . we present new analytic methods for calculating the branch cut on a schwarzschild black hole for _ arbitrary _ values of the frequency . the branch cut yields a power - law tail decay for late times in the response of a black hole to an initial perturbation . we determine explicitly the first three orders in the power - law and show that the branch cut also yields a new logarithmic behaviour @xmath0 for late times . before the tail sets in , the quasinormal modes dominate the black hole response . for electromagnetic perturbations , the quasinormal mode frequencies approach the branch cut at large overtone index @xmath1 . we determine these frequencies up to @xmath2 and , formally , to _ arbitrary _ order . highly - damped quasinormal modes are of particular interest in that they have been linked to quantum properties of black holes . the retarded green function for linear field perturbations in black hole spacetimes is of central physical importance in classical and quantum gravity . an understanding of the make - up of the green function is obtained by performing a fourier transform , thus yielding an integration just above the real - frequency ( @xmath3 ) axis . in his seminal paper , leaver @xcite deformed this real-@xmath3 integration in the case of schwarzschild spacetime into a contour on the complex-@xmath3 plane . he thus unraveled three contributions making up the green function : ( 1 ) a high - frequency arc , ( 2 ) a series over poles of the green function ( quasinormal modes qnms ) , and ( 3 ) an integral of modes around a branch cut originating at @xmath4 and extending down the negative imaginary axis ( nia ) , which we refer to as branch cut modes ( bcms ) . the three contributions dominate the black hole response to an initial perturbation at different time regimes . the high - frequency arc yields a ` direct ' contribution which is expected to vanish after a certain finite time @xcite . the qnm contribution to the green function dominates the black hole response during ` intermediate ' times and it has been extensively investigated ( e.g. , @xcite for a review ) . at ` late times ' the qnm contribution decays exponentially , with a decay rate given by the overtone number @xmath5 . qnms have also triggered numerous interpretations in different contexts in classical and quantum physics , ranging from astrophysical ` ringdown ' @xcite to hawking radiation @xcite , the ` gauge - gravity duality ' ( @xcite for schwarzschild black holes which are asymptotically anti - de sitter and @xcite for asymptotically flat ones ) , black hole area quantization @xcite and structure of spacetime at the shortest length scales @xcite . the quantum interpretations are given in the highly - damped limit , i.e , for large @xmath1 . the highly - damped qnm frequencies in schwarzschild have been calculated up to next - to - leading order in . despite all the efforts , the leading order of the real part of the frequencies for electromagnetic perturbations has remained elusive ( only in @xcite they find numerical indications that it goes like @xmath6 ) . the contribution from the bcms , on the other hand , remains largely unexplored . the technical difficulties of its analysis mean that most of the studies have been constrained to large radial coordinate as well as small @xmath7 along the nia . an exception is a large-@xmath8 asymptotic analysis of the bcms in @xcite ( and near the algebraically - special frequency in @xcite ) solely for gravitational perturbations . the small-@xmath8 bcms are known to give rise to a power - law tail decay at ` late ' times of an initial perturbation @xcite . in general , however , there is an appreciable time interval between when the qnm contribution becomes negligible and when the power - law tail starts @xcite . the calculation of the bcms for general values of the frequency ( i.e. , not in the asymptotically small nor large regimes ) , to the best of our knowledge has only been attempted in @xcite where the radial functions were calculated off the nia via a numerical integration of the radial ode ( [ eq : radial ode ] ) followed by extrapolation to the nia , and only for the gravitational case . in this letter we present the following new results : a new analytic method for the calculation of the bcms directly _ on _ the nia and valid for _ any _ value of @xmath8 . in particular , this method provides analytic access for the first time to the ` mid'-@xmath8 regime . a consistent expansion up to @xmath9th order for small-@xmath8 of the bcms for arbitrary value of the radial coordinate . we explicitly derive a new logarithmic behaviour @xmath0 at late times . a large-@xmath8 asymptotic analysis of the bcms . it shows a formal divergence , which is expected to be cancelled out by the other contributions to the green function . a new asymptotic analysis for large-@xmath1 of the electromagnetic qnms . the analysis is formally valid up to _ arbitrary _ order in @xmath1 ; we explicitly calculate the corresponding frequencies up to @xmath2 . methods in ( 1)(3 ) provide the first full analytic account of the bcms and they are valid for any spin @xmath10 ( scalar ) , @xmath11 ( electromagnetic ) and @xmath12 ( gravitational ) of the field perturbation . for the qnm calculation we focus on spin-1 as this is the least well understood case . we note that spin-1 perturbations are acquiring increasing importance @xcite , although it is expected that only the lowest overtones of the qnms are astrophysically relevant . we present details in @xcite and @xcite . we take units @xmath13 , where @xmath14 is the mass of the black hole .

1205.6592

observational manifestations of magnetic fields in intermediate- and high - mass stars with radiative envelopes differ considerably from the magnetism of solar - type and low - mass stars . as directly observed for the sun and inferred for many late - type stars , vigorous envelope convection and differential rotation give rise to ubiquitous intermittent magnetic fields , which evolve on relatively short time - scales and generally exhibit complex surface topologies . although details of the dynamo operation in late - type stars , in particular the relative importance of the convective and tachocline dynamo mechanisms is a matter of debate @xcite and probably depends on the position in the h - r diagram , it is understood that essentially every cool star is magnetic . chromospheric and x - ray emission and surface temperature inhomogeneities , which are responsible for characteristic photometric variability , provide an indirect evidence of the surface magnetic fields in cool stars . in contrast , stars hotter than about mid - f spectral type and more massive than @xmath01.5@xmath1 are believed to lack a sizable convective zone near the surface and therefore are incapable of generating observable magnetic fields through a dynamo mechanism . nevertheless , about 10% of o , b , and a stars exhibit very strong ( up to 30 kg ) , globally organized ( axisymmetric and mostly dipolar - like ) magnetic fields that appear to show no intrinsic temporal variability whatsoever . this phenomenon is usually attributed to the so - called fossil stellar magnetism a hitherto unknown process ( possibly related to initial conditions of stellar formation or early stellar mergers ) by which a fraction of early - type stars become magnetic early in their evolutionary history . by far the most numerous among the early - type magnetic stars are the a and b magnetic chemically peculiar ( ap / bp ) stars . these stars were the first objects outside our solar system in which the presence of magnetic field was discovered @xcite . ap / bp stars are distinguished by slow rotation @xcite and are easy to recognize spectroscopically by the abnormal line strengths of heavy elements in their absorption spectra . these spectral peculiarities are related to distinctly non - solar surface chemical composition of these stars and non - uniform horizontal ( e.g. * ? ? ? * ; * ? ? ? * ) and vertical distributions of chemical elements ( e.g. * ? ? ? * ; * ? ? ? these chemical structures are presumably formed by the magnetically - controlled atomic diffusion @xcite operating in stable atmospheres of these stars . the chemical spot distributions and magnetic field topologies of ap stars remain constant ( frozen in the atmosphere ) . yet , all these stars show a pronounced and strictly periodic ( with periods from 0.5 d to many decades ) spectroscopic , magnetic and photometric variability due to rotational modulation of the aspect angle at which stellar surface is seen by a distant observer . a subset of cool magnetic ap stars rapidly oscillating ap ( roap ) stars also varies on much shorter time scales ( @xmath010 min ) due to the presence of @xmath2-mode oscillations aligned with the magnetic field @xcite . a large field strength and lack of intrinsic variability facilitates detailed studies of the field topologies of individual magnetic ap stars and statistical analyses of large stellar samples . in this review i outline common methodologies applied to detecting and modeling surface magnetic fields in early - type stars and summarize main observational results . closely related contributions to this volume include an overview of massive - star magnetism ( wade , grunhut ) , a discussion of the stability and interior structure of fossil magnetic fields ( braithwaite ) , and an assessment of the chemical peculiarities and magnetism of pre - main sequence a and b stars ( folsom ) . with a few exceptions , investigations of the magnetism of cool stars have to rely on high - resolution spectropolarimetry and to engage in a non - trivial interpretation of the complex polarization signatures inside spectral lines in order to characterize the field topologies @xcite . in contrast , a key advantage of the magnetic field studies of early - type stars with stable global fields is availability of a wide selection of magnetic observables that are simple to derive and interpret , but are still suitable for a coarse analysis of the surface magnetic field structure . the simplest approach to detecting the presence of the field in early - type stars is to perform spectroscopic observation with a zeeman analyzer equipped with a quarter - wave retarder plate and a beamsplitter . the resulting pair of left- and right - hand circularly polarized spectra will exhibit a shift proportional to the land factors of individual spectral lines and to _ the mean longitudinal magnetic field _ the line - of - sight field component averaged over the stellar disk . various versions of this longitudinal field diagnostic technique have been applied by @xcite , @xcite , and @xcite to medium - resolution spectra . @xcite and @xcite have extended it to , respectively , photopolarimetric and low - resolution spectropolarimetric measurements of polarization in the wings of hydrogen lines . the mean longitudinal magnetic field represents a particular example of an integral measurement derived from a moment of stokes @xmath3 profile ( the first moment in this case ) . @xcite have generalized the moment technique to other stokes @xmath4 and @xmath3 profile moments . in practice , only _ the mean quadratic field _ ( the second moment of stokes @xmath4 ) and _ crossover _ ( the second moment of stokes @xmath3 ) , in addition to longitudinal field , were systematically studied by mathys and collaborators using medium - resolution observations of ap stars ( e.g. * ? ? ? observations are offset vertically . clear circular and linear polarization signatures are evident in many individual spectral lines . detailed analysis of these observations has been published by @xcite . ] the three aforementioned magnetic observables can be related to the disk - averaged properties of stellar magnetic field under a number of simplifying and restrictive assumptions ( weak lines , weak field , no chemical spots ) . at the same time , an entirely assumption - free method to diagnose magnetic fields in early - type stars is to measure a separation of the resolved zeeman - split components in the intensity profiles of magnetically sensitive spectral lines . the resulting _ mean field modulus _ measurements have been obtained for many ap stars showing strong magnetic fields in combination with a particularly slow rotation @xcite . a different approach can be applied to diagnose the transverse magnetic field components which give rise to linear polarization . cooler ap stars exhibit measurable _ net linear polarization _ due to differential saturation of the @xmath5 and @xmath6 components of the strong spectral lines . the resulting net @xmath7 and @xmath8 signals can be detected with a broad - band photopolarimetric technique and related to disk - averaged characteristics of the transverse magnetic field @xcite . with the advent of high - resolution spectropolarimeters at the 24 m - class telescopes it became possible to directly record and interpret the circular and linear polarization signatures in individual spectral lines . a multi - line lsd ( least - squares deconvolution ) technique @xcite is often used in conjunction with such observations to obtain very high signal - to - noise ( s / n ) ratio mean intensity and polarization profiles . lsd analysis greatly facilitates detection of weak magnetic fields ( e.g. * ? ? ? * ) and allows to derive the mean longitudinal field , net linear polarization and other profile moments for direct comparison with historic studies . the first high - resolution full stokes vector investigations were carried out for ap stars with now decommissioned musicos spectropolarimeter @xcite . more recent analyses took advantage of espadons , narval @xcite and harpspol @xcite instruments . an example of exceptionally high quality ( @xmath9 , @xmath10 , 16 rotation phases , coverage of 38006910 wavelength region ; see @xcite for details ) harpspol stokes spectra of the roap star hd24712 is illustrated in fig . these observations represent the highest quality full stokes vector spectra available for any star other than the sun while covering a much wider wavelength domain than typical for solar polarization observations . fitting the phase curves of one or several magnetic observables constitutes the basic method of constraining the stellar magnetic field parameters . a limited information content of integral observables and their relatively simple sinusoidal variation in most ap stars justifies describing the stellar magnetic field topology with a small number of free parameters . by far the most common approximation is a simple rigidly rotating dipolar field geometry @xcite , characterized by an inclination angle of the stellar rotational axis @xmath11 , magnetic obliquity @xmath12 , and a polar field strength @xmath13 . observations of the phase variation of the mean longitudinal magnetic field alone allows one to constrain @xmath13 and @xmath12 , provided that @xmath11 is known and not too close to 90@xmath14 . in the latter case longitudinal field measurements constrain only the product @xmath15 . occasional deviations of the longitudinal field curves from the sinusoidal shape expected for a dipolar field and the requirement to fit simultaneous measurements of the longitudinal and mean surface fields led to the development of more complex field geometry models , described with additional free parameters . different low - order multipolar field parameterizations have been considered in the literature . this included a dipolar field offset along its axis ( e.g. * ? ? ? * ) , an arbitrary offset dipole @xcite , an axisymmetric combination of the aligned dipole , quadrupole , and octupole components @xcite , a general non - axisymmetric quadrupolar field @xcite , and a potential field geometry formed by a superposition of an arbitrary number of point - like magnetic sources @xcite . the choice between these different multipolar parameterizations is typically subjective . stellar observations themselves frequently do not allow one to make a clear - cut distinction between multipolar models established in the framework of different parameterizations , even when several magnetic observables are available for a given star . indeed , it was demonstrated that the same set of observed magnetic curves can be successfully interpreted with very different actual surface magnetic field distributions , depending on which multipolar parameterization is used @xcite . nevertheless , systematic applications of multipolar modeling to a large number of stars allowed to reach interesting conclusions . using centered dipole fits to the longitudinal field curves , @xcite established the existence of a lower field limit of @xmath16 g for ap stars . this threshold of global fossil field strength is likely to be of fundamental importance for understanding the magnetism of intermediate - mass stars ( see lignires , this volume ) . among other notable findings one can mention the work by @xcite , who demonstrated that magnetic field axis tends to be more aligned with the stellar rotation axis for ap stars with long ( @xmath17 d ) rotation periods . @xcite confirmed this result using a different multipolar field parameterization . they also found a certain dependence of the relative orientation of the dipolar and quadrupolar components on the stellar rotation rate . both studies fitted the observed curves of the mean field modulus , longitudinal field , crossover , and quadratic field . despite the overall statistical agreement , in many individual cases the surface field maps resulting from the application of landstreet s and bagnulo s parameterizations appear very different for the same stars . furthermore , some of the observables are poorly reproduced by either multipolar model , which can be ascribed to the presence of more complex field structures , an unaccounted influence of chemical abundance spots or to shortcomings of the basic assumptions of the moment technique or to a combination of all these effects . some applications of the multipolar fitting procedure have incorporated detailed polarized radiative transfer ( prt ) modeling of the zeeman - split stokes @xmath4 profiles into solving for the surface field geometry @xcite . this approach enables an independent validation of the magnetic field topology and makes possible to deduce a schematic horizontal distribution of chemical spots in addition to studying the field geometry . however , a feedback of chemical spots on the magnetic observables is not taken into account by these studies . the most sophisticated and well - constrained non - axisymmetric multipolar models were developed by @xcite for the ap stars @xmath12 crb and 53 cam using all integral magnetic observables available from the stokes @xmath18 spectra together with the broad - band linear polarization measurements . however , even such detailed models do not guarantee a satisfactory description of the same stokes parameter spectra from which the magnetic observables are obtained . as found by @xcite , the multipolar models derived from magnetic observables provide a rough qualitative reproduction of the phase variation of the stokes @xmath3 profiles but sometimes fail entirely in matching the stokes @xmath19 signatures observed in individual metal lines . this problem points to a significant limitation of the multipolar models : a successful fit of the phase curves of all magnetic observables is often non - unique and is generally insufficient to guarantee an adequate description of the high - resolution polarization spectra . on the other hand , discrepancies between the model predictions and observations in partially successful multipolar fits can not be easily quantified in terms of deviations from the best - fit magnetic field geometry model . modeling of high - resolution observations of polarization signatures in individual spectral lines or in mean line profiles represents the ultimate method of extracting information about stellar magnetic field topologies . the wide - spread usage of the lsd processing of high - resolution polarization spectra stimulated development of various stokes @xmath3 profile fitting methodologies @xcite . these studies usually deal with weak - field early - type stars without prominent chemical spots ( e.g. magnetic massive or herbig ae / be stars , but not typical ap stars ) . the observed lsd profiles are approximated with a dipolar field topology , using a simplified analytical treatment of the polarized line formation . eventual variations caused by chemical spots or other surface features are not considered . so far , this modeling approach has been applied to a few stars , but it has a potential of providing constraints on magnetic field and other stellar parameters ( inclination , @xmath20 ) beyond what can be obtained from the longitudinal field curves @xcite . a more rigorous approach to the problem of finding the stellar surface magnetic field geometry from spectropolarimetric observations is to perform a full magnetic inversion known as magnetic ( zeeman ) doppler imaging ( mdi ) . in the mdi methodology developed by @xcite and @xcite the time - series observations in two or four stokes parameters are interpreted with detailed prt calculations , taking surface chemical inhomogeneities into account . simultaneous reconstruction of the magnetic field topology and chemical spot distributions is carried out by solving a regularized inverse problem . regularization limits the range of possible solutions and is needed to stabilize the iterative optimization process and to exclude small - scale surface structures not justified by the data . different versions of regularization have been applied for magnetic mapping of early - type stars . @xcite needed only the local tikhonov regularization ( imposing a correlation between neighboring surface pixels ) to achieve a reliable reconstruction of an arbitrary magnetic field map from full stokes vector spectra . however , a more restrictive multipolar regularization @xcite or a spherical harmonic field expansion @xcite is required to reconstruct a low - order multipolar field in the case when only stokes @xmath4 and @xmath3 observations are available . magnetic imaging of ap stars was recently coupled to a calculation of the atmospheric models that take into account horizontal variations of the atmospheric structure due to chemical spots @xcite . however , while the self - consistency between spots and atmospheric models is critical for magnetic mapping of cool stars @xcite and may be needed for mapping he inhomogeneities in he - rich stars , it is generally unnecessary for treating metal spots in ap stars . it should be emphasized that , in contrast to temperature or chemical spot imaging from intensity spectra , the mdi with polarization data is not limited to rapid rotators . polarization is strongly modulated by the stellar rotation even for magnetic stars with negligible @xmath20 . numerical experiments and studies of real stars demonstrated that this modulation is sufficient for recovering the field structure at least at the largest spatial scales ( e.g. * ? ? ? * ; * ? ? ? several studies @xcite applied mdi to high - quality time - series circular polarization spectra of several early - type magnetic stars . these stokes @xmath3 analyses did not find any major deviations from dipolar field topologies . at the same time , they found numerous examples of chemical spot maps showing diverse and complex distributions of chemical elements , often not correlating in any meaningful way to the underlying simple magnetic field geometry . these results are difficult to explain in the framework of atomic diffusion theory because the latter expects a very similar behavior for different elements and a definite correlation between the spots and magnetic field @xcite . a couple of other studies have attempted to examine the surface magnetic field structure in b - type stars with fields deviating significantly from dipolar geometry . a study of the he - peculiar star hd37776 @xcite has simultaneously interpreted a longitudinal field curve and moderate - resolution stokes @xmath3 spectra . this analysis inferred a decisively non - axisymmetric , complex and strong ( up to 30 kg locally ) magnetic field , but ruled out a record @xmath0100 kg quadrupolar field proposed for this star by previous longitudinal field curve fits . an mdi study of the early b - type star @xmath21 sco @xcite revealed the presence of weak complex magnetic field configuration , which exhibits no appreciable temporal variation @xcite . these two studies have proven that stable complex fields can exist in early - type stars and tend to be found in the most massive objects . despite these impressive mdi results , it should be kept in mind that the stokes @xmath18 inversions are intrinsically non - unique and their outcome is highly sensitive to additional constraints adopted to stabilize inversions . details of the magnetic field maps of hd37776 and @xmath21 sco are likely to change if different regularizations or different forms of spherical harmonic expansion are adopted for magnetic imaging . numerical tests of mdi inversions @xcite have concluded that reconstruction of stellar magnetic field topologies from the full stokes vector data should be considerably more reliable and resistant to cross - talk and non - uniqueness problems in comparison to the stokes @xmath18 imaging . in particular , a four stokes parameter inversion is able to recover the field structure without imposing any _ a priori _ constraints on the global field geometry . the first stokes @xmath22 mdi studies exploiting this possibility were carried out for the ap stars 53 cam @xcite and @xmath23 cvn @xcite using the musicos spectra collected by @xcite . both studies succeeded in reproducing the phase variation of the circular and linear polarization signatures in metal lines with the magnetic maps containing small - scale deviations from the dominant dipolar - like field component . interestingly , it was the inclusion of stokes @xmath19 profiles in the magnetic inversions that allowed to ascertain the presence of complex fields . the deviations from dipolar field configurations occur on much smaller spatial scales than can be described by a quadrupolar field . thus , the widely adopted dipole+quadrupole expansion may not be particularly useful for interpreting the stokes @xmath22 spectra of ap stars . cvn reconstructed by silvester et al . ( 2013 , submitted ) using mdi . these magnetic inversions were based on a set of high - resolution four stokes parameter spectra described by @xcite . the star is shown at five different rotation phases and an inclination angle @xmath24 . the spherical maps show a ) surface distribution of the magnetic field strength , b ) distribution of the radial magnetic field component , and c ) vector plot of magnetic field . the field complexity is evident , especially in the field strength map . ] cvn with a dipolar field model ( red thin line ) and a non - axisymmetric dipole plus quadrupole magnetic configuration ( blue thick line ) . it is evident that none of the models fits the @xmath19 observations at all rotational phases , thus requiring a more structured field geometry for this star ( see fig . [ fig2 ] ) . however , the presence of this small - scale field can not be recognized from the stokes @xmath18 data alone . ] the limited resolution , s / n ratio , and wavelength coverage of the musicos spectra allowed us to model the stokes @xmath22 profiles of only 23 saturated metal lines . a new generation of mdi studies is currently underway , taking advantage of the higher - quality stokes profile data available from espadons , narval , and harpspol spectropolarimeters . in particular , silvester et al . ( submitted ) have reassessed the magnetic field topology of @xmath23 cvn using new observations and extending the prt mdi modeling to a large number of weak and strong fe and cr lines . the resulting magnetic maps ( fig . [ fig2 ] ) show some dependence of the mapping results on the spectral line choice but generally demonstrate a very good agreement with the magnetic topology found by @xcite from observations obtained about 10 years earlier . thus , the small - scale magnetic features discovered in ap stars by mdi studies do not exhibit any temporal evolution . the new four stokes parameter observations of @xmath23 cvn also demonstrate very clearly the necessity of going beyond a low - order multipolar field model and the role of stokes @xmath19 spectra in recognizing this field complexity . as illustrated by fig . [ fig3 ] , an attempt to reproduce the observations of @xmath23 cvn with either a pure dipole or dipole+quadrupole geometries fails for stokes @xmath19 while providing a reasonable fit to stokes @xmath18 . despite an improved sensitivity to complex fields , not all four stokes parameter magnetic inversions point to local deviations from dipolar field topologies . the ongoing harpspol study of the cool ap star hd24712 does not reveal any significant non - dipolar field component ( @xcite and in this volume ) . the preliminary conclusion of this work is that the previous mdi analysis of this star carried out by @xcite using the stokes @xmath4 and @xmath3 data and assuming a dipolar field did not miss any significant aspects of the field topology . hd24712 is much cooler , hence less massive and/or older than 53 cam and @xmath23 cvn , raising an intriguing possibility of the mass and/or age dependence of the degree of magnetic field complexity in early - type stars . the presence of very complex non - dipolar fields only in relatively massive b - type magnetic stars ( hd37776 , @xmath21 sco ) agrees with this trend . modeling of magnetic fields in early - type stars with radiative envelopes has traditionally assumed low - order multipolar field configurations and focused on interpretation of the phase curves of the mean longitudinal magnetic field and other integral magnetic observables derived from moderate - quality circular polarization spectra . the studies based on this methodology have reached several important statistical conclusions about the nature of magnetic fields in ap and related stars . this includes the discovery of a lower threshold of the surface magnetic field strength , demonstration of an alignment of the magnetic and rotational axes in stars with long rotation periods , and confirmation of long - term stability stability of fossil magnetic fields . as observational material improves and high - resolution spectra in several stokes parameters become more widely available , the focus of magnetic field modeling studies gradually shifts to direct interpretation of the polarization signatures in spectral line profiles . the most powerful version of this methodology magnetic doppler imaging inversions based on detailed calculation of polarized spectra has been applied to a handful of ap stars observed in all four stokes parameters . these studies revealed significant local deviations from a dominant dipolar field topology , suggesting that the magnetic field structure of early - type stars with fossil fields is more complex than thought before and that the degree of field complexity increases with stellar mass . with the exception of a couple of massive stars with distinctly non - dipolar fields , these small - scale field structures could be recognized and fully characterized only using spectropolarimetric observations in all four stokes parameters .

stars with radiative envelopes , specifically the upper main sequence chemically peculiar ( ap ) stars , were among the first objects outside our solar system for which surface magnetic fields have been detected . currently magnetic ap stars remains the only class of stars for which high - resolution measurements of both linear and circular polarization in individual spectral lines are feasible . consequently , these stars provide unique opportunities to study the physics of polarized radiative transfer in stellar atmospheres , to analyze in detail stellar magnetic field topologies and their relation to starspots , and to test different methodologies of stellar magnetic field mapping . here i present an overview of different approaches to modeling the surface fields in magnetic a- and b - type stars . in particular , i summarize the ongoing efforts to interpret high - resolution full stokes vector spectra of these stars using magnetic doppler imaging . these studies reveal an unexpected complexity of the magnetic field geometries in some ap stars .

1310.2605

generally , the observed radio spectra of most pulsars can be modelled as a power law with negative spectral indices of about -1.8 ( @xcite ) . if a pulsar can be observed at frequencies low enough ( i.e. ) , it may also show a low - frequency turnover in its spectrum ( @xcite ; @xcite ) . on the other hand , lorimer ( 1995 ) mentioned three pulsars which have positive spectral indices in the frequency range 300 - 1600 mhz . later , maron ( 2000 ) re - examined spectra of these pulsars taking into account the data obtained at higher frequencies ( above 1.6 ghz ) and consequently were the first to demonstrate a possible existence of spectra with turnover at high frequencies , about 1 ghz . kijak ( 2011a ) provided a definite evidence for a new type of pulsar radio spectra . these spectra show the maximum flux above 1 ghz , while at higher frequencies the spectra look like a typical pulsar spectrum . at lower frequencies ( below 1 ghz ) , the observed flux decreases , showing a positive spectral index ( @xcite ) . they called these objects the gigahertz - peaked spectra ( gps ) pulsars . a frequency at which such a spectrum shows the maximum flux was called the peak frequency . kijak et al . ( 2011a ) also indicated that the gps pulsars are relatively young objects , and they usually adjoin such interesting environments as hii regions or compact pulsar wind nebulae . additionally , some of them seem to be coincident with the known but sometimes unidentified x - ray sources from third egret catalogue or hess observations . we can assume that the gps appearance owes to the environmental conditions around the neutron stars rather than to the radio emission mechanism . psr b1259 - 63 was also listed by lorimer ( 1995 ) as a pulsar with positive spectral index . therefore , it seems a natural candidate to be classified as the gps pulsar . this pulsar is in an unique binary with a massive main - sequence be star . has a short period of 48 ms and a characteristic age of 330 kyr . its average dispersion measure ( dm ) is about 147 pc @xmath0 and the corresponding distance is about 2.75 kpc . the companion star ls 2883 is a 10-mag massive be star with a mass of about 10m@xmath1 and a radius of 6r@xmath1 . be stars are generally believed to have a hot tenuous polar wind and a cooler high - density equatorial disc . the psr b1259 - 63/ls 2883 emits unpulsed non - thermal emission over a wide range of frequencies ranging between radio and @xmath2rays , and its flux varies with orbital phase . [ cols="^,^ " , ] the flux at the given frequency apparently changes with orbital phases . when the pulsar is close to periastron , the flux generally decreases at all observed frequencies and the most drastic decrease is observed at the lowest frequency . moreover , we noticed all types of radio pulsar spectra . psr b1259 - 63 is a object with relatively high dispersion measure which means that its transition frequency is very high this implies that we definitely have to take into consideration both refractive ( riss ) and diffractive ( diss ) scintillations when analysing spectra for a given day . we used diffractive bandwidth @xmath3 and timescale @xmath4 from scintillation observations of the pulsar made far from periastron at 4.8 ghz and 8.4 ghz ( @xcite ) to estimate values of these parameters at 1.4 ghz and 2.4 ghz assuming @xmath5 , where @xmath6 and @xmath7 denote frequency and distance respectively . we estimated the values of @xmath4 to be ranging from 40 s at 1.4 ghz to 360 s at 8.4 ghz which suggests that diffractive scintillations should not affect the average flux measurements ( observing sessions was usually 4 hours long ) . roughly estimated refractive timescales vary from 12 hours at 8.4 ghz to more than 20 days at 1.4 ghz . however , for lower frequencies the modulation index is relatively small which means lower uncertainty estimates when measuring flux . high frequency observations will be affected by refractive scintillations what leads to conclusion that flux values should be averaged over epochs and/or orbital phase intervals to be more reliable . close to the periastron point the spectra of b1259 - 63 resemble those of the gps pulsars . the spectrum for the orbital epochs further from the periastron point are more consistent with typical pulsar spectra ( i.e. power - law and broken ) . moreover , detailed study of psr b1259 - 63 spectra revealed the appearance of all types of spectral shapes , including a flat spectrum ( see fig . [ detailed ] ) . we believe that the case of b1259 - 63 can be treated as a key factor to our understanding of not only the gps phenomenon ( observed for the solitary pulsars with interesting environments ) but also other types of untypical spectra as well ( e.g. flat or broken spectra ) . this in turn would suggest , that the appearance of various non - standard spectra shapes in the general population of pulsars can be caused by peculiar environmental conditions . md is a scholar within sub - measure 8.2.2 regional innovation strategies , measure 8.2 transfer of knowledge , priority viii regional human resources for the economy human capital operational programme co - financed by european social fund and state budget . , t. w. , johnston , s. , manchester , r. n. & mcconnell , d. 2002 , _ mnras _ , 336 , 1201 , s. , manchester , r. n. , mcconnell , d. & campbell - wilson , d. , 1999 , _ mnras _ , 302 , 277 , s. , ball , l. , wang , n. & manchester , r. n. , 2005 , _ mnras _ , 358 , 1069 , j. , lewandowski , w. , maron , o. , gupta , y. & jessner , a. , 2011a , _ a&a _ , 531 , a16 , j. , dembska , m. , lewandowski , w. , melikidze , g. & sendyk , m. , 2011b , _ mnras _ , 418 , l114 , d. r. , yates , j. a. , lyne , a. g. & gould , d. m. , 1995 , _ mnras _ , 273 , 411 , v. m. , gil , j. a. , jessner , a. , et al . 1994 , _ a&a _ , 285 , 201 , o. , kijak , j. , kramer , m. & wielebinski , r. , 2000 , _ a&a s _ , 147 , 195 , n. m. , johnston , s. , stinebring , d. r. & nicastro , l. , _ apj _ , 1998 , 492 , l49 , w. , 1973 , _ a&a _ , 28 , 237

we studied the radio spectrum of psr b1259 - 63 in an unique binary with be star ls 2883 and showed that the shape of the spectrum depends on the orbital phase . we proposed a qualitative model which explains this evolution . we considered two mechanisms that might influence the observed radio emission : free - free absorption and cyclotron resonance . recently published results have revealed a new aspect in pulsar radio spectra . there were found objects with turnover at high frequencies in spectra , called gigahertz - peaked spectra ( gps ) pulsars . most of them adjoin such interesting environments as hii regions or compact pulsar wind nebulae ( pwn ) . thus , it is suggested that the turnover phenomenon is associated with the environment than being related intrinsically to the radio emission mechanism . having noticed the apparent resemblance between the b1259 - 63 spectrum and the gps , we suggest that the same mechanisms should be responsible for both cases . therefore , the case of b1259 - 63 can be treated as a key factor to explain the gps phenomenon observed for the solitary pulsars with interesting environments and also another types of spectra ( e.g. with break ) .

1212.0503

an electron incident on a superconductor from a normal metal , with an energy smaller than the superconducting energy gap , can not propagate into the superconductor and thus should be perfectly reflected . however , andreev discovered a mechanism for transmission , in which an electron may form a cooper pair with another electron and be transmitted across the superconductor . as a consequence of charge conservation a hole must be left behind , which , as a result of momentum conservation , should propagate in a direction opposite to that of the incident electron . this process is termed andreev reflection @xcite . apart from providing a confirmation for the existence of cooper pairs and superconductor energy gaps @xcite , this process may also have applications in spintronics . it has been suggested that point contact andreev reflection can be used to probe spin polarization of ferromagnets by fabricating ferromagnet - superconductor nanojunctions @xcite . materials - specific modelling of such experiments , however , is complex and so far it has been somehow unsatisfactory . for instance tight - binding based scattering theory @xcite and green s functions theory @xcite calculations found poor fits to the experimental data for ferromagnet - superconductor junctions , while produced excellent fitting to normal metal - superconductor junctions results . based on this observation , xia and co - workers suggested that there may be an interaction between the ferromagnet and superconductor which is not accounted for in the blonder - tinkham - klapwijk ( btk ) model @xcite . consequently , the simple interpretation and two - parameter btk model fitting of experimental data to extract the spin polarization of various ferromagnets , was also called into question . more recently , chen , tesanovic and chien proposed a unified model for andreev reflection at a ferromagnet - superconductor interface @xcite . this is based on a partially polarized current , where the andreev reflection is limited by minority states and the excess majority carriers provide an evanescent contribution . however , this model has also been called into doubt by eschrig and co - workers @xcite . in particular , they pointed out that the additional evanescent component is introduced in an _ ad - hoc _ manner , and that the resulting wavefunction violates charge conservation . so , the debate about the correct model to describe andreev reflection at a ferromagnet - superconductor junction seems far from being settled . among other mesoscopic systems , andreev reflection has also been measured in carbon nanotubes ( cnts ) @xcite . there has been a theoretical study of normal metal - molecule - superconductor junction from density functional theory based transport calculations @xcite . in this study it was shown that the presence of side groups in the molecule can lead to fano resonances in andreev reflection spectra . topological insulators , a very recent and exciting development in condensed matter physics , have also been shown to be characterized by perfect andreev reflection @xcite . wang and co - authors have recently suggested performing a self - consistent calculation of the scattering potential to study andreev reflection at normal metal - superconductor junctions @xcite . they calculated the conductance for carbon chains sandwiched between a normal and a superconducting al electrode and found different values depending on whether or not the calculation was carried out self - consistent over the hartree and exchange - correlation potential . however , the theoretical justification for such a self - consistent procedure is at present not clear . in particular , it is difficult to argue that the variational principle , which underpins the hohenberg - kohn theorems , is still obeyed when a pairing energy is added _ by hand _ to the kohn - sham potential . in principle a rigorous self - consistent treatment should use the superconducting version of density functional theory @xcite , which probably remains computationally too expensive for calculating the interfaces needed to address a scattering problem . given such theoretical landscape and the fact that a non self - consistent approach to density functional theory based transport calculations has shown excellent agreement to experimental results for normal metal - superconductor junctions , we follow this methodology in the present work . in this paper , we study andreev reflection in normal - superconductor junctions , including all - metal junctions and carbon nanotubes sandwiched between normal and superconducting electrodes . we take into account the atomistic details of the junction by using density functional theory to obtain the underlying electronic structure , and then employ an extended btk model to solve the normal - superconductor scattering problem . our transverse momentum resolved calculations allow identifying the contributions to conductance from different parts of the brillouin zone . we also study the variation of conductance as a function of an applied potential difference between the electrodes for various normal metal - superconductor junctions , by performing approximate finite bias calculations . after this introduction , the rest of our paper is organized as follows : in section [ formulation ] we summarize the extended btk model and beenakker s formula , which we employ in this work . in the subsequent section [ results ] , we present our results for cu - pb , co - pb and au - al junctions , as well as al - cnt - al junctions . we also include the computational details in each of these subsections . finally , we conclude and summarize our findings in section [ conclusions ] . for the sake of completeness , here we briefly summarize the extended btk model @xcite that we use to study andreev reflection at a normal metal - superconductor interface . following refs . [ ] , we begin with the bogoliubov - de gennes equation @xmath0 where @xmath1 is the single particle hamiltonian for majority ( @xmath2 ) and minority ( @xmath3 ) spins , @xmath4 is the pairing potential and @xmath5 and @xmath6 are respectively the electron and hole wavefunctions . the energy @xmath7 sets the reference to the fermi energy , @xmath8 . we follow the approach of beenakker consisting in inserting a layer of superconductor in its normal state between the metal - superconductor interface . this ensures that at the fictitious normal metal - superconductor interface the only scattering process is andreev scattering . , indicated by the arrow . self - energies are used to simulate the effect of semi - infinite leads attached to the edge of the scattering region . ] other scattering processes are accounted for at the junction between the normal metal the and superconductor in its normal state . at this interface the scattering matrix can be written as @xmath9 here the superscripts @xmath10 and @xmath11 denote the right- and left - going states and the subscripts @xmath12 and @xmath13 refer to the normal and fictitious normal metal regions , respectively . the normal state scattering matrix reads @xmath14 now at the fictitious normal metal - superconductor interface @xmath15 where the factor @xmath16 is @xmath17 , \quad |\varepsilon|<\delta \nonumber \\ & = & \frac{1}{\delta}[\varepsilon-\mathrm{sign}(\varepsilon)\sqrt{\varepsilon^{2}-\delta^{2 } } ] , \quad |\varepsilon|>\delta\:.\end{aligned}\ ] ] the states in the normal metal are given by @xmath18 then the reflection coefficients for the complete system are @xmath19 and @xmath20 finally the conductance of the system is given by @xmath21 the implicit assumptions in the above derivation are that the superconducting order parameter is switched on abruptly as a step function ( i.e. , there are no proximity effects ) and the order parameter is much smaller than the fermi energy ( the so - called andreev approximation ) . a great simplification occurs if one considers scattering at fermi energy , namely @xmath22 , and the presence of time reversal symmetry , i.e. , the normal metal is not a ferromagnet . the above expression for conductance reduces to @xmath23 where the eigenvalues of the transmission matrix product @xmath24 are @xmath25 . this is the beenakker s formula @xcite . notice that all the dependence on the superconductor pairing has dropped out and the conductance depends on the normal state transmission eigenvalues . in this case superconductivity enters implicitly in the form of a boundary condition . in our first - principles transport code smeagol @xcite , we construct the full scattering matrix and then use the expressions in equations ( [ ree ] ) and ( [ rhe ] ) to evaluate the conductance from equation ( [ gnsfull ] ) . for the special case of @xmath26 , we construct the transmission matrix , @xmath24 . it is then straightforward to obtain its eigenvalues by numerical diagonalization . these are then interted into the beenakker s formula [ equation ( [ gnsbeenakker ] ) ] to obtain @xmath27 , while a direct summation of the eigenvalues yields @xmath28 . to compute the current , @xmath29 , at a bias @xmath30 , we use @xmath27 from equation ( [ gnsfull ] ) and calculate @xmath31g(\varepsilon)\:,\ ] ] and the finite bias conductance is evaluated from @xmath32 here @xmath33 can either be the normal state conductance , @xmath34 , or the normal metal - superconductor conductance , @xmath27 , and @xmath35 is the fermi function . we begin by presenting our results for cu - pb junctions , which have also been investigated experimentally in the past @xcite . we choose cu @xmath36 and @xmath37 and pb @xmath38 and @xmath39 as valence electrons and the effect of other core electrons are described by troullier - martins norm - conserving pseudopotentials . the local density approximation with the ceperley - alder parametrization was employed for the exchange - correlation functional . we choose an energy cutoff of 400 rydberg for the real space mesh , and a double-@xmath40 polarized basis set . the lattice constants of cu ( @xmath41 ) and pb ( @xmath42 ) are quite different , however a matching is obtained by rotating cu unit cell by a @xmath43 angle . in this geometry a small strain ( @xmath44% ) exists on both cu ( compressive ) and pb ( tensile ) . for the self - consistent calculation we use a @xmath45 in plane monkhorst - pack grid , while transport quantities are evaluated over a much denser @xmath46 @xmath47 grid . , between cu and pb . note that @xmath48 remains positive for all the distances investigated here . ] plane ( orthogonal to transport direction , @xmath49 ) periodic boundary conditions are employed . ] the scattering region for a cu - pb junction is shown in fig . [ cu - pb - setup ] . we use periodic boundary conditions in the plane orthogonal to the transport direction , and open boundary conditions along the direction of transport . we plot the available channels for both electrodes resolved over the brillouin zone ( bz ) at the fermi energy in fig . [ cu - pb - kp](a ) . for the left electrode ( cu ) four channels are available in quadrants centered at the edge of the bz , with a residual region around the zone center in which either three or two channels are available . for the right electrode ( pb ) around the bz center there exists a rectangular region with three open channels , while at the bz corners there are small pockets of reduced available channels , which even drop down to zero . the normal conductance , @xmath34 , is large over almost the entire bz , along with small pockets of lower transmission at the edges of the bz , which are inherited from the reduced channel pockets in the pb electrode , as shown in fig . [ cu - pb - kp](b ) . another small conductance pocket is present at the zone center , which originates from the distribution of open channels across the bz in the cu electrode . the overall conductance remains largely unchanged as the cu - pb distance is increased from @xmath50 1.5 to 3.0 . next we show the normal metal - superconductor junction conductance , @xmath27 , in fig . [ cu - pb - kp](c ) . at @xmath50 1.5 , the pockets of small conductance at the zone edges are more prominent , as compared to @xmath34 . moreover , the region around @xmath51 , with reduced conductance is also larger . on increasing the distance to 2 , these low conductance pockets shrink in size and the overall conductance increases . at larger distances , a broader region of low conductance develops and this reduces overall @xmath27 . in table [ cu - pb - tab ] , we provide the @xmath47-averaged value of the conductance above ( @xmath34 ) and below ( @xmath27 ) the pb superconducting temperature . for both quantities a maximum is obtained at @xmath50 2 . we also tabulate the ratio @xmath52 , which is the quantity expressing the zero - bias suppression due to andreev reflection . for a single channel btk model describing an ideal interface this ratio is exactly two , however when one takes into account the band structure mismatch and the underlying electronic structure of the electrodes a much lower value for this ratio can be obtained . for the cu - pb equilibrium distance ( @xmath53 ) we find @xmath52 close to 1.4 , which is in excellent agreement with the experimental value of 1.38 reported in ref . . , between the two constituents . ] .cu - pb junction : normal conductance , @xmath34 , normal - superconductor conductance , @xmath27 , and their ratio at different cu / pb distances . [ cols="^,^,^,^,^ " , ] finally , we study the variation of normalized conductance as a function of an applied bias , which is plotted in fig . [ al - cnt - bias ] . at low bias , for al - cnt(3,0)-al junction the normal metal - superconductor conductance is greater than the normal conductance . interestingly , the situation is reversed at a voltage of 0.1 mv . in contrast , for al - cnt(4,0)-al junction , the normalized conductance remains negative for voltages less than the superconducting gap . in conclusion , we have studied andreev reflection in normal - superconductor junctions using density functional theory based transport calculations . this approach allowed us to include the atomistic details of the junction electronic structure in the extended blonder - tinkham - klapwijk model . we studied au - al and cu - pb all metal junctions and calculated the normal and normal - superconductor conductances for different separations of the two materials at the interface . our transverse momentum resolved analysis has allowed us to identify contributions to these quantities from different parts of the brillouin zone . we found that the conductances for junctions in the superconducting state follows a similar @xmath47-point dependence as the normal state conductance . in other words , andreev reflection is higher in brillouin zone regions , where transmission is also high . we have also investigated co - pb ferromagnet - superconductor junctions . in this case , while at zero bias , our results satisfactorily match the experimental reports , a discrepancy was revealed at a finite bias , particularly at voltages close to the superconductor gap . this could possibly be attributed to stray magnetic fields from the ferromagnet or to proximity effects , both causes which are not included in the extended blonder - tinkham - klapwijk model . we further studied andreev reflection from carbon nanotubes sandwiched between normal metal and superconducting electrodes and found @xmath52 ratios to lie on opposite sides of unity for @xmath54 ( higher than one ) and @xmath55 ( lesser than one ) carbon nanotubes . this highlights the sensitivity of such calculations to details and the need for a truly atomistic theory for tackling this problem . concerning the potential outlook for future studies , our work provides a stepping stone for analyzing with first - principles methods the experimental setups needed to investigate and detect majorana fermions . these particles , which are their own anti - particles , are expected to play a crucial role in topological quantum computing and have recently garnered significant attention in the condensed matter community . after several theoretical proposals , signatures of this particle were found experimentally in large spin - orbit nanowires in proximity with superconductors @xcite . however , a number of issues remain unresolved and important questions need to be answered to confirm that indeed majoranas were observed . our implementation of the phenomenology of andreev reflection in a first - principles approach can be quite useful to study such a setup , in particular , by taking into account the underlying electronic structure of the nanowires . when combined with the order-@xmath56 implementation of our smeagol code @xcite , which allows us treating thousands of atoms , it opens the opportunity of recreating theoretically the aforementioned experiments in an _ ab inito _ manner , which till now have been modelled empirically . an is financially supported by irish research council under the embark initiative . ir and ss acknowledge additional support by kaust ( acrab project ) . the computational resources have been provided by trinity centre for high performance computing .

we study andreev reflection in normal metal - superconductor junctions by using an extended blonder - tinkham - klapwijk model combined with transport calculations based on density functional theory . starting from a parameter - free description of the underlying electronic structure , we perform a detailed investigation of normal metal - superconductor junctions , as the separation between the superconductor and the normal metal is varied . the results are interpreted by means of transverse momentum resolved calculations , which allow us to examine the contributions arising from different regions of the brillouin zone . furthermore we investigate the effect of a voltage bias on the normal metal - superconductor conductance spectra . finally , we consider andreev reflection in carbon nanotubes sandwiched between normal and superconducting electrodes .

1410.7178

the recent discovery of superconductivity in lafeas[o , f ] has intrigued tremendous interest in layered feas systems.@xcite intensive studies have revealed that , by substituting la with ce , sm , nd , pr , and gd , @xcite the superconducting temperature ( @xmath6 ) can be raised from 26 up to 53.3 k , and even higher ( about 55 k ) under high pressure.@xcite as we know , the parent compound of the these superconductors has a tetrahedral zrcusias - type structure with alternate stacking of tetrahedral feas layers and tetrahedral lao layers , and favors a stripe like antiferromagnetic ( afm ) ground state . the parent compound is not a superconductor but a poor metal with high density of states and low carrier density . @xcite the ground state of the parent compound is supposed to be a spin density wave ( sdw ) ordered state with a stripe like afm configuration . @xcite superconducting occurs when the sdw instability is suppressed by replacing of o with f or importing o vacancies ( electron doping ) , or sr substituting of la ( hole doping).@xcite very recently , the family of feas - based supercondutors has been extended to double layered rfe@xmath2as@xmath2 ( r = sr , ba , ca ) . @xcite the electronic structure of the parent compound has been studied both experimentally @xcite and theoretically . @xcite the density of states of rfe@xmath2as@xmath2 is very similar to that of refeaso around the fermi level , so does the fermi surface . the magnetic order of bafe@xmath2as@xmath2 has been revealed by experiment,@xcite and the magnetic moment on fe is 0.87 @xmath5 . besides , sdw anomaly has also been found in the rfe@xmath2as@xmath2 systems.@xcite although the superconducting mechanism of these new superconductors is still unclear , the peculiar properties of the feas layers , especially the magnetic properties , are believed to be very important for understanding the origin of the superconductivity in these compounds . although theoretical works have been reported for the double layered feas superconductors , the doping structure , magnetic coupling , as well as the the electronic structure after doping have not been thoroughly investigated . besides , the magnetic moment on fe atom obtained from previous theoretical studies is much larger than the experimental value ( cal . 2.67 @xmath5 v.s . exp . 0.87 @xmath5 ) . @xcite similar problem has been encountered for the single layered refeaso superconductors , and it was suggested that a negative on - site energy @xmath4 should be applied to such systems . @xcite it is interesting to see if such a remedy also works for bafe@xmath2as@xmath2 . although the use of a negative u is counterintuitive , it is physically possible . as suggested in a very recent work , @xcite in itinerant systems , for d@xmath7 configuration as fe@xmath8 is , the exchange - correlation effect may cause charge disproportionation ( 2d@xmath7 @xmath9 @xmath10 ) and lead to @xmath11 . in this paper , we report the theoretical electronic and magnetic properties of ba@xmath0k@xmath1fe@xmath2as@xmath2 ( @xmath3 = 0.00 , 0.25 , 0.50 , 0.75 , and 1.00 ) from first - principles calculations in the framework of generalized gradient approximation(gga)+u . with a negative @xmath4 , we obtain a magnetic moment per fe atom for bafe@xmath2as@xmath2 equal to 0.83 @xmath5 . by comparing the total energies , we predict the most favorable doping structure . moreover , we find slight doping ( @xmath3 near or small than 0.25 ) tends to enhance the magnetic instability , while medium and heavy dopings ( @xmath3 near or larger than 0.5 ) tend to suppress it . bafe@xmath2as@xmath2 exhibits the thcr@xmath2si@xmath2-type structure ( space group @xmath12 ) , where feas layers are separated by single ba layers along the c axis as shown in fig.[fig1 ] ( a ) . the feas layers are formed by edge - shared feas@xmath13 tetrahedra , similar to that in refeaso . in the calculation , we adopt a @xmath14 supercell , which contains four ba atoms , eight fe atoms , and eight as atoms . all structures are fully optimized until force on each atom is smaller than 0.01 ev / . during all the optimizations and static calculations , the lattice parameters are fixed to the experimental values @xmath15 and @xmath16 .@xcite although the lattice constants are different at different doping levels , the variations are very small , and we think they will not significantly change the electronic structures of the systems . to simulate doping , we replace one , two , three , and four ba atoms with k atoms , which corresponds to 25% , 50% , 75% , and 100% doping , respectively . the electronic structure calculations are carried out using the vienna _ ab initio _ simulation package@xcite within gga+u.@xcite the electron - ion interactions are described in the framework of the projected augment waves method and the frozen core approximation.@xcite the energy cutoff is set to 400 ev . for density of states ( dos ) calculation , we use a 12@xmath1712@xmath176 monkhorst dense grid to sample the brillouin zone , while for geometry optimization , a 6@xmath176@xmath173 monkhorst grid have been used . the on - site coulomb repulsion is treated approximately within a rotationally invariant approach , so only an effective u , defined as @xmath18=u j needs to be determined , where u is the on - site coulomb repulsion ( hubbard u ) and j is the atomic - orbital intra - exchange energy ( hund s parameter)@xcite . here we adopt a negative @xmath18 of -0.5 ev , and if not specially mentioned , all the discussions in the results are based on @xmath19 ev . bafe@xmath2as@xmath2 supercell . ( b ) the two fe planes in the supercell . red arrows show the afm4 configuration.,width=321 ] bafe@xmath2as@xmath2 with different u@xmath20.,width=321 ] bafe@xmath2as@xmath2 . since the spin - up and spin - down states are degenerated for afm states , we plot the spin - up channel only.,width=321 ] first , we focus on the electronic properties of the mother compound bafe@xmath2as@xmath2 . in order to describe the electronic structures with different magnetic orderings , the fe atoms in two planes are numbered as in fig.[fig1 ] ( b ) . except for the nonmagnetic(nm ) and ferromagnetic ( fm ) states , the system have six possible afm states : square - like in - plane afm with fe atoms directly above each other in the c - direction aligned parallelly , afm1 ( ,,+,+,,,+,+ ) , and antiparallelly , afm2 ( ,,+,+,+,+,, ) ; stripe - like in - plane afm with fe atoms directly above each other in the c - direction aligned parallelly afm3 ( ,+,,+,+,,+, ) , and antiparallelly afm4 ( ,+,,+,,+,,+,,+ ) ; one plane with square - like afm and the other with stripe - like afm , afm5 ( ,,+,+,+,,+, ) ; and in - plane fm with two planes aligned antiparallelly , afm6 ( ,,,,+,+,+,+ ) . we initialize the systems with these nm , fm and six afm orderings . after scf calculations , the afm1 , afm2 , and afm3 states converge to the nm state . the instabilities of nm state to other magnetic states , and the corresponding magnetic moment of fe in these magnetic states are listed in table [ table1 ] and [ table2 ] . we find very weak instabilities from nm to fm and afm6 , a stronger one to afm5 , and the strongest instability to afm4 , which is the ground state . this ground state is consistent with the previous experimental result@xcite and other calculations@xcite , where the ground state of bafe@xmath2as@xmath2 was found to be stripe - like afm with fe atoms aligned antiparallelly to each other in c - direction . the magnetic moment we obtained for the ground state is about 0.83 @xmath5/fe , comparing with that in other calculations ( about 2.67 @xmath5/fe ) , our result agrees much better with the experimental one ( 0.87 @xmath5/fe ) . [ cols="^,^,^,^,^,^,^,^,^,^",options="header " , ] we have tested the effects of different u@xmath20 on the magnetic moments and total energies . as shown in fig . [ fig2 ] , the magnetic moment changes monotonously with u@xmath20 , and a slight change of the u@xmath20 will significantly alter the magnetic moment . similar results have been found in refeaso , @xcite a negative effective u thus may be a common feature of the feas layers . , width=321 ] the density of states ( dos ) of the afm4 state is shown in fig . [ fig3]a , similar to that in refeaso , the contributions from fe and as dominate the dos near the fermi level , and the density of states at the fermi level ( n@xmath22 ) is 5.65 . the band structure of afm4 is illustrated in fig . [ fig3]b , the small dispersions along c axis ( from @xmath23 to z and a to m ) indicate the interactions between layers are small . there are three bands cross the fermi level , one electron band around @xmath23 to x , and two hole bands around @xmath23 to z and m to @xmath23 , which indicates the multi - band feature of the system . k@xmath24fe@xmath2as@xmath2.,width=321 ] next , we turn to the doping effects on the electronic structure of the system . in the case of one k replace of ba ( ba@xmath25k@xmath21fe@xmath2as@xmath2 ) , the k site has two choices , one is in the layer of z=0.0 ( case1 ) , and the other is in the layer of z=0.5 ( case2 ) , where z is the direct coordinate along the c axis of the supercell . the total energies of these two cases are very close , for nm state , the total energy of case 1 is about 0.2 mev higher than that of case2 . the results of magnetic instabilities and the magnetic moment on fe of these two cases are listed in table [ table1 ] and [ table2 ] . we find the afm3 state disappears in case1 , while in case2 , it is the state has the lowest energy , and the magnetic moment on fe is not significantly changed in the states with the in - plane stripe - like afm . the dos and band structures of afm4 of case1 and afm3 of case2 are illustrated in fig . [ fig4 ] . compared with the parent compound , the shape of the dos does not alter significantly , but the states near the fermi level are shifted up , resulted in an increase of the n@xmath22 , which is 9.46 in afm4 of case1 , and 9.18 in afm3 of case2 . in the band structures , the changes of the states near the fermi level is much clearer , in both afm4 of case1 and afm3 of case2 , only two hole bands across the fermi level , this is accord with the experimental results where hole pockets at @xmath23 sites become larger after doping.@xcite k@xmath25fe@xmath2as@xmath2.,width=321 ] then , we go to the case of ba@xmath24k@xmath24fe@xmath2as@xmath2 . there are three possible ways of substitution for k , that is two in the layer z=0.0 ( case 1 ) , two in the layer z=0.5 ( case 2 ) and one in each layer ( case 3 ) . the calculations show case 3 is the most favorable in the energy point of view , for nm state , the total energy of case 3 is about 0.19 ev lower than that of case 1 and case 2 per supercell . thus , here we discuss the magnetic and electronic properties of case3 only . as shown in table [ table1 ] and [ table2 ] , although afm3 is the state with the lowest energy , afm4 is very close to it , and the magnetic moments on fe of these two states are almost the same . so afm3 and afm4 may co - exist at this doping level , competing with each other . the dos and band structure of afm3 and afm4 are given in fig . [ fig5 ] , though the shape of the dos is similar to former cases , the states are further slightly shifted up with n@xmath22 of 12.84 for afm3 and 13.13 for afm4 . in the band structure , there are 3 bands across the fermi level . , width=321 ] for ba@xmath21k@xmath25fe@xmath2as@xmath2 , k atoms have two choices , one is two atoms in layer z=0.0 , and the other one in layer z=0.5 ( case 1 ) , the other choice is two in layer z=0.5 , and the other one in layer z=0.0 ( case 2 ) . in spin - unpolarized calculations , these two cases have almost the same total energy ( case 1 lower about 3.65 mev than case 2 ) , so these two structures may co - exist at this doping level . although with very close energy , their magnetic structures are different . in table [ table1 ] , we can see the afm3 state is not exist in case1 , while in case2 , it has the lowest energy . and again , we find the magnetic moments on fe of the states with the in - plane stripe - like afm are almost the same . besides , at this doping level , we find the instability from nm to fm is increased , close to that of nm to afm4 or afm3 . so here , the system may have competing orders of fm , afm3 , and afm4 . in all states of case 1 and case 2 , the afm3 of case 2 has the lowest energy , the dos and band structure of this state are plotted in fig . the n@xmath22 in this case is increased to 15.24 , and the bands near the fermi level are moved up , exhibiting 5 bands across the fermi level . lastly , although the 100% percent doping is hard to achieve in experiments , we still investigate this kfe@xmath2as@xmath2 case for consistency . the magnetic instabilities and magnetic moments on fe atoms are given in table [ table1 ] and [ table2 ] , we find the nm to afm6 has the strongest instability here , with afm1 the next . this is different with the above cases , where the states with in - plane stripe - like afm is always the state with the lowest energy . the dos and band structure are shown in fig . [ fig7 ] . comparing with the dos of ba@xmath21k@xmath25fe@xmath2as@xmath2 , the n@xmath22 is slightly decreased to 14.70 , and 5 bands cross the fermi level here . from the results illustrated above , we find the magnetic properties of ba@xmath0k@xmath1fe@xmath2as@xmath2 is very sensitive to the doping geometry , and the magnetic moments highly depend on the ordering . these properties imply that the magnetism of these compounds is of itinerant character . @xcite no matter the interlayer alignment of the states with in - plane stripe - like afm , they have almost the same magnetic moments on fe atoms . doping does not change the nature that the dos near the fermi level is dominated by the feas layer . it results in an increase of the n@xmath22 by shifting up the bands across the fermi level . in conclusion , we have performed first - principles calculation for ba@xmath0k@xmath1fe@xmath2as@xmath2 systems within the gga+u method . using a negative @xmath18 of @xmath26 ev , we find the same sdw ground state with experiment for the parent compound , and the magnetic moment on fe is very close to the experimental value ( cal . 0.83 @xmath5 v.s . exp . 0.87 @xmath5 ) . we predict the most favorable doping geometries from the energy point of view . besides , we find that the magnetic instability is enhanced with x=0.25 , and then start to decrease . moreover , in our result , the magnetic structure is very sensitive to the geometry , and the magnetic moment on fe highly depends on ordering , especially the in - plane ordering . this work was partially supported by the national natural science foundation of china under grant nos . 20773112 , 10574119 , 50121202 , and 20533030 , by national key basic research program under grant no . 2006cb922004 , by the ustc - hp hpc project , and by the sccas and shanghai supercomputer center . y. kamihara , t. watanabe , m. hirano , and h. hosono , j. am . soc . * 130 * , 3296 ( 2008 ) . x. h. chen _ nature doi : 10.1038/nature07045 ( 2008 ) ; arxiv0803.3603 ( 2008 ) g. f. chen _ arxiv:0803.4384 ( 2008 ) z. a. ren _ _ arxiv:0803.4283 ( 2008 ) g. f. chen _ arxiv:0803.3790 ( 2008 ) j. yang _ et al . _ arxiv:0804.3727 ( 2008 ) z. a. ren _ arxiv:0803.2053 ( 2008 ) y. kamihara _ et al . _ nature * 453 * , 376 ( 2008 ) d. j. singh _ lett . * 100 * , 237003 ( 2008 ) j. dong _ arxiv:0803.3246 ( 2008 ) c. de la cruz _ et al . _ arxiv:0804.0795 ( 2008 ) h .- h . wen , gang mu , lei fang , huan yang and xiyu zhu , eur . lett . * 82 * , 179009 ( 2008 ) . z. a. ren _ _ arxiv:0804.2582 ( 2008 ) m. rotter , m. tegel and d. johrendt arxiv:0805.4630 ( 2008 ) k. sasmal _ et al . _ arxiv:0806.1301 ( 2008 ) g. f. chen _ arxiv:0806.1209 ( 2008 ) g. wu _ arxiv:0806.4279 ( 2008 ) n. ni _ et al . _ arxiv:0806.4328 ( 2008 ) f. ronning _ et al . _ arxiv:0806.4599 ( 2008 ) m. rotter _ arxiv:0805.4021 ( 2008 ) l. x. yang _ arxiv:0806.2627 ( 2008 ) h. y. liu _ arxiv:0806.4806 ( 2008 ) f. j. ma , zhong - yi lu , and t. xiang arxiv:0806.3526 ( 2008 ) q. huang _ et al . _ arxiv:0806.2776 ( 2008 ) c. liu _ arxiv:0806:3453 ( 2008 ) c. krellner _ arxiv:0806.1043 ( 2008 ) i. a. nekrasov _ _ arxiv:0806.2630 ( 2008 ) h. nakamura _ arxiv:0806.4804 ( 2008 ) h. katayama - yoshida _ arxiv:0807.3770 ( 2008 ) g. kresse and d. joubert , phys . b * 59 * , 1578 ( 1999 ) ; g. kresse and j. furthmuller , phys . rev . b * 54 * , 11169 ( 1996 ) . p. e. blhl , phys . b * 50 * , 17953 ( 1994 ) j. p. perdew , k. burke , and m. ernzerhof , phys . lett . * 77 * , 3865 ( 1996 ) s. l. dudarev , g. a. botton , s. y. savrasov , c. j. humphreys and a. p. sutton , phys . rev . b * 57 * , 1505 ( 1998 ) d. j. singh , arxiv:0807.2643 ( 2008 )

we report a systematic first - principles study on the recent discovered superconducting ba@xmath0k@xmath1fe@xmath2as@xmath2 systems ( @xmath3 = 0.00 , 0.25 , 0.50 , 0.75 , and 1.00 ) . previous theoretical studies strongly overestimated the magnetic moment on fe of the parent compound bafe@xmath2as@xmath2 . using a negative on - site energy @xmath4 , we obtain a magnetic moment 0.83 @xmath5 per fe , which agrees well with the experimental value ( 0.87 @xmath5 ) . k doping tends to increase the density of states at fermi level . the magnetic instability is enhanced with light doping , and is then weaken by increasing the doping level . the energetics for the different k doping sites are also discussed .

0808.0065

the exponentially growing number of known extrasolar planets now enables statistical analyses to probe their formation mechanism . two theoretical frameworks have been proposed to account for the formation of gas giant planets : the slow and gradual core accretion model @xcite , and the fast and abrupt disk fragmentation model @xcite . the debate regarding their relative importance is still ongoing . both mechanisms may contribute to planet formation , depending on the initial conditions in any given protoplanetary disk ( * ? ? ? * and references therein ) . by and large , our understanding of the planet formation process is focused on the case of a single star+disk system . yet , roughly half of all solar - type field stars , and an even higher proportion of pre - main sequence ( pms ) stars , possess a stellar companion ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? since the disk and multiplicity phenomena are associated with similar ranges of distances from the central star , the dynamical influence of a companion on a disk may be dramatic . theory and observations agree that stellar companions can open large gaps in disks , or truncate them to much smaller radii than they would otherwise have ( e.g. , * ? ? ? * ; * ? ? ? the consequences for planet formation are still uncertain , however . observations of protoplanetary disks among pms stars have revealed that tight binaries generally show substantially reduced ( sub)millimeter thermal emission @xcite as well as a much rarer presence of small dust grains in regions a few au from either component @xcite . both trends can be qualitatively accounted for by companion - induced disk truncation , which can simultaneously reduce the disk s total mass , outer radius and viscous timescale . these observational facts have generally been interpreted as evidence that binaries tighter than @xmath0au are much less likely to support gas giant planet formation . however , follow - up imaging surveys have identified some 50 planet - host stars that possess at least one stellar companion ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? in particular , it it is worth noting that about 20% of all known planets in binary systems have a stellar companion within less 100au , so that planet formation in such an environment can not be considered a rare occurrence . in this _ letter _ , i review several key statistical properties of pms and field binary systems that provide insight on the planet formation process ( sections[sec : ci ] and [ sec : end ] ) . i then discuss the implications for the main mechanisms of planet formation in binary systems as a function of their projected separation ( section[sec : implic ] ) . in this study , i only consider binaries in the 51400au separation range , for which current pms multiplicity surveys are reasonably complete . the tightest binary system known to host a planet has a 19au separation . stellar companions beyond 1400au are not expected to have much influence on planet formation . in order to draw a broad and homogeneous view of the initial conditions for planet formation , i compiled a sample of 107 pms binaries for which deep ( sub)millimeter continuum observations and/or near- to mid - infrared colors are available in the literature . the ( sub)millimeter data are taken from the work of @xcite ; for almost all targets , a 1@xmath1 sensitivity of 15mjy or better at 850@xmath2 m and/or 1.3 mm is achieved . the median projected separation in this sample is 92au . i also defined a comparison sample of 222 pms stars for which no companion has ever been detected . i focus here on the taurus and ophiuchus star forming regions , the only ones for which high - resolution multiplicity , photometric and millimeter surveys have a high completeness rate . the two clouds contribute an almost equal number of binaries to the sample . furthermore , both regions have similar stellar age distributions ( median age around 1myr , ophiuchus being probably slighter younger on average than taurus ) and their mass function fully samples the 0.11.5@xmath3 range ( e.g. , * ? ? ? * ; * ? ? ? finally , taurus represents an instance of distributed star formation , while ophiuchus is a more clustered environment . these two clouds therefore offer a global view of the early stages of planet formation among solar - type and lower - mass stars . i first address the question of the presence of dust in the planet - forming region , namely the innermost few au around each component , within binary systems . to probe the presence of an optically thick dusty inner disk , i used near- to mid - infrared colors . i selected the following standard thresholds to conclude that a circumstellar disk is present : @xmath4-[8.0 ] \ge 0.8$]mag , @xmath5mag , @xmath6mag , @xmath7 ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? about 80% of the pms binaries considered here have _ spitzer_/irac colors , which are used whenever available . @xcite have demonstrated that tighter binaries have a much lower probability of hosting circumstellar dust . the same effect is observed here in a somewhat smaller sample . the median separation of binaries with an inner disk in this sample is about 100au , whereas that of disk - free binaries is 40au . the simplest interpretation of this trend is that disks in tight binaries are dissipated much faster than in wide systems ( * ? ? ? * kraus et al . , in prep . ) . to extend upon this previous analysis , i used the two - sided fischer exact test to determine the probability that wide and tight binaries have a different proportion of diskless systems , using a sliding threshold to split the sample . as shown in figure[fig : proba ] , the difference is significant at the 2@xmath1 level or higher for a wide range of threshold separations . in particular , this analysis reveals that _ the observed reduced disk lifetime in binaries only applies to systems that are tighter than about 100au_. on the other hand , there is no statistical difference between binaries wider than 100au and single stars . while near- and mid - infrared emission best traces the presence of dust within a few au of star , only long - wavelength flux measurements can probe the total dust mass of protoplanetary disks ( e.g. , * ? ? ? * ) . from the sample defined above , i selected those objects which show evidence of an optically thick inner disk ( as defined above ) and have been observed in the ( sub)millimeter . the median separation in this subsample of 44 binaries is 130au . while the 850@xmath2 m survey of ophiuchus is not yet as complete as that of taurus , the existing 1.3 mm observations of pms stars are generally less sensitive to cold dust . since using both wavelengths yield similar conclusions but with lower significance for the 1.3 mm one , i focus here on 850@xmath2 m measurements . as has long been known , tight binaries have a different distribution of submillimeter fluxes than wide ones , with a much lower median flux ( 13mjy vs 50mjy at 850@xmath2 m using a 100au separation threshold ) and only very few high - flux systems @xcite . i compared the distributions of 850@xmath2 m fluxes for tight and wide binaries defined by the same sliding threshold as above using the conservative survival analysis peto - pentrice generalized wilcoxon test to account for upper limits . i find that wide and tight binaries are different at the 2@xmath1 level or higher if the separation threshold is in the 75 - 300au range ( see figure[fig : proba ] ) . i therefore conclude that _ binaries with a projected separation smaller than 300au have a substantially reduced submillimeter flux_. on the other hand , the distribution of 850@xmath2 m fluxes for wide binaries is indistinguishable from that of single stars . in past studies , it has been assumed that a reduced ( sub)millimeter flux necessarily implies a reduced total dust mass independently of the disk properties ( for instance , see the prescription used by * ? ? ? while this is true in general , it is unclear whether this assumption is valid for severely truncated disks for which optical depth effects may become important . the model constructed by @xcite seems to support this hypothesis , but these authors assumed that tight binaries are always surrounded by a massive circumbinary structure , which we now know is rare . to revisit this issue , i have computed a grid of radiative transfer models using the mcfost code @xcite to compute the 850@xmath2 m flux of a disk with a typical @xmath8 surface density profile , an 0.1au inner radius and a flaring power law @xmath9 . emission from the central star is modeled as a 4000k , 2@xmath10 photosphere and a distance of 140pc is assumed . the dust is assumed to be made of astronomical silicates with a @xmath11 power law size distribution ranging from 0.03@xmath2 m to 1 mm . the only variables in the model are the disk outer radius , @xmath12 , and the total dust mass , @xmath13 . figure[fig : diskmass ] demonstrates that the proportionality between total dust mass and submillimeter flux observed for large disks breaks down for @xmath14au as the disk becomes optically thick to its own emission . disk truncation by an outer stellar component is dependent on the orbital parameters and mass ratio of the binary system @xcite . it is therefore not possible to uniquely associate a binary separation with a tidally - set value of @xmath12 . the ratio between these quantities is typically in the broad 2.55 range . systems whose separation is less than 100au are therefore expected to possess disks whose outer radius is 40au or less . in this configuration , total disk masses of at least @xmath15 are necessary to produce 850@xmath2 m fluxes as low as @xmath1630mjy . in the sample studied here , about a third ( 6 out of 19 ) of all binaries that are tighter than 100au and possess an inner disk have an 850@xmath2 m flux that is higher than 30mjy . therefore , _ a significant fraction of the circumstellar disks in tight binaries ( @xmath17au ) are massive enough to potentially form gas giant planets , despite their much lower ( sub)millimeter fluxes . _ les us turn our attention to mature planetary systems . as of this writing , there are 38 exoplanets that are in a system with at least 2 stellar components ( using 1400au as the upper limit for binary separation ) , including 5 systems with a stellar companions within 25au ( * ? ? ? * and references therein ) . most of these planet - bearing stars are of solar type . the overall detection rates of gas giant planets in binary systems and in single stars are undistinguishable @xcite . there is marginal evidence that planets in binaries tighter than @xmath0au may be somewhat less frequent than one would assume based on the frequency of planets in wider binaries ( by [email protected]% , * ? ? ? however , the small sample size , adverse selection biases and incompleteness of current multiplicity surveys are such that it is premature to reach definitive conclusions . in any case , we can use this sample to test whether the separation of the stellar binary has any influence on planet properties . despite an earlier claim for a distinct period - mass distribution @xcite , @xcite have shown that there is essentially no difference in the properties of planetary systems around single stars and in binary systems . however , a previously unrecognized trend is evident in figure[fig : planets ] . while planets covering two orders of magnitude in mass can be found in wide binaries ( as around single stars ) , systems tighter than @xmath0au appear to host only high - mass , @xmath19 , planets . to quantify this effect , i used the two - sided fischer exact test to determine whether close and wide binaries ( with the usual sliding threshold ) have different proportion of high- and low - mass planets . i used 1.6@xmath20 , the median for all planets known to date , to separate low- from high - mass planets . figure[fig : proba ] confirms that _ binary systems tighter than about 100au produce a distribution of planets that is strongly biased towards the highest masses . _ this conclusion is significant at the 3@xmath1 level . it is important to test whether this trend is not a mere consequence of a selection bias , as a close stellar companion can alter the detectability of a planet - induced radial velocity signal . to evaluate this possibility , i build on the `` uniform detectability '' sample defined by @xcite which contains all stars for which close - in planets as low mass as 0.3@xmath20 ( well below the apparent cut - off in mass for planets in tight binaries ) , as well as 1@xmath20 planets on a 4yr orbit , could be detected . in the current sample of binary planet hosts , the proportions of stars that belong to the uniform detectability sample among binaries tighter and wider than 100au are indistinguishable ( 5/9 and 24/40 , respectively ) . i therefore conclude that the trend discussed above is unlikely to be the consequence of a selection bias or of observational limitations . an indirect signpost of planet formation is the debris disk phenomenon . in these systems , small dust grains are produced via the collisions of large solid bodies @xcite . @xcite observed 69 a- and f - type known binaries with _ spitzer _ and found debris disks in systems spanning 6 decades in separation . they further suggested that intermediate separation ( 330au ) binaries are substantially less likely to host a debris disks than either tighter or wider systems , although the formal significance of this difference is marginal at best . no such trend was found by @xcite , who included 24 a- through m - type binaries in their own _ spitzer _ survey . this latter survey focused on targets that are more similar in mass to exoplanet hosts and the pms population discussed in the previous section . i used the two - sided fischer test to determine whether the occurrence of debris disks is indeed different in tight and wide binaries , using the same sliding threshold as above ( see figure[fig : proba ] ) . there is no significant difference for any value of the threshold in the sample from @xcite , nor in a combined sample that also includes systems from @xcite . the combined sample contains 52 binaries in the 51400au range , with a median separation of 50au , an increase of 15 sources from the sole sample of @xcite . in addition , the occurrence rates of debris disks in binary systems and single stars are very similar @xcite . in other words , _ any 0.52@xmath3 star , irrespective of the presence of a companion ( within the 51400au range studied here ) , may experience the early phases of planet formation up to the planetesimal stage_. this analysis has revealed a clear dichotomy between tight and wide binaries . systems with separation @xmath21au are indistinguishable from single stars as far as the initial conditions and end product of planet formation are concerned . the only caveat to this statement is the possibility of mild disk truncation in 100 - 300au systems , but most disks in these systems retain a mass reservoir that is sufficient to build up gas giant planets . on the other hand , planet formation in binaries with separations @xmath22au is characterized by a much shorter clearing timescale for the protoplanetary disks and a strong bias towards high - mass planets . despite these differences , planetesimals and mature planetary systems appear to form at roughly the same frequency as around other stars . furthermore , while protoplanetary disks are more compact in tight systems because of truncation , a significant fraction of them possess large mass reservoirs ( at least several times @xmath20 ) . taken together , _ these results suggest that planet formation in binaries tighter than 100au proceeds through a different , but not much less frequent , mechanism compared to wide binaries and single stars . _ the shorter disk lifetime in tight binaries makes it extremely difficult to form gas giant planets through the core accretion model , especially if the final planets are particularly massive . rather , this combination of observed trends supports an abrupt process to form planets in tight binaries , such as the disk fragmentation model . indeed , this mechanism can be extremely efficient in the case of a compact , massive protoplanetary disk which is naturally prone to gravitational instability . furthermore , gravitational perturbations induced by a close stellar companion can trigger the instability even though the disk itself is not unstable to its own gravity @xcite . on the other hand , considering the long survival timescale and slim chances of gravitational instabilities , disks located within wide binaries and around single stars are good candidates to form planets via the core accretion model in their inner regions ( e.g. , * ? ? ? while a violent process is most likely responsible for the formation of planets in tight binaries , it is however unclear whether all planets in wide binaries form through a single mechanism . indeed , it is also conceivable that high - mass planets ( @xmath23 ) mostly form via disk fragmentation , while lower mass planets are preferentially the result of core accretion . this scenario would naturally alleviate the difficulty of the core accretion model to form the highest - mass planets in less than a few myr . this hypothesis has the additional advantage that it could also apply to tight binaries . indeed , since a stellar companion located within less than 100au dramatically shortens the disk lifetime , core accretion is essentially prevented from occurring , accounting for the absence of low - mass ( @xmath24 ) gas giant planets in tight binaries . planetesimals can presumably form in either scenario , accounting for the observations regarding the debris disks phenomenon . in summary , it remains to be determined whether the trends discussed here indicate an actual dichotomy between the main planet formation theories or a mere change of the relative importance of the two models as a function of the location of the stellar companion . improving the statistical significance of the various trends discussed here and determining the exact properties of disks within tight pms binaries will help shed further light these two possibilities . i am grateful to silvia alencar and jane gregorio - hetem for organizing and inviting me to `` special session 7 '' at the iau 27th general assembly held in rio de janeiro , where this work was first presented , as well as to anne eggenberger , deepak raghavan , david rodriguez and peter plavchan for invaluable input regarding exoplanets and debris disks . the work presented here has been funded in part by the agence nationale de la recherche through contract anr-07-blan-0221 . andrews , s. m. , & williams , j. p. 2005 , , 631 , 1134 andrews , s. m. , & williams , j. p. 2007 , , 671 , 1800 artymowicz , p. , & lubow , s. h. 1994 , , 421 , 651 beckwith , s. v. w. , sargent , a. i. , chini , r. s. , & guesten , r. 1990 , , 99 , 924 boley , a. c. 2009 , , 695 , l53 bonavita , m. , & desidera , s. 2007 , , 468 , 721 bontemps , s. , et al . 2001 , , 372 , 173 boss , a. p. 2006 , , 641 , 1148 cieza , l. a. , et al . 2009 , , 696 , l84 chauvin , g. , lagrange , a .- m . , udry , s. , fusco , t. , galland , f. , naef , d. , beuzit , j .- l . , & mayor , m. 2006 , , 456 , 1165 duchne , g. , et al . , 2010 , apj , submitted duchne , g. , delgado - donate , e. , haisch , k. e. , jr . , loinard , l. , & rodrguez , l. f. 2007 , protostars and planets v , 379 duquennoy , a. , & mayor , m. 1991 , , 248 , 485 durisen , r. h. , boss , a. p. , mayer , l. , nelson , a. f. , quinn , t. , & rice , w. k. m. 2007 , protostars and planets v , 607 eggenberger , a. , udry , s. , chauvin , g. , beuzit , j .- l . , lagrange , a .- m . , sgransan , d. , & mayor , m. 2007 , , 474 , 273 eggenberger , a. , udry , s. , chauvin , g. , beuzit , j. l. , lagrange , a. m. , & mayor , m. 2008 , astronomical society of the pacific conference series , 398 , 179 fischer , d. a. , & valenti , j. 2005 , , 622 , 1102 ghez , a. m. , neugebauer , g. , & matthews , k. 1993 , , 106 , 2005 guilloteau , s. , dutrey , a. , & simon , m. 1999 , , 348 , 570 hartmann , l. , megeath , s. t. , allen , l. , luhman , k. , calvet , n. , dalessio , p. , franco - hernandez , r. , & fazio , g. 2005 , , 629 , 881 ireland , m. j. , & kraus , a. l. 2008 , , 678 , l59 jensen , e. l. n. , & akeson , r. l. 2003 , , 584 , 875 jensen , e. l. n. , mathieu , r. d. , & fuller , g. a. 1996 , , 458 , 312 kenyon , s. j. , & hartmann , l. 1995 , , 101 , 117 lissauer , j. j. , & stevenson , d. j. 2007 , protostars and planets v , 591 luhman , k. l. , whitney , b. a. , meade , m. r. , babler , b. l. , indebetouw , r. , bracker , s. , & churchwell , e. b. 2006 , , 647 , 1180 luhman , k. l. 2000 , , 544 , 1044 luhman , k. l. , & rieke , g. h. 1999 , , 525 , 440 mathieu , r. d. 1994 , , 32 , 465 mccabe , c. , ghez , a. m. , prato , l. , duchne , g. , fisher , r. s. , & telesco , c. 2006 , , 636 , 932 mugrauer , m. , & neuhuser , r. 2009 , , 494 , 373 pinte , c. , mnard , f. , duchne , g. , & bastien , p. 2006 , , 459 , 797 patience , j. , et al . 2002 , , 581 , 654 pierens , a. , & nelson , r. p. 2008 , , 483 , 633 plavchan , p. , werner , m. w. , chen , c. h. , stapelfeldt , k. r. , su , k. y. l. , stauffer , j. r. , & song , i. 2009 , , 698 , 1068 quintana , e. v. , adams , f. c. , lissauer , j. j. , & chambers , j. e. 2007 , , 660 , 807 raghavan , d. , henry , t. j. , mason , b. d. , subasavage , j. p. , jao , w .- c . , beaulieu , t. d. , & hambly , n. c. 2006 , , 646 , 523 trilling , d. e. , et al . 2007 , , 658 , 1289 zucker , s. , & mazeh , t. 2002 , , 568 , l113 zuckerman , b. 2001 , , 39 , 549 llcccccl target & alt . name & sep . & ir color & inner disk ? & @xmath25 & @xmath26 & references + & & [ au ] & [ mag ] & & [ mjy ] & [ mjy ] & + + cz tau & & 46 & 3.41 & y & @xmath27 & @xmath28 & 1,2 + dd tau & & 79 & 1.99 & y & @xmath29 & 17 & 1,2 + df tau & & 13 & 1.37 & y & 8.8 & @xmath30 & 3,2 + di tau & & 17 & 0.76 & n & & & 3 + dk tau & & 350 & 1.27 & y & 80 & 35 & 3,2 +

in this article , i examine several observational trends regarding protoplanetary disks , debris disks and exoplanets in binary systems in an attempt to constrain the physical mechanisms of planet formation in such a context . binaries wider than about 100au are indistinguishable from single stars in all aspects . binaries in the 5100au range , on the other hand , are associated with shorter - lived but ( at least in some cases ) equally massive disks . furthermore , they form planetesimals and mature planetary systems at a similar rate as wider binaries and single stars , albeit with the peculiarity that they predominantly produce high - mass planets . i posit that the location of a stellar companion influences the relative importance of the core accretion and disk fragmentation planet formation processes , with the latter mechanism being predominant in binaries tighter than 100au .

0912.3025

the self - avoiding walk ( saw ) model is an important model in statistical physics @xcite . it models the excluded - volume effect observed in real polymers , and exactly captures universal features such as critical exponents and amplitude ratios . it is also an important model in the study of critical phenomena , as it is the @xmath4 limit of the @xmath5-vector model , which includes the ising model ( @xmath6 ) as another instance . indeed , one can straightforwardly simulate saws in the infinite volume limit , which makes this model particularly favorable for the calculation of critical parameters . exact results are known for self - avoiding walks in two dimensions @xcite and for @xmath7 ( mean - field behavior has been proved for @xmath8 @xcite ) , but not for the most physically interesting case of @xmath9 . the pivot algorithm is a powerful and oft - used approach to the study of self - avoiding walks , invented by lal @xcite and later elucidated and popularized by madras and sokal @xcite . the pivot algorithm uses pivot moves as the transitions in a markov chain which proceeds as follows . from an initial saw of length @xmath0 , such as a straight rod , new @xmath0-step walks are successively generated by choosing a site of the walk at random , and attempting to apply a lattice symmetry operation , or pivot , to one of the parts of the walk ; if the resulting walk is self - avoiding the move is accepted , otherwise the move is rejected and the original walk is retained . thus a markov chain is formed in the ensemble of saws of fixed length ; this chain satisfies detailed balance and is ergodic , ensuring that saws are sampled uniformly at random . one typical use of the pivot algorithm is to calculate observables which characterize the size of the saws : the squared end - to - end distance @xmath10 , the squared radius of gyration @xmath11 , and the mean - square distance of a monomer from its endpoints @xmath12 . to leading order we expect the mean values of these observables over all saws of @xmath0 steps , with each saw is given equal weight , to be @xmath13 ( @xmath14 ) , with @xmath15 a universal critical exponent . for @xmath0-step saws , the implementation of the pivot algorithm due to madras and sokal has estimated mean time per attempted pivot of @xmath16 on @xmath17 and @xmath18 on @xmath19 ; performance was significantly improved by kennedy @xcite to @xmath20 and @xmath21 respectively . in this article , we give a detailed description of a new data structure we call the saw - tree . this data structure allows us to implement the pivot algorithm in a highly efficient manner : we present a heuristic argument that the mean time per attempted pivot is @xmath1 on @xmath17 and @xmath19 , and numerical experiments which show that for walks of up to @xmath22 steps the algorithmic complexity is well approximated by @xmath3 . this improvement enables the rapid simulation of walks with many millions of steps . in a companion article @xcite , we describe the algorithm in general terms , and demonstrate the power of the method by applying it to the problem of calculating the critical exponent @xmath15 for three - dimensional self - avoiding walks . thus far the saw - tree has been implemented for @xmath17 , @xmath19 , and @xmath23 , but it can be straightforwardly adapted to other lattices and the continuum , as well as polymer models with short - range interactions . other possible extensions would be to allow for branched polymers , confined polymers , or simulation of polymers in solution . we intend to implement the saw - tree and associated methods as an open source software library for use by researchers in the field of polymer simulation . madras and sokal @xcite demonstrated , through strong heuristic arguments and numerical experiments , that the pivot algorithm results in a markov chain with short integrated autocorrelation time for global observables . the pivot algorithm is far more efficient than markov chains which utilize local moves ; see @xcite for detailed discussion . the implementation of the pivot algorithm by madras and sokal utilized a hash table to record the location of each site of the walk . they showed that for @xmath0-step saws the probability of a pivot move being accepted is @xmath24 , with @xmath25 dimension - dependent but close to zero ( @xmath26 ) . as accepted pivots typically result in a large change in global observables such as @xmath10 , this leads to the conclusion that the pivot algorithm has integrated autocorrelation time @xmath27 , with possible logarithmic corrections . in addition , they argued convincingly that the cpu time per successful pivot is @xmath28 for their implementation . throughout this article we work with the mean time per attempted pivot , @xmath29 , which for the madras and sokal implementation is @xmath30 . madras and sokal argued that @xmath28 per successful pivot is best possible because it takes time @xmath28 to merely write down an @xmath0-step saw . kennedy @xcite , however , recognized that it is _ not _ necessary to write down the saw for each successful pivot , and developed a data structure and algorithm which cleverly utilized geometric constraints to break the @xmath28 barrier . in this paper , we develop methods which further improve the use of geometric constraints to obtain a highly efficient implementation of the pivot algorithm . we have efficiently implemented the pivot algorithm via a data structure we call the saw - tree , which allows rapid monte carlo simulation of saws with millions of steps . this new implementation can also be adapted to other models of polymers with short - range interactions , on the lattice and in the continuum , and hence promises to be widely useful . the heart of our implementation of the algorithm involves performing intersection tests between `` bounding boxes '' of different sub - walks when a pivot is attempted . in @xcite we generated large samples of walks with up to @xmath31 steps , but for the purpose of determining the complexity of our algorithm we have also generated smaller samples of walks of up to @xmath32 steps . for @xmath33 , the mean number of intersection tests needed per attempted pivot is remarkably low : 39 for @xmath17 , 158 for @xmath19 , and 449 for @xmath23 . in sec . [ sec : complexity ] we present heuristic arguments for the asymptotic behavior of the mean time per attempted pivot for @xmath0-step saws , @xmath29 , and test these predictions with computer experiments for @xmath34 . we summarize our results in table [ tab : performance ] ; note that @xmath35 indicates @xmath36 is bounded above by @xmath37 asymptotically , @xmath38 indicates @xmath37 dominates @xmath36 , @xmath39 indicates @xmath36 dominates @xmath37 , and @xmath40 indicates @xmath37 bounds @xmath36 both above and below . for comparison , we also give the algorithmic complexity of the implementations of madras and sokal @xcite , and kennedy @xcite . in sec . [ sec : complexityhighdim ] , we develop an argument for the complexity of our algorithm on @xmath23 ; this same argument leads to an estimate for the performance of the implementation of madras and sokal on @xmath23 . we do not know the complexity of kennedy s implementation for @xmath23 and @xmath41 with @xmath42 , but we suspect it is @xmath43 with @xmath44 , with possible logarithmic corrections . ccccc lattice & madras and sokal & kennedy & + & & & predicted & observed ' '' '' + @xmath17 & @xmath16 & @xmath20 & @xmath1 & @xmath2 ' '' '' + @xmath19 & @xmath18 & @xmath21 & @xmath1 & @xmath3 + @xmath23 & @xmath45 & ? & @xmath2 & @xmath46 + @xmath41 , @xmath42 & @xmath28 & ? & @xmath47 & ? ' '' '' + our implementation is also fast in practice : for simulations of walks of length @xmath48 on @xmath19 , our implementation is almost 400 times faster when compared with kennedy s , and close to four thousand times faster when compared with that of madras and sokal . we have measured @xmath29 for each implementation over a wide range of @xmath0 on @xmath17 , @xmath19 , and @xmath23 , and report these results in sec . [ sec : comparison ] . in sec . [ sec : implementation ] , we give a detailed description of the saw - tree data structure and associated methods which are required for implementing the pivot algorithm . in sec . [ sec : complexity ] we present heuristic arguments that @xmath29 for self - avoiding walks on @xmath17 and @xmath19 is @xmath1 , and numerical evidence which shows that for walks of up to @xmath22 steps @xmath29 is @xmath2 for @xmath17 and @xmath3 for @xmath19 . we also discuss the behavior of our implementation for higher dimensions . in sec . [ sec : initialization ] we discuss initialization of the markov chain , including details of how many data points are discarded . we also explain why it is highly desirable to have a procedure such as * pseudo_dimerize * for initialization ( pseudo - code in sec . [ sec : implementationhigh ] ) when studying very long walks , and show that the expected running time of * pseudo_dimerize * is @xmath49 . in sec . [ sec : autocorrelation ] we discuss the autocorrelation function for the pivot algorithm , and show that the batch method for estimating confidence intervals is accurate , provided the batch size is large enough . this confirms the accuracy of the confidence intervals for our data published in @xcite . finally , in sec . [ sec : comparison ] we compare the performance of our implementation with previous implementations of the pivot algorithm @xcite . we show that the saw - tree implementation is not only dramatically faster for long walks , it is also faster than the other implementations for walks with as few as 63 steps . self - avoiding walks ( saws ) are represented as binary trees ( see e.g. @xcite ) via a recursive definition ; we describe here the saw - tree data structure and associated methods using pseudo - code . these methods can be extended to include translations , splitting of walks , joining of walks , and testing for intersection with surfaces . indeed , for saw - like models ( those with short range interactions ) , it should be possible to implement a wide variety of global moves and tests for saws of @xmath0 steps in time @xmath3 or better . it is also possible to parallelize code by , for example , performing intersection testing for a variety of proposed pivot moves simultaneously . parallelization of the basic operations is also possible , but would be considerably more difficult to implement . in this section we give precise pseudo - code definitions of the data structure and algorithms . for reference , r - trees @xcite and bounding volume hierarchies ( see e.g. @xcite ) are data structures which arise in the field of computational geometry which are related to the saw - tree . for self - avoiding walks , the self - avoidance condition is enforced on sites rather than bonds , and this means that the saw - tree is naturally defined in terms of sites . this representation also has the advantage that the basic objects , sites , have physical significance as they correspond to the monomers in a polymer . the only consequences of this choice are notational : a saw - tree of @xmath5 sites has @xmath50 steps . we adopt this notation for the remainder of this section . when discussing the complexity of various algorithms we will still use @xmath0 rather than @xmath5 in order to be consistent with the companion article and other sections of the present work . an @xmath5-site saw on @xmath41 is a mapping @xmath51 with @xmath52 for each @xmath53 ( @xmath54 denotes the euclidean norm of @xmath55 ) , and with @xmath56 for all @xmath57 . saws may be either rooted or unrooted ; our convention is that the saws are rooted at the site which is at @xmath58 ( unit vector in the first coordinate direction ) , i.e @xmath59 . this convention simplifies some of the algebra involved in merging sub - walks , and is represented visually , e.g. in fig . [ fig : example ] , by indicating a dashed bond from the origin to the first site of the walk . ( 0,0 ) circle ( 3pt ) ; ( 1,0 ) circle ( 3pt ) ; ( 1,1 ) circle ( 3pt ) ; ( 2,1 ) circle ( 3pt ) ; ( 3,1 ) circle ( 3pt ) ; ( 3,0 ) circle ( 3pt ) ; ( 0,0 ) ( 1,0 ) ; ( 1,0 ) ( 1,1 ) ( 2,1 ) ( 3,1 ) ( 3,0 ) ; we denote the group of symmetries of @xmath41 as @xmath60 , which corresponds to the dihedral group for @xmath61 , and the octahedral group for @xmath9 . this group acts on coordinates by permuting any of the @xmath62 coordinate directions ( @xmath63 choices ) , and independently choosing the orientation of each of these coordinates ( @xmath64 choices ) ; thus @xmath60 has @xmath65 elements . the group of lattice symmetries for @xmath19 therefore has 48 elements , and we use all of them except the identity as potential pivot operations ; other choices are possible . we can represent the symmetry group elements as @xmath66 orthogonal matrices , and the symmetry group elements act on the coordinates written as column vectors . we also define the ( non - unique ) pivot sequence representation of a self - avoiding random walk on @xmath41 as a mapping from the integers to @xmath60 , @xmath67 . the sequence elements @xmath68 represent changes in the symmetry operator from site @xmath69 to site @xmath53 , while @xmath70 represent absolute symmetry operations , i.e. relative to the first site of the walk . we can relate this to the previous definition of a self - avoiding walk in terms of sites via the recurrence relations @xmath71 with @xmath72 , and initial conditions @xmath73 , @xmath74 , and @xmath59 . as noted by madras and sokal ( footnote 10 , p132 in @xcite ) , for the pivot sequence representation it is possible to perform a pivot of the walk in time @xmath1 by choosing a site @xmath53 uniformly at random , and multiplying @xmath68 by a ( random ) symmetry group element . however , the pivot sequence representation does no better than the hash table implementation of madras and sokal if we wish to determine if this change results in a self - intersection , or if we wish to calculate global observables such as @xmath10 for the updated walk . forgetting for the moment the self - avoidance condition , and using the fact that @xmath60 has @xmath75 elements , we see that for random walks of @xmath5 sites there are @xmath76 possible pivot sequences , while there are only @xmath77 random walks . this suggests that each random walk is represented by @xmath78 pivot sequences . this can be derived directly by noting that given a pivot sequence @xmath79 , we can insert a pivot @xmath80 which preserves the vector @xmath58 , between two elements @xmath81 and @xmath82 as follows @xmath83 without altering the walk . the number of symmetry group elements which preserve @xmath58 is @xmath84 , and there are @xmath50 locations where these symmetry group elements can be inserted , leading to @xmath78 equivalent pivot representations for a random walk of @xmath5 sites . for @xmath61 , given @xmath85 the recurrence relations in eqs . [ eq : qrecurrence ] and [ eq : omegarecurrence ] only fix one of the two non - zero elements in @xmath68 , leaving the choice of sign for the other non - zero element free . for our example walk @xmath86 we have @xmath87 we give three of the 16 equivalent choices for the pivot representation of @xmath86 , the first involving only proper rotations , the second with improper rotations for @xmath68 with @xmath88 , and the third with proper and improper rotations alternating : @xmath89 the non - uniqueness of the pivot representation for saws is due to the fact that the monomers ( occupied sites ) are invariant under the symmetry group @xmath60 , i.e. it is not possible to distinguish the different orientations of a single site . the non - uniqueness is of no practical concern , but perhaps hints that it may be possible to derive a more succinct and elegant representation of walks than the mapping to @xmath60 defined here . the merge operation is the fundamental operation on saws which allows for the binary tree data structure we call the saw - tree . this is related to the concatenation operation defined , for example , in sec . 1.2 of @xcite ; for concatenation the number of bonds is conserved , whereas for the merge operation the number of sites is preserved . merging two saws with @xmath5 and @xmath90 sites respectively results in a saw with @xmath91 sites . it is convenient to also include a pivot operation , @xmath80 , when merging the walks , and the result of merging two walks @xmath92 and @xmath93 is @xmath94 the merge operation is represented visually in fig . [ fig : merge ] . to merge two sub - walks , pin the open circle of the left - hand sub - walk to the origin , and then pin the open circle of the right - hand sub - walk to the tail end of the left - hand sub - walk . finally , apply the symmetry @xmath80 to the right - hand sub - walk , using the second pin as the pivot . ( 0,0 ) circle ( 3pt ) ; ( 1,0 ) circle ( 3pt ) ; ( 1,1 ) circle ( 3pt ) ; ( 0,0 ) ( 1,0 ) ; ( 1,0 ) ( 1,1 ) ; ( 1.5,0 ) node[q ] @xmath95 ; ( 2,0 ) circle ( 3pt ) ; ( 3,0 ) circle ( 3pt ) ; ( 3,-1 ) circle ( 3pt ) ; ( 2,0 ) ( 3,0 ) ; ( 3,0 ) ( 3,-1 ) ; ( 3.5,0.0 ) node @xmath96 ; ( 4,0 ) circle ( 3pt ) ; ( 5,0 ) circle ( 3pt ) ; ( 5,1 ) circle ( 3pt ) ; ( 6,1 ) circle ( 3pt ) ; ( 6,0 ) circle ( 3pt ) ; ( 4,0 ) ( 5,0 ) ; ( 5,0 ) ( 6,1 ) ( 6,0 ) ; ( 0,0 ) circle ( 3pt ) ; ( 1,0 ) circle ( 3pt ) ; ( 1,1 ) circle ( 3pt ) ; ( 0,1 ) circle ( 3pt ) ; ( 0,0 ) ( 1,0 ) ; ( 1,0 ) ( 1,1 ) ( 0,1 ) ; ( 1.5,0.5 ) node[q ] @xmath97 ; ( 1.5,0.58 ) node @xmath98 ; ( 2,0 ) circle ( 3pt ) ; ( 3,0 ) circle ( 3pt ) ; ( 3,1 ) circle ( 3pt ) ; ( 3,2 ) circle ( 3pt ) ; ( 2,0 ) ( 3,0 ) ; ( 3,0 ) ( 3,2 ) ; ( 3.5,0.5 ) node @xmath96 ; ( 5,0 ) circle ( 3pt ) ; ( 6,0 ) circle ( 3pt ) ; ( 6,1 ) circle ( 3pt ) ; ( 5,1 ) circle ( 3pt ) ; ( 4,1 ) circle ( 3pt ) ; ( 4,0 ) circle ( 3pt ) ; ( 4,-1 ) circle ( 3pt ) ; ( 5,0 ) ( 6,0 ) ; ( 6,0 ) ( 4,-1 ) ; here we define various quantities which are necessary for implementing our data structure and for calculating observables such as the mean - square end - to - end distance , @xmath10 . we first define various quantities which will be used to calculate observables which measure the size of a walk : @xmath99 a bounding box of a walk is a convex shape which completely contains the walk . the obvious choice of shape for @xmath41 is the rectangular prism with faces formed from the coordinate planes @xmath100 , @xmath101 , with the constants chosen so that the faces of the prism touch the walk , i.e. the bounding box has minimum extent . other choices are possible , e.g. other planes can be used such as @xmath102 , @xmath103 , and have the advantage of matching the shape of the walk more closely , but at the expense of more computational overhead and memory consumption . with closer fitting bounding boxes , fewer intersection tests need to be performed to ascertain whether two walks intersect . however , in practice , the coordinate plane rectangular prism implementation was fastest on our computer hardware ( by a narrow margin ) , and has the benefit that it is straightforward to implement . the choice of bounding box for continuum models is not as obvious ; possibilities include spheres and oriented rectangular prisms . we note that the choice of bounding box shape determines the maximum number of sites , @xmath104 , a saw can have so that it is guaranteed that its bounding box contains the sites of the saw and no others . suppose we are given two saws for which the bounding boxes overlap : if each of the walks has @xmath104 or fewer sites , we can be certain that the two walks intersect , while if at least one of the walks has more than @xmath104 sites , it may be that the walks do not intersect . the value of @xmath104 determines the cut - off for intersection testing for the function * intersect * in sec . [ sec : implementationuser ] . for @xmath41 with @xmath105 , the bounding box with faces formed from the coordinate planes leads to the maximum number of sites being two , as there are counter - examples with three sites ( e.g. the bounding box of @xmath106 also contains @xmath107 ) . for the bounding box with the faces being the coordinate planes and @xmath102 , the maximum number of sites is three ( as the bounding box of @xmath108 also contains @xmath109 ) . it is possible to push this one step further so that the maximum number of sites is four , but five is not possible as we can see that @xmath86 in fig . [ fig : example ] has five sites , and an unvisited site on its convex hull , which must also therefore be interior to any bounding box . we write bounding boxes as a product of closed intervals , in the form @xmath110 $ ] , where the product is taken over @xmath111 . consider a walk @xmath112 , with bounding box @xmath113 , which is split into left- and right - hand sub - walks , @xmath114 and @xmath115 , with bounding boxes @xmath116 $ ] , and @xmath117 $ ] respectively . we can then define the union operation on bounding boxes , @xmath118\cup[c_i , d_i]\right ) \nonumber \\ & = \times [ \inf\{a_i , c_i\},\sup\{b_i , d_i\}].\end{aligned}\ ] ] the intersection operation is defined as @xmath119\cap[c_i , d_i]\right ) \nonumber \\ & = \times [ \sup\{a_i , c_i\},\inf\{b_i , d_i\}].\end{aligned}\ ] ] there is no guarantee that @xmath120 , and we adopt the convention that an interval @xmath121 $ ] is considered empty if @xmath122 . if any interval is empty , then the corresponding bounding box is also empty as it contains no interior sites . a quantity associated with the bounding box which we will find useful is the sum of the dimensions of the bounding box , * perim*. if @xmath123 $ ] , then we define @xmath124 for @xmath86 ( in fig . [ fig : example ] ) we have the following values for the various parameters : @xmath125\times[0,1 ] ; \\ \mathbf{x}_{\mathrm{e}}(\omega_a ) & = ( 3,0 ) ; \\ \mathbf{x}(\omega_a ) & = ( 1,0 ) + ( 1,1 ) + ( 2,1 ) + ( 3,1 ) + ( 3,0 ) \nonumber \\ & = ( 10,3);\\ x_2(\omega_a ) & = ( 1,0)\cdot(1,0 ) + ( 1,1)\cdot(1,1 ) + ( 2,1)\cdot(2,1 ) \nonumber \\ & \mathrel{\phantom{= } } + ( 3,1)\cdot(3,1 ) + ( 3,0)\cdot(3,0 ) \nonumber \\ & = 1 + 2 + 5 + 10 + 9 \nonumber \\ & = 27.\end{aligned}\ ] ] the observables @xmath126 , with @xmath127 , may be straightforwardly calculated from @xmath128 , @xmath129 , and @xmath130 . we give expressions for @xmath131 with @xmath132 , and note that higher euclidean - invariant moments can be obtained via @xmath133 ( @xmath134 ) ( these moments are calculated for @xmath17 in @xcite and for @xmath19 in @xcite ) . in addition we introduce another observable , @xmath135 , which measures the mean - square deviation of the walk from the endpoint @xmath136 . @xmath137 \nonumber \\ & = \frac{1}{2 } + \frac{1}{2}\mathbf{x}_{\mathrm{e } } \cdot \mathbf{x}_{\mathrm{e } } - \frac{1}{n } \hat{\mathbf{x}}_1 \cdot \mathbf{x } - \frac{1}{n } \mathbf{x}_{\mathrm{e } } \cdot \mathbf{x } + \frac{1}{n } x_2 \\ \mathcal{r}_{\mathrm{m}}^2 & = \frac{1}{n}\sum_{i=0}^{n-1 } |\omega(i)-\omega(n-1)|^2 \nonumber \\ & = \mathbf{x}_{\mathrm{e } } \cdot \mathbf{x}_{\mathrm{e } } - \frac{2}{n } \mathbf{x}_{\mathrm{e } } \cdot \mathbf{x } + \frac{1}{n } x_2 \label{eq : calrm}\end{aligned}\ ] ] in @xcite , we chose to calculate @xmath135 rather than @xmath12 , as it has a slightly simpler expression , and relied on the identity @xmath138 . compared with @xmath12 , @xmath135 has larger variance but smaller integrated autocorrelation time ( for the pivot algorithm ) . before performing the computational experiment in @xcite , we believed that given the same number of pivot attempts the confidence intervals for @xmath139 and @xmath140 would be comparable . we have since confirmed that working directly with @xmath12 results in a standard error which is of the order of 17% smaller for @xmath9 , an amount which is not negligible ; in future experiments we will calculate @xmath12 directly . here follow some comments to aid in the interpretation of the pseudo - code description of the saw - tree data structure and associated algorithms . * all calls are by value , following the c programming language convention . data structures are passed to methods via pointers . * pointers : the walk @xmath141 is a data structure whose member variables can be accessed via pointers , e.g. the vector for the end - to - end distance for the walk @xmath141 is @xmath142 . the left - hand sub - walk of @xmath141 is indicated by @xmath143 , and the right - hand sub - walk by @xmath144 . this notation is further extended by indicating @xmath145 for the left - hand sub - walk of @xmath143 , @xmath146 for the right - hand sub - walk of @xmath143 , etc .. * suggestive notation for member variables used to improve readability ; all quantities , such as `` @xmath128 '' ( the end - to - end vector ) must correspond to a particular walk @xmath141 . e.g. @xmath147 , @xmath148 ( i.e. , superscript @xmath149 indicates that @xmath150 is the end - to - end vector for the left sub - walk ) , @xmath151 , @xmath152 . * variables with subscript @xmath153 are used for temporary storage only . * comments are enclosed between the symbols / * and * / following the c convention . * boolean negation is indicated via the symbol `` ! '' , e.g. ! true = false . the key insight which has enabled a dramatic improvement in the implementation of the pivot algorithm is the recognition that _ sequences _ of sites and pivots can be replaced by _ binary trees_. the @xmath5 leaves of the tree are individual sites of the walk , and thus encode no information , while each of the @xmath50 ( internal ) nodes of the tree contain aggregate information about all sites which are below them in the tree . we call this data structure the saw - tree , which may be defined recursively : a saw - tree of @xmath5 sites either has @xmath154 and is a leaf , or has a left child saw - tree with @xmath155 sites , and a right child saw - tree with the remaining @xmath156 sites . our implementation of the saw - tree node is introduced in table [ tab : definition ] . a saw - tree consists of one or more saw - tree nodes in a binary tree structure ; the pointers @xmath143 and @xmath144 allow traversal from the root of the tree to the leaves , while @xmath157 allows for traversal from the leaves of the tree to the root . saw - trees are created by merging other saw - trees , with a symmetry operation acting on the right - hand walk . in particular , any _ internal node _ @xmath141 may be expressed in terms of its left child @xmath143 , a symmetry operation @xmath158 , and its right child @xmath144 via a merge operation : @xmath159 lll + type & name & description ' '' '' + _ integer _ & @xmath5 & number of sites ' '' '' + _ saw - tree ptr _ & @xmath157 & parent + _ saw - tree ptr _ & @xmath143 & left - hand sub - walk + _ saw - tree ptr _ & @xmath144 & right - hand sub - walk + _ matrix _ & q & symmetry group element + _ vector _ & @xmath128 & @xmath136 + _ vector _ & @xmath129 & @xmath160 + _ integer _ & @xmath130 & @xmath161 + _ bounding box _ & @xmath162 & convex region ' '' '' + the leaves of the saw - tree correspond to sites in a saw , and are thus labeled from 0 to @xmath50 . a binary tree with @xmath5 leaves has @xmath50 internal nodes , and we label these nodes from 1 to @xmath50 , so that the symmetry @xmath68 is to the left of @xmath163 . the symmetry @xmath164 is not part of the saw - tree as it is applied to the whole walk , and thus can not be used in a merge operation . for some applications it may be necessary to keep track of @xmath164 , e.g. when studying polymers in a confined region , but in @xcite this was not necessary . assume that the end - to - end vectors , @xmath128 , and symmetry group elements , @xmath68 , for a saw - tree and its left and right children are given . if we know the location of the anchor site of the parent node , @xmath165 , along with the overall _ absolute _ symmetry group element @xmath166 being applied to the walk , we can then find the same information for the left and right children as follows : @xmath167 thus @xmath85 can be determined for any site @xmath53 by iteratively performing this calculation while following the ( unique ) path from the root of the saw - tree to the appropriate leaf . n.b . : @xmath165 must be updated before @xmath166 . we give explicit examples of saw - trees in appendix [ sec : examplesawtrees ] . in fig . [ fig : sawtree_sequence ] , we give a saw - tree representation of a saw with @xmath5 sites which is precisely equivalent to the pivot sequence representation . we also give two equivalent representations of @xmath86 ( shown in fig . [ fig : example ] ) in figs . [ fig : sawtree_exampleaa ] and [ fig : sawtree_exampleab ] . conceptually we distinguish single - site walks ( individual sites ) , which reside in the leaves of the tree , from multi - site walks . in particular , the symmetry group element of a single site has no effect , and in the case where all monomers are identical then all single sites are identical . if the saw - tree structure remains fixed it is not possible to rotate part of the walk by updating a single symmetry group element , in contrast to the pivot sequence representation . this is because when we change a symmetry group element in a given node , it only alters the position of sites which are in the right child of the node . to rotate the part of the walk with sites labeled @xmath168 and greater , we choose the @xmath169 internal node of the saw - tree from the left . we then need to alter the symmetry group element of this node , and also all nodes which are above and to the right of it in the saw - tree . if we select a random node then it will likely be near the leaves of the tree , and assuming that the saw - tree is balanced this means that on average @xmath3 symmetry group elements will need to be altered . however , we note that the root node at the top of the tree has no parents , and therefore only one symmetry group element needs to be altered to rotate the right - hand part of the walk in this case . by utilizing tree - rotation operations , which alter the structure of the tree while preserving node ordering , it is possible to move the @xmath169 node to the root of the saw - tree . once this has been done , it _ is _ then possible to implement a rotation of part of the walk by updating a single symmetry group element . on average , @xmath3 of these tree - rotation operations are required . binary trees are a standard data structure in computer science . by requiring trees to be balanced , i.e. so that the height of a tree with @xmath0 nodes is bounded by a constant times @xmath170 , optimal bounds can be derived for operations such as insertion and deletion of nodes from the tree . we refer the interested reader to sedgewick @xcite for various implementations of balanced trees , such as red - black balanced trees . we have the advantage that our saw - tree is , essentially , static , which means that we can make it perfectly balanced without the additional overhead of maintaining a balanced tree . included in this subsection are the primitive operations , which would generally not be called from the main program . left and right tree - rotations are modified versions of standard tree operations ; for binary trees , only ordering needs to be preserved , while for saw - trees the sequence of sites needs to be preserved , which means that symmetry group elements and other variables need to be modified . * procedure : * [ cols="<,^ , < " , ] child node[leaf ] @xmath171 child node[leaf ] @xmath172 ; 0 * acknowledgments * i thank ian enting , tony guttmann , gordon slade , alan sokal , and two anonymous referees for useful comments on the manuscript . i would also like to thank an anonymous referee for comments on an earlier version of this article which led to deeper consideration of the algorithmic complexity of * shuffle_intersect*. i am grateful to tom kennedy for releasing his implementation of the pivot algorithm under the gnu gplv2 licence . computations were performed using the resources of the victorian partnership for advanced computing ( vpac ) . financial support from the australian research council is gratefully acknowledged . 0 lawler , g.f . , schramm , o. , werner , w. : on the scaling limit of planar self - avoiding walk . in : fractal geometry and applications : a jubilee of benoit mandelbrot , part 2 . pure math . 339364 . am . , providence ( 2004 ) sergio caracciolo , anthony j. guttmann , iwan jensen , andrea pelissetto , andrew n. rogers , and alan d. sokal , _ correction - to - scaling exponents for two - dimensional self - avoiding walks _ , j. stat . * 120 * ( 2005 ) , 10371100 . james t. klosowski , martin held , joseph s. b. mitchell , henry sowizral , and karel zikan , _ efficient collision detection using bounding volume hierarchies of @xmath173-dops _ , ieee t. vis . gr . * 4 * ( 1998 ) , 2136 . gregory f. lawler , oded schramm , and wendelin werner , _ on the scaling limit of planar self - avoiding walk _ , fractal geometry and applications : a jubilee of benoit mandelbrot , part 2 . pure math . soc . , providence , 2004 , pp .

the pivot algorithm for self - avoiding walks has been implemented in a manner which is dramatically faster than previous implementations , enabling extremely long walks to be efficiently simulated . we explicitly describe the data structures and algorithms used , and provide a heuristic argument that the mean time per attempted pivot for @xmath0-step self - avoiding walks is @xmath1 for the square and simple cubic lattices . numerical experiments conducted for self - avoiding walks with up to 268 million steps are consistent with @xmath2 behavior for the square lattice and @xmath3 behavior for the simple cubic lattice . our method can be adapted to other models of polymers with short - range interactions , on the lattice or in the continuum , and hence promises to be widely useful . 0 + + * keywords * self - avoiding walk ; polymer ; monte carlo ; pivot algorithm 0 0

1005.1444

let @xmath8 and @xmath1 be the space of @xmath2-frames in @xmath3 ( i.e. the space of @xmath2-tuples of linearly independent vectors in @xmath3 ) , @xmath9 . the group @xmath10 acts on this space as follows : @xmath11 the action is transitive for @xmath12 . let @xmath5 be a lattice in @xmath10 ; that is , a discrete subgroup in @xmath10 such that the factor space @xmath13 has finite volume ( e.g. @xmath14 ) . the main result of this paper concerns distribution of @xmath5-orbits in @xmath1 . when @xmath15 , every orbit of @xmath5 is discrete the situation becomes much more interesting for @xmath12 . let us recall known results : * ( dani , raghavan @xcite ) * [ th_dr ] let @xmath14 , and @xmath16 be an @xmath2-frame in @xmath3 , @xmath4 . then the orbit @xmath17 is dense in @xmath1 iff the space spanned by @xmath18 contains no nonzero rational vectors . * ( veech @xcite ) * [ th_ve ] if @xmath5 is a cocompact lattice in @xmath10 , then every orbit of @xmath5 in @xmath1 , @xmath4 , is dense . theorems [ th_dr ] and [ th_ve ] provide examples of dense @xmath5-orbits in @xmath1 . here we show that dense @xmath5-orbits are uniformly distributed with respect to an explicitly described measure on @xmath1 . this measure is @xmath19 , where @xmath20 is the lebesgue measure on @xmath21 , and @xmath22 is the @xmath2-dimensional volume of the frame @xmath23 . note that the measure @xmath20 is @xmath10-invariant , and it is unique up to a constant . however , orbits of @xmath5 are equidistributed with respect to the measure @xmath19 , which is not @xmath10-invariant . this phenomenon was already observed by ledrappier @xcite . define a norm on @xmath24 by @xmath25 for @xmath26 , @xmath27 , @xmath28 , put @xmath29 we determine asymptotic behavior of @xmath30 as @xmath31 . this result gives a quantitative strengthening of theorems [ th_dr ] and [ th_ve ] , and it can be interpreted as uniform distribution of dense orbits of @xmath5 in @xmath1 . [ th_frames00 ] let @xmath5 be a lattice in @xmath32 . let @xmath33 be an @xmath2-frame in @xmath3 such that @xmath34 is dense in @xmath35 . let @xmath36 be a relatively compact borel subset of @xmath35 such that @xmath37 . then @xmath38 where @xmath39 is a constant ( which is computed in ( [ eq_anl ] ) below ) , and @xmath40 is a @xmath10-invariant measure on @xmath13 ( which is defined in ( [ eq_mubar ] ) below ) . the term @xmath41 in ( [ eq_f_main00 ] ) comes from the asymptotics of the volume of the set @xmath42 in the stabilizer @xmath43 of @xmath44 with respect to the measure on @xmath43 which is determined by the choice of the haar measures on @xmath10 and @xmath45 ( see section [ sec_ttt ] ) . for @xmath46 and @xmath47 , this theorem was proved by ledrappier @xcite for general @xmath5 and by nogueira @xcite for @xmath48 and @xmath49-norm using different methods . combining theorems [ th_dr ] and [ th_frames00 ] , we get : [ th_frames ] let @xmath14 . let @xmath50 be an @xmath2-frame in @xmath3 such that the space @xmath51 contains no nonzero rational vectors . let @xmath36 be a relatively compact borel subset of @xmath35 such that @xmath37 . then @xmath52 where @xmath53 is a constant ( which is computed in ( [ eq_bnl ] ) below ) . figure [ pic1 ] shows a part of the the orbit @xmath54 for @xmath55 . by the result of ledrappier , this orbit is uniformly distributed in @xmath56 with respect to the measure @xmath57 . [ pic1 ] dani and raghavan also considered orbits of frames under @xmath58 . denote @xmath59 j= ( [ cols="^,^",options="header " , ] ) , @xmath59 and @xmath60 is a continuous function depending only on the @xmath61-components of @xmath62 . we can use proposition [ pro_assym ] with @xmath63 , @xmath64 , and @xmath65 . since @xmath34 is dense in @xmath66 , @xmath67 is dense in @xmath10 . by ( [ eq_last ] ) , the condition ( [ eq_h1 ] ) holds for @xmath63 . since @xmath68 is unipotent , the condition ( [ eq_h2 ] ) for @xmath63 holds too @xcite . applying proposition [ pro_assym ] , we get @xmath69 as @xmath31 , where @xmath70 is defined in ( [ eq_dh ] ) . thus , by ( [ eq_last ] ) , @xmath71 where @xmath72 to find the constant @xmath73 , we calculate measures of the set @xmath74 denote by @xmath75 the lebesgue measure of a @xmath76-dimensional unit ball . recall that @xmath77 clearly , @xmath78 for @xmath79 , @xmath80 , and @xmath81 , @xmath82 iff @xmath83 for @xmath84 . we have @xmath85 let as introduce new coordinates on @xmath86 : @xmath87 , @xmath88 . the haar measure on @xmath86 ( [ eq_da ] ) is given by @xmath89 . by ( [ eq_nnn ] ) , the set of @xmath90 such that @xmath91 is described by conditions : @xmath92 thus , @xmath93 in the last step , we have used ( [ eq_vball ] ) and the well - known identity for @xmath5-function and @xmath61-function . finally , by ( [ eq_vol1 ] ) and ( [ eq_vol2 ] ) , @xmath94 let @xmath95 for @xmath96 , and @xmath97 for @xmath98 , define @xmath99 note that @xmath100 . thus , it is enough to compute asymptotics of the function @xmath101 as @xmath102 . by tauberian theorem ( v , theorem 4.3 ) , it can be deduced from asymptotics of the function @xmath103 as @xmath104 . it is more convenient to work with the function @xmath105 let @xmath106 . one can check that @xmath107 for @xmath108 . ( in fact , each of the integral defines a right haar measure on @xmath109 . ) consider mellin transform of the function @xmath110 : @xmath111 using that @xmath112 , we get @xmath113 making substitution @xmath114 , we get @xmath115 by mellin inversion formula , for sufficiently large @xmath116 , @xmath117 since @xmath5-function decays fast on vertical strips , we can shift the line of integration to the left . by ( [ eq_fz ] ) , the first pole of @xmath118 occurs at @xmath119 . therefore , it follows from ( [ eq_fff ] ) that @xmath120 by ( [ eq_psi2 ] ) , @xmath121 finally , the asymptotic estimate for @xmath122 as @xmath102 follows from tauberian theorem ( * ? ? ? * ch . v , theorem 4.3 ) . we have @xmath123 this proves the lemma . note that @xmath127 we use the formula ( [ eq_rho_l ] ) for @xmath128 and make the change of variables @xmath129 for @xmath130 . the formula ( [ eq_rbtc ] ) follows from the fact that the volume of a unit ball in @xmath131 is @xmath132 . for @xmath133 , put @xmath134 we claim that @xmath135 as @xmath31 . if @xmath136 , then @xmath137 for every @xmath138 . then as in lemma [ lem_btc ] , @xmath139 now the claim follows from ( [ eq_btasy ] ) . since @xmath140 we have @xmath141 therefore , @xmath142 as @xmath31 . by theorem [ th_dr ] , @xmath143 is dense in @xmath1 . by theorem [ th_frames00 ] , ( [ eq_ntasy ] ) holds . the volume of @xmath13 was computed by minkowski . for the measure @xmath144 , we have @xmath145 ( see ( * ? ? ? * theorem 5.6 ) ) . 30 s. g. dani , g. a. margulis , _ limit distributions of orbits of unipotent flows and values of quadratic forms_. i. m. gelfand seminar , 91137 , adv . soviet math . , 16 , part 1 , ams , providence , ri , 1993 .

we study distribution of orbits of a lattice @xmath0 in the the space @xmath1 of @xmath2-frames in @xmath3 ( @xmath4 ) . examples of dense @xmath5-orbits are known from the work of dani , raghavan , and veech . we show that dense orbits of @xmath5 are uniformly distributed in @xmath1 with respect to an explicitly described measure . we also establish analogous result for lattices in @xmath6 that act on the space of isotropic @xmath7-frames .

math0310235

the casimir force , as exhibited between neutral metallic parallel plates , was discovered theoretically in 1948 @xcite . the casimir torque between asymmetric materials was first studied in 1973 @xcite . recently , theoretical study of the lateral casimir force between corrugated parallel plates was pioneered and developed by the mit group in @xcite . in particular , in @xcite , the authors evaluated analytic expressions for the lateral casimir force , to the leading order , between two corrugated parallel plates perturbatively . experimentally , the casimir interaction between corrugated surfaces was explored during the same period of time by roy and mohideen in @xcite . this experiment measured the lateral casimir force between a plate , with small sinusoidal corrugations , and a large sphere with identical corrugations . the motivation was to study the nontrivial boundary dependence in the casimir force . the experimental parameters in our notation are ( see figure [ corru ] ) : @xmath1 , @xmath2 , and @xmath3 , where @xmath4 is the height of the corrugations , @xmath5 is the wavelength of the corrugations , and @xmath6 is the mean distance between the plates . the corresponding dimensionless quantities are : @xmath7 , @xmath8 , and @xmath9 , where @xmath10 is the wavenumber related to the spatial wavelength of the corrugations . experimental data was analyzed based on the theoretical results obtained from the proximity force approximation ( pfa ) , and has been presented in @xcite . the validity of the pfa in the above analysis has been the topic of a recent debate and controversy , see @xcite . theoretical results based on perturbative approximations as done in @xcite do not settle the issue because the error keeping only the leading order may be high . it is generally believed that the next - to - leading - order calculation will be able to throw light on the issue . we carry out this calculation for the case of scalar fields . the analogous calculation for the electromagnetic case should now be straightforward . this paper in principle is an offshoot of @xcite where we shall deal with corrugated cylinders to study non - contact gears . while evaluating the leading order for the case of corrugated cylinders it was noticed that it would be possible to extend the calculation to the next - to - leading order . this led to the study in the present paper . in this installment we present the next - to - leading - order calculation for the case of corrugated parallel plates . the leading order calculation for the corrugated cylinders , which in itself is a significant result , will form the sequel @xcite of this paper . the next - to - leading - order calculation for the corrugated cylinders is in progress . in this section we shall describe the formalism and derive the key formula used for calculating the casimir energy . this has been done in various papers before , ( see @xcite , @xcite , @xcite , and references therein ) . we hope our derivation using schwinger s quantum action principle techniques will be illuminating . in an earlier paper @xcite describing the multiple scattering formalism it was mentioned that the use of the scattering matrix , @xmath11 , was equivalent to using the full green s function , and required the same computational effort . as a justification of this comment we exclusively use the full green s function in this article . let us consider a scalar field , @xmath12 , interacting with a scalar background potential , @xmath13 , described by the lagrangian density @xmath14 in terms of the source function , @xmath15 , corresponding to the scalar field , we write the action for this description to be @xmath16 = \int d^4x \big [ k(x)\phi(x ) + { \cal l}(\phi(x ) ) \big ] . \label{wpk}\ ] ] the vacuum to vacuum persistence amplitude , @xmath17 = \langle 0_+|0_- \rangle^k,\ ] ] which generates the amplitudes for all the physical processes , satisfies schwinger s quantum action principle , @xmath18 = i\,\langle 0_+|\,\delta w[\phi;k ] \,|0_- \rangle^k . \label{qap}\ ] ] our immediate task will be to get a formal solution for the vacuum amplitude , @xmath19 $ ] , in the form @xmath17 = e^{i \,w[k ] } , \label{z = eiw}\ ] ] where @xmath20 $ ] , which is not an action anymore , is dependent only on the source function . note that the action , @xmath21 $ ] in eq . , which satisfied the action principle , was described in terms of both the ( operator ) scalar field and the source function . variation with respect to the source function in the quantum action principle in eq . allows us to write @xmath22 } = \frac{1}{z[k ] } \frac{1}{i } \frac{\delta z[k]}{\delta k(x ) } , \label{eff - field}\ ] ] where the redefined scalar field , on the left of the above expression , is an effective field . this can be used to replace operator fields with functional derivatives with respect to the sources . variation with respect to the scalar field in eq . gives us @xmath23 \ , \frac{1}{i } \frac{\delta z[k]}{\delta k(x ) } = k(x ) z[k ] , \label{ginz = kz}\ ] ] which can be immediately inverted and written in the form , after using eq . , @xmath24 } \frac{1}{i } \frac{\delta z[k]}{\delta k(x ) } = \int d^4 x^\prime g(x , x^\prime ) k(x^\prime ) , \label{zinz = gz}\ ] ] where we defined the inverse of the differential operator , the green s function , as @xmath23 \ , g(x , x^\prime ) = \delta^{(4 ) } ( x - x^\prime ) . \label{green}\ ] ] the solution to eq . is a gaussian in the functional sense , and when written in the desired form in eq . , lets us identify @xmath25 = q[v ] + \frac{1}{2 } \int d^4 x \int d^4 x^\prime k(x ) g(x , x^\prime ) k(x^\prime),\ ] ] where @xmath26 $ ] is a functional of the background potential alone . for the case when the background potential is switched off , described by @xmath27 , we have @xmath28 = \text{exp}(i w_0[k])$ ] , where @xmath29 = q[0 ] + \frac{1}{2 } \int d^4 x \int d^4 x^\prime k(x ) g_0(x , x^\prime ) k(x^\prime ) , \label{free - act}\ ] ] where , @xmath30 is the corresponding green s function in eq . for the case when @xmath27 . now , in the absence of a source function the vacuum should not decay , which amounts to the statement @xmath31 . this implies that @xmath32 = 0 $ ] which when used in eq . lets us conclude that @xmath33 = 0 $ ] . variation with respect to the background potential in eq . yields @xmath18 = - \frac{i}{2 } \int d^4x \ , \delta v(x ) \,\langle 0_+| \,\phi(x ) \phi(x)\ , |0_- \rangle,\ ] ] where we can replace the operator fields with functional derivatives and then write the formal solution as @xmath17 = e^{- \frac{i}{2 } \int d^4x \ , v(x ) \frac{1}{i } \frac{\delta}{\delta k(x ) } \frac{1}{i } \frac{\delta}{\delta k(x ) } } e^{\frac{i}{2 } \int d^4 x \int d^4 x^\prime k(x ) g_0(x , x^\prime ) k(x^\prime)}.\ ] ] using the standard identity , we have @xmath25 = - \frac{i}{2 } \,\text{tr } \,\text{ln } \,g g_0^{-1 } + \frac{1}{2 } \int d^4 x \int d^4 x^\prime k(x ) g(x , x^\prime ) k(x^\prime ) , \label{action}\ ] ] which uses @xmath34 . using eq . in eq . we observe that in the presence of a background the vacuum to vacuum transition amplitude is not unity , instead it evaluates to @xmath35 } = e^{\frac{1}{2 } \,\text{tr } \,\text{ln } \,g g_0^{-1}}. \label{z = iw0}\ ] ] for the case when the process under investigation is time independent , denoting @xmath36 to be the time associated with the physical process , we can evolve the vacuum state using the hamiltonian of the system and thus conclude @xmath37 where we assumed the vacuum to be an eigenstate of the hamiltonian . comparing the two forms , eq . , we thus identify the energy of the vacuum in the presence of a background to be @xmath38 which serves as the central formula for calculating the casimir energy . further , for the case of a time independent situation , making a fourier transformation in the time variable and using translational independence in time , we can formally write @xmath39 + \frac{i}{2 } \int \frac{d\omega}{2\pi } \,\text{tr } \,\text{ln } \big [ - \omega^2 - \nabla^2 \big]\ ] ] where we used eq . to define the fourier transformed green s function in the form @xmath40 $ ] . integrating by parts and throwing away the boundary terms we derive @xmath41 which is an alternate expression for evaluating the casimir energy . this latter form has been used in studies @xcite related to surface divergences which requires the evaluation of the energy density rather than the total energy . we consider two semitransparent , corrugated plates , parallel to the @xmath42 plane , described by the potentials , @xmath43 where @xmath44 , are labels that identify the individual plates , and we define @xmath45 . the functions @xmath46 are designed to describe the corrugations associated with the individual plates . let us define the function @xmath47 which measures the relative corrugations between the two plates . we shall define the corrugations @xmath46 so that the mean of the relative corrugations is @xmath6 . in general we require @xmath48 , which is a restriction on the corrugations . translational invariance is assumed in the @xmath49 direction . the change in energy due to the change in the mean distance between the plates , @xmath6 , leads to the conventional casimir force which points in the direction perpendicular to the surface of the plates , and is expressed as @xmath50 where @xmath51 is the total casimir energy associated with the corrugated plates including the divergent contributions associated with the single plates . the divergent contributions being independent of @xmath6 , do not contribute to the casimir force . when the plates have corrugations on them we expect to have a change in the total energy due to a shift of one of the plates parallel ( lateral ) to the other plate . the force corresponding to the change in the total energy due to this shift acts in the lateral direction and is called the lateral casimir force . the shift is mathematically described by a translation in the @xmath52-coordinate , @xmath53 , which corresponds to a lateral shift of one plate with respect to the other . the lateral force is expressed as @xmath54 where @xmath55 is the measure of the translational shift . we note that there will be no lateral force between the plates if the corrugations are switched off by setting @xmath56 , @xmath57 and @xmath58 . the physical quantities associated with this configuration will thus act as a background , and a reference , and we shall find it convenient to denote them by the superscript @xmath59 to mean zeroth order . the potential for the background will thus be described by @xmath60 which has no @xmath52 dependence . the total casimir energy associated with the background , due to the two uncorrugated plates , will be denoted as @xmath61 , which will include the divergent contributions associated with the single plates . this background energy will be independent of the displacement @xmath55 due to the @xmath52 independence of this configuration . thus , we can conclude that @xmath62 we further note that there will be no lateral force between the plates if either one of the plates have their corrugations switched off by setting @xmath56 , @xmath57 or @xmath58 . the casimir energy associated with this configuration can be written as @xmath63 where @xmath64 is the additional contribution to the energy with respect to the background energy due to the presence of the corrugations on one of the plates . throughout this article we shall use @xmath65 to mean the deviation from the background . for example , we will have @xmath66 . using the argument of @xmath52 independence , or translational symmetry , we can again conclude that @xmath67 in light of the above observations we are led to write the total casimir energy , for the case when both the plates have their corrugations switched on , in the form @xmath68 where @xmath69 is the contribution to the total energy due to the interaction between the corrugations in the plates . only this part of the total energy contributes to the lateral casimir force . thus , we conclude that @xmath70 our central problem will be to evaluate @xmath69 using eq . . using the central formula derived in the multiple scattering formalism in eq to evaluate eq . we have @xmath71 where @xmath72 is the free green s function introduced in eq . . note that @xmath72 cancels in the above expression and the reference is now with respect to the uncorrugated plates . the green s function @xmath73 satisfies eq . with potentials described by eq . , which in symbolic notation reads @xmath74 g = 1,\ ] ] and the corresponding green s function associated with the background satisfies the differential equation , @xmath75 g^{(0 ) } = 1 . \label{g0eqn}\ ] ] the above two equations can be used to deduce @xmath76^{-1}.\ ] ] next , one makes the observation that the above expression can be rewritten in the form @xmath77^{-1 } g_1 { g^{(0)}}^{-1 } , \label{gg0 - 1}\ ] ] where @xmath78 ( @xmath44 ) are the green s functions for the parallel plates when only one of the plates has corrugations on it . the differential equations for @xmath78 s are @xmath79g_i = 1,\ ] ] which together with eq . can be used to deduce @xmath80 using eq . in eq . we we immediately obtain eq . . the last term in eq . , @xmath69 , which is the only term that contributes to the lateral force between the two corrugated plates , can be read out from eq . to be given by the expression @xmath81 . \label{de12}\ ] ] we could have written this down immediately , but we hope it was instructive to go through the steps because it clarifies our notation , which would anyhow have required us to write many of the above equations , and also to emphasize that the configuration due to parallel plates is treated as a background from the outset . formally expanding the logarithm in the above expression and using eq . to expand @xmath82 in terms of @xmath83 we can write @xmath84^m . \label{formal - series}\ ] ] our potentials in eq . can be formally expanded in powers of @xmath85 in the form @xmath86^n}{n ! } \frac{\partial^n}{\partial z^n } v_i^{(0)}(z ) = \lambda_i \big[e^{- h_i(y ) \frac{\partial}{\partial z}}-1\big]\delta ( z - a_i ) , \label{pot - series}\ ] ] so we can further write the series expansion as @xmath87\delta_1 \right\}^{n_1 } \left\ { g^{(0 ) } \big[e^{- h_2 \partial } - 1\big]\delta_2 \right\}^{n_2 } \bigg]^m,\ ] ] where we again use symbolic notation and suppress the variable dependence in @xmath85 , @xmath88 , and the delta functions . in this section we shall derive the leading order contribution to the casimir energy when the corrugation amplitude is small in comparison to the corrugation wavelength . this has been evaluated for the dirichlet scalar case and the electromagnetic case in @xcite . we obtain the result for scalar @xmath0-function potentials as a warm up exercise in preparation for the calculation for the next - to - leading order . in this section we shall illustrate our methodology which will be further used in the higher order calculation . for the particular case when the corrugations can be treated as small perturbations we can approximate the potentials by keeping a few terms in the expansion in eq . where we use the superscripts @xmath89 to represent the @xmath90-th order perturbation in a quantity . thus , to the leading order the interaction energy of the corrugations in eq . takes the form @xmath91 . \label{de12 - 2}\ ] ] we observe that the potentials in the @xmath90-th order are given by derivatives acting on @xmath0-functions in eq . which can be transferred over to the green s functions after integration by parts . the background green s function , which is a solution to eq . , can be written as @xmath92 where @xmath93 , with @xmath94 , where @xmath95 and @xmath96 are the fourier variables corresponding to the space - time coordinates @xmath97 and @xmath98 respectively . the reduced green s function , @xmath99 , satisfies the equation @xmath100 g^{(0)}(z , z^\prime;\kappa ) = \delta ( z - z^\prime ) . \label{g0zzp}\ ] ] in terms of the reduced green s function defined above we can write eq . as @xmath101 where @xmath102 is a large length in the @xmath49 direction and where @xmath103 are the fourier transforms of the functions @xmath46 , which describe the corrugations on the parallel plates , @xmath104 the kernel @xmath105 in eq . is suitably expressed in the form @xmath106 where we have switched to imaginary frequencies by a euclidean rotation , @xmath107 , and defined @xmath108 \bigg|_{\bar{z}=a_1 , z = a_2 } , \label{i2}\ ] ] where @xmath109 . using the reciprocal symmetry in the green s function in eq . we deduce that @xmath110 we evaluate the derivatives in the expression for @xmath111 , using the prescription described in appendix [ ddz - green ] , as @xmath112 , \label{i2-gen}\end{aligned}\ ] ] where @xmath113 s are given by eq . after replacing @xmath114 . for the case of the dirichlet limit ( @xmath115 ) the expression for @xmath111 in eq . takes on the relatively simple form @xmath116 where the subscript @xmath117 stands for dirichlet limit . using the above expression in eq . we have @xmath118 we describe the correspondence of the expression for the interaction energy , in eq . , in the dirichlet limit , to the result in @xcite in appendix [ app - emig ] . we shall digress a little to answer how the above result is related to the dirichlet green s function . we start by recalling the reduced green s function corresponding to eq . in the dirichlet limit , which can be derived by taking the @xmath119 limit in @xmath120 in eq . defined in region 2 ( @xmath121 ) in figure [ regions ] , @xmath122 where @xmath123 and @xmath124 stand for @xmath125 and @xmath126 respectively . it is trivially verified that the above function satisfies the dirichlet boundary conditions , @xmath127 and @xmath128 . using eq . we can evaluate @xmath129_{z_<=a_1,z_>=a_2 } = - \frac{\kappa}{\sinh \kappa a}.\ ] ] using the above result eq . can be expressed as the product of derivatives of two dirichlet green s functions . it is also worth mentioning that the result in eq . can also be derived by exclusively using @xmath120 in eq . after taking the limit @xmath119 . this is expected because the result in the dirichlet limit should depend only on the quantities between the plates . thus , all the results related to the dirichlet limit can be derived without relying on the averaging prescription for taking derivatives described in appendix [ ddz - green ] . however , for the more general case being considered here , we require the averaging prescription to derive eq . . for the weak coupling limit ( @xmath130 ) the expression for @xmath111 in eq . takes the very simple form @xmath131 where the subscript @xmath132 stands for the weak limit . we point out that in the case of weak coupling the averaging prescription for the evaluation of the @xmath133-kernels was not necessary because only the free green s functions come in . using the above expression in eq . we can evaluate the corresponding @xmath134-kernel as @xmath135 , \label{l2w}\end{aligned}\ ] ] where in the evaluation of the integral we used the change of variables @xmath136 , and the corresponding relation @xmath137 . for the particular case of sinusoidal corrugations , as described in figure [ corru ] , we will have @xmath138 , \\ h_2(y ) & = & h_2 \sin [ k_{0 } y],\end{aligned}\ ] ] [ sin - p ] where @xmath139 is the wavenumber corresponding to the corrugation wavelength @xmath5 . we get nonzero contributions , to the leading order , only for the case when both plates have the same corrugation wavelength . the fourier transforms @xmath103 for sinusoidal corrugations get contributions in the form of delta functions @xmath140 . \label{h - til}\ ] ] using the above expression in eq . , and after interpreting @xmath141 to be the infinite length in the @xmath52 direction , we write @xmath142 where we have used the symmetry property in the @xmath143-kernel , noted in eq . , and performed suitable rescaling in the integration variables . we have used the notations @xmath144 , and @xmath145 . the @xmath143-kernel in the above expression is provided by eq . . versus @xmath146 . , width=302 ] in the dirichlet limit , where the @xmath143-kernel is given by eq . , the interaction energy in eq . can be expressed in the form @xmath147 where the function @xmath148 is normalized such that @xmath149 , which becomes a convenience when we compare the results with those obtained by the proximity force approximation . the expression for @xmath148 is @xmath150 where the @xmath151-kernel , and the corresponding @xmath152-kernel , in the dirichlet limit were given in eq . and eq . respectively using the notations , @xmath153 , @xmath154 , and @xmath155 , we can write @xmath156 this expression can be numerically evaluated and has been plotted in figure [ a11-d - t0-fig ] . the value of the function at @xmath157 can be evaluated exactly by rewriting the integral in terms of spherical polar coordinates to yield @xmath158 ^ 2 = 1.\ ] ] it should be mentioned that the result corresponding to eq . for the electromagnetic case and the scalar dirichlet case have been evaluated exactly in @xcite . the lateral casimir force can be evaluated using eq . in the definition of lateral force in eq . which yields @xmath159 where @xmath160 is the magnitude of the normal scalar casimir force between two uncorrugated parallel dirichlet plates given as @xmath161 versus @xmath146 . , width=302 ] let us start by recalling the expression for the casimir energy and casimir force between uncorrugated parallel plates in the weak limit : @xmath162 in the weak limit we have seen that it is possible to evaluate the @xmath134-kernel in eq . without any effort . using this in eq . we have @xmath163,\ ] ] which after performing the integral can be written in the form @xmath164 where we have introduced the function @xmath165 = e^{-t_0 } \sum_{m=0}^2 \ , \frac{t_0^m}{m ! } = \frac{e_2(t_0)}{e^{t_0}},\ ] ] where @xmath166 is the the truncated exponential function which approximates to unity for @xmath167 . @xmath168 has been plotted in figure [ a2-w - t0-fig ] . using eq . we evaluate the lateral casimir force in this perturbation order as @xmath169 we start by collecting the terms in eq . in the form @xmath170 to economize typographical space and for bookkeeping purposes we introduce the notation @xmath171^{(m_1,m_2,\ldots , m_n)}_{(i_1,i_2,\ldots , i_n ) } = \text{tr } \big [ g^{(0 ) } \delta v_{i_1}^{(m_1 ) } g^{(0 ) } \delta v_{i_2}^{(m_2 ) } \cdots g^{(0 ) } \delta v_{i_n}^{(m_n ) } \big ] , \label{gv - not}\ ] ] where the superscripts @xmath172 denote the @xmath172-th order contribution in the power series expansion of the potential as defined in eq . . the subscripts @xmath173 identifies the potential that is contributing . in the case under consideration we have only two potentials coupling and thus @xmath174 for any @xmath90 . as an illustrative example we have @xmath171^{(2,1,1)}_{(1,1,2 ) } = \text{tr } \big [ g^{(0 ) } \delta v_1^{(2 ) } g^{(0 ) } \delta v_1^{(1 ) } g^{(0 ) } \delta v_2^{(1 ) } \big].\ ] ] using the above notation we can write @xmath175^{(2,1)}_{(1,2 ) } + [ g^{(0 ) } \delta v]^{(1,2)}_{(1,2 ) } - [ g^{(0 ) } \delta v]^{(1,1,1)}_{(1,1,2 ) } - [ g^{(0 ) } \delta v]^{(1,1,1)}_{(1,2,2 ) } \big\}.\end{aligned}\ ] ] we argue that for the case of sinusoidal corrugations of the same wavelength on both plates the third order perturbation does not contribute . ( contributions from third order are nonzero if the wavelength of the corrugations of one plate is double that of the other plate . ) since the plates under consideration are infinite in extent , the interaction energy is independent of translations in the @xmath52 direction . let us now make a translation of the amount @xmath176 in the potentials in eq . . this amounts to replacing @xmath177 . thus invariance under translation requires that the total power of @xmath85 s in the result should be even . this rules out the third order perturbation from contributing . starting from eq . and keeping terms contributing to the fourth order we have @xmath178 where the superscripts represent the powers of @xmath85 s . for example , the superscript @xmath179 represents terms involving @xmath180 . all terms except one gets contribution from @xmath181 in the logarithm expansion in eq . . since the @xmath182 term is distinct in structure we further separate it out by defining @xmath183 where the term involving the superscript @xmath184 is the contribution from @xmath182 in . using the notation in eq . we can thus collect the terms contributing to eq . and eq . as @xmath185^{(1,1,1,1)}_{(1,2,1,2 ) } \big\ } , \\ e_{12}^{(2,2)b } & = & \frac{i}{2\tau } \big\ { [ g^{(0)}\delta v]^{(1,1,1,1)}_{(1,1,2,2 ) } - [ g^{(0)}\delta v]^{(1,1,2)}_{(1,1,2 ) } - [ g^{(0)}\delta v]^{(2,1,1)}_{(1,2,2 ) } + [ g^{(0)}\delta v]^{(2,2)}_{(1,2 ) } \big\ } , \\ e_{12}^{(3,1 ) } & = & \frac{i}{2\tau } \big\ { [ g^{(0)}\delta v]^{(1,1,1,1)}_{(1,1,1,2 ) } - [ g^{(0)}\delta v]^{(1,2,1)}_{(1,1,2 ) } - [ g^{(0)}\delta v]^{(2,1,1)}_{(1,1,2 ) } + [ g^{(0)}\delta v]^{(3,1)}_{(1,2 ) } \big\ } , \\ e_{12}^{(1,3 ) } & = & \frac{i}{2\tau } \big\ { [ g^{(0)}\delta v]^{(1,1,1,1)}_{(1,2,2,2 ) } - [ g^{(0)}\delta v]^{(1,2,1)}_{(1,2,2 ) } - [ g^{(0)}\delta v]^{(1,1,2)}_{(1,2,2 ) } + [ g^{(0)}\delta v]^{(1,3)}_{(1,2 ) } \big\}.\end{aligned}\ ] ] [ de = gv - not ] in terms of the reduced green s function defined in eq . and eq . , and the fourier transform of the corrugations in eq . , we can write each of the terms in eq . in terms of a corresponding @xmath134-kernel . in terms of the notation introduced in eq . these will read as @xmath186^{(m_1,\ldots , m_n)}_{(i_1,\ldots , i_n ) } = l_x \int \frac{dk_1}{2\pi } \cdots \frac{dk_n}{2\pi } \,\tilde{h}_{i_1}^{m_1}(k_1-k_2 ) \cdots \tilde{h}_{i_n}^{m_n}(k_n - k_1 ) \,l^{(m_1,\ldots , m_n)}_{(i_1,\ldots , i_n)}(k_1,\ldots , k_n),\ ] ] where implicitly we have interpreted the powers of the fourier transformed corrugations as @xmath187 observe that each term contributing to the @xmath188-th order ( @xmath189 ) has exactly @xmath188 @xmath4 s . also , the total number of variables inside the @xmath134-kernel is @xmath190 which is in general less than or equal to @xmath188 . it might be appropriate to call them @xmath190-point kernels . proceeding further , in the spirit of the second order calculation in eq . , we introduce the corresponding @xmath133-kernels for the @xmath134-kernels as @xmath191 where the @xmath133-kernels are expressed in terms of derivatives operating on the reduced green s function defined in eq . as @xmath192 \bigg|_{z_n = a_{i_n}}. \label{imi}\ ] ] for clarification we illustrate the evaluation of a particular term which should also serve as an illustration of the notation . the term we consider is @xmath193^{(1,2,1)}_{(1,1,2 ) } & = & l_x \int \frac{dk_1}{2\pi } \frac{dk_2}{2\pi } \frac{dk_3}{2\pi } \,\tilde{h}_1(k_1-k_2 ) \tilde{h}_1 ^ 2(k_2-k_3 ) \tilde{h}_2(k_3-k_1 ) \,l^{(1,2,1)}_{(1,1,2)}(k_1,k_2,k_3 ) \nonumber \\ & = & l_x \int \frac{dk_1}{2\pi } \frac{dk_2}{2\pi } \frac{dk_3}{2\pi } \frac{dk_4}{2\pi } \,\tilde{h}_1(k_1-k_2 ) \,\tilde{h}_1(k_2-k_3 ) \tilde{h}_1(k_3-k_4 ) \,\tilde{h}_2(k_4-k_1 ) \,l^{(1,2,1)}_{(1,1,2)}(k_1,k_2,k_4 ) . \hspace{5mm}%\nonumber\end{aligned}\ ] ] notice how the particular @xmath194 dependence in the @xmath134-kernel is unambiguously specified . the corresponding @xmath133-kernel using eq . and eq . is given as @xmath195 \bigg|_{{\scriptstyle z_1=a_1 , z_2=a_1 , z_3=a_2}}.\ ] ] using the reciprocal symmetry in the green s functions we can learn the following symmetries in the @xmath133-kernels associated with the terms in eq . : @xmath196 [ i4sym ] we have not listed the 2-point @xmath133-kernels in the above list because they have the symmetry of the kind in eq . . in fact any 2-point kernel will have the following symmetry : @xmath197 in the above discussion involving very general notations we have in principle expressed each term in eq . . specific evaluation of each term involves the evaluation of the corresponding @xmath133-kernels which are given in terms of the derivatives of the green s functions . the derivatives are evaluated using the prescription described in appendix [ ddz - green ] . we can thus collect the terms in eq . into four distinct @xmath134-kernels in the form @xmath198 [ de12=l4 ] where it should be noted that different @xmath134 s combine with specific combination of @xmath4 s . the factor of one - half in eq . can be traced back to the coefficient of the second term in the expansion of logarithm in eq . . the respective kernels @xmath199 above are related to their corresponding @xmath200 by eq . , which are given by @xmath201 [ i4-cons ] proceeding in the spirit of section [ leading - order ] has brought us to the point of evaluation of the thirteen @xmath133-kernels on the right hand side of eq . . using the prescription described in appendix [ ddz - green ] we have evaluated all the thirteen kernels and explicit expressions analogous to eq . have been derived . this would have involved a lot more labor and bookkeeping if not for the facilitation achieved by the use of mathematica . we shall not display the explicit expressions here because they are too long . but , as in section [ leading - order ] , these expressions simplify considerably in the dirichlet limit and the weak coupling limit . observe that the reduced green s function defined by eq . has a well defined dirichlet limit . in the light of this observation in conjunction with the expression for the @xmath133-kernels in eq . taken at face value suggests that these kernels might not have a well - defined finite dirichlet limit . however , we evaluated the second order contribution in eq . . in fact it can be verified that all 2-point kernels have a well - defined finite dirichlet limit . further , @xmath202 also has a well - defined finite dirichlet limit . the rest of the nine @xmath133-kernels on the right hand side of eq . do not have a finite dirichlet limit . but , the sums of the @xmath133-kernels listed in eq . have finite dirichlet limits . the higher powers in @xmath203 in the numerator of each of these sums cancel identically to give a well - defined limit as @xmath119 . this seems to be a generic phenomena . here we list the @xmath133-kernels in eq . evaluated in the dirichlet limit : @xmath204 , \\ i^{(1,3)}_d(\kappa_1,\kappa_2,\kappa_3,\kappa_4 ) & = & -\frac{\kappa_1}{\sinh \kappa_1 a } \frac{\kappa_2}{\sinh \kappa_2 a } \frac{1}{4 } \left [ 4 \frac{\kappa_3}{\tanh \kappa_3 a } \frac{\kappa_4}{\tanh \kappa_4 a } + \frac{\kappa_1 ^ 2 + \kappa_2 ^ 2}{3 } - \kappa_3 ^ 2 - \kappa_4 ^ 2 \right].\end{aligned}\ ] ] [ i4d ] we can similarly evaluate the @xmath133-kernels in the weak limit by keeping terms to order @xmath205 in the general expressions for the @xmath133-kernels . but , we find it instructive , and simpler , to start over from eq . . we observe that only the 2-point kernels contribute in the weak limit . we further notice that only the contributions from the free green s function ( which is obtained by switching off the couplings @xmath203 in eq . ) contribute in the evaluation of the weak limit . these observations allows us to write the 2-point kernels in the weak limit as @xmath206_{z_1 = a_1 , z_2=a_2 } \nonumber \\ & = & \frac{(-1)^{m_1}}{m_1 ! \ , m_2 ! } \ , \frac{\lambda_1 \lambda_2}{2\kappa_1 2\kappa_2 } \ , ( \kappa_1 + \kappa_2)^{m } \ , e^{-a ( \kappa_1 + \kappa_2)},\end{aligned}\ ] ] where @xmath207 . note that the derivatives in the above expressions are well defined because they are evaluated at a point where @xmath208 . in particular using these in eq . with the observation that only the 2-point kernels contribute in the weak limit we get @xmath209 and @xmath210 [ i4w ] where notice how they differ in the particular dependence on the variables . we again ( see comments after eq . ) point out that the averaging prescription for the evaluation of the @xmath133-kernels was not necessary to arrive at the above expressions . using the change of variables introduced in evaluating eq . we can similarly evaluate the @xmath134-kernels using eq . in the weak limit as @xmath211 , \label{l2w - gen}\ ] ] which reproduces the result in eq . for @xmath212 we shall now specialize to the particular case of sinusoidal corrugations described by eq . . using the fourier transforms in eq . all but one of the @xmath194 integrals in eq . can be immediately performed . the symmetries listed in eq . lead to simplifications in the expressions . as in section [ leading - order ] the expressions boil down to two integrals in variables @xmath194 and @xmath213 . in particular eq . takes the form @xmath214 , \\ \frac{e^{(3,1)}_{12}}{l_xl_y } & = & - \cos ( k_0y_0 ) \ , \frac{h_1 ^ 3 h_2}{64\pi^2 } \int_{-\infty}^{\infty } dk\int_0^\infty \bar{\kappa } \,d\bar{\kappa } \big [ i^{(3,1)}(\kappa,\kappa_+,\kappa,\kappa_+ ) + i^{(3,1)}(\kappa,\kappa_+,\kappa,\kappa_- ) + i^{(3,1)}(\kappa,\kappa_-,\kappa,\kappa_+ ) \big ] , \hspace{8 mm } \\ \frac{e^{(1,3)}_{12}}{l_xl_y } & = & - \cos ( k_0y_0 ) \ , \frac{h_1 h_2 ^ 3}{64\pi^2 } \int_{-\infty}^{\infty } dk\int_0^\infty \bar{\kappa } \,d\bar{\kappa } \big [ i^{(1,3)}(\kappa,\kappa_+,\kappa,\kappa_+ ) + i^{(1,3)}(\kappa,\kappa_+,\kappa,\kappa_- ) + i^{(1,3)}(\kappa,\kappa_-,\kappa,\kappa_+ ) \big],\end{aligned}\ ] ] [ de12=i4 ] where the @xmath133-kernels are given from eq . . we use the notations @xmath215 and @xmath216 introduced after eq . . we note the factor of @xmath58 in the argument of cosine function of the first term above is a mark of the fourth order in perturbation theory . it should be mentioned here that in the above expressions we have omitted finite terms not having any dependence in the lateral shift variable @xmath55 . these terms do not contribute to the lateral force . we presented the expressions for the @xmath133-kernels in the dirichlet limit in eq . . using these in eq . and then adding the contributions from the three terms in eq . we can write the total contribution to interaction energy due to the presence of corrugations as @xmath217,\ ] ] where we have introduced the functions @xmath218 , \hspace{7 mm } \\ a^{(2,2)}_d(t_0 ) & = & \frac{1}{\pi^4 } \int_0^\infty \bar{s}\,d\bar{s } \int_{-\infty}^{\infty } dt \left [ \frac{s^2}{\sinh^2 s } \frac{s_-^2}{\sinh^2 s_- } + 2 \frac{s^2}{\tanh^2 s } \frac{s_+}{\sinh s_+ } \frac{s_-}{\sinh s_- } \right].\end{aligned}\ ] ] [ cols="^,^ " , ] a recent debate , see @xcite , involved the comparison of the pfa result and the perturbative result . we earlier noted that the pfa limit is obtained by taking the limit @xmath219 while keeping the ratio @xmath220 fixed . in figure [ pfa - and - exact - w - versus - hoa ] we plot the lateral force in the pfa limit and compare it with the non - perturbative result for various values of @xmath146 . this justifies our presumption that the pfa is a good approximation for @xmath221 . we note that the error in the pfa is less than 1% for @xmath221 for arbitrary @xmath220 . for @xmath222 , we observe that the pfa is a very good approximation for @xmath223 which satisfies @xmath224 . after viewing the plots for various values of @xmath225 we note that in general the pfa is a good approximation for @xmath223 and further beyond for offsets @xmath226 . it is , in fact , plausible that the pfa holds for large amplitude corrugations for small offsets because the corrugations fit together like fingers in a glove . we thank jef wagner for extensive collaborative assistance throughout this project . kvs would like to thank osama alkhouli , subrata bal , pravin chaubey , james dizikes , david hartnett , k. v. jupesh , sai krishna rao , s. shankar , and stphane valladier , for discussions , comments , and help with programming in mathematica . we thank steve fulling for constructive comments on the manuscript . we thank the us national science foundation ( grant no . phy-0554926 ) and the us department of energy ( grant no . de - fg02 - 04er41305 ) for partially funding this research . icp would like to thank the french national research agency ( anr ) for support through carnot funding . evaluation of eq . involves taking derivatives of the green s function which as we shall see involves evaluating a function at a point where it has jump discontinuities in the derivatives . we start by noting that the differential equation in eq . can be solved in terms of exponential functions in the different regions described in figure [ regions ] . explicitly the solutions are given by the following piecewise - defined functions , with subscripts denoting the regions they are defined in , @xmath227 , \\ g_2^{(0)}(z , z^\prime;\kappa ) & = & \frac{1}{2\kappa } \,e^{-\kappa |z - z^\prime| } - \frac{1}{\delta } \frac{1}{2\kappa } \left [ \frac{\lambda_2}{2\kappa } \big ( 1 + \frac{\lambda_1}{2\kappa } \big ) \,e^{\kappa ( z + z^\prime - 2a_2 ) } + \frac{\lambda_1}{2\kappa } \big ( 1 + \frac{\lambda_2}{2\kappa } \big ) \,e^{-\kappa ( z + z^\prime - 2a_1 ) } \right . \nonumber \\ & & \hspace{35 mm } \left . - \frac{\lambda_1}{2\kappa } \frac{\lambda_2}{2\kappa } \,e^{\kappa ( z - z^\prime - 2a ) } - \frac{\lambda_1}{2\kappa } \frac{\lambda_2}{2\kappa } \,e^{-\kappa ( z - z^\prime + 2a ) } \right ] , \label{g20zzp - sol } \\ g_3^{(0)}(z , z^\prime;\kappa ) & = & \frac{1}{2\kappa } \,e^{-\kappa |z - z^\prime| } - \frac{1}{\delta } \frac{1}{2\kappa } \,e^{-\kappa ( z+z^\prime ) } \left [ \frac{\lambda_1}{2\kappa } \,e^{2\kappa a_1 } + \frac{\lambda_2}{2\kappa } \,e^{2\kappa a_2 } - \frac{\lambda_1}{2\kappa } \frac{\lambda_2}{2\kappa } \big ( e^{2\kappa a_1 } - e^{2\kappa a_2 } \big ) \right ] , \\ g_4^{(0)}(z , z^\prime;\kappa ) & = & \frac{1}{\delta } \frac{1}{2\kappa } \left [ \big ( 1 + \frac{\lambda_2}{2\kappa } \big ) \,e^{\kappa ( z - z^\prime ) } - \frac{\lambda_2}{2\kappa } \,e^{\kappa ( z + z^\prime - 2a_2 ) } \right ] , \\ g_5^{(0)}(z , z^\prime;\kappa ) & = & \frac{1}{\delta } \frac{1}{2\kappa } \,e^{\kappa ( z - z^\prime ) } , \\ g_7^{(0)}(z , z^\prime;\kappa ) & = & \frac{1}{\delta } \frac{1}{2\kappa } \left [ \big ( 1 + \frac{\lambda_1}{2\kappa } \big ) \,e^{\kappa ( z - z^\prime ) } - \frac{\lambda_1}{2\kappa } \,e^{-\kappa ( z + z^\prime - 2a_1 ) } \right],\end{aligned}\ ] ] [ g0zzp - sol ] where @xmath228 using the reciprocal symmetry of the green s function we further have @xmath229 , @xmath230 , and @xmath231 . the above piecewise solution gives the complete green s function . domain on which the piecewise functions contributing to the green s function are defined.,width=188 ] it is of relevance to observe that the green s function represented above is continuous everywhere in the @xmath232 domain while its first derivatives have simple ( jump ) discontinuities along the lines , @xmath233 , @xmath234 , and @xmath235 . since the evaluation of @xmath133-kernels in eq . , and in eq . , involves taking derivatives of the green s function at the lines of discontinuity , a definite prescription for the evaluation of the derivatives is called for . for this purpose , we interpret the value of a function at the point where it has a jump discontinuity to be the average of all the possible limiting values . for example , the derivative of the green s function at the point @xmath236 in figure [ regions ] takes on four different values when approached from the four regions 2 , 6 , 8 , and 9 . the value of the derivative is defined as the average of these four values . we provide the following supporting arguments for the averaging prescription . firstly , we note that eq . was derived from eq after integrating over the complete @xmath237 domain . the potentials in eq involved derivatives of delta functions , and after integration by parts the integrals got contributions from a single point in the @xmath237 domain . since an integral is a sum we expect it to support the averaging prescription . the second justification we provide is in the way of verification for known examples . we made checks by using this prescription to analyze the green s function for the case of a single plate . we do not provide the details of the exercise here . evaluation of the derivatives using the averaging prescription is made convenient when we define the green s function around a point as the average of the suitable piecewise functions . thus , we evaluate @xmath238 = \frac{1}{\delta } \frac{1}{2\kappa } \,e^{-\kappa a } , \nonumber \\ g^{(0)}(a_2,a_1;\kappa ) & = & \frac{1}{4 } \big [ g_2^{(0)}(a_2,a_1;\kappa ) + g_6^{(0)}(a_2,a_1;\kappa ) + g_8^{(0)}(a_2,a_1;\kappa ) + g_9^{(0)}(a_2,a_1;\kappa ) \big ] = \frac{1}{\delta } \frac{1}{2\kappa } \,e^{-\kappa a } , \nonumber \\ g^{(0)}(a_1,a_1;\kappa ) & = & \frac{1}{4 } \big [ g_1^{(0)}(a_1,a_1;\kappa ) + g_2^{(0)}(a_1,a_1;\kappa ) + g_4^{(0)}(a_1,a_1;\kappa ) + g_6^{(0)}(a_1,a_1;\kappa ) \big ] = \frac{1}{\delta } \frac{1}{2\kappa } \left [ 1 + \frac{\lambda_2}{2\kappa } \,\big ( 1 - e^{-2\kappa a } \big ) \right ] , \nonumber \\ g^{(0)}(a_2,a_2;\kappa ) & = & \frac{1}{4 } \big [ g_2^{(0)}(a_2,a_2;\kappa ) + g_3^{(0)}(a_2,a_2;\kappa ) + g_7^{(0)}(a_2,a_2;\kappa ) + g_9^{(0)}(a_2,a_2;\kappa ) \big ] = \frac{1}{\delta } \frac{1}{2\kappa } \left [ 1 + \frac{\lambda_1}{2\kappa } \,\big ( 1 - e^{-2\kappa a } \big ) \right ] . \nonumber\end{aligned}\ ] ] the first derivatives evaluate to be @xmath239 where @xmath240 is derivative with respect to @xmath232 . the second derivatives involving two distinct variables evaluate to be @xmath241 \partial_z \partial_{z^\prime } \,g^{(0)}(z , z^\prime;\kappa ) \big|_{z = a_1,z^\prime = a_1 } & = & - \frac{1}{\delta } \frac{\kappa^2}{2\kappa } \left [ \big ( 1 + \frac{\lambda_1}{2\kappa } \big ) \big ( 1 + \frac{\lambda_2}{2\kappa } \big ) + \frac{\lambda_2}{2\kappa } \,e^{-2 \kappa a } \right ] , \\[2 mm ] \partial_z \partial_{z^\prime } \,g^{(0)}(z , z^\prime;\kappa ) \big|_{z = a_2,z^\prime = a_2 } & = & - \frac{1}{\delta } \frac{\kappa^2}{2\kappa } \left [ \big ( 1 + \frac{\lambda_1}{2\kappa } \big ) \big ( 1 + \frac{\lambda_2}{2\kappa } \big ) + \frac{\lambda_1}{2\kappa } \,e^{-2 \kappa a } \right].\end{aligned}\ ] ] the second derivatives involving the same variables evaluate to @xmath242 which when compared with eq . tells us that the averaging prescription practically throws away the delta function contributions in eq . . a more detailed study of this issue is being sought . since the second derivative returns the green s function back , all higher order derivatives are obtained in terms of the above expressions . here we wish to explicitly compare our results in the leading orderwith those in @xcite . in particular we show how the dirichlet limit of our expression for the interaction energy in the leading order [ eq . with eq . inserted ] matches with the results in @xcite [ eq . ( 11 ) with eqs . ( 13 ) and ( 17 ) there ] . to this end we begin from eq . and write the interaction energy in the form @xmath243 where we have used the notation introduced in @xcite : @xmath244 , @xmath245 , and switched to euclidean time by replacing @xmath246 . the @xmath247-kernel introduced above , following @xcite , is given as @xmath248 \bigg|_{z^\prime = a_1,z = a_2 } , \label{q = gg}\ ] ] where @xmath249 and @xmath250 is given in terms of eq . after switching to euclidean time as @xmath251 where @xmath252 and @xmath253 , @xmath254 . in the dirichlet limit the above expression should correspond to the @xmath247-kernel introduced in @xcite [ eq . ( 13 ) there ] . note that in @xcite the interaction energy is not isolated from the outset unlike in our eq . . this is achieved in @xcite by identifying and subtracting the contributions to the energy from the single plates . we identify the presence of our @xmath133-kernel [ see eq . ] in eq . which lets us express the dirichlet limit of the @xmath247-kernel , using eq . , in the form @xmath255 ^ 2 = \frac{1}{2 } \bigg [ \frac{1}{a^4 } p\big ( \frac{|{\bf y}-{\bf y}^\prime|}{a } \big ) \bigg]^2\ ] ] in terms of the integral @xmath256 this exact form for the @xmath247-kernel in the dirichlet limit reproduces the result in @xcite . this exact form for the @xmath247-kernel in the dirichlet limit suggests that we can consider a class of corrugations for which exact results , in the perturbative approximation , might be achievable . like the exact results achieved in @xcite for the weak coupling limit , we should be able to explore geometries in the dirichlet limit in the perturbative approximation starting from eq . . further , it should also be possible to extend these explorations to the next - to - leading orders . and , for very special geometries it might just be possible to transcend the perturbative approximation and get exact results in the dirichlet limit alone . we hope to be able to address some of these explorations in our forthcoming publications . h. b. g. casimir , kon . wetensch . proc . * 51 * , 793 ( 1948 ) . s. barash , izv . . radiofiz . * 16 * , 1086 ( 1973 ) [ sov . * 16 * , 945 ( 1973 ) ] . r. golestanian and m. kardar , phys . lett . * 78 * , 3421 ( 1997 ) [ arxiv : quant - ph/9701005 ] . r. golestanian and m. kardar , phys . a * 58 * , 1713 ( 1998 ) . t. emig , a. hanke , r. golestanian and m. kardar , phys . lett . * 87 * , 260402 ( 2001 ) [ arxiv : cond - mat/0106028 ] . t. emig , a. hanke , r. golestanian and m. kardar , phys . . a * 67 * , 022114 ( 2003 ) . r. buscher and t. emig , phys . rev . a * 69 * , 062101 ( 2004 ) [ arxiv : cond - mat/0401451 ] . a. roy and u. mohideen , phys . lett . * 82 * , 4380 ( 1999 ) . f. chen , u. mohideen , g. l. klimchitskaya , and v. m. mostepanenko , phys . lett . * 88 * , 101801 ( 2002 ) . f. chen , u. mohideen , g. l. klimchitskaya , and v. m. mostepanenko , phys . rev . a * 66 * , 032113 ( 2002 ) . r. b. rodrigues , p. a. maia neto , a. lambrecht , and s. reynaud , phys . lett . * 96 * , 100402 ( 2006 ) . f. chen , u. mohideen , g. l. klimchitskaya , and v. m. mostepanenko , phys . * 98 * , 068901 ( 2007 ) . r. b. rodrigues , p. a. maia neto , a. lambrecht , and s. reynaud , phys . * 98 * , 068902 ( 2007 ) . i. cavero - pelez , k. a. milton , p. parashar and k. v. shajesh , arxiv:0805.2777 [ hep - th ] . t. emig , n. graham , r. l. jaffe and m. kardar , phys . 025005 ( 2008 ) [ arxiv:0710.3084 [ cond-mat.stat-mech ] ] . o. kenneth and i. klich , phys . lett . * 97 * , 160401 ( 2006 ) [ arxiv : quant - ph/0601011 ] . k. a. milton and j. wagner , j. phys . a * 41 * , 155402 ( 2008 ) [ arxiv:0712.3811 [ hep - th ] ] . k. a. milton , j. phys . a * 37 * , 6391 ( 2004 ) [ arxiv : hep - th/0401090 ] . k. a. milton , p. parashar and j. wagner , in preparation . k. a. milton , p. parashar and j. wagner , arxiv:0806.2880 [ hep - th ] .

we calculate the lateral casimir force between corrugated parallel plates , described by @xmath0-function potentials , interacting through a scalar field , using the multiple scattering formalism . the contributions to the casimir energy due to uncorrugated parallel plates is treated as a background from the outset . we derive the leading- and next - to - leading - order contribution to the lateral casimir force for the case when the corrugation amplitudes are small in comparison to corrugation wavelengths . we present explicit results in terms of finite integrals for the case of the dirichlet limit , and exact results for the weak - coupling limit , for the leading- and next - to - leading - orders . the correction due to the next - to - leading contribution is significant . in the weak coupling limit we calculate the lateral casimir force exactly in terms of a single integral which we evaluate numerically . exact results for the case of the weak limit allows us to estimate the error in the perturbative results . we show that the error in the lateral casimir force , in the weak coupling limit , when the next - to - leading order contribution is included is remarkably low when the corrugation amplitudes are small in comparison to corrugation wavelengths . we expect similar conclusions to hold for the dirichlet case . the analogous calculation for the electromagnetic case should reduce the theoretical error sufficiently for comparison with the experiments .

0805.2776

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

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

1205.3804

the formation of jets is the most prominent feature of perturbative qcd in @xmath0 annihilation into hadrons . jets can be visualized as large portions of hadronic energy or , equivalently , as a set of hadrons confined to an angular region in the detector . in the past , this qualitative definition was replaced by quantitatively precise schemes to define and measure jets , such as the cone algorithms of the weinberg sterman @xcite type or clustering algorithms , e.g. the jade @xcite or the durham scheme ( @xmath1 scheme ) @xcite . a refinement of the latter one is provided by the cambridge algorithm @xcite . equipped with a precise jet definition the determination of jet production cross sections and their intrinsic properties is one of the traditional tools to investigate the structure of the strong interaction and to deduce its fundamental parameters . in the past decade , precision measurements , especially in @xmath0 annihilation , have established both the gauge group structure underlying qcd and the running of its coupling constant @xmath2 over a wide range of scales . in a similar way , also the quark masses should vary with the scale . a typical strategy to determine the mass of , say , the bottom - quark at the centre - of - mass ( c.m . ) energy of the collider is to compare the ratio of three - jet production cross sections for heavy and light quarks @xcite . at jet resolution scales below the mass of the quark , i.e. for gluons emitted by the quark with a relative transverse momentum @xmath1 smaller than the mass , the collinear divergences are regularized by the quark mass . in this region mass effects are enhanced by large logarithms @xmath3 , increasing the significance of the measurement . indeed , this leads to a multiscale problem since in this kinematical region also large logarithms @xmath4 appear such that both logarithms need to be resummed simultaneously . a solution to a somewhat similar two - scale problem , namely for the average sub - jet multiplicities in two- and three - jet events in @xmath0 annihilation was given in @xcite . we report here on the resummation of such logarithms in the @xmath1-like jet algorithms @xcite and provide some predictions for heavy quark production . a preliminary comparison with next - to - leading order calculations of the three - jet rate @xcite is presented . a clustering according to the relative transverse momenta has a number of properties that minimize the effect of hadronization corrections and allow an exponentiation of leading ( ll ) and next - to - leading logarithms ( nll ) @xcite stemming from soft and collinear emission of secondary partons . jet rates in @xmath1 algorithms can be expressed , up to nll accuracy , via integrated splitting functions and sudakov form factors @xcite . for a better description of the jet properties , however , the matching with fixed order calculations is mandatory . such a matching procedure was first defined for event shapes in @xcite . later applications include the matching of fixed - order and resummed expressions for the four - jet rate in @xmath0 annihilation into massless quarks @xcite . a similar scheme for the matching of tree - level matrix elements with resummed expressions in the framework of monte carlo event generators for @xmath0 processes was suggested in @xcite and extended to general collision types in @xcite . we shall recall here the results obtained in @xcite for heavy quark production in @xmath0 annihilation . in the quasi - collinear limit @xcite , the squared amplitude at tree - level fulfils a factorization formula , where the splitting functions @xmath5 for the branching processes @xmath6 , with at least one of the partons being a heavy quark , are given by @xmath7~ , \nnb \\ p_{gq}(z , q ) \!\!\!\ ! & = & \!\!\!\ ! t_r \left [ 1 - 2z(1-z ) + \frac{2z(1-z)m^2}{q^2+m^2 } \right]~ , \label{eq : pqq}\end{aligned}\ ] ] where @xmath8 is the usual energy fraction of the branching , and @xmath9 is the space - like transverse momentum . as expected , these splitting functions match the massless splitting functions in the limit @xmath10 for @xmath9 fixed . the splitting function @xmath11~.\end{aligned}\ ] ] obviously does not get mass corrections at the lowest order . branching probabilities are defined through @xcite [ full ] & & _ q(q , q , m ) = _ q / q^1-q / q dz p_qq(z , q ) + & & = _ q(q , q , m=0 ) + c_f , + & & _ f(q , q , m ) = _ q / q^1-q / q dz p_gq(z , q ) + & & = t_r , + & & _ g(q , q ) = _ q / q^1-q / q dz p_gg(z ) = 2c_a(- ) , with [ gamdef ] _ q(q , q , m=0 ) & = & 2c_f(-34 ) , and the sudakov form factors , which yield the probability for a parton experiencing no emission of a secondary parton between transverse momentum scales @xmath12 down to @xmath13 , read [ suddef ] & & _ q(q , q_0 ) = , + & & _ g(q , q_0 ) = + & & , + & & _ f(q , q_0 ) = ^2/_g(q , q_0 ) , where @xmath14 accounts for the number @xmath15 of active light or heavy quarks . jet rates in the @xmath1 schemes can be expressed by the former branching probabilities and sudakov form factors . for the two- , three- and four - jet rates [ jetrates ] _ 2 & = & ^2 , + _ 3 & = & 2 ^ 2 + & & _ q_0^q _ q(q , q)_g(q , q_0 ) , + _ 4 & = & 2 ^ 2 + & & \{^2 + & + & _ q_0^q + & + & _ q_0^q } , where @xmath12 is the c.m . energy of the colliding @xmath0 , and @xmath16 plays the role of the jet resolution scale . single - flavour jet rates in eq . ( [ jetrates ] ) are defined from the flavour of the primary vertex , i.e. events with gluon splitting into heavy quarks where the gluon has been emitted off primary light quarks are not included in the heavy jet rates but would be considered in the jet rates for light quarks . in order to catch which kind of logarithmic corrections are resummed with these expressions it is illustrative to study the above formulae in the kinematical regime such that @xmath17 . expanding in powers of @xmath2 , jet rates can formally be expressed as _ n = _ n2 + _ k = n-2^ ( ) ^k _ l=0 ^ 2k c^(n)_kl , where the coefficients @xmath18 are polynomials of order @xmath19 in @xmath20 and @xmath21 . the coefficients for the first order in @xmath2 are given by [ expand1 ] c^(2)_12 = -c^(3)_12 & = & -12 c_f ( l_y^2-l_m^2 ) , + c^(2)_11 = -c^(3)_11 & = & 32 c_f l_y + 12 c_f l_m . [ coef1 ] for second order @xmath2 , with @xmath22 active flavours at the high scale , the ll and nll coefficients read [ expand2 ] c^(2)_24 & = & 18 c_f^2 ( l_y^2 - l_m^2)^2 , + c^(3)_24 & = & -14 c_f^2 ( l_y^2 - l_m^2)^2 + & - & c_fc_a(l_y^4-l_m^4 ) , + c^(4)_24 & = & 18 c_f^2 ( l_y^2 - l_m^2)^2 + & + & c_fc_a(l_y^4-l_m^4 ) , + + c^(2)_23 & = & -c_f^2 ( l_y^2-l_m^2 ) ( 34 l_y - 14 l_m ) + & -&13_n c_f ( l_y^3 - 32 l_yl_m^2 + 12l_m^3 ) + & - & 13 ( _ n-_n-1 ) c_f l_m^3 , + c^(3)_23 & = & 12 c_f^2 ( l_y^2-l_m^2 ) ( 3 l_y - l_m ) + & + & 12_n c_f l_y ( l_y^2 - l_m^2 ) + & + & c_f c_a(3 l_y^3 - l_m^3 ) + & + & 16 ( _ n-_n-1 ) c_f l_m ( l_y^2 - l_y l_m + 2 l_m^2 ) , + c^(4)_23 & = & -c_f^2 ( l_y^2-l_m^2 ) ( 34 l_y - 14 l_m ) + & - & 16_n c_f ( l_y^3-l_m^3 ) + & - & 18 c_f c_a(l_y^3 - 13 l_m^3 ) + & - & 16 ( _ n-_n-1 ) c_f l_yl_m ( l_y - l_m ) . [ coef2 ] terms @xmath23 in the nll coefficients , where the @xmath24-function @xmath25 for @xmath22 active quarks is given by _ n = , are due to the combined effect of the gluon splitting into massive quarks and of the running of @xmath2 below the threshold of the heavy quarks , with a corresponding change in the number of active flavours . with our definition of jet rates with primary quarks the jet rates add up to one at nll accuracy . this statement is obviously realized in the result above order by order in @xmath2 . the corresponding massless result @xcite is obtained from eqs . ( [ coef1 ] ) and ( [ coef2 ] ) by setting @xmath26 . notice that eqs . ( [ coef1 ] ) and ( [ coef2 ] ) are valid only for @xmath27 and therefore @xmath10 does not reproduce the correct limit , which has to be smooth as given by eq.([jetrates ] ) . let us also mention that for @xmath28 there is a strong cancellation of leading logarithms and therefore subleading effects become more pronounced . an approximate way of including mass effects in massless calculations , that is sometimes used , is the `` dead cone '' @xcite approximation . the dead cone relies on the observation that , at leading logarithmic order , there is no radiation of soft and collinear gluons off heavy quarks . this effect can be easily understood from the splitting function @xmath29 in eq . ( [ eq : pqq ] ) . for @xmath30 this splitting function is not any more enhanced at @xmath31 . this can be expressed via the modified integrated splitting function [ deadcone ] _ q^d.c.(q , q , m ) & = & _ q(q , q , m=0 ) + & + & 2 c_f ( ) ( m - q ) . to obtain this result the massless splitting function has been used , which is integrated with the additional constraint @xmath32 . we also compare our results with this approximation . the impact of mass effects can be highlighted by two examples , namely by the effect of the bottom quark mass in @xmath0 annihilation at the @xmath33-pole , and by the effect of the top quark mass at a potential linear collider operating in the tev region . with @xmath34 gev , @xmath35 gev , and @xmath36 , the effect of the @xmath37-mass at the @xmath33-pole on the two- and three - jet rates is depicted in fig . [ bb91 ] ( left ) . the result obtained in the dead cone approximation is shown in fig . [ bb91 ] ( right ) . clearly , by using the full massive splitting function , the onset of mass effects in the jet rates is not abrupt as in the dead cone case and becomes visible much earlier . already at the rather modest value of the jet resolution parameters of @xmath38 , the two - jet rate , including mass effects , is enhanced by roughly @xmath39 with respect to the massless case , whereas the three - jet rate is decreased by roughly @xmath40 . for even smaller jet resolution parameters , the two - jet rate experiences an increasing enhancement , whereas the massive three - jet rate starts being larger than the massless one at values of the jet resolution parameters of the order of @xmath41 . the curves have been obtained by numerical integration of eq . ( [ jetrates ] ) . furthermore , in order to obtain physical result the branching probabilities have been set to one whenever they exceed one or to zero whenever they become negative . while in the case of bottom quarks at lep1 energies the overall effect of the quark mass is at the few - per - cent level , this effect becomes tremendous for top quarks at the linear collider ( fig . [ t1 ] ) . in fig . [ b91 ] , leading order ( lo ) and next - to - leading order ( nlo ) predictions for three - jet rates are compared with the nll result showed in the previous plots . fixed order predictions for @xmath37-quark production clearly fail at very low values of @xmath42 , by giving unphysical values for the jet rate , while the nll predictions keep physical and reveal the correct shape . the latter is an indication of the necessity for performing such kind of resummations . fixed order predictions work well for top production at the linear collider , a consequence of the strong cancellation of leading logarithmic corrections , and are fully compatible with our nll result . sudakov form factors involving heavy quarks have been employed to estimate the size of mass effects in jet rates in @xmath0 annihilation into hadrons . these effects are sizeable and therefore observable in the experimentally relevant region . a preliminary comparison with fixed order results have been presented , and showed good agreement . matching between fixed - order calculations and resummed results is in progress @xcite . it is a pleasure to thank the organizers of this meeting for the stimulating atmosphere created during the workshop , and m. mangano for very useful comments . g.r . acknowledges partial support from generalitat valenciana under grant ctidib/ 2002/24 and mcyt under grants fpa-2001 - 3031 and bfm 2002 - 00568 . y. l. dokshitzer , g. d. leder , s. moretti and b. r. webber , jhep * 9708 * ( 1997 ) 001 [ hep - ph/9707323 ] . g. rodrigo , a. santamaria and m. s. bilenky , phys . * 79 * ( 1997 ) 193 [ hep - ph/9703358 ] . p. abreu _ et al . _ [ delphi collaboration ] , phys . b * 418 * ( 1998 ) 430 . f. krauss and g. rodrigo , hep - ph/0303038 ; hep - ph/0309325 . g. rodrigo , nucl . phys . proc . suppl . * 54a * ( 1997 ) 60 [ hep - ph/9609213 ] . m. s. bilenky , s. cabrera , j. fuster , s. marti , g. rodrigo and a. santamaria , phys . d * 60 * ( 1999 ) 114006 [ hep - ph/9807489 ] . g. rodrigo , m. s. bilenky and a. santamaria , nucl . phys . b * 554 * ( 1999 ) 257 [ hep - ph/9905276 ] ; nucl . . suppl . * 64 * ( 1998 ) 380 [ hep - ph/9709313 ] . s. catani , l. trentadue , g. turnock and b. r. webber , nucl . b * 407 * ( 1993 ) 3 . l. j. dixon and a. signer , phys . d * 56 * ( 1997 ) 4031 [ hep - ph/9706285 ] . z. nagy and z. trocsanyi , phys . d * 59 * ( 1999 ) 014020 [ erratum - ibid . d * 62 * ( 2000 ) 099902 ] [ hep - ph/9806317 ] . s. catani , f. krauss , r. kuhn and b. r. webber , jhep * 0111 * ( 2001 ) 063 [ hep - ph/0109231 ] . f. krauss , jhep * 0208 * ( 2002 ) 015 [ hep - ph/0205283 ] . s. catani , s. dittmaier and z. trocsanyi , phys . b * 500 * ( 2001 ) 149 [ hep - ph/0011222 ] . s. catani , s. dittmaier , m. h. seymour and z. trocsanyi , nucl . b * 627 * ( 2002 ) 189 [ hep - ph/0201036 ] . y. l. dokshitzer , v. a. khoze and s. i. troian , j. phys . g * 17 * ( 1991 ) 1602 .

expressions for sudakov form factors for heavy quarks are presented . they are used to construct resummed jet rates in @xmath0 annihilation . predictions are given for production of bottom quarks at lep and top quarks at the linear collider .

hep-ph0309326

the physical origin of spatially extended ( tens to hundreds of kiloparsecs ) luminous ( @xmath4erg / s ) ly@xmath0 sources , also known as ly@xmath0 blobs ( labs ) first discovered more than a decade ago ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , remains a mystery . by now several tens of labs have been found ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? one fact that has confused the matter considerably is that they appear to be associated with a very diverse galaxy population , including regular lyman break galaxies ( lbgs ) ( e.g. , * ? ? ? * ) , ultra - luminous infrared galaxies ( ulirgs ) and sub - millimeter galaxies ( smgs ) ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , unobscured ( e.g. , * ? ? ? * ; * ? ? ? * ) and obscured quasars ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , or either starbursts or obscured quasars ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . an overarching feature , however , is that the vast majority of them are associated with massive halos or rich large - scale structures that reside in dense parts of the universe and will likely evolve to become rich clusters of galaxies by @xmath5 ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? another unifying feature is that labs are strong infrared emitters . for instance , most of the 35 labs with size @xmath6 kpc identified by @xcite in the ssa 22 region have been detected in deep spitzer observations @xcite . many physical models of labs have been proposed . a leading contender is the gravitational cooling radiation model in which gas that collapses inside a host dark matter halo releases a significant fraction of its gravitational binding energy in ly@xmath0 line emission ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? the strongest observational support for this model comes from two labs that appear not to be associated with any strong agn / galaxy sources @xcite , although lack of sub - mm data in the case of @xcite and a loose constraint of @xmath7yr@xmath8 ( @xmath9 ) in the case of @xcite both leave room to accommodate agn / galaxies powered models . another tentative support is claimed to come from the apparent positive correlation between velocity width ( represented by the full width at half maximum , or fwhm , of the line ) and ly@xmath0 luminosity @xcite , although the observed correlation fwhm @xmath10 appears to be much steeper than expected ( approximately ) fwhm @xmath11 for virialized systems . other models include photoionization of cold dense , spatially extended gas by obscured quasars ( e.g. , * ? ? ? * ; * ? ? ? * ) , by population iii stars ( e.g. , * ? ? ? * ) , or by spatially extended inverse compton x - ray emission ( e.g. , * ? ? ? * ) , emission from dense , cold superwind shells ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , or a combination of photoionization and gravitational cooling radiation ( e.g. , * ? ? ? the aim of this writing is , as a first step , to explore a simple star formation based model in sufficient details to access its physical plausibility and self - consistency , through detailed @xmath1 radiative transfer calculations utilizing a large set of massive ( @xmath12 ) starbursting galaxies from an ultra - high resolution ( @xmath13pc ) , cosmological , adaptive mesh refinement ( amr ) hydrodynamic simulation at @xmath14 . the most critical , basically the only major , free parameter in our model is the magnitude of dust attenuation . adopting the observationally motivated trend that higher sfr galaxies have higher dust attenuation , with an overall normalization that seems plausible ( e.g. , we assume that @xmath15% of @xmath1 photons escape a galaxy of @xmath16 yr@xmath8 ) , the model can successfully reproduce the global @xmath1 luminosity function and the luminosity - size relation of labs . to our knowledge this is the first model that is able to achieve this . the precise dependence of dust attenuation on sfr is not critical , within a reasonable range , and hence the results are robust . in this model we show that labs at high redshift correspond to proto - clusters containing the most massive galaxies / halos in the universe . within each proto - cluster , all member galaxies contribute collectively to the overall @xmath1 emission , giving rise to the diverse geometries of the apparent contiguous large - area lab emission , which is further enhanced by projection effects due to other galaxies that are not necessarily in strong gravitational interactions with the main galaxy ( or galaxies ) , given the strong clustering environment of massive halos in a hierarchical universe . this prediction that labs should correspond to the most overdense regions in the universe at high redshift is fully consistent with the observed universal association of labs with high density peaks ( see references above ) . the relative contribution to the overall @xmath1 emission from each individual galaxy depends on a number of variables , including dust attenuation of @xmath1 photons within the galaxy and propagation and diffusion processes through its complex circumgalactic medium and the intergalactic medium . another major predictions of this model is that a large fraction of the stellar ( and agn ) optical and ultraviolet ( uv ) radiation ( including ly@xmath0 photons ) is reprocessed by dust and emerges as infrared ( ir ) radiation , consistent with observations of ubiquitous strong infrared emission from labs . we should call this model simply starburst model " ( sbm ) , encompassing those with or without contribution from central agns . this model automatically includes emission contribution from gravitational cooling radiation , which is found to be significant but sub - dominant compared to stellar radiation . interestingly , we also find that @xmath1 emission originating from nebular emission ( rather than the stellar emission ) , which includes contribution from gravitational binding energy due to halo collapse , is more centrally concentrated than that from stars . one potentially very important prediction is that in this model the ly@xmath0 emission from photons that escape to us is expected to contain significant polarization signals . although polarization radiative transfer calculations will be performed to detail the polarization signal in a future study , we briefly elaborate the essential physics and latest observational advances here . one may broadly file all the proposed models into two classes in terms of the spatial distribution of the underlying energy source : central powering or in situ . starburst galaxy and agn powered models belong to the former , whereas gravitational cooling radiation model belongs to the latter . a smoking gun test between these two classes of models is the polarization signal of the ly@xmath0 emission . in the case of a central powering source ( not necessarily a point source ) the ly@xmath0 photons diffuse out , spatially and in frequency , through optically thick medium and escape by a very large number of local resonant scatterings in the ly@xmath0 line profile core and a relatively smaller number of scatterings in the damping wings with long flights . upon each scattering a ly@xmath0 photon changes its direction , location and frequency , dependent upon the geometry , density and kinematics of the scattering neutral hydrogen atoms . in idealized models with central powering significant linear polarizations of tens of percent on scales of tens to hundreds of kiloparsecs are predicted and the polarization signal strength increases with radius ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? on the other hand , in situ radiation from the gravitational cooling model is not expected to have significant polarizations ( although detailed modeling will be needed to quantify this ) or any systematic radial trend , because thermalized cooling gas from ( likely ) filaments will emit ly@xmath0 photons that are either not scattered significantly or have no preferential orientation or impact angle with respect to the scattering medium . an earlier attempt to measure polarization of labd05 at @xmath17 produced a null detection @xcite . a more recent observation by @xcite , for the first time , detected a strong polarization signal tangentially oriented ( almost forming a complete ring ) from lab1 at @xmath18 , whose strength increases with radius from the lab center , a signature that is expected from central powering ; they found the polarized fraction ( p ) of 20 percent at a radius of 45 kpc . @xcite convincingly demonstrate their detection and , at the same time , explain the consistency of their result with the non - detection by @xcite , if the emission from labd05 is in fact polarized , thanks to a significant improvement in sensitivity and spatial resolution in @xcite . this latest discovery lends great support to models with central powering , including sbm , independent of other observational constraints that may or may not differentiate between the two classes of models or between models in each class . but we stress that detailed polarization calculations will be needed to enable statistical comparisons . the outline of this paper is as follows . in 2.1 we detail simulation parameters and hydrodynamics code , followed by a description of our ly@xmath0 radiative transfer method in 2.2 . results are presented in 3 with conclusions given in 4 . we perform cosmological simulations with the amr eulerian hydro code , enzo @xcite . first we ran a low resolution simulation with a periodic box of @xmath19mpc ( comoving ) on a side . we identified a region centered on a cluster of mass of @xmath20 at @xmath5 . we then resimulate with high resolution of the chosen region embedded in the outer @xmath21mpc box to properly take into account large - scale tidal field and appropriate boundary conditions at the surface of the refined region . the refined region has a comoving size of @xmath22mpc@xmath23 and represents @xmath24 matter density fluctuation on that volume . the dark matter particle mass in the refined region is @xmath25 . the refined region is surrounded by three layers ( each of @xmath26mpc ) of buffer zones with particle masses successively larger by a factor of @xmath27 for each layer , which then connects with the outer root grid that has a dark matter particle mass @xmath28 times that in the refined region . we choose the mesh refinement criterion such that the resolution is always better than @xmath29pc ( physical ) , corresponding to a maximum mesh refinement level of @xmath30 at @xmath5 . the simulations include a metagalactic uv background @xcite , and a model for shielding of uv radiation by neutral hydrogen @xcite . they include metallicity - dependent radiative cooling @xcite . our simulations also solve relevant gas chemistry chains for molecular hydrogen formation @xcite , molecular formation on dust grains @xcite , and metal cooling extended down to @xmath31k @xcite . star particles are created in cells that satisfy a set of criteria for star formation proposed by @xcite . each star particle is tagged with its initial mass , creation time , and metallicity ; star particles typically have masses of @xmath32@xmath33 . supernova feedback from star formation is modeled following @xcite . feedback energy and ejected metal - enriched mass are distributed into 27 local gas cells centered at the star particle in question , weighted by the specific volume of each cell , which is to mimic the physical process of supernova blastwave propagation that tends to channel energy , momentum and mass into the least dense regions ( with the least resistance and cooling ) . we allow the entire feedback processes to be hydrodynamically coupled to surroundings and subject to relevant physical processes , such as cooling and heating . the total amount of explosion kinetic energy from type ii supernovae for an amount of star formed @xmath34 with a chabrier initial mass function ( imf ) is @xmath35 ( where @xmath36 is the speed of light ) with @xmath37 . taking into account the contribution of prompt type i supernovae , we use @xmath38 in our simulations . observations of local starburst galaxies indicate that nearly all of the star formation produced kinetic energy is used to power galactic superwinds ( e.g. , * ? ? ? supernova feedback is important primarily for regulating star formation and for transporting energy and metals into the intergalactic medium . the extremely inhomogeneous metal enrichment process demands that both metals and energy ( and momentum ) are correctly modeled so that they are transported in a physically sound ( albeit still approximate at the current resolution ) way . the kinematic properties traced by unsaturated metal lines in damped lyman - alpha systems ( dlas ) are extremely tough tests of the model , which is shown to agree well with observations @xcite . we use the following cosmological parameters that are consistent with the wmap7-normalized @xcite @xmath39cdm model : @xmath40 , @xmath41 , @xmath42 , @xmath43 , @xmath44 and @xmath45 . this simulation has been used @xcite to quantify partitioning of stellar light into optical and infrared light , through ray tracing of continuum photons in a dusty medium that is based on self - consistently computed metallicity and gas density distributions . we identify galaxies in our high resolution simulations using the hop algorithm @xcite , operated on the stellar particles , which is tested to be robust and insensitive to specific choices of concerned parameters within reasonable ranges . satellites within a galaxy are clearly identified separately . the luminosity of each stellar particle at each of the sloan digital sky survey ( sdss ) five bands is computed using the gissel stellar synthesis code @xcite , by supplying the formation time , metallicity and stellar mass . collecting luminosity and other quantities of member stellar particles , gas cells and dark matter particles yields the following physical parameters for each galaxy : position , velocity , total mass , stellar mass , gas mass , mean formation time , mean stellar metallicity , mean gas metallicity , star formation rate , luminosities in five sdss bands ( and various colors ) and others . at a spatial resolution of @xmath2pc ( physical ) with nearly 5000 well resolved galaxies at @xmath46 , this simulated galaxy catalog presents an excellent ( by far , the best available ) tool to study galaxy formation and evolution . the amr simulation resolution is @xmath2pc at @xmath47 . for each galaxy we produce a cylinder of size @xmath48 on a uniform grid of cell size @xmath49pc , where @xmath50 is the virial radius of the host halo . the purpose of using the elongated geometry is to incorporate the line - of - sight structures . subsequently , in our @xmath1 radiative transfer calculation , the line - of - sight direction is set to be along the longest dimension of the cylinder . in each cell of a cylinder @xmath1 photon emissivities are computed , separately from star formation and cooling radiation . the luminosity of @xmath1 produced by star formation is computed as @xmath51\ , { \rm erg\ , s^{-1}}$ ] @xcite , where sfr is the star formation rate in the cell . the @xmath1 emission from cooling radiation is computed with the gas properties in the cell by following the rates of excitation and ionization . with @xmath1 emissivity , neutral hydrogen density , temperature , and velocity in the simulations , a monte carlo code @xcite is adopted to follow the @xmath1 radiative transfer . the code has been recently used to study @xmath1 emitting galaxies @xcite . in our radiative transfer calculation , the number of @xmath1 photons drawn from a cell is proportional to the total @xmath1 luminosity in the cell , with a minimum number of 1000 , and each photon is given a weight in order to reproduce the luminosity of the cell . @xmath1 photons associated with star formation and cooling radiation are tracked separately so that we can study their final spatial distributions . for each photon , the scattering with neutral hydrogen atoms and the subsequent changes in frequency , direction , and position are followed until it escapes from the simulation cylinder . more details about the code can be found in @xcite and @xcite . the pixel size of the @xmath1 images from the radiative transfer calculation is chosen to be equal to @xmath49pc , corresponding to @xmath52 . we smooth the @xmath1 images with 2d gaussian kernels to match the resolutions in @xcite for detecting and characterizing labs from observation . in @xcite , the area of an lab is the isophotal area with a threshold surface brightness @xmath53 in the narrowband image smoothed to an effective seeing of fwhm 1.4@xmath54 ( slightly different from @xcite , where fwhm=1@xmath54 ) , while the @xmath1 luminosity is computed with the isophotal aperture in the fwhm=1@xmath54 image . we define labs in our model by applying a friends - of - friends algorithm to link the pixels above the threshold surface brightness in the computed @xmath1 images , with the area and luminosity computed from smoothed images with fwhm=1.4@xmath54 and fwhm=1@xmath54 , respectively . the sbm model that we study here in great detail may appear at odds with available observations at first sight . in particular , the labs often lack close correspondence with galaxies in the overlapping fields and their centers are often displaced from the brightest galaxies in the fields . as we show below , these puzzling features are in fact exactly what are expected in the sbm model . the reasons are primarily three - fold . first , labs universally arise in large halos with a significant number of galaxies clustered around them . second , dust attenuation renders the amount of ly@xmath0 emission emerging from a galaxy dependent substantially sub - linearly on star formation rate . third , the observed ly@xmath0 emission , in both amount and three - dimensional ( 3d ) location , originating from each galaxy depends on complex scattering processes subsequently . -0.0 cm 0.1 cm -0.0 cm -0.0 cm 0.1 cm -0.0 cm ( 0 ) we find that large - scale structure and clustering of galaxies play a fundamental role in shaping all aspects of labs , including two - dimensional line - of - sight velocity structure , line profile and @xmath1 image in the sky plane . to illustrate this , figure [ fig : map2 ] shows ly@xmath0 surface brightness maps ( after the radiative transfer calculation ) for four randomly selected galaxies with virial mass of the central galaxy exceeding @xmath55 at @xmath14 . we find that ly@xmath0 emission stemming from stellar radiation dominate over the gas cooling by about 10:1 to 4:1 in all relevant cases . we also find that the ly@xmath0 emission due to gas cooling is at least as centrally concentrated as from the stellar emission for each galaxy . from this figure it has become clear that large - scale structure and projection effects are instrumental to rendering the appearance of labs in all aspects ( image as well as spectrum ) . one could see that , for example in the top - left panel of figure [ fig : map2 ] , the approximately linear structure aligned in the direction of lower - left to upper - right is composed of three additional galaxies that are well outside the virial radius of the primary galaxy but from projected structures . at the @xmath56 detection isophotal contours of @xcite and @xcite for labs , the entire linear structure may be identified as a single lab . this rather random example is strikingly reminiscent of the observed lab structures ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? interestingly , depending on which galaxy is brighter and located on the front or back , the overall ly@xmath0 emission of the lab may show a variety of line profiles . for example , it could easily account for a broad / brighter blue side in the line profile , as noted by @xcite for some of the observed labs , which was originally taken as supportive evidence for the gravitational cooling radiation model . furthermore , it is not difficult to envision that the overall velocity width of an lab does not necessarily reflect the virial velocity of a virialized system and may display a wide range from small ( masked by caustics effect ) to large ( caused by either large virial velocities , infall velocities , or hubble expansion ) . a detailed spectral analysis will be presented elsewhere . for the results shown in figure [ fig : map2 ] we have not included dust effect , contributions from small galaxies ( @xmath57 ) that are not properly captured in our simulation due to finite resolution , and instrumental noise . we now describe how we include these important effects . although the resolution of our simulations is high , it is still finite and small sources are incomplete . we find that the star formation rate ( sfr ) function in the simulation flattens out at 3@xmath58 toward lower sfr at @xmath59 @xcite , which likely means that sources with sfr@xmath60 are unresolved / under - resolved and hence incomplete in the simulations . since these low sfr sources that cluster around large galaxies contribute to the @xmath1 emission of labs , it is necessary to include them in our modeling . for this purpose , we need to sample their sfr distribution and spatial distribution inside halos . first , we need to model the luminosity or sfr distribution of the faint , unresolved sources . in each lab - hosting halo in the simulations , the number of ( satellite ) sources with sfr@xmath613@xmath58 is found to be proportional to the halo mass @xmath62 . observationally , the faint end slope @xmath0 of the uv luminosity function of star forming galaxies is @xmath63 ( e.g. , * ? ? ? given this faint end slope , the contribution due to faint , unresolved sources is weakly convergent . as a result , the overall contribution from faint sources do not strongly depend on the faint limit of the correction procedure . we find that the conditional sfr function @xmath64 of faint sources ( sfr@xmath65 ) in halos can be modeled as @xmath66 where @xmath67 represents the sfr and @xmath68 , @xmath69 , and @xmath70 . this conditional sfr function allows us to draw sfr for faint sources to be added in our model . we now turn to the spatial distribution of faint sources . in the simulation the spatial distribution ( projected to the sky plane ) of satellite sources in halos is found to closely follow a power - law with a slope of @xmath71 . this is in good agreement with the observed small scale slope of the projected two - point correlation function of lbgs @xcite . there is some direct observational evidence that there are faint uv sources distributed within the lab radii . @xcite perform stacking analysis of @xmath72 @xmath1 emitters and protocluster lbgs , showing diffuse @xmath1 profile in the stacked @xmath1 image . interestingly , the profiles in the stacked uv images appear to be extended to scales of tens of kpc ( physical ) for the most luminous @xmath1 sources or for sources in protoclusters , suggesting contributions from faint , starforming galaxies . we add the contribution from faint sources to post - processed unsmoothed @xmath1 images from radiative transfer modeling as follows . for each model lab , we draw the number and sfrs of faint sources in the range of 0.013@xmath73 based on the conditional sfr distribution in equation ( [ eqn : csfrf ] ) . then we distribute them in the unsmoothed @xmath1 image in a radial range of 0.011@xmath50 by following the power - law distribution with slope @xmath71 . the faint sources can be either added as point or extended sources in @xmath1 emission . if added as point sources , they would be smoothed with a 2d gaussian kernel of fwhm=1.4@xmath54 or 1@xmath54 when defining lab size and luminosity . in our fiducial model , each faint source is added as an exponential disk with scale length of 3@xmath54 to approximate the radiative transfer effect , which is consistent with the observed diffuse emission profile of star - forming galaxies @xcite . we find that our final conclusion does not sensitively depend on our choice of the faint source @xmath1 profile . in figure [ fig : isophot ] , panel ( a ) shows the surface brightness and the @xmath53 isphotal contour for a model lab without including the faint sources , while panel ( b ) is the case with faint sources . we see that the size of the lab defined by the isophotal aperture does not change much . if the @xmath1 emission of each faint sources is more concentrated , e.g. , close to a point source in the unsmoothed image , the lab size can increase a little bit . therefore , in both panels ( a ) and ( b ) , the size is mainly determined by the central bright source . however , as will be described in the next subsection , including the effect of dust extinction will suppress the contribution of the central source and relatively boost that of the faint sources in determining the lab size . in the cases shown in figure [ fig : map2 ] , the central galaxies each have sfr that exceeds @xmath74 yr@xmath8 and is expected to be observed as a luminous infrared galaxy ( lirg ) or ulirg @xcite . this suggests that dust effects are important and have to be taken into account . in general , there are two types of effects of dust on @xmath1 emission from star - forming galaxies . the first one is related to the production of @xmath1 phtons . dust attenuates ionizing photons in star - forming galaxies . since @xmath1 photons come from reprocessed ionizing photons , the attenuation by dust leads to a lower @xmath1 luminosity in the first place . second , after being produced , @xmath1 photons can be absorbed by dust during propagation . a detailed investigation needs to account for both effects self - consistently , and we reserve that for a future study . in @xcite the dust obscuration / absorption is considered in a self - consistent way , with respect to luminosity functions observed in uv and fir bands . the modelling uses detailed ray tracing with dust obscuration model based on that of our own galaxy @xcite and extinction curve taken from @xcite . while the simultaneous match of both uv and fir luminosity functions at @xmath75 without introducing additional free parameters is an important validation of the physical realm of our simulations , it is not necessarily directly extendable to the radiative transfer of @xmath1 photons . nevertheless , it is reasonable to adopt a simple optical depth approach as follows for our present purpose , normalized by relevant observations , as follows . for each galaxy we suppress the initial intrinsic ly@xmath0 emission , by applying a mapping @xmath76 to @xmath77}$ ] , where the effective " optical depth @xmath78 is intended to account for extinction of @xmath1 photons as a function of sfr . we stress that this method is approximate and its validation is only reflected by the goodness of our model fitting the observed properties of labs . we adopt @xmath79^{0.6}$ ] . in reality , in addition , it may be that there is a substantial scatter in @xmath78 at a fixed sfr . we ignore such complexities in this treatment . the adopted trend that higher sfr galaxies have larger optical depths is fully consistent with observations ( e.g. , * ? ? ? * ) . at intrinsic @xmath80 the escaped @xmath76 luminosity is equivalent to @xmath81 , whereas at intrinsic @xmath82 the escaped @xmath76 luminosity is equivalent to @xmath83 . it is evident that the scaling of the emerging @xmath76 luminosity on intrinsic sfr is substantially weakened with dust attenuation . in fact , it may be common that , due to dust effect , the optical luminosity of a galaxy does not necessarily positively correlate with its intrinsic sfr , or the most luminous source in @xmath1 does not necessarily correspond to the highest sfr galaxy within an lab . as a result , a variety of image appearance and mis - matches between the lab centers and the most luminous galaxies detected in other bands may result , seemingly consistent with the anecdotal observational evidence mentioned in the introduction . the effect of dust on the surface brightness distribution for a model lab is shown in panel ( c ) of figure [ fig : isophot ] . compared to panel ( a ) , which is the model without dust effect , we see that surface brightness of the central source is substantially reduced and the isophotal area for the threshold @xmath53 also reduces . the case in panel ( c ) does not include the contribution from faint sources . in general , taking into account dust effect in our ly@xmath0 radiative transfer calculation , the central galaxies tend to make reduced ( absolutely and relative to other smaller nearby galaxies ) contributions to the ly@xmath0 surface brightness maps and in fact the center of each lab may or may not coincide with the primary galaxy that would likely be a ulirg in these cases , which is again reminiscent of some observed labs . in the next subsection , we describe the modeling results of combining all the above effects . by accounting for the line - of - sight structures , the unresolved faint sources , and the dust effect , we find that the observed properties of labs can be reasonably reproduced by our model . in panel ( d ) of figure [ fig : isophot ] , we add the faint sources and apply the dust effect . compared with the case in panel ( c ) , where no faint sources are added , the isophotal area increases . the central source has a substantially reduced surface brightness because of extinction . there appears to be another source near the central source , which corresponds to a source of lower sfr seen in panel ( a ) but with lower extinction than the central source . from figure [ fig : isophot ] , we see that the overall effect is that dust helps reduce the central surface brightness and faint sources help somewhat enlarge the isophotal area . 0.0 cm -3.0 cm 0.0 cm -3.0 cm to test the model and see the effect of different assumptions on extinction and faint sources , we compare the model predictions with observational properties of labs , shown in figure [ fig : pilot2 ] . in the top panels , we compare the luminosity - size relation defined by the isophot with surface brightness @xmath53 . the observed data points are taken from @xcite ( open squares ) and @xcite ( open circles ) , which has been supplemented with new , yet unpublished data ( matsuda 2012 , private communications ) . note that the isophotal area is defined with fwhm=1@xmath54 and 1.4@xmath54 images in @xcite and @xcite , respectively . this may partly explain that the lab sizes are somewhat larger with the @xcite data points . however , the difference is not substantial . our model data points follow @xcite in defining the luminosity and size . in the bottom panels of figure [ fig : pilot2 ] , we show the cumulative @xmath1 luminosity function or abundance of labs . the data points from @xcite and @xcite ( supplemented with new unpublished data ; matsuda 2012 , private communications ) have a large offset ( @xmath321 dex at the luminous end ) from each other , suggesting large sample variance . the survey volumes of @xcite and @xcite are @xmath84 and @xmath85 , respectively . for comparison , the volume of our parent simulation from which we choose our lab sample is only @xmath86 , much smaller than the volume probed by observation . the red points in top panel ( a ) of figure [ fig : pilot2 ] come from our model without extinction and faint sources . compared to the observational data , the model predicts more or less the correct slope in the luminosity - size relation . however , the overall relation has an offset , which means that the model either overpredicts the luminosity or underpredicts the size , or both . from the bottom panel ( a ) , the model greatly over - predicts the lab abundance , showing as a vertical shift . but it can also be interpreted as an overprediction of the lab luminosity , leading to a horizontal shift , which is more likely . because the central sources are bright , adding faint sources only slightly changes the sizes , as shown in panel ( b ) , which leads to little improvement in solving the mismatches in the luminosity - size relation and in the abundance . once the dust extinction effect is introduced , the situation greatly improves . panel ( c ) of figure [ fig : pilot2 ] shows the case with extinction but without adding faint sources . with the extinction included , the luminosity of the predicted labs drops , and at the same time , the size becomes smaller . now the model points agree well with observations at the lower end of the range of lab luminosity ( @xmath87erg / s ) and size ( 15 - 30 arcsec@xmath88 ) , the predicted luminosity - size relation conforms to and extends the observed one to still lower luminosity and smaller size . the predicted abundance is much closer to the observed one , as well . finally , panel ( d ) shows the case with both extinction and faint sources included . adding faint sources helps enlarge the size of an lab , because faint sources extends the isophot to larger radii . the luminosity also increases by including the contribution from faint sources . as a whole , the model data points appear to slide over the luminosity - size relation towards higher luminosity and larger size . the model luminosity - size relation , although still at the low luminosity end , is fully overlapped with the observed relation . the abundance at the high - luminosity end from the model is within the range probed by observation and shows a similar slope as that in @xcite . the agreement of the luminosity function between simulations and @xcite is largely fortuitous , reflecting that the overall bias of our simulation box over the underlying matter happens to be similar to that of the @xcite volume over matter , provided that the model universe is a reasonable statistical representation of the real universe . limited by the simulation volume , we are not able to directly simulate the full range of the observed luminosity and size of labs . our model , however , reproduces the luminosity - size relation and abundance in the low luminosity end . the most important ingredient in our model to achieve such an agreement with the observation is the dust extinction , which drives the apparent @xmath1 luminosity down into the right range . accounting for the contribution of faint , unresolved sources in the simulation also plays a role in further enhancing the sizes and , to a less extent , the luminosities of labs . to rectify the lack of high luminosity , large size labs in our simulations due to the limited simulation volume , we perform the following exercise . figure [ fig : halomass ] shows the @xmath1 luminosity and lab size as a function of halo mass from our model labs in figure [ fig : pilot2](d ) . both quantities correlate with halo mass , but there is a large scatter , which is caused by varying sfrs as well as different environmental effects for halos of a given mass . the largest labs fall into the range probed by the observational data and they reside in halos above @xmath89 . the model suggests that the vast majority of the observed labs should reside in proto - clusters with the primary halos of mass above @xmath89 at @xmath46 and on average larger labs correspond to more massive halos . note that the sources with halo mass below @xmath89 is highly incomplete here . our results suggest an approximate relation between the halo mass of the central galaxy and the apparent @xmath1 luminosity of the lab : @xmath90 which is shown as the solid curve in the left panel of figure [ fig : halomass ] . this relation should provide a self - consistency test of our model , when accurate halo masses hosting labs or spatial clustering of labs can be measured , interpreted in the context of the @xmath39cdm clustering model . we also find that the area - halo mass relation : @xmath91 shown as the solid curve in the right panel of figure [ fig : halomass ] . equations ( [ eqn : ml ] ) and ( [ eqn : ma ] ) lead to the following luminosity - size relation @xmath92 which matches the observed one , nothing new in this except as a self - consistency check . 0.0 cm -9.0 cm by extrapolating the above relations ( 2,3 ) to higher halo mass and using the analytic halo mass function @xcite , we can obtain the global @xmath1 lf expected from our model . in detail , we draw halo masses based on the analytic halo mass function . for each halo , we compute @xmath76 from equation ( [ eqn : ml ] ) . a scatter in @xmath93 is added following a gaussian distribution with 1@xmath94 deviation of 0.28dex ( indicated by the dotted lines in the left panel of figure [ fig : halomass ] ) . then equation ( [ eqn : al ] ) is used to assign the area , and a gaussian scatter of 0.11dex is added to approximately reproduce the scatter seen in the observed luminosity - size relation . the implied scatter in the area - halo mass relation is the sum of the above two scatters in quadrature , i.e. , about 0.30 dex , which is indicated by the dotted lines in the right panel of figure [ fig : halomass ] . finally , we adopt the same area cut ( @xmath6115 @xmath95 ) used in observations @xcite to define labs . our computed global @xmath1 lf of labs is shown as the blue curve in the bottom panel ( d ) . the agreement between our predicted global lf and that from the larger - survey - volume observations of @xcite is striking . given still substantial uncertainties involved in our model assumptions , the precise agreement is not to be overstated . however , the fact that the relative displacement between lf from our simulated volume and global lf is in agreement with that between @xcite and @xcite is quite encouraging , recalling that we have no freedom to adjust any cosmological parameters . this is also indicative of the survey volume of @xcite having becoming a fair sample of the universe for labs in question . the blue dots in top panel ( d ) show that the predicted luminosity - area relation is simultaneously in agreement with observations , now over the entire luminosity and size range , suggesting that our derived relations in equations ( [ eqn : ml ] ) , ( [ eqn : ma ] ) , and ( [ eqn : al ] ) are statistically applicable to labs of luminosities higher than those probed by the current simulations . we present a new model , termed star - burst model ( sbm ) , for the spatially extended ( tens to hundreds of kiloparsecs ) luminous ( @xmath96erg / s ) ly@xmath0 blobs . the sbm model is the first model to successfully reproduce both the global @xmath1 luminosity function and the luminosity - size relation of the observed labs @xcite . in the sbm model @xmath1 emission from both stars and gravitational sources ( such as gravitational binding energy released from structure collapse ) is included , although it is found that the nebular @xmath1 emission sourced by those other than stars , while significant , is sub - dominant compared to stellar emission . it is also found that @xmath1 emission originating from sources rather than stars is at least as centrally concentrated as that from stars within each galaxy . our modeling is based on a high - resolution large - scale cosmological hydrodynamic simulation of structure formation , containing more than @xmath97 galaxies with halo mass @xmath98 and more than @xmath99 galaxies with @xmath100 at @xmath14 , all resolved at a resolution of @xmath2pc or better . detailed 3d @xmath1 radiative transfer calculation is applied to sub - volumes centered on each of the 40 most massive star - bursting galaxies in the simulation box with @xmath101yr@xmath8 . a self - consistent working model emerges , if proper dust attenuation trend is modeled in that @xmath1 emission from higher sfr galaxies are more heavily attenuated by dust than lower sfr galaxies , which is empirically motivated by observations . for the results shown , we adopt an effective @xmath1 optical depth @xmath79^{0.6}$ ] , which translates to escape fractions of ( @xmath102 , @xmath103 ) for @xmath1 photons at @xmath104 , respectively . the dust attenuation model has two parameters , a normalization and a powerlaw index . the powerlaw index actually follows the slope of the metal column density dependence on sfr in the simulation . this thus leaves us with the normalization as the only free parameter . in practice , changing the powerlaw index does not sensitively change the results , as long as the normalization is adjusted such that the attenuation at high sfr end ( @xmath105yr@xmath8 ) is approximately the same as the adopted value , making the model rather robust . also very encouraging is that the model is in broad agreeement with other observed properties of labs , in addition to the simultaneous reproduction of the observed global @xmath1 luminosity function and the luminosity - size relation aforenoted . among them , we predict that labs at high redshift correspond to proto - clusters containing the most massive galaxies / halos in the universe and ubiquitous strong infrared emitters , with the most luminous member galaxies mostly copious in fir emission , fully consistent with extant observations ( e.g. , * ? ? ? * ; * ? ? ? it seems inevitable that some of the galaxies would contain active galactic nuclei ( agn ) at the epoch of peak agn formation in the universe ( e.g. , * ? ? ? while it is straight - forward to include , the results shown do not include agn , partly because , to the zero - th order , we may simply absorb " that by adjusting the dust attenuation effect and partly because observations indicate agn contribution is subdominant ( e.g. , * ? ? ? * ; * ? ? ? the most massive halos in the standard cold dark matter universe also tend be the most strongly clustered in the universe , among all types of galaxies , and we predict that there should be numerous galaxies clustered around labs ( e.g. , * ? ? ? @xcite use high - resolution _ hubble space telescope _ imaging to resolve galaxies within a giant lab at @xmath106 . they find many compact , low - luminosity galaxies . their observation becomes incomplete below @xmath107 , and with extrapolation there would be about 80 sources above @xmath108 within a radius of 7@xmath54 . their lab has @xmath109 and an isophotal area @xmath110 , falling well onto the observed luminosity - size relation shown in figure [ fig : pilot2 ] . extrapolating from our model , the lab is predicted to reside in a halo with mass of @xmath111 ( fig . [ fig : halomass ] ) . the number of faint sources within 7@xmath54 above @xmath108 from our model would be about 100 , in agreement with the observation . with the availability of alma , observers could make use of its superb capabilities to confirm the generic prediction of this model that there should be fir sources in each lab with the most luminous fir source likely representing the center of the proto - cluster . in combination with optical and other observations , this will potentially provide extremely useful information on the formation of galaxies in the most overdense regions of the universe when star formation is most vigorous and clusters have yet to be assembled . we highlight here that a potentially very discriminating signature of this model lies in the expected , significant polarization strength of the @xmath1 emission at large scales ( @xmath3210100kpc ) , which is not expected in some competing models for labs , such as those sourcing primarily gravitational binding energy on large scales due to massive halo formation . we plan to quantify this signal with detailed polarization radiative transfer calculations of @xmath1 photons . it is mentioned in passing that our model suggests the trends seen in labs , in terms of the global @xmath1 luminosity function and the luminosity - size relation of the observed labs , are continuously extended to less luminous @xmath1 emitters ( laes ) . consequently , we predict that laes , less luminous than labs , have smaller sizes compared to those of labs at a fixed isophotal level and should also be less strongly clustered than labs , forming an extension of the observed lab luminosity - size relation as well as the lab luminosity and correlation functions . finally , it is reassuring to note that the cosmological simulations themselves have already been subject to and passed a range of tests concerning a variety of observables of galaxies and the intergalactic medium , including properties of dlas at @xmath112 @xcite , o vi absorbers in the circumgalactic and intergalactic medium in the local universe @xcite , global evolution of star formation rate density and cosmic downsizing of galaxies @xcite , galaxy luminosity functions from @xmath5 to @xmath47 @xcite , and properties of galaxy pairs as a function of environment in the low-@xmath113 universe @xcite , among others . we would like to thank dr . yuichi matsuda for kindly providing and allowing us to use new observational data before publication . computing resources were in part provided by the nasa high - end computing ( hec ) program through the nasa advanced supercomputing ( nas ) division at ames research center . is supported in part by grant nnx11ai23 g . z.z . is supported in part by nsf grant ast-1208891 . the simulation data are available from the authors upon request . , c. r. , blain , a. , borys , c. j. k. , petty , s. , benford , d. , eisenhardt , p. , farrah , d. , griffith , r. l. , jarrett , t. , stanford , s. a. , stern , d. , tsai , c .- w . , wright , e. l. , & wu , j. 2012 , arxiv e - prints , j. e. , alexander , d. m. , lehmer , b. d. , smail , i. , matsuda , y. , chapman , s. c. , scharf , c. a. , ivison , r. j. , volonteri , m. , yamada , t. , blain , a. w. , bower , r. g. , bauer , f. e. , & basu - zych , a. 2009 , , 700 , 1 , e. , smith , k. m. , dunkley , j. , bennett , c. l. , gold , b. , hinshaw , g. , jarosik , n. , larson , d. , nolta , m. r. , page , l. , spergel , d. n. , halpern , m. , hill , r. s. , kogut , a. , limon , m. , meyer , s. s. , odegard , n. , tucker , g. s. , weiland , j. l. , wollack , e. , & wright , e. l. 2010 , arxiv e - prints , y. , nakamura , y. , morimoto , n. , smail , i. , de breuck , c. , ohta , k. , kodama , t. , inoue , a. k. , hayashino , t. , kousai , k. , nakamura , e. , horie , m. , yamada , t. , kitamura , m. , saito , t. , taniguchi , y. , tanaka , i. , & hibon , p. 2009 , , 400 , l66 , y. , yamada , t. , hayashino , t. , tamura , h. , yamauchi , r. , ajiki , m. , fujita , s. s. , murayama , t. , nagao , t. , ohta , k. , okamura , s. , ouchi , m. , shimasaku , k. , shioya , y. , & taniguchi , y. 2004 , , 128 , 569 , y. k. , suzuki , r. , tokoku , c. , ichikawa , t. , konishi , m. , yoshikawa , t. , omata , k. , nishimura , t. , yamada , t. , tanaka , i. , kajisawa , m. , akiyama , m. , matsuda , y. , yamauchi , r. , & hayashino , t. 2008 , , 60 , 683

we present a new model for the observed ly@xmath0 blobs ( labs ) within the context of the standard cold dark matter model . in this model , labs are the most massive halos with the strongest clustering ( proto - clusters ) undergoing extreme starbursts in the high - z universe . aided by calculations of detailed radiative transfer of @xmath1 photons through ultra - high resolution ( @xmath2pc ) large - scale ( @xmath3mpc ) adaptive mesh - refinement cosmological hydrodynamic simulations with galaxy formation , this model is shown to be able to , for the first time , reproduce simultaneously the global @xmath1 luminosity function and luminosity - size relation of the observed labs . physically , a combination of dust attenuation of @xmath1 photons within galaxies , clustering of galaxies , and complex propagation of @xmath1 photons through circumgalactic and intergalactic medium gives rise to the large sizes and frequently irregular isophotal shapes of labs that are observed . a generic and unique prediction of this model is that there should be strong far - infrared ( fir ) sources within each lab , with the most luminous fir source likely representing the gravitational center of the proto - cluster , not necessarily the apparent center of the @xmath1 emission of the lab or the most luminous optical source . upcoming alma observations should unambiguously test this prediction . if verified , labs will provide very valuable laboratories for studying formation of galaxies in the most overdense regions of the universe at a time when global star formation is most vigorous . = 1

1210.3600

investigations of sheath formation in front of a floating plate have hitherto been restricted to fluid studies on the ion time scale [ 1 ] . by contrast , the response of the plasma in the very early stages of sheath formation is not well known . in this paper , we present pic simulations of the plasma dynamics over just a few electron plasma periods after the beginning of the process . these simulations have been performed by means of the bit1 code [ 2 ] , developed on the basis of the xpdp1 code from u. c. berkeley [ 3 ] . a floating plate is placed in contact with a uniform , quasi - neutral plasma , which is assumed to be infinitely extended on one side . due to the higher thermal velocity of the electrons , the plate starts charging up negatively , so that electrons are gradually repelled , ions are attracted , and a positive - space - charge sheath begins to form . an electron plasma wave is observed the properties of which strongly depend on the plasma characteristics ( electron and ion temperatures , plasma density , etc . ) . our pic simulations are performed with different numerical set - ups and plasma characteristics . a full set of simulation diagnostics is used to measure the properties of the electron waves . we consider a one - dimensional system . the planar conducting plate and the ( artificial ) right - hand boundary of the systems are placed at positions @xmath5 and @xmath6 , respectively . the length @xmath7 is to be chosen large enough for this system to reasonably approximate a semi - infinite plasma ( @xmath8 , with @xmath9 the electron debye length ) . in order to have adequate resolution in space , the length of the grid cells has been selected as @xmath10 . at the initial time @xmath11 the electron and ion densities are equal @xmath12 , the distribution functions of both particle species are fully maxwellian , and the electric potential is zero @xmath13 everywhere in the system , including the plate surface . throughout the entire simulation , the following boundary conditions are applied to the particles : at the plate , all particles impinging are absorbed and no particles are injected into the plasma . at the right - hand boundary , on the other hand , all particles impinging are absorbed but new particles with half maxwellian distribution functions are injected at a constant rate . the system is floating , i.e. , the sum of particle plus displacement currents equals zero . according to these conditions we observe the following behavior . in the unperturbed plasma region ( i.e. , for @xmath14 ) the electron velocity distribution function will not change appreciably ( so that @xmath15 ) , whereas at the plate it will acquire a cut - off form . this is because the negative - velocity electrons are absorbed by the plate and charge it negatively ; during this process , the ions can be considered to be at rest . with increasing negative surface charge , the negative potential drop in the region close to the plate becomes higher and more and more electrons are reflected towards the plasma . after some time this perturbation propagates into the system . the shape of the distribution function essentially depends on the potential drop at the plate . due to the loss of particles by absorption at the plate , the total number of particles in the system is dropping all the time . however , this aspect is not of great concern here because the total loss of particles during the entire simulation presented is negligible . in the following tables we present the parameters used for our simulation . the ( electron and ion ) particle fluxes corresponding to the unperturbed plasma region are : @xmath16 these expressions are used to calculate the particle injection fluxes from the right - hand boundary . * parameter * & * value * & * remarks * + @xmath17 & @xmath18 & + @xmath19 & @xmath20 & + @xmath21 & @xmath22 & at @xmath23 + @xmath24 & @xmath25 & + @xmath26 & @xmath27 & + @xmath28 & @xmath29 & + @xmath30 & @xmath31 & + @xmath32 & @xmath33 & electron plasma frequency + @xmath34 & @xmath35 & ion plasma frequency + @xmath36 & @xmath37 & proton mass + @xmath9 & @xmath38 & + * parameter * & * value * & * remarks * + @xmath39 & @xmath40 & grid - cell length @xmath41 + @xmath7 & @xmath42 & system lenght + @xmath43 & @xmath44 & plate aria + @xmath45 & @xmath46 & time step + @xmath47 & @xmath48 & total simulation time + r0.5 figure [ pot_strat_sursa ] shows the potential profile close to the plate at @xmath49 s. the potential drop at the beginning of the sheath evolution is monotonic in space . after quick acquisition of negative charge , the plate repels the electrons in the form of a pulse leaving behind a positive - space charge region . as a result , the potential close to the plate becomes slightly positive . in front of this region , the negative space charge produced by the primary - pulse electrons leads to a potential minimum ( `` virtual cathode '' ) , which gradually reflects more and more slower electrons back into the plasma . these latter electrons spend a long time in the region of the virtual cathode and hence deepen its potential further . according to figures . [ evol_rho ] and [ cimp_strat ] , this first potential perturbation ( consisting of a potential hill and a potential well ) propagates into the unperturbed plasma , with additional similar wave structures forming behind it . r0.5 to verify that these waves are physical and not just due to numerical effects , we have performed other simulations with different parameters . in particular , we concentrated on the electron temperature . we know that the debye length is proportional to the square root of the electron temperature . hence , if we increase the temperature by a factor of four , the debye length must increase by a factor of two . since , in addition , there is a relation between the wavelength of the electron waves and the debye length , the variation of the electron temperature should also have an effect on the wavelength . this is clearly illustrated in + figure [ comparare ] , where the wavelength is seen to increase with the square root of the electron temperature . and @xmath50 _ , scaledwidth=70.0% ] this work represents the beginning of a self - consistent kinetic study of sheath formation , taking into account both electron and ion dynamics . here , during the short simulation time considered , the ions are practically immobile , and only the electrons take part in the process . in the next step , the effect of ion dynamics on sheath formation will be considered as well . this work was supported by the austrian science fund ( fwf ) projects p15013-n08 and p16807-n08 , ceepus network a103 , and erasmus / socrates grant 2004 - 2005 . * [ 1 ] * j.w . cipolla , jr . , and m. b. silevitch , on the temporal development of a plasma sheath , j. plasma phys . 25 , 373 - 89 ( jun 1981 ) * [ 3 ] * j. p. verboncoeur , m. v. alves , v. vahedi , and c. k. birdsall , simultaneous potential and circuit solution for 1d bounded plasma particle simulation codes , j. comput . 104 ( 2 ) , 321 ( 1993 ) . abstract submittal form

the problem of sheath formation in front of a conductive planar plate inserted into the plasma is formulated . initially , the plate is assumed to be neutral . it is shown that the charging - up process of the plate is accompanied by the excitation of electron plasma waves . _ @xmath0 plasma physics department , faculty of physics , al . i. cuza university , ro-700506 iasi , romania , + @xmath1j . stefan institute , university of ljubljana , jamova 39 , slo-1000 ljubljana , slovenia , + @xmath2 association euratom - oaw , department of theoretical physics , university of innsbruck , a-6020 innsbruck , austria , + @xmath3 permanent address : institute of physics , georgian academy of sciences , 380077 tbilisi , georgia , + @xmath4association euratom - oaw , department of ion physics , university of innsbruck , a-6020 innsbruck , austria _

physics0410242

studies of nuclei far from stability have long been a goal of nuclear science . nuclei on either side of the stability region , either neutron - rich or deficient , are being produced at new radioactive beam facilities across the world . at these facilities , and with the help of advances in nuclear many - body theory , the community will address many of the key physics issues including : mapping of the neutron and proton drip lines , thus exploring the limits of stability ; understanding effects of the continuum on weakly bound nuclear systems ; understanding the nature of shell gap modifications in very neutron - rich systems ; determining nuclear properties needed for astrophysics ; investigating deformation , spin , and pairing properties of systems far from stability ; and analyzing microscopically unusual shapes in unstable nuclei . the range and diversity of nuclear behavior , as indicated in the above list of ongoing and planned experimental investigations , have naturally engendered a host of theoretical models . short of a complete solution to the many - nucleon problem , the interacting shell model is widely regarded as the most broadly capable description of low - energy nuclear structure , and the one most directly traceable to the fundamental many - body problem . difficult though it may be , solving the shell - model problem is of fundamental importance to our understanding of the correlations found in nuclei . one avenue of research during the past few years has been in the area of the nuclear shell model solved not by diagonalization , but by integration . in what follows , we will describe the shell - model monte carlo ( smmc ) method and discuss several recent and interesting results obtained from theory . these include calculations in @xmath0-@xmath1-shell neutron - rich nuclei , a discussion of electron - capture rates in @xmath2-shell nuclei , pairing correlations in medium - mass nuclei near n@xmath3z , and studies of level densities in rare - earth nuclei . in the following we briefly outline the formalism of the smmc method . we begin with a brief description of statistical mechanics techniques used in our approach , then discuss the hubbard - stratonovich transformation , and end with a discussion of monte carlo sampling procedures . we refer the reader to previous works @xcite for a more detailed exposition . smmc methods rely on an ability to calculate the imaginary - time many - body evolution operator , @xmath4 , where @xmath5 is a real @xmath6-number . the many - body hamiltonian can be written schematically as @xmath7 where @xmath8 is a density operator , @xmath9 is the strength of the two - body interaction , and @xmath10 is a single - particle energy . in the full problem , there are many such quantities with various orbital indices that are summed over , but we omit them here for the sake of clarity . while the smmc technique does not result in a complete solution to the many - body problem in the sense of giving all eigenvalues and eigenstates of @xmath11 , it can result in much useful information . for example , the expectation value of some observable , @xmath12 , can be obtained by calculating @xmath13 here , @xmath14 is interpreted as the inverse of the temperature @xmath15 , and the many - body trace is defined as @xmath16 where the sum is over many - body states of the system . in the canonical ensemble , this sum is over all states with a specified number of nucleons ( implemented by number projection @xcite ) , while the grand canonical ensemble introduces a chemical potential and sums over _ all _ many - body states . in the limit of low temperature ( @xmath17 or @xmath18 ) , the canonical trace reduces to a ground - state expectation value . alternatively , if @xmath19 is a many - body trial state not orthogonal to the exact ground state , @xmath20 , then @xmath21 can be used as a filter to refine @xmath22 to @xmath20 as @xmath5 becomes large . an observable can be calculated in this `` zero temperature '' method as @xmath23 if @xmath24 is the hamiltonian , then ( [ eq_d ] ) at @xmath25 is the variational estimate of the energy , and improves as @xmath5 increases . of course , the efficiency of the refinement for any observable depends upon the degree to which @xmath22 approximates @xmath20 . beyond such static properties , @xmath21 allows us to obtain some information about the dynamical response of the system . for an operator @xmath12 , the response function , @xmath26 , in the canonical ensemble is defined as @xmath27 where @xmath28 is the imaginary - time heisenberg operator . interesting choices for @xmath12 are the annihiliation operators for particular orbitals , the gamow - teller , @xmath29 , or quadrupole moment , etc . inserting complete sets of @xmath30-body eigenstates of @xmath11 ( @xmath31 with energies @xmath32 ) shows that @xmath33 where @xmath34 is the partition function . thus , @xmath26 is the laplace transform of the strength function @xmath35 : @xmath36 hence , if we can calculate @xmath26 , @xmath35 can be determined . short of a full inversion of the laplace transform ( which is often numerically difficult ) , the behavior of @xmath26 for small @xmath37 gives information about the energy - weighted moments of @xmath38 . in particular , @xmath39 is the total strength , @xmath40 is the first moment ( the prime denotes differentiation with respect to @xmath37 ) . it is important to note that we usually can not obtain detailed spectroscopic information from smmc calculations . rather , we can calculate expectation values of operators in the thermodynamic ensembles or the ground state . occasionally , these can indirectly furnish properties of excited states . for example , if there is a collective @xmath41 state absorbing most of the @xmath42 strength , then the centroid of the quadrupole response function will be a good estimate of its energy . but , in general , we are without the numerous specific excitation energies and wave functions that characterize a direct diagonalization . this is both a blessing and a curse . the former is that for the very large model spaces of interest , there is no way in which we can deal explicitly with all of the wave functions and excitation energies . indeed , we often do nt need to , as experiments only measure average nuclear properties at a given excitation energy . the curse is that comparison with detailed properties of specific levels is difficult . in this sense , the smmc method is complementary to direct diagonalization for modest model spaces , but is the only method for treating very large problems . it remains to describe the hubbard - stratonovich `` trick '' by which @xmath21 is managed . in broad terms , the difficult many - body evolution is replaced by a superposition of an infinity of tractable one - body evolutions , each in a different external field , @xmath43 . integration over the external fields then reduces the many - body problem to quadrature . to illustrate the approach , let us assume that only one operator @xmath44 appears in the hamiltonian ( [ eq_a ] ) . then all of the difficulty arises from the two - body interaction , that term in @xmath11 quadratic in @xmath8 . if @xmath11 were solely linear in @xmath8 , we would have a one - body quantum system , which is readily dealt with . to linearize the evolution , we employ the gaussian identity @xmath45 here , @xmath46 is a one - body operator associated with a @xmath6-number field @xmath43 , and the many - body evolution is obtained by integrating the one - body evolution , @xmath47 , over all @xmath43 with a gaussian weight . the phase , @xmath48 , is @xmath49 if @xmath50 , or @xmath51 if @xmath52 . equation ( [ eq_j ] ) is easily verified by completing the square in the exponent of the integrand ; since we have assumed that there is only a single operator @xmath8 , there is no need to worry about non - commutation . for a realistic hamiltonian , there will be many non - commuting density operators , @xmath53 , present , but we can always reduce the two - body term to diagonal form . thus for a general two - body interaction in a general time - reversal invariant form , we write @xmath54 where @xmath55 is the time reverse of @xmath56 . since , in general , @xmath57\not=0 $ ] , we must split the interval @xmath5 into @xmath58 `` time slices '' of length @xmath59 , @xmath60^{n_t } , \label{eq_l}\ ] ] and for each time slice @xmath61 perform a linearization similar to eq . [ eq_j ] using auxiliary fields @xmath62 . note that because the various @xmath63 need not commute , the representation of @xmath64 must be accurate through order @xmath65 to achieve an overall accuracy of order @xmath66 . we are now able to write expressions for observables as the ratio of two field integrals . thus expectations of observables can be written as @xmath67 this is , of course , a discrete version of a path integral over @xmath43 . because there is a field variable for each operator at each time slice , the dimension of the integrals @xmath68 can be very large , often exceeding @xmath69 . the errors in eq . [ eq_m ] are of order @xmath66 , so that high accuracy requires large @xmath58 and perhaps extrapolation to @xmath70 ( @xmath71 ) . thus , the many - body observable is the weighted average ( weight @xmath72 ) of the observable @xmath73 calculated in an ensemble involving only the one - body evolution @xmath74 . similar expressions involving two @xmath43 fields ( one each for @xmath75 and @xmath76 ) can be written down for the response function ( [ eq_e ] ) , and all are readily adapted to the canonical or grand canonical ensembles or to the zero - temperature case . an expression of the form ( [ eq_m ] ) has a number of attractive features . first , the problem has been reduced to quadrature we need only calculate the ratio of two integrals . second , all of the quantum mechanics ( which appears in @xmath73 ) is of the one - body variety , which is simply handled by the algebra of @xmath77 matrices . the price to pay is that we must treat the one - body problem for all possible @xmath43 fields . we employ the metropolis , rosenbluth , rosenbluth , teller , and teller algorithm @xcite to generate the field configurations , @xmath43 , which requires only the ability to calculate the weight function for a given value of the integration variables . this method requires that the weight function @xmath72 must be real and non - negative . unfortunately , many of the hamiltonians of physical interest suffer from a sign problem , in that @xmath72 is negative over significant fractions of the integration volume . the fractional variance of a given expectation value becomes unacceptably large as the average sign approaches zero . it was shown @xcite that for even - even and @xmath78 nuclei there is no sign problem for hamiltonians if all @xmath79 . such forces include reasonable approximations to the realistic hamiltonian like pairing - plus - multipole interactions . however , for an arbitrary hamiltonian , we are not guaranteed that all @xmath80 ( see , for example , alhassid _ et al . _ . however , we may expect that a _ realistic _ hamiltonian will be dominated by terms like those of the schematic pairing - plus - multipole force ( which is , after all , why the schematic forces were developed ) so that it is , in some sense , close to a hamiltonian for which the mc is directly applicable . thus , the `` practical solution '' to the sign problem presented in alhassid _ et al . _ @xcite is based on an extrapolation of observables calculated for a `` nearby '' family of hamiltonians whose integrands have a positive sign . success depends crucially upon the degree of extrapolation required . empirically , one finds that for all of the many realistic interactions tested in the @xmath0- and @xmath1-shells , the extrapolation required is modest , amounting to a factor - of - two variation in the isovector monopole pairing strength . based on the above observation , it is possible to decompose @xmath11 in eq . [ eq_k ] into its `` good '' and `` bad '' parts , @xmath81 . the `` good '' hamiltonian , @xmath82 , includes , in addition to the one - body terms , all the two - body interactions with @xmath83 , while the `` bad '' hamiltonian , @xmath84 , contains all interactions with @xmath85 . by construction , calculations with @xmath82 alone have @xmath86 and are thus free of the sign problem . we define a family of hamiltonians , @xmath87 , that depend on a continuous real parameter @xmath88 as @xmath89 , so that @xmath90 , and @xmath91 is a function with @xmath92 and @xmath93 that can be chosen to make the extrapolations less severe . ( in practical applications , @xmath94 with @xmath95 , and applied only to the two - body terms in @xmath96 has been found to be a good choice . ) if the @xmath97 that are large in magnitude are `` good , '' we expect that @xmath98 is a reasonable starting point for the calculation of an observable @xmath99 . one might then hope to calculate @xmath100 for small @xmath101 and then to extrapolate to @xmath102 , but typically @xmath103 collapses even for small positive @xmath88 . however , it is evident from our construction that @xmath87 is characterized by @xmath86 for any @xmath104 , since all the `` bad '' @xmath105 are replaced by `` good '' @xmath106 . we can therefore calculate @xmath107 for any @xmath108 by a monte carlo sampling that is free of the sign problem . if @xmath109 is a smooth function of @xmath88 , it should then be possible to extrapolate to @xmath102 ( i.e. , to the original hamiltonian ) from @xmath108 . we emphasize that @xmath110 is not expected to be a singular point of @xmath109 ; it is special only in the monte carlo evaluation . the extrapolation methods we employ have been tested against standard shell - model diagonalizations in many cases , and have , in general , been shown to work very well @xcite . studies of extremely neutron - rich nuclei have revealed a number of intriguing new phenomena . two sets of these nuclei that have received particular attention are those with neutron number @xmath111 in the vicinity of the @xmath112 and @xmath113 shell closures ( @xmath114 and @xmath115 ) . experimental studies of neutron - rich mg and na isotopes indicate the onset of deformation , as well as the modification of the @xmath116 shell gap for @xmath117 mg and nearby nuclei @xcite . inspired by the rich set of phenomena occurring near the @xmath116 shell closure when @xmath118 , attention has been directed to nuclei near the @xmath119 ( sub)shell closure for a number of s and ar isotopes @xcite where similar , but less dramatic , effects have been seen as well . in parallel with the experimental efforts , there have been several theoretical studies seeking to understand and , in some cases , predict properties of these unstable nuclei . both mean - field @xcite and shell - model calculations @xcite have been proposed . the latter require a severe truncation to achieve tractable model spaces , since the successful description of these nuclei involves active nucleons in both the @xmath0- and the @xmath1-shells . the natural basis for the problem is therefore the full @xmath0-@xmath1 space , which puts it out of reach of exact diagonalization on current hardware . smmc methods offer an alternative to direct diagonalization when the bases become very large . though smmc provides limited detailed spectroscopic information , it can predict , with good accuracy , overall nuclear properties such as masses , total strengths , strength distributions , and deformation precisely those quantities probed by the recent experiments . it thus seems natural to apply smmc methods to these unstable neutron - rich nuclei . two questions will arise center - of - mass motion and choice of the interaction that are not exactly new , but demand special treatment in very large spaces . these questions were addressed in detail in ref . we present a brief selection of results here . there is limited experimental information about the highly unstable , neutron - rich nuclei under consideration . in many cases only the mass , excitation energy of the first excited state , the @xmath120 to that state , and the @xmath5-decay rate is known , and not even all of this information is available in some cases . from the measured @xmath120 , an estimate of the nuclear deformation parameter , @xmath121 , has been obtained via the usual relation @xmath122 with @xmath123 fm and @xmath120 given in @xmath124@xmath125 . much of the interest in the region stems from the unexpectedly large values of the deduced @xmath121 , results which suggest the onset of deformation and have led to speculations about the vanishing of the @xmath126 and @xmath119 shell gaps . the lowering in energy of the 2@xmath127 state supports this interpretation . the most thoroughly studied case , and the one which most convincingly demonstrates these phenomena , is @xmath117 mg with its extremely large @xmath128@xmath125 and corresponding @xmath129 @xcite ; however , a word of caution is necessary when deciding on the basis of this limited information that we are in the presence of well - deformed rotors : for @xmath130 mg , we would obtain @xmath131 , even more spectacular , and for @xmath132c , @xmath133 , well above the superdeformed bands . 1.5 in most of the measured observables can be calculated within the smmc framework . it is well known that in _ deformed _ nuclei the total @xmath120 strength is almost saturated by the @xmath134 transition ( typically 80% to 90% of the strength lies in this transition ) . thus the total strength calculated by smmc should only slightly overestimate the strength of the measured transition . in fig . [ fig_cm ] the smmc computed @xmath135 values are compared to the experimental @xmath136 values . reasonable agreement with experimental data across the space is obtained when one chooses effective charges of @xmath137 and @xmath138 . using these same effective charges , the usd values for the @xmath139 of the @xmath0-shell nuclei @xmath117 mg and @xmath140ne are 177.1 and 143.2 @xmath124@xmath125 , respectively , far lower than the full @xmath0-@xmath1 calculated and experimental values . all of the theoretical calculations require excitations to the @xmath1-shell before reasonable values can be obtained . we note a general agreement among all calculations of the @xmath120 for @xmath141ar , although they are typically larger than experimental data would suggest . we also note a somewhat lower value of the @xmath120 in this calculation as compared to experiment and other theoretical calculations in the case of @xmath142s @xcite . also shown in fig . [ fig_cm ] are the differences between experimental and theoretical binding energies for nuclei in this region . agreement is quite good overall . further details of the interaction and results may be found in @xcite . .comparisons of the smmc electron capture rates with the total ( @xmath143 ) and partial gamow - teller ( @xmath144 ) rates as given in ref . physical conditions at which the comparisons were made are @xmath145 , @xmath146 , and @xmath147 for the upper part of the table , and @xmath148 , @xmath149 , and @xmath150 for the lower part . [ cols="^,^,^,^ " , ] the impact of nuclear structure on astrophysics has become increasingly important , particularly in the fascinating , and presently unsolved , problem of type - ii supernovae explosions . one key ingredient of the precollapse scenario is the electron - capture cross section on nuclei @xcite . an important contribution to electron - capture cross sections in supernovae environments is the gamow - teller ( gt ) strength distribution . this strength distribution , calculated in smmc using eqs . ( 3 and 4 ) above , is used to find the energy - dependent cross section for electron capture . in order to obtain the electron - capture rates , the cross section is then folded with the flux of a degenerate relativistic electron gas @xcite . note that the gamow - teller distribution is calculated at the finite nuclear temperature which , in principle , is the same as the one for the electron gas . it is important to calculate the gt strength distributions reasonably accurately for both the total strength and the position of the main gt peak in order to have a quantitative estimate for the electron - capture rates . for astrophysical purposes , calculating the rates to within a factor of two is required . we concentrate here on mid-@xmath2-shell results for the electron - capture cross sections@xcite . the kuo - brown interaction @xcite , modified in the monopole terms by zuker and poves @xcite , was used throughout these @xmath1-shell calculations . this interaction reproduces quite nicely the ground- and excited - state properties of mid-@xmath2-shell nuclei @xcite , including the total gamow - teller strengths and distributions , where the overall agreement between theory and experiment @xcite is quite reasonable . the smmc technique allows one to probe the complete @xmath151 @xmath2-shell region without any parameter adjustments to the hamiltonian , although the gamow - teller operator has been renormalized by the standard factor of 0.8 . do the electron - capture rates presented here indicate potential implications for the precollapse evolution of a type ii supernova ? to make a judgment on this important question , one should compare in table i the smmc rates for selected nuclei with those currently used in collapse calculations @xcite . for the comparison , we choose the same physical conditions as assumed in tables 46 in aufderheide _ table i also lists the partial electron - capture rate which has been attributed to gamow - teller transitions @xcite . note that for even - parent nuclei , the present rate approximately agrees with the currently recommended _ total _ rate . a closer inspection , however , shows significant differences between the present rate and the one attributed to the gamow - teller transition in aufderheide _ et al . _ the origin of this discrepancy is due to the fact that fuller _ et al . _ @xcite places the gamow - teller resonance for even - even nuclei systematically at too high an excitation energy . this shortcoming has been corrected in fuller _ et al . _ @xcite and aufderheide _ et al . _ @xcite by adding an experimentally known low - lying strength in addition to the one attributed to gamow - teller transitions . however , the overall good agreement between the smmc results for even - even nuclei and the recommended rates indicates that the smmc approach also accounts correctly for this low - lying strength . this has already been deduced from the good agreement between smmc gamow - teller distributions and data including the low - energy regime @xcite . thus , for even - even nuclei , the smmc approach is able to predict the _ total _ electron - capture rate rather reliably , even if no experimental data are available . note that the smmc rate is somewhat larger than the recommended rate for @xmath152fe and @xmath153ni . in both cases , the experimental gamow - teller distribution is known and agrees well with the smmc results @xcite . while the proposed increase of the rate for @xmath153ni is not expected to have noticeable influence on the pre - collapse evolution , the increased rate for @xmath152fe makes this nucleus an important contributor in the change of @xmath154 during the collapse ( see table 15 of aufderheide _ et al . _ @xcite ) . for electron capture on odd-@xmath30 nuclei , observe that the smmc rates , derived from the gamow - teller distributions , are significantly smaller than the recommended total rate . this is due to the fact that for odd-@xmath30 nuclei the gamow - teller transition peaks at rather high excitation energies in the daughter nucleus . the electron - capture rate on odd-@xmath30 nuclei is therefore carried by weak transitions at low excitation energies . comparing the smmc rates to those attributed to gamow - teller transitions in fuller _ et al._@xcite and aufderheide _ et al._@xcite reveals that the latter have been , in general , significantly overestimated , which is caused by the fact that the position of the gamow - teller resonance is usually put at too low excitation energies in the daughter . the smmc calculation implies that the gamow - teller transitions should not contribute noticeably to the electron - capture rates on odd-@xmath30 nuclei at the low temperatures studied in tables 1416 in aufderheide _ et al._@xcite . thus , the rates for odd-@xmath30 nuclei given in these tables should generally be replaced by the non - gamow - teller fraction . we would now like to turn to the subject of pair correlations in nuclei , and calculations aimed at their understanding . nuclei near n@xmath3z offer a unique place to study proton - neutron pairing , particularly in the isospin t=1 channel . in fact , most heavy odd - odd , n@xmath3z nuclei beyond @xmath155ca have total spin j@xmath30 , t@xmath31 ground states . theoretical studies have shown that many of these nuclei have enhanced t@xmath31 proton - neutron correlations when compared to their even - even counterparts . these correlations are , to a lesser extent , present in even - even systems , but tend to decrease as one moves away from n@xmath3z . in at least one nucleus in the mass 70 region , @xmath156rb , there is experimental evidence for a ground state @xmath157 band @xcite . 0.5 in experimentally , pair correlations can best be measured by pair transfer on nuclei . although total cross sections are typically underpredicted when one employs spectroscopic factors computed from the shell model , relative two - nucleon spectroscopic factors within one nucleus are more reliable . therefore , it is necessary for one to calculate and measure pair transfer from both the ground and excited states in a nucleus . the smmc method may be used to calculate the strength distribution of the pair annihilation operator @xmath158 , as defined in koonin _ et al._@xcite . the total strength of these pairing operators , i.e. the expectation @xmath159 , has been studied previously as a function of mass , temperature @xcite , and rotation @xcite . we would like to briefly present here the strength distributions of the pair operators as calculated in smmc . the strength distribution for the pair transfer spectroscopic factors is proportional to @xmath160 and is calculated by the inversion of eq . ( 4 ) . in future work , we will discuss the strength distributions in detail . here we would like to briefly conclude by demonstrating that the smmc results and the direct diagonalization results agree very nicely for the proton pair strength distributions in the ground state of @xmath141v . this is demonstrated in the left panel of fig . 2 . shown in the right panel is the isovector proton - neutron pairing strength distribution with respect to the daughter nucleus . notice that the overall strength is much larger in the proton - neutron channel , as discussed previously in langanke _ @xcite , and that the peak is several mev lower in excitation relative to the like - particle channel . in both cases the strength distribution in @xmath141v differs significantly from that found in @xmath161cr , where one finds that the dominant component is a ground - state to ground - state transition involving mainly particles in the @xmath113 single - particle state . in both odd - odd n@xmath3z channels , the distribution is fairly highly fragmented . we have recently applied smmc techniques to survey rare - earth nuclei in the dy region . this extensive study formed the thesis topic of j.a . white @xcite , whose goal was to examine how the phenomenologically motivated `` pairing - plus - quadrupole '' interaction compares in exact shell - model solutions with other methods . we also examined how the shell - model solutions compare with experimental data ; static path approximation ( spa ) calculations were also performed . there have been efforts recently by others to use spa calculations , since they are simpler and faster ( see @xcite as examples ) . however , we found that spa results are not consistently good . this study was also designed to investigate whether the phenomenological pairing - plus - quadrupole - type interactions can be used in exact solutions for large model spaces , and whether the interaction parameters require significant renormalization when using spa . we discuss here one particular aspect of that work , namely level density calculations . details may be found in @xcite . we used the kumar - baranger hamiltonian with parameters appropriate for this region . our single - particle space included the 50 - 82 subshell for the protons and the 82 - 126 shell for the neutrons . while several interesting aspects of these systems were studied in smmc , we limit our discussion here to the level densities obtained for @xmath162dy . smmc is an excellent way to calculate level densities . @xmath163 is calculated for many values of @xmath5 which determine the partition function , @xmath164 , as @xmath165=-\int_0^\beta d\beta'e(\beta')\ ] ] @xmath166 is the total number of available states in the space . the level density is then computed as an inverse laplace transform of @xmath164 . here , the last step is performed with a saddle point approximation with @xmath167 : @xmath168 smmc has been used recently to calculate level density in iron region nuclei @xcite , and here we demonstrate its use in the rare - earth region . the comparison of smmc density in @xmath162dy with the tveter et al . @xcite data is displayed in fig . [ fig : dy162 ] . the experimental method can reveal fine structure , but does not determine the absolute density magnitude . the smmc calculation is scaled by a factor to facilitate comparison . in this case , the factor has been chosen to make the curves agree at lower excitation energies . from 1 - 3 mev , the agreement is very good . from 3 - 5 mev , the smmc density increases more rapidly than the data . this deviation from the data can not be accounted for by statistical errors in either the calculation or measurement . near 6 mev , the measured density briefly flattens before increasing and this also appears in the calculation , but the measurement errors are larger at that point . 0.5 in the measured density includes all states included in the theoretical calculation plus some others , so that one would expect the measured density to be greater than or equal to the calculated density and never smaller . we may have instead chosen our constant to match the densities for moderate excitations and let the measured density be higher than the smmc density for lower energies ( 1 - 3 mev ) . comparing structure between smmc and data is difficult for the lowest energies due to statistical errors in the calculation and comparison at the upper range of the smmc calculation , i.e. , @xmath169 mev , is unfortunately impossible since the data only extend to about @xmath170 mev excitation energy . in these proceedings , we have used four specific examples ( there are several others ) for which the smmc calculations have proven very useful in understanding the properties of nuclei in systems where the number of valence particles prohibits the use of more traditional approaches . the method has proven to be a valuable tool for furthering our understanding of nuclear structure and astrophysics . continued developments in both creating useful interactions and shell - model technology should continue to enhance our ability to understand nuclei far from stability in the coming years . oak ridge national laboratory ( ornl ) is managed by lockheed martin energy research corp . for the u.s . department of energy under contract number de - ac05 - 96or22464 . this work was supported in part through grant de - fg02 - 96er40963 from the u.s . department of energy . this work was supported in part by the national science foundation , grants no . phy-9722428 , phy-9420470 , and phy-9412818 .

the shell - model monte carlo ( smmc ) technique transforms the traditional nuclear shell - model problem into a path - integral over auxiliary fields . we describe below the method and its applications to four physics issues : calculations of @xmath0-@xmath1- shell nuclei , a discussion of electron - capture rates in @xmath1-shell nuclei , exploration of pairing correlations in unstable nuclei , and level densities in rare earth systems .

nucl-th9812031

recent advanced material technologies have made it possible to access low - dimensional quantum systems . furthermore , material synthesis has offered a great opportunity to explore more intriguing lower - dimensional spin systems rather than well - understood conventional spin systems @xcite . in such a low - dimensional system , for instance , alternating bond interactions and/or less symmetry interactions in spin lattices can be realizable in synthesizing two different magnetic atoms . of particular importance , therefore , is understanding quantum phase transitions in which one - dimensional spin systems are unlikely found naturally . normally , quantum fluctuations in a low - dimensional spin system are stronger than higher dimensional spin systems @xcite . quantum phase transitions driven by stronger quantum fluctuations then exhibit more interesting and novel quantum phenomena in low - dimensional spin systems . the effects of alternating bond interactions , especially , have been intensively studied theoretically in spin systems such as antiferromagnetic heisenberg chains @xcite , heisenberg chains with next - nearest - neighbor bond alternations @xcite , a tetrameric heisenberg antiferromagnetic chain @xcite , and two - leg spin ladders @xcite . a recent experiment has demonstrated a realization of a bond - alternating chain by applying magnetic fields in a spin-1/2 chain antiferromagnet @xcite . in this study , we will consider one - dimensional ising - type spin chains with an alternating exchange coupling . actually , this bond alternation can not destroy the antiferromagnetic phase of the uniform bond case but just quantitatively changes the ground state properties originating from a dimerization of the spin lattice . then , a less symmetric interaction can play a significant role to induce a quantum phase transition . to see a quantum phase transition , we will employ a dzyaloshinskii - moriya ( dm ) interaction @xcite which results from the spin - orbit coupling . based on the ground state fidelity @xcite with the imps presentation @xcite , we discuss the quantum criticality in the system . it is shown that a uniform dm interaction can destroy the antiferromagnetic phase , which is a continuous quantum phase transition , and its critical value is inversely proportional to the alternating exchange coupling strength . let us start with a spin-1/2 ising chain with antisymmetric anisotropic , and alternative bond interactions on the infinite - size lattice . our system can be described by the spin hamiltonian @xmath1 where @xmath2 are the spin operators acting on the @xmath3-th site . the exchange interaction is chosen as @xmath4 and the alternative bond interaction is characterized by the relative strength @xmath5 of exchange coupling for the even and odd lattice sites . to describe an antisymmetric anisotropic exchange coupling between the two spins on the lattice , we employ a uniform dm interaction @xmath6 , that is characterized by the dm vector @xmath7 . for @xmath8 and @xmath9 , ( [ hamt ] ) is reduced to the conventional ising chain hamiltonian . if @xmath8 and @xmath10 , eq . ( [ hamt ] ) can be mapped onto the xxz spin chain model which has a quantum phase transition from the gapped neel or antiferromagnetic ( afm ) phase to the gapless luttinger liquid ( ll ) phase at the critical point @xmath11 @xcite . this study will then be focused on the antiferromagnetic exchange interaction @xmath12 , i.e. , @xmath13 , and a transverse dm interaction denoting @xmath14 . the hamiltonian in eq . ( [ hamt ] ) is actually invariant under the transformation @xmath15 with @xmath16 for @xmath17-th site and @xmath18 for ( @xmath19)-th site . our model hamiltonian then possesses a @xmath20 symmetry generated by the transformation @xmath21 . the ground state of the system may undergo a spontaneous @xmath20 symmetry breaking which gives rise to a quantum phase transition between an ordered phase and a disordered phase . for a quantum spin system with a finite @xmath22 lattice site , its wave function with the periodic boundary condition can be expressed in the matrix product state ( mps ) representation @xcite as @xmath23}a^{[2 ] } \cdots a^{[n]}\right ] \ , |s^{[1]}s^{[2 ] } \cdots s^{[n]}\rangle$ ] , where @xmath24}$ ] is a site - dependent @xmath25 matrix with the truncation dimension @xmath26 of the local hilbert space at the @xmath3-th site , @xmath27}\rangle$ ] is a basis of the local hilbert space at the @xmath3-th site , and the physical index @xmath28 takes value @xmath29 with the dimension @xmath30 of the local hilbert space . this mps representation for a finite lattice system can be extended to describe an infinite lattice system . to do this , for an infinite lattice , one may replace the matrix @xmath24}$ ] with @xmath31}\lambda^{[i]}$ ] @xcite , where @xmath31}$ ] is a three - index tensor and @xmath32}$ ] is a diagonal matrix at the @xmath3-th site , which is called the _ canonical infinite matrix product state _ ( imps ) representation . if system hamiltonian is translational invariant for an infinite lattice , for instance , our system hamiltonian describe by eq . ( [ hamt ] ) has a two - site translational invariance , the two - site translational invariance allows us to reexpress the hamiltonian as @xmath33}$ ] , where @xmath34}$ ] is the nearest - neighbor two - body hamiltonian density . in such a case , one can introduce a two - site translational invariant imps representation , i.e. , for the even ( odd ) sites a ( b ) , only two three - index tensors @xmath35 and two diagonal matrices @xmath36 can be applied in representing a system wave function : @xmath37}\ } } \cdots \gamma_{a}\lambda_{a}\gamma_{b}\lambda_{b}\gamma_{a } \lambda_{a}\gamma_{b}\lambda_{b } \cdots |\cdots s^{[i]}s^{[i+1]}s^{[i+2]}s^{[i+3 ] } \cdots \rangle . \label{wave}\ ] ] note that , actually , for an infinite lattice sites , the diagonal elements of the matrix @xmath38 are the normalized schmidt decomposition coefficients of the bipartition between the semi - infinite chains @xmath39 and @xmath40 . in order to find a ground state of our system in the imps representation , the infinite time - evolving block decimation ( itebd ) algorithm introduced by vidal @xcite is employed . for a given initial state @xmath41 and the hamiltonian @xmath42 , a ground - state wave function can be yield by the imaginary time evolution @xmath43|\psi(0)\rangle /|\exp(-h\tau)|\psi(0)\rangle|$ ] for a large enough @xmath44 . to realize the imaginary time evolution operation numerically , the imaginary time @xmath44 is divided into the time slices @xmath45 and a sequence of the time slice evolution gates approximates the continuous time evolution . meanwhile , the time evolution operator @xmath46 } \delta\tau\right]$ ] for @xmath47 is expanded to a product of the evolution operators acting on two successive @xmath3 and @xmath48 sites through the suzuki - trotter decomposition @xcite . after absorbing a time - slice evolution gate , in order to recover the imps representation , a singular value decomposition of a matrix is performed , which contracted from @xmath49 , @xmath50 , one @xmath51 , two @xmath52 and the evolution operators . then only the @xmath26 largest singular values are retained . this procedure yields the new tensors @xmath49 , @xmath50 and @xmath51 that are used to update the tensors for all the sites . as a result , the translational invariance under two - site shifts is recovered . repeating the above procedure until the ground - state energy converges yields the ground - state wave function of the system in the imps representation . one can define a fidelity @xmath53 from a ground state wavefunction @xmath54 . a fidelity per lattice site ( fls ) @xmath30 @xcite can be defined as @xmath55 where @xmath22 is the system size . remarkably , fls is well defined and plays a similar role to an order parameter although @xmath56 trivially becomes zero in the thermodynamic limit when @xmath22 approaches infinity . the fls satisfies the properties inherited from fidelity @xmath56 : ( i ) normalization @xmath57 , ( ii ) symmetry @xmath58 , and ( iii ) range @xmath59 . as a function of @xmath60 for various truncation dimensions @xmath26 with @xmath61 . for higher truncation dimension , a bifurcation point @xmath62 moves to saturate to its critical value . inset : extrapolation of bifurcation point @xmath62 . from the numerical fitting function @xmath63 with @xmath64 and @xmath65 , the critical point is estimated as @xmath66.,scaledwidth=45.0% ] on adapting the transfer matrix approach @xcite , the imps representation of the ground - state wave - functions allows us to calculate the fidelity per lattice site ( fls ) @xmath67 . let us choose @xmath68 as a reference state for the fls @xmath67 . for @xmath69 , in fig . [ fidelitybifur ] , the fls @xmath67 is plotted as a function of the dm interaction parameter @xmath60 for various values of the truncation dimensions with a randomly chosen initial state in the imps representation . figure [ fidelitybifur ] shows a singular behavior of the fls @xmath70 , which indicates that there occurs a quantum phase transition across the singular point . a bifurcation behavior of the fls @xmath70 is also shown when the interaction parameter @xmath60 becomes smaller than its characteristic singular value that can be called a ` bifurcation point ' @xmath71 . the bifurcation points depend on the truncation dimension @xmath26 , i.e. , @xmath72 . as the truncation dimension @xmath26 increases , the bifurcation occurs starting at the lower value of @xmath60 . for @xmath73 , the bifurcation point @xmath74 at which a bifurcation starts to occur can be extrapolated . in the inset of fig . [ fidelitybifur ] , we use an extrapolation function @xmath75 , characterized by the coefficients @xmath76 , @xmath77 , and @xmath78 being a positive real number , which guarantees that @xmath74 becomes a finite value . the numerical fitting gives @xmath79 , @xmath64 , and @xmath65 . from the extrapolation , the bifurcation point is shown to saturate to @xmath80 which can be regarded as a critical point @xmath81 @xcite . actually , the critical point also corresponds to the pinch point of the fls in the thermodynamics limit , i.e. , @xmath73 . consequently , a fls bifurcation point @xmath62 plays the role of a pseudo phase transition point for a given finite truncation dimension @xmath26 in the mps representation . in addition , the continuous function behavior of the fls across the bifurcation point implies that a continuous quantum phase transition occurs at the critical point @xcite . in fig . [ fidelitybifur ] , the bifurcation occurs for @xmath82 , which is captured in the imps representation with a randomly chosen initial state . in fact , the bifurcation behavior of the fls means that there are two possible ground states for @xmath83 while there is a single ground state for @xmath84 . such a property of the ground states can be understood by the @xmath20 symmetry of the hamiltonian from the invariant transformation @xmath85 . in the thermodynamic limit , there are two possible ground states satisfying @xmath85 , that is , @xmath86 or @xmath87 . once a spontaneous symmetry breaking happens , the system can choose one of two possible ground states @xmath86 or @xmath87 , which indicate a broken symmetry phase . for symmetric phase , the system has a single ground state , being a linear combination of two possible ground state , which should satisfy the transformation invariance . in fig . [ fidelitybifur ] , then , the fls is plotted from the two fidelities , i.e. , @xmath88 ( upper lines ) and @xmath89 ( lower lines ) . then , the bifurcation point is the transition point between the symmetry phase and the broken - symmetry phase . from the imps with the numerical extrapolation of the bifurcation points , in fig . [ phasedigram ] , we draw the ground - state phase diagram in the interaction parameter @xmath90 plane . below the phase boundary ( red solid line ) the system is in an antiferromagnetic phase while above the boundary the system is in a disordered phase . the phase diagram shows that the @xmath91 is inversely proportional to the @xmath92 . a best fitting function ( dotted line ) of the critical points @xmath93 is given by @xmath94 with a single parameter @xmath95 . the characteristic phase boundary can be understood as follows : if @xmath96 and @xmath97 , the lattice sites are dimerised and the hamiltonian in eq . ( [ hamt ] ) has a two - site translational invariance due to the alternating bond coupling @xmath5 . as assumed , for the antiferromagnetic exchange interaction @xmath98 , i.e. , for @xmath99 , the ising chain with the alternating bond coupling is in an antiferromagnetic state even though the lattice sites are strongly dimerised . if @xmath100 , the antisymmetric anisotropic dm interaction can destroy the antiferromagnetic order originating from the antiferromagnetic exchange interaction @xmath101 . the antiferromagnetic order may be destroyed more easily by the uniform antisymmetric anisotropic dm interaction for the dimerised lattice sites than for the ising chain without the alternating bond coupling because the antiferromagnetic correlation between the sites becomes weaker due to the dimerised lattice sites . then , to destroy the antiferromagnetic order , a stronger dimerisation of the lattice sites ( bigger @xmath5 ) requires a much weaker uniform antisymmetric anisotropic interaction ( much smaller @xmath60 ) . consequently , the phase boundary separating the antiferromagnetic phase and a disordered phase might have a inversely proportional relation between @xmath91 and @xmath92 . is inversely proportional to the @xmath102 . a best fitting is denoted by the dotted line . this is in sharp contrast with the results from the real space renormolaztion group ( rsrg ) method @xcite . , scaledwidth=45.0% ] very recently , a real space renormalization group ( rsrg ) approach @xcite has been applied in the same model and has shown that a symmetric phase boundary is given by @xmath103 ( blue dashed line in fig . [ phasedigram ] ) . compared with the rsrg approach , our results from the imps show that there is a quite significant discrepancy in the phase boundary line because , contrast to the imps , the rsrg is an approximate method based on low - energy states , which implies that it does not capture properly a contribution from relevant higher energy states . then , the ground state phase diagram from our fidelity approach based on the imps is more reliable and accurate . in order to understand more clearly the quantum phase transition in our system , let us consider quantum entanglement that can also detect a quantum phase transition @xcite . to quantify the quantum entanglement , we employ the von neumann entropy , which is a good measure of bipartite entanglement between two subsystems of a pure state @xcite , because our ground states are in a pure state . then , the spin chain can be partitioned into the two parts denoted by the left semi - infinite chain @xmath104 and the right semi - infinite chain @xmath105 . the von neumann entropy is defined as @xmath106 in terms of the reduced density matrix @xmath107 or @xmath108 of the subsystems @xmath104 and @xmath105 . in the imps representation , the von neumann entropy for the semi - infinite chains @xmath104 or @xmath105 becomes @xmath109 where @xmath110 s are diagonal elements of the matrix @xmath111 that could be directly obtained in the imps algorithm . this is because , when one partitions the two semi - infinite chains @xmath112 and @xmath113 , one gets the schmidt decomposition @xmath114 . from the spectral decomposition , @xmath115 are actually eigenvalues of the reduced density matrices for the two semi - infinite chains @xmath104 and @xmath105 . in our two - site translational invariant imps representation , there are two schmidt coefficient matrices @xmath51 and @xmath52 that describe two possible ways of the partitions , i.e. , one is on the odd sites , the other is on the even sites . from the @xmath51 and @xmath52 , one can obtain the two von neumann entropies depending on the odd- or even - site partitions . between left and right halves of a chain as a function of @xmath60 for @xmath61 and @xmath116 . both the von neumann entropies for odd and even bonds show a singularity at @xmath117.,scaledwidth=45.0% ] in fig . [ entanglement ] , we plot the von neumann entropy @xmath118 as a function of the dm interaction strength @xmath60 for even ( @xmath17 site ) and odd ( ( @xmath19 ) site ) bonds with @xmath61 . the von neumann entropy for the odd bonds is always larger than those for the even bonds because , by the definition , the odd - site exchange interaction @xmath119 is stronger that the even - site exchange interaction @xmath120 . furthermore , it is shown that both the entropies for the even and odd bonds have a singularity at the same value of the dm interaction strength @xmath60 . note that the singularities of the entropies occur at the critical point @xmath121 . this result then shows clearly that both the fls @xmath30 and the von neumann entropy @xmath118 give the same phase transition point . as a consequence , in fact , the von neumann entropy @xmath118 gives the same phase diagram from the fls in fig . [ phasedigram ] . as discussed , for the antiferromagnetic state of our system , there are two possible ground states that are connected by the unitary transformation @xmath21 . from the bifurcation of fls , then , one might expect a bifurcation in the von neumann entropy too . however , contrary to the fls @xmath30 , in fig . [ entanglement ] , no bifurcation is seen in the von neumann entropy @xmath118 for the antiferromagnetic state even though the initial state is randomly chosen in the imps representation . the reason for the absence of bifurcation in the von neumann entropy is why the singular values @xmath38 in eq . ( [ entropy ] ) do not depend on the unitary transformation because the unitary transformation @xmath21 acts only on a single site of the spin lattice in the imps representation . at a critical point , characteristic singular behaviors of thermodynamics system properties depend only on few features such as dimensionality and symmetry , which can be classified by the concept of universality classes . especially , the central charge can be used for the classification of universality classes @xcite . owing to implement the imps representation , we can obtain a central charge @xmath78 and a so - called finite - entanglement scaling exponent @xmath122 numerically via the unique behaviors of the correlation length @xmath123 and the von neumann entropy @xmath118 with respect to the truncation dimension @xmath26 at a critical point @xcite , i.e. , @xmath124 ( color online ) ( a ) correlation length @xmath123 as a function of the truncation dimension @xmath26 at the critical point . the power curve fitting @xmath125 yields @xmath126 and @xmath127 . ( b ) scaling of the von neumann entropy @xmath118 with the truncation dimension @xmath26 at the critical point . for @xmath127 from ( a ) , the linear fitting @xmath128 yields the central charge @xmath129 . here , the alternating bond strength is chosen as @xmath61 . , scaledwidth=45.0% ] in fig . [ fig4 ] , the correlation length @xmath123 and the von neumann entropy @xmath118 as a function of the truncation dimension @xmath26 at the critical point @xmath91 for @xmath61 . here , the truncation dimensions are taken as @xmath130 , and @xmath131 . it is shown that both the correlation length @xmath123 and the von neumann entropy @xmath118 diverges as the truncation dimension @xmath26 increases . from a power - law fitting on the correlation length @xmath123 , we have @xmath132 and @xmath133 . as shown in fig . [ fig4 ] ( b ) , our numerical result demonstrates a linear scaling behavior , which gives a central charge @xmath134 with @xmath132 . our central charge is close to the exact value @xmath135 . consequently , the quantum phase transition in our system is in the same universality class as the quantum transverse field ising model . quantum phase transitions have been investigated in the ising chain with the dzyaloshinskii - moriya interaction as well as the alternating bond - coupling . the fls and its bifurcation have clearly shown a characteristic singular point as a signature of the quantum phase transition and behaves as a continuous function , which shows a continuous phase transition occurring at the critical point . the phase diagram was obtained from the fls and the von neumann entropy . with a finite - entanglement scaling of the von neumann entropy with respect to the truncation dimension in the imps representation , a central charge was estimated to be @xmath0 , which shows that the system is in the same universality class with the quantum transverse field ising model . we thank huan - qiang zhou for helpful discussions . this work was supported by the fundamental research funds for the central universities ( project no . cdjzr10100027 ) . syc acknowledges the support from the nsfc under grant no.10874252 . 99 mallah t , thiebaut s , verdaguer m and veillet p 1993 _ science _ * 262 * 1554 + sato o , lyoda t , fujishima a and hashimoto k 1996 _ science _ * 272 * 704 + verdaguer m , gleizes a , renard j p and seiden j 1984 _ phys . rev . _ b * 29 * 5144 + kahn o , pei y , verdaguer m , renard j p and sletten j 1988 _ j. am . soc . _ * 110 * 782 + koningsbruggen p j van , kahn o , nakatani k , pei y , renard j p , drillon m and leggol p 1990 _ inorg . chem . _ * 29 * 3325 + zheludev a , maslov s , yokoo t , raymond s , nagler s e and akimitsu j 2001 _ j. phys . : condes * 13 * r525 + hagiwara m , minami k , narumi y , tatani k and kindo k 1998 _ j. phys . jpn . _ * 67 * 2209 + yamamoto s , 2000 _ phys . _ b * 61 * r842 + culp j t , park j h , meisel m w and talham d r 2003 _ inorg . chem . _ * 42 * 2842 perk j h h and capel h w 1976 _ phys . lett . _ a * 58 * 115 + jafari r , kargarian m , langari a and siahatgar m 2008 _ phys . _ b * 78 * 214414 + kadar z and zimboras z 2010 _ phys . rev . _ a * 82 * 032334 + soltani m r , mahdavifar s , akbari a and masoudi a a 2010 _ j. supercond . * 23 * 1369

a systematic analysis is performed for quantum phase transitions in a bond - alternative one - dimensional ising model with a dzyaloshinskii - moriya ( dm ) interaction by using the fidelity of ground state wave functions based on the infinite matrix product states algorithm . for an antiferromagnetic phase , the fidelity per lattice site exhibits a bifurcation , which shows spontaneous symmetry breaking in the system . a critical dm interaction is inversely proportional to an alternating exchange coupling strength for a quantum phase transition . further , a finite - entanglement scaling of von neumann entropy with respect to truncation dimensions gives a central charge @xmath0 at the critical point .

1105.0533

it is now an established experimental fact that there are events with large rapidity gaps in the hadronic final state in which there is a large momentum transfer across the gap . such events have been observed at both the tevatron @xcite and hera @xcite in the rapidity gaps between jets process suggested for study by bjorken @xcite . the issue now for experimentalists and theorists alike is to address the question of what underlying dynamical process is responsible for such striking events . it is clear that conventional regge phenomenology can not provide an answer , since the soft pomeron contribution has died away at much lower @xmath4 values due to shrinkage . the two best developed models currently available are the bfkl pomeron @xcite , calculated within the leading logarithmic approximation ( lla ) by mueller and tang @xcite and implemented into the herwig monte carlo @xcite , and the soft colour rearrangement model @xcite . the recent gaps between jets analysis by the d0 collaboration @xcite favoured the soft colour model to the bfkl pomeron , although conclusions from gaps between jets measurements may be difficult to draw due to the uncertainties in the role of multiple interactions , which are poorly understood theoretically at the present time @xcite . furthermore , gaps between jets measurements at both hera and the tevatron are limited by the requirement that two jets are observed in the detector , severely restricting the accessible gap size . since the bfkl cross section is predicted to rise exponentially with @xmath5 , whilst soft colour is not , this is a severe restriction . at hera , measurements of high @xmath4 vector meson production @xcite have provided access to larger rapidity gaps in a well defined kinematic range , although the rate is low . with these issues in mind , cox and forshaw @xcite suggested the study of the more inclusive double dissociative process @xmath0 at high @xmath4 . in this paper we report the first measurement of this process , based on h1 data taken during 1996 . the photon and proton dissociative systems , @xmath1 and @xmath2 respectively , are separated by finding the largest rapidity gap in the event ( the procedure used by the h1 collaboration in previous diffractive measurements @xcite ) . the process , shown schematically in figure [ diffplot ] , is considered in terms of the kinematic variables @xmath6 @xmath7 where @xmath8 and @xmath2 are the 4-vectors of the photon , proton and x and y systems respectively . @xmath9 is the @xmath10 center of mass energy and @xmath11 is the four momentum transfer across the rapidity gap . in this study we present measurements of the differential cross section @xmath12 in the range @xmath13 , @xmath14 , @xmath15 , @xmath16 . the data for this analysis were collected with the h1 detector during the 1996 running period , when hera collided @xmath17 positrons with @xmath18 protons , with an integrated luminosity of 6.7 @xmath19 . photoproduction events were selected by detecting the scattered positron in the electron tagger , 33 m down the beam pipe in the scattered electron direction . this restricts the virtuality of the photon to @xmath20 gev@xmath21 . the reconstruction of the @xmath1 and @xmath2 system 4-vectors has been optimised by combining tracking and calorimeter information . techniques are applied to minimise the effects of detector noise . precise details can be found elsewhere @xcite . losses in the forward and backward directions are , however , unavoidable , making the measurement of the invariant masses of the systems problematic . for this reason , we introduce the kinematic variables @xmath22 and @xmath23 , reconstructed using the expressions @xmath24 where @xmath25 and @xmath26 are the proton and photon beam energies respectively , and the quantity @xmath27 ( @xmath28 ) is summed over all hadrons reconstructed backward ( forward ) of the largest rapidity gap in the event . this quantity has the property that it is insensitive to losses down the beam pipe , for which @xmath29 ( @xmath30 ) . in order to ensure that the systems @xmath1 and @xmath2 are clearly separated , only events with a rapidity gap between the two systems of at least 1.5 units of rapidity are selected . these events are specified by @xmath31 , and hence our sample is defined in the kinematic range @xmath32 and @xmath15 . and @xmath2 systems must be @xmath33 is not part of the hadron level cross section definition . any losses due to this cut are included in the acceptance corrections ] the reconstruction of @xmath11 is more problematic . it is measured as the negative squared transverse momentum of the @xmath1 system , @xmath34 , and is sensitive to losses down the backward beam pipe , particularly for low values of @xmath4 . for this reason we choose to define our sample for @xmath35 . the events selected by the criteria described in section 2 are used to determine the cross section @xmath36 in the kinematic range defined in section 1 . the herwig monte carlo , including bfkl pomeron exchange , is used to correct for losses and migration effects in @xmath22 , @xmath23 and @xmath11 . in the bfkl formalism at leading order , it does not make sense to run the coupling , and therefore @xmath37 is fixed in the herwig generation at @xmath38 . this corresponds at leading order to a hard pomeron intercept of @xmath39 , where @xmath40 . the dominant contribution to the statistical error comes from the limited number of data events in the sample . systematic uncertainties are calculated on a bin by bin basis , and added in quadrature . the dominant error is due to the limited number of data events available to calculate the trigger efficiency , contributing a systematic error of approximately @xmath41 in each bin . the @xmath22 distribution , corrected for detector effects , is shown in figure [ xpom_fixw ] . the inner error bars are statistical and the outer error bars are the quadratic sum of the statistical and systematic errors . the solid line is the prediction from the herwig generator for all non - singlet exchange photoproduction processes . a significant excess above the expectation from the standard photoproduction model is observed . the dashed line shows the herwig prediction with the lla bfkl prediction added . good agreement is observed in both normalisation and shape . care must be taken , however , in the interpretation of this result . there is a large theoretical uncertainty in the overall normalisation of the lla bfkl cross section prediction . the agreement in normalisation may well therefore be fortuitous . it should also be noted that the shape of the @xmath22 distribution in this region of phase space is not only determined by the underlying dynamics of the interaction , but also by kinematic effects . there is a kinematic limit on the lowest possible value of @xmath22 , set by the requirement that @xmath42 and @xmath43 gev , of @xmath44 x @xmath45 ( see equation ( 3 ) ) . this forces the cross section down in the lowest @xmath22 bin . the good agreement in shape with the bfkl monte carlo prediction , however , implies that the data are consistent with a value of @xmath46 within this model . despite these limitations , however , with higher statistics the outlook for the future is promising . this measurement demonstrates that it is possible to extend greatly the reach in rapidity allowed by the gaps between jets approach . with the improved statistics already collected in the 1997 hera running period , and higher luminosity in the future , a much more precise determination of the dependence of the cross section on @xmath22 , i.e. the energy dependence , will be possible . s. abachi et al ( d0 collaboration ) , phys . 72 ( 1994 ) 2332 ; phys . 76 ( 1996 ) 734 ; b. abbott et al ( d0 collaboration ) , phys . lett . b440 ( 1998 ) 189 . f. abe et al ( cdf collaboration ) , phys . 74 ( 1995 ) 855 ; phys . lett 80 ( 1998 ) 1156 ; phys . rev . 81 ( 1998 ) 5278 . m. derrick et al ( zeus collaboration ) , phys . b369 ( 1996 ) 55 . h1 collaboration , `` rapidity gaps between jets in photoproduction at hera '' , contribution to the international europhysics conference on high energy physics , august 1997 , jerusalem , israel . bjorken , phys . d47 ( 1992 ) 101 . e.a.kuraev , l.n.lipatov and v.s.fadin , sov.phys.jetp 45 ( 1977 ) 199 . + ya.ya.balitsky and l.n.lipatov , sov.j.nucl.phys 28 ( 1978 ) 822 . + l.n.lipatov , sov.phys.jetp 63 ( 1986 ) 904 . a. h. mueller and w .- k . tang , phys . b284 ( 1992 ) 123 . g.marchesini et al . , comp.phys.comm . 67 ( 1992 ) 465 . m.e.hayes , bristol university , phd thesis ( 1998 ) . o. j. p. eboli , e. m. gregores and f. halzen , mad / ph-96 - 965 ( 1997 ) j. r. forshaw : `` high @xmath4 diffraction '' , in these proceedings . b. e. cox , j. r. forshaw and l. lnnblad , in preparation collaboration : `` production of @xmath47 mesons with large @xmath4 at hera '' contribution to the international europhysics conference on high energy physics , august 1997 , jerusalem , israel . j. crittenden : `` recent results from decay - angle analyses of @xmath48 photoproduction at high momentum transfer from zeus '' , in these proceedings . b. e. cox and j. r. forshaw , phys . lett . * b434 * ( 1998 ) 133 - 140 . collaboration : c.adloff et al . c74 ( 1997 ) 221 .

the double dissociation photoproduction cross section for the process @xmath0 , in which the systems @xmath1 and @xmath2 are separated by a large rapidity gap , is measured at large 4-momentum transfer squared @xmath3 by the h1 collaboration at hera . this measurement provides for the first time a direct measurement of the energy dependence of the gap production process at high @xmath4 .

hep-ph9906203

one of the most basic questions that arises in trying to understand the nonperturbative structure of string theory concerns the classification of vector bundles over real and complex manifolds . in the presence of d - branes one encounters gauge theories in spacetime dimensionalities up to ten . already more than 20 years ago , bps - type equations in higher dimensions were proposed @xcite as a generalization of the self - duality equations in four dimensions . for nonabelian gauge theory on a khler manifold the most natural bps condition lies in the donaldson - uhlenbeck - yau equations @xcite , which arise , for instance , in compactifications down to four - dimensional minkowski spacetime as the condition for at least one unbroken supersymmetry . while the criteria for solvability of these bps equations are by now very well understood , in practice it is usually quite difficult to write down explicit solutions of them . one recent line of attack has been to consider noncommutative deformations of these field theories @xcite@xcite . in certain instances , d - branes can be realized as noncommutative solitons @xcite , which is a consequence @xcite of the relationship between d - branes and k - theory @xcite@xcite . all celebrated bps configurations in field theories , such as instantons @xcite , monopoles @xcite and vortices @xcite , have been generalized to the noncommutative case , originally in @xcite , in @xcite and in @xcite , respectively ( see @xcite for reviews and further references ) . solution generating techniques such as the adhm construction @xcite , splitting @xcite and dressing @xcite methods have also been generalized to the noncommutative setting in @xcite and in @xcite . solutions of the generalized self - duality equations @xcite were investigated in @xcite , for example . noncommutative instantons in higher dimensions and their interpretations as d - branes in string theory have been considered in @xcite@xcite . in all of these constructions the usual worldvolume description of d - branes emerges from the equivalence between analytic and topological formulations of k - homology . in this paper we will complete the construction initiated in @xcite of multi - instanton solutions of the yang - mills equations on the manifold which is the product of noncommutative euclidean space @xmath3 with an ordinary two - sphere @xmath4 . we consider both bps and non - bps solutions , and extend previous solutions to those which are explicitly @xmath1-equivariant for any value of the dirac monopole charge characterizing the gauge field components along the @xmath4 directions . dimensional reduction techniques are used to establish an equivalence between multi - instantons on @xmath0 and nonabelian vortices on @xmath2 . the configurations can be interpreted in type iia superstring theory as _ chains _ of branes and antibranes with higgs - like open string excitations between neighbouring sets of d - branes . the equivalence between instantons and vortices may then be attributed to the decay of an unstable configuration of d@xmath5-branes into a state of d0-branes ( there are no higher brane charges induced because @xmath6 is equivariantly contractible ) . the d0-brane charges are classified by @xmath1-equivariant k - theory and the low - energy dynamics may be succinctly encoded into a simple quiver gauge theory . unlike the standard brane - antibrane systems , the effective action can not be recast using the formalism of superconnections @xcite but requires a more general formulation in terms of new geometrical entities that we call `` graded connections '' . this formalism makes manifest the interplay between the assignment of k - theory classes to the explicit instanton solutions and their realization in terms of a quiver gauge theory . the organisation of this paper is as follows . the material is naturally divided into two parts . sections 25 deal with _ ordinary _ gauge theory on a generic khler manifold of the form @xmath7 in order to highlight the geometric structures that arise due to dimensional reduction and which play a prominent role throughout the paper . sections 610 are then concerned with the noncommutative deformation @xmath8 and they construct explicit solutions of the dimensionally reduced yang - mills equations , emphasizing their interpretations in the context of equivariant k - theory , quiver gauge theory , and ultimately as states of d - branes . in section 2 we introduce basic definitions and set some of our notation , and present the field equations that are to be solved . in section 3 we write down an explicit ansatz for the gauge field which is used in the @xmath1-equivariant dimensional reduction . in section 4 we describe three different interpretations of the ansatz as configurations of d - branes , as charges in equivariant k - theory , and as field configurations in a quiver gauge theory ( later on these three descriptions are shown to be equivalent ) . in section 5 the dimensional reduction mechanism is explained in detail in the new language of graded connections and the resulting nonabelian vortex equations , arising from reduction of the donaldson - uhlenbeck - yau equations , are written down . in section 6 we introduce the noncommutative deformations of all these structures . in section 7 we find explicit bps and non - bps solutions of the noncommutative yang - mills equations and show how they naturally realize representations of the pertinent quiver . in section 8 we develop an @xmath1-equivariant generalization of the ( noncommutative ) atiyah - bott - shapiro construction , which provides an explicit and convenient representation of our solution in terms of k - homology classes . in section 9 we compute the topological charge of our instanton solutions directly in the noncommutative gauge theory , and show that it coincides with the corresponding k - theory charge , which then allows us to assign d0-brane charges to the solutions from a worldvolume perspective . finally , in section 10 we construct some novel bps solutions in the vacuum sectors of the noncommutative field theory and describe their relation to stable states of brane - antibrane systems . in this section we will introduce the basic definitions and notation that will be used throughout this paper , as well as the pertinent field equations that we will solve . * the manifold @xmath9 . * let @xmath10 be a real @xmath11-dimensional lorentzian manifold with nondegenerate metric of signature @xmath12 , and @xmath13 the standard two - sphere of constant radius @xmath14 . we shall consider the manifold @xmath15 with local real coordinates @xmath16 on @xmath10 and coordinates @xmath17 $ ] , @xmath18 $ ] on @xmath4 . in these coordinates the metric on @xmath15 reads @xmath19 where hatted indices @xmath20 run over @xmath21 while primed indices @xmath22 run through @xmath23 . we use the einstein summation convention for repeated spacetime indices . * the khler manifold @xmath24 . * as a special instance of the manifold @xmath10 we shall consider the product @xmath25 of dimension @xmath26 with metric @xmath27 here @xmath28 is a khler manifold of real dimension @xmath29 with local real coordinates @xmath30 , where the indices @xmath31 run through @xmath32 . the cartesian product @xmath33 is also a khler manifold with local complex coordinates @xmath34 and their complex conjugates , where @xmath35 while @xmath36 are stereographic coordinates on the northern hemisphere of @xmath4 . in these coordinates the metric on @xmath37 takes the form @xmath38 while the khler two - form @xmath39 is given by @xmath40 * yang - mills equations . * consider a rank @xmath41 hermitean vector bundle @xmath42 with gauge connection @xmath43 of curvature @xmath44 . in local coordinates , wherein @xmath45 , the two - form @xmath46 has components @xmath47 $ ] , where @xmath48 . both @xmath49 and @xmath50 take values in the lie algebra @xmath51 . for the usual yang - mills lagrangian can be introduced via the redefinition @xmath52 . ] @xmath53 the equations of motion are @xmath54=0 \ , \ ] ] where @xmath55 . the curvature two - form can be written in local coordinates on @xmath56 as @xmath57 and the yang - mills lagrangian becomes l^ _ ym&=&- tr^ _ kk . [ lagrprod ] * donaldson - uhlenbeck - yau equations . * for static field configurations in the temporal gauge @xmath58 , the yang - mills equations ( [ ym ] ) on @xmath59 reduce to equations on @xmath37 . their stable solutions are provided by solutions of the donaldson - uhlenbeck - yau ( duy ) equations which can be formulated on any khler manifold @xcite . the importance of these equations derives from the fact that they yield the bps solutions of the full yang - mills equations . the duy equations on @xmath7 are @xmath60 where @xmath61 is the hodge duality operator and @xmath62 is the khler decomposition of the gauge field strength . in the local complex coordinates @xmath63 these equations take the form @xmath64\label{duy2 } { { { \cal{f}}}}_{{{\bar{z}}}^{{{\bar{a}}}}{{\bar{z}}}^{{{\bar{b}}}}}&=&0~=~{{\cal{f}}}_{z^az^b } \ , \\[4pt]\label{duy3 } { { { \cal{f}}}}_{{{\bar{z}}}^{{{\bar{a}}}}{{\bar{y}}}}&=&0~=~{{\cal{f}}}_{z^ay } \ , \end{aligned}\ ] ] where the indices @xmath65 run through @xmath66 . ( [ duy1 ] ) is a hermitean condition on the gauge field strength tensor , while eqs . ( [ duy2 ] ) and ( [ duy3 ] ) are integrability conditions implying that the bundle @xmath67 endowed with a connection @xmath43 is holomorphic . it is easy to show that any solution of these @xmath68 equations also satisfies the full yang - mills equations . in this section we shall write down the fundamental form of the gauge potential @xmath43 on @xmath56 that will be used later on to dimensionally reduce the yang - mills equations for @xmath43 to equations on @xmath10 . this will be achieved by prescribing a specific @xmath69 dependence for @xmath43 , which we proceed to describe first . * monopole bundles . * consider the hermitean line bundle @xmath70 over the sphere with @xmath71 and unique @xmath1-invariant unitary connection @xmath72 having , in the local complex coordinate @xmath73 on @xmath69 , the form @xmath74 where @xmath75 is an integer . the curvature of this connection is @xmath76 the topological charge of this gauge field configuration is given by the first chern number ( equivalently the degree ) of the associated complex line bundle as @xmath77 in terms of the spherical coordinates @xmath78 the configuration ( [ f1],[f2 ] ) has the form @xmath79 it describes @xmath80 dirac monopoles or antimonopoles sitting on top of each other . the @xmath75-monopole bundle is classified by the hopf fibration @xmath81 . for each @xmath82 there is a one - dimensional representation @xmath83 of the circle group @xmath84 defined by _ m : v v=^mv s^1 v . [ u1irreps]we denote this irreducible @xmath85-module by @xmath86 . regarding the sphere as the homogeneous space @xmath87 , the @xmath1-equivariant line bundle @xmath88 corresponds to the representation @xmath89 in the sense that it can be expressed as l^m = su(2)_u(1)_m , [ monmexpl]where the quotient on @xmath90 is by the @xmath91 action @xmath92 for @xmath93 , @xmath94 and @xmath95 . the action of @xmath1 on @xmath90 given by @xmath96 descends to an action on ( [ monmexpl ] ) . any @xmath1-equivariant hermitean vector bundle over the sphere is a whitney sum of bundles ( [ monmexpl ] ) . there is an alternative description in terms of the holomorphic line bundle @xmath97 defined as the @xmath75-th power of the tautological bundle over the complex projective line . the universal complexification of the lie group @xmath1 is @xmath98 , and we may regard the sphere as a projective variety through the natural diffeomorphism @xmath99 , where @xmath100 is the parabolic subgroup of lower triangular matrices in @xmath98 . the @xmath1 action on ( [ monmexpl ] ) lifts to a smooth @xmath98 action , and the complexification of ( [ monmexpl ] ) is realized as the @xmath98-equivariant line bundle o(m)=sl(2,)_p_m [ monmholexpl]over @xmath69 . only the cartan subgroup @xmath101 of non - zero complex numbers acts non - trivially on the modules @xmath102 , with the @xmath103 action defined analogously to ( [ u1irreps ] ) . the two descriptions are equivalent after the introduction of a hermitean metric on the fibres of @xmath104 . this holomorphic line bundle has transition function @xmath105 transforming sections from the northern hemisphere to the southern hemisphere of @xmath4 . however , the monopole connection ( [ f1 ] ) is transformed on the intersection of the two patches covering @xmath69 via the transition function @xmath106 , which is the unitary reduction of the holomorphic transition function @xmath105 . thus the bundle @xmath104 regarded as a hermitean line bundle has transition function @xmath106 and can be substituted for the monopole bundle @xmath107 . * su(2)-invariant gauge potential . * the form of our ansatz for the gauge connection on @xmath56 is fixed by imposing invariance under the @xmath1 isometry group of @xmath69 acting through rigid rotations of the sphere . let @xmath108 be an @xmath1-equivariant @xmath109-bundle , with the group @xmath1 acting trivially on @xmath10 and in the standard way on @xmath110 . let @xmath43 be a connection on @xmath111 . imposing the condition of @xmath1-equivariance means that we should look for representations of the group @xmath1 inside the @xmath109 structure group , i.e. for homomorphisms @xmath112 . the ansatz for @xmath43 is thus given by @xmath41-dimensional representations of @xmath1 . up to isomorphism , for each positive integer @xmath113 there is a unique irreducible @xmath1-module @xmath114 of dimension @xmath113 . therefore , for each positive integer @xmath75 , the module = _ i=0^m _ k_i _ i=0^mk_ik [ genrepsu2uk]gives a representation @xmath115 of @xmath1 inside @xmath116 . the total number of such homomorphisms is the number of partitions of the positive integer @xmath117 into @xmath118 components . the original @xmath109 gauge symmetry is then broken down to the centralizer subgroup of @xmath119 in @xmath120 as ( k ) _ i=0^mu(k_i ) . [ gaugebroken ] it is natural to allow for gauge transformations to accompany the @xmath1 action @xcite , and so some `` twisting '' can occur in the reduction of the connection @xmath43 on @xmath56 . the @xmath69 dependence in this case is uniquely determined by the above @xmath1-invariant dirac monopole configurations @xcite . the @xmath51-valued gauge potential @xmath43 thus splits into @xmath121 blocks @xmath122 , @xmath123 where the indices @xmath124 run over @xmath125 , @xmath126 and [ f5 ] ^ii&=&a^i(x)1 + _ k_ia_m-2i(y ) , + [ f6 ] ^ii+1 = : ^ _ i+1&= & ^ _ i+1(x)|(y ) , + [ f7 ] ^i+1i = -(^ii+1)^ = -(^ _ i+1 ) ^&=&-_i+1^ ( x)(y ) , + [ f9 ] ^ii+l&=&0 = ^i+li l 2 . here @xmath127 are the unique covariantly constant , @xmath128-invariant forms of type @xmath129 and @xmath130 such that the khler @xmath131-form on @xmath69 is @xmath132 . they respectively take values in the bundles @xmath133 and @xmath134 . it is easy to see that the gauge potential @xmath43 given by ( [ f5])([f9 ] ) is anti - hermitean and @xmath1-invariant . note that we do not use the einstein summation convention for the repeated indices @xmath135 labelling the components of the irreducible representation @xmath136 of the group @xmath1 . thus the gauge potential @xmath137 decomposes into gauge potentials @xmath138 with @xmath139 and a multiplet of scalar fields @xmath140 with @xmath141 transforming in the bi - fundamental representations @xmath142 of the subgroup @xmath143 of the original @xmath109 gauge group . all fields @xmath144 depend only on the coordinates @xmath145 . every @xmath1-invariant unitary connection @xmath43 on @xmath56 is of the form given in ( [ f4])([f9 ] ) @xcite , which follow from the fact that the complexified cotangent bundle of @xmath69 is @xmath146 . this ansatz amounts to an equivariant decomposition of the original rank @xmath41 @xmath1-equivariant bundle @xmath147 in the form = _ i_k_il^m-2i , [ caleansatz]where @xmath148 is a hermitean vector bundle of rank @xmath149 with typical fibre the module @xmath150 , and @xmath151 is the bundle with fibres @xmath152 . by regarding @xmath153 for @xmath154 and defining @xmath155 , we can summarize our ansatz through the following chain of bundles : @xmath156 * field strength tensor . * the calculation of the curvature ( [ curvprod ] ) for @xmath43 of the form ( [ f4])([f9 ] ) yields @xmath157 where [ f11 ] ^ii&=&f^i+ f_m-2i+ ( ^ _ i+1^_i+1 - _ i^^ _ i)| , + [ f12 ] ^ii+1&=&d _ i+1| , + [ f13 ] ^i+1i = - ( ^ii+1)^&=&- ( d _ i+1)^ , + [ f14 ] ^ii+l&=&0 = ^i+li l 2 . here we have defined @xmath158 and introduced the bi - fundamental covariant derivatives @xmath159 from ( [ f11])([f14 ] ) we find the non - vanishing field strength components [ f15 ] ^ii_&=&f_^i , + [ f16 ] ^ii+1_|y&= & d_ _ i+1 = -(^i+1i_y)^ , + [ fyyb ] ^ii_y|y&= & - ( m-2i+_i^^ _ i - ^ _ i+1^_i+1 ) . in this section we shall clarify some features of the ansatz constructed in the previous section from three different points of view . to set the stage for the string theory interpretations of the solutions that we will construct later on , we begin by indicating how the ansatz can be interpreted in terms of configurations of d - branes in type ii superstring theory . this leads into a discussion of how the ansatz is realized in topological k - theory , which classifies the ramond - ramond charges of these brane systems , and we will derive the decomposition ( [ caleansatz ] ) directly within the framework of @xmath1-equivariant k - theory . we will then explain how seeking explicit realizations of our ansatz is equivalent to finding representations of the @xmath160 quiver . one of the goals of the subsequent sections will be to establish the precise link between these three descriptions , showing that they are all equivalent . * physical interpretation . * before entering into the formal mathematical characterizations of the ansatz of the previous section , let us first explain the physical situation which they will describe . our ansatz implies an equivalence between brane - antibrane systems on @xmath10 and wrapped branes on @xmath56 . in the standard d - brane interpretation , our initial rank @xmath41 hermitean vector bundle @xmath161 corresponds to @xmath41 coincident d(@xmath11 + 1)-branes wrapping the worldvolume manifold @xmath162 . the condition of @xmath1-equivariance imposed on this bundle fixes the dependence on the coordinates of @xmath69 and breaks the gauge group u(@xmath41 ) as in ( [ gaugebroken ] ) . the rank @xmath149 sub - bundle @xmath148 of this bundle is twisted by the dirac multi - monopole bundle @xmath163 . the system of @xmath41 coincident d(@xmath11 + 1)-branes thereby splits into blocks of @xmath164 coincident d(@xmath11 + 1)-branes , associated to irreducible representations of @xmath1 and wrapping a common sphere @xmath69 with the monopole fields . this system is equivalent to a system of @xmath165 d(@xmath1661)-branes carrying different magnetic fluxes on their common worldvolume @xmath10 . the d(@xmath1661)-branes which carry negative magnetic flux have opposite orientation with respect to the d(@xmath1661)-branes with positive magnetic flux , i.e. they are antibranes . this will become evident from the k - theory formalism , which will eventually lead to an explicit worldvolume construction , and also from the explicit calculation of the topological charges of the instanton solutions . in addition to the usual chan - paton gauge field degrees of freedom @xmath167 living on each block of branes , the field content on the brane configuration contains bi - fundamental scalar fields @xmath168 corresponding to massless open string excitations between neighbouring blocks of @xmath149 and @xmath169 d(@xmath1661)-branes . other excitations are suppressed by the condition of @xmath1-equivariance . however , as we shall see explicitly in the following , the fields @xmath170 should not be regarded as tachyon fields , but rather only as ( holomorphic ) higgs fields responsible for the symmetry breaking ( [ gaugebroken ] ) . only the brane - antibrane pairs whose constituents carry equal and opposite monopole charges are neutral and can thus annihilate to the vacuum , which carries no monopole charge ( although it can carry a k - theory charge from the virtual chan - paton bundles over @xmath10 ) . other brane pairs are stable because their overall non - vanishing chern number over @xmath69 is an obstruction to decay , and the monopole bundles thereby act as a source of flux stabilization for such brane pairs by giving them a conserved topological charge . in particular , neighbouring blocks of d@xmath171-branes are marginally bound by the massless open strings stretching between them . in this sense , the @xmath1-invariant reduction of d - branes on @xmath56 induces brane - antibrane systems on @xmath10 . note that while the system on @xmath10 is generically unstable , the original brane configuration on @xmath56 can be nonetheless stable . * k - theory charges . * given that the charges of configurations of d - branes in string theory are classified topologically by k - theory @xcite , let us now seek the k - theory representation of the above physical situation . the one - monopole bundle @xmath172 is a crucial object in establishing the bott periodicity isomorphism k(m_qp^1)=k(m_q ) [ bottper]in topological k - theory . the isomorphism is generated by taking the k - theory product of the tachyon field @xmath173 of a virtual bundle @xmath174\in{{\rm k}}({{\cal m}}_q)$ ] with that of the class of the line bundle @xmath172 which represents the bott generator of @xmath175 @xcite . the topological equivalence ( [ bottper ] ) then implies the equivalence of brane - antibrane systems on @xmath56 and @xmath10 , with the brane and antibrane systems each carrying a single unit of monopole charge . when they carry @xmath176 units of charge , the isomorphism breaks down , and it is necessary to introduce the notion of `` d - operations '' to establish the relationship @xcite . while these operations are natural , they are not isomorphisms and they reflect the fact that the explicit solutions in this setting are not @xmath128-invariant , so that the equivalence breaks down due to spurious moduli dependences of the system of branes on the @xmath69 factor . in what follows we will derive a modification of the relation ( [ bottper ] ) in _ equivariant _ k - theory which will naturally give the desired isomorphism , reflecting the equivalence of the brane - antibrane systems for arbitrary monopole charge , and bypass the need for introducing d - operations this is only possible by augmenting the basic brane - antibrane system to a _ chain _ of @xmath177 branes and antibranes with varying units of monopole charge as described above , and we will thereby arrive at an independent purely k - theoretic derivation of our ansatz . the representation ring @xmath178 of a group @xmath179 @xcite is the grothendieck ring of the category of finite dimensional representations of @xmath179 , with addition induced by direct sum of vector spaces , @xmath180+[\,\underline{v'}\,]:= [ \,\underline{v}\oplus\underline{v'}\,]$ ] , and multiplication induced by tensor product of modules , @xmath180\cdot[\,\underline{v'}\,]:= [ \,\underline{v}\otimes\underline{v'}\,]$ ] . as an abelian group it is generated by the irreducible representations of @xmath179 alternatively , since the isomorphism class of a @xmath179-module @xmath181 is completely determined by its character @xmath182 , the map @xmath183 identifies @xmath178 as a subring of the ring of @xmath179-invariant functions on @xmath179 . if @xmath10 is a @xmath179-space , then the grothendieck group of @xmath179-equivariant bundles over @xmath10 is called the @xmath179-equivariant k - theory group @xmath184 . this group unifies ordinary k - theory with group representation theory , in the sense that for the trivial space @xmath185 is the representation ring of @xmath179 , while for the trivial group @xmath186 is the ordinary k - theory of @xmath10 . the former property implies that @xmath184 is an @xmath178-module and the coefficient ring in equivariant k - theory is @xmath178 , rather than just @xmath187 as in the ordinary case . if the @xmath179-action on @xmath10 is trivial , then any @xmath179-equivariant bundle @xmath188 may be decomposed as a finite whitney sum e= _ ( g)hom^ _ g(11^ _ , e ) 11^ _ [ trivialedecomp]where @xmath189 is the trivial bundle over @xmath10 with fibre the irreducible @xmath179-module @xmath181 . it follows that for trivial @xmath179-actions the equivariant k - theory takes the simple form k_g(m_q)=k(m_q)r_g . [ kgxqtrivialg]the @xmath190-functor behaves analogously to the ordinary @xmath191-functor , and in addition @xmath190 is functorial with respect to group homomorphisms . a useful computational tool is the equivariant excision theorem . if @xmath192 is a closed subgroup of @xmath179 and @xmath10 is an @xmath192-space , then the inclusion @xmath193 induces an isomorphism @xcite ^*:k_g(g_fm_q ) k_f(m_q ) , [ excision]where the quotient on @xmath194 is by the @xmath192-action @xmath195 for @xmath196 , @xmath145 and @xmath197 . the @xmath179-action on @xmath198 descends from that on @xmath194 given by @xmath199 . let us specialize to our case of interest by taking @xmath200 , @xmath201 and the trivial action of @xmath1 on the space @xmath10 . using ( [ kgxqtrivialg ] ) and ( [ excision ] ) we may then compute k_su(2)(m_qp^1)&=&k_su(2)(su(2)_u(1)m_q ) + & = & k_u(1)(m_q ) = k(m_q)r_u(1 ) . [ excisionslc]this k - theoretic equality asserts a one - to - one correspondence between classes of @xmath1-equivariant bundles over @xmath56 and classes of @xmath91-equivariant bundles over @xmath10 with @xmath91 acting trivially on @xmath10 . the isomorphism ( [ excisionslc ] ) of equivariant k - theory groups is constructed explicitly as follows @xcite . given an @xmath1-equivariant bundle @xmath147 , we can induce a @xmath91-equivariant bundle @xmath202 by restriction to the slice @xmath203 . conversely , if @xmath188 is a @xmath91-equivariant bundle , then @xmath204 is an @xmath1-equivariant bundle , where the quotient on @xmath205 is by the action of @xmath91 on both factors , @xmath206 for @xmath207 , and the action of @xmath208 on @xmath209 descends from that on @xmath205 given by @xmath210 . this construction defines equivalence functors between the categories of @xmath1-equivariant vector bundles over @xmath56 and @xmath91-equivariant vector bundles over @xmath10 , and hence the corresponding grothendieck groups coincide , as in ( [ excisionslc ] ) . the role of the representation ring @xmath211 is unveiled by setting @xmath212 in ( [ excisionslc ] ) to get k_su(2)(p^1)=r_u(1 ) , [ kslccp1rp]which establishes a one - to - one correspondence between classes of homogeneous vector bundles over the sphere @xmath69 and classes of finite - dimensional representations of @xmath91 . since the corresponding irreducible representations are the @xmath89 given by ( [ u1irreps ] ) , the representation ring of @xmath91 is the ring of formal laurent polynomials in the variable @xmath213 , @xmath214 $ ] . using ( [ monmexpl ] ) we can associate the monopole bundle @xmath172 to the generator @xmath213 , and thereby identify ( [ kslccp1rp ] ) as the laurent polynomial ring k_su(2)(p^1)= . [ kslccp1mon]in particular , the relationship ( [ excisionslc ] ) can be expressed as k_su(2)(m_qp^1)=k(m_q ) . [ botteq]this is the appropriate modification of the bott periodicity isomorphism ( [ bottper ] ) to the present setting . the crucial difference now is that virtual bundles over @xmath10 are multiplied by arbitrary powers of the one - monopole bundle , allowing us to extend the equivalence to arbitrary monopole charges @xmath82 . in the equivariant setting , there is no need to use external twists of the monopole bundle , nor the ensuing k - theory product as done in @xcite . the monopole fluxes are now naturally incorporated by the coefficient ring @xmath211 of the @xmath91-equivariant k - theory , superseding the need for introducing d - operations . it is instructive to see precisely how the correspondence ( [ botteq ] ) works . for this , it is convenient to work instead in the category of holomorphic @xmath98-equivariant bundles @xcite . if @xmath215 is an @xmath1-equivariant vector bundle over @xmath56 , then the action of @xmath1 can be extended to an @xmath98 action . everything we have said above carries through by replacing the group @xmath1 with its complexification @xmath98 and the cartan torus @xmath216 with the subgroup @xmath217 of lower triangular matrices . we are then interested in @xmath100-equivariant bundles over @xmath10 with @xmath100 acting trivially on @xmath10 . the lie algebra @xmath218 is generated by the three pauli matrices _ 3= 1&0 + 0&-1 , _ + = 0&1 + 0&0 _ -= 0&0 + 1&0 [ sl2cmatrices]with the commutation relations = 2_=_3 . [ sl2clie]the lie algebra of the subgroup @xmath100 is generated by the elements @xmath219 and @xmath220 , while the cartan subgroup @xmath101 is generated by the element @xmath219 with the corresponding irreducible representations being the @xmath89 given by ( [ u1irreps ] ) . since the manifold @xmath10 carries a trivial action of the subgroup @xmath103 , any @xmath103-equivariant bundle @xmath221 can be written using ( [ trivialedecomp ] ) as a finite whitney sum e=_l(e)e_l^ _ l , [ ewhitneysl]where @xmath222 is the set of eigenvalues for the @xmath103-action on @xmath223 and @xmath224 are bundles carrying the trivial @xmath103-action . the rest of the @xmath100-equivariant structure is determined by the generator @xmath220 . since @xmath225=-2\,\sigma_-$ ] , the action of @xmath220 on @xmath226 corresponds to holomorphic bundle morphisms @xmath227 and the trivial @xmath220-action on the irreducible @xmath103-modules @xmath228 . thus every indecomposable @xmath100-equivariant bundle @xmath221 has weight set of the form @xmath229 for some @xmath230 with @xmath231 . after an appropriate twist by a @xmath103-module and a relabelling , the @xmath219-action is given by the @xmath103-equivariant decomposition e=_i=0^me_k_i_m-2i [ hactione]while the @xmath220-action is determined by a _ chain _ 0 e_k_m e_k_m-1 e_k_1 e_k_0 0 [ holchain]of holomorphic bundle maps between consecutive @xmath232 s . we can now consider the underlying @xmath91-equivariant hermitean vector bundle defined by the unitary @xmath109 reduction of the @xmath233 structure group of the holomorphic bundle ( [ hactione ] ) , after introducing a hermitean metric on its fibres . then the corresponding bundle @xmath234 is given by = su(2)_u(1)e . [ caleinduction]using ( [ monmexpl ] ) one finds that ( [ caleinduction ] ) coincides with the original equivariant decomposition ( [ caleansatz ] ) . conversely , given an @xmath1-equivariant bundle @xmath234 , its restriction @xmath235 defines a @xmath91-equivariant bundle over @xmath10 which thereby admits an isotopical decomposition of the form ( [ hactione ] ) and @xmath67 may be recovered from ( [ caleinduction ] ) . * quiver gauge theory . * the ansatz for the gauge potential on @xmath56 , represented symbolically by the bundle chain ( [ bundlechain ] ) , corresponds to the disjoint union of two copies of the quiver [ aquiver]with the second copy obtained from ( [ aquiver ] ) by reversing the directions of the arrows and replacing @xmath236 with @xmath237 for each @xmath154 . the vertices of the quiver are labelled by the degrees of the monopole bundles @xmath238 , while the arrows correspond to module morphisms @xmath239 ( locally at each point @xmath240 ) . equivalently , the vertices may be labelled by irreducible chiral representations of the group @xmath100 . thus our ansatz determines a representation of the quiver @xmath241 in the category of complex vector bundles over the manifold @xmath10 such a representation is called an @xmath241-bundle . many properties of the explicit solutions that we construct later on find their most natural explanation in the context of such a quiver gauge theory , which provides a more refined description of the brane configurations than just their k - theory charges . this framework encompasses the algebraic and representation theoretic aspects of the problem @xcite . the quiver graph ( [ aquiver ] ) is identical to the dynkin diagram of the lie algebra @xmath160 . the adjacency matrix of the quiver has matrix elements specifying the number of links between each pair of vertices @xmath242 , and in the case ( [ aquiver ] ) it is given by @xmath243 . the matrix elements @xmath244 are then identical to those of the cartan matrix @xmath245 , where @xmath246 , @xmath247 are the simple roots of @xmath160 . corresponding to the gauge symmetry breaking ( [ gaugebroken ] ) , the dimension vector @xmath248 can be regarded as a positive root of @xmath160 associated with the cartan matrix @xmath249 by writing it as k^ _ = _ i=0^mk_i e_i one - to - one correspondence between the isomorphism classes of indecomposable representations of the quiver @xmath241 and the set of positive roots of the lie algebra @xmath160 . this property is a consequence of the @xmath1-invariance of our ansatz . let us focus for a while on the case @xmath212 . in this case eq . ( [ caleansatz ] ) , with the @xmath75-monopole bundles @xmath107 substituted everywhere by the holomorphic line bundles ( [ monmholexpl ] ) , gives a relation between the categories of homogeneous holomorphic vector bundles over @xmath250 and of finite - dimensional chiral representations of @xmath100 , while the quiver representation further gives a relation with the abelian category of finite - dimensional representations of @xmath241 @xcite . to describe this latter category , it is convenient to introduce the notion of a path @xmath251 in @xmath241 , which is generally defined as a sequence of arrows of the quiver which compose . in the present case any path is of the form [ path]with @xmath252 . we will denote it by the formal vector @xmath253 . the non - negative integer @xmath254 is the length of the path ( [ path ] ) . the trivial path of length @xmath255 based at a single vertex @xmath256 is denoted @xmath257 . the path algebra @xmath258 of the quiver ( [ aquiver ] ) is then defined as the algebra generated by all paths @xmath251 of @xmath241 , i.e. as the vector space a_m+1=_^m|m_0, ,m_1 ) [ pathalg]together with the @xmath259-linear multiplication induced by ( left ) concatenation of paths where possible , |m_0, ,m_1)|n_0, ,n_1)= _ m_1n_0 |m_0, [ pathalgmult]this makes @xmath258 into a finite - dimensional quasi - free algebra . the path algebra has a natural @xmath260-grading by path length , a_m+1=_i=0^m(a_m+1)_i ( a_m+1)_i=_m_0=-m^m-2i |m_0, ,m_0 + 2i ) , [ pathgrading]and can thereby be alternatively described as the tensor algebra over the ring c_0=_i=0^m|m-2i ) ^m+1 [ c0ring]of the @xmath261-bimodule c_1=_i=0^m|m-2i , m-2i+2 ) . [ c1bimodule ] the importance of the path algebra stems from the fact that the category of representations of the quiver @xmath241 is equivalent to the category of ( left ) @xmath258-modules @xcite . given a representation @xmath262 , @xmath154 , of @xmath241 , the associated @xmath258-module @xmath263 is = _ i=0^m_m-2i [ calwdef]with multiplication extended @xmath259-linearly from the definitions |m-2i)w_j=_ij w_j |m-2i , m-2i+2)w_j=_i , j+1 _ j(w_j ) [ quivalgmod]for @xmath264 . conversely , given a left @xmath258-module @xmath263 , we can set @xmath265 for @xmath247 and define @xmath266 for @xmath154 by _ i(w_i)=|m-2i , m-2i+2)w_i . [ etaiwi]one can further show that morphisms of representations of @xmath241 correspond to @xmath258-module homomorphisms @xcite . thus , the problem of determining finite - dimensional representations of the quiver @xmath241 , or equivalently homogeneous vector bundles over @xmath69 , is equivalent to finding representations of its path algebra . as an example , consider the @xmath267 quiver [ a2quiver]it represents the standard brane - antibrane system , and as expected @xmath1-equivariance implies that it can only carry @xmath268 unit of monopole charge @xcite . the corresponding path algebra is a_2=|-1 ) |+1 ) |-1,+1)= & + 0 & . [ a2pathalg]representations of this algebra yield the standard superconnections characterizing the low - energy field content on the worldvolume of a brane - antibrane system @xcite . in the next section we will show how to generalize the superconnection formalism to account for representations of generic path algebras ( [ pathalg ] ) . later on we shall write down explicit solutions with generic monopole charge @xmath82 that also correspond to the basic brane - antibrane system . our technique for generating d - branes from a quiver gauge theory on @xmath10 arises via a quotient with respect to a generalized @xmath1-action on chan - paton bundles over @xmath56 . this new construction is rather different from the well - known quiver gauge theories that arise from orbifolds with respect to the action of a _ discrete _ group @xmath179 @xcite . in the latter case the nodes of a quiver represent the irreducible representation fractional branes into which a regular representation d - brane decays into when it is taken to an orbifold point of @xmath269 , and they can be thought of in terms of a projection of branes sitting on the leaves of the covering space @xmath10 . while our quiver gauge theory is fundamentally different , it shares many of the physical features of orbifold theories of d - branes . for instance , the blowing up of vortices on @xmath10 into instantons on @xmath56 is reminescent of the blowing up of fractional d@xmath171-branes into d@xmath270-branes wrapping a non - contractible @xmath69 that is used to resolve the orbifold singularity in @xmath269 . our solutions provide explicit realizations of this blowing up phenomenon , but in a completely smooth setting . the condition of @xmath1-equivariance uniquely prescribes a specific @xmath69 dependence for the gauge potential @xmath43 and reduces the yang - mills equations ( [ ym ] ) on @xmath162 to equations on @xmath10 . in this section we will formulate this reduction in detail and relate it to representations of the path algebra ( [ pathalg ] ) . this will be done by developing a new formalism of @xmath260-graded connections which describes the field content corresponding to the bundle chains ( [ bundlechain ] ) and ( [ holchain ] ) , and which generalizes the standard superconnection field theories on the worldvolumes of brane - antibrane systems @xcite . this formalism will be the crux to merging together the three interpretations of the previous section . * reduction of the yang - mills functional . * the dimensional reduction of the yang - mills equations can be seen at the level of the yang - mills lagrangian ( [ lagr ] ) . substituting ( [ f15])([fyyb ] ) into ( [ lagrprod ] ) and performing the integral over @xmath69 we arrive at the action , @xmath247 can be introduced via the redefinitions @xmath271 . ] s^ _ ym&:=&_m_q ^q+2x l_ym^ + & = & r^2_m_q^qx _ i=0^mtr^ _ k_ik_i , [ symred]where @xmath272 . in the remainder of this paper we shall only consider static field configurations on @xmath25 in the temporal gauge @xmath58 . in this case one can introduce the corresponding energy functional [ ef ] e^ _ _ m_2n^2nx _ i=0^m tr^ _ k_ik_i , where @xmath273 . the functional ( [ ef ] ) is non - negative . * graded connections . * the energy functional ( [ ef ] ) is analysed most efficiently by introducing a framework specific to connections on the rank @xmath274 @xmath260-graded vector bundle e:=_i=0^me_k_i [ gradedbundle]over @xmath28 whose typical fibre is the module ( [ genrepsu2uk ] ) . the endomorphism algebra bundle corresponding to ( [ gradedbundle ] ) is given by the direct sum decomposition ( e)=_i=0^mend(e_k_i ) _ ^m hom(e_k_i , e_k_j ) . [ endgraded]we may naturally associate to ( [ endgraded ] ) a distinguished representation of the @xmath160 quiver . for this , we note that the path algebra @xmath258 is itself a @xmath258-module , and that the elements @xmath275 define a complete set of orthogonal projectors of the path algebra , i.e. @xmath276 for @xmath277 with @xmath278 . analogously to the construction of ( [ calwdef])([etaiwi ] ) , we may thereby define a _ projective _ @xmath258-module @xmath279 for each @xmath247 @xcite , which is the subspace of @xmath280 generated by all paths which start at the @xmath135-th vertex of the quiver @xmath241 . then @xmath281 is the vector space generated by the path from the @xmath135-th vertex to the @xmath282-th vertex , and the corresponding dimension vector is k^ _ _ i=_j = i^me_j . [ pidimvec]the modules @xmath283 , @xmath247 are exactly the set of all indecomposable projective representations of the @xmath284 quiver @xcite , with a_m+1=_i=0^m_i . [ pathalgmoddecomp ] the importance of this path algebra representation stems from the fact that , for any quiver representation ( [ genrepsu2uk ] ) , there is a natural isomorphism @xcite ( _ i , ) _ [ hompathnatural]we may thereby identify @xmath285 in terms of appropriate combinations of the spaces ( _ j,_i ) |m-2j)a_m+1|m-2i ) . [ hompij]this is the vector space generated by the path from the @xmath135-th vertex to the @xmath282-th vertex of @xmath241 . a natural representation of this path is by a matrix of dimension @xmath286 with @xmath287 in its @xmath288-th entry and @xmath255 s everywhere else . the path algebra ( [ pathalgmoddecomp ] ) is thereby identified with the algebra of upper triangular @xmath286 complex matrices @xcite . for a given quiver representation ( [ genrepsu2uk ] ) , this algebra may be represented by assembling the chiral higgs fields @xmath289 into the @xmath290 matrix ^ _ ( m):= 0&_1&0& &0 + 0&0&_2& &0 + & & & & + 0&0&0& &_m + 0&0&0& &0 [ mgradedphidef]with respect to the decomposition ( [ gradedbundle ] ) . this object generates a representation of the path algebra in the category of complex vector bundles over @xmath28 , corresponding to the off - diagonal @xmath291 components of the decomposition ( [ endgraded ] ) . the finite dimensionality of @xmath280 is reflected in the property that generically ^ _ ( m ) , ( ^ _ ( m))^2 , , ( ^ _ ( m))^m 0 ( ^ _ ( m))^m+10 . [ mphi0s]the field configuration ( [ mgradedphidef ] ) generates the basic zero - form component of a geometric object that we shall refer to as a `` @xmath260-graded connection '' on @xmath28 . for @xmath268 it corresponds to a standard superconnection @xcite , while for @xmath176 it is the appropriate entity that constructs representations corresponding to the enlargement of the path algebra @xmath280 . its matrix form is similar to ( [ f4])([f9 ] ) , but without the one - forms on @xmath69 . to formulate the definition precisely , we note that the algebra @xmath292 of differential forms on @xmath28 with values in the bundle ( [ gradedbundle ] ) has a natural @xmath293 grading , where the @xmath187-grading is by form degree . we can thereby induce a total @xmath260-grading by the decomposition _ ( m_2n , e)=_p=0^m_(p)(m_2n , e ) _ ( p)(m_2n , e)= _ i+j^ _ m+1p^i(m_2n , e_k_j ) , [ omegagrading]where @xmath294 denotes congruence modulo @xmath177 . by using ( [ endgraded ] ) and the usual tensor product grading , this induces a @xmath260-grading on the corresponding endomorphism algebra as _ ( m_2n , end e)=_p=0^m_(p)(m_2n , end e ) [ omegaenddecomp]with _ ( p)(m_2n , end e)= _ i_a^ _ m+1(p - a ) ^i_a(m_2n)(e_k_i , e_k_i+a ) . [ omegaendgrading]a _ graded connection _ on ( [ omegagrading ] ) is defined to be a linear operator @xmath295 which shifts the total @xmath260-grading by @xmath287 modulo @xmath177 , i.e. an element of _ ( 1)(m_2n , end e)&=&_i=0^m ( _ i_1^ _ m+11^i_1(m_2n)(e_k_i ) . + & & . _ m+10 ^i_0(m_2n)(e_k_i , e_k_i+1 ) ) , [ omega1conn]and which satisfies the usual leibniz rule on @xmath296 . as in the standard cases , the @xmath260-graded connections form an affine space modelled on a set of local operators . in our case we retain only the @xmath297 and @xmath298 components of ( [ omega1conn ] ) corresponding to the lowest lying massless degrees of freedom on the given configuration of d - branes . from the leibniz rule it follows that the pertinent graded connections are then of the form @xmath299 , where ^(m):=_i=0^ma^i_i [ madef]and @xmath300 are the canonical orthogonal projections of rank @xmath287 , _ i_j=_ij _ i , [ piortho]which may be represented , with respect to the decomposition ( [ gradedbundle ] ) , by diagonal matrices @xmath301 of unit trace . in this geometric framework all @xmath302 are assumed to anticommute with a given local basis @xmath303 of the cotangent bundle of the khler manifold @xmath28 , as if they were @xmath75 basic odd complex elements of a superalgebra . this requisite property may be explicitly realized by extending the graded connection formalism to @xmath7 . for this , we rewrite the ansatz ( [ f4])([f8 ] ) in terms of the above field configurations as _ & = & ( ^(m))_1 , [ calagradedmu ] + _ y&=&_k ( ^(m))_y- ( ^ _ ( m))^_y , [ calagradedy ] + _ |y&=&_k ( ^(m))_|y+ ( ^ _ ( m))|_|y , [ calagraded]where ^(m):=_i=0^ma_m-2i_i [ madef]and @xmath304 are the canonical projections on ( [ caleansatz ] ) . the coupling of @xmath305 to @xmath306 in ( [ calagraded ] ) yields the desired anticommutativity with @xmath303 . alternatively , we may use the canonical isomorphism @xmath307 to map the cotangent basis @xmath308 onto the generators of the clifford algebra ^^+^^= -2g^ _ 2^n+1 , 1, [ 2n2cliffalg]the gamma - matrices in ( [ 2n2cliffalg ] ) may be decomposed as \{^}=\{^,^y , ^|y } ^=^_2 , ^y=^y ^|y=^|y , [ gamma2n2decomp]where the @xmath309 matrices @xmath310 act on the spinor module @xmath311 over the clifford algebra @xmath312 , ^^+^^=-2g^ _ 2^n , 1, ,2n , [ 2ncliffalg]while = _ _ 1_2n ^_1^_2n ( ) ^2=_2^n ^=-^[chiralityop]is the corresponding chirality operator . here @xmath313 is the levi - civita symbol with @xmath314 . the action of the clifford algebra @xmath315 on the spinor module @xmath316 is generated by ^y=1r^2(r^2+y|y)^y ^|y=1r^2(r^2+y|y)^|y[cp1cliffalg]with constant @xmath317 pauli matrices @xmath318 and @xmath319 obeying @xmath320=-\sigma_3 $ ] . the gauge potential ( [ f4])([f9 ] ) may then be written in an algebraic form as & : = & ^ _ + & = & ^ ( ^(m))__2 + ( ^ _ ( m ) ) ^|y|_|y- ( ^ _ ( m))^ ^y_y + & & + ( ^y ( ^(m))_y+^|y ( ^(m))_|y ) , [ calagammas]and the coupling of ( [ mgradedphidef ] ) with the chirality operator ( [ chiralityop ] ) realizes the desired anticommutativity with the one - form representatives @xmath321 . note that the products ( ^ _ ( m))^|y|_|y= 1r ( ^ _ ( m))^|y ( ^ _ ( m))^^y_y= 1r ( ^ _ ( m))^^y [ cp1indepprods]are independent of the coordinates @xmath322 . the curvature @xmath323 of the graded connection is also most elegantly expressed through dimensional reduction from @xmath7 . from ( [ f10])([fyyb ] ) it is given by & : = & _ + & = & ( ^(m))__2 - 1r(^ d _ ^ _ ( m))^^y-1r(^ d _ ^ _ ( m))^|y + & & + 12r^2 ( ^ _ ( m)+ ( ^ _ ( m))^ ( ^ _ ( m))- ( ^ _ ( m ) ) ( ^ _ ( m))^ ) _ 2^n_3 [ gradedcurv]where @xmath324 and ^ _ ( m):=_i=0^m(m-2i ) _ i . [ mupdef]the contribution ( [ mupdef ] ) is generated by the monopole connection on @xmath69 in ( [ calagammas ] ) , while the higgs potentials in ( [ gradedcurv ] ) are produced by ( [ cp1indepprods ] ) . the graded curvature is independent of @xmath325 , and the standard gamma - matrix trace formulas tr^ _ ^2^n+1 ( ^^_2)&=&-2^n+1 g^ , [ trgammaid1 ] + tr^ _ ^2^n+1 ( ^^ ^^_2)&=&2^n+1 ( g^g^+g^g^- g^g^ ) , [ trgammaid2 ] + tr^ _ ^2^n+1 ( _ 2)&=&2^n+3(g^g^-g^ g^ ) , [ trgammaid3 ] + tr^ _ ^2^n+1 ( ^ ^^|y^y ) & = & -2^ng^ = tr^ _ ^2^n+1(^ ^^y^|y ) [ trgammaid4]imply that the energy functional ( [ ef ] ) can be compactly written in terms of ( [ gradedcurv ] ) as e^ _ ym=_m_2n^2nx tr_kk^ tr^ _ ^2^n+1 ^2 . [ efgraded ] * nonabelian coupled vortex equations . * let us now examine the reduction of the duy equations on @xmath37 for a gauge potential of the form proposed in section 3 ( with static configurations in the gauge @xmath326 ) . substituting ( [ f11])([f14 ] ) into ( [ duy1])([duy3 ] ) , we obtain [ f24 ] g^a|bf^i_a|b&=&(m-2i+_i^ ^ _ i -^ _ i+1^_i+1 ) , + [ f240 ] f_|a|b^i&=&0 = f_ab^i , + [ f25 ] |^ _ |a^ _ i+1 + a^i_|a^ _ i+1 - ^ _ i+1a^i+1_|a&=&0 for each @xmath247 , where @xmath327 . recall that there is no summation over @xmath135 in these equations . we have abbreviated @xmath328 etc . , and defined the derivatives @xmath329 and @xmath330 with @xmath331 . we shall call ( [ f24])([f25 ] ) the nonabelian coupled vortex equations . ( [ f240 ] ) implies that the vector bundles @xmath332 are holomorphic , while eq . ( [ f25 ] ) implies that the higgs fields @xmath333 are holomorphic maps . by using a bogomolny - type transformation @xcite one can show that solutions to these equations realize absolute minima of the energy functional ( [ ef ] ) . these field configurations describe supersymmetric bps states of d - branes . * seiberg - witten monopole equations . * for @xmath334 , @xmath268 and @xmath335 ( so that @xmath336 ) , the equations ( [ f24])([f25 ] ) coincide with the perturbed abelian seiberg - witten monopole equations on a khler four - manifold @xmath337 @xcite . in this case we have @xmath338 and the equations ( [ f24])([f25 ] ) reduce to [ f26 ] g^a|bf_a|b&=&(1-| ) , + f_|a|b&=&0 = f_ab , + [ f27 ] |_|a+ 2a_|a&=&0 . the perturbation , i.e. the term @xmath339 in ( [ f26 ] ) , is needed whenever @xmath337 has non - negative scalar curvature in order to produce a non - trivial and non - singular moduli space of finite energy @xmath340-solutions . it is usually introduced into the seiberg - witten equations by hand . in the present context , it arises automatically from the extra space @xmath69 and the reduction from @xmath341 to @xmath337 . to build further on the interpretation of our ansatz in terms of configurations of d - branes as described in section [ ansatzdescr ] , we should now proceed to construct explicit solutions of the reduced yang - mills equations on @xmath28 . unfortunately , even solutions of the vortex equations ( [ f24])([f25 ] ) are difficult to come by and there is no known general method for explicitly constructing the appropriate field configurations . as we will demonstrate in the following , explicit realizations of these d - brane states are possible in the context of _ noncommutative _ gauge theory , which can be mapped afterwards onto commutative worldvolume configurations . for this , we will now specialize the khler manifold @xmath37 to be @xmath342 with metric tensor @xmath343 on @xmath6 and pass to a noncommutative deformation of the flat part of the space , i.e. @xmath344 . note that the @xmath69 factor remains a commutative space throughout this paper . then we will deform the yang - mills , duy and nonabelian coupled vortex equations , and in the subsequent sections construct various solutions of them . * noncommutative deformation . * field theory on @xmath345 may be realized in an operator formalism which turns schwartz functions @xmath346 on @xmath6 into compact operators @xmath347 acting on the @xmath348-harmonic oscillator fock space @xmath349 @xcite . the noncommutative space @xmath345 is then defined by declaring its coordinate functions @xmath350 to obey the heisenberg algebra relations @xmath351 = { \,\mathrm{i}\,}\th^{\mu\nu}\ ] ] with a constant real antisymmetric tensor @xmath352 . via an orthogonal transformation of the coordinates , the matrix @xmath353 can be rotated into its canonical block - diagonal form with non - vanishing components @xmath354 for @xmath355 . we will assume for definiteness that all @xmath356 . the noncommutative version of the complex coordinates ( [ zz ] ) has the non - vanishing commutators @xmath357 \= -2\,\de^{a{{\bar{b}}}}\,\th^a \ = : \ \th^{a{{\bar{b } } } } \= -\th^{{{\bar{b}}}a}\ \le\ 0 \ .\ ] ] taking the product of @xmath345 with the commutative sphere @xmath69 means extending the noncommutativity matrix @xmath358 by vanishing entries along the two new directions . the fock space @xmath349 may be realized as the linear span @xmath359 are connected by the action of creation and annihilation operators subject to the commutation relations @xmath360 = \de^{a{{\bar{b } } } } \ .\ ] ] in the weyl operator realization @xmath361 , coordinate derivatives are given by inner derivations of the noncommutative algebra according to @xmath362 \ = : \ \pa_{{\hat{z}}^a } { \hat{f}}\qquad\textrm{and}\qquad \widehat{\pa_{{{\bar{z}}}^{{{\bar{a } } } } } f}\=\th_{{{\bar{a}}}b}\,\big[{\hat{z}}^b \,,\ , { \hat{f}}\,\big ] \ \pa_{{\hat{\bar{z}}}^{\,{{\bar{a } } } } } { \hat{f}}\ , \ ] ] where @xmath363 is defined via @xmath364 so that @xmath365 . on the other hand , integrals are given by traces over the fock space @xmath349 as @xmath366 the transition to the noncommutative yang - mills and duy equations is trivially achieved by going over to operator - valued objects everywhere . in particular , vector bundles @xmath367 whose typical fibres are complex vector spaces @xmath181 are replaced by the corresponding ( trivial ) projective modules @xmath368 over @xmath3 . the field strength components along @xmath2 in ( [ ym ] ) and ( [ duy1])([duy3 ] ) read @xmath369 $ ] , where @xmath370 are simultaneously @xmath51 and operator valued . to avoid a cluttered notation , we drop the hats over operators from now on . thus all our equations have the same form as previously but are considered now as operator equations . * noncommutative coupled vortex equations . * by reducing the noncommutative version of the duy equations on @xmath371 to @xmath345 we obtain the noncommutative nonabelian coupled vortex equations . instead of working with the gauge potentials @xmath372 we shall use the operators @xmath373 defined by @xmath374 in terms of these operators the field strength tensor reads @xmath375 + \th_{a{{\bar{b } } } } \ , ~~ f_{{{{\bar{a}}}}{{{\bar{b}}}}}^{i}\ = \ \big[x_{{{{\bar{a}}}}}^{i}\,,\ , x_{{{\bar{b}}}}^{i}\,\big ] \quad\textrm{and}\quad f_{{a}{b}}^{i}\ = \ \big[x_{{a}}^{i}\,,\ , x_{b}^{i}\,\big]\ , \label{fieldstrengthx}\ ] ] while the bi - fundamental covariant derivatives become d_|a^ ^ _ i+1x_|a^i^ _ i+1 x_|a^i+1 d_a^ ^ _ i+1x_a^i ^ _ i+1x_a^i+1 . [ bifundx ] the nonabelian vortex equations ( [ f24])([f25 ] ) can then be rewritten as [ ddd1 ] ^a|b(+ _ a|b ) & = & ( m-2i+_i^^ _ i - ^ _ i+1^_i+1 ) , + [ ddd ] & = & 0 = , + [ ddd2 ] x_|a^i^ _ i+1 - ^ _ i+1x_|a^i+1&=&0 for @xmath247 . note that for @xmath268 we obtain the equations [ f28 ] ^a|bf^0_a|b&=&(1-^ _ 1 _ 1^ ) f_|a|b^0 = 0 = f_ab^0 , + [ f29 ] ^a|bf_a|b^1&=&-(1-_1^ ^ _ 1 ) f_|a|b^1 = 0 = f_ab^1 , + [ f30 ] |^ _ |a^ _ 1 + a^0_|a^ _ 1 - ^ _ 1a^1_|a&=&0 which are considered in @xcite . in particular , for @xmath334 and @xmath335 the equations ( [ f28])([f30 ] ) coincide with the perturbed seiberg - witten @xmath376 monopole equations on @xmath377 as considered in @xcite . we are now ready to construct solutions to the yang - mills equations on @xmath378 . we shall first present the generic non - bps solutions of the full yang - mills equations , and then proceed to solve the nonabelian coupled vortex equations ( [ ddd1])([ddd2 ] ) , and thus the duy equations on @xmath378 , which describe the stable bps states . our technique will make use of appropriate partial isometry operators @xmath379 in the noncommutative space . * ansatz for explicit solutions . * let us fix a monopole charge @xmath380 and an arbitrary integer @xmath381 . consider the ansatz [ ansatz3 ] x_a^i & = & _ a|bt^ _ n_i|z^|bt_n_i^ x_|a^i = ^ _ |ab t^ _ n_i z^bt_n_i^ , + [ ansatz3p ] ^ _ i+1 & = & ^ _ i+1t^ _ n_it_n_i+1^ ^_i+1 = |^ _ n_i+1t_n_i^ for @xmath247 , where @xmath382 are some constants with @xmath383 . denoting by @xmath349 the @xmath348-oscillator fock space , the toeplitz operators @xmath384 are partial isometries described by _ rectangular _ @xmath385 matrices ( with operator entries acting on @xmath349 ) possessing the properties @xmath386 where @xmath387 is a hermitean projector of finite rank @xmath388 on the fock space @xmath389 so that p_n_i^2p^ _ n_ip^_n_i tr^ _ _ k_ih p^ _ n_in_i . [ pnitrace]from ( [ ansatz4 ] ) it follows that the operator @xmath379 has a trivial kernel , while the kernel of @xmath390 is the @xmath388-dimensional subspace of @xmath389 corresponding to the range of @xmath391 . thus t^ _ n_i0 t^_n_in_i . [ dimkertni ] substituted into ( [ fieldstrengthx ] ) this ansatz yields the gauge field strength @xmath392 while from ( [ bifundx ] ) one finds the covariant derivatives d_|a_i+10d_a_i+1 . [ ansatzcovconst]thus our ansatz describes _ holomorphic _ fields , and the projector @xmath391 defines a noncommutative gauge field configuration of rank @xmath388 and constant curvature in the subspace @xmath393 . in particular , the higgs fields @xmath170 are covariantly constant with ^_i^ _ i=|_i|^2 ( ^ _ k_i- p^ _ n_i ) ^ _ i+1^_i+1=|_i+1|^2 ( ^ _ k_i - p^ _ n_i ) . [ phiphidags ] the ranks @xmath388 are generically non - negative integers . if some @xmath394 , then we should formally set @xmath395 , @xmath396 and @xmath397 in the @xmath135-th component of the ansatz . then x_a^i=_a|b|z^|bx^i_|a=^ _ |abz^b [ xni0]which leads to the vacuum gauge field configuration a^i0 f^i0 . [ afni0]these matter fields correspond to open strings with one end on a d - brane and the other end on the closed string vacuum . our ansatz has a natural interpretation in quiver gauge theory . consider the module : = _ i=0^mt_n_i^k^ _ = _ i=0^mn_i e_i [ vnam]over the quiver @xmath241 , which is a finite - dimensional submodule of the infinite - dimensional representation @xmath398 of @xmath241 given by the noncommutative quiver bundle . let us fix an integer @xmath399 , and take @xmath400 for all @xmath401 and @xmath394 for all @xmath402 . the quiver representation ( [ vnam ] ) is a combination of the indecomposable projective representations @xmath283 of @xmath241 that we encountered in section [ dimred ] . the @xmath283 s form a complete set of projective representations in the sense that any quiver representation has a projective resolution in terms of sums of them @xcite . in particular , the canonical ringel resolution of ( [ vnam ] ) is given by the exact sequence 0 _ i=1^s_i-1t_n_i^ _ i=0^s _ it_n_i^ 0 . [ ringelres ] * solving the yang - mills equations . * we shall now demonstrate that the field configurations ( [ ansatz3])([ansatz4 ] ) yield solutions of the full yang - mills equations on @xmath403 for any values of @xmath75 , @xmath404 and @xmath405 . for this , we write the ansatz in the form [ chia ] _ a- _ a|b|z^|b & = & _ i=0^m x^i_a^ _ i = _ a|b_i=0^m t^ _ n_i|z^|bt_n_i^ ^ _ i , + [ chiab ] _ |ab z^b & = & _ i=0^m x^i_|a^ _ i = _ |a b_i=0^m t^ _ n_iz^bt_n_i^ ^ _ i . we also have _ y^ii&= & _ k_i , [ ayii ] + _ |y^ii&=&- _ k_i , [ aybii ] + [ avt ] _ |y^i i+1 & = & ^ _ i+1 = t^ _ n_it_n_i+1^ , + [ avp ] _ y^i+1 i & = & - ^_i+1 = -t^ _ n_i+1t_n_i^ , with _ j j i , i+1 . [ av0]thus for the ansatz ( [ ansatz3])([ansatz4 ] ) the field strength tensor is given by [ cfabb ] _ a|b & = & _ a|b _ i=0^m p^ _ n_i^ _ i , + [ cfvtvp ] _ y|y & = & - _ i=0^m(m-2i+(|^ _ i|^2 -|^ _ i+1|^2)(_k_i - p^ _ n_i ) ) ^ _ i , with all other components of @xmath406 vanishing . let us now insert these expressions into the yang - mills equations ( [ ym ] ) ( for static configurations with @xmath407 ) . it is enough to consider the cases @xmath408 and @xmath409 , since the cases @xmath410 and @xmath411 can be obtained by hermitean conjugation of ( [ ym ] ) due to the anti - hermiticity of @xmath412 and @xmath413 . for @xmath408 , eq . ( [ ym ] ) becomes @xmath414\bigr)=0\ ] ] which is equivalent to @xmath415=0\ .\ ] ] substituting ( [ chiab ] ) and ( [ cfabb ] ) , we see that ( [ rym ] ) is satisfied due to the identities ( [ piortho ] ) and @xmath416 in the case @xmath417 , eq . ( [ ym ] ) simplifies to @xmath418=0\ ] ] with @xmath419 . substituting ( [ ayii ] ) , ( [ avp ] ) , ( [ av0 ] ) and ( [ cfvtvp ] ) , we find that ( [ rrym ] ) is also satisfied due to the identities ( [ piortho ] ) and ( [ iden ] ) . hence , the yang - mills equations on @xmath378 are solved by our choice of ansatz . * finite - energy solutions . * the arbitrary coefficients @xmath420 can be fixed ( up to a phase ) by demanding that the solution ( [ ansatz3])([ansatz4 ] ) yield finite - energy field configurations . for this , we evaluate the energy functional ( [ ef ] ) using ( [ intnc ] ) . from ( [ ansatzfieldstrength ] ) we may compute ( f_^i)^(f^i)= 8^a|c^d|bf_a|b^if_d|c^i2 ( _ a=1^n1(^a)^2 ) p^ _ n_i , [ efncfieldstrength]and combining this with ( [ ansatzcovconst ] ) and ( [ phiphidags ] ) we find the noncommutative yang - mills energy e^ _ ym&=&2r^2(_a=1^n2^a ) _ i=0^mtr^ _ _ [ efncgen]because of the trace over the infinite - dimensional fock space @xmath349 , the constant terms in ( [ efncgen ] ) which are not proportional to the projectors @xmath391 must all vanish in order for the energy to be finite . this leads to the finite - energy conditions @xmath421 for each @xmath247 . with @xmath383 , the constraints ( [ cnstr1 ] ) are solved by @xmath422 ) can thereby be written as e^ _ ym=2r^2(_a=1^n2^a ) _ i=0^m2(n_i+n_m - i ) . [ efncfinite]we have naturally split the sum over nodes @xmath135 into contributions from dirac monopoles and antimonopoles , which for each @xmath247 have the same yang - mills energy on the sphere @xmath69 . later on we will see that this splitting corresponds to a @xmath423-grading of the chain of d - branes into brane - antibrane pairs . the monopole independent terms in ( [ efncfinite ] ) can be interpreted as the tension of @xmath424 d0-branes inside a d@xmath5-brane @xcite in the seiberg - witten decoupling limit @xcite . * bps solutions . * the solutions we have described generically yield non - bps solutions of the full yang - mills equations on @xmath403 . on the other hand , the duy equations on @xmath403 are bps conditions for the yang - mills equations . inserting ( [ ansatz3])([ansatz4 ] ) and ( [ ansatzfieldstrength])([phiphidags ] ) into our nonabelian vortex equations ( [ ddd1])([ddd2 ] ) , we find that ( [ ddd ] ) and ( [ ddd2 ] ) are automatically satisfied . the vanishing of the constant term ( not proportional to @xmath391 ) in ( [ ddd1 ] ) is precisely the finite - energy constraint ( [ cnstr1 ] ) , whose solution is given in ( [ 7.8 ] ) . equating the coefficients of @xmath391 in ( [ ddd1 ] ) for each @xmath400 leads to the additional constraints @xmath425 for @xmath426 the conditions ( [ cnstr2 ] ) are incompatible with one another , implying that the ansatz ( [ ansatz3])([ansatz4 ] ) with @xmath426 does not allow for bps configurations . for @xmath427 , the equation ( [ cnstr2 ] ) relates the radius @xmath14 of the sphere to the noncommutativity parameters @xmath428 of @xmath3 . in this case we obtain the explicit solutions of the noncommutative vortex and duy equations parametrized by the partial isometry operators @xmath429 as [ explsl0 ] x^0_a&=&_a|bt^ _ n_0|z^|b t_n_0^ ^ _ 1=^ _ 1t^ _ n_0 , + x^i_a&=&_a|b|z^|b ^ _ i=^ _ i _ k_1 0 < i m . [ explsl1]the bps conditions ( [ ddd1])([ddd2 ] ) force us to take @xmath430 corresponding to the gauge symmetry breaking @xmath431 , so that @xmath432 , @xmath433 with @xmath434 and @xmath435 . the configurations with @xmath436 correspond to the vacuum gauge fields ( [ afni0 ] ) with trivial bundle maps @xmath302 given as multiplication by the complex numbers @xmath437 satisfying ( [ 7.8 ] ) . using ( [ cnstr2 ] ) and ( [ efncfinite ] ) we find that the energies of these bps states are given by e^ _ bps=2(2)^n+1r^2(_^n _ ^n^a ) n_0 . [ ebps ] these solutions have a natural physical interpretation along the lines described in section [ ansatzdescr ] . the original noncommutative duy equations are fixed by the positive integers @xmath348 and @xmath41 . our ansatz ( [ f4])([f9 ] ) and ( [ ansatz3])([ansatz4 ] ) is labelled by the collection of positive integers @xmath438 with @xmath439 . according to the standard identification of d - branes as noncommutative solitons @xcite , the configuration ( [ explsl0],[explsl1 ] ) with @xmath427 describes a collection of @xmath440 bps d0-branes as a stable bound state ( i.e. a vortex - like solution on @xmath3 ) in a system of @xmath441 d(2@xmath348 ) branes and antibranes . but from the point of view of the initial branes wrapped on @xmath403 , they are spherical @xmath442 d2-branes . this means that instantons on @xmath403 are the spherical extensions of vortices which are points in @xmath2 . for @xmath426 the configuration ( [ ansatz3])([ansatz4 ] ) describes an unstable system of @xmath443 d0-branes ( vortices ) in a d(2@xmath348 ) brane - antibrane system , because @xmath444 for each @xmath445 . again they form a system of spherical d2-branes ( i.e. an su(2)-symmetric multi - instanton ) in the initial brane - antibrane system on @xmath403 . their orientation depends on the sign of the magnetic charge @xmath446 for each @xmath439 , which determines whether we have d2-branes or d2-antibranes . if more than one @xmath447 then the ansatz either describes pairs of d0-branes with overall non - vanishing monopole charges , or both d0-branes and anti - d0-branes . such systems can not be stable , i.e. the corresponding configuration ( [ ansatz3])([ansatz4 ] ) can not satisfy the noncommutative vortex and duy equations . the distinction between bps versus non - bps solutions is very natural in quiver gauge theory . the bps configurations are described by the simple schur representations @xmath448 , @xmath247 of the @xmath160 quiver given by a one - dimensional vector space at vertex @xmath135 with all maps equal to @xmath255 , i.e. the @xmath241-module with @xmath449 and dimension vector @xmath450 . the bps states constructed above then correspond to the quiver representations @xmath451 . together with the projective modules @xmath283 , the schur modules @xmath448 admit the projective resolutions 0 _ 0 _ 0 & & 0 , [ plres0 ] + 0 _ i-1 _ i _ ,s [ plresi]and satisfy the relations @xcite ( _ i,_j)=_ij hom(_i,_j ) . [ plhomrels ] the resolutions ( [ ringelres ] ) and ( [ plres0],[plresi ] ) exhibit a sharp homological distinction between bps and non - bps solutions . the constituent d - branes at the vertices of the quiver @xmath241 are associated with the basic representations @xmath448 . sums @xmath452 for fixed @xmath135 correspond to bps states , associated generally with the symmetry breaking @xmath453 , which are constructed analogously to ( [ explsl0],[explsl1 ] ) but with the vacuum higgs configurations @xmath454 for @xmath455 and @xmath456 for @xmath457 . a generic non - bps state , associated to the quiver representation ( [ vnam ] ) , corresponds to the decay of the original @xmath1-symmetric branes wrapped on @xmath403 into the collection of constituent branes @xmath458 in @xmath3 . for @xmath426 this collection is unstable . in the quiver gauge theory , we have thereby arrived at a natural construction of the unstable d - brane configurations in terms of stable bps states of d - branes , which may be succinctly summarized through the sequence of distinguished triangles of quiver representations ( _ 0)^n_0 = & _ 0 & & _ 1 & & _ m-1 & & _ m&= + & & & & & & & & + & & ( _ 1)^n_1 & & & & ( _ m)^n_m & & [ bpstriangles]where @xmath459 and the horizontal maps are the canonical inclusions of submodules . this exact sequence expresses the fact that , for each @xmath460 , the non - bps module @xmath461 is an extension of the bps module @xmath462 by the non - bps module @xmath463 . in this section we shall construct an explicit realization of the basic partial isometry operators @xmath464 which will be particularly useful for putting the d - brane interpretation of our noncommutative multi - instanton solutions on firmer ground . it is based on an @xmath1-equivariant generalization of the ( noncommutative ) atiyah - bott - shapiro ( abs ) construction of tachyon field configurations @xcite@xcite . * equivariant abs construction . * if @xmath179 is a group and @xmath465 , we denote by @xmath466 the grothendieck group of isomorphism classes of finite - dimensional @xmath423-graded @xmath467 modules , i.e. clifford modules possessing an even ( @xmath423-degree preserving ) @xmath179-action which commutes with the @xmath468-action . more precisely , we consider representations of @xmath469\otimes{{{\rm c}\ell}}_{2n}$ ] with @xmath469 $ ] the group ring of @xmath179 . the inclusion @xmath470 of clifford algebras induces a restriction map _ g(2n)^*:r_spin_g(2n+1 ) r_spin_g(2n ) [ restrmapg]on equivariant clifford modules . following the standard abs construction @xcite , we may then obtain the @xmath179-equivariant k - theory @xmath471 ( with compact support ) through the descendent isomorphism k_g(^2n ) = coker_g(2n)^ * = r_spin_g(2n)/ _ g(2n)^*r_spin_g(2n+1 ) . [ descisog]the image of @xmath472 in @xmath466 contains classes of clifford modules @xmath180 $ ] which admit a @xmath467-equivariant involution @xmath473 , where @xmath474 is the clifford module @xmath181 with its @xmath423-parity reversed . in our case , we take @xmath475 acting trivially on @xmath6 , and thereby consider @xmath476-modules with the @xmath91-action commuting with the clifford action . any such module is a direct sum of tensor products of a @xmath91-module and a spinor module , and hence r_spin_u(1)(2n)r_spin(2n)r_u(1)_u(1)(2n)^*=(2n)^ * . [ spinortrivdecomp]since from the standard abs construction one has the isomorphism @xcite k(^2n ) = coker(2n)^ * = r_spin(2n)/ ( 2n)^*r_spin(2n+1 ) [ usualabsiso]of abelian groups , we can reduce ( [ descisog ] ) for @xmath477 to the isomorphism k_u(1)(^2n)=k(^2n)r_u(1)[eqabsiso]of @xmath211-modules , where @xmath478 ( note that the isomorphism @xmath479 also follows from the fact that @xmath6 is equivariantly contractible to a point ) . we may describe the isomorphism ( [ eqabsiso ] ) along the lines explained in section [ ansatzdescr ] . in particular , the spinor module @xmath480 admits the isotopical decomposition _ i hom^ _ u(1)(_m-2i,_2n ) [ spinmoddecomp]obtained by restricting @xmath481 to representations of @xmath482 . the @xmath483 s in ( [ spinmoddecomp ] ) are the corresponding multiplicity spaces . the most instructive and useful way to explicitly realize the decomposition ( [ spinmoddecomp ] ) is to use the equivariant excision theorem ( [ excisionslc ] ) directly and consider the @xmath1-invariant dimensional reduction of spinors from @xmath484 to @xmath6 . for this , we introduce the twisted dirac operator on @xmath484 using the graded connection formalism of section [ dimred ] to write the @xmath260-graded clifford connection : = ^d _ = ^d__2 + ( ^ _ ( m))^|y_|y- ( ^ _ ( m))^^y _ y+_p^1 , [ diracgradeddef]where _ p^1 : = ^yd_y+^|yd_|y = ^y(_y+_y+ ( ^(m))_y)+ ^|y(_|y+_|y+ ( ^(m))_|y ) [ diracs2def]and @xmath485 are the components of the levi - civita spin connection on the tangent bundle of @xmath69 . from ( [ diracgradeddef ] ) we see that the monopole charges @xmath446 in the yang - mills energy functional ( [ ef ] ) can be understood as originating from the dirac operator ( [ diracs2def ] ) on @xmath69 . the operator ( [ diracgradeddef ] ) acts on spinors @xmath486 which are sections of the bundle = ^+ + ^- _ i=0^m(e_k_i_2n ) l^m-2i+1 + l^m-2i-1 [ spinortotgen]over @xmath484 , where @xmath487 are the twisted spinor bundles of rank @xmath488 over the sphere @xmath69 . we are therefore interested in the twisted spinor module @xmath489 over the clifford algebra @xmath490 which is the product of the spinor module @xmath491 with the fundamental representation ( [ genrepsu2uk ] ) of the gauge group @xmath109 broken as in ( [ gaugebroken ] ) . the symmetric fermions on @xmath6 that we are interested in correspond to @xmath1-invariant spinors on @xmath484 . they belong to the kernel of the dirac operator ( [ diracs2def ] ) on @xmath69 and will be massless on @xmath6 . one can write _ p^1=_i=0^m_m-2i= _ i=0^m 0&^-_m-2i + ^+_m-2i&0 , [ diracs2decomp]where ^+_m-2i&=&1r^2 , [ diracs2p ] + ^-_m-2i&=&-1r^2 . [ diracs2m]the operator ( [ diracs2decomp ] ) acts on sections of the bundle ( [ spinortotgen ] ) which we write with respect to this decomposition as = _ ( m-2i)^+ + _ ( m-2i)^- , [ psidecomp]where @xmath492 are sections of @xmath493 taking values in @xmath494 with coefficients depending on @xmath495 . to describe the kernel of the dirac operator ( [ diracs2decomp ] ) , we need to solve the differential equations ^+_m-2i^+_(m-2i)0 ^-_m-2i^-_(m-2i)0 [ diracpkernel]for the positive and negative chirality spinors @xmath496 and @xmath497 in @xmath498 and @xmath499 . by recalling the form of the transition functions for the monopole bundles from section [ invgauge ] , one easily sees that the only solutions of these equations which are regular on both the northern and southern hemispheres of @xmath4 are of the form _ ( m-2i)^+=1(r^2+y|y)^t_i/2 _ = 0^t_i^+_(m-2i)(x ) y^_(m-2i)^-0 m-2i < 0 [ solschargeneg]and _ ( m-2i)^-=1(r^2+y|y)^t_i/2 _ = 0^t_i^-_(m-2i)(x ) |y^_(m-2i)^+0 m-2i > 0 . [ solschargepos]here @xmath500 and the component functions @xmath501 on @xmath6 with @xmath502 form the irreducible representation @xmath503 of the group @xmath1 . thus the chirality grading is by the sign of the magnetic charges . this analysis is valid when the monopole charge @xmath75 is an even or odd integer . however , when @xmath75 is even there is precisely one term in ( [ spinortotgen ] ) with @xmath504 for which the sub - bundle @xmath505 is twisted by the ordinary spinor bundle @xmath506 of vanishing magnetic charge . this bundle admits an infinite - dimensional vector space of symmetric @xmath507-sections comprised of spinor harmonics @xmath508 with @xmath509 , @xmath510 and @xmath511 @xcite . the spectrum of the ( untwisted ) dirac operator @xmath512 consists of the eigenvalues @xmath513 , each of even multiplicity @xmath514 . after dimensional reduction , this produces an infinite tower of massive spinors on @xmath6 , and such fermions of zero magnetic charge have no immediate interpretation in the present context . however , one has @xmath515 , and this will be enough for our purposes . we will therefore fix one of these vector spaces , such that after integration over @xmath69 it corresponds to the space _ p ^2^p+1 p1,3,5 , . [ spinorharm]all of our subsequent results will be independent of the particular choice of eigenspace ( [ spinorharm ] ) . we have thereby shown that the @xmath1-equivariant reduction of the twisted spinor representation of @xmath490 decomposes as a @xmath423-graded bundle giving _ ( ^2np^1 ) ^su(2)=_2n(_^+ _ ^- ) m , [ twistedspingrad]where _ ^+=_i = m_+^m _ k_i_|m-2i| _ ^-= _ i=0^m_- _ k_i_m-2i [ twistedspinpm]with @xmath516 and @xmath517 . when @xmath75 is an even integer , one should also couple the eigenspace ( [ spinorharm ] ) giving _ ( ^2np^1 ) ^su(2)=_2n(_^+ ( _ k_m2_p ) _ ^- ) m [ twistedspingradeven]with @xmath518 and @xmath517 . it remains to work out the corresponding action of clifford multiplication ^ _ : _ ^- _ ^+ . [ cliffmultv]for this , we recall from section [ ansatzdescr ] that the action of the generators of the parabolic subgroup @xmath217 on the equivariant decomposition ( [ twistedspingrad],[twistedspinpm ] ) is given by @xmath519 and @xmath520 . since the clifford action is required to commute with this action , the map ( [ cliffmultv ] ) is thereby uniquely fixed on the isotopical components in the form ^ _ ^ _ i:_k_i_|m-2i| _ k_i+m2 + 1 _ |m-2i| i0,1, [ cliffmultisot]furthermore , since @xmath521 for all @xmath522 , the space of spinor harmonics must lie in the kernel of the clifford map and one has ^ _ _ m20 m . [ cliffordmulteven ] it is also illuminating to formulate this equivariant dimensional reduction from a dynamical point of view , as we did for the gauge fields in section [ dimred ] . using the gauged dirac operator ( [ diracgradeddef ] ) we may define a fermionic energy functional on the space of sections of the bundle ( [ spinortotgen ] ) by e^ _ d:=_^2np^1 ^ 2n+2x g ^ . [ eddef]one has ^ ( ( ^ _ ( m))^|y- ( ^ _ ( m))^^y)= ( ^+)^&(^-)^ ( ^ _ ( m))^(^- ) + ( ^ _ ( m))(^+ ) . [ psiphiid]substituting ( [ diracs2decomp])([solschargepos ] ) , we see that ( [ psiphiid ] ) vanishes on symmetric spinors and after integration over @xmath69 the energy functional ( [ eddef ] ) for @xmath75 odd becomes e^ _ d&=&4r^2_^2n^2nx . [ efred]the symmetric fermion energy functional for @xmath75 even also contains mass terms for fermions of vanishing magnetic charge which are proportional to the multiplicity @xmath523 of the spinor harmonics . * explicit form of the operators @xmath524 . * the operators @xmath525 parametrizing the solutions of the previous section may be realized explicitly by appealing to a noncommutative version of the above construction . for this , we first note that the ( trivial ) action of @xmath216 on @xmath6 induces an action on functions @xmath346 on @xmath6 by @xmath526 for @xmath95 . this in turn defines an action of @xmath91 on the noncommutative space @xmath527 through automorphisms @xmath528 of the weyl operator algebra , i.e. a representation of @xmath91 in the automorphism group of the algebra . we will assume that the fock space ( [ fockspace ] ) carries a unitary representation of @xmath91 . we can then decompose it into its isotopical components in the usual way as h=_i=0^mh_i_m-2i . [ fockisotop]for @xmath95 we denote the corresponding unitary operator on @xmath349 by @xmath529 . if we demand that the representations of @xmath91 above are covariant with respect to each other @xcite , f^-1= , [ covariantreps]then they define a representation of the crossed - product of the algebra of weyl operators with the group @xmath91 . this defines the ( trivial ) noncommutative @xmath91-space @xmath530 , and equivariant field configurations are operators belonging to the commutant of @xmath91 in the crossed - product algebra . in quiver gauge theory , the pertinent representation of @xmath241 thus labels isotopical components of the hilbert space of the noncommutative gauge theory . since the @xmath91-action is trivial here , the isotopical components of the fock space ( [ fockisotop ] ) are given by @xmath531 for each @xmath247 . note that one has an isomorphism @xmath532 by the usual hilbert hotel argument . we will now construct a representation on ( [ fockisotop ] ) of the partial isometry operators @xmath379 in @xmath530 . for this , let us put @xmath533 and consider the operators @xcite @xmath534 where @xmath535 , @xmath536 and the @xmath537 matrices @xmath538 are subject to the anticommutation relations @xmath539 eq . ( [ sigmaanti ] ) implies that the matrices @xmath540 generate the clifford algebra @xmath468 . note that for @xmath541 we have @xmath542 , @xmath543 and @xmath544 , which yields @xmath545 and we obtain the standard shift operator @xmath546 on the fock space @xmath349 in this case . generally , the operators ( [ tt ] ) obey @xmath547 where @xmath548 is a projector of rank @xmath388 on the vector space @xmath549 , and @xmath550 are the irreducible chiral spinor modules of dimension @xmath551 ( with @xmath552 ) on which the matrices @xmath553 act . the partial isometry operators @xmath554 in @xmath527 do not act on the isotopical decomposition ( [ fockisotop ] ) and thus do not properly incorporate the @xmath1-equivariant reduction of the original system of d - branes . the desired operators @xmath464 in @xmath555 are obtained by first projecting these partial isometries onto constituent brane subspaces . with @xmath556 the rank @xmath287 projector onto the @xmath135-th isotopical component in ( [ fockisotop ] ) , we thereby define the @xmath537 matrices @xmath557 the operator @xmath558 acts as the shift operator @xmath554 on @xmath559 and as the identity operator @xmath560 on @xmath561 for all @xmath562 . it is easy to see that these matrices satisfy the equations ( t^(0)_n_i)^(t^(0)_n_i)=_r ( t^(0)_n_i)(t^(0)_n_i ) ^=_r - p_n_i^(0 ) [ tni0eqs]with @xmath563 a projector of rank @xmath388 on the fock space @xmath549 . they also satisfy the algebra ( t^(0)_n_i)^nt^(0)_n_in t^(0)_n_it_n_j^(0)t^(0)_n_i+t_n_j^(0)-^ _ r= t^(0)_n_jt_n_i^(0 ) i j . [ tnitnj]the operator ( [ 8 ] ) may be regarded as a linear map @xmath564 in particular , the map @xmath565 has a trivial kernel , while @xmath566 has a one - dimensional kernel which is spanned by the vector @xmath567 where @xmath568 denotes the lowest - weight spinor of so(@xmath29 ) . finally , the desired _ rectangular _ @xmath385 toeplitz operators @xmath379 may be realized in terms of the partial isometries ( [ 8 ] ) by appealing to the hilbert hotel argument . for this , we introduce a lexicographic ordering @xmath569 on the fock space @xmath349 so that @xmath570 with @xmath571 , and fix an orthonormal basis @xmath572 of the chiral spinor representation space @xmath573 . then @xmath574 , @xmath575 is an orthonormal basis for @xmath576 and there is a one - to - one correspondence @xmath577 of basis states . similarly , by fixing an orthonormal basis @xmath578 of the @xmath1 representation space @xmath579 , there is a one - to - one correspondence latexmath:[$\vec\lambda_{a_i}^{\,i}\otimes|q_i\rangle\leftrightarrow corresponding orthonormal basis of @xmath389 . let us now introduce unitary isomorphisms @xmath581 and @xmath582 by the formulas [ uipdef ] u^ _ i&=&_a=0^r-1 _ a_i=0^k_i-1 _ ^|k_iq_i+a_irq+a| + & = & _ a=0^r-1 _ a_i=0^k_i-1 _ ^^i_a_i ^_a^|q_iq| , + u^_i&=&_a=0^r-1 _ a_i=0^k_i-1 _ ^|rq+ak_iq_i+a_i| + & = & _ a=0^r-1 _ a_i=0^k_i-1 _ ^_a^ ^i_a_i^|qq_i| . [ uimdef]by using the shift operators ( [ 8 ] ) , we then define the operators t^ _ n_i^ _ i(t_n_i^(0 ) ) t_n_i^=(t_n_i^(0))^u^_i [ tniuidef]on @xmath583 and @xmath584 . they satisfy the requisite equations ( [ ansatz4 ] ) , with the @xmath585 matrix p^ _ n_i = u_i^ ( p^ _ n_i^ _ i ) u_i^[pnifinal]a projector of rank @xmath388 on the fock space @xmath389 . notice that the rank @xmath551 used in this construction is an even integer for @xmath586 . to work with odd ranks @xmath587 one may introduce the @xmath588 matrices @xmath589 where @xmath590 is defined as above and @xmath591 is a shift operator on the fock space @xmath349 . then the operators ( [ 10 ] ) satisfy the equations ( [ tni0eqs ] ) with @xmath592 a projector of rank @xmath388 on the fock space @xmath593 , where @xmath594 . in this case the toeplitz operators @xmath464 are obtained by substituting ( [ 10 ] ) into ( [ tniuidef ] ) with the replacement @xmath595 . note also that the partial isometry operator t^(0)&:=&_i=0^mt^(0)_n_i + & = & _ r^ + _ i=0^m ( t^(0)_n_i-_r^ ) = _ r^ + _ i=0^m(()^n_i-_r^ ) , [ totaltn]together with the above representations of the @xmath91 group on the weyl operator algebra of @xmath3 and on the fock space @xmath349 , defines a cycle in the @xmath91-equivariant analytic k - homology @xmath596 . after a twisting appropriate to the inclusion of the pertinent magnetic monopole bundles , it describes the @xmath1-invariant configurations of d - branes as branes on the ( trivial ) quotient space @xmath597 . the charge of this class is the same as that of the cocycle @xmath598 $ ] built earlier in the topological k - theory ( [ eqabsiso ] ) from the standard abs brane - antibrane class @xmath599 $ ] which is the generator of ( [ usualabsiso ] ) @xcite@xcite,@xcite . the computation of the topological charge , as well as the equivalence between the commutative ( topological ) and noncommutative ( analytic ) k - homology descriptions of the d - brane configurations , will be presented in the next section . * moduli space of solutions . * the realization ( [ 8 ] ) can be generalized in order to introduce @xmath600 real moduli into the solution which specify the locations of the various noncommutative solitons in @xmath6 @xcite . for this , one first has to introduce `` shifted ground states '' centered at @xmath601 , @xmath602 for each @xmath247 . the operators ( [ 8 ] ) are rewritten as t^(0)_n_i=^ _ i)+(^i_1^i_2 ^i_n_i)^ _ i , [ tnirewrite]where each @xmath603 , @xmath604 , @xmath247 is of the form of the shift operator @xmath605 in ( [ tt ] ) but with the coordinates @xmath606 shifted to @xmath607 . they behave just like @xmath605 except that now the kernel of @xmath608 is spanned by the vector @xmath609 , where @xmath610 is the fermionic ground state and the shifted ground state @xmath611 is a coherent state in the @xmath348-oscillator fock space @xmath349 , i.e. @xmath612 . the states @xmath613 and @xmath614 for @xmath615 span the kernel of the operator @xmath616 given by ( [ tnirewrite ] ) , and we find that the equations ( [ tni0eqs ] ) are obeyed with @xmath617 the orthogonal projection onto @xmath618 . the space of partial isometries ( [ totaltn ] ) may thereby be described as the complex manifold @xmath619 . after a quotient by the appropriate discrete symmetry group , the moduli space for the full solution consisting of the rectangular toeplitz operators ( [ tniuidef ] ) is given by ( n;k^ _ , k^ _ ) = q(k^ _ ) _ i=0^mhilb^n_i ( ^n ) , [ partisospace]where @xmath620 is the moduli space of @xmath388 noncommutative solitons on @xmath2 @xcite which is given as the ( singular ) hilbert scheme of @xmath388 points in @xmath621 , i.e. the set of ideals @xmath622 of codimension @xmath388 in the polynomial ring @xmath623 $ ] . the factor @xmath624 is the moduli space of isomorphism classes of quiver representations ( [ genrepsu2uk ] ) of dimension @xcite ( k^ _ ) = 1-k^ _ c k^ _ 1+_i=0^mk_i(k_i+1-k_i ) . [ dimcalq]note that real roots ( having @xmath625 ) correspond to rigid representations of the quiver @xmath241 with no moduli , while imaginary roots ( having @xmath626 ) carry moduli associated to the gauge symmetry breaking ( [ gaugebroken ] ) . the points of the moduli space ( [ partisospace ] ) label the positions of well - separated d - branes , and it coincides in the low - energy limit with the moduli space of the commutative brane description @xcite . in this section we will compute the topological charge of our multi - instanton solutions in essentially two distinct ways . the first one is a direct field theoretic calculation of the @xmath627-th chern number of our gauge field configurations on @xmath0 , which can also be computed using the @xmath260-graded connection formalism of section [ dimred ] . the second one is a homological calculation of the index class of our solutions in k - theory , which is also equivalent to the euler - ringel character of the pertinent representations of the quiver @xmath241 . the equivalence of these two calculations will then lead us directly into a worldvolume description whereby we can interpret the topological charge in terms of cycles in topological equivariant k - homology , yielding the claimed d - brane interpretation of our solutions . the results of this section bridge together the descriptions presented in section [ ansatzdescr ] and justify the brane interpretations that have been given throughout this paper thus far . * field theory calculation . * we will first compute the topological charge of the configurations ( [ ansatz3])([ansatz4 ] ) . for this , it is convenient to parametrize the two - sphere by the angular coordinates @xmath628 and @xmath629 defined in ( [ zn1 ] ) . in these coordinates @xmath630 and we have @xmath631 { { { \cal{f}}}}^{~}_{\vt\vp } \= & -{\,\mathrm{i}\,}\,\frac{\sin\vt}{2}\ , \sum^m_{i=0}\,(m-2i)~p^{~}_{n_i}\otimes\pi^{~}_i\end{aligned}\ ] ] giving @xmath632 \=&\,(-{\,\mathrm{i}\,})^{n+1}\,\frac{\sin\vt}{2\,\prod\limits^n_{a=1}\th^a}\ , \sum^m_{i=0}\,(m-2i)~p^{~}_{n_i}\otimes\pi^{~}_i \ , \end{aligned}\ ] ] where we have used the definitions ( [ pnitrace ] ) and ( [ piortho ] ) of the projectors @xmath391 and @xmath633 . the instanton charge is then given by the @xmath627-th chern number @xmath634 \= & \,\bigl(\frac{{\,\mathrm{i}\,}}{2\pi}\bigr)^{n+1}\ , \frac{(-{\,\mathrm{i}\,})^{n+1}}{2\,\prod\limits^n_{b=1}\th^b}\ , \bigl(\,\prod_{a=1}^n{2\pi\,\th^a}\bigr)\ , \sum^m_{i=0}\,(m-2i)\,{n_i}\ , \int_{s^2 } \sin\vt~{\mathrm{d}}\vt\wedge{\mathrm{d}}\vp \ . \label{chernnum}\end{aligned}\ ] ] after splitting the sum over @xmath135 into contributions from monopoles and antimonopoles analogously to ( [ efncfinite ] ) , this becomes q=^m2_i=0(m-2i ) ( n_i - n_m - i ) , [ topchargeym]where we recall that @xmath635 for @xmath247 . the formula ( [ topchargeym ] ) clarifies the d - brane interpretation of the configuration ( [ ansatz3])([ansatz4 ] ) . it describes a collection of @xmath636 d0-branes for @xmath637 and @xmath638 anti - d0-branes for @xmath639 as a bound state ( i.e. a vortex - like configuration on @xmath2 ) in a system of @xmath640 d(2@xmath348 ) branes and antibranes . however , from the point of view of the initial brane - antibrane system on @xmath0 , they are spherical @xmath641 d2-branes or d2-antibranes depending on the sign of the monopole charge @xmath446 . note that the vortices with @xmath642 , which always exist for even @xmath75 , have vanishing instanton charge since they couple with the trivial line bundle @xmath643 . thus they are not extended to instantons on @xmath0 , but are rather unstable and simply decay into the vacuum . the topological charge can be alternatively computed within the graded connection formalism of section [ dimred ] . recalling the equivariant abs construction ( [ twistedspingrad])([twistedspingradeven ] ) , we note that the @xmath260-graded vector space ( [ genrepsu2uk ] ) ( the fibre of the @xmath260-graded bundle ( [ gradedbundle ] ) ) also has a _ natural _ @xmath423-grading by the sign of the magnetic charge , i.e. by the involution @xmath644 defined by @xmath645 for @xmath646 , where throughout we use the convention @xmath647 . the corresponding supertrace is given by str^ _ kk x : = tr_kk^ ( x ) = _ i=0^msgn(m-2i ) tr_k_ik_i^ x_i [ supertracedef]for any linear operator @xmath648 with block - diagonal components @xmath649 . this extends to a supertrace @xmath650 which we may use to express the chern number in terms of the graded curvature ( [ gradedcurv ] ) as q= ( ) ^n+1 ( _ a=1^n2^a ) str^ _ tr^ _ ^2^n+1 ( ^n+1 ) _ asym , [ topchargegraded]where @xmath651 and the antisymmetrized product of gamma - matrices ( ^_1^_q)_asym : = 1q!_s_q sgn ( ) ^_(1)^_(q ) [ asymgamma]mimicks the algebraic structure of the exterior product of differential forms . the formula ( [ topchargeym ] ) follows from ( [ topchargegraded ] ) upon repeated application of the clifford algebra and the trace identities ( [ trgammaid1])([trgammaid4 ] ) , with the supertrace ( [ supertracedef ] ) giving the appropriate sign alternations . * k - theory calculation . * the origin of the topological charge lies in the _ graded chern character _ @xmath652 . standard transgression arguments can be used to show that the cohomology class defined by this closed differential form is independent of the choice of graded connection @xcite . in particular , we may either compute it by setting the off - diagonal higgs fields @xmath653 or by setting the diagonal gauge fields @xmath654 . it is instructive to recall how this works in the case @xmath268 corresponding to the basic brane - antibrane system represented by the chain ( [ a2quiver ] ) @xcite . in the former case we would obtain the difference @xmath655 of topological charges on the branes and antibranes . in the latter case we would compute the index of the tachyon field @xmath656 , or equivalently the euler characteristic of the two - term complex @xmath657 . the virtual euler class generated by the cohomology of this complex is the analytic k - homology class @xmath658\in{{\rm k}}^{\rm a}({{\mathbb{r}}}^{2n})$ ] of the brane configuration . the equivalence of the two computations is asserted by the index theorem . the situation for @xmath176 is more subtle . the action of the graded connection zero - form ( [ mgradedphidef ] ) on the bundle ( [ gradedbundle ] ) produces the holomorphic chain ( [ holchain ] ) . in general this is _ not _ a complex because , according to ( [ mphi0s ] ) , @xmath659 for @xmath176 , i.e. @xmath660 . the only physical instance in which such a chain generates a complex is when it corresponds to an alternating sequence of branes and antibranes @xcite . but if one has a tachyon field which is a holomorphic map from an antibrane to a brane , then the adjoint map is antiholomorphic . recalling ( [ f25 ] ) , we see that in our chain ( [ holchain ] ) all maps @xmath302 are _ holomorphic _ and thus do not represent tachyon fields between pairs of branes and antibranes . furthermore , the maps @xmath302 obtained as solutions of the vortex equations , which can be associated with the @xmath160 quiver and are obtained by @xmath1-invariant reduction , can never satisfy the constraints @xmath661 @xcite . the solution to this problem is to fold the given holomorphic chain into maps between branes and antibranes . let us first carry out the calculation in the case that the monopole chern number @xmath75 is an odd integer . by using the @xmath423-grading @xmath662 introduced above , we explicitly decompose ( [ genrepsu2uk ] ) as a @xmath423-graded module into the @xmath663 eigenspaces of the involution @xmath664 giving = _ + _ - _ + = _ i=0^m_- _ k_i _ -= _ i = m_+^m_k_i . [ calvz2module]using ( [ mgradedphidef ] ) and ( [ mphi0s ] ) we now introduce the operator t^ _ ( m):= ( ^ _ ( m))^m2 + 1 . [ mtdef]with respect to the @xmath423-grading ( [ calvz2module ] ) , it is an odd map ^ _ ( m):_-h _ + h ( ^ _ ( m))^20 . [ mtmap]thus the triple @xmath665 $ ] defines a two - term complex and represents a brane - antibrane system with tachyon field given in terms of the graded connection by ( [ mtdef ] ) . the corresponding index class @xmath666\in{{\rm k}}^{\rm a}({{\mathbb{r}}}^{2n})$ ] is thus the analytic k - homology class of our configuration of d - branes . in particular , on isotopical components one has ^ _ ( m)^ _ i+1+m2 = ^ _ i+1^ _ i+1+m2=(^ _ i+1^ _ i+1+m2 ) t^ _ n_it^_n_i+1+m2 [ mtisotopexpl]while @xmath667 , where @xmath668 . the tachyon field is thus a holomorphic map between branes of equal and opposite magnetic charge , ^ _ ( m)^ _ i+m2 + 1 : _ k_i+m2 + 1h _ k_i h , [ mtisotopmap]and from ( [ dimkertni ] ) it follows that it has a finite dimensional kernel and cokernel with ( ^ _ ( m)^ _ i+m2 + 1 ) n_i+m2 + 1 ( ^ _ ( m)^ _ i+m2 + 1 ) ^n_i . [ dimkermt ] to incorporate the twistings by the magnetic monopole bundles , we use the abs construction ( [ twistedspingrad])([cliffordmulteven ] ) to define the tachyon field ^ _ ( m ) : = ^ _ ( m):_^+h _ ^-h . [ mcaltdef]it behaves like a noncommutative version of clifford multiplication @xmath669 in ( [ cliffmultv],[cliffmultisot ] ) . since @xmath670 , from ( [ dimkermt ] ) it follows that the index of the tachyon field ( [ mcaltdef ] ) is given by ^ _ ( m)&= & ( ^ _ ( m))- ( ^ _ ( m))^ + & = & _ i = m_+^m|m-2i|n_i-_i=0^m_- |m-2i|n_i =- q . [ indexmcalt]thus the k - theory charge of the noncommutative soliton configuration ( [ ansatz3])([ansatz4 ] ) coincides with the yang - mills instanton charge ( [ chernnum],[topchargeym ] ) on @xmath0 . when the monopole charge @xmath75 is even , we introduce the tachyon field @xmath671 by the same formula ( [ mtdef ] ) . the only difference now is that the subspace @xmath672 is annihilated by both operators @xmath673 and @xmath674 so that _ k_m2h ( ^ _ ( m ) ) ( ^ _ ( m))^ . [ mtvm20]according to ( [ twistedspingradeven ] ) , this subspace should be coupled to the eigenspace ( [ spinorharm ] ) of spinor harmonics on @xmath69 when defining the extended tachyon field ( [ mcaltdef ] ) . analogously to ( [ cliffordmulteven ] ) , one then has ( ^ _ ( m)^ _ m2)= ( ^ _ ( m)^ _ m2)^= _ k_m2_ph . [ kermcaltm2]with a suitable regularization of the infinite dimensions of the kernel and cokernel of the operator @xmath675 , these subspaces will make no contribution to the index ( [ indexmcalt ] ) . this statement will be justified below by the fact that @xmath676 and that the index class of the noncommutative tachyon field coincides with that of the twisted @xmath1-invariant dirac operator on @xmath484 . we can give a more detailed picture of how the topological charge of the system of d - branes arises by relating the index to a homological computation in the corresponding quiver gauge theory , which shows precisely how the original brane configuration folds itself into branes and antibranes . consider the @xmath241-module ( [ vnam ] ) defined by a generic ( non - bps ) solution of the yang - mills equations on @xmath403 , and let = _ i=0^m_i k^ _ = _ i=0^mw_i e_i [ anywrep]be any quiver representation . applying the functor @xmath677 to the projective resolution ( [ ringelres ] ) gives a complex whose cohomology in the @xmath522-th position defines the extension groups @xmath678 , with @xmath679 and @xmath680 . we may then define the relative euler character between these two representations through the corresponding euler form ( , ) : = _ p0(-1)^p ^p ( , ) . [ eulerformgen]since the @xmath160 quiver has no relations , one has @xmath681 for all @xmath682 in the present case @xcite . by using ( [ hompathnatural ] ) , the resolution ( [ ringelres ] ) induces an exact sequence of extension groups given by 0 ( , ) & & _ i=0^s ( t_n_i^,_i ) + & & _ i=0^s-1 ( t_n_i+1^,_i ) ( , ) 0 [ extexactseq]from which we may compute the euler form ( [ eulerformgen ] ) explicitly to get ( , ) & = & ( , ) - ( , ) + & = & _ i=0^s(t_n_i^ , _ i)-_i=0^s-1 ( t_n_i+1^,_i ) + & = & _ i=0^mn_iw_i-_i=0^m-1n_i+1w_i . [ eulerwgen]thus the relative euler character depends only on the dimension vectors of the corresponding representations and coincides with the ringel form @xmath683 on the representation ring @xmath684 of the @xmath160 quiver @xcite . the map @xmath685\mapsto\vec k^{~}_{\underline{\cal w}}$ ] gives a linear map @xmath686 which is an isomorphism of abelian groups since @xmath684 is generated by the schur modules @xmath448 , @xmath247 . by using ( [ ansatz3p ] ) and ( [ dimkertni ] ) we can write this bilinear pairing in the suggestive form ( , ) = -_i=0^mw_i index(_i+1 ) . [ eulerindexphi ] the appropriate representation @xmath263 to couple with in the present case is dictated by the correct incorporation of magnetic charges . as before , the fact that the higgs fields @xmath170 in ( [ eulerindexphi ] ) themselves are not tachyonic , i.e. do not generate a complex , means that we have to fold the @xmath1 representations @xmath687 appearing in the abs construction ( [ twistedspinpm ] ) appropriately . the correct folding is expressed by the collection of distinguished triangles ( [ bpstriangles ] ) which shows that we should couple an increasing sequence @xmath688 of representations as we move along the chain of constituent d - branes of the quiver , so that the @xmath1-module @xmath689 gives an extension of the monopole field carried by the elementary brane state at vertex @xmath135 by the @xmath1-module @xmath690 . thus we take @xmath691 and embed its class into the representation ring @xmath692 using the @xmath423-grading above as the element = _ j=0^isgn(m-2j ) sgn(m-2i ) + [ widef]of virtual dimension w_i=_j=0^i(m-2j)(i+1)(m - i ) [ wiwidef]for each @xmath247 . in this case the euler - ringel form ( [ eulerwgen ] ) becomes ( , ) = _ i=0^m(i+1)(m - i)(n_i - n_i+1)=_i=0^m(m-2i ) n_iq [ eulertopcharge]and it also coincides with the instanton charge of the gauge field configurations on @xmath0 . the equivalence of the relative euler character with the index of the tachyon field above is a consequence of the grothendieck - riemann - roch theorem . * worldvolume construction . * we can now present a very explicit geometric description of the equivalence between the brane configurations on @xmath484 and on @xmath6 . the crux of the formulation is the well - known map in k - theory between analytic ( noncommutative ) and topological ( commutative ) descriptions @xcite . if @xmath693 is the usual dirac operator on @xmath6 , then its index coincides with that of the noncommutative abs configuration ( [ tt ] ) giving = index . [ indexsigmadirac]this coincides with the k - theory charge of the bott class @xmath599 \in{{\rm k}}({{\mathbb{r}}}^{2n})$ ] given by the ordinary abs construction @xcite , where @xmath694 is clifford multiplication by @xmath495 . in particular , the dirac operator itself can be used to represent the analytic k - homology class @xmath695=[\dirac]$ ] described by the noncommutative abs field . let us represent a system of @xmath41 type iia d - branes wrapped on @xmath484 with virtual chan - paton bundle @xmath696 by the k - cycle @xmath697 $ ] in the topological k - homology @xmath698 . its equivalence class is invariant under the usual relations of bordism , direct sum and vector bundle modification @xcite . there is an isomorphism @xmath699 of abelian groups which sends this k - cycle to the analytic k - homology class @xmath700 $ ] defined by the corresponding twisted dirac operator on @xmath484 . similarly , if @xmath701 and @xmath702 is the slice induced by the inclusion @xmath703 of groups , then the topological k - cycle @xmath704\in{{\rm k}}^{\rm t}({{\mathbb{r}}}^{2n}\times{{\mathbb{c}}}p^1)$ ] corresponds to the analytic k - homology class @xmath705\in{{\rm k}}^{\rm a}({{\mathbb{r}}}^{2n}\times{{\mathbb{c}}}p^1)$ ] , where @xmath706 is the twisted dirac operator on @xmath6 . now consider the @xmath1-equivariant reduction of these cycles . from the construction of the previous section with @xmath707 and the equivariant excision theorem of section [ ansatzdescr ] we have the equality ^su(2)=_*^u(1)[diracsu2eq]in @xmath708 which leads to = su(2)_u(1)[kcyclesu2eq]in the left - hand side of ( [ kcyclesu2eq ] ) corresponds to the class of d@xmath5 brane - antibrane pairs wrapping @xmath6 , while the right - hand side corresponds to d@xmath710 brane - antibrane pairs wrapping @xmath484 . this is just the equivalence between instantons on @xmath484 and vortices on @xmath6 . we note that in the case @xmath268 , the monopole field is automatically spherically symmetric on @xmath69 and one can formulate the equivalence ( [ kcyclesu2eq ] ) using only the requirement of vector bundle modification in _ ordinary _ topological k - homology @xcite , which is equivalent to bott periodicity ( [ bottper ] ) . in contrast , for @xmath176 one must appeal to an @xmath1-equivariant framework and the identification ( [ kcyclesu2eq ] ) of k - cycles is far more intricate . in this case it is a result of the equivariant excision theorem , and _ not _ of bott periodicity in equivariant k - theory . it is this intricacy that leads to a more complicated brane - antibrane system when @xmath176 . using the equivariant abs construction of the previous section , the k - homology class of the multi - instanton solution ( [ ansatz3])([ansatz4 ] ) is given by the left - hand side of ( [ kcyclesu2eq ] ) with = , [ ximultiinst]where _ e^+=_i = m_+^m e_k_i_|m-2i| _ e^-= _ i=0^m_- e_k_i_m-2i [ twistedebundles]while ^ _ n_0,n_1, ,n_m=_i=0^m_- ( ^ _ e^ _ i)^n_i_j = m_+^m ( _ e^_j^ ) ^n_j [ muns]with @xmath711 acting fibrewise as clifford multiplication ( [ cliffmultv],[cliffmultisot ] ) . the class ( [ ximultiinst ] ) is the k - theory class of the noncommutative soliton field ( [ totaltn ] ) . the relation ( [ kcyclesu2eq ] ) equates the resulting k - homology class with that defined by = , [ ximultiinst]where the projection @xmath712 is a left inverse to the inclusion @xmath713 , i.e. @xmath714 . through the standard process of tachyon condensation on the system of d@xmath710 branes and antibranes wrapping @xmath6 , the right - hand side of ( [ kcyclesu2eq ] ) then describes @xmath715 d2-branes and @xmath716 d2-antibranes . on the left - hand side of ( [ kcyclesu2eq ] ) , these are instead d0-branes corresponding to vortices left over from condensation in the transverse space @xmath6 . one can also compute the topological charge in this worldvolume picture and explicitly demonstrate that the k - theory charges on both sides of ( [ kcyclesu2eq ] ) are the same . the natural charge of branes defined by elements of equivariant k - theory is given by the equivariant index @xmath717 , which may be computed by using the @xmath1-index theorem @xcite ^ _ su(2 ) _ = -_^2np^1 ^ _ su(2)()(^2np^1 ) , [ equiindexthm]where @xmath718 is the equivariant chern character taking values in @xmath1-equivariant rational cohomology . since this index depends only on the equivariant k - homology class of the dirac operator on @xmath484 , we may explicitly use ( [ diracsu2eq ] ) and perform the dimensional reduction to write the index ( [ equiindexthm ] ) as ^ _ su(2 ) _ = -_^2n^ _ su(2 ) ( ) . [ equiindexred ] since the chern character in ( [ equiindexred ] ) is a ring homomorphism between @xmath719 and @xmath720 , upon substitution of ( [ ximultiinst],[twistedebundles ] ) we can use its additivity and multiplicativity to compute ^ _ su(2)()=^ _ su(2 ) ( _ e^+_e^-)=_i = m_+^m ( e_k_i)^ _ _ |m-2i|-_i=0^m_- ( e_k_i)^ _ _ |m-2i| , [ chernmultadd]where @xmath721 are the characters of the @xmath1 representations @xmath722 . this enables us to write the equivariant index on @xmath484 in terms of ordinary indices on @xmath6 to get ^ _ su(2 ) _ = _ i=0^m_- index(_e_k_i ) ^ _ _ |m-2i| -_i = m_+^mindex(_e_k_i)^ _ _ |m-2i| . [ indexord]we can turn ( [ indexord ] ) into a linear map @xmath723 by composing it with the projection @xmath724 onto the trivial representation . acting on the character ring this gives _ 0(^ _ _ |m-2i|)= ^ _ _ |m-2i|(id ) = _ |m-2i||m-2i| [ pi0compose]and one finally arrives at _ 0(index^ _ su(2 ) _ ) = _ i=0^m_- |m-2i| index(_e_k_i ) -_i = m_+^m|m-2i| index(_e_k_i ) . [ pi0indexfinal ] alternatively , one may arrive at the same formula by directly computing the _ ordinary _ index of the dirac operator ( [ diracgradeddef ] ) with @xmath707 using ( [ diracs2decomp ] ) and ( [ diracpkernel])([solschargepos ] ) . since _ m-2i=^+_m-2i- ^-_m-2i=-(m-2i ) , [ indexcp1]the index of ( [ diracgradeddef ] ) acting on sections of the bundle ( [ spinortotgen ] ) coincides with ( [ pi0indexfinal ] ) . for a gauge field configuration appropriate to the k - theory class defined by the tachyon field ( [ muns ] ) , these topological charges coincide with ( [ topchargeym ] ) . the extremal cases for which the higgs fields have the configurations @xmath725 and @xmath726 , fall outside of the general scope of the previous analysis and are worth special consideration . they correspond to vacuum sectors of the noncommutative gauge theory and are associated with indecomposable representations of the quiver @xmath241 that have no arrows . nevertheless , these vacuum sectors admit non - trivial bps solutions which signal the presence of stable d - branes attached to the closed string vacuum after condensation on the brane - antibrane system . we shall now study them in some detail . * monopole vacuum . * let us first look at the case @xmath727 . the nonabelian coupled vortex equations ( [ ddd1])([ddd2 ] ) then imply @xmath728 which is only possible in the equal rank case @xmath729 corresponding to the gauge symmetry breaking pattern @xmath730 with @xmath731 . thus we take @xmath732 and @xmath733 with @xmath734 , where @xmath735 are given in ( [ 7.8 ] ) . in quiver gauge theory , the bps conditions in this sector thus correspond to the representation of @xmath241 which is @xmath587 copies of the indecomposable quiver representation @xmath736 . they also require @xmath737 which are simply the duy equations on @xmath2 . note that ( [ fyyb ] ) implies @xmath738 in this case , giving the trivial dimensional reduction to @xmath2 . after switching to matrix form via ( [ x ] ) , we obtain @xmath739 + \de^{a{{\bar{b}}}}\,\th_{a{{\bar{b}}}}\=0 \qquad\textrm{and}\qquad \big[x_{{{\bar{a}}}}\,,\,x_{{{\bar{b}}}}\big]\=0\= \big[x_a\,,\,x_b\big ] \ .\ ] ] the obvious solution to ( [ mf ] ) is the trivial one with @xmath740 , giving @xmath741 . this sector can be understood physically as the endpoint of tachyon condensation , wherein the higgs fields @xmath140 have rolled to their minima at @xmath742 and the fluxes have been radiated away to infinity . here the d0-branes have been completely dissolved into the d(2@xmath348)-branes . however , non - trivial solutions of the equations ( [ mf ] ) also exist . for this , let us restrict ourselves to the abelian case @xmath542 and simplify matters by taking @xmath743 for all @xmath744 . we fix an integer @xmath745 and consider the ansatz @xcite @xmath746 where @xmath346 is a real function of the `` total number operator '' @xmath747 with the property that @xmath748 for @xmath749 . the shift operator @xmath750 in ( [ ans ] ) is defined to obey @xmath751 with @xmath752 where @xmath753 with @xmath754 . note that @xmath755 and @xmath756 projects all states with @xmath757 out of the fock space @xmath758 . one easily sees that ( [ ans ] ) fulfills the homogeneous equations in ( [ mf ] ) . remembering that @xmath759 , we also obtain @xmath760\ & = \ \th_{a{{\bar{c}}}}\,\th_{{{\bar{b}}}d}\,\sigma_{l}^\+\,\left\ { f({{\cal n}}\,)\,{{{\bar{z}}}}^{{{\bar{c}}}}\,(1{-}p^{~}_{l})\,z^d\,f ( { { \cal n}}\,)-z^d\,f({{\cal n}}\,)\,(1{-}p^{~}_{l})\,f({{\cal n}}\ , ) \,{{\bar{z}}}^{{{\bar{c}}}}\right\}\,\sigma^{~}_{l } \\[4pt ] \label{mf4 } & = \ -\frac{1}{4\,\th^2}\,\de_{a{{{\bar{c}}}}}\,\de_{d{{{\bar{b}}}}}\,\sigma_{l}^\+\,\left\ { f^2({{\cal n}}\,)\,{{{\bar{z}}}}^{{{\bar{c } } } } z^d - f^2({{\cal n}}{-}1)\,z^d { { { \bar{z}}}}^{{{\bar{c}}}}\right\}\,\sigma^{~}_{l}\end{aligned}\ ] ] with the help of the identities @xmath761 where @xmath762 . we have also used @xmath763 substituting ( [ mf4 ] ) into ( [ mf ] ) , we employ @xmath764 to find the conditions @xmath765 + \de^{a{{\bar{b}}}}\,\th_{a{{\bar{b } } } } \\[4pt ] \nonumber & \=-\frac{1}{2\,\th}\ , \sigma_{l}^\+\,\biggl\ { f^2({{\cal n}}\,)\,({{\cal n}}{+}n)- f^2({{\cal n}}{-}1)\,{{\cal n}}\biggr\}\,\sigma^{~}_{l}+ \frac{n}{2\,\th } \\[4pt ] \label{mf5 } & \=\frac{1}{2\,\th}\,\sigma_{l}^\+\,\biggl\ { { { \cal n}}\,f^2({{\cal n}}{-}1)- ( { { \cal n}}{+}n)\,f^2({{\cal n}}\,)+ n\biggr\}\,\sigma^{~}_{l}\end{aligned}\ ] ] on the operator @xmath346 . with the initial conditions @xmath766 and the finite - energy condition @xmath767 as @xmath768 , these recursion relations are solved by @xmath769 where @xmath770 is the number of states in @xmath349 with @xmath771 , i.e. the number of states removed by the operator @xmath772 . we arrive finally at the non - trivial gauge field configuration given by @xmath773 the field strength @xmath192 on @xmath774 obtained from ( [ xsol ] ) has finite @xmath348-th chern number @xmath775 @xcite . the topological charge @xmath775 given by ( [ qpm ] ) is calculated here via an integral over @xmath2 . however , the @xmath776-th chern number for this configuration considered as a gauge field on @xmath378 with @xmath777 vanishes . moreover , this configuration has finite energy ( [ ef ] ) proportional to the topological charge @xcite , e^ _ bps=(2)^n+1r^2n(n-1)q , [ ebpsm0]as usual for a bps instanton solution . * higgs vacuum . * the choice @xmath778 for all @xmath734 is somewhat more interesting since from ( [ fyyb ] ) and ( [ mupdef ] ) we then have @xmath779 with @xmath780 this configuration gives the local maximum of the higgs potential corresponding to the open string vacuum containing d - branes . in this case the vortex equations ( [ ddd1])([ddd2 ] ) reduce to @xmath781 after switching to matrix form via ( [ x ] ) we obtain @xmath782 + \de^{a{{\bar{b}}}}\,\bigl(1- \frac{(m-2i)\,\th}{2n\,r^2}\,\bigr)\,\th_{a{{\bar{b}}}}\=0 \qquad\textrm{and}\qquad \big[x^i_{{{\bar{a}}}}\,,\,x^i_{{{\bar{b}}}}\big]\=0 \=\big[x^i_a\,,\,x^i_b\big]\ , \ ] ] where we have used the formula @xmath783 . recall that there is no summation over the index @xmath247 in the equations ( [ mf1 ] ) . by comparing ( [ mf1 ] ) and ( [ mf ] ) , we conclude that ( [ mf1 ] ) can be solved for each @xmath135 by the same ansatz as for ( [ mf ] ) . for this , let us restrict ourselves again to the abelian case for all nodes @xmath247 ( so that @xmath784 ) , and fix @xmath785 positive integers @xmath786 . we take @xmath787 analogously to ( [ ans])([projl ] ) . producing then the same calculations as before , we obtain the gauge field configuration @xmath788 where @xmath789 and @xmath790 we have chosen the radius @xmath14 of the sphere so that @xmath791 . the solutions ( [ xso ] ) coincide with those given by ( [ xsol ] ) if one assigns different noncommutativity parameters @xmath792 to the worldvolumes of d(@xmath29)-branes carrying different magnetic fluxes proportional to @xmath446 . then the field strength @xmath793 on @xmath794 obtained from ( [ xso ] ) will have finite topological charge @xmath795 given by ( [ qi ] ) and corresponding finite bps energy analogous to ( [ ebpsm0 ] ) , and the configuration thus described extends to instantons on @xmath403 . the interesting idea of introducing distinct noncommutativity parameters on multiple coincident d - 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we construct explicit bps and non - bps solutions of the yang - mills equations on the noncommutative space @xmath0 which have manifest spherical symmetry . using @xmath1-equivariant dimensional reduction techniques , we show that the solutions imply an equivalence between instantons on @xmath0 and nonabelian vortices on @xmath2 , which can be interpreted as a blowing - up of a chain of d0-branes on @xmath2 into a chain of spherical d2-branes on @xmath0 . the low - energy dynamics of these configurations is described by a quiver gauge theory which can be formulated in terms of new geometrical objects generalizing superconnections . this formalism enables the explicit assignment of d0-brane charges in equivariant k - theory to the instanton solutions . hep - th/0504025 + itp uh05/05 + hwm0503 + empg0504 + 1.5 cm * quiver gauge theory of nonabelian vortices + and noncommutative instantons in higher dimensions * alexander d. popov + _ institut fr theoretische physik , universitt hannover + appelstrae 2 , 30167 hannover , germany _ + and + _ bogoliubov laboratory of theoretical physics , jinr + 141980 dubna , moscow region , russia _ + email : [email protected] + richard j. szabo + _ department of mathematics , heriot - watt university + colin maclaurin building , riccarton , edinburgh eh14 4as , u.k . _ + email : [email protected]

hep-th0504025

it may be worth recalling some of the main properties of the milky way open cluster system . the census of open clusters is still highly incomplete beyond distances of a few kpc from the sun , although the situation is improving with new surveys such as 2mass ( see e.g. the contributions by carpenter and hanson in this volume ) . the luminosity function of milky way open clusters was analysed by @xcite , who found it to be well modelled by a power - law @xmath1 over the range @xmath2 . however , they also noted that extrapolation of this luminosity function would predict about 100 clusters as bright as @xmath3 in the galaxy , clearly at odds with observations , and thus suggested some flattening of the lf slope at higher luminosities . the brightest known young clusters ( e.g. ngc 3603 , @xmath4 and @xmath5 per ) have absolute @xmath6 magnitudes of @xmath7 , corresponding to total masses of several thousand . recently , there have been claims that the cyg ob2 association might be an even more massive cluster @xcite , but this object is probably too diffuse to be a bound star cluster ( though it does have a compact core ) . there are , however , a number of old ( @xmath8 gyr ) open clusters in the milky way with masses of @xmath9 @xcite . these objects are likely to have lost a significant fraction of their total mass over their lifetimes , and may thus originally have been even more massive . they serve to illustrate that , even in the milky way , the distinction between globular and open clusters is not always clear - cut . it has been recognized for about a century that the magellanic clouds , and the lmc in particular , host a number of `` blue globular clusters '' @xcite . among the most massive of these is ngc 1866 , with a mass of around @xmath10 and an age of @xmath11 myr @xcite . an older example is ngc 1978 with similar mass but an age of 23 gyr , clearly demonstrating that at least some such clusters can survive for several gyrs . the interaction between the lmc and the milky way has probably affected the star formation history of the lmc , which is known to be bursty with major peaks in the star formation rate correlating with perigalactic passages @xcite . one might argue , then , that the formation of ymcs in the lmc could be induced by interaction with the milky way . however , the lmc is not the only example even in the local group of a galaxy that hosts ymcs . another well - known example is m33 , which does not display evidence for a bursty cluster ( and , presumably star- ) formation history @xcite . @xcite have identified many more star clusters in this galaxy , though not all are particularly massive . with the launch of hst it became possible to investigate more crowded and/or distant systems in detail and attention started to shift towards more extreme starbursts , including a large number of merger galaxies ( e.g. whitmore , this volume ) . it is now clear that luminous , young star clusters often form in very large numbers in such galaxies , and this has led to suggestions that formation of `` massive '' star clusters might require special conditions such as large - scale cloud - cloud collisitions @xcite . however , the question remains to be answered why some non - interacting galaxies also contain ymcs , whereas apparently the milky way does not . ymcs are now being found in an increasing number of non - interacting galaxies , posing a severe challenge for formation scenarios which require special conditions . during the 1980s , some studies had already identified ymcs in a few galaxies beyond the local group @xcite . we undertook a systematic , ground - based study of 21 nearby spirals , aiming at identifying cluster systems and further investigating which factors might lead to the formation of ymcs @xcite . generally lacking sufficient resolution to identify clusters as spatially resolved objects , our candidate lists were compiled based on @xmath12 photometry , selecting compact objects with @xmath13 and @xmath14 brighter than @xmath15 ( for @xmath16 ) or @xmath17 ( for @xmath18 ) . we also required that the objects had no h@xmath19 emission . the @xmath20 limit excluded most foreground stars , while the @xmath14 limit was designed to minimise the risk that individual , luminous stars in the galaxies would contaminate the sample . as the mass - to - light ratios of star clusters are highly age dependent , the magnitude cut does not translate to a well - defined mass limit , but most clusters selected in this way have masses @xmath21 . our survey would probably pick up a few clusters in the milky way . in the lmc , 8 clusters in the @xcite catalogue pass our selection criteria . we found a surprising variety in the numbers of ymcs in the galaxies . some galaxies , such as ngc 45 , ngc 300 and ngc 3184 contained hardly any clusters passing our selection criteria , but in others we found more than a hundred . the two most cluster - rich galaxies were ngc 5236 ( m83 ) and ngc 6946 , both of which are also known for their very high supernova rates and surface brightnesses , indicative of very active star formation . following @xcite , we defined the _ specific luminosity _ of young star clusters as @xmath22 where @xmath23 and @xmath24 are the total @xmath25-band luminosities of clusters and their host galaxy . the @xmath26 turned out to correlate strongly with the host galaxy area - normalised star formation rate @xcite , as if bound star clusters form more efficiently in higher - sfr environments . here , it is important to note that our sample excludes the very youngest clusters , which are often located in crowded regions in spiral arms where they are difficult to identify with ground - based imaging . therefore , it is probably better to think of @xmath26 as a _ survival- _ rather than a formation efficiency . in fact , most stars probably form in clusters , both in normal galaxies such as the milky way ( * ? ? ? * carpenter this volume ) and in the mergers like the antennae ( fall , this volume ) . the fraction of those clusters which remain bound may vary , however . while @xmath26 may be a useful measure of the overall richness of a cluster system , it does not provide any information about possible variations in the cluster mass distributions , and in particular , whether some galaxies form a higher proportion of _ massive _ clusters than others . this question still remains largely unanswered , because mass distributions are difficult to derive observationally . in order to convert observed cluster luminosities to masses , the m / l ratios need to be known . these , in turn , depend strongly on the cluster ages , which can not be reliably estimated without photometry in ultraviolet passbands . this is mostly a consequence of the fact that the colours of young clusters are dominated by hot stars , where the rayleigh - jeans approximation applies for passbands centered at optical wavelengths . a `` poor man s '' solution is to look at luminosity- rather than mass functions , bearing in mind the first are not necessarily identical to the latter . in @xcite , archive hst data were used to analyse the luminosity functions ( lfs ) of star clusters in 6 nearby spirals which had previously been studied from the ground . the lfs were generally found to be consistent with power - laws with slopes between @xmath27 and @xmath28 , somewhat steeper than the value found for milky way open clusters by @xcite , though not extending nearly as deep . for young clusters in m66 , @xcite found a lf slope of @xmath29 . if the luminosity function has a universal power - law form which is populated at random , one would predict a strong correlation between the total number of clusters in a galaxy and the luminosity of the brightest cluster . such a relation is indeed observed and has a slope , normalisation and scatter similar to those expected from sampling statistics arguments @xcite . however , it should be noted that a few galaxies do stand out , having clusters that are much too bright for the total number of clusters in those galaxies . notable examples are ngc 1569 and ngc 1705 , both of which are dominated by 12 highly luminous clusters . the possibility remains open that these clusters formed by a special mechanism , but the issue needs to be investigated in more detail . hst archive data for a larger sample ( 17 ) of nearby spirals were analysed in @xcite . the main aim here was to study the structural parameters and investigate possible correlations with age , mass or other cluster properties . the clusters were modelled using ` eff ' profiles of the type shown by @xcite to fit lmc clusters : @xmath30^{-\gamma/2 } \label{eq : eff}\ ] ] structural parameters were obtained from fits to wfpc2 images and combined with @xmath31 photometry from ground - based imaging . [ fig : reff_mass ] shows the cluster half - light radii ( ) versus masses estimated from bruzual & charlot ssp models . only clusters with @xmath32 are included in this plot , as is undefined for @xmath33 . the majority of clusters have half - light radii of 34 pc albeit with a fairly large scatter . interestingly , this is similar to the effective radii of galactic and extragalactic old globular clusters . the dashed line shows the relation corresponding to a constant cluster density ( @xmath34 ) , while the solid line is a least - squares fit to the data . no strong correlation between and mass is observed . a formal fit to the data yields @xmath35 , similar to the @xmath36 relation found for young clusters in the merger remnant ngc 3256 by @xcite . the observation that no strong correlation exists between cluster size and mass implies that high - mass clusters generally have much higher stellar _ densities _ than low - mass clusters , a fact that may have important implications for theories for cluster formation . if the cluster ages are known , luminosities can be converted to masses using simple stellar population models and assuming a stellar initial mass function ( imf ) . an alternative approach is to obtain dynamical mass estimates by measuring the internal velocity dispersions and cluster sizes and applying the virial theorem . the dynamically derived m / l ratios can then be compared with ssp models for different imfs , providing a potentially useful method to constrain the imf . the line - of - sight velocity dispersion @xmath37 , mass @xmath38 and projected half - light radius are related as @xmath39 where @xmath40 . in practice , however , there are many caveats to this method , both theoretical ones ( assumption of velocity isotropy , virial equilibrium , effects of mass segregation and binaries ) , and practical ones : for a mass of @xmath10 and @xmath41 pc , the line - of - sight velocity dispersion is less than 4 km / s . the red supergiants which provide most of the lines useful for velocity dispersion measurements have macroturbulent velocities on the order of 10 km / s , with a scatter of perhaps 12 km / s @xcite . since the velocity dispersions usually have to be derived from integrated light , it is clear that this method is limited to relatively massive objects . even if spectra of sufficient resolution to resolve the line broadening ( @xmath42 ) and s / n could be obtained , an exact match between the template stars used to derive the velocity dispersions and those present in the cluster becomes increasingly critical as the cluster mass decreases . additionally , the clusters have to be close enough that reasonably reliable size estimates can be obtained , although these are less critical since @xmath38 scales only linearly with . several groups have obtained dynamical mass estimates for extragalactic young star clusters , sometimes with hints of non - standard imfs ( e.g. @xcite ; see also mengel , this volume ) . in our sample of spiral galaxies , we found a few clusters for which dynamical mass estimates appeared feasible . critical selection criteria were that the clusters be reasonably well isolated , so that the spectroscopic observations would not be contaminated by neighbouring objects , and that they have hst imaging for reliable size measurements . objects which satisfy these criteria include one cluster in ngc 6946 which was observed with the hires spectrograph on the keck i telescope @xcite , and two clusters in ngc 5236 ( fig . [ fig : cl5236 ] ) , observed with uves on the eso vlt @xcite . structural parameters are available for all clusters from hst imaging , and ages and reddenings were estimated by comparing ground - based @xmath31 colours with bruzual & charlot ssp models . basic properties for the three clusters are summarised in table [ tab : cprop ] . they all have masses greater than @xmath43 , well in excess of those of the most massive young lmc clusters . even if our reference frame had been the lmc rather than the milky way , we would still have characterised these clusters as `` massive '' . in fig . [ fig : pmtol ] , the observed @xmath6-band m / l ratios are compared with ssp model predictions for various imfs . these ssps were computed by populating stellar isochrones from the padua group @xcite according to the imfs indicated in the figure legend , i.e. a @xcite imf and @xcite - type imfs with lower mass cut - offs at 0.01 , 0.1 and 1.0 . the kroupa imf nominally extends down to 0.01 , but the lower cut - off is of little importance as the slope below 0.08 is very shallow . within the error bars , the three clusters all appear consistent with a `` standard '' kroupa - like imf . in particular , there is no evidence for an excess of high - mass stars in any of these clusters . we checked the curves in fig . [ fig : pmtol ] against the bruzual & charlot models , which are available for salpeter imf truncated at 0.1 @xmath44 and found very similar results , with differences at the level of 0.1 mag at most . it should be emphasized that this method does not constrain the exact shape of the imf . power - law imfs with a shallow slope , for example , would mimick the effect of a salpeter imf with a high - mass cut - off . it is becoming increasingly clear that `` massive '' star clusters can form in a wide variety of galaxies , and not just in mergers or otherwise disturbed galaxies . with the possible exception of some dwarf galaxies , the luminosity distributions of young star clusters generally appear to be power - laws . if cluster luminosities are sampled at random from a power - law distribution , the most luminous clusters will naturally be rare , but so far there is no evidence for a statistically significant upper cut - off . in other words , very luminous ( and massive ) clusters appear to form whenever clusters form in large numbers . this is illustrated by the fact that young star clusters with masses up to @xmath0 have been identified in the disks of several apparently normal , isolated spiral galaxies with rich cluster systems . these galaxies , such as ngc 5236 , ngc 6946 are characterised by high star formation rates , but these do not generally appear to be triggered by interactions with other galaxies . dynamical mass estimates are now available for a small number of these clusters , and the mass - to - light ratios are compatible with standard kroupa - type imfs . there is every reason to be optimistic that important clues to the formation of classical globular clusters may be obtained by studying their younger counterparts in the local universe . bica , e. , clari , j. j. , dottori , h. , santos jr . , j. f. c. , piatti , a. e. , 1996 , apj suppl . , 102 , 57 billett , o. a. , hunter , d. a. , & elmegreen , b. g. , 2002 , aj , 123 , 1454 chandar , r. , bianchi , l. , & ford , h. c. , 1999 , apjs , 122 , 431 chandar , r. , bianchi , l. , & ford , h. c. , 1999 , a&a , 366 , 498 christian , c. a. & schommer , r. a. , 1982 , apjs , 49 , 405 christian , c. a. & schommer , r. a. , 1988 , aj , 95 , 704 dolphin , a. e. , & kennicutt , r. c. , 2000 , aj , 123 , 207 elson , r. a. w. , fall , s. m. , & freeman , k. c. , 1987 , apj , 323 , 54 fischer , p. , welch , d. l. , ct , p. , et al . , 1992 , aj , 103 , 857 friel , e. d. , 1995 , ara&a 33 , 381 girardi , l. , bressan , a. , bertelli , g. , & chiosi , c. , 2000 , a&as 141 , 371 gray , d. f. , & toner , c. g. , 1987 , apj , 322 , 360 harris , w. e. , 1991 , ann . 29 , 543 jog , c. j. , & solomon , p. m. , 1992 , apj , 387 , 152 kennicutt , r. c. & chu , y - h . , 1988 , aj , 95 , 720 kndlseder , j. , 2000 , a&a , 360 , 539 kroupa , p. 2002 , science , 295 , 82 lada , c. j. , & lada , e. a. , 2003 , ara&a 41 , 57 larsen , s. s. , brodie , j. p. , elmegreen , b. g. , efremov , y. n. , hodge , p. w. , & richtler , t. 2001 , apj , 556 , 801 larsen , s. s. , 2002 , aj , 124 , 1393 larsen , s. s. , 2004 , a&a , in press larsen , s. s. & richtler , t. , 1999 , a&a 345 , 59 larsen , s. s. & richtler , t. , 2000 , a&a 354 , 836 larsen , s. s. & richtler , t. , 2004 , a&a , in preparation salpeter , e. e. 1955 , , 121 , 161 shapley , h. , 1930 , `` star clusters '' , mcgraw - hill ( new york ) smecker - hane , t. a. , cole , a. a. , gallagher iii , j. s. , & stetson , p. b. , 2002 , apj , 566 , 239 smith , l. j. , gallagher , j. s. 2001 , mnras , 326 , 1027 van den bergh , s. , 1999 , pasp , 111 , 1248 van den bergh , s. , & lafontaine , a. , 1984 , aj , 89 , 1822 whitmore , b. c. , in : `` a decade of hubble space telescope science '' , eds . m. livio , k. noll , m. stiavelli , uk : cambridge university press zepf , s. e. , ashman , k. m. , english , j. , freeman , k. c. , sharples , r. m. , 1999 , aj , 118 , 752

young star clusters with masses similar to those of classical old globular clusters are observed not only in starbursts , mergers or otherwise disturbed galaxies , but also in normal spiral galaxies . some young clusters with masses as high as @xmath0 have been found in the disks of isolated spirals . dynamical mass estimates are available for a few of these clusters and are consistent with kroupa - type imfs . the luminosity ( and possibly mass- ) functions of young clusters are usually well approximated by power - laws . thus , massive clusters at the tail of the distribution are naturally rare , but appear to be present whenever clusters form in large numbers . while bound star clusters may generally form with a higher efficiency in environments of high star formation rate , many of the apparent differences between clusters in starbursts and `` normal '' galaxies might be simply due to sampling effects . it is a human habit to characterise those things with which we are most familiar as normal . although large spiral galaxies are not the most common type of galaxy in the universe , we happen to live within one and many astronomers would probably tend to characterise the milky way as a fairly normal galaxy . thus , at least for the purpose of this paper , `` normal '' galaxies mostly refer to non - interacting star forming disk galaxies . our location within the milky way gives us a unique perspective from which we can study many of its properties in great detail , and it naturally provides a benchmark for comparison with other galaxies . nevertheless , we should question whether it is justified to apply results obtained from studies of our own galaxy to other galaxies which may appear superficially similar to it . in the context of this workshop , it is of particular relevance to ask how similar the cluster system in the milky way is to those in other galaxies . an increasing amount of observational evidence is pointing to the conclusion that many spirals host `` young massive clusters '' ( ymcs ) or `` super star clusters '' similar to those observed in large numbers in starburst galaxies . the definition of a ymc is rather vague and varies from one author to another , but the term generally seems to refer to young clusters that are more massive than the most massive open clusters in the milky way . however , giving a meaningful definition of a massive cluster may eventually be as difficult as distinguishing galaxies that are normal from those that are not .

astro-ph0403244

finite temperature instantons ( calorons ) have a rich structure if one allows the polyakov loop , @xmath1 in the periodic gauge @xmath2 , to be non - trivial at spatial infinity ( specifying the holonomy ) . it implies the spontaneous breakdown of gauge symmetry . for a charge one @xmath3 caloron , the location of the @xmath4 constituent monopoles can be identified through : i. points where two eigenvalues of the polyakov loop coincide , which is where the @xmath5 symmetry is partially restored to @xmath6 . ii . the centers of mass of the ( spherical ) lumps . iii . the dirac monopoles ( or rather dyons , due to self - duality ) as the sources of the abelian field lines , extrapolated back to the cores . if well separated and localised , all these coincide @xcite . here we study the case of two constituents coming close together for @xmath7 , with an example for @xmath0 . the eigenvalues of @xmath8 can be ordered by a constant gauge transformation @xmath9 & & -3 mm w_^w_== , + & & -3mm_1 _n_n+11+_1 , with @xmath10 . the constituent monopoles have masses @xmath11 , where @xmath12 ( using the classical scale invariance to put the extent of the euclidean time direction to one , @xmath13 ) . in the same way we can bring @xmath14 to this form by a _ local _ gauge function , @xmath15 . we note that @xmath16 ( unique up to a residual abelian gauge rotation ) and @xmath17 will be smooth , except where two ( or more ) eigenvalues coincide . the ordering shows there are @xmath4 different types of singularities ( called defects @xcite ) , for each of the _ neighbouring _ eigenvalues to coincide . the first @xmath18 are associated with the basic monopoles ( as part of the inequivalent @xmath19 subgroups related to the generators of the cartan subgroup ) . the @xmath20 defect arises when the first and the last eigenvalue ( still neighbours on the circle ) coincide . its magnetic charge ensures charge neutrality of the caloron . the special status @xcite of this defect also follows from the so - called taubes winding @xcite , supporting the non - zero topological charge @xcite . to analyse the lump structure when two constituents coincide , we recall the simple formula for the @xmath3 action density @xcite . & & -6mmf_^2(x)=_^2_^2 , + & & -6mm_m(r_m&|y_m -y_m+1| + 0&r_m+1 ) ( c_m&s_m + s_m&c_m ) , with @xmath21 the center of mass location of the @xmath22 constituent monopole . we defined @xmath23 , @xmath24 , @xmath25 , as well as @xmath26 , @xmath27 . we are interested in the case where the problem of two coinciding constituents in @xmath3 is mapped to the @xmath28 caloron . for this we restrict to the case where @xmath29 for some @xmath30 , which for @xmath0 is _ always _ the case when two constituents coincide . since now @xmath31 , one easily verifies that @xmath32 $ ] , describing a _ single _ constituent monopole ( with properly combined mass ) , reducing eq . ( 2 ) to the action density for the @xmath28 caloron , with @xmath33 constituents . the topological charge can be reduced to surface integrals near the singularities with the use of @xmath34 , where @xmath35 . if one assumes _ all _ defects are pointlike , this can be used to show that for each of the @xmath4 types the ( net ) number of defects has to equal the topological charge , the type being selected by the branch of the logarithm ( associated with the @xmath4 elements in the center ) @xcite . one might expect the defects to merge when the constituent monopoles do . a triple degeneracy of eigenvalues for @xmath0 implies the polyakov loop takes a value in the center . yet this can be shown _ not _ to occur for the @xmath0 caloron with _ unequal _ masses . we therefore seem to have ( at least ) one more defect than the number of constituents , when @xmath36 . we will study in detail a generic example in @xmath0 , with @xmath37 . we denote by @xmath38 the position associated with the @xmath22 constituent where two eigenvalues of the polyakov loop coincide . in the gauge where @xmath39 ( see eq . ( 1 ) ) , we established numerically @xcite that p_1=p(z_1)=(e^-i_3 , e^-i_3,e^2i_3 ) , + p_2=p(z_2)=(e^2i_1 , e^-i_1,e^-i_1 ) , + p_3=p(z_3)=(-e^-i_2 , e^2i_2,-e^-i_2).this is for _ any _ choice of holonomy and constituent locations ( with the proviso they are well separated , i.e. their cores do not overlap , in which case to a good approximation @xmath40 ) . here we take @xmath41 , @xmath42 and @xmath43 . the limit of coinciding constituents is achieved by @xmath44 . with this geometry it is simplest to follow for changing @xmath45 the location where two eigenvalues coincide . in very good approximation , as long as the first two constituents remain well separated from the third constituent ( carrying the taubes winding ) , @xmath46 will be constant in @xmath45 and the @xmath0 gauge field @xcite of the first two constituents will be constant in time ( in the periodic gauge ) . thus @xmath47 for @xmath48 , greatly simplifying the calculations . when the cores of the two approaching constituents start to overlap , @xmath49 and @xmath50 are no longer diagonal ( but still block diagonal , mixing the lower @xmath51 components ) . at @xmath52 they are diagonal again , but @xmath50 will be no longer in the fundamental weyl chamber . a weyl reflection maps it back , while for @xmath53 a more general gauge rotation back to the cartan subgroup is required to do so , see fig . 1 . at @xmath52 , _ each _ @xmath54 ( and @xmath55 ) lies on the dashed line , which is a direct consequence of the reduction to an @xmath19 caloron . to illustrate this more clearly , we give the expressions for @xmath54 ( which we believe to hold for any non - degenerate choice of the @xmath56 ) when @xmath57 : p_1=p(z_1)=(e^2i_2 , e^2i_2,e^-4i_2 ) , + p_2=p(z_2)=(e^-i_2 , e^2i_2,e^-i_2 ) , + p_3=p(z_3)=(-e^-i_2 , e^2i_2,-e^-i_2).these can be factorised as @xmath58 , where @xmath59 describes an overall @xmath60 factor . in terms of @xmath61 , @xmath62 and @xmath63 the @xmath19 embedding in @xmath0 becomes obvious . it leads for @xmath64 to the trivial and for @xmath65 to the non - trivial element of the center of @xmath19 ( appropriate for the latter , carrying the taubes winding ) . on the other hand , @xmath66 corresponds to @xmath67 , which for the @xmath19 caloron is not related to coinciding eigenvalues . for @xmath44 , fig . 2 shows that @xmath68 gets `` stuck '' at a _ finite _ distance ( 0.131419 ) from @xmath69 . the @xmath19 embedding determines the caloron solution for @xmath70 , with constituent locations @xmath71 and @xmath72 , and masses @xmath73 and @xmath74 . the best proof for the spurious nature of the defect is to calculate its location purely in terms of this @xmath19 caloron , by demanding the @xmath19 polyakov loop to equal @xmath75 . for this we can use the analytic expression @xcite of the @xmath19 polyakov loop along the @xmath76-axis . the location of the spurious defect , @xmath77 , is found by solving @xmath78 $ ] . for our example , @xmath79 indeed verifies this equation . with the @xmath19 embedded result at hand , we find that only for @xmath80 the defects merge to form a triple degeneracy . using @xmath81 , this is so for coinciding constituent monopoles of _ equal _ mass . for _ unequal _ masses the defect is always spurious , but it tends to stay within reach of the non - abelian core of the coinciding constituent monopoles , except when the mass difference approaches its extremal values @xmath82 , see fig . 2 ( bottom ) . at these extremal values one of the @xmath0 constituents becomes massless and _ delocalised _ , which we excluded for @xmath53 . however , the limit @xmath44 is singular due to the _ global _ decomposition into @xmath83 at @xmath52 . gauge rotations @xmath84 in the global @xmath19 subgroup do not affect @xmath59 , and therefore any @xmath85 gives rise to the _ same _ accidental degeneracy . in particular solving @xmath86 $ ] ( corresponding to the weyl reflection @xmath87 ) yields @xmath88 for @xmath89 ( isolated point in fig . 2 ( top ) ) . indeed , @xmath90 traces out a ( nearly spherical ) _ shell _ where two eigenvalues of @xmath91 coincide ( note that for @xmath80 this shell collapse to a single point , @xmath92 ) . a perturbation tends to remove this accidental degeneracy . abelian projected monopoles are not always what they seem to be , even though required by topology . _ topology _ can not be localised , no matter how tempting this may seem for smooth fields . i am grateful to andreas wipf for his provocative question that led to this work . i thank jan smit , and especially chris ford , for discussions . 9 t. c. kraan and p. van baal , nucl . b * 533 * ( 1998 ) 627 [ hep - th/9805168 ] . p. van baal , in : lattice fermions and structure of the vacuum , eds . v.mitrjushkin and g. schierholz ( kluwer , dordrecht , 2000 ) p. 269 [ hep - th/9912035 ] . c. ford , t. tok and a. wipf , nucl . b * 548 * ( 1999 ) 585 [ hep - th/9809209 ] ; phys . b * 456 * ( 1999 ) 155 [ hep - th/9811248 ] . t. c. kraan and p. van baal , phys . b * 435 * ( 1998 ) 389 [ hep - th/9806034 ] . m. n. chernodub , t. c. kraan and p. van baal , nucl . * 83 * ( 2000 ) 556 [ hep - lat/9907001 ] . c. taubes , in : progress in gauge field theory , eds . g.t hooft e.a . , ( plenum press , new york , 1984 ) p.563 . m.garca prez , a.gonzlez-arroyo , a. montero and p. van baal , jhep * 9906 * ( 1999 ) 001 [ hep - lat/9903022 ] .

we analyse what happens with two merging constituent monopoles for the @xmath0 caloron . identified through degenerate eigenvalues ( the singularities or defects of the abelian projection ) of the polyakov loop , it follows that there are defects that are not directly related to the actual constituent monopoles . -1 cm

hep-lat0108027

bressan , granato & silva @xcite have suggested that the presence of dusty circumstellar envelopes around asymptotic giant branch(agb ) stars should leave a signature , a clear excess at 10 @xmath0 m , in the mid infrared ( mir ) spectral region of passively evolving stellar systems . early detections of such an excess were suspected in m32 ( impey et al . 1986 ) from ground based observations , and in a few ellipticals observed with isocam ( bregman et al . the first unambiguous confirmation of the existence of this feature , though barely resolved , was found in the iso cvf spectrum of ngc 1399 ( bressan et al . 2001 ) . since agb stars are luminous tracers of intermediate age and old stellar populations , an accurate analysis of this feature has been suggested as a complementary way to disentangle age and metallicity effects among early type galaxies ( bressan et al . 1998 ; 2001 ) . more specifically , bressan et al.s models show that a degeneracy between metallicity and age persists even in the mir , since both age and metallicity affect mass - loss and evolutionary lifetimes on the agb . while in the optical age and metallicity need to be anti - correlated to maintain a feature unchanged ( either colour or narrow band index ) , in the mir it is the opposite : the larger dust - mass loss of a higher metallicity simple stellar population ( ssp ) must be balanced by its older age . thus a detailed comparison of the mir and optical spectra of passively evolving systems constitutes perhaps one of the cleanest ways to remove the degeneracy . besides this simple motivation and all other aspects connected with the detection of evolved mass - losing stars in passive systems ( e.g. athey et al . 2002 ) , a deep look into the mid infrared region may reveal even tiny amounts of activity . in this letter we present the detection of extended silicate features in a sample of virgo cluster early type galaxies , observed with the irs instrument ( houck et al . 2004 ) of the _ spitzer space telescope . _ ( werner et al . 2004 ) . .virgo galaxies observed with irs [ cols="<,^,^,^,^,^,^ " , ] standard staring mode short sl1 ( [email protected]@xmath0 m ) , sl2 ( [email protected]@xmath0 m ) and long ll2 ( [email protected]@xmath0 m ) , low resolution ( r@xmath164 - 128 ) irs spectral observations of 17 early type galaxies , were obtained during the first _ spitzer _ general observer cycle . the galaxies were selected among those that define the colour magnitude relation of virgo cluster ( bower , lucy & ellis 1992 ) . the observing log is given in table 1 . we also report , in columns 4 and 5 , the number of cycles of 60 and 120 seconds exposures performed with sl1/2 and ll2 , respectively . the spectra were extracted within a fixed aperture ( 3@xmath218 " for sl ) and calibrated using custom made software , tested against the smart software package ( higdon et al . 2004 ) . the on - target exposures in each sl segment ( e.g. sl1 ) also provide @xmath180 offset sky spectra in the complementary module ( e.g. sl2 ) that were used to remove the sky background from the source spectrum in the corresponding segment . since for the ll module we have obtained only ll2 observations , ll2 spectra were sky - subtracted by differencing observations in the two nod positions . the irs pipeline version s12 ( and older versions ) is designed for point source flux extraction . we present here an alternative procedure that exploits the large degree of symmetry that characterizes the light distribution in early type galaxies . we first obtained the real e@xmath3/sec to jy conversion following the procedure outlined by kennicutt et al . we have corrected the conversion table provided for point sources by applying the corrections for aperture losses ( alcf ) and slit losses ( slcf ) . the alcf is due to the residual flux falling outside the aperture selected in the standard calibration pipeline . to estimate the alcf we used 4 calibration stars ( hr 2194 , hr 6606 , hr 7341 and hr 7891 ) observed with _ spitzer _ irs . using the _ spitzer _ science center spice software package we have evaluated the correction resulting from the average ratio of the fluxes extracted within the standard aperture and within twice the standard aperture . the slcf correction is applied to retrieve the real flux of an observed point source that hits the slit , accounting for the slit losses due to the point spread function of the optical combination ( slcf ) . it is defined as the wavelength dependent ratio between the whole flux of a point source on the field of view and the flux selected by the slit to hit the detector . to obtain this correction we have simulated the point spread function of the system ( psf ) using the spitzer - adapted tiny tim code and adopting a `` hat '' beam transmission function of the slit . after the alcf and slcf corrections were applied we obtained the flux _ received _ by the slit within a given aperture . the estimate of the flux _ emitted _ by an extended source within the selected angular aperture of the slit involves the deconvolution of the received flux with the psf of the instrument . this correction is important to obtain the shape of the intrinsic spectral energy distribution ( sed ) of the galaxy , because from the slcf we have estimated that for a point source the losses due to the psf amount to about 20% at 5@xmath4 m and to about 40% at 15@xmath4 m . conversely , a uniform source will not suffer net losses . in order to recover the intrinsic sed we have convolved a surface brightness profile model with the psf , and we have simulated the corresponding observed linear profile along the slits , taking into account the relative position angles of the slits and the galaxy . the adopted profile is a wavelength dependent two dimensional modified king s law ( elson at al . 1987 ) : @xmath5^{-\gamma/2 } \label{elson}\ ] ] where @xmath6 and @xmath7 are the coordinates along the major and minor axis of the galaxies , @xmath8 is the axial ratio taken from the literature . @xmath9 , @xmath10 and @xmath11 are free parameters that are functions of the wavelength and are obtained by fitting the observations with the simulated profile . in order to get an accurate determination of the parameters of the profiles several wavelength bins have been co - added . this procedure has a twofold advantage because it allows us to ( i ) reconstruct the intrinsic profile and the corresponding sed and , ( ii ) to recognise whether a particular feature is resolved or not . the spectrum , extracted in a fixed width around the maximum intensity , is corrected by the ratio between the intrinsic and observed profile . since for the ll2 segment the above procedure is generally not as stable as for sl segments , we have preferred to fix @xmath10 to the corresponding value derived in the nearby wavelength region of the sl segment . an estimate of the signal to noise ( s / n ) ratio was performed by considering two sources of noise : the instrumental plus background noise and the poissonian noise of the source . the former was evaluated by measuring the variance of pixel values in background - subtracted coadded images far from the source . the poissonian noise of the sources was estimated as the square root of the ratio between the variance of the number of e@xmath3 extracted per pixel in each exposure , and the number of the exposures . the total noise was obtained by summing the two sources in quadrature and by multiplying by the square root of the extraction width in pixels . the corresponding s / n ratio at 6@xmath4 m is shown in column 6 of table 1 . finally we notice that the overall absolute photometric uncertainty of irs is 10% , while the slope deviation within a single segment ( affecting all spectra in the same way ) is less than 3% ( see the spitzer observer manual ) . the final flux calibrated spectra of the selected virgo cluster early type galaxies are shown in figures [ passive ] and [ active ] . in figure [ passive ] we have collected the thirteen galaxies ( 76% of the sample ) whose irs spectra are characterised by the presence of a broad emission feature around @xmath1210@xmath4 m that extends toward longer wavelengths . these galaxies show neither pah features nor emission lines . the observed spectra ( solid lines ) are superimposed on old ssp from bressan et al . ( 1998 ) normalized at @xmath125.3@xmath4 m . the dotted line is a 10 gyr , z=0.02 ( solar metallicity ) ssp computed without accounting for dusty circumstellar envelopes . dashed lines from bottom to top refer to 10 gyr ssps with increasing metallicity z=0.008 , z=0.02 and z=0.05 , computed with dusty silicate circumstellar envelopes . the models that account for dusty circumstellar envelopes show an extended feature due to silicate emission , which is very similar to that observed . the feature gets stronger at decreasing age and/or at increasing metallicity due to the corresponding higher dust mass loss rate of the ssp . since , in addition to the match with the models , the analysis of the intrinsic spatial profile indicates that the whole spectrum is extended , we argue that the observed features are of stellar origin and most likely arise from dusty circumstellar envelopes of mass - losing , evolved stars . to corroborate this possibility we compare , in fig . [ passive ] , the normalized continuum subtracted 10@xmath4 m silicate emission of the _ mean outflow _ oxygen - rich agb star ( molster et al . 2002 ) and of u cam ( a carbon rich star with sic emission , sloan et al . 1998 ) with that of ngc 4365 . though early type galaxies are expected to harbour carbon stars , given the wide metallicity spread within a galaxy , it seems that the dominant contribution comes from evolved m giants . more detailed models , fully accounting for the expected mixture of evolved stars , will be presented in a forthcoming paper . the mir view of early type galaxies proves to be a strong diagnostic for the population content of these galaxies . recently temi et al . ( 2005 ) noticed that the irac flux ratios at 8@xmath4 m and 3.6@xmath4 m or mips 24@xmath4 m to irac 3.6@xmath4 m flux ratio remain fairly constant in early type galaxies which otherwise show different h@xmath13 strength . they conclude that this disagreement supports a small rejuvenation episode . although this is one of the possibilities invoked by bressan et al . ( 1996 ) to explain early type galaxies with strong balmer line absorptions , caution must be paid before drawing definite conclusions . indeed , we show in figure [ passive ] that a young ( 5 gyr ) more metal poor ( z=0.008 ) ssp , dot dashed line , is very similar to an old , more metal rich one . thus even the mid infrared spectral region is degenerate and in order to break the age - metallicity degeneracy in passively evolved systems a careful combined optical ( including possibly nir ) and mir analysis is required . to further illustrate the strength of this kind of analysis , we show in figure [ comp ] a comparison of the irs spectra of ngc 4551 and ngc 4365 . a recent optical spectroscopic study ( yamada et al . 2006 ) indicates that ngc 4551 is significantly younger and more metal rich than ngc 4365 . in this case we would expect ngc 4551 to be richer in bright mass - losing agb stars than ngc 4365 , and its silicate features to be more prominent . however , the opposite is observed , suggesting that ngc 4551 is either older or more metal poor ( or both ) than ngc 4365 . evidently , the effects of degeneracy in the optical can be strong ( see e.g. denicol et al . 2005 , annibali et al . 2006 ) . we finally notice that ngc 4473 was observed by iso ( xilouris et al . 2004 ) and shows spatially extended emission at 6.7 and 15@xmath4 m . these authors measured a 15@xmath4 m excess with respect to ssp models _ without _ dusty circumstellar envelopes . the excess was interpreted as due to hot diffuse interstellar dust . irs spectra , such as those presented here , permit the disentangling of the contribution of evolved agb stars and the presence of interstellar dust . the remaining four galaxies ( 24% of the sample ) display different signatures of _ activity _ in the mir spectra ( figure [ active ] ) . these galaxies are classified as active from optical studies ( from agn to transition liner - hii ) at odds with the former group . the spectra of ngc 4636 and ngc 4486 ( m 87 ) show emission lines ( [ arii]7@xmath4 m , [ neii]12.8@xmath4 m , [ neiii]15.5@xmath4 m and [ siii]18.7@xmath4 m ) possibly of non - stellar origin . the broad continuum feature at 10@xmath4 m in ngc 4486 is not spatially extended and likely due to silicate emission from the dusty torus ( siebenmorgen et al . 2005 ; hao et al . 2005 ) . line emission in ngc 4636 falls on top of the circumstellar emission sed and , as for ngc 4473 , its excess at 15@xmath4 m is of stellar origin and not due to emission by hot diffuse dust as suggested by ferrari et al . ( 2002 ) . the spectrum of ngc 4550 shows pah emissions features ( at 6.2 , 7.7 , 8.6 , 11.3 and 12.7 @xmath4 m ) and the h@xmath14 s(5 ) 6.9@xmath4 m and s(3 ) 9.66@xmath4 m emission lines . ngc 4435 shows a typical star - forming spectrum . a preliminary interpretation suggests that an unresolved starburst is dominating the mir emission ( panuzzo et al . in preparation ) . we presented _ spitzer _ mir irs spectra of early type galaxies selected along the colour - magnitude relation of the virgo cluster . we have reconstructed the intrinsic sed of these galaxies from the observed spatial profile sampled by the slits , via a careful analysis of psf effects . in this way we are also able to differentiate between spatially resolved and unresolved regions within the spectrum . this provides independent support for the interpretation of their nature . most of the galaxies ( 76% ) show an excess at 10@xmath4 m and longward which appears spatially extended and is likely due to silicate emission . this class of spectra do not show any other emission features . we argue that the 10@xmath4 m excess arises from mass - losing evolved stars , as predicted by adequate ssp models . a detailed modelling of these features together with the analysis of combined optical , nir and mir spectra will be presented in a forthcoming paper . in the remaining smaller fraction ( 24% ) we detect signatures of _ activity _ at different levels . we observe line emission superimposed on the stellar silicate features in ngc 4636 , unresolved line and silicate emission in m 87 that likely originate in the dusty torus and unresolved pah emission in ngc 4550 and ngc 4435 . the latter galaxy displays the main characteristics of a nuclear starburst ( panuzzo et al . in preparation ) . if we exclude m 87 , which is a well known agn , only two out of 16 early - type galaxies observed show pahs , which corresponds to quite a low fraction ( @xmath112% ) of the observed sample . it is premature to conclude that such a low fraction of galaxies with pahs is representative of the cluster early - type galaxy population , especially if we consider that our investigation is limited to the brightest cluster members ( the upper two magnitudes of the colour - magnitude relation ) . a detailed comparison of our results with those obtained for field galaxies will cast light on the role of environment in the galaxy evolution process . this work is based on observations made with the spitzer space telescope , which is operated by the jpl , caltech under a contract with nasa . we thank j.d.t . smith for helpful suggestions on the irs flux calibration procedure and the anonymous referee for useful suggestions . a. b. , g.l . g. and l. s. thank inaoe for warm hospitality . annibali , f. , bressan , a. , rampazzo , r. danese , l. , & zeilinger , w.w . 2006 , , submitted athey , a. , bregman , j. , bregman , j. , temi , p. , & sauvage , m. 2002 , , 571 , 272 bregman , j. n. , athey , a. e. , bregman , j.d . , & temi , p. 1998 , aas , 193 , 0903 bower , r. g. , lucey , j. r. , & ellis , r. s. 1992 , mnras , 254 , 601 bressan , a. , chiosi , c. , & tantalo , r. 1996 , , 311 , 425 bressan , a. , granato , g.l . , & silva , l. 1998 , aa , 332 , 135 bressan , a. et al . 2001 , , 277 , 251 denicol , g. , terlevich , r. , terlevich , e. , forbes , d. a. , & terlevich , a. 2005 , , 358 , 813 elson , r. a. w. , fall , s. m. , & freeman , k. c. 1987 , , 323 , 54 ferrari , f. , pastoriza , m. g. , macchetto , f. d. , bonatto , c. , panagia , n. , & sparks , w. b. 2002 , , 389 , 355 hao , l. et al . 2005 , , 625 , l75 higdon , s.j.u . 2004 , pasp , 116 , 975 houck , j.r . 2004 , apjs , 154 , 18 impey , c.d . , wynn - williams , c.g . , & becklin , e.e . 1986 , apj , 309 , 572 kennicutt , r. c. et al . 2003 , , 115 , 928 molster , f. j. , waters , l. b. f. m. , & tielens , a. g. g. m. 2002 , , 382 , 222 siebenmorgen , r. , haas , m. , krgel , e. , & schulz , b. 2005 , , 436 , l5 sloan , g. c. , little - marenin , i. r. , & price , s. d. 1998 , , 115 , 809 temi , p. , brighenti , f. , & mathews , w.g . 2005 , , 635 , l25 werner , m.w . 2004 , apjs , 154 , 1 xilouris , e. m. et al . 2004 , , 416 , 41 yamada , y. , arimoto , n. , vazdekis , a. , & peletier , r. f. 2006 , , 637 , 200

we present high signal to noise ratio _ spitzer _ infrared spectrograph observations of 17 virgo early - type galaxies . the galaxies were selected from those that define the colour - magnitude relation of the cluster , with the aim of detecting the silicate emission of their dusty , mass - losing evolved stars . to flux calibrate these extended sources we have devised a new procedure that allows us to obtain the intrinsic spectral energy distribution and to disentangle resolved and unresolved emission within the same object . we have found that thirteen objects of the sample ( 76% ) are passively evolving galaxies with a pronounced broad silicate feature which is spatially extended and likely of stellar origin , in agreement with model predictions . the other 4 objects ( 24% ) are characterized by different levels of activity . in ngc 4486 ( m 87 ) the line emission and the broad silicate emission are evidently unresolved and , given also the typical shape of the continuum , they likely originate in the nuclear torus . ngc 4636 shows emission lines superimposed on extended ( i.e. stellar ) silicate emission , thus pushing the percentage of galaxies with silicate emission to 82% . finally , ngc 4550 and ngc 4435 are characterized by polycyclic aromatic hydrocarbon ( pah ) and line emission , arising from a central unresolved region . a more detailed analysis of our sample , with updated models , will be presented in a forthcoming paper . # 1 # 2 _ mem . soc . astron . it . _ * # 1 * , # 2 # 1 # 2 _ the messenger _ # 1 , # 2 # 1 # 2 astron . nach . # 1 , # 2 # 1 # 2 a&a # 1 , # 2 # 1 # 2 a&a # 1 , l#2 # 1 # 2 a&ar # 1 , # 2 # 1 # 2 a&as # 1 , # 2 # 1 # 2 aj # 1 , # 2 # 1 # 2 ara&a # 1,#2 # 1 # 2 apj # 1 , # 2 # 1 # 2 apj # 1 , l#2 # 1 # 2 apjs # 1 , # 2 # 1 # 2 ap&ss # 1 , # 2 # 1 # 2 adv . space res . # 1 , # 2 # 1 # 2 bull . astron . inst . czechosl . # 1 , # 2 # 1 # 2 j. quant . spectrosc . radiat . transfer # 1 , # 2 # 1 # 2 mnras # 1 , # 2 # 1 # 2 mem . r. astr . soc . # 1 , # 2 # 1 # 2 phys . lett . rev . # 1 , # 2 # 1 # 2 publ . astron . soc . japan # 1 , # 2 # 1 # 2 publ . astr . soc . pacific # 1 , # 2 # 1 # 2 nat # 1 , # 2 # 1 # 2 acta astron . # 1 , # 2

astro-ph0602014

the identification of dark matter is one of the major open questions in physics , astrophysics , and cosmology . recent cosmological observations together with constraints from primordial nucleosynthesis point to the presence of non - baryonic dark matter in the universe . the nature of this non - baryonic dark matter is still unknown . one of the preferred candidates for non - baryonic dark matter is a weakly interacting massive particle ( wimp ) . substantial efforts have been dedicated to wimp searches in the last decades @xcite . a particularly active area @xcite are wimp direct searches , in which low - background devices are used to search for the nuclear recoil caused by the elastic scattering of galactic wimps with nuclei in the detector @xcite . in these searches , characteristic signatures of a wimp signal are useful in discriminating a wimp signal against background . a wimp signature which was pointed out very early @xcite is an annual modulation of the direct detection rate caused by the periodic variation of the earth velocity with respect to the wimp `` sea '' while the earth goes around the sun . the typical amplitude of this modulation is 5% . a modulation with these characteristics was observed by the dama collaboration @xcite , but in light of recent results @xcite , its interpretation as a wimp signal is currently in question . different , and possibly clearer , wimp signatures would be beneficial . a stronger modulation , with an amplitude that may reach 100% , was pointed out by spergel in 1988 @xcite . spergel noticed that because of the earth motion around the sun , the most probable direction of the nuclear recoils changes with time , describing a full circle in a year . in particular this produces a strong forward - backward asymmetry in the angular distribution of nuclear recoils . unfortunately it has been very hard to build wimp detectors sensitive to the direction of the nuclear recoils . a promising development is the drift detector @xcite . the drift detector consists of a negative ion time projection chamber , the gas in the chamber serving both as wimp target and as ionization medium for observing the nuclear recoil tracks . the direction of the nuclear recoil is obtained from the geometry and timing of the image of the recoil track on the chamber end - plates . a 1 m@xmath0 prototype has been successfully tested , and a 10 m@xmath0 detector is under consideration . in addition to merely using directionality for background discrimination , what can be learned about wimp properties from the directionality of wimp detectors ? it is obvious that different wimp velocity distributions give rise to different recoil distributions in both energy and recoil direction . copi , heo , and krauss @xcite , and then copi and krauss @xcite , have examined the possibility of distinguishing various wimp velocity distributions using a likelihood analysis of the resulting recoil spectra , which they generated through a monte carlo program . they have concluded that a discrimination among common velocity distributions is possible with a reasonable number of detected events . here we want to gain insight into the properties of the nuclear recoil spectra in energy and direction . for this purpose , we develop a simple formalism that relates the wimp velocity distribution to the distribution of recoil momenta . we find that the recoil momentum spectrum is the radon transform of the velocity distribution ( see eq . ( [ eq : main ] ) below ) . we apply this analytical tool to a series of velocity distributions , and discover for example how the recoil momentum spectrum of a stream of wimps differs from that of a maxwellian velocity distribution . with our gained insight , we suggest that if a wimp signal is observed in directional detectors in the future , it may be possible to invert the measured recoil momentum spectrum and reconstruct the wimp velocity distribution from data . in section [ sec : ii ] we describe the general kinematics of elastic wimp - nucleus scattering , and in section [ sec : iii ] we obtain our main formula for the nuclear recoil momentum spectrum . sections [ sec : iv ] and [ sec : v ] contain general considerations and examples of radon transforms of velocity distributions . finally , section [ sec : inv ] discusses the possibility of inverting the recoil momentum spectrum to recover the wimp velocity distribution . the appendices contain useful mathematical formulas for the computation and inversion of 3-dimensional radon transforms . consider the elastic collision of a wimp of mass @xmath1 with a nucleus of mass @xmath2 in the detector ( see fig . [ fig : kinem ] ) . let the arrival velocity of the wimp at the detector be @xmath3 , and neglect the initial velocity of the nucleus . after the collision , the wimp is deflected by an angle @xmath4 to a velocity @xmath5 , and the nucleus recoils with momentum @xmath6 and energy @xmath7 . let @xmath8 denote the angle between the initial wimp velocity @xmath3 and the direction of the nuclear recoil @xmath6 . energy and momentum conservation impose the following relations : @xmath9 eliminating @xmath4 by summing the squares of eqs . ( [ em2 ] ) and ( [ em3 ] ) , @xmath10 and using this expression to eliminate @xmath11 from eq . ( [ em1 ] ) , gives @xmath12 where @xmath13 is the reduced wimp - nucleus mass . we deduce that the magnitude @xmath14 of the recoil momentum , and the recoil energy @xmath15 , vary in the range @xmath16 eq . ( [ eq : costheta ] ) will be exploited in the following section to express the recoil momentum distribution in a simple mathematical form . for this purpose , we also need the expression for the wimp - nucleus scattering cross section . we write the differential wimp - nucleus scattering cross section as @xmath17 where @xmath18 is the total scattering cross section of the wimp with a ( fictitious ) point - like nucleus , and @xmath19 is a nuclear form factor normalized so that @xmath20 . ( both @xmath21 and @xmath22 are confusingly called form factors . ) eq . ( [ eq : sigma ] ) is valid for both spin - dependent and spin - independent wimp - nucleus interactions , although @xmath18 and @xmath22 have different expressions in the two cases . for example , for spin - independent interactions with a nucleus with @xmath23 protons and @xmath24 neutrons , @xmath25 ^ 2,\ ] ] where @xmath26 and @xmath27 are the scalar four - fermion couplings of the wimp with pointlike protons and neutrons , respectively ( see ref . if the nucleus can be approximated by a sphere of uniform density , its form factor is @xmath28 ^ 2 } { ( qr)^6 } , \ ] ] where @xmath29 \times 10^{-13 } { \rm cm}\ ] ] is ( an approximation to ) the nuclear radius . more realistic expressions for spin - independent form factors , and formulas for spin - dependent cross sections , can be found , e.g. , in refs . eqs . ( [ eq : costheta ] ) and ( [ eq : sigma ] ) can be combined to give the differential recoil spectrum in both energy and direction , i.e. the recoil _ momentum _ spectrum . we define it as the double differential event rate , in events per unit time per unit detector mass , differentiated with respect to the nuclear recoil energy @xmath15 and the nuclear recoil direction @xmath30 , @xmath31 where @xmath32 denotes an infinitesimal solid angle around the direction @xmath30 . the double differential rate follows from the double differential cross section @xmath33 first through the change of differentials @xmath34 , and then through multiplication by the number @xmath35 of nuclei in the detector , division by the detector mass @xmath36 , and multiplication by the flux of wimps with velocities @xmath3 in the velocity space element @xmath37 , @xmath38 here @xmath39 is the wimp number density , @xmath40 is the wimp mass density , and @xmath41 is the wimp velocity distribution in the frame of the detector , normalized to unit integral . the double differential cross section is obtained as follows . azimuthal symmetry of the scattering around the wimp arrival direction gives @xmath42 . the relation between @xmath43 and @xmath14 in eq . ( [ eq : costheta ] ) , @xmath44 , can be imposed through a dirac @xmath45 function , @xmath46 . thus @xmath47 this is correctly normalized as can be seen by integration of the expression in the middle over @xmath48 . summarizing , the double differential event rate per unit time per unit detector mass is @xmath49 we write it as @xmath50 here @xmath51 is the minimum velocity a wimp must have to impart a recoil momentum @xmath14 to the nucleus , or equivalently to deposit an energy @xmath52 , as can be seen from eq . ( [ eq : range ] ) . moreover , @xmath53 is the 3-dimensional radon transform of the velocity distribution function @xmath41 . we note in passing that @xmath54 has units of inverse speed . ( [ eq : main ] ) is the main result of this paper . it states that , apart from a normalizing factor , the recoil momentum spectrum is the radon transform of the wimp velocity distribution . the radon transform is a linear integral transform ( see refs . @xcite ) , which was introduced in two dimensions by radon in 1917 @xcite . the radon transform has been widely studied for its use in solving differential equations , and especially in two - dimensions , for its medical applications in computer tomography . geometrically , @xmath55 is the integral of the function @xmath41 on a plane orthogonal to the direction @xmath56 at a distance @xmath57 from the origin . for reference , some mathematical properties of the radon transform are given in the appendices . as a check of our formalism , we show that integrating our basic equation ( [ eq : main ] ) over recoil directions reproduces the usual expression for the recoil energy spectrum @xmath58 . applying eq . ( [ eq : etaave ] ) in appendix a to our expression for the differential rate , we find @xmath59 this is the usual expression of the recoil energy spectrum ( cfr . ( 8.3 ) in ref . we have cast the nuclear recoil momentum spectrum in terms of a radon transform . now we can take advantage of the properties of radon transforms , some of which are listed in the appendices , to compute recoil momentum spectra analytically . in this section we give some general considerations , and in the next section we give explicit examples of analytic recoil momentum spectra . when the wimp velocity distribution is isotropic , @xmath60 , the recoil spectrum is also isotropic , @xmath61 . from the definition of radon transform , eq . ( [ eq : eta ] ) , @xmath62 we would have obtained the same result starting from eq . ( [ eq : etaave ] ) . wimp velocity distributions are often given in the galactic rest frame , while we are interested in the recoil momentum spectrum in the laboratory frame of the detector . the change of velocity frame can be performed either on the velocity distribution before computing the radon transform or on the radon transform computed in the galactic rest frame . the latter is often easier to compute , and the change of reference frame can be done simply as follows . the wimp velocities @xmath63 and @xmath64 in the laboratory and galactic rest frames , respectively , are related by @xmath65 where @xmath66 is the velocity of the laboratory with respect to the galactic rest frame . this velocity transformation is a translation in velocity space , and we can use eq . ( [ eq : transl ] ) in appendix a to relate the radon transforms in the galactic and laboratory frames , @xmath67 thus the recoil momentum spectrum in the laboratory frame is given directly in terms of the radon transform @xmath68 of the wimp velocity distribution in the galactic rest frame by @xmath69 with @xmath70 as before . if we rotate the coordinate system , we see from eq . ( [ eq : rotate ] ) in appendix a that the recoil momentum spectrum is simply rotated , with the magnitude of the recoil momentum remaining the same , as expected . we give some examples of recoil momentum spectra corresponding to common velocity distributions . we obtain the recoil spectra for streams of particles and for isotropic and anisotropic gaussian distributions with and without bulk velocities . the simplest case is that of a particle stream in which all wimps in the stream move with the same velocity @xmath71 . in this case , @xmath72 and @xmath73 the recoil spectrum of a stream with velocity @xmath71 is concentrated on a sphere of radius @xmath74 , centered in @xmath75 and passing through the origin . the stream velocity @xmath71 is a diameter of the sphere . [ fig : stream ] shows the @xmath76 section of the recoil momentum spectrum of a stream of wimps arriving from the left with velocity @xmath77 km / s . the full distribution is obtained through a rotation around the @xmath78 axis . the pattern of recoil momenta forms a sphere . a maxwellian distribution with velocity dispersion @xmath79 , @xmath80,\ ] ] is a particular case of isotropic distribution , and we can use eq . ( [ eq : iso ] ) above to compute its radon transform . we find @xmath81 .\ ] ] if the detector has velocity @xmath82 , we can use eq . ( [ eq : moving ] ) to find the radon transform in the laboratory frame , @xmath83 ^ 2 } { 2\sigma_v^2 } } \right ] .\ ] ] notice that @xmath84 is the projection of the velocity of the observer in the direction of the nuclear recoil . this expression coincides with , but is simpler than , the analogous expression obtained by elementary methods in ref . @xcite ( @xmath85 in ref . @xcite is @xmath86 ) . the recoil momentum distribution for a maxwellian distribution is shown in fig . [ fig : maxwell ] , assuming a velocity dispersion of 300 km / s and an observer moving at 220 km / s in direction @xmath87 . the distribution is symmetric around the observer velocity . the figure shows the section in the @xmath76 plane only . the full distribution can be obtained by symmetry . ( [ eq : gauss ] ) illustrates the reason for writing @xmath55 instead of @xmath88 ( see section [ sec : notation ] for more details ) . the function @xmath89 assumes different values for different directions @xmath56 ; the function @xmath90 would therefore be multivalued at the origin . we may truncate a maxwellian distribution at the escape speed @xmath91 , @xmath92 , & v < v_{\rm esc } \cr 0 , & { \rm otherwise } \end{cases},\ ] ] with @xmath93 .\ ] ] then we have @xmath94 - \exp\left [ { -\frac{v_{\rm esc}^2}{2\sigma_v^2 } } \right ] \right\}.\ ] ] the recoil - momentum spectrum corresponding to an anisotropic gaussian distribution can also be obtained analytically . an anisotropic gaussian distribution with variance matrix @xmath95 and mean velocity @xmath71 is given by @xmath96.\ ] ] we are using matrix notation , @xmath97 being the transpose of @xmath3 , etc . using the fourier slice theorem , actually eq . ( [ eq : fst2 ] ) , we find the radon transform of the anisotropic gaussian to be @xmath98 ^ 2 } { 2{\widehat{\bf w}}^t \ ! \boldsymbol{\sigma}^2 \ , { \widehat{\bf w } } } } \right ] .\ ] ] this is another example of a function which assumes different values at @xmath99 according to the direction @xmath56 . the recoil spectrum of a stream and a maxwellian velocity distribution are very different : a sphere the first , a smooth distribution the second . this suggests that it may be possible to distinguish different kinds of wimp velocity distributions just by examining the pattern of recoil momenta . subtle differences among velocity distributions may be revealed by a maximum likelihood analysis of the corresponding recoil spectra @xcite . more ambitiously , we may think of recovering the wimp velocity distribution by inverting the measured recoil momentum spectrum . indeed , if we know the nuclear form factor of the detector nuclei , then for any fixed wimp mass we can estimate the radon transform of the wimp velocity distribution from the measured recoil momentum spectrum , modulo a normalization constant @xmath100 . ( [ eq : main ] ) can in fact be written as @xmath101 enabling us to obtain a measurement of the radon transform @xmath102 of the wimp velocity distribution from the measured recoil spectrum @xmath103 . we may be able to invert this radon transform and obtain the wimp velocity distribution @xmath41 , again modulo a normalization constant . finally , we may be able to fix the normalization constant either by normalizing @xmath41 to unit integral or better by examining the detector efficiency as a function of wimp velocity . there are several analytic formulas for the inversion of three - dimensional radon transforms . some of these formulas are collected in appendix b for convenience . most of the analytical inversion formulas can be converted into numerical algorithms . however , any inversion algorithm we were able to find in the literature is suited only to a large amount of data in recoil momentum space , since they all assume that it would be possible to define a discretized version of @xmath55 . this is _ not _ the case for directional dark matter searches , where the total number of events is not under the control of the experimentalist and is expected to be rather small . new inversion algorithms suited to small numbers of events are therefore needed if one wants to reconstruct the wimp velocity distribution using data from directional detectors . as a first attempt in this direction , we have devised the following simple algorithm . divide the wimp velocity space into small cells @xmath104 , @xmath105 , and assume that the wimp velocity distribution @xmath41 is constant over each of these small cells , with value @xmath106 in cell @xmath104 . to each recorded event @xmath107 with measured recoil momentum @xmath108 , @xmath109 , associate the plane @xmath110 in wimp velocity space defined by the equation @xmath111 @xmath110 is the plane orthogonal to the recoil direction @xmath112 and at a distance @xmath113 from the origin . velocity vectors on this plane are all the wimp velocities that can produce the observed nuclear recoil . let @xmath114 the area of the intersection of the plane @xmath110 with the cell @xmath104 ( see appendix c for an explicit expression ) . for each event @xmath107 , assign weight @xmath115 to the @xmath1-th cell . sum the weights over the events , @xmath116 , essentially counting how many planes cross any given cell . take the discrete laplacian of the sum of the weights , and keep only those cells whose values exceed a predetermined threshold . the resulting distribution of cell values is our estimate of the wimp velocity distribution . to test the capabilities of our algorithm , we simulated the recoil spectrum due to two streams of wimps arriving at the detector from opposite directions , with velocities @xmath117 and @xmath118 ( in arbitrary units ) . we generated 100 events , and applied the previous algorithm with @xmath119 cells in velocity space and a threshold of @xmath120 . we found that only two cells in velocity space are above threshold , and they correspond exactly to the location of the simulated streams . [ fig : pignose ] plots the @xmath121 section of the reconstructed velocity distribution . it is impressive that we were able to recover this velocity distribution with only 100 events . we leave further studies of our simple algorithm , and the development of other algorithms , to future work . directional detectors for wimp dark matter searches will be able to measure not only the energy but also the direction of the nuclear recoils caused by the elastic scattering of galactic wimps with nuclei in the detector . this directional capability will help in separating a wimp signal from background , and will also provide a measurement of the recoil momentum spectrum as compared to just the recoil energy spectrum . to gain insight into the properties of recoil momentum spectra , we have devised a simple formalism for the analytic computation of recoil momentum spectra from wimp velocity distributions . mathematically , the recoil momentum spectrum is the 3-dimensional radon transform of the velocity distribution . well - established mathematical properties of the radon transform allow the computation of analytical expressions for recoil spectra associated to several common wimp velocity distributions . as examples we presented recoil spectra for a wimp stream , a maxwellian , a truncated maxwellian , and a non - isotropic gaussian . we found in particular that a stream of wimps produces a characteristic spherical pattern of nuclear recoils . a maxwellian distribution gives instead a smooth recoil pattern . other velocity distributions lead to more complicated spectra . the analytic expressions we found for the nuclear recoil spectra will facilitate the discrimination of different velocity distributions through likelihood analysis . in addition , it may be possible to invert the measured momentum spectrum to reconstruct the local wimp velocity distribution from data . for this purpose , we have presented an algorithm to recover the velocity distribution from a small number of recorded events . we have successfully recovered a simulated velocity distribution with just 100 generated events . we expect that the tools we have presented will be useful for the design and analysis of directional wimp detectors . this research was supported in part by the national science foundation under grant no . phy99 - 07949 at the kavli institute for theoretical physics , university of california , santa barbara . in this appendix we collect some useful mathematical properties of the 3-dimensional radon transform . we denote the 3-dimensional radon transform of a function @xmath41 by @xmath122 . it is defined by @xmath123 it is easy to see that the radon transform is linear , @xmath124 one may be tempted to write @xmath125 for @xmath55 , after all @xmath126 . this notation may however be ambiguous and should be used with care . indeed , one must keep in mind that the radon transform as defined in eq . ( [ eq : def ] ) is a function of the magnitude @xmath57 and the direction @xmath56 separately . in other words , one may have @xmath127 for @xmath128 . namely , @xmath129 may assume different values for different directions . this will not be reflected in the notation @xmath125 . the latter would read @xmath130 at the origin , independently of the direction @xmath56 . in other words , @xmath125 would be a multiple - valued function at the origin . mathematically , the distinction between @xmath55 and @xmath125 is important , and is expressed by saying that @xmath55 is defined on @xmath131 while @xmath125 is defined on @xmath132 . for our application , however , the distinction is of little concern , since the problematic origin @xmath99 corresponds to the region of vanishingly small recoil momenta , which is experimentally inaccessible . we have nevertheless used the mathematically correct notation throughout for clarity . under linear transformations of the coordinate axes , @xmath3 transforms as @xmath133 where @xmath134 is a @xmath135 non - singular matrix and @xmath136 is a constant vector . a velocity distribution function @xmath41 transforms so as to keep the number of particles in a volume @xmath37 invariant : @xmath137 hence , @xmath138 where @xmath139 is the determinant of @xmath134 and @xmath140 is the inverse of @xmath134 . to find the relation between the radon transforms of @xmath41 and @xmath141 , we change integration variable from @xmath5 to @xmath3 in the definition , eq . ( [ eq : def ] ) , @xmath142 where @xmath143 is the transpose of @xmath134 . thus @xmath144 in particular , under a pure rotation @xmath145 , @xmath146 and under a pure translation @xmath147 , @xmath148 the following relations hold for derivatives of the radon transform ( here @xmath149 and @xmath150 ) @xmath151 we find the following expression for the integral of the radon transform @xmath152 over the directions @xmath153 : @xmath154 \ , f({{\bf v } } ) \ , d^3 v \nonumber\\ & = & \int \left [ 2 \pi \int_{-1}^{1 } \delta ( v \cos\gamma - w ) \ , d\!\cos\gamma \right ] \ , f({{\bf v } } ) \ , d^3 v = \int \frac{2 \pi}{v } \ , \theta(v-|w| ) \ , \ , f({{\bf v } } ) \ , d^3 v \nonumber\\ & = & 2 \pi \int_{v>|w| } \frac{f({{\bf v}})}{v } \,\ , d^3v .\end{aligned}\ ] ] there is a connection between the radon transform and the fourier transform . taking the fourier transform of the definition , eq . ( [ eq : def ] ) , with respect to @xmath57 at fixed @xmath56 gives @xmath155 this equation goes under the name of fourier slice theorem . the right hand side is just the fourier transform of @xmath41 evaluated at @xmath156 , while the left hand side is the fourier transform of @xmath157 at fixed @xmath153 . inverting the fourier transform in the left hand side of the fourier slice theorem , we have @xmath158 this alternative expression of the radon transform actually serves as its definition when functions are replaced by distributions ( in the mathematical sense , see ref . @xcite ) . let us expand @xmath41 and its radon transform @xmath55 into spherical harmonics @xmath159 and @xmath160 , respectively . we have @xmath161 and @xmath162 the coefficients of the spherical harmonic expansions are related by @xmath163 where @xmath164 is a legendre polynomial of order @xmath165 . these expressions are useful when the velocity distributions are not isotropic . ( [ eq : etalm ] ) can be proven using the decomposition of the @xmath45-function in legendre polynomials @xmath166 the addition theorem for spherical harmonics @xmath167 and the orthogonality of the spherical harmonics @xmath168 which lead to the relation @xmath169 an inversion formula for the radon transform is @xmath170 where @xmath171 is the laplacian in @xmath3 . it can also be written in terms of @xmath172 as @xmath173 the inversion formula ( [ eq : inv1 ] ) can be proven by inverting the fourier transform of @xmath41 in the fourier slice theorem , eq . ( [ eq : fst ] ) , then integrating separately in @xmath174 and @xmath56 , and finally using the relation @xmath175 where @xmath176 is the second derivative of the dirac @xmath45-function . another inversion method is through an expansion in spherical harmonics . referring to eqs . ( [ eq : sph1 ] ) and ( [ eq : sph2 ] ) , one can prove the following inversion formula @xmath177 where @xmath164 is a legendre polynomial and @xmath178 , the second derivative of @xmath179 with respect to the modulus of @xmath57 . ( [ eq : flm ] ) is proven along the same lines as eq . ( [ eq : etalm ] ) , starting from eq . ( [ eq : inv1b ] ) written as @xmath180 the fourier slice theorem , eq . ( [ eq : fst ] ) , can be made into an algorithm for the numerical evaluation of the inverse radon transform . typically one would use fast fourier transforms . ( [ eq : inv1 ] ) can also be made into an algorithm . for each given @xmath3 , the integration in @xmath181 amounts to an integration over the sphere of diameter @xmath3 and passing through the origin ( a `` stream sphere '' ) , with integration measure @xmath182 in spherical coordinates centered at the center of the sphere and north pole in @xmath3 . the final laplacian can be computed numerically as the difference between the central value and the average value of its six nearest neighbors . an algebraic inversion method is the following @xcite . suppose that the values @xmath183 , @xmath184 , corresponding to the points @xmath185 are known . in medical applications , the points @xmath185 form a grid or other structure in space , and the @xmath183 s are the measured signal intensities . in our case , the number of detected events may be quite small , in which case we may let @xmath185 be the actual measurement of a nuclear recoil momentum , with @xmath107 varying over the number of events , and @xmath186 , where @xmath187 is the efficiency for detecting event @xmath107 . by definition of radon transform we have @xmath188 where the integral is taken over the plane in @xmath3-space defined by the equation @xmath189 @xmath110 is the plane orthogonal to the recoil direction @xmath56 and at a distance @xmath57 from the origin . now suppose that @xmath41 has compact support , meaning that it vanishes for @xmath190 something . this is a technical simplification that is valid in practice since real velocity distributions are always truncated at some large velocity ( e.g. at the escape speed from the galaxy ) . divide the @xmath3-space into small cells @xmath104 , @xmath105 , and assume that @xmath41 is constant over each of these small cells , with value @xmath106 in cell @xmath104 . this is the discretizing approximation . let @xmath114 be the area of the intersection of the plane @xmath110 with the cell @xmath104 ( see appendix c for an explicit expression ) . a discretized version of eq . ( [ eq : etaj ] ) is then @xmath191 in matrix form @xmath192 where @xmath193 is an @xmath194 matrix , @xmath195 and @xmath196 . this is a system of linear equations for @xmath197 that can be solved by inverting @xmath198 . since few @xmath115 differ from zero , @xmath198 is a sparse matrix , and it is convenient to solve this system iteratively . fix @xmath199 , @xmath200 . let the initial guess be @xmath201 and the @xmath202-th update be @xmath203 . from @xmath203 compute the following vectors successively @xmath204 here @xmath205 and @xmath206 . finally let the next update be @xmath207 . @xcite attributes this method to kaczmarz . for future reference , we give here the expression for the area of the intersection of a plane with a rectangular cell . assume the @xmath208 space is divided into rectangular cells of sides @xmath209 , @xmath210 , and @xmath211 along @xmath212 , @xmath213 , and @xmath214 , respectively . let the @xmath215-th cell @xmath216 be centered in @xmath217 , @xmath218 let @xmath219 be the plane defined by @xmath220 then the area of the intersection of the plane @xmath219 with the @xmath215-th cell @xmath216 is @xmath221 where @xmath222 and @xmath223 , @xmath224 , and @xmath225 are the quantities @xmath226 , @xmath227 , and @xmath228 sorted in order of increasing magnitude , @xmath229 . the last case in eq . ( [ eq : explicitarea ] ) becomes @xmath230 in the limit of small @xmath223 . c. j. martoff _ et al . 2nd int . workshop on the identification of dark matter ( idm 98 ) , _ buxton , england , 1998 ; d. snowden - ifft _ et al . _ , _ ibid . _ ; m. j. lehner _ _ , in _ dark matter in astro and particle physics ( dark98 ) , _ heidelberg , germany , 1998 [ astro - ph/9905074 ] ; d. p. snowden - ifft , c. j. martoff and j. m. burwell , phys . d * 61 * , 101301 ( 2000 ) ; d. p. snowden - ifft , in _ sources and detection of dark matter and dark energy , _ marina del rey , ca , 2002 ; c. j. martoff , _ ibid._. , assuming a stream of wimps with velocity @xmath231 . the full @xmath232 distribution can be obtained by revolution around the @xmath78 axis . the recoil momenta describe a sphere in recoil space.,scaledwidth=80.0% ] , assuming a maxwellian velocity distribution of wimps with velocity dispersion 300 km / s , and a detector moving with velocity @xmath233 . lighter areas have higher probability . the full @xmath232 distribution can be obtained by revolution around the @xmath78 axis.,scaledwidth=80.0% ]

directional dark matter detectors will be able to record the recoil momentum spectrum of nuclei hit by dark matter wimps . we show that the recoil momentum spectrum is the radon transform of the wimp velocity distribution . this allows us to obtain analytic expressions for the recoil spectra of a variety of velocity distributions . we comment on the possibility of inverting the recoil momentum spectrum and obtaining the three - dimensional wimp velocity distribution from data .

hep-ph0209110

study of geodesics has been one of the main tools in the investigation of the physical properties of spacetimes since the very establishment of general relativity for a general review of both geodesics and electrogeodesics for the reissner - nordstrm as well as other spacetimes , see , for example , the classical book by chandrasekhar @xcite . besides revealing the causal structure of the spacetime , geodesics help us build our intuition about the solution and connect it to the classical newtonian case , if possible . it may also be possible to interpret parameters appearing in the solution and restrict their ranges based on physically reasonable requirements on the behavior of geodesics . until this day , new papers are still appearing on geodesics in kerr @xcite , which is certainly very important for astrophysical applications , or even in schwarzschild @xcite . with the advent of the ads / cft conjecture there has been renewed interest in the geodetical structure of spacetimes involving the cosmological constant @xcite . in these cases , there is usually some test - particle motion which is not allowed in the schwarzschild spacetime or the interval of admissible radii extends farther , which is also the case here as we can reach the axis . the different allowed ranges could perhaps enable us to distinguish between the various solutions through direct observation . the general method is to find the appropriate effective potential and study its properties and that is also the approach we adopt in the present paper . the maxwell field possibly present in the spacetime influences test - particle motion in two ways : firstly by deforming the spacetime , which also has an effect on neutral particles ( or even photons @xcite ) , and , secondly , by generating lorentz force acting upon charged particles . we focus here on the second effect , which was also studied in @xcite in the case of kerr - newmann solution , where there are two different angular frequencies for a circular equatorial orbit of a given radius due to co- and counterrotation of test particles . papers @xcite and @xcite investigate charged electrogeodesics in reissner - nordstrm with and without the cosmological constant , respectively , but they do not discuss the existence of double frequencies of circular orbits unlike @xcite where the two different solutions are discussed for the pure reissner - nordstrm spherically symmetric spacetime . circular orbits are important in astrophysics where they determine the appearance of accretion disks around compact objects . it is thus of interest that a spacetime admits orbits of the same radius but different angular velocities . in principle , the double frequencies could show in observations of plasma orbiting charged sources . in this paper we introduce charged test particles to an axially symmetric static spacetime consisting of two black holes of charges equal to their masses to ensure a static equilibrium between their gravitational attraction and electrostatic repulsion . that is , these are in fact two extremally charged black holes . this is a special case of the majumdar - papapetrou class of solutions @xcite , which admit point charges with a flat asymptotic region @xcite . there are two independent scales characterizing the spacetime : it is determined by the masses of the two black holes and their distance but we can always rescale the masses by the distance so there are only two free parameters . we find static positions of the test particles and compare them both to the geodesic case studied previously @xcite and the classical non - relativistic solution . we then specifically investigate linear and circular motion of these charged test particles , focussing on analytic solutions while also using numerical approach as required in the study of geodesic deviation . although the spacetime is not physically realistic due to its extremal charge , it is an interesting exact solution exhibiting axial symmetry . in fact , with multiple black holes of this kind , one can prescribe any discrete symmetry or even remove it altogether . perhaps unrealistic but the studied spacetime is still mathematically rewarding since this is an exact solution of einstein - maxwell equations and we can use analytic approach throughout most of our paper . the paper is structured as follows : in section [ newtonian case ] we review the newtonian case of two charged point masses in static equilibrium and study motion of charged test particles in their field to be able to check and compare our later results . the background field is the classical analog of the extreme reissner - nordstrm di - hole metric , which we introduce in section [ the spacetime ] . we then investigate static electrogeodesics ( section [ static electrogeodesics ] ) and test particles oscillating along the @xmath0-axis ( section [ oscillation along z ] ) . in section [ circular electrogeodesics ] we study general circular electrogeodesics to concentrate on the equatorial case in section [ circular electrogeodesics within the equatorial plane ] . in the final section [ deviation of geodesics ] we briefly look at the stability of geodesics from the point of view of geodesic deviation . let us start with the newtonian case of two static massive point charges with their gravitational attraction balanced by their electrostatic repulsion and then let us add a charged massive test particle to study its motion . suppose that the sources of the field have masses @xmath1 and charges @xmath2 ( of the same sign ) chosen in such a way that the particles are in a static equilibrium regardless of their positions . we have the relation ( in ` cgs ` ) : @xmath3 where @xmath4 is newton s gravitational constant in the following we use the geometrized units @xmath5 .. the particular choice @xmath6 is required for three or more sources of the field and in gr . ] without loss of generality we choose both charges to be positive and put the particles to @xmath7 ( @xmath8 ) and @xmath9 ( @xmath10 ) , respectively , using the standard cylindrical coordinate system @xmath11 and @xmath10 in terms of @xmath12 so that @xmath13 . ] . the electrostatic and gravitational potentials , @xmath14 and @xmath15 , read @xmath16 , \varphi_g = -\varphi_e.\ ] ] the lagrangian @xmath17 of the test particle of charge - to - mass ratio @xmath18 is @xmath19 where the dot denotes derivative with respect to newtonian absolute time while in the relativistic case ( from section [ the spacetime ] onward ) it denotes derivative with respect to proper time . we thus obtain a set of equations @xmath20\right\rbrace - \ddot{\rho } , \\ \fl 0 & = & - \rho \left(\rho \ddot{\phi } + 2 \dot{\rho } \dot{\phi } \right),\\ \fl 0 & = & ( 1-q ) \left\lbrace\frac{m_1 ( a - z)}{\left[\rho ^2+(z - a)^2\right]^{3/2}}-\frac{m_2 ( a+z)}{\left[\rho ^2+(z+a)^2\right]^{3/2}}\right\rbrace - \ddot{z}. \label{z - direction}\end{aligned}\ ] ] looking for circular orbits with @xmath21 we have @xmath22 and thus define @xmath23 , finding several solutions . firstly , the static case @xmath24 with the test particle located anywhere ( we can immediately see from ( [ lagrangian ] ) and ( [ potential ] ) that this is equivalent to a free particle , generally moving along a straight line at a constant velocity ) . additionally , if @xmath25 , we need to solve ( [ z - direction ] ) only to get static points located along the @xmath0-axis at @xmath26 where @xmath27 and the test particle is allowed to have an arbitrary charge - to - mass ratio @xmath18 . finally , we have a solution of the form @xmath28^{3/2},\\ \omega ^2 & = & \left(1-q\right ) \left[\frac{m_1}{\left((z - a)^2+\rho ^2\right)^{3/2}}+\frac{m_2}{\left((a+z)^2+\rho ^2\right)^{3/2}}\right].\end{aligned}\ ] ] it follows that @xmath29 and @xmath30 so the circular orbits may only occur in planes parallel to @xmath31 between the two point masses . it is of interest that there are also periodic solutions lying entirely within the @xmath0-axis . these necessarily entail the condition @xmath32 ) outside of the interval @xmath33 $ ] to find out there is only a single turning point in @xmath34 $ ] and a single turning point in @xmath35 $ ] so that no periodic motion is possible there . ] , using which , the equation of motion can be rewritten as @xmath36 integrating the last relation , we find @xmath37 we will not give here the full solution as it leads to a complicated expression involving elliptic integrals and , instead , we will simply find the turning points with @xmath38 : @xmath39 we conclude that , for @xmath40 there are generally periodic orbits with purely axial motion . it seems that the position of the turning points is independent of the test particle s charge . however , @xmath41 where @xmath42 is energy per unit mass of the particle . we now proceed to discuss the relativistic situation . when discussing the newtonian limit of the relativistic solutions , we must bear in mind that the above solutions are only relevant for slow motion and weak fields , i.e. , at distances from both point sources much greater than their masses . the extreme reissner - nordstrm di - hole metric reads @xmath43 with @xmath44 where @xmath45 denotes the spatial position of point sources in cartesian coordinates . the geometry describes a system consisting of two static black holes with charges equal to their masses and located on the @xmath0-axis at @xmath46 . their electromagnetic field is described by the 4-potential @xmath47 the geometry is axially symmetric so it is useful to replace the cartesian coordinates with the standard cylindrical ones , @xmath48 . the metric , function @xmath49 , and the maxwell 2-form , @xmath50 , then take the form @xmath51 consider a test particle with a charge - to - mass ratio @xmath18 . its lagrangian reads @xmath52,\ ] ] yielding the following equations of motion @xmath53,\\ \label{eq : elgeodmotion2 } \fl 0 & = & \left(\ddot{\rho}-\rho \dot{\phi}^2\right)+u^{-5}(qu\dot{t}-\dot{t}^2){{\partial}_{\rho}{u}}+\frac{1}{u}\left[2\dot{z}\dot{\rho}{{\partial}_{z}{u}}+(\dot{\rho}^2-\rho ^2 \dot{\phi}^2-\dot{z}^2){{\partial}_{\rho}{u}}\right],\\ \label{eq : elgeodmotion3 } \fl 0 & = & \frac{1}{\rho u}\left[u \left(2 \dot{\rho } \dot{\phi}+\rho \ddot{\phi}\right)+2\rho \dot{\phi}\left(\dot{z}{{\partial}_{z}{u}}+\dot{\rho}{{\partial}_{\rho}{u}}\right)\right],\\ \label{eq : elgeodmotion4 } \fl 0 & = & \frac{1}{u^5}\left\ { u^5 \ddot{z}+\left(qu\dot{t}-\dot{t}^2\right){{\partial}_{z}{u}}+u^4\left[(\dot{z}^2-\rho ^2 \dot{\phi}^2-\dot{\rho}^2){{\partial}_{z}{u}}+2\dot{z}\dot{\rho}{{\partial}_{\rho}{u}}\right]\right\}.\end{aligned}\ ] ] the coordinates @xmath54 and @xmath55 are cyclic so the integrals of motion are @xmath56 these are the energy of the particle , @xmath42 , and projection of its angular momentum on the @xmath0-axis , @xmath57 . for a vanishing charge of the test particle , these relations are consistent with those for an uncharged test particle @xcite . for simplicity , we first investigate the static solutions @xmath58 with spatial coordinates independent of the proper time , @xmath59 , to obtain @xmath60 a particularly simple solution with arbitrary @xmath61 reads @xmath62 and requires @xmath63 . in fact , since the charge - to - mass ratio of the test particle is the same as for the two black holes , this corresponds to an exact solution of the full einstein - maxwell equations apart from the field due to this additional point source itself . there are , however , additional static solutions , requiring @xmath64 and @xmath65 which means there is a single point @xmath66 on the @xmath0-axis where a test particle can be static . it can have an arbitrary charge since @xmath18 drops out of the equations . from @xmath67 we infer @xmath68 with the solution @xmath69 this relation , plotted in fig . [ fig : zequ ] , is identical to the newtonian expression ( [ eq : zequklas ] ) . for equal - mass black holes , the stationary point is located at the center while for @xmath70 it lies closer to the less massive black hole and it is identical to that of an uncharged particle as calculated in @xcite . -axis as a function of the black holes mass ratio @xmath71.,width=283 ] similarly to the classical case , there are periodic solutions limited to the @xmath0-axis when the equations of motion reduce to the conservation of energy and normalization of the 4-velocity as follows @xmath72 unlike the classical case , the test particle can reach the non - singular points @xmath73 now and cross the horizon . this , however , does not produce a periodic solution since the particle can never come back . as in the classical case , we need to discuss the ranges @xmath74 $ ] , @xmath75 $ ] , and @xmath76 separately . we define @xmath77 and search for the turning points with @xmath78 . we always have @xmath79 . we are thus looking for neighboring turning points separated by an interval with @xmath80 which means we are looking for local maxima . we now rescale the two masses and @xmath0 by @xmath12 so that @xmath81 , @xmath82 , and @xmath83 and drop the tildes . let us begin with @xmath84 where we can write @xmath85 ^ 2.\ ] ] we find @xmath86.\ ] ] the first bracket never vanishes while the second has no extremum as its derivative yields again the first bracket . therefore , the second bracket has at most one root within @xmath87 $ ] and @xmath88 has a single extremum here , which is then necessarily a minimum due to the fact it diverges to @xmath89 at @xmath90 . we thus can not have periodic motion of test particles here . the same line of reasoning also applies to the case @xmath91 so that we now proceed directly to the case @xmath92 $ ] . we have @xmath93 ^ 2\ ] ] and @xmath94.\ ] ] the first bracket has two roots now but only one of them lies within @xmath95 $ ] . similarly , the second bracket only has a single local extremum here and thus at most two roots . we conclude that @xmath88 has at most three local extrema yielding a single possible interval for periodic motion around the middle value . to summarize , oscillatory @xmath0-motion of test particles is only allowed between the hypersurfaces @xmath96 . in we give an example of such motion . investigating now ( [ between rescaled ] ) , we can express the turning points as follows @xmath97}}{2q_k } , k = 1 \ldots 4,\\ \delta & \equiv & m_1- m_2 , \sigma \equiv m_1 + m_2 , q_{1,2}=e - q+1,q_{3,4}=e - q-1.\end{aligned}\ ] ] as @xmath98 , the condition for oscillation is that all the turning points must lie between the black - hole horizons @xmath99 and @xmath100 , which is not always the case . this is generally a complicated system so we restrict the situation to black holes of equal masses . we find that @xmath101 . the oscillation takes place between the two turning points nearest to the origin . sums up the two possible cases . .regions of parameters where we can have @xmath0-axis oscillations for black holes of equal masses , @xmath102 . [ cols="<,<,<,<",options="header " , ] in the limit @xmath103 all the turning points approach @xmath104 where their distance from the singularity is of the same order as the mass of the black holes and , therefore , there is no newtonian limit and we can not compare ( [ relativistic_turning_points ] ) to ( [ classical_turning_points ] ) . we now assume @xmath105 constant so that the normalization and electrogeodesic equations read @xmath106 equations ( [ eq : kruhcas ] , [ eq : uhelcas ] ) can be easily integrated to yield @xmath107 and @xmath108 . if we require the bracket in ( [ eq : deudez ] ) to vanish and combine this with ( [ eq : rhocas ] ) , we only get the previous static solution with @xmath63 . therefore , we must have @xmath109 , which results in the following relation @xmath110^{3/2},\ ] ] which can be inverted ( for @xmath111 , see the next section ) as @xmath112 as in the newtonian case , these orbits only exist for @xmath113 . moreover , they only admit @xmath114 ( @xmath115 for equality ) and @xmath116 ( @xmath117 for equality ) . finally , we get the following formulae for @xmath118 and @xmath119 @xmath120 since these equations are generally quadratic , we expect up to two solutions for @xmath118 and @xmath119 ( the sign of @xmath118 only describes clockwise or counterclockwise motion ) . however , ( [ eq : kvadr1 ] ) and ( [ eq : kvadr2 ] ) become linear for the special case of @xmath121 , yielding @xmath122 with @xmath123 . this gives a particular set of radii for possible orbits which , interestingly , coincide with the positions of null circular geodesics as discussed in @xcite so we have the same radius with a photon or a charged massive particle ( at different velocities , of course ) . on the other hand , the general solution of ( [ eq : kvadr1 ] ) and ( [ eq : kvadr2 ] ) reads @xmath124 we conclude that there can be two different values of angular velocity for a given orbital radius since the electrogeodesic equation contains both linear and quadratic terms in @xmath119 with the linear one stemming from the lorentz force . this is very different from the case of a neutral test particle and also from the classical newtonian case with a charged particle where in both cases the orbital radius determines a single angular velocity . in the following discussion we specialize to trajectories within the equatorial plane . this fixes @xmath31 but , on the other hand , @xmath125 can be arbitrary and we ultimately have 3 independent parameters . if , instead , we left the equatorial plane then @xmath125 would be fixed by ( [ orbital radius ] ) but the masses of the two holes would be independent , resulting in 4 parameters which would render the discussion even more complicated without bringing any new type of solutions into the play and we thus only investigate the simpler case of equatorial orbits . we now investigate the special case of @xmath126 . it follows from ( [ eq : kruhprvni ] ) that this is only possible if the two black holes have equal masses , @xmath102 . the function @xmath49 then simplifies to @xmath127 in this case , the orbital radius , @xmath125 , can be arbitrary as ( [ eq : kruhprvni ] ) is an identity and ( [ orbital radius ] ) does not apply . in the asymptotically flat region @xmath128 ( the newtonian limit ) we find @xmath129 ^ 2\right).\end{aligned}\ ] ] and near the axis with @xmath130 ( strong relativistic effects if @xmath131 ) we have @xmath132 ^ 2\right),\\ a^2 \omega_{\pm}^2 & \approx & \frac{2 \frac{m}{a } ( 1\mp q)}{(1 + 2 \frac{m}{a})^3 } + o\left(\left[\frac{\rho}{a}\right]^2\right).\end{aligned}\ ] ] as we require @xmath133 , we only have a single solution with the upper sign in both asymptotic regions . the leading order of the angular velocity requires @xmath134 and it is consistent with the classical solution ( near the axis we need to assume a weak field with @xmath135 ) . therefore , asymptotically , a given radius of orbit only corresponds to a single orbital frequency . this , however , is not the case generally as we will discuss in the following section . this might have an observable effect for particles orbiting compact charged objects . it is of interest that this does not occur in the strongest field along the axis but farther outside . we proceed by discussing the regions where both solutions ( [ eq : kruhb ] ) , ( [ eq : kruhomegapm ] ) exist . this also occurs in reissner - nordstrm geometry as discussed in @xcite and mentioned in @xcite . our requirements are that @xmath136 be positive and @xmath137 non - negative . can be written as @xmath138 and we thus have a stronger condition @xmath139 since @xmath49 and @xmath136 are positive . the discussion is very complicated since we have 3 independent parameters appearing in our expressions ( after rescaling everything by @xmath12 ) . for this reason , we just summarize the results in three tables below . @xmath140 @xmath141 @xmath142 the values used in the tables are defined as follows @xmath143 @xmath144}{\left(4 a^4 - 4 a^2 \rho ^2+q^2 \rho ^4\right)^2}}.\ ] ] these are not constants as opposed to @xmath145but rather functions of @xmath125 , @xmath12 , and possibly @xmath18 ( in fact , we can again rescale everything using @xmath12 as our basic unit ) . we give below plots of these functions to understand the conditions appearing in the tables . as all the @xmath145 s diverge for @xmath146 we only get a limited range of admissible radii apart from the second row in . if @xmath147 we get the special case discussed in ( [ eq : special_case_gamma ] , [ eq : special_case_omega ] ) . the most interesting option is summarized in . these are the regions in the parameter space where we get both solutions and , therefore , two different frequencies for the same radius of a circular orbit . this , however , can only occur between @xmath148 and a finite radius given by the last column in . we thus conclude that we can only get both solutions simultaneously in the vicinity of the axis but not directly upon it . to examine the stability of circular orbits of neutral test particles and thus , e.g. , the possible existence of accretion disks in this system , we now investigate the geodesic deviation . we assume equatorial trajectories and study numerically the evolution of a ring of particles centered on the exact circular geodesic . the general equation of geodesic deviation is @xmath149 with @xmath150 the deviation from the central geodesic . in our case this now yields a set of four equations as follows @xmath151{\delta \rho } -\rho \omega u^5 \left(\rho u_{,\rho } + u \right){\delta \dot{\phi}}+\\ \fl + u \left ( u^5{\delta \ddot{\rho } } -\gamma u_{,\rho } { \delta \dot{t } } \right ) = 0 , \label{geodevrho}\\ \fl \gamma ~{\delta \ddot{t}}-\rho ^2 \omega u^4 { \delta \ddot{\phi } } = 0 , \label{geodevphi}\\ \fl { \delta \ddot{z}}=0 . \label{geodevz}\end{aligned}\ ] ] here , we already used the equation of a circular geodesic and combined equations for @xmath152 and @xmath153 . let us further assume @xmath154 , which means the ring stays within the equatorial plane @xmath31 . we investigate numerically the deformation of a ring with zero initial velocities in both @xmath55 and @xmath125 directions . we thus parametrize the initial conditions as @xmath155 which defines a circle in cylindrical coordinates with radius @xmath156 , centered at a distance @xmath125 from the origin along the @xmath125 axis . the remaining initial deviation values are set to zero . to depict deformations of the ring , we first transform the resulting deviations to a non - rotating frame @xmath157with the @xmath158 axis along the @xmath125 axis at zero proper time and then to a co - rotating cartesian frame @xmath159 . the new deviations are defined as @xmath160 thus , @xmath161 represents locally the @xmath125 direction and @xmath162 the @xmath55 direction at each point along the central geodesic . + + inspecting , we can see that the upper left and upper right orbits are stable while the other two diverge from the original configuration . this is just a hint at the underlying dynamics and one would need to explore a general perturbation of the geodesics . it is , however , obvious that there are both stable and unstable trajectories in the present spacetime . related to this , we remark that the hamilton - jacobi equation is separable for the classical motion in prolate spheroidal coordinates , which however does not translate into the general relativistic situation . we have investigated paths of charged test - particles in the background field of two extremally charged black holes held in equilibrium by their electromagnetic field . the spacetime is static and axially symmetric . we note that some of the trajectories we studied do not admit a newtonian limit since they do not avoid the strong - field regions near the black - hole horizons . it is of interest that as opposed to the newtonian case , there are regions with two different angular velocities for a single radius of the orbit , which might have observable consequences . this may be the case for all charged , static , asymptotically flat spacetimes as these approximate the reissner - nordstrm . however , it may happen that the double - frequency region is empty and one would need to study these cases separately . it would then be rather interesting to study the same problem in the kerr - newman stationary spacetime . it is to be seen whether the region of double frequencies would be stable under perturbations or not . jr was supported by student faculty grant of faculty of mathematics and physics , charles university in prague . mz was supported by the albert einstein center , project of excellence no . 14 - 37086 g funded by the czech science foundation . 99 chandrasekhar s 1983 _ the mathematical theory of black holes _ oxford university press , new york fujita r and hikida w 2009 * 26 * 135002 hod s 2013 b * 718 * 1552 kosti u 2012 * 44 * 1057 cruz n , olivares m and villanueva j r 2005 * 22 * 1167 hackmann e and lmmerzahl c 2008 * 100 * 171101 hackmann e and lmmerzahl c 2008 d * 78 * , 024035 villanueva j r , saavedra j , olivares m and cruz n 2013 _ astrophysics and space science _ * 344 * 437 bik j , stuchlk z and balek v 1989 _ bulletin of the astronomical institutes of czechoslovakia _ * 40 * 65 balek v , bik j and stuchlk z 1989 _ bulletin of the astronomical institutes of czechoslovakia _ * 40 * 133 olivares m , saavedra j , leiva c and villanueva j r 2011 _ modern physics letters a _ * 26 * 2923 grunau s and kagramanova v 2011 d * 83 * 044009 pugliese d , quevedo h and ruffini r 2011 d * 83 * 104052 majumdar s d 1947 * 72 * 930 papapetrou a 1947 _ proc . irish acad . _ * a51 * 191 hartle j b and hawking s w 1972 _ communications in mathematical physics _ * 26 * 87 wnsch a , mller t , weiskopf d and wunner g 2013 d * 87 * 024007

we investigate the ( electro-)geodesic structure of the majumdar - papapetrou solution representing static charged black holes in equilibrium . we assume only two point sources , imparting thus the spacetime axial symmetry . we study electrogeodesics both in and off the equatorial plane and explore the stability of circular trajectories via geodesic deviation equation . in contrast to the classical newtonian situation , we find regions of spacetime admitting two different angular frequencies for a given radius of the circular electrogeodesic . we look both at the weak- and near - field limits of the solution . we use analytic as well as numerical methods in our approach . _ keywords _ : electrogeodesic , majumdar - papapetrou , black hole , extreme reissner - nordstrm

1510.02314

the parallel chip - firing game or candy - passing game is a periodic automaton on graphs in which vertices , each of which contains some nonnegative number of chips , `` fire '' exactly one chip to each of their neighbors if possible . formally , let @xmath4 be an undirected graph with vertex set @xmath5 and edge set @xmath6 . define the _ parallel chip - firing game _ on @xmath4 to be an automaton governed by the following rules : * at the beginning of the game , @xmath7 chips are placed on each vertex @xmath8 in @xmath4 , where @xmath7 is a nonnegative integer . position _ of the parallel chip - firing game , denoted by @xmath9 , be the ordered pair @xmath10 containing the graph and the number of chips on each vertex of the graph . * at each _ move _ or _ step _ of the game , if a vertex @xmath8 has at least as many chips as it has neighbors , it will give ( _ fire _ ) exactly one chip to each neighbor . such a vertex is referred to as _ firing _ ; otherwise , it is _ non - firing_. all vertices fire simultaneously ( in parallel ) . we employ the notation of levine @xcite . let @xmath11 denote the step operator ; that is , @xmath12 is the position resulting after one step is performed on @xmath9 . let @xmath13 , and @xmath14 . we refer to @xmath15 as the position occurring _ after @xmath16 steps_. for simplicity , we limit our discussion to connected graphs . as the number of chips and number of vertices are both finite , there are a finite number of positions in this game . additionally , since each position completely determines the next position , it follows that for each initial position @xmath9 , there exist some positive integers @xmath17 such that for large enough @xmath18 , @xmath19 . we refer to the minimal such @xmath17 as the _ period _ @xmath20 of @xmath9 , and we refer to the set @xmath21 as one period of @xmath9 . also , we call the minimal such @xmath18 the _ transient length _ @xmath22 of @xmath9 . the parallel chip - firing game was introduced by bitar and goles @xcite in 1992 as a special case of the general chip - firing game posited by bjrner , lovsz , and shor @xcite in 1991 . they @xcite showed that the period of any position on a tree graph is 1 or 2 . in 2008 , kominers and kominers @xcite further showed that all connected graphs satisfying @xmath23 have period 1 ; they further established a polynomial bound for the transient length of positions on such graphs . their result @xcite that the set of all `` abundant '' vertices @xmath24 with @xmath25 stabilizes is particularly useful in simplifying the game . it was conjectured by bitar @xcite that @xmath26 for all games on all graphs @xmath4 . however , kiwi et . @xcite constructed a graph on which there existed a position whose period was at least @xmath27 , disproving the conjecture . still , it is thought that excluding particular graphs constructed to force long periods , most graphs still have periods that are at most @xmath28 . in 2008 , levine @xcite proved this for the complete graph @xmath1 . the parallel chip - firing game is a special case of the more general chip - firing game , in which at each step , a vertex is chosen to fire . the general chip - firing game , in turn , is an example of an _ abelian sandpile _ @xcite , and has been shown to have deep connections in number theory , algebra , and combinatorics , ranging from elliptic curves @xcite to the critical group of a graph @xcite to the tutte polynomial @xcite . bitar and goles @xcite observed that the parallel chip - firing game has `` nontrivial computing capabilities , '' being able to simulate the and , not , and or gates of a classical computer ; later , goles and margenstern @xcite showed that it can simulate any two - register machine , and therefore solve any theoretically solvable computational problem . finally , the parallel chip - firing game can be used to simulate a pile of particles that falls whenever there are too many particles stacked at any point ; this important problem in statistical physics is often referred to as the _ deterministic fixed - energy sandpile _ @xcite . the fixed - energy sandpile , in turn , is a subset of the more general study of the so - called _ spatially extended dynamical systems _ , which occur frequently in the physical sciences and even economics @xcite . such systems demonstrate the phenomenon of _ self - organized criticality _ , tending towards a `` critical state '' in which slight perturbations in initial position cause large , avalanche - like disturbances . self - organized critical models such as the abelian sandpile tend to display properties of real - life systems , such as @xmath29 noise , fractal patterns , and power law distribution @xcite . finally , the parallel chip - firing game is an example of a cellular automaton , the study of which have implications from biology to social science . in section 2 , we establish some lemmas about parallel chip - firing games on general simple connected graphs . we bound the number of chips on any single vertex in games with nontrivial period , define the notion of a complement position @xmath30 of @xmath9 and show that it has the same behavior as @xmath9 , and find a necessary and sufficient condition for a period to occur . then , in section 3 , we find , with proof , every possible period for the complete bipartite graph @xmath31 . we do so by first showing the only possible periods are of length @xmath32 or @xmath33 for @xmath34 , and then constructing games with such periods , proving our main result . finally , in section 4 , we construct positions on the complete @xmath2-partite graph @xmath35 with period @xmath17 for all @xmath36 . consider a simple connected graph @xmath4 . for each vertex @xmath8 in @xmath4 , let @xmath37 denote the number of firing neighbors @xmath38 of @xmath8 ; that is , the number of vertices @xmath38 neighboring @xmath8 satisfying @xmath39 . a step of the parallel chip - firing game on @xmath4 is then defined as follows : @xmath40 define a _ terminating _ position to be a position in which no vertices fire after finitely many moves . we begin our investigation by proving some lemmas limiting the number of chips on each vertex in a game with nontrivial period ( period greater than 1 ) . [ lem:2n-1 ] for sufficiently large @xmath18 , @xmath41 for all @xmath42 in all games with nontrivial period on a connected graph @xmath4 . kominers and kominers @xcite showed that if a vertex @xmath42 satisfies @xmath43 , then @xmath44 . they then showed that if , after sufficiently many steps @xmath18 , there still exists a vertex @xmath8 with @xmath45 , then all vertices must be firing from that step onward . since the period of a position is 1 if and only if either all or no vertices in @xmath4 are firing @xcite , @xmath46 is true for any game on @xmath4 with nontrivial period and sufficiently large @xmath18 . we further bound the number of chips on each vertex by generalizing a result of levine @xcite : [ lem : confined ] consider a vertex @xmath8 in position @xmath9 such that @xmath47 . then @xmath48 either @xmath49 or not . we consider the cases individually . if @xmath50 , then @xmath51 . so @xmath52 if instead @xmath53 , then @xmath54 . hence @xmath55 @xmath56 if a vertex @xmath8 satisfies @xmath57 , we call it _ confined_. furthermore , call a position confined if all vertices in the position are confined . note that for confined @xmath8 , @xmath58 lemmas [ lem:2n-1 ] and [ lem : confined ] imply that if @xmath59 , then @xmath60 is confined if @xmath61 , where @xmath22 is the transient length of @xmath9 ; that is , once the game reaches a position which repeats periodically , all subsequent positions are confined . we generally limit our discussion to confined positions to exclude positions with trivial periods . next , we define @xmath62 to be the indicator function of whether a vertex @xmath8 fires at step @xmath18 . we prove a lemma about positions that are equivalent , or have the same behavior , when acted upon by the step operator @xmath11 . [ lem : complement ] let the _ complement _ @xmath30 of a confined position @xmath9 be the position that results after replacing the @xmath7 chips on each vertex @xmath63 with @xmath64 chips . then @xmath65 . we begin by noticing that since @xmath9 is confined , each vertex @xmath8 has at most @xmath66 chips , so each vertex in @xmath30 has a nonnegative number of chips . observe that a vertex @xmath8 fires in @xmath30 exactly when it did not fire in @xmath9 . hence , @xmath67 , and all but @xmath37 neighbors will fire in @xmath68 . so @xmath69 this lemma means we may treat @xmath9 and @xmath30 as equivalent positions , as at any point during their firing , we may transform one into the other . this implies the following corollary : for all positions @xmath9 on @xmath4 , @xmath70 . next , we prove a proposition that characterizes a period of the game on any connected graph @xmath4 . for each position @xmath9 and vertex @xmath63 , let @xmath71 be the number of times @xmath8 fires in the first @xmath18 steps . [ prop : alleq ] the position @xmath9 on @xmath4 satisfies @xmath72 if and only if each vertex has fired the same number of times within those @xmath18 steps ; that is , iff for all vertices @xmath73 , @xmath74 if equation holds , then by equation , @xmath75 for all @xmath8 , so @xmath72 . conversely , if @xmath72 , consider the vertex @xmath76 such that @xmath77 is maximal . then , since @xmath78 for all vertices @xmath38 neighboring @xmath8 , @xmath79 but as @xmath80 , we see that @xmath81 must hold for all @xmath38 neighboring @xmath8 . since the graph is connected , we continue inductively through the entire graph to obtain equation . recall that a complete bipartite graph @xmath82 may be partitioned into two subsets of vertices , @xmath83 and @xmath84 , such that no edges exist among vertices in the same set , but every vertex in @xmath83 is connected to every vertex in @xmath84 . we refer to the sets @xmath85 as the _ sides _ of @xmath4 . define @xmath86 and @xmath87 . as stated above , bitar and goles @xcite showed that if no vertices or all vertices are firing , the period is 1 . we consider only games whose period is greater than 1 ; that is , at least one vertex is firing every turn , and not all vertices fire every turn . let @xmath88 and @xmath89 denote the number of vertices in @xmath83 and @xmath84 , respectively , that fire in @xmath9 . then , @xmath90 if @xmath91 , and @xmath92 if @xmath93 . notice that @xmath89 is the number of vertices in @xmath84 with at least @xmath94 chips , and @xmath88 is the number of vertices in @xmath83 with at least @xmath95 chips . let @xmath96 be the number of times any of the vertices in @xmath83 have fired in the first @xmath18 steps starting from , and including , @xmath15 , and define @xmath97 similarly . define @xmath98 and @xmath99 . without loss of generality , we prove facts about the vertices in @xmath83 , which also hold for vertices in @xmath84 . in the first @xmath18 steps , a vertex @xmath8 in @xmath83 fires a total of @xmath100 chips and receives @xmath101 chips . hence , @xmath102 next , we prove a lemma that bounds the number of times a vertex has fired once the position is confined . [ lem : diff1 ] let @xmath103 . if @xmath9 is confined , and @xmath104 , then for all @xmath105 , @xmath106 we prove this by induction on @xmath18 . the base case , @xmath107 , is straightforward : vertices @xmath8 and @xmath38 have each fired either 0 or 1 times . if @xmath8 fires after step @xmath108 , then @xmath109 chips , and @xmath38 also fires . now , assume @xmath110 if @xmath111 , then by equation , @xmath112 @xmath113 thus , if @xmath8 is ready to fire after step @xmath18 , then @xmath38 must be ready to fire also . it follows that @xmath114 otherwise , @xmath115 . then , since @xmath60 is confined from lemma [ lem : confined ] , by equation , @xmath116 @xmath117 by lemma [ lem : confined ] , since the degrees of both @xmath8 and @xmath38 are @xmath95 . so , if @xmath38 is ready to fire after step @xmath18 , so is @xmath8 , and equation again holds . from the above lemma , we can deduce the following : [ lem : ndiv ] if @xmath9 is confined and @xmath118 , then @xmath119 for all @xmath120 . let @xmath76 be the vertex in @xmath83 with @xmath121 minimal . by lemma [ lem : diff1 ] , for all @xmath91 , @xmath122 where @xmath123 . if @xmath124 is the number of vertices @xmath125 with @xmath126 , then @xmath127 since @xmath128 , we have @xmath129 . then @xmath130 because @xmath131 ; so @xmath132 for all @xmath91 . since @xmath133 , this implies @xmath119 for all @xmath91 . clearly , if all @xmath94 vertices in @xmath83 have fired the same number of times , then @xmath118 ; so we have found a necessary and sufficient condition for all vertices on the same side to fire the same number of times . but by proposition [ prop : alleq ] , a period is completed when all @xmath63 have fired the same number of times ; thus , we desire a relation between the sides that forces every vertex on both sides to fire the same number of times . our first step is the following lemma . [ lem : t1 ] if @xmath9 is confined , and @xmath134 for some positive integer @xmath32 , then @xmath135 for all @xmath136 . if @xmath93 is firing , then @xmath137 since @xmath9 is confined and @xmath8 is firing , @xmath138 ; and since @xmath139 is confined by lemma [ lem : confined ] , we have @xmath140 these two inequalities together imply that , for firing vertices @xmath8 , @xmath141 if @xmath136 is instead non - firing , then @xmath142 chips . @xmath143 is confined by lemma [ lem : confined ] , so @xmath144 ; since @xmath145 because @xmath8 is non - firing , we then deduce , similarly as above , that @xmath146 for non - firing vertices @xmath8 as well . therefore , for all @xmath136 , we have that @xmath147 so @xmath148 for all @xmath136 . if @xmath149 , then we can compute @xmath150 ; hence @xmath8 does not fire after step @xmath18 , and @xmath151 . if instead @xmath152 , then @xmath153 , so @xmath8 fires after step @xmath18 , and @xmath154 , and we are done . note that applying this lemma to @xmath15 also means @xmath155 . next , recalling the definition of @xmath156 in equation , we define @xmath157 for nonnegative integers @xmath16 and positive integers @xmath18 . note that by definition , @xmath158 applying lemmas [ lem : ndiv ] and [ lem : t1 ] to the position @xmath15 , we find that if @xmath159 , then @xmath160 for all vertices @xmath91 and all vertices @xmath161 . now , we give a sufficient condition for a period of a position on @xmath31 to occur . [ lem : periodt ] if @xmath9 is confined , and for some @xmath162 and @xmath163 , @xmath159 and @xmath164 for all @xmath91 , then @xmath165 if @xmath18 is chosen to be as small as possible . by lemma [ lem : ndiv ] applied to @xmath166 , since @xmath159 , @xmath167 for all @xmath91 . if for some @xmath162 , @xmath164 for all @xmath91 , then @xmath168 for all @xmath91 . but by equation , @xmath169 for all @xmath91 , @xmath161 . hence , @xmath170 for all vertices @xmath171 , which by proposition [ prop : alleq ] applied to @xmath172 implies @xmath173 , or @xmath174 . but @xmath18 is taken to be as small as possible , so @xmath165 . using this fact , we limit which periods are possible for games on @xmath31 . [ prop : t2 t ] if @xmath9 is confined and @xmath134 , then @xmath165 or @xmath175 . by iteratively applying lemma [ lem : t1 ] @xmath176 times , along with lemma [ lem : ndiv ] , we find that @xmath177 for all vertices @xmath91 and nonnegative even integers @xmath16 . expanding the sums @xmath178 and @xmath179 in terms of @xmath180 , we obtain @xmath181 for all nonnegative even integers @xmath16 and vertices @xmath91 . if @xmath164 for all @xmath91 and some @xmath182 , then the period is @xmath18 by lemma [ lem : periodt ] . otherwise , let @xmath183 be the set of vertices @xmath184 satisfying @xmath185 for all @xmath186 . the range of @xmath187 is @xmath188 , so this condition is equivalent to @xmath189 for all @xmath186 . we show that the period of @xmath9 is then @xmath190 . * case 1 . * @xmath18 is odd . consider some nonnegative integer @xmath191 , and let @xmath192 . if @xmath191 is even , let @xmath193 ; otherwise , let @xmath194 . if @xmath195 then @xmath196 and @xmath197 . but by equation , @xmath198 contradicting the assumption that @xmath185 for all @xmath186 . hence , @xmath199 for all nonnegative @xmath191 . this implies @xmath200 for all @xmath201 ; hence @xmath202 for all @xmath192 . since @xmath203 for other vertices @xmath204 , by lemma [ lem : periodt ] , @xmath175 . * case 2 . * @xmath18 is even . let @xmath192 . by equation , @xmath205 but by equation , @xmath206 for all @xmath192 . then as above , @xmath175 by lemma [ lem : periodt ] . more specifically , the following corollary holds : [ cor : lea ] let @xmath9 be a position on @xmath31 . if @xmath20 is odd , @xmath207 ; and if @xmath20 is even , @xmath208 . without loss of generality , let @xmath209 . if @xmath210 , @xmath211 . otherwise , @xmath59 . since @xmath212 , we may replace @xmath9 by @xmath213 and assume @xmath9 is confined by lemma [ lem : confined ] . by the pigeonhole principle , there must exist steps @xmath214 with @xmath215 . but then @xmath216 , so by proposition [ prop : t2 t ] applied to @xmath217 , @xmath218 or @xmath33 , where @xmath219 hence , if @xmath20 is odd , @xmath211 , and if @xmath20 is even , @xmath220 . finally , we characterize all possible periods for @xmath9 . [ prop : existpos ] there exist positions @xmath9 on @xmath82 with period @xmath32 and @xmath33 for all @xmath221 . without loss of generality , let @xmath209 . let @xmath222 be the vertices in @xmath83 , and @xmath223 be the vertices in @xmath84 . let @xmath32 be a positive integer such that @xmath224 . we represent each position @xmath9 on @xmath4 by two vectors @xmath225 @xmath226 consider the following position @xmath227 , which we claim has period @xmath32 : @xmath228 @xmath229 @xmath230 and @xmath231 fire , so @xmath232 is represented by @xmath233 @xmath234 we can see that the vertices @xmath235 with @xmath236 satisfy @xmath237 for @xmath238 . so , @xmath239 follows upon applying proposition [ prop : alleq ] , noting that after @xmath32 steps , each vertex has fired exactly once . hence , @xmath227 has period @xmath32 . next , consider the following position @xmath240 , which we claim has period @xmath33 : @xmath241 @xmath242 note that , if at any point @xmath243 , then @xmath244 , because @xmath245 and @xmath246 have the same neighbors . so , @xmath247 is represented by @xmath248 @xmath249 and @xmath250 is represented by @xmath251 @xmath234 we can see that for @xmath252 , the vertex in @xmath4 that fires ( has @xmath95 chips if it is in @xmath83 , or @xmath94 chips if it is in @xmath84 ) in position @xmath253 is @xmath254 so , after @xmath33 steps , every vertex will have fired once , and by proposition [ prop : alleq ] , @xmath255 . it remains to construct initial positions with period 1 or 2 . the trivial game with no chips on any vertex has period 1 , while the initial position where each vertex in @xmath83 has @xmath95 chips , and each vertex in @xmath84 has 0 chips , can be easily checked to have period 2 . thus , all periods @xmath256 , and @xmath257 , are achievable . combining our results , we obtain our main theorem . a nonnegative integer @xmath17 is a possible period of a position @xmath9 of the parallel chip - firing game on @xmath31 if and only if @xmath258 by corollary [ cor : lea ] , no period lengths may lie outside the sets in ; and in proposition [ prop : existpos ] , we have constructed positions with all such periods . we again use the vector notation from above to represent the positions of a parallel chip - firing game on the complete @xmath2-partite graph @xmath259 formed by joining the anticliques @xmath260 ; let the vertices in @xmath261 be @xmath262 for each @xmath263 . without loss of generality , we will assume @xmath264 . as above , we represent a position on @xmath4 by the set of vectors @xmath265 below is a representation of a position which has period @xmath266 for all @xmath267 and @xmath268 . note that a vertex in @xmath269 fires when it has at least @xmath270 chips ; here @xmath271 is the degree of any vertex in @xmath269 . for our construction , we let @xmath272 we now show that this position indeed has period @xmath266 . let @xmath273 be the set of all firing vertices in @xmath274 . it can be checked that @xmath275 @xmath276 for @xmath277 ; and @xmath278 for @xmath279 , encompassing steps @xmath280 through @xmath281 ; @xmath282 for @xmath283 ; and @xmath284 for @xmath285 . ( in fact , each vertex @xmath63 fires exactly when it contains @xmath286 chips . ) the latter two categories describe which vertices fire during steps @xmath287 through @xmath288 . but after this @xmath289 step , every vertex in @xmath4 has fired exactly once ; the last to fire is @xmath290 . hence , @xmath291 by proposition [ prop : alleq ] , and the period is @xmath292 as desired . this means all periods from @xmath293 to @xmath294 are achievable , as @xmath295 ranges from @xmath108 to @xmath296 and @xmath32 ranges from @xmath293 to @xmath297 . as an example , consider the following position on the graph @xmath298 with period @xmath299 : @xmath300 for this position , @xmath301 , and its predicted period length is @xmath302 as desired . for several graphs , a proof of bitar s conjecture that @xmath26 for all parallel chip - firing games on those graphs would be interesting ; we proved the conjecture for the complete bipartite graph . though we have constructed many periods of games on complete @xmath2-partite graphs in section 4 , there exist periods longer than those detailed . for example , take the following position on @xmath303 , which has period 5 : @xmath304 though positions with these larger periods are more difficult to characterize generally , bitar s conjecture still appears to be true for complete @xmath2-partite graphs . moreover , bounding the periods of positions on vertex - regular graphs and more general bipartite graphs are directions for further research . by doubling the length of each cycle in the graph used in the counterexample by kiwi et . @xcite , we find a counterexample on a graph containing only even cycles , that is , for the general bipartite graph . we would also like to determine which periods less than the bound are possible . levine @xcite related period lengths of games on the complete graph to the _ activity _ , defined as @xmath305 . on the other hand , we believe that period lengths are related to lengths of subcycles ( closed paths ) of the graph @xmath4 ; in particular , we conjecture that any period length of a game on @xmath4 is either a divisor of the order of some subcycle of @xmath4 , or perhaps the least common multiple of the orders of some disjoint subcycles of @xmath4 . this agrees with known results for the tree graph @xcite , complete graph @xcite , and now the complete bipartite graph . our numerical experiments have also verified this conjecture for cycle graphs and complete @xmath32-partite graphs ; in fact , my correspondence with zhai @xcite has produced a proof of this conjecture for the cycle graph . another interesting direction to pursue is observing the implications of `` reducing '' the parallel chip - firing game by removing as many chips as possible from each vertex without affecting their firing pattern ( without changing @xmath156 for all @xmath42 and @xmath306 ) . this reduction may simplify some games into being more approachable by induction . besides studying period lengths of parallel chip - firing games , an examination of the transient length of games on certain graphs would be useful in modeling real - world phenomena . studying transient positions would also help uncover what attributes determine whether a position is within a period or not , and bounding the transient length would make for more efficient computation of the period length of games on complex graphs . chip - firing games on lattices and tori have been used as cellular automaton models of the deterministic fixed - energy sandpile ( see @xcite ) . since most studies of sandpiles have been concerned with asymptotic measures such as the `` activity , '' bounding the period length of such models could serve as a measure of the fidelity of the model to the real world . the author would like to thank my mentor , yan zhang of the massachusetts institute of technology , for his teaching , guidance , and support , the center for excellence in education and the research science institute for sponsoring my research , and dr . john rickert for tips on writing and lecturing . the author would also like to thank dr . lionel levine , scott kominers , paul kominers , daniel vitek , brian hamrick , dr . dan teague , patrick tenorio , alex zhai , and dr . ming ya jiang for valuable pointers and advice on this problem and this paper .

the parallel chip - firing game is a periodic automaton on graphs in which vertices `` fire '' chips to their neighbors . in 1989 , bitar conjectured that the period of a parallel chip - firing game with @xmath0 vertices is at most @xmath0 . though this conjecture was disproven in 1994 by kiwi et . al . , it has been proven for particular classes of graphs , specifically trees ( bitar and goles , 1992 ) and the complete graph @xmath1 ( levine , 2008 ) . we prove bitar s conjecture for complete bipartite graphs and characterize completely all possible periods for positions of the parallel chip - firing game on such graphs . furthermore , we extend our construction of all possible periods for games on the bipartite graph to games on complete @xmath2-partite graphs , @xmath3 , and prove some pertinent lemmas about games on general simple connected graphs . tian - yi jiang

1003.0943

being able to read news from other countries and written in other languages allows readers to be better informed . it allows them to detect national news bias and thus improves transparency and democracy . existing online translation systems such as _ google translate _ and _ _ bing translator _ _ are thus a great service , but the number of documents that can be submitted is restricted ( google will even entirely stop their service in 2012 ) and submitting documents means disclosing the users interests and their ( possibly sensitive ) data to the service - providing company . for these reasons , we have developed our in - house machine translation system onts . its translation results will be publicly accessible as part of the europe media monitor family of applications , @xcite , which gather and process about 100,000 news articles per day in about fifty languages . onts is based on the open source phrase - based statistical machine translation toolkit moses @xcite , trained mostly on freely available parallel corpora and optimised for the news domain , as stated above . the main objective of developing our in - house system is thus not to improve translation quality over the existing services ( this would be beyond our possibilities ) , but to offer our users a rough translation ( a `` gist '' ) that allows them to get an idea of the main contents of the article and to determine whether the news item at hand is relevant for their field of interest or not . a similar news - focused translation service is `` found in translation '' @xcite , which gathers articles in 23 languages and translates them into english . `` found in translation '' is also based on moses , but it categorises the news after translation and the translation process is not optimised for the news domain . europe media monitor ( emm ) gathers a daily average of 100,000 news articles in approximately 50 languages , from about 3,400 hand - selected web news sources , from a couple of hundred specialist and government websites , as well as from about twenty commercial news providers . it visits the news web sites up to every five minutes to search for the latest articles . when news sites offer rss feeds , it makes use of these , otherwise it extracts the news text from the often complex html pages . all news items are converted to unicode . they are processed in a pipeline structure , where each module adds additional information . independently of how files are written , the system uses utf-8-encoded rss format . inside the pipeline , different algorithms are implemented to produce monolingual and multilingual clusters and to extract various types of information such as named entities , quotations , categories and more . onts uses two modules of emm : the named entity recognition and the categorization parts . named entity recognition ( ner ) is performed using manually constructed language - independent rules that make use of language - specific lists of trigger words such as titles ( president ) , professions or occupations ( tennis player , playboy ) , references to countries , regions , ethnic or religious groups ( french , bavarian , berber , muslim ) , age expressions ( 57-year - old ) , verbal phrases ( deceased ) , modifiers ( former ) and more . these patterns can also occur in combination and patterns can be nested to capture more complex titles , @xcite . in order to be able to cover many different languages , no other dictionaries and no parsers or part - of - speech taggers are used . to identify which of the names newly found every day are new entities and which ones are merely variant spellings of entities already contained in the database , we apply a language - independent name similarity measure to decide which name variants should be automatically merged , for details see @xcite . this allows us to maintain a database containing over 1,15 million named entities and 200,000 variants . the major part of this resource can be downloaded from http://langtech.jrc.it/jrc-names.html all news items are categorized into hundreds of categories . category definitions are multilingual , created by humans and they include geographic regions such as each country of the world , organizations , themes such as natural disasters or security , and more specific classes such as earthquake , terrorism or tuberculosis , articles fall into a given category if they satisfy the category definition , which consists of boolean operators with optional vicinity operators and wild cards . alternatively , cumulative positive or negative weights and a threshold can be used . uppercase letters in the category definition only match uppercase words , while lowercase words in the definition match both uppercase and lowercase words . many categories are defined with input from the users themselves . this method to categorize the articles is rather simple and user - friendly , and it lends itself to dealing with many languages , @xcite . in this section , we describe our statistical machine translation ( smt ) service based on the open - source toolkit moses @xcite and its adaptation to translation of news items . * which is the most suitable smt system for our requirements ? * the main goal of our system is to help the user understand the content of an article . this means that a translated article is evaluated positively even if it is not perfect in the target language . dealing with such a large number of source languages and articles per day , our system should take into account the translation speed , and try to avoid using language - dependent tools such as part - of - speech taggers . inside the moses toolkit , three different statistical approaches have been implemented : _ phrase based statistical machine translation _ ( pbsmt ) @xcite , _ hierarchical phrase based statistical machine translation _ @xcite and _ syntax - based statistical machine translation _ @xcite . to identify the most suitable system for our requirements , we run a set of experiments training the three models with europarl v4 german - english @xcite and optimizing and testing on the news corpus @xcite . for all of them , we use their default configurations and they are run under the same condition on the same machine to better evaluate translation time . for the syntax model we use linguistic information only on the target side . according to our experiments , in terms of performance the hierarchical model performs better than pbsmt and syntax ( 18.31 , 18.09 , 17.62 bleu points ) , but in terms of translation speed pbsmt is better than hierarchical and syntax ( 1.02 , 4.5 , 49 second per sentence ) . although , the hierarchical model has the best bleu score , we prefer to use the pbsmt system in our translation service , because it is four times faster . * which training data can we use ? * it is known in statistical machine translation that more training data implies better translation . although , the number of parallel corpora has been is growing in the last years , the amounts of training data vary from language pair to language pair . to train our models we use the freely available corpora ( when possible ) : europarl @xcite , jrc - acquis @xcite , dgt - tm , opus @xcite , se - times @xcite , tehran english - persian parallel corpus @xcite , news corpus @xcite , un corpus @xcite , czeng0.9 @xcite , english - persian parallel corpus distributed by elra and two arabic - english datasets distributed by ldc . this results in some language pairs with a large coverage , ( more than 4 million sentences ) , and other with a very small coverage , ( less than 1 million ) . the language models are trained using 12 model sentences for the content model and 4.7 million for the title model . both sets are extracted from english news . for less resourced languages such as farsi and turkish , we tried to extend the available corpora . for farsi , we applied the methodology proposed by @xcite , where we used a large language model and an english - farsi smt model to produce new sentence pairs . for turkish we added the movie subtitles corpus @xcite , which allowed the smt system to increase its translation capability , but included several slang words and spoken phrases . * how to deal with named entities in translation ? * news articles are related to the most important events . these names need to be efficiently translated to correctly understand the content of an article . from an smt point of view , two main issues are related to named entity translation : ( 1 ) such a name is not in the training data or ( 2 ) part of the name is a common word in the target language and it is wrongly translated , e.g. the french name `` bruno le maire '' which risks to be translated into english as `` bruno mayor '' . to mitigate both the effects we use our multilingual named entity database . in the source language , each news item is analysed to identify possible entities ; if an entity is recognised , its correct translation into english is retrieved from the database , and suggested to the smt system enriching the source sentence using the xml markup option in moses . this approach allows us to complement the training data increasing the translation capability of our system . * how to deal with different language styles in the news ? news title writing style contains more gerund verbs , no or few linking verbs , prepositions and adverbs than normal sentences , while content sentences include more preposition , adverbs and different verbal tenses . starting from this assumption , we investigated if this phenomenon can affect the translation performance of our system . we trained two smt systems , @xmath0 and @xmath1 , using the europarl v4 german - english data as training corpus , and two different development sets : one made of content sentences , news commentaries @xcite , and the other made of news titles in the source language which were translated into english using a commercial translation system . with the same strategy we generated also a title test set . the @xmath1 used a language model created using only english news titles . the news and title test sets were translated by both the systems . although the performance obtained translating the news and title corpora are not comparable , we were interested in analysing how the same test set is translated by the two systems . we noticed that translating a test set with a system that was optimized with the same type of data resulted in almost 2 blue score improvements : title - testset : 0.3706 ( @xmath1 ) , 0.3511 ( @xmath0 ) ; news - testset : 0.1768 ( @xmath1 ) , 0.1945 ( @xmath0 ) . this behaviour was present also in different language pairs . according to these results we decided to use two different translation systems for each language pair , one optimized using title data and the other using normal content sentences . even though this implementation choice requires more computational power to run in memory two moses servers , it allows us to mitigate the workload of each single instance reducing translation time of each single article and to improve translation quality . to evaluate the translation performance of onts , we run a set of experiments where we translate a test set for each language pair using our system and google translate . lack of human translated parallel titles obliges us to test only the content based model . for german , spanish and czech we use the news test sets proposed in @xcite , for french and italian the news test sets presented in @xcite , for arabic , farsi and turkish , sets of 2,000 news sentences extracted from the arabic - english and english - persian datasets and the se - times corpus . for the other languages we use 2,000 sentences which are not news but a mixture of jrc - acquis , europarl and dgt - tm data . it is not guarantee that our test sets are not part of the training data of google translate . each test set is translated by google translate - translator toolkit , and by our system . bleu score is used to evaluate the performance of both systems . results , see table [ results ] , show that google translate produces better translation for those languages for which large amounts of data are available such as french , german , italian and spanish . surprisingly , for danish , portuguese and polish , onts has better performance , this depends on the choice of the test sets which are not made of news data but of data that is fairly homogeneous in terms of style and genre with the training sets . the impact of the named entity module is evident for arabic and farsi , where each english suggested entity results in a larger coverage of the source language and better translations . for highly inflected and agglutinative languages such as turkish , the output proposed by onts is poor . we are working on gathering more training data coming from the news domain and on the possibility of applying a linguistic pre - processing of the documents . .[results ] automatic evaluation . [ cols="<,^,^ " , ] the translation service is made of two components : the connection module and the moses server . the connection module is a servlet implemented in java . it receives the rss files , isolates each single news article , identifies each source language and pre - processes it . each news item is split into sentences , each sentence is tokenized , lowercased , passed through a statistical compound word splitter , @xcite , and the named entity annotator module . for language modelling we use the kenlm implementation , @xcite . according to the language , the correct moses servers , title and content , are fed in a multi - thread manner . we use the multi - thread version of moses @xcite . when all the sentences of each article are translated , the inverse process is run : they are detokenized , recased , and untranslated / unknown words are listed . the translated title and content of each article are uploaded into the rss file and it is passed to the next modules . the full system including the translation modules is running in a 2xquad - core with intel hyper - threading technology processors with 48 gb of memory . it is our intention to locate the moses servers on different machines . this is possible thanks to the high modularity and customization of the connection module . at the moment , the translation models are available for the following source languages : arabic , czech , danish , farsi , french , german , italian , polish , portuguese , spanish and turkish . our translation service is currently presented on a demo web site , see figure [ fig::demo ] , which is available at http://optima.jrc.it / translate/. news articles can be retrieved selecting one of the topics and the language . all the topics are assigned to each article using the methodology described in [ cat ] . these articles are shown in the left column of the interface . when the button `` translate '' is pressed , the translation process starts and the translated articles appear in the right column of the page . the translation system can be customized from the interface enabling or disabling the named entity , compound , recaser , detokenizer and unknown word modules . each translated article is enriched showing the translation time in milliseconds per character and , if enabled , the list of unknown words . the interface is linked to the connection module and data is transferred using rss structure . in this paper we present the optima news translation system and how it is connected to europe media monitor application . different strategies are applied to increase the translation performance taking advantage of the document structure and other resources available in our research group . we believe that the experiments described in this work can result very useful for the development of other similar systems . translations produced by our system will soon be available as part of the main emm applications . the performance of our system is encouraging , but not as good as the performance of web services such as google translate , mostly because we use less training data and we have reduced computational power . on the other hand , our in - house system can be fed with a large number of articles per day and sensitive data without including third parties in the translation process . performance and translation time vary according to the number and complexity of sentences and language pairs . the domain of news articles dynamically changes according to the main events in the world , while existing parallel data is static and usually associated to governmental domains . it is our intention to investigate how to adapt our translation system updating the language model with the english articles of the day . the authors thank the jrc s optima team for its support during the development of onts . c. callison - burch , and p. koehn and c. monz and k. peterson and m. przybocki and o. zaidan . 2009 . . proceedings of the joint fifth workshop on statistical machine translation and metricsmatr , pages 1753 . uppsala , sweden . p. koehn and f. j. och and d. marcu . proceedings of the 2003 conference of the north american chapter of the association for computational linguistics on human language technology , pages 4854 . edmonton , canada . p. koehn and h. hoang and a. birch and c. callison - burch and m. federico and n. bertoldi and b. cowan and w. shen and c. moran and r. zens and c. dyer and o. bojar and a. constantin and e. herbst 2007 . . proceedings of the annual meeting of the association for computational linguistics , demonstration session , pages 177180 . columbus , oh , usa . r. steinberger and b. pouliquen and a. widiger and c. ignat and t. erjavec and d. tufi and d. varga . proceedings of the 5th international conference on language resources and evaluation , pages 21422147 . genova , italy . m. turchi and i. flaounas and o. ali and t. debie and t. snowsill and n. cristianini . proceedings of the european conference on machine learning and knowledge discovery in databases , pages 746749 . bled , slovenia .

we propose a real - time machine translation system that allows users to select a news category and to translate the related live news articles from arabic , czech , danish , farsi , french , german , italian , polish , portuguese , spanish and turkish into english . the moses - based system was optimised for the news domain and differs from other available systems in four ways : ( 1 ) news items are automatically categorised on the source side , before translation ; ( 2 ) named entity translation is optimised by recognising and extracting them on the source side and by re - inserting their translation in the target language , making use of a separate entity repository ; ( 3 ) news titles are translated with a separate translation system which is optimised for the specific style of news titles ; ( 4 ) the system was optimised for speed in order to cope with the large volume of daily news articles .

1401.2943

the acceleration of charged particles to high energies in the solar corona is related to flares , which reveal the dissipation of magnetically stored energy in complex magnetic field structures of the low corona , and to coronal mass ejections ( cmes ) , which are large - scale , complex magnetic - field - plasma structures ejected from the sun . cmes can drive bow shocks , and their perturbation of the coronal magnetic field can also give rise to magnetic reconnection , where energy can be released in a similar way as during flares . when several cmes are launched along the same path , a faster cme may overtake a slower preceding one , and the two cmes can merge into a single structure . for this phenomenon @xcite introduced the term _ cme cannibalism_. the cme - cme interaction was found associated with a characteristic low - frequency continuum radio emission . @xcite interpreted this type of activity as the radio signature of non - thermal electrons originating either during reconnection between the two cmes or as the shock of the second , faster cme travels through the body of the first ( see * ? ? ? * for a numerical study of two interacting coronal mass ejections ) . in this paper we use radio diagnostics to study electron acceleration during a complex solar event broadly consisting of two stages , each associated with a distinct episode of a flare and with a fast cme , which occurred in close temporal succession on 17 january 2005 . the cmes interacted at a few tens of r@xmath0 . both the flare / cme events and the cme interaction were accompanied by radio emission , which is used here to study electron acceleration scenarios . energetic electrons in the corona and interplanetary space are traced by their dm - to - km - wave radio emission , mostly excited at or near the electron plasma frequency . the emission provides a diagnostic of the type of the exciter and its path from the low corona ( cm - dm wavelengths ) to 1 au ( km wavelengths ) . radio emissions from exciters moving through the corona appear in dynamic spectra as structures exhibiting a drift in the time frequency domain . the drift rate depends on their speed and path , resulting in a variety of bursts . type iii bursts trace the path of supra thermal electrons guided by magnetic structures . they appear , on dynamic spectra , as short ( lasting from a fraction of a second at dm - waves to a few tens of minutes at km - waves ) structures with fast negative drift , ( @xmath1 ; see for example * ? ? ? this corresponds to anti - sunward propagation of the electrons through regions of decreasing ambient density at speeds @xmath2 ( e.g. , * ? ? ? sunward travelling beams produce reverse drift bursts ( rs bursts ) , and beams propagating in closed loops emit type u or j bursts comprising a succession of an initial drift towards lower frequencies and a more or less pronounced rs burst . type ii bursts are more slowly drifting bursts ( @xmath3 ; see , for example , table a.1 in * ? ? ? * ) excited by electrons accelerated at travelling shocks and emitting in their upstream region . finally broadband dm - m wave continuum emission that may last over several minutes or even hours ( type iv burst ) is ascribed to electrons confined in closed coronal magnetic structures . the reader is referred to the reviews in @xcite , @xcite , @xcite and @xcite for more detailed accounts of the radio emission by non thermal electrons in the corona . lllll * event * & * time * & * characteristics * & * remarks * + & * ut * & & + sxr start & 06:59 & & ar10720 ( n15@xmath4 w25@xmath4 ) + type iv & 08:40 & 3.0 - 630 mhz & ar10720 + cme@xmath5 & 09:00 & & lift - off + * sxr stage 1 * & 09:05 & & + first cm & 09:05 & & rstn 15400 mhz + burst start & & & + type iii@xmath5 & 09:07 - 09:28 & 0.2 - 630 mhz & ar10720 + type ii@xmath5 & 09:11 & 0.2 - 5 mhz & ar10720 + h@xmath6 start & 09:13 & 3b & kanz , ar10720 + cme@xmath5 & 09:30 & 2094 km sec@xmath7 & on c2 + hxr start & 09:35:36 & & rhessi number 5011710 + cme@xmath8 & 09:38 & & lift - off + * sxr stage 2 * & 09:42 & & end sxr stage 1 + second cm & 09:43 & & rstn 15400 mhz + burst start & & & + type iii@xmath8 & 09:43 - 09:59 & 0.2 - 630 & ar10720 + hxr peak & 09:49:42 & 7865 counts sec@xmath7 & + type ii@xmath8 & 09:48 & 0.2 - 8 mhz & ar10720 + sxr peak & 09:52 & x3.8 & end sxr stage 2 + cme@xmath8 & 09:54 & 2547 km sec@xmath7 & on c2 + first rise & 10:00 & 38 - 315 kev & ace / epam + electron flux & & & + sxr end & 10:07 & & ar720 + hxr end & 10:38:52 & 53152112 total counts & rhessi + second rise & 12:00 & 38 - 315 kev & ace / epam + electron flux & & & + type iii@xmath9 & 11:37 & 0.5 mhz & cme@xmath5 , cme@xmath8 merge at 37 r@xmath0 + & & & type ii@xmath8 overtakes type ii@xmath5 + h@xmath6 end & 11:57 & & kanz + type iv end & 15:24 & 3.0 - 630 mhz & ar10720 + line centre ( top left ) and in the wing , observed at kanzelhhe observatory ( courtesy m. temmer ) . solar north is at the top , west on the right . the two snapshots at the top show the active region before the flare under discussion , the two bottom images show two instants during the stages 1 and 2 , respectively . these stages were associated with the disappearance of the filaments labelled ` f1 ' and ` f2 ' . ] . bottom : two frames of the 09:54:05 halo cme with back - extrapolated lift off at 09:38:25 ut and plane - of - the - sky speed 2547 km sec@xmath7 . solar north is at the top , west on the right . ] the 17 january 2005 event consisted of a complex flare , two very fast coronal mass ejections ( cmes ) , and intense and complex soft x - ray ( sxr ) and radio emission . in all radiative signatures two successive stages can be distinguished . the cmes were launched successively from neighbouring regions of the corona and interacted in interplanetary space . the sequence of the observed energetic phenomena is summarized in table [ t ] and described , in detail , in the following subsections . figure [ fig_kanz ] displays snapshots in the h@xmath6 line obtained from the kanzelhhe solar observatory ( courtesy m. temmer ; see also @xcite , their figure 2 , for details on the evolution of the h@xmath6 flare ribbons ) . the only major active region on the disk is noaa 10720 in the north - western quadrant ( n15@xmath4 w25@xmath4 ) . it displayed nearly uninterrupted activity since the early hours of 17 january 2005 . the most conspicuous event was a 3b h@xmath6 flare reported by kanzelhhe 09:16 - 11:57 ut . this flare proceeded successively in two different parts of ar 10720 , as shown in the two snapshots of the bottom panel . the first part of the event , referred to as stage 1 " ( illustrative snapshot at 09:13 ut ) , is seen in the eastern part of the active region , close to the sunspots . it is associated with the temporary disappearance or eruption of the filament labelled ` f1 ' in the upper right panel . two major flare ribbons are distinguished in the snapshot at 09:13 ut : a narrow band essentially in the east - west direction and a broader north - southward oriented region . the significant offset of the two ribbons with respect to the neutral line shows the eruption of a strongly sheared magnetic field . after about 09:35 ut the brightest emission is seen in the western part of the active region ( stage 2 " ; see snapshot at 09:54 ut ) , together with the eruption of another filament ` f2 ' ( or of a different part of the filament whose northern section erupted before ) . the brightening consisted of two essentially parallel flare ribbons , which were connected by post flare loops in later snapshots ( not shown here ) . these two stages of the event were also found in the soft x - ray ( sxr ) and radio emissions , as will be discussed below . two cmes were observed in close succession . a sequence of difference images from the large angle and spectrometric coronagraph ( lasco ) aboard the soho spacecraft @xcite is displayed in figure [ cmes ] : the first cme ( henceforth cme@xmath5 ) is seen in the image at 09:30 ut in the north - western quadrant . while it travelled through the corona , the second , broader cme ( cme@xmath8 ) appeared underneath ( image at 09:54 ut ) . the most conspicuous features of both cmes are seen above the north - western limb , but both were labelled halo cmes in the lasco cme catalog @xcite . speeds of , respectively , 2094 and 2547 km s@xmath7 were derived from linear fits to the trajectories of their fronts published in the cme catalogue . formally the cme fronts described by the fits intersected near 12:32 ut at a heliocentric distance of about 38 r@xmath0 . the statistical error of the speeds of the cme fronts and their liftoff times , derived from the abovementioned linear least - squares fit to the measured heliocentric distances , leads to an uncertainty of @xmath103 h in the time of intersection . this uncertainty stems from the fact that the two height - time trajectories are nearly parallel . we will show in sect . [ cme2cme ] that cme interaction actually occurred well before the formal time of intersection . of course a single instant of interaction between two complex cmes is fictitious anyway . an overview of the complex radio event is given in figure [ f3 ] . there we present the dynamic flux density spectrum of the radio bursts in the 650 mhz-20 khz range ( heliocentric distance @xmath11 1.1 r@xmath0to 1 au ) using combined recordings of the _ appareil de routine pour le traitement et lenregistrement magntique de linformation spectrale _ ( artemis - iv ) solar radio - spectrograph @xcite and the _ wind_/waves experiment @xcite . several other time histories are superposed on the dynamic spectrum : * dashed lines display the approximate frequency - time trajectories of the two cme fronts , using the density model of @xcite , which describes well the coronal density behavior in the large range of distances from low corona to interplanetary space : @xmath12 + the linear fits to the height - time trajectories of the cme fronts in the lasco images were converted to frequency - time tracks of fundamental ( black line ) and harmonic ( red line ) plasma emission . * the solid blue curve displays the sxr time history ( 0.1 - 0.8 nm ) , using goes on line data ( http//www.sel.noaa.gov / ftpmenu / indices ) , describing thermal emission from the flare - heated plasma . * the red curve is the microwave time history at 15.4 ghz , produced by non thermal electrons ( energies @xmath13100 kev ) in magnetic fields of a few hundred g ; these were obtained from the san vito solar observatory of the radio solar telescope network ( rstn ) @xcite . the two stages of the flare identified in the h@xmath6 observations in figure [ fig_kanz ] correspond to two distinct events of energy release seen in the sxr and microwave time profiles ( figure [ f3 ] ) . the sxr time profile had an initial smooth increase between 06:59 ut and 09:05 ut . subsequently the sxr flux rose slightly faster until 09:45 ut ( stage 1 ) , and even faster ( stage 2 ) until the x3.8 peak at 09:52 ut . the gradual rise in stage 1 and the faster rise in stage 2 were each accompanied by strong microwave bursts . the second burst was also observed in hard x - rays by rhessi @xcite . the dominant features in the dynamic spectrum observed by _ wind_/waves at frequencies below 2 mhz are two groups of type iii bursts , labelled iii@xmath5 and iii@xmath8 . they occurred in association with the sxr and microwave emissions of stages 1 and 2 , respectively , and with the two different parts of the h@xmath6 flare . the two type iii groups occurred near the extrapolated liftoff times of the two cmes . radio images taken by the nanay radioheliograph ( nrh ; * ? ? ? * ) show that the sources are located in the north - western quadrant near the flaring active region . hence both flare episodes were efficient accelerators of electrons that escaped to the interplanetary space along open magnetic field lines rooted at or near the flare site . the second type iii group ( type iii@xmath8 ) was followed by a more slowly drifting narrow - band burst ( type ii , labelled ii@xmath8 ) produced by a coronal shock wave . upon closer inspection the spectrum suggests that similar drifting features can also be associated with the first flare episode , although the association is less evident . we label these bursts ii@xmath5 in figure [ f3 ] . since the two cmes are extremely fast , they are expected to drive shock waves in the corona . the observed type ii emission can be compared with the dashed curves in figure [ f3 ] , which track fundamental ( black ) and harmonic ( red ) emission expected from the trajectory of the cme front and the coronal density model . it is clear that this density model is only indicative , especially in the perturbed corona through which travels the second cme ( see discussion in subsection [ m ] ) . we therefore associate type ii@xmath5 and ii@xmath8 to the bow shocks of the two cmes , although other interpretations , like shocks on the flanks or shocks from a driver related to the flare , are not excluded . the dm - m wave emission consisted of a type iv continuum , the metre wave counterparts of the dekametre - hectometre ( dh ) type iii groups and of the type ii bursts . the type iv continuum started near 08:40 ut during the initial smooth increase of the sxr flux before stage 1 . it was first visible as a grey background in the dynamic spectrum , and became progressively more intense . it dominated the metre wave spectrum during and after type iii@xmath8 , and gradually penetrated to lower frequencies , down to 5 mhz . images in the eit 195 channel @xcite and in the 164 - 432 mhz range taken by the nrh indicate that the thermal ( soft x - rays ) and non thermal ( radio ) emissions all originated near noaa ar 10720 . in the time interval from the start of the type iv burst to the start of stage 1 a wealth of fine structures was recorded ( see * ? ? ? * ) . from the high - resolution observations in the 200 - 500 mhz range ( see figures [ fs1 ] , [ fs2 ] for example ) it appears that most bursts are broadband pulsations . other fine structures of type iv emission such as spikes , fiber bursts and zebra pattern appear occasionally ( see * ? ? ? * for a description of fine structure of type iv emission ) . during type iii@xmath5 the spectral character of the radio emission was clearly different at frequencies below and above the inferred frequency - time track of the cme@xmath5 ( see fig . [ f3 ] ) . on the low - frequency side of the track strong type iii bursts were prominent after about 09:22 ut . they were preceded by a less regular emission , which @xcite label complex type iii bursts " because of its varying flux density across the spectrum . the metre wave counterpart on the high - frequency side of the estimated cme track consisted of a succession of spectral fine structures on the time scale of seconds , with different spectral characteristics superposed on the type iv continuum , followed after 09:11 ut by the high - frequency extension of the dekametre - hectometre ( dh ) type iii group iii@xmath5 . a more detailed view of the difference spectrum is given in the top panel of figure [ f4 ] , while high resolution images of the fine structures are in figure [ fs1 ] . among these fine structures were broadband pulsations , bursts with ordinary and reverse drift , and fiber bursts due to whistlers travelling upwards in the corona ( see figure [ fs1 ] , _ e.g. _ , 09:16:20 - 09:16:45 ut ) . the variety of these bursts shows the acceleration and partial trapping of electron populations in the corona well behind the front of the cme . indeed , few of the well - identified bursts above 100 mhz seem to continue into the 30 - 70 mhz range . it was only near the end of type iii@xmath5 ( @xmath11 08:18 ut ) that metre wave type iii bursts appeared as systematic high - frequency extensions of the type iii bursts observed below 2 mhz . type iii@xmath8 started at 09:43 ut , together with the second microwave burst , near the back - extrapolated lift - off of cme@xmath8 ( 09:38 ut ) and the onset of stage 2 of the sxr burst . a close look at the dynamic spectrum ( bottom panel of figure [ f4 ] ) reveals negative overall drifts below 100 mhz , while burst groups with positive overall drift prevailed above 130 mhz . the high - resolution spectrogram in the 300 - 400 mhz range ( fig . [ fs2 ] ) shows a wealth of individual bursts with different drift rates and zebra pattern . these bursts show again , like in stage 1 of the event , that the high - frequency bursts are produced by an accelerator below the cme front , while low - frequency bursts show the start of the prominent dh type iii bursts . well after the decay of the sxr and microwave emission a third group of bursts ( iii@xmath9 in figure [ f3 ] ) is identified ( near 11:30 ut ) , with unusually low starting frequency ( 0.6 mhz ) , pointing to an acceleration of the emitting electrons at unusually great height . a more detailed view of the low - frequency radio spectrum of this burst group and the preceding groups iii@xmath5 and iii@xmath8 is given by the dynamic spectrum as observed by the _ thermal noise receiver _ ( tnr ) of _ wind_/waves in the left panel of figure [ fig_tnr_20050117 ] . the narrow - band short bursts near 32 khz are langmuir wave packets . together with the fainter continuous band on which they are superposed they indicate how the electron plasma frequency evolves at the _ wind _ spacecraft . at the time of iii@xmath9 it is about 35 khz . using a standard interplanetary density model , where the electron plasma frequency decreases as the inverse of the heliocentric distance @xmath14 , the starting frequency of iii@xmath9 implies @xmath15 r@xmath0 for fundamental plasma emission , and @xmath16 r@xmath0 for the harmonic . from the lasco observations and the uncertainties resulting from the straight - line fits to the cme front trajectories , the heliocentric distances of the cme fronts at 11:30 ut are , respectively , @xmath17 r@xmath0 and @xmath18 r@xmath0 . the burst group iii@xmath9 is hence consistent with harmonic emission from the vicinity of the cme fronts . this points to a close relationship of this episode of electron acceleration with the interaction of the two cmes . comparison of the three groups of type iii bursts in the tnr spectrum of figure [ fig_tnr_20050117 ] shows that type iii@xmath9 is much shorter than the previous type iii bursts . it has intrinsic structure that indicates a group of bursts . the low - frequency cutoff is near the plasma frequency at the spacecraft at that time . more details are seen in the selected time profiles in the right panel , plotted together with the peak times of the burst at each frequency in the 35 - 256 khz range ( open triangles ) . the time profiles show that the peak times are not distinguishable over a large part of the frequency spectrum with 1-min integrated data , but that the centre of gravity of the brightest feature shifts to later times at the lower frequencies . we determined the maximum of the burst at each of the tnr frequencies where it is well defined , using a parabolic interpolation between the observed maximum and its two neighbours . it is this interpolated time which is plotted by an open triangle . the peak time spectrum resembles a type iii burst especially at the lower frequencies . the peak time delay is merely 1 min between 250 and 50 khz , but becomes clear at frequencies below 50 khz . for comparison , the peak time delay between 50 and 250 khz is 42 min during the previous burst iii@xmath8 . the frequency drift rate is hence faster than 3 khz / s during iii@xmath9 , as compared to 0.08 khz / s during iii@xmath8 . because of the morphological similarity in the dynamic spectrum , and despite the different drift rates , we assume in the following that the type iii@xmath9 bursts are indeed produced by electron beams travelling in the anti - sunward direction from the acceleration region . since the emission extends rapidly to the plasma frequency at the spacecraft , we conclude that the electron beams do not travel within the cmes , but escape rapidly from the acceleration region in the vicinity of the cme fronts to 1 au . this means that they must travel along pre - existing open solar wind field lines . to the extent that drift rates reflect the speed of the exciter , the fast frequency drift of the type iii@xmath9 bursts implies that the exciter speed is higher than during the preceding groups iii@xmath5 and iii@xmath8 . the total and differential radio spectrum observed by _ wind_/waves are shown in figure [ iii3 ] . the spectrum shows a chain of narrow - band emissions with negative frequency drift , indicating the type ii bursts , followed by the high - frequency part of type iii@xmath9 between 11:28 and 11:40 ut . the spectrum in figure [ f3 ] leaves it open if this is the continuation of the first type ii burst ( ii@xmath5 ) , presumably associated with cme@xmath5 , or whether it contains contributions from both cmes . the starting frequency of the type iii bursts is similar to the type ii frequency when extrapolated to the time of the type iii bursts . this is consistent with the type iii electron beams radiating in the upstream region , like the shock - acclerated electrons emitting the type ii burst . one may go one step further and consider this coincidence as a hint that the electron beams are accelerated at the shock , as argued in cases where type iii bursts clearly emanate from type ii lanes ( see * ? ? ? * ; * ? ? ? we will come back to this problem in the discussion . the energetic particle data were obtained from the _ advanced composition explorer _ ( ace ) spacecraft . we use high - resolution intensities of magnetically deflected electrons ( de ) in the energy range 38 - 315 kev measured by the b detector of the ca60 telescope of the epam experiment ( _ electron , proton and alpha monitor _ ; * ? ? ? * ) on board ace , and measurements of the angular distributions in the energy range 45 - 312 kev detected by the sunward looking telescope lefs60 . in figure [ o1 ] ( top ) an overview of the 20-min averaged differential intensities of four channels is presented for the interval 15 - 20 january 2005 . ar 10720 produced numerous solar events prior to as well as on 17 january 2005 @xcite ; in response to this solar activity , a sequence of energetic electron intensity enhancements was observed . the electron intensities are observed to reach their maximum values during this period following the solar events on 17 january 2005 . figure [ o1 ] ( bottom ) shows 1-min averaged deflected electron intensities ( 38 - 315 kev ) for the time interval 04:00 - 20:00 ut on 17 january 2005 . the intensities measured during the time interval 04:00 - 08:00 ut for each electron channel have been averaged to obtain a pre - event background ( denoted by horizontal lines in figure [ o1 ] ) . we defined the onset time of the event at ace for all energy channels as the time when the intensities get @xmath19 above the background and continue to rise from then on . using this criterion , we found the first significant rise of the electron intensities to occur at 10:00 ut . no velocity dispersion was observed , probably because the high pre - event ambient intensities ( see top panel figure [ o1 ] ) mask the onset of the electron event ( see * ? ? ? * for a similar case ) . the spiky increase observed at about 10:40 ut is probably due to x - ray contamination . we found no evidence of a magnetic structure influencing the intensity profiles , which indicates the observed time intensity changes are not due to spatial structures crossing over the spacecraft , but are most likely dominated by temporal effects . twenty - minute averaged representative snapshots of pitch angle distributions ( pads ) are shown as inserts in the bottom panel of figure [ o1 ] . normalized differential electron intensity is plotted versus the cosine of the pitch angle . statistically significant pads are detected first at about 11:00 ut . the pad snapshot denoted as * a * in figure [ o1 ] indicates that immediately after the onset of the event unidirectional electron anisotropies are observed . based on the observations available we can not distinguish whether the electrons were directed sunward or antisunward , since the magnetic field ( not shown ) was directed dominantly transverse to the radial direction during this period ( see figure 7 in * ? ? ? * for a similar case ) . however , it is highly likely that the observed electrons are streaming away from the sun in response to the intense solar activity during this period . furthermore the type iii bursts clearly indicate that electrons stream away from the sun ( towards regions of lower density ) . we can not be certain that the electron population measured at ace / epam is the high - energy counterpart of the electron beams emitting the radio waves , yet the overall timing suggests this . in the work by @xcite a detailed analysis of the plasma and magnetic field measurements at 1 au by ace during the period 16 - 26 january 2005 was carried out . this includes the period under study in the present paper . a forward shock was detected at 07:12 ut on 17 january 2005 . we have denoted the arrival time of this shock by a vertical solid line in figure [ o1 ] ( bottom ) . the analysis has shown that after the passage of this shock an unusually extended region exhibiting sheath - like characteristics is observed for @xmath11 1.5 day with highly variable magnetic field magnitude and directions and typical to high proton temperatures ( see figure 3 of * ? ? ? * also ruth skoug , ace / swepam pi team , private communication ) . this region is probably related to two cmes ejected in close temporal sequence at the sun on 15 january ( see figure 2 of * ? ? ? subsequently , at @xmath11 23:00 ut on 18 january 2005 the arrival of an icme at earth is detected , ending at about 02:30 ut on 20 january . the energetic electrons observed at 1 au analyzed in this work were thus detected in the region with disturbed magnetic field characteristics following the shock on 17 january 2005 . for the purposes of this work , as an approximation , we calculated that the nominal parker spiral for the measured solar wind speed of 620 @xmath20 ( ace / swepam ) at the time of the rise of the electron intensity had a length of about 1.05 au and was rooted near w 37@xmath4 on the hypothetical solar wind source surface at 2.5 r@xmath0 . this longitude is not contradictory with an active region at w 25@xmath4 , because non - radial coronal field lines can easily establish a connection @xcite . supposing that the early rise of the intensities was produced by the faster electrons in an energy channel moving along the interplanetary field line with 0 pitch angle , we estimate a travel time of about 15 min , which indicates the electrons were released from about 09:45 ut at the sun . this corresponds to a photon arrival time at 09:53 ut . given that our estimate of the electron rise gives only an upper limit , we consider that this electron release is related to type iii@xmath5 and type iii@xmath8 ( table [ t ] ) , but can not give a more detailed identification . the electron intensities are subsequently observed to exhibit a significant and more abrupt rise at all energies . extrapolation of this second rise to the pre - event background intensities indicates a start at @xmath11 12:00 ut . the electron pads ( inset * b * in figure [ o1 ] ) indicate stronger unidirectional anisotropies are observed in association with this electron enhancement , which provides evidence for fresh injection of energetic electrons between the sun and the spacecraft . the outstanding radio emission near this time is the group of fast type iii bursts during the cme interaction , type iii@xmath9 . if the electrons are accelerated at a heliocentric distance of about 25 r@xmath0 , the path travelled to the spacecraft along the nominal parker spiral is 0.92 au for the solar wind speed measured at the time of type iii@xmath9 ( 800 km / s ) . the inferred upper limit of the solar release time is 11:46 ut for 100 kev electrons . photons released at that time at 25 r@xmath0(0.12 au ) will reach the earth about 7 min later . since the high background implies that our estimations of the electron rise times are upper limits , we consider that this timing is consistent with the type iii@xmath9 burst group near 11:37 ut ( table [ t ] ) . this process is hence accompanied by the acceleration of copious amounts of electrons that escape to the vicinity of the earth . on 17 january 2005 two flare / cme events occurred in close temporal succession in the same active region . both cmes had very high projected speed , above 2000 km s@xmath7 , but the second one was faster than the first and eventually overtook it . the cmes were associated with two successive filament eruptions and sxr enhancements in the same active region . since the filament eruptions occurred at neighbouring places in the parent active region , the cmes probably resulted from the eruption of neighbouring parts of the same overall magnetic configuration . the soft x - ray characteristics of the two successive events were different : a slow monotonic rise to moderate flux during the first event , and a more impulsive rise to the x3.8 level ( @xmath21 ) during the second . both events had a conspicuous microwave burst , but the first one was stronger than the second , contrary to the soft x - rays . the second burst was also seen in hard x - rays by rhessi , which was in the earth s shadow during the first burst . [ [ evidence - for - evolving - acceleration - regions - in - the - corona - during - the - flares ] ] evidence for evolving acceleration regions in the corona during the flares ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ the decametre - to - hectometre wave spectra in the two stages looked similar , with bright groups of type iii bursts signalling the escape of electron beams at heliocentric distances beyond 2 @xmath22 . but radio emission from lower heights shows distinctive differences that point to an evolving acceleration region , _ i.e _ either an acceleration region which progresses through the corona or a number of acceleration sites activated in succession . shock acceleration was clearly at work during both stages of the flare , as shown by the type ii emission . the strong type iii bursts at the low - frequency side of the estimated spectral track of the cme front could also be ascribed to the acceleration at the type ii shock . the presence of type iii bursts with negative drift at higher frequencies and the fine structures of the type iv continuum show , however , that at the time when the shock traveled through the high corona , other acceleration regions were active at lower altitude , as is usually the case during complex type iii bursts at decametre and longer waves ( see * ? ? ? * and references therein ) . the type iii bursts ( iii@xmath5 ) might then not start near the cme front , but at higher frequencies , and be interrupted by interactions of the electron beams with the turbulence near the front of cme@xmath5 . this is a frequently quoted interpretation of complex features in type iii bursts , both at kilometric @xcite and decametric wavelengths ( e.g. * ? ? ? * ) . in the second type iii group ( iii@xmath8 ) the overall frequency drift of the low - resolution spectrum ( figure [ f4 ] ) was positive . the persistence of the metric type iv burst , which suggested acceleration in the lower corona rather than at the shock during the first stage , is again a likely indication of an accelerator that was distinct from the cme shock , and acted in addition to the shock , at lower altitude . this is consistent with an interpretation of the electron acceleration in terms of reconnection in the corona behind the cme @xcite . new evidence for this interpretation has recently been provided by @xcite using uv and white light coronographic diagnostics along with radio data . decametric - hectometric radio emission as a signature of cme interaction was discussed in some detail in two event studies @xcite . in both events the radio emission had a limited bandwidth , and was referred to as a continuum . in the present case this emission is likely a set of type iii bursts and was therefore labelled type iii@xmath9 . the starting frequency and the timing of these bursts are consistent with the idea that the electrons are accelerated while the faster following cme catches up with the slower preceding one . the association of cme interaction with particle acceleration has been ascribed by @xcite to acceleration by the shock of the second cme as it traverses the previous one . problems of this interpretation were discussed by @xcite . another important question is how the preceding cme could lead to a strengthening of the shock of the following one . this problem is still more evident in the 17 january 2005 event , because here the two cmes are already extremely fast and likely to drive strong shocks even in the ambient solar wind . their relative speed , however , is rather slow , so that efficient acceleration by the shock of the second cme is not expected in the first cme . another important feature is that the type iii@xmath9 bursts extend to the plasma frequency at the spacecraft . the electrons hence can not propagate in closed magnetic structures related to the cmes . the accelerator must release the electron beams onto open solar wind - type field lines . an alternative scenario to acceleration at the cme shock is again magnetic reconnection . one can surmise that these rapid cmes were preceded by sheath regions with strong magnetic fields of interplanetary origin , draped around the cme front . these regions are favourable for magnetic reconnection ( see the overview by * ? ? ? * and references therein ) . the high pressure in the sheath of the second cme will be further enhanced when its progression is slowed down by the previous cme . this makes the configuration favourable to magnetic reconnection involving open solar wind field lines and strong magnetic fields , and allows one to understand qualitatively why accelerated electrons escape immediately towards the outer heliosphere . in this scenario the type iii emission is expected to start close to the cme , in the upstream region . this can explain why the starting frequency is close to the frequency of the type ii burst , without implying that the electron beams were themselves accelerated at the shock . the flare / cme events under discussion were clearly related with enhanced fluxes of near - relativistic electrons at 1 au . the peak intensity measured by ace , of order @xmath23 in the 38 - 53 kev range , make the event comparable to the most intense ones of the sample studied by @xcite , as seen in their figure 3a . the cme speed is well above the speeds of the cmes identified in that sample @xcite . since the energetic electrons were observed at 1 au within the region exhibiting sheath - like characteristics following the shock on 17 january 2005 , it is difficult to estimate electron travel times and to relate the _ in situ _ measurements to solar processes . but the observations strongly suggest that successive intensity increases are first due to the coronal acceleration in the flare / cme event , and then to an episode during the interaction of the two cmes . the escape of these electrons to ace confirms the view discussed above that the electrons can not have been accelerated in the body of the first cme , even if a shock driven by the second one passed through it . neither can they originate from reconnection between closed magnetic field lines of the two cmes . the electrons must rather be accelerated in regions from where they have ready access to solar wind magnetic field lines . this is consistent with a common acceleration of the mildly relativistic electrons and the electron beams at lower energies that produce the type iii@xmath9 emission . this work was supported in part by the university of athens research center ( elke / ekpa ) . the authors appreciate discussions with and assistance of c. caroubalos , c. alissandrakis . and they would also like to thank an anonymous referee for many useful comments on the original manuscript h@xmath6 data were provided by the kanzelhhe observatory , university of graz , austria by m. temmer . the soho / lasco data used here are produced by a consortium of the naval research laboratory ( usa ) , max - planck - institut fuer aeronomie ( germany ) , laboratoire dastronomie ( france ) , and the university of birmingham ( uk ) . the soho / lasco cme catalog is generated and maintained at the cdaw data center by nasa and the catholic university of america in cooperation with the naval research laboratory . soho is a project of international cooperation between esa and nasa . klk acknowledges the kind hospitality of the solar radio astronomy group at the university of athens .

on 17 january 2005 two fast coronal mass ejections were recorded in close succession during two distinct episodes of a 3b / x3.8 flare . both were accompanied by metre - to - kilometre type - iii groups tracing energetic electrons that escape into the interplanetary space and by decametre - to - hectometre type - ii bursts attributed to cme - driven shock waves . a peculiar type - iii burst group was observed below 600 khz 1.5 hours after the second type iii group . it occurred without any simultaneous activity at higher frequencies , around the time when the two cmes were expected to interact . we associate this emission with the interaction of the cmes at heliocentric distances of about 25 r@xmath0 . near - relativistic electrons observed by the epam experiment onboard ace near 1 au revealed successive particle releases that can be associated with the two flare / cme events and the low - frequency type - iii burst at the time of cme interaction . we compare the pros and cons of shock acceleration and acceleration in the course of magnetic reconnection for the escaping electron beams revealed by the type iii bursts and for the electrons measured _ in situ_.

1101.5759

biometric authentication systems are becoming prevalent in access control and in consumer technology . in such systems , the user submits their user name and his / her biometric sample , which is compared to the stored biometric template associated with this user name ( one - to - one matching ) . the popularity of biometric - based systems stems from a popular belief that such authentication systems are more secure and user friendly than systems based on passwords . at the same time , the use of such systems raises concerns about the security and privacy of the stored biometric data . unlike passwords , replacing a compromised biometric trait is impossible , since biometric traits ( e.g. , face , fingerprint , and iris ) are considered to be unique . therefore , the security of biometric templates is an important issue when considering biometric based systems . moreover , poor protection of the biometric templates can have serious privacy implications on the user , as discussed in previous work @xcite . various solutions have been proposed for protecting biometric templates ( e.g , @xcite ) . the most prominent of them are secure sketch @xcite and fuzzy extractors @xcite . unfortunately , these solutions are not well adopted in practice . the first reason for this is the tradeoff between security and usability due to the degradation in recognition rates @xcite . the second reason is related to the use of tokens that are required for storing the helper data , thus affecting usability . finally , these mechanisms rely on assumptions which are hard to verify ( e.g. , the privacy guarantees of secure sketch assume that the biometric trait is processed into an almost full entropy string ) . in this work we propose a different approach for protecting biometric templates called _ honeyfaces_. in this approach , we hide the real biometric templates among a very large number of synthetic templates that are indistinguishable from the real ones . thus , identifying real users in the system becomes a very difficult ` needle in a haystack ' problem . at the same time , honeyfaces does not require the use of tokens nor does it affect recognition rate ( compared to a system that does not provide any protection mechanism ) . furthermore , it can be integrated with other privacy solutions ( e.g. , secure sketch ) , offering additional layers of security and privacy . for the simplicity of the discussion , let us assume that all biometric templates ( real and synthetic ) are stored in a _ biometric `` password file''_. our novel approach enables the size of this file to be increased by several orders of magnitudes . such inflation offers a 4-tier defense mechanism for protecting the security and privacy of biometric templates with no usability overhead . namely , honeyfaces : * reduces the risk of the biometric password file leaking ; * increases the probability that such a leak is detected online ; * allows for post - priori detection of the ( biometric ) password file leakage ; * protects the privacy of the biometrics in the case of leakage ; in the following we specify how this mechanism works and its applications in different settings . the very large size of the `` password file '' improves the * resilience of system against its exfiltration*. we show that one can inflate a system with 270 users ( 180 kb `` password file '' ) into a system with up to @xmath0 users ( 56.6 tb `` password file '' ) . obviously , exfiltrating such a huge amount of information is hard . moreover , by forcing the adversary to leak a significantly larger amount of data ( due to the inflated file ) he either needs significantly more time , or has much higher chances of being caught by intrusion detection systems . thus , the file inflation facilitates in * detecting the leakage * while it happens . the advantages of increasing the biometric `` password file '' can be demonstrated in networks whose outgoing bandwidth is very limited , such as air - gap networks ( e.g. , those considered in @xcite ) . such networks are usually deployed in high - security restricted areas , and thus are expected to employ biometric authentication , possibly in conjunction with other authentication mechanisms . once an adversary succeeds in infiltrating the network , he usually has a very limited bandwidth for exfiltration , typically using a physical communication channel of limited capacity ( with a typical bandwidth of less than 1 kbit / sec ) . in such networks , inflating the size of the database increases the resilience against exfiltration of the database . namely , exfiltrating 180 kb of information ( the size of a biometric `` password file '' in a system with 270 users ) takes a reasonable time even in low bandwidth channels compared with 56.6 tb ( the size of the inflated biometric `` password file '' ) , which takes more than 5.2 days for exfiltration in 1 gbit / sec , 14.4 years in 1 mbit / sec , or about 14,350 years from an air - gaped network at the speed of 1 kbit / sec . similarly to honeywords @xcite , the fake accounts enable * detection of leaked files*. namely , by using two - server authentication settings , each authentication query is first sent to the server that contains the inflated password file . once the first server authenticates the user , it sends a query to the second server that contains only the legitimate accounts , thus detecting whether a fake account was invoked with the `` correct '' credentials . this is a clear evidence that despite the hardness of exfiltration , the password file ( or a part of it ) was leaked . all the above guarantees heavily rely on the inability of the adversary to isolate the real users from the fake ones . we show that this task is nearly impossible in various adversarial settings ( when the adversary has obtained access to the password file ) . we also show that running membership queries to identify a real user by matching a facial image from an external source to the biometric `` password file '' is computationally infeasible . we analyze the robustness of the system in the worst case scenario in which the adversary has the facial images of all users except one and he tries to locate the unknown user among the synthetic faces . we show that the system protects the privacy of the users in this case too . to conclude , honeyfaces * protects the biometric templates of real users * in all settings that can be protected . the addition of a large number of synthetic faces may raise a concern about the degradation of the authentication accuracy . however , we show that this is not the case . the appearance of faces follows a multivariate gaussian distribution , which we refer to in this article as _ face - space _ , the parameters of which are learned from a set of real faces , including the faces of the system users . we sample synthetic faces from the same generative model constraining them to be at a certain distance from real and other synthetic faces . we selected this distance to be sufficiently large that new samples of real users would not collide with the synthetic ones . even though such a constraint limits the number of faces the system could produce , the number remains very large . using a training set of 500 real faces to build the generative face model , we successfully created @xmath0 synthetic faces . our honeyfaces system requires a method for generating synthetic faces which satisfies three requirements : * the system should be able to generate a ( very ) large number of unique synthetic faces . * these synthetic faces should be indistinguishable from real faces . * the synthetic faces should not affect the authentication accuracy of real users . these requirements ensure that the faces of the real users can hide among the synthetic ones , without affecting recognition accuracy . there are two lines of research related to the ideas introduced in this paper . one of them is honeyobjects , discussed in section [ sec : sub : sub : honeyobjects ] . the second one , discussed in section [ sec : sub : sub : biometric_synthesis ] , is the synthesis of biometric traits . honeyobjects are widely used in computer security . the use of honeypot users ( fake accounts ) is an old trick used by system administrators . login attempts to such accounts are a strong indication that the password file has leaked . later , the concept of honeypots and honeynets was developed @xcite . these tools are used to lure adversaries into attacking decoy systems , thus exposing their tools and strategies . honeypots and honeynets became widely used and deployed in the computer security world , and play an important role in the mitigation of cyber risks . recently , juels and rivest introduced honeywords @xcite , a system offering decoy passwords in addition to the correct one . a user first authenticates to the main server using a standard password - based authentication in which the server can keep track of the number of failed attempts . once one of the stored passwords is used , the server passes the query to a second server which stores only the correct password . identification of the use of a decoy password by the second server , suggests that the password file has leaked . obviously , just like in honeypots and honeynets , one needs to make sure that the decoy passwords are sampled from the same space as the real passwords ( or from a space as close as possible ) . to this end , there is a need to model passwords correctly , a non - trivial task , which was approached in several works @xcite . interestingly , we note that modeling human faces was extensively studied and very good models exist ( see the discussion in section [ sec : sub : generating ] ) . in honeywords it is a simple matter to change a user s password once if it has been compromised . clearly it is not practicable to change an individual s facial appearance . thus , when biometric data is employed , the biometric `` password file '' itself should be protected . honeyfaces protects the biometric data by inflating the `` password file '' such that it prevents leaks , which is a significant difference between honeywords and honeyfaces . another decoy mechanism suggested recently , though not directly related to our work , is honey encryption @xcite . this is an encryption procedure which generates ciphertexts that are decrypted to different ( yet plausible ) plaintexts when decrypted under one of a few wrong keys , thus making ciphertext - only exhaustive search harder . artificial biometric data are understood as biologically meaningful data for existing biometric systems @xcite . biometric data synthesis was suggested for different biometric traits , such as faces ( e.g. , @xcite ) , fingerprints ( e.g , @xcite ) and iris ( e.g. , @xcite ) . the main application of biometrics synthesis has been augmenting training sets and validation of biometric identification and authentication systems ( see @xcite for more information on synthesis of biometrics ) . synthetic faces are also used in animation , facial composite construction , and experiments in cognitive psychology . making realistic synthetic biometric traits has been the main goal of all these methods . however , the majority of previous work did not address the question of distinguishing the synthetic samples from the real ones . the work in iris synthesis @xcite analyses the quality of artificial samples by clustering synthetic , real , and non - iris images into two clusters iris / non - iris . such a problem definition is obviously sub - optimal for measuring indistinguishability . supervised learning using real and synthetic data labels has much better chances of success in separating between real and synthetic samples than unsupervised clustering ( a weaker learning algorithm ) into iris / non - iris groups . these methods also used recognition experiments , in which they compare the similarity of the associated parameters derived from real and synthetic inputs . again , this is an indirect comparison that shows the suitability of the generation method for evaluating the quality of the recognition algorithm , but it is not enough for testing the indistinguishability between real and synthetic samples . in fingerprints , it was shown that synthetic samples generated by different methods could be distinguished from the real ones with high accuracy @xcite . subsequent methods for synthesis @xcite showed better robustness against distinguishing attacks that use statistical tests based on @xcite . several methods for synthetic facial image generation @xcite provide near photo - realistic representations , but to the best of our knowledge , the question of indistinguishability between real and synthetic faces has not been addressed before . section [ sec : sub : generating ] describes , with justification , the method we use for generating honeyfaces . in section [ sec : system ] we present our setup for employing honeyfaces in a secure authentication system . the privacy analysis of honeyfaces , discussed in section [ sec : privacyanalysis ] , shows that the adversary can not obtain private biometric information from the biometric `` password file '' . section [ sec : securityanalysis ] analyses the additional security offered by inflating the `` password file '' . we conclude the paper in section [ sec : summary ] . biometric systems take a raw sample ( usually an image ) and process it to extract features or a representation vector , robust ( as much as possible ) to changes in sampling conditions . in the honeyfaces system , we have an additional requirement the feature space should allow sampling of artificial `` outcomes '' ( faces ) in _ large numbers_. these synthetic faces will be used as the passwords of the fake users . different models have been proposed for generating and representing faces including , active appearance models @xcite , 3d deformable models @xcite , and convolutional neural networks @xcite . such models have been used in face recognition , computer animation , facial composite construction ( an application in law enforcement ) , and experiments in cognitive psychology . among these models we choose the active appearance model @xcite for implementing the honeyfaces concept . an active appearance model is a parametric statistical model that encodes facial variation , extracted from images , with respect to a mean face . this work has been extended and improved in many subsequent papers ( e.g. , @xcite ) . in this context the word ` active ' refers to fitting the appearance model ( am ) to an unknown face to subsequently achieve automatic face recognition @xcite . am can also be used with random number generation to create plausible , yet completely synthetic , faces . these models achieve near photo - realistic representations that preserve identity , although are less effective at modeling hair and finer details , such as birth marks , scars , or wrinkles which exhibit little or no spatial correspondence between individuals . our choice of using the am for honeyfaces is motivated by two reasons : 1 ) the representation of faces within an am is consistent with human visual perception and hence also consistent with the notion of face - space @xcite . in particular , perceptual similarity of faces is correlated with distance in am space @xcite . 2 ) am is a well understood model used previously in face synthesis ( e.g. @xcite ) . alternative face models may also be considered , provided a sufficient number of training images ( as the functions of the representation length ) is available to adequately model the facial variation within the population of real faces . recent face recognition technology uses deep learning ( dl ) methods as they provide very good representations for verification . however , the image reconstruction quality from dl representation is still far from being satisfactory for our application . ams describe the variation contained within the training set of faces , used for its construction . given that this set spans all variations associated with identity changes , the am provides a good approximation to any desired face . this approximation is represented by a point ( or more precisely , localized contiguous region ) within the face - space , defined by the _ am _ _ coefficients_. the distribution of am coefficients of faces belonging to the same ethnicity are well approximated by an independent , multivariate , gaussian probability density function @xcite ( for example , see figure [ fig : distr_ex ] that presents the distribution of the first 21 am coefficients for a face - space constructed from 500 faces . ) new instances of facial appearance , the synthetic faces , can be obtained by randomly sampling from such a distribution . for simplicity , hereafter we assume that faces belong to a single ethnicity . to accommodate faces from different ethnic backgrounds , the same concept could be used with the mixture of gaussians distribution . we follow the procedure for am construction , proposed in @xcite . the training set of facial images , taken under the same viewing conditions , is annotated using a point model that delineates the face shape and the internal facial features . in this process , 22 landmarks are manually placed on each facial image . based on these points , 190 points of the complete model are determined ( see @xcite for details ) . for each face , landmark coordinates are concatenated to form a shape vector , @xmath1 . the data is then centered by subtracting the mean face shape , @xmath2 , from each observation . the shape principle components @xmath3 are derived from the set of mean subtracted observations ( arranged as columns ) using pca . the synthesis of a face shape ( denoted by @xmath4 ) from the _ shape model _ is done as follows , @xmath5 where @xmath6 is a vector in which the first @xmath7 elements are normally distributed parameters that determine the linear combination of shape principal components and the remaining elements are equal to zero . we refer to @xmath6 as the _ shape coefficients_. before deriving the texture component of the am , training images must be put into correspondence using non - rigid shape alignment procedure . each shape normalized and centered rgb image of a training face is then rearranged as a vector @xmath8 . such vectors for all training faces form a matrix which is used to compute the texture principle components @xmath9 by applying pca . a face texture ( denoted by @xmath10 ) is reconstructed from the _ texture model _ as follows , @xmath11 where @xmath12 are the _ texture coefficients _ which are also normally distributed and @xmath13 is the mean texture . the final model is obtained by a pca on the concatenated shape and texture parameter vectors . let @xmath14 denote the principal components of the concatenated space . the am coefficients ( @xmath15 ) are obtained from the corresponding shape ( @xmath1 ) and texture ( @xmath8 ) as follows , @xmath16\equiv q^t\left [ \begin{array}{c } w p_s^t(x-\bar{x})\\ p_g^t(g-\bar{g})\\ \end{array } \right]\ ] ] where @xmath17 is a scalar that determines the weight of shape relative to texture . figure [ fig_amm_example ] illustrates the shape vector @xmath1 ( center image ) and the shape free texture vector @xmath8 ( on the right ) used to obtain the am coefficients . am coefficients of a * real face * are obtained by projecting its shape @xmath1 and texture @xmath8 onto the shape and texture principal components correspondingly and then combining the shape and texture parameters into a single vector and projecting it onto the am principal components . in order to create the * synthetic faces * , we first estimate a @xmath18-dimensional gaussian distribution @xmath19 of the am coefficients using the training set of real faces . then am coefficients of synthetic faces are obtained by directly sampling from this distribution , discarding the samples beyond @xmath20 standard deviations . such that all training samples are within @xmath20 standard deviations from the mean . ] theoretically , the expected distance between the samples from am distribution to its center is about @xmath21 standard deviation units . we observed that the distance of real faces from the center is indeed close to @xmath21 standard deviation units . in other words , am coefficients are most likely to lie on the surface of an @xmath18-dimensional ellipsoid with radii @xmath22 , where @xmath23 . hence to sample synthetic faces , we use the following process : sample @xmath24 from a @xmath18-dimensional gaussian @xmath25 , normalize @xmath24 to the unit length and multiply coordinate - wise by @xmath26 . to minimize the differences between the am representations of real and synthetic faces , we apply the same normalization process to the am coefficients of the real faces as well . the biometric `` password file '' of the honeyfaces system is composed of records , containing the am coefficients of either real or synthetic faces . the coefficients are sufficient for the authentication process without reconstructing the face . however , we use reconstructed faces in our privacy and security analysis , thus in the following , we show how to reconstruct faces from their corresponding am coefficients . first , the shape and texture coefficients are obtained from the am coefficients as follows , @xmath27 and @xmath28 , where @xmath29^t$ ] is the am basis . then the texture and shape of the face are obtained via eq . ( [ eq : shape_bit ] ) and ( [ eq : texture_bit ] ) . finally , the texture @xmath10 is warped onto the shape @xmath4 , resulting in a facial image . figure [ fig : face_samples ] shows several examples of reconstructed real faces and synthetic faces , sampled from the estimated distribution of am coefficients . to prevent exfiltration and protect privacy of the users , we create a very large number of synthetic faces . these faces can be incorporated in the authentication system in different ways . for example , the honeywords @xcite method stores a list of passwords ( one of which is correct and the rest are fake ) per account . in our settings , both the number of synthetic faces and the ratio of synthetic to real faces should be large . thus , the configuration , in which the accounts are created solely for real users , requires a very large number of synthetic faces to be attached to each account . hence , in such an implementation , during the authentication process , a user s face needs to be compared to a long list of candidates ( all fake faces stored with the user name ) . this would increase the authentication time by a factor equal to the synthetic - to - real ratio , negatively affecting the usability of the system and leading to an undesirable trade off between privacy and usability . another alternative is creating many fake accounts with a single face as a password . this does not change the authentication time of the system ( compared to a system with no fake accounts ) . since most real systems have very regular user names ( e.g. , the first letter of the given name followed by the family name ) , it is quite easy to generate fake accounts following such a convention . as we show in section [ sec : sub : blowup ] , this allows inflating the password file to more than 56.6 tb ( when disregarding the storage of user names ) . one can also consider a different configuration , aimed to fool an adversary that knows the correct user names , but not the real biometrics . specifically , we can store several faces in each account ( instead of only one ) in addition to the fake accounts ( aimed at adversaries without knowledge of user names ) . the faces associated with a fake account are all synthetic . the faces associated with a real account include one real face of that user and the rest are synthetic one . in such a configuration the authentication time does not increase significantly , but the total size of the `` biometric data '' and the ratio of real - to - synthetic faces remains large . moreover , the adversary that knows the user name still needs to identify the real face among several synthetic faces . in this work we implemented and analyzed the configuration in which real and decoy users have an account with a single password the majority of the users are fake in order to hide the real ones . each user ( both real and fake ) has an account name and a password composed of 80 am coefficients . these coefficients are derived from the supplied facial image for real users or artificially generated for decoy ones . the number of training subjects for the face - space construction should be larger than the number of system users . this provides a better modeling of the facial appearance , allowing a large number of synthetic faces to be created , and protecting the privacy of system s users as discussed in section [ sec : privacyanalysis ] . we used a set of 500 subjects to train the am . 270 of them were the users of the honeyfaces system . all images in the training set were marked with manual landmarks using a tool similar to the one used in @xcite . we computed a 50-dimensional shape model and a 350-dimensional texture model as described in section [ sec : sub : generating ] and we reduced the dimension of the am parameters to 80 . we note that this training phase is done once , and needs to contain the users of the system mainly for optimal authentication rates . however , as we later discuss in section [ sec : sub : facespace ] , extracting biometric information of real users from the face - space is infeasible . all representations used in the system were normalized to unit norm and then multiplied by 7 standard deviations . this way we forced all samples ( real and synthetic ) to have the same norm , making the distribution of distances of real and synthetic faces very similar to each other ( see figure [ fig : distr_ex ] ) . we used the resulting face - space to generate synthetic faces . we discarded synthetic faces that fall closer than a certain distance from the real or previously created synthetic faces . the threshold on the distance between faces of different identities was set to 4,800 , thereby minimizing the discrepancy between the distance distributions of real and synthetic faces . this minimum separation distance prevents collisions between faces and thus the addition of synthetic faces does not affect the authentication accuracy of the original system ( prior to inflation ) . the process of synthetic face generation is very efficient and takes only @xmath30 seconds on average using matlab . using 500 training faces we were able to create about @xmath0 synthetic faces , with sufficient distance from each other . we strongly believe that more faces can be generated ( especially if the size of the training set is increased ) , but @xmath0 faces that occupy 56.6 tb seems sufficient for proof of concept . the authentication process of most biometric systems is composed of the user supplying the user name and her or his facial image . this image ( hereafter the test image ) is aligned to conform with the reference image stored for that user . after the registration , the distance between the test and reference templates are computed and compared to some predefined threshold . to find the registration between the test and the reference facial templates in our system , we first reconstruct the facial shape of the corresponding subject in the database from the am coefficients ( as shown in section [ sec : sub : recon ] ) . we then run an automatic landmark detector on the test image ( using face++ landmark detector @xcite ) and use these landmarks and the corresponding locations in the reference shape to find the scaling , rotation , and translation transformations between them . then we apply this transformation to the reference shape to put it into correspondence with the coordinate frame of the test image . the am coefficients of the test image are computed using the transformed reference shape and the test image itself ( as shown in section [ sec : sub : coefficients ] ) and then compared to the stored am coefficients the password , using the l2 norm . the threshold on the l2 distance was set to 3,578 which corresponds to 0.01% of far . note that the threshold is smaller than the distance between the faces ( 4,800 ) used for synthetic face generation . figure [ fig : lmk_samples ] illustrates the authentication process for the genuine and imposter attempts . we ran 270 genuine attempts , comparing the test image with the corresponding reference image , and about 4,200,000 impostor attempts ( due to the access to face++ ) . for a threshold producing an far of 0.01% , our system showed the true acceptance rate ( 100-frr ) of 93.33% . figure [ fig : roc ] shows the corresponding roc curve . our tests showed no degradation in frr / far after the inclusion of synthetic faces . finding landmarks in a test image , using the face++ landmark detector @xcite , takes 1.42 seconds per subject on average . we note that the implementation of the landmark detector is kept at the face++ server and thus the reported times include network communications . running the detector locally will significantly reduce the running time . obtaining the am coefficients of a test image and comparing them to those of the target identity in the database takes additional 0.53 seconds on average . this brings us to a total of 1.95 seconds ( on average ) for a verification attempt . the system was implemented and tested in matlab r2014b 64-bit on windows 7 , in 64-bit os environment with intel s i7 - 4790 3.60ghz cpu and 16 gb ram . local implementation that uses c is expected to improve the running times significantly ( though not faster than 1 ms ) . our privacy analysis targets an adversary with access to the inflated biometric `` password file '' , and is divided into three cases . the first scenario , discussed in section [ sec : sub : noprior ] , is an adversary who has no prior knowledge about the users of the system . such an adversary tries to identify the real users out of the fake ones . the second scenario , discussed in section [ sec : sub : out_source ] , concerns an adversary that tries to achieve the same goal , but has access to a comprehensive , external source of facial images that adequately represents the world wide variation ( population ) in facial appearance but does not know who the users are . the last scenario assumes that the adversary obtained the biometric data of all but one out of the system s users , and wishes to use this to find the biometrics of the remaining user . we discuss this case in section [ sec : sub : facespace ] . we first discuss the scenario in which the adversary has the full database ( e.g. , after breaking into the system ) and wishes to identify the real users but has no prior knowledge concerning the real users . more explicitly , this assumption means that the adversary does not have a candidate list and their biometrics , to check if they are in the database . an inflated password file is a file that contains @xmath31 facial templates , @xmath32 of which correspond to real faces and remaining @xmath33 are synthetic faces sampled from the same face - space as the real faces . a simulated password file is a file that contains @xmath31 facial templates , all of which are synthetic faces sampled from the same face - space . an adversary that can distinguish between an inflated password file and a simulated password file , can be transformed into an adversary that extracts all the real users . similarly , an adversary that can extract real users from a password file can be used for distinguishing between inflated and simulated password files . we start with the simpler case transforming an adversary that can extract the real faces into a distinguisher between the two files . the reduction is quite simple . if the adversary can extract real faces out of the password file ( and even only a single real face ) , we just give it the password file we have received . if the adversary succeeds in extracting any face out of it , we conclude that we received the inflated password file . otherwise , we conclude that we received a simulated password file . it is easy to see that the running time of the distinguishing attack and its success rate are exactly the same as that of the original extraction adversary . now , assume that we are given an adversary that can distinguish between an inflated password file and a simulated one with probability @xmath34 . we start by recalling that the advantage of distinguishing between two simulated ones is necessarily zero . hence , one can generate a hybrid argument , of replacing one face at a time in the file . when we replace a synthetic face with a different synthetic face , we have not changed the distribution of the file . thus , the advantage drops only when we replace a real face with a synthetic face , which suggests that if there are @xmath32 real users in the system , and @xmath31 total users in the system , we can succeed in identifying at least one of the real users of the system with probability greater than or equal to @xmath35 and running time of at most @xmath31 times the running time of the distinguishing adversary . [ cor_1 ] if the distributions of the inflated password file and the simulated password file are statistically indistinguishable , an adversary with no prior knowledge ( of either user s biometrics or user names ) can not identify the real users . theoretically , synthetic and real faces are sampled from the same distribution and thus are indistinguishable according to corollary [ cor_1 ] . however , in practice , synthetic faces are sampled from a parametric distribution which is estimated from real faces . the larger the set of faces , used to estimate the distribution , the closer these distributions will be . in practice , the number of training faces is limited which could introduce some deviations between the distributions . our following analysis shows that these deviations are too small to distinguish between the distributions of real and synthetic faces either by statistical tests or by human observers . the first part of the analysis performs a statistical test of the am coefficients of the real and the synthetic faces and shows that these distributions are indeed very close to each other . the second part studies the distribution of mutual distances among real and synthetic faces and reaches the same conclusion . finally , we perform a human experiment on the reconstructed and simulated faces , showing that even humans can not distinguish between them . the am coefficients are well approximated by a gaussian distribution in all dimensions @xcite . therefore , sampling am coefficients for synthetic faces from the corresponding distribution is likely to produce representations that can not be distinguished by standard hypothesis testing from real identities . the examples of real and synthetic distributions for the first 21 dimensions are depicted in figure [ fig : distr_ex ] and the following analysis verifies this statement . first , we show that coefficients of real and synthetic faces can not be reliably distinguished based on two sample kolmogorov smirnov ( ks ) test . to this end , we sampled a subset of 500 synthetic samples from 80-dimensional am and we compare it to the 500 vectors of coefficients of training images . we ran the ks test on these two sets for each of the 80 dimensions and recorded the result of the hypothesis test and the corresponding p - value . we repeated this test 50 times , varying the set of synthetic faces . the ks tests supported the hypothesis that the two samples come from the same distributions in 98.72% of the cases with a mean p - value 0.6 ( over 50 runs and 80 components , i.e. , 4000 tests ) . these results show that am coefficients of real and synthetic faces are indistinguishable using a two - sample statistical test . we analyzed the distributions of distances between the real faces , synthetic ones , and a mixture of both . figure [ fig : dist_distr ] shows that these distributions , both in the case of euclidean distances and in the case of angular distances , are very close . hence , the statistical distance between them is negligible , suggesting that attacks trying to use mutual distances are expected to be ineffective . we conducted a human experiment containing two steps . in the first step , the participants were shown a real face , not used in the experiment , and its reconstruction . in the second step of the experiment , each participant was presented with the same set of 16 faces ( 11 of which were synthetic and 5 of which were real ) and was asked to classify them as real or fake . we also allowed the users to avoid answering in the case of uncertainty or fatigue . the 11 synthetic faces were chosen at random from all the @xmath0 synthetic faces we generated , and the 5 real ones were chosen at random from the 500 real faces . for the real faces , we computed the am coefficients for each real image and then used the method described in section [ sec : sub : recon ] to generate the real faces and synthetic faces from the model . examples of real and synthetic faces are provided in the second and third rows , respectively , of figure [ fig : face_samples ] . out of 179 answers we have received , 97 were correct , showing a success rate of 54.19% . the fake faces received 120 answers , of which 66 were correct ( 55% ) . the real faces received 59 answers , of which 31 were correct ( 52.5% ) . our analysis shows that the answers for each face are distributed very similarly to the outcome of a binomial random variable with a probability of success at each trial of 0.5 . next , we analyze the case where an adversary has access to the inflated `` password file '' and to an extensive external source of facial images ( e.g. the internet ) . we consider two attack vectors : the first tries to use membership queries with random facial images to match real users of the system , the second attempts to distinguish between real and synthetic faces using a training process on a set of real facial images unrelated to the users of the system . an adversary could use a different source of facial images to try and run a membership query against the honeyfaces system to obtain the biometric of the real users . to match a random image from an external source of facial images , the adversary must run the authentication attempt with all users of the system ( including the fake ones ) . our experiments show that the current implementation takes about 2 seconds per authentication attempt ( mostly due to the landmarking via face++ ) . even under the unrealistic assumption that the authentication time could be reduced to 1 ms , it would take about @xmath36 seconds ( slightly more than 2 cpu years ) , to run the matching of a single facial image against @xmath37 fake faces . we note that one can not use a technique to speed up this search and comparison ( such as kd - trees ) as the process of comparison of faces requires aligning them ( based on the landmarks ) , which can not be optimized ( to the best of our knowledge ) . one can try to identify the membership of a person in the system by projecting his / her image onto the face - space of the system and analyzing the distance from the projection to the image itself . if the face - space was constructed from system users only , a small distance could reveal the presence of the person in the face - space . such an attack can be easily avoided by building the face - space from a sufficiently large ( external ) source of faces . such a face - space approximates many different appearances ( all combinations of people in the training set ) and thus people unrelated to the users of the system will also be close to the face - space . we conclude that a membership attack to obtain the real faces from the data base is impractical . the task of the adversary who obtained the inflated `` biometric password file '' is to distinguish the real faces from the synthetic ones . he can consider using a classifier that was trained to separate real faces from the fake ones . to this end the adversary needs to construct a training set of real and synthetic faces . synthetic faces can be generated using the system s face - space . however , the real faces of the system are unavailable to the adversary . one way an adversary might approach this problem is by employing a different set of real faces ( a substitute set ) to construct the face - space . he then can create a training set by generating synthetic faces using that space and reconstructing the real faces from the substitute set following the algorithms described in section [ sec : sub : generating ] . a trained classifier could then be used to classify the faces in the biometric `` password file '' . the substitute training set is likely to have different characteristics than the original one . the adversary could try to combine the system s face - space with the substitute set in attempt to improve the similarity of the training set to the biometric `` password file '' . then , the adversary can construct the training set of real faces by projecting the images from the substitute set on the mixed face - space and reconstructing them as described in section [ sec : sub : recon ] . to create a training set of synthetic faces , the adversary can either use the mixed face - space or the system s face - space . deep learning and , in particular , convolutional neural networks ( cnn ) , showed close to human performance in face verification @xcite . it is a common believe that the success of the cnn in recognition tasks is due to its ability to extract good features . moreover , it was shown that cnn features can be successfully transferred to perform recognition tasks in similar domains ( e.g. @xcite ) . such techniques are referred to as fine tuning or transfer learning . it proceeds by replacing the upper layers of the fully trained dl network ( that solves a related classification problem ) by layers that fit the new recognition problem ( the new layers are initialized randomly ) . the updated network is then trained on the new classification problem with the smaller data set . note , that most of the network does not require training , only slight tuning to fit the new classification task , and the last layer can be well trained using good cnn features and a smaller data set . following this strategy , we took the vgg - face deep network @xcite that was trained to recognize 2,622 subjects and applied the transfer learning method to train a dl network to classify between real and synthetic faces . to this end , we replaced the last fully connected 2,622 size layer with a fully connected layer of size 2 and trained this new architecture in the following settings . in all experiments we split the training set for training and validation of the network . then we applied the trained network on a subset of system s data set to classify the images into real and synthetic . the subset included all real faces and a subset of the synthetic faces ( same size as real set to balance the classification results ) . setting 1 : a face - space was constructed from 500 faces belonging to the substitute set . the training set included 400 reconstructed real faces and 400 synthetic faces , generated using the substitute face - space . the validation set included 100 reconstructed real faces and 100 synthetic faces from the same domain , not included in the training set . the results on the substitute validation set showed that the dl network classifies 62.5% of faces correctly . the results on the system s set dropped to 53.33% , which is close to random . setting 2 : a face - space was constructed by combining the system s face - space with the substitute set . the training set included 400 real faces projected and reconstructed using the mixed face - space and 400 synthetic faces , generated using the mixed face - space . the validation set included 100 reconstructed real faces and 100 synthetic faces from the same domain , not included in the training set . the results on the validation set showed good classification : 75% of synthetic faces were classified as synthetic and 93% of real faces were classified as real . however , the same network classified all faces of the system s face set as synthetic . this result shows that using a mixed face - space to form a training set is not effective . the prime reason for this is the artifacts in synthetic images due to variation in viewing conditions between the sets . setting 3 : the real training and validation sets were the same as in setting 2 . the synthetic training and validation sets were formed by generating synthetic faces using system s face - space . here the classifier was able to perfectly classify the validation set , but it classified all system s faces as synthetic . this shows that using real and synthetic faces from different face - spaces introduces even more differences between them , which do not exist in system s biometric `` password file '' . to conclude , the state - of - the - art deep learning classifier showed accuracy of 53.33% in distinguishing between real and synthetic faces in the system s biometric `` password file '' . this result is close to random guessing . an adversary who obtains the facial images of all but one of the real users of the system can try and use it for extracting information about the remaining user from the password file . if the training set used for constructing the face - space contains only the users of the system , the following simple attack will work : recall that the authentication procedure requires removing the mean face from the facial image obtained in the authentication process . thus , the mean of all faces in the training set is stored in the system . the adversary can find the last user by computing the mean of the users he holds and solving a simple linear equation . to mitigate this attack , and to allow better modeling of the facial appearance , the training set should contain a significant amount of training faces that are not users of the system . note that these additional faces must be discarded after the face - space is constructed . assuming that the training set for the face - space construction was not limited to a set of system users ( as is the case in our implementation ) , the adversary could try the following attack . create @xmath38 face - spaces by adding each unknown face from the biometric `` password file '' in turn to the @xmath39 real faces that are in the possession of the adversary . @xmath40 is equal to the number of synthetic faces in the biometric `` password file '' plus one real face . then compare these @xmath40 face - spaces to the one stored in the system ( using statistical distance between the distributions ) . such comparison provides a ranking of unknown faces to be the @xmath32th real face . if the attack is effective , we expect the face - space including the @xmath32th user to be highly ranked ( i.e. , to appear in a small percentile ) . however , if the distribution of the rankings associated with the face - space including the @xmath32th real face over random splits of @xmath39 known and 1 unknown face is ( close to ) uniform , then we can conclude that the adversary does not gain any information about the last user using this attack . in our implementation of the attack , we assume that the adversary knows 269 faces of real users and he tries to identify the last real user among the synthetic ones . running the attack with all synthetic faces is time consuming . to get statistics of rankings we can use a much smaller subset of synthetic faces . specifically , we used 100 synthetic faces and ran the experiment over 100 randomized splits into 269 known and 1 unknown faces . figure [ fig : dist_rankings ] shows the histogram of rankings associated with the face - space including the last real user in 100 experiments . the histogram confirms that the distribution of rankings is indeed uniform , which renders the attack ineffective . an alternative approach , that the adversary may take , is to analyze the effects of a single face on the face - space distribution . however , our experiments show that the statistical distances between neighboring distributions ( i.e. , generated from training sets that differ by a single face ) are insignificant . specifically , the average statistical distance between the distribution estimated from the full training set ( of 500 real faces ) and all possible sets of 499 faces ( forming 500 neighboring sets , each composed of a different subset of 499 faces ) is @xmath41 and the maximal distance is @xmath42 . these distances are negligible compared to the standard deviations of the face - space gaussians ( the largest std is 6,473.7 and the smallest is 304.1717 ) . these small differences suggest that one can use differential privacy mechanisms with no ( or marginal ) usability loss ( for example , by using ideas related to @xcite ) to mitigate attacks that rely on prior knowledge of the system s users . we leave the implementation and evaluation of this mechanism for future research . to conclude , the honeyfaces system protects the privacy of users even in the extreme case when the adversary learned all users but one , assuming that the training set for constructing the face - space contains a sufficiently large set of additional faces . we now discuss the various scenarios in which honeyfaces improve the security of a biometric data . we start by discussing the scenario of limited outgoing bandwidth networks ( such as air - gaped networks ) , and showing the affects of the increased file size on the exfiltration times . we follow by discussing the effects honeyfaces has on the detection of the exfiltration process . we conclude the analysis of the security offered by our solution in the scenario of partial exposure of the database . the time needed to exfiltrate a file is easily determined by the size of the file to be exfiltrated and the bandwidth . when the exfiltration bandwidth is very slow ( e.g. , in the air - gap networks studied in @xcite ) , a 640-byte representation of a face ( or 5,120-bit one ) takes between 5 seconds ( at 1,000 bits per second rate ) to 51 seconds ( in the more realistic 100 bits per second rate ) . hence , leaking even a 1 gbyte database takes between 92.6 to 926 days ( assuming full bandwidth , and no need for synchronization or error correction overhead ) . the size of the password file can be inflated to contain all the @xmath0 faces we created , resulting in a 56.6 tbytes file size ( whose leakage would take about 14,350 years in the faster speed ) . a possible way to decrease the file size is to compress the file . our experiments show that linux s zip version 3.0 , could squeeze the password file by only 4% . it is highly unlikely that one could devise a compression algorithm that succeeds in compressing significantly more . in other words , compressing the face file reduces the number of days to exfiltrate 1 gbyte to 88.9 days ( in the faster speed ) . one can consider a lossy compression algorithm , for example by using only the coefficients associated with the principle components ( carrying most information ) . we show in section [ sec : sub : rates ] that this approach requires using many coefficients for identification . hence , we conclude that if the bandwidth is limited , exfiltration of the full database in acceptable time limit is infeasible . the improved leakage detection stems from two possible defenses : the use of intrusion detection systems ( and data loss prevention products ) and the use of a two - server settings as in honeywords . intrusion detection systems , such as snort , monitor the network for suspicious activities . for example , a high outgoing rate of dns queries may suggest an exfiltration attempt and raise an alarm @xcite . similar exfiltration attempts can also increase the detection of data leakage ( such as an end machine which changes its http footprint and starts sending a large amount of information to some external server ) . hence , an adversary who does not take these tools into account is very likely to get caught . on the other hand , an adversary who tries to `` lay low '' is expected to have a reduced exfiltration rate , preventing quick leakage and returning to the scenario discussed in the previous section . the use of honeyfaces also allows for a two - server authentication setting similarly to honeywords @xcite . the first server uses a database composed of the real and the synthetic faces . after a successful login attempt is made into this system , a second authentication query is sent to the second server , which holds only the real users of the system . a successful authentication to the first server that uses a fake account , is thus detected at the second server , raising an alarm . we showed that exfiltrating the entire password file in acceptable time is infeasible if the bandwidth is limited . hence , the adversary can decide to pick one of two approaches ( or combine them ) when trying to exfiltrate the file either leak only partial database ( possibly with an improved ratio of real to synthetic faces ) , or to leak partial representations such as the first 10 am coefficients out of 80 per user . as we showed in the privacy analysis ( section [ sec : privacyanalysis ] ) , statistical tests or machine learning methods fail to identify the real faces among the synthetic ones . using membership queries to find real faces in the database is computationally infeasible without prior knowledge of the real user names . we conclude that reducing the size of the data set by identifying the real users or significantly improving the real to synthetic ratio is impossible . the second option is to leak a smaller number of the coefficients ( a partial representation ) . leaking a smaller number of coefficients can be done faster than the entire record , and allow the adversary to run on his system ( possibly with greater computational power ) , any algorithm he wishes for the identification of the real users . in the following , we show that partial representations ( that significantly decrease the size of the data set ) do not provide enough information for successful membership queries . we experimented with 10 coefficients ( i.e. , assume that the adversary leaked the first 10 am coefficients of all users ) . as the adversary does not know the actual threshold for 10 coefficients , he can try and approximate this value using the database . our proposed method for this estimation is based on computing the distance distribution for 30,000 faces from the database , and setting a threshold for authentication corresponding to the 0.01% `` percentile '' of the mutual distances . we then take test sets of real users faces and of outsiders faces and for each face from these sets , computed the minimal distance from this face to all the faces in the reduced biometric password file . we assume that if this distance is smaller than the threshold , then the face was in the system , otherwise we conclude that the face was not in it . our experiments show that for the 0.01% threshold , 98.90% of the outsider set and 99.26% of the real users were below the threshold . in other words , there is almost no difference between the chance of determining that a user of the system is indeed a user vs. determining that an outsider is a user of the system . this supports the claim that 10 coefficients are insufficient to distinguish between real users and outsiders . we also used a smaller threshold which tried to maximize the success rate of an outsider to successfully match to a real face . for this smaller threshold , 71.08% of the outsiders were below it compared with 74.07% of the real users . to further illustrate the effects of partial representation on the reconstructed face , we show in figure [ fig : grad_rec ] the reconstruction of faces from 80 , 30 , and 10 coefficients , compared with the real face . as can be seen , faces reconstructed from 30 coefficients are somewhat related to the original face , but faces reconstructed from 10 , bare little resemblance to the original . although it is possible to match a degraded face to the corresponding original when a small number of faces are shown ( figure [ fig : grad_rec ] ) , visual matching is impossible among @xmath0 faces . thus , an adversary wishing to leak partial information about an image , needs to leak more than 10 coefficients . to conclude , exfiltrating even a partial set of faces ( or parts of the records ) does not constitute a plausible attack vector against the honeyfaces system . in this paper we explored the use of synthetic faces for increasing the security and privacy of face - based authentication schemes . we have proposed a new mechanism for inflating the database of users ( honeyfaces ) which guarantees users privacy with no usability loss . furthermore , honeyfaces offers improved resilience against exfiltration ( both the exfiltration itself and its detection ) . we also showed that this mechanism does not interfere with the basic authentication role of the system and that the idea allows the introduction of a two - server authentication solution as in honeywords . future work can explore the application of the honeyfaces idea to other biometric traits ( such as iris and fingerprints ) . we believe that due to the similar nature of iris codes ( that also follow multi - dimensional gaussian distribution ) , the application of the concept is going to be quite straightforward . the funds received under the binational uk engineering and physical sciences research council project ep / m013375/1 and israeli ministry of science and technology project 3 - 11858 , `` improving cyber security using realistic synthetic face generation '' allowed this work to be carried out . j. l. araque , m. baena , b. e. chalela , d. navarro , and p. r. vizcaya . synthesis of fingerprint images . in _ pattern recognition , 2002 . proceedings . 16th international conference on _ , volume 2 , pages 422425 . ieee , 2002 . v. blanz and t. vetter . a morphable model for the synthesis of 3d faces . in _ proceedings of the 26th annual conference on computer graphics and interactive techniques _ , pages 187194 . acm press / addison - wesley publishing co. , 1999 . j. cui , y. wang , j. huang , t. tan , and z. sun . an iris image synthesis method based on pca and super - resolution . in _ 17th international conference on pattern recognition , icpr 2004 , cambridge , uk , august 23 - 26 , 2004 . _ , pages 471474 , 2004 . j. donahue , y. jia , o. vinyals , j. hoffman , n. zhang , e. tzeng , and t. darrell . decaf : a deep convolutional activation feature for generic visual recognition . in _ international conference in machine learning ( icml ) _ , 2014 . g. j. edwards , t. f. cootes , and c. j. taylor . . in _ computer vision - eccv98 , 5th european conference on computer vision , freiburg , germany , june 2 - 6 , 1998 , proceedings , volume ii _ , pages 581595 , 1998 . c. imdahl , s. huckemann , and c. gottschlich . towards generating realistic synthetic fingerprint images . in _ 9th international symposium on image and signal processing and analysis , ispa 2015 , zagreb , croatia , september 7 - 9 , 2015 _ , pages 7882 , 2015 . a. juels and m. wattenberg . . in j. motiwalla and g. tsudik , editors , _ ccs 99 , proceedings of the 6th acm conference on computer and communications security , singapore , november 1 - 4 , 1999 . _ , pages 2836 . acm , 1999 . m. weir , s. aggarwal , b. de medeiros , and b. glodek . . in _ 30th ieee symposium on security and privacy ( s&p 2009 ) , 17 - 20 may 2009 , oakland , california , usa _ , pages 391405 . ieee computer society , 2009 . s. n. yanushkevich , v. p. shmerko , a. stoica , p. s. p. wang , and s. n. srihari . introduction to synthesis in biometrics . in _ image pattern recognition - synthesis and analysis in biometrics _ , pages 530 . l. zhang , l. lin , x. wu , s. ding , and l. zhang . end - to - end photo - sketch generation via fully convolutional representation learning . in _ proceedings of the 5th acm on international conference on multimedia retrieval _ , pages 627634 . acm , 2015 .

one of the main challenges faced by biometric - based authentication systems is the need to offer secure authentication while maintaining the privacy of the biometric data . previous solutions , such as secure sketch and fuzzy extractors , rely on assumptions that can not be guaranteed in practice , and often affect the authentication accuracy . in this paper , we introduce honeyfaces : the concept of adding a large set of synthetic faces ( indistinguishable from real ) into the biometric `` password file '' . this password inflation protects the privacy of users and increases the security of the system without affecting the accuracy of the authentication . in particular , privacy for the real users is provided by `` hiding '' them among a large number of fake users ( as the distributions of synthetic and real faces are equal ) . in addition to maintaining the authentication accuracy , and thus not affecting the security of the authentication process , honeyfaces offer several security improvements : increased exfiltration hardness , improved leakage detection , and the ability to use a two - server setting like in honeywords . finally , honeyfaces can be combined with other security and privacy mechanisms for biometric data . we implemented the honeyfaces system and tested it with a password file composed of 270 real users . the `` password file '' was then inflated to accommodate up to @xmath0 users ( resulting in a 56.6 tb `` password file '' ) . at the same time , the inclusion of additional faces does not affect the true acceptance rate or false acceptance rate which were 93.33% and 0.01% , respectively . biometrics ( access control ) , face recognition , privacy

1611.03811

thanks to a fortunate coincidence of observations by agile , _ fermi _ , and _ swift _ satellites , together with the optical observations by the vlt / fors2 and the nordic optical telescope , it has been possible to obtain an unprecedented set of data , extending from the optical - uv , through the x - rays , all the way up to the high energy ( gev ) emission , which allowed detailed temporal / spectral analyses on grb 090510 @xcite . in contrast with this outstanding campaign of observations , a theoretical analysis of the broadband emission of grb 090510 has been advanced within the synchrotron / self - synchrotron compton ( ssc ) and traditional afterglow models ( see , e.g. , sections 5.2.1 and 5.2.2 in * ? ? ? * ) . paradoxically , this same methodology has been applied in the description of markedly different type of sources : e.g. , @xcite for the low energetic long grb 060218 , @xcite for the high energetic long grb 130427a , and @xcite for the s - grf 051221a . in the meantime , it has become evident that grbs can be subdivided into a variety of classes and sub - classes @xcite , each of them characterized by specific different progenitors which deserve specific theoretical treatments and understanding . in addition every sub - class shows different episodes corresponding to specifically different astrophysical processes , which can be identified thanks to specific theoretical treatments and data analysis . in this article , we take grb 090510 as a prototype for s - grbs and perform a new time - resoved spectral analysis , in excellent agreement with the above temporal and spectral analysis performed by , e.g. , the _ fermi _ team . now this analysis , guided by a theoretical approach successfully tested in this new family of s - grbs @xcite , is directed to identify a precise sequence of different events made possible by the exceptional quality of the data of grb 090510 . this include a new structure in the thermal emission of the p - grb emission , followed by the onset of the gev emission linked to the bh formation , allowing , as well , to derive the structure of the circumburst medium from the spiky structure of the prompt emission . this sequence , for the first time , illustrates the formation process of a bh . already in february 1974 , soon after the public announcement of the grb discovery @xcite , @xcite presented the possible relation of grbs with the vacuum polarization process around a kerr - newman bh . there , evidence was given for : a ) the formation of a vast amount @xmath2-baryon plasma ; b ) the energetics of grbs to be of the order of @xmath11 erg , where @xmath12 is the bh mass ; c ) additional ultra - high energy cosmic rays with energy up to @xmath13 ev originating from such extreme process . a few years later , the role of an @xmath2 plasma of comparable energetics for the origin of grbs was considered by @xcite and it took almost thirty years to clarify some of the analogies and differences between these two processes leading , respectively , to the alternative concepts of fireball " and fireshell " @xcite . in this article we give the first evidence for the formation of a kerr newman bh , in grb 090510 , from the merger of two massive nss in a binary system . grbs are usually separated in two categories , based on their duration properties . short grbs have a duration @xmath14 s while the remaining ones with @xmath15 s are traditionally classified as long grbs . short grbs are often associated to ns - ns mergers ( see e.g. @xcite ; see also @xcite for a recent review ) : their host galaxies are of both early- and late - type , their localization with respect to the host galaxy often indicates a large offset @xcite or a location of minimal star - forming activity with typical circumburst medium ( cbm ) densities of @xmath16@xmath17 @xmath18 , and no supernovae ( sne ) have ever been associated to them . the progenitors of long grbs , on the other hand , have been related to massive stars @xcite . however , in spite of the fact that most massive stars are found in binary systems @xcite , that most type ib / c sne occur in binary systems @xcite and that sne associated to long grbs are indeed of type ib / c @xcite , the effects of binarity on long grbs have been for a long time largely ignored in the literature . indeed , until recently , long grbs have been interpreted as single events in the jetted _ collapsar _ fireball model ( see e.g. @xcite and references therein ) . multiple components evidencing the presence of a precise sequence of different astrophysical processes have been found in several long grbs ( e.g. @xcite , @xcite ) . following this discovery , further results led to the introduction of a new paradigm expliciting the role of binary sources as progenitors of the long grb - sn connection . new developments have led to the formulation of the induced gravitational collapse ( igc ) paradigm @xcite . the igc paradigm explains the grb - sn connection in terms of the interactions between an evolved carbon - oxygen core ( co@xmath19 ) undergoing a sn explosion and its hypercritical accretion on a binary ns companion @xcite . the large majority of long bursts is related to sne and are spatially correlated with bright star - forming regions in their host galaxies @xcite with a typical cbm density of @xmath20 @xmath18 @xcite . a new situation has occurred with the observation of the high energy gev emission by the _ fermi_-lat instrument and its correlation with both long and short bursts with isotropic energy @xmath21 erg , which has been evidenced in @xcite and @xcite , respectively . on the basis of this correlation the occurrence of such prolonged gev emission has been identified with the onset of the formation of a bh @xcite . as recalled above , the long grbs associated to sne have been linked to the hypercritical accretion process occurring in a tight binary system when the ejecta of an exploding co@xmath19 accretes onto a ns binary companion ( see , e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? when the hypercritical accretion occurs in a widely separated system with an orbital separation @xmath22 cm @xcite , the accretion is not sufficient to form a bh . for these softer systems with rest - frame spectral peak energy @xmath23 kev the upper limit of their observed energy is @xmath24 erg , which corresponds to the maximum energy attainable in the accretion onto a ns @xcite . such long a burst corresponds to an x - ray flash ( xrf ) . the associated x - ray afterglow is also explainable in terms of the interaction of the prompt emission with the sn ejecta ( fryer et al . , in preparation ) . in these systems no gev emission is expected in our theory and , indeed , is not observed . interestingly , a pioneering evidence for such an x - ray flash had already been given in a different context by @xcite , @xcite , and @xcite . for tighter binaries ( @xmath25 cm , @xcite ) , the hypercritical accretion onto the companion ns leads to the formation of a bh . for these harder systems with @xmath26 kev the lower limit of their observed energy is @xmath24 erg , which necessarily needs the accretion process into a bh . an associated prolonged gev emission occurs after the p - grb emission and at the beginning of the prompt emission , and originates at the onset of the bh formation @xcite . these more energetic events are referred to as binary - driven hypernovae ( bdhne ) . specific constant power - law behaviors are observed in their high energy gev , x - rays , and optical luminosity light curves @xcite . in total analogy , the formation of a bh can occur in short bursts , depending on the mass of the merged core of the binary system . when the two ns masses are large enough , the merged core can exceed the ns critical mass and the bh formation is possible . in the opposite case , a massive ns ( mns ) is created , possibly , with some additional orbiting material to guarantee the angular momentum conservation . we then naturally expect the existence of two short bursts sub - classes : authentic short grbs ( s - grbs ) , characterized by the formation of a bh @xcite , with @xmath21 erg , a harder spectrum ( see section [ epeakeiso ] ) and associated with a prolonged gev emission ( see section [ gevemission ] ) ; short gamma - ray flashes ( s - grfs ) , producing a mns @xcite , with @xmath27 erg . in this second sub - class , of course , the gev emission should not occur and , indeed , is never observed . , where @xmath28 is the critical value for vacuum polarization and @xmath29 is the electric field strength . the plot assumes a black hole mass energy @xmath30 . figure reproduced from @xcite with their kind permission . ] following the discovery of the first prototype of this s - grb class , namely grb 090227b @xcite , the first detailed analysis of such a genuine short grb originating from a binary ns merger leading to a bh was done for grb 140619b by @xcite , determining as well the estimated emission of gravitational waves . the latter has been estimated following the method applied by @xcite for grb 090227b . from the spectral analysis of the early @xmath31 s , they inferred an observed temperature @xmath32 kev of the @xmath2 plasma at transparency ( p - grb ) , a theoretically derived redshift @xmath33 , a total burst energy @xmath34 erg , a rest - frame peak energy @xmath35 mev , a baryon load @xmath36 , and an average cbm density @xmath37 @xmath18 . we turn in this article to the most interesting case of grb 090510 which has , in addition to very similar properties of the members of this new class of s - grb sources , a spectroscopically determined value of the redshift and represents one of the most energetic sources of this family both in the @xmath38-ray and in the gev ranges . actually , a first attempt to analyze grb 090510 was made by interpreting this source as a long grb @xcite . an unusually large value of the cbm density was needed in order to fit the data : this interpretation was soon abandoned when it was noticed that grb 090510 did not fulfill the nesting conditions of the late x - ray emission typical of long grbs @xcite , see also section [ xray ] and figure [ episode3 ] . in light of the recent progress in the understanding of the fireshell theory , we address the interpretation of grb 090510 as the merging of a binary ns . we give clear evidence for the validity of this interpretation . in view of the good quality of the data both in @xmath38- rays and in the gev range , we have performed a more accurate description of the p - grb , best fitted by a convolution of thermal spectra . this novel feature gives the first indication for the existence of an axially symmetric configuration of the dyadotorus emitting the @xmath2 plasma which had been previously theoretically considered and attentively searched for . this gives the first indication that indeed the angular momentum plays a role and a dyadotorus is formed , as theoretically predicted in a series of papers ( see * ? ? ? * and figure [ dyadotorus ] ) . this naturally leads to the evidence for the formation of a rotating bh as the outcome of the gravitational collapse . we turn then to the main new feature of grb 090510 which is the high energy @xmath6@xmath7 gev emission ( see figure [ 090510gev ] ) . the direct comparison of the gev emission in this source and in the bdhne 130427a shows the remarkable similarities of these two gev components ( see figure [ 090510gev ] ) . the fact that the s - grb 090510 originates from a binary ns merger and the bdhn 130427a from the igc of a sn hypercritical accretion process onto a companion ns clearly points to the bh as originating this gev emission , the reason being that these two astrophysical systems are different in their progenitors and physical process and have in the formation of a bh their unique commonality . this paper is structured as follows : in section 2 we summarize the relevant aspects of the fireshell theory and compare and contrast it with alternative approaches . in section 3 we discuss the recent progress on the ns equilibrium configuration relevant for s - grbs and bdhne . in section 4 we move on to describe the observations of grb 090510 and their analysis . the s - grb nature of grb 090510 is justified in section 5 , and we offer an interpretation of our results in section 6 . section 7 concludes this work . a standard flat @xmath39cdm cosmological model with @xmath40 and @xmath41 km s@xmath42 mpc@xmath42 is adopted throughout the paper . the fireshell scenario @xcite , has been initially introduced to describe a grb originating in a gravitational collapse leading to the formation of a kerr - newman bh . a distinct sequence of physical and astrophysical events are taken into account : * an optically thick pair plasma the fireshell of total energy @xmath43 is considered . as a result , it starts to expand and accelerate under its own internal pressure @xcite . the baryonic remnant of the collapsed object is engulfed by the fireshell the baryonic contamination is quantified by the baryon load @xmath44 where @xmath45 is the mass of the baryonic remnant @xcite . * after the engulfment , the fireshell is still optically thick and continues to self - accelerate until it becomes transparent . when the fireshell reaches transparency , a flash of thermal radiation termed proper - grb ( p - grb ) is emitted @xcite . * in grbs , the @xmath2-baryon plasma evolves from the ultra - relativistic region near the bh all the way reaching ultra - relativistic velocities at large distances . to describe such a dynamics which deals with unprecedentedly large lorentz factors and also regimes sharply varying with time , in @xcite it has been introduced the appropriate relative spacetime transformation paradigm . this paradigm gives particular attention to the constitutive equations relating four time variables : the comoving time , the laboratory time , the arrival time , and the arrival time at the detector corrected by the cosmological effects . this paradigm is essential for the interpretation of the grb data : the absence of adopting such a relativistic paradigm in some current works has led to a serious misinterpretation of the grb phenomenon . * in compliance with the previous paradigm , the interactions between the ultra - relativistic shell of accelerated baryons left over after transparency and the cbm have been considered . they lead to a modified blackbody spectrum in the co - moving frame @xcite . the observed spectrum is however non - thermal in general ; this is due to the fact that , once the constant arrival time effect is taken into account in the equitemporal surfaces ( eqts , see * ? ? ? * ; * ? ? ? * ) , the observed spectral shape results from the convolution of a large number of modified thermal spectra with different lorentz factors and temperatures . * all the above relativistic effects , after the p - grb emission , are necessary for the description of the prompt emission of grbs , as outlined in @xcite . the prompt emission originates in the collisions of the accelerated baryons , moving at lorentz factor @xmath46@xmath47 , with interstellar clouds of cbm with masses of @xmath48@xmath49 g , densities of @xmath50@xmath3 @xmath18 and size of @xmath51@xmath52 cm , at typical distances from the bh of @xmath53@xmath54 cm ( see , e.g. , @xcite for long bursts ) . our approach differs from alternative tratments purporting late activities from the central engine ( see , e.g. , the _ collapsar _ model in @xcite , @xcite , @xcite and references therein , and the _ magnetar _ model in @xcite , @xcite , @xcite , @xcite , @xcite , and references therein ) . * @xmath43 and @xmath55 are the only two parameters that are needed in a spherically symmetric fireshell model to determine the physics of the fireshell evolution until the transparency condition is fulfilled . three additional parameters , all related to the properties of the cbm , are needed to reproduce a grb light curve and its spectrum : the cbm density profile @xmath56 , the filling factor @xmath57 that accounts for the size of the effective emitting area , and an index @xmath58 that accounts for the modification of the low - energy part of the thermal spectrum @xcite . they are obtained by running a trial - and - error simulation of the observed light curves and spectra that starts at the fireshell transparency . * a more detailed analysis of pair cration process around a kerr - newman bh has led to the concept of dyadotorus @xcite . there , the axially symmetric configuration with a specific distribution of the @xmath2 , as well as its electromagnetic field , have been presented as function of the polar angle . the total spectrum at the transparency of the @xmath2plasma is a convolution of thermal spectra at different angles . this formalism describing the evolution of a baryon - loaded pair plasma is describable in terms of only three intrinsic parameters : the @xmath2 plasma energy @xmath59 , the baryon load @xmath55 , and the specific angular momentum @xmath60 of the incipient newly - formed bh . it is , therefore , independent of the way the pair plasma is created . in addition to the specific case , developed for the sake of example , of the dyadotorus created by a vacuum polarization process in an already formed kerr - newman bh , more possibilities have been envisaged in the meantime : * the concept of dyadotorus can be applied as well in the case of a pair plasma created via the @xmath61 mechanism in a ns merger as described in @xcite , @xcite , @xcite , @xcite , assuming that the created pair plasma is optically thick . the relative role of neutrino and weak interactions vs. the electromagnetic interactions in building the dyadotorus is currently topic of intense research . * equally important are the relativistic magneto - hydrodynamical process leading to a dyadotorus , indicated in the general treatment of @xcite , and leading to the birth of a kerr - newman bh , surrounded by an opposite charged magnetosphere in a system endowed with global charge neutrality . active research is ongoing . * progress in understanding the ns equilibrium configuration imposing the global charge neutrality condition , as opposed to the local charge neutrality usually assumed @xcite . a critical mass for a non - rotating ns @xmath62 has been found for the nl3 nuclear equation of state @xcite . the effects of rotation and of the nuclear equation of state on the critical mass is presented in @xcite and in @xcite . the existence of electromagnetic fields close to the critical value has been evidenced in the interface between the core and the crust in the above global neutrality model , as well as very different density distributions in the crust and in the core , which could play an important role during the ns ns mergers ( see figure [ nanda ] and * ? ? ? ( _ middle panel _ ) in the core - crust transition layer normalized to the @xmath63-meson compton wavelength @xmath64 fm . _ lower panel _ : density profile inside a ns star with central density @xmath65 , where @xmath66 is the nuclear density , from the solution of the tov equations ( locally neutral case ) and the globally neutral solution presented in @xcite . the density at the edge of the crust is the neutron drip density @xmath67 g @xmath68 . reproduced from @xcite with their kind permission . ] the above three possibilities have been developed in recent years , but they do not have to be considered exaustive for the formation of a dyadotorus endowed by the above three parameters . in conclusion the evolution in the understanding of the grb phenomenon , occurring under very different initial conditions , has evidenced the possibility of using the dyadotorus concept for describing sources of an optically thick baryon - loaded @xmath2 plasma within the fireshell treatment in total generality . the key role neutrino emission in the hypercritical accretion process onto a ns has been already examined in the literature ( see , e.g. , * ? ? ? * ; * ? ? ? the problem of hypercritical accretion in a binary system composed of a co@xmath19 and a companion ns has been studied in @xcite ( see also references therein ) . the energy released during the process , in form of neutrinos and photons , is given by the gain of gravitational potential energy of the matter being accreted by the ns and depends also on the change of binding energy of the ns while accreting both matter and on the angular momentum carried by the accreting material ( see , e.g. , @xcite and @xcite ) . for a typical ns mass of @xmath70 m@xmath71 , a value observed in galactic ns binaries , and a ns critical mass @xmath72 in the range from @xmath73 m@xmath71 up to @xmath74 depending on the equations of state and angular momentum ( see * ? ? ? * ; * ? ? ? * ; * ? ? ? * for details ) , the accretion luminosity can be as high as @xmath75@xmath76 erg s@xmath42 for accretion rates @xmath77@xmath78 s@xmath42 ( see * ? ? ? * ; * ? ? ? * for details ) . for binary systems with a separation @xmath79 cm ( @xmath80 min ) , our numerical simulations indicate that : a ) the accretion process duration lasts @xmath81 s ( see , e.g. , * ? ? ? * ; * ? ? ? * ) , b ) the ns collapses to a bh , and c ) a total energy larger than @xmath69 erg is released during the hypercritical accretion process . these systems correspond to the bdhne @xcite . for systems with larger separations the hypercritical accretion is not sufficient to induce the collapse of the ns into a bh and the value of @xmath69 erg represents a theoretical estimate of the upper limit to the energy emitted by norm in the hypercritical accretion process . this sub - class of sources corresponds to the xrfs @xcite . the same energetic considerations do apply in the analysis of the hypercritical accretion occurring in a close binary ns system undergoing merging @xcite . therefore , in total generality , we can conclude that the energy emitted during a ns ns merger leading to the formation of a bh should be larger than @xmath69 erg ( see figure [ rt_gw ] ) . the limit of @xmath69 erg clearly depends on the initial ns mass undergoing accretion , by norm assumed to be @xmath70 m@xmath71 , and on the yet unknown value of @xmath72 , for which only an absolute upper limit of @xmath82 m@xmath71 has been established for the non - rotating case @xcite . as already pointed out in @xcite , for ns ns mergers , the direct determination of the energy threshold of @xmath69 erg dividing s - grfs and s - grbs , as well as xrfs and bdhne , provides fundamental informations for the determination of the actual value of @xmath83 , for the minimum mass of the newly - born bh , and for the mass of the accreting ns . ] ) . the binary orbit gradually shrinks due to energy loss through gravitational waves emission ( yellow - brown ) . at point a , the merger occurs : the fireshell ( in red ) is created and starts its expansion . it reaches transparency at point b , emitting the p - grb ( light purple ) . the prompt emission ( deep purple ) then follows at point c. the dashed lines represent the gev emission ( delayed relative to the start of the grb ) originating in the newly - born bh . this space - time diagram well illustrates how the gev emission originates in the newly - born bh and follows a different space - time path from the prompt emission , contrary to what stated in @xcite . the prompt emission originates from the interactions of the baryons , accelerated to ultrarelativistic lorentz factors during the pair - baryon electromagnetic pulse , with the clumpy circumburst medium ( see section 2 ) . the analysis of the spiky structure of the prompt emission allows to infer the structure of the circumburst medium ( see figure [ 090510simlc ] ) . there is the distinct possibility that the gev emission prior to @xmath84 s in the arrival time may interact with the prompt emission . in this sense the work by @xcite may become of interest . ] in this section , we summarize the observations of grb 090510 as well as the data analysis . we used _ ( gbm and lat ) and _ swift_/xrt data for the purposes of this work . the _ fermi_/gbm instrument @xcite was triggered at @xmath8500:22:59.97 ut on may 10 , 2009 by the short and bright burst grb 090510 ( @xcite , trigger 263607781 / 090510016 ) . the trigger was set off by a precursor emission of duration 30 ms , followed @xmath86 0.4 s later by a hard episode lasting @xmath87 s. this grb was also detected by swift @xcite , _ fermi_/lat @xcite , agile @xcite , konus - wind @xcite , and suzaku - wam @xcite . the position given by the gbm is consistent with that deduced from _ swift _ and lat observations . during the first second after lat trigger at 00:23:01.22 ut , _ fermi_/lat detected over 50 events ( respectively over 10 ) with an energy above 100 mev ( respectively above 1 gev ) up to the gev range , and more than 150 ( respectively more than 20 ) within the first minute @xcite . this makes grb 090510 the first bright short grb with an emission detected from the kev to the gev range . observations of the host galaxy of grb 090510 , located by vlt / fors2 , provided a measurement of spectral emission lines . this led to the determination of a redshift @xmath88 @xcite . the refined position of grb 090510 obtained from the nordic optical telescope @xcite is offset by 0.7 " relative to the center of the host galaxy in the vlt / fors2 image . at @xmath89 , this corresponds to a projected distance of 5.5 kpc . the identified host galaxy is a late - type galaxy of stellar mass @xmath90 , with a rather low star - forming rate sfr @xmath91 ( @xcite and references therein ) . . lower panel : comptonized + power law best fit of the corresponding spectrum ( from @xmath92 to @xmath93 s).,title="fig : " ] . lower panel : comptonized + power law best fit of the corresponding spectrum ( from @xmath92 to @xmath93 s).,title="fig : " ] [ cols="<,^,^,^,^,^,^,^",options="header " , ] in order to determine the profile of the cbm , a simulation of the prompt emission following the p - grb has been performed . the simulation starts at the transparency of the fireshell with the parameters that we determined above . a trial - and - error procedure is undertaken , guided by the necessity to fit the light curve of grb 090510 . the results of this simulation ( reproduction of the light curve and spectrum , in the time interval from @xmath94 to @xmath95 s , and cbm profile ) are shown in figure [ 090510simlc ] . the average cbm density is found to be @xmath96 . this low value , typical of galactic halo environments , is consistent with the large offset from the center of the host and further justifies the interpretation of grb 090510 as a short grb originating in a binary ns merger . our theoretical fit of the prompt emission ( see red line in the middle plot of figure [ 090510simlc ] ) predicts a cut - off at @xmath97 mev . the spectrum at energy @xmath98 mev could be affected by the onset of the high energy power - law component manifested both in the data of the mini - calorimeter on board agile ( see top panel of figure 4 in @xcite ) and in the data points from the _ fermi_-gbm bgo - b1 detector . there is a weak precursor emission about 0.4 s before the p - grb ( or @xmath99 s in the cosmological rest frame ) . two gev photons have been detected during the precursor emission . precursors are commonly seen in long bursts : @xcite found that @xmath100 20% of them show evidence of an emission preceding the main emission by tens of seconds . short bursts are less frequently associated with precursors . no significant emission from the grb itself is expected prior to the p - grb since it marks the transparency of the fireshell but the precursor may be explainable in the context of a binary ns merger by invoking the effects of the interaction between the two nss just prior to merger . indeed , it has been suggested that precursor emission in short bursts may be caused by resonant fragmentation of the crusts @xcite or by the interaction of the ns magnetospheres @xcite . the timescale ( @xmath100 0.21 s between the precursor and the p - grb ) is consistent with a pre - merger origin of the precursor emission . from its formation to its transparency , the fireshell undergoes a swift evolution . the thermalization of the pair plasma is achieved almost instantaneously ( @xmath101 s , @xcite ) ; and the @xmath102 plasma of grb 090510 reaches the ultra - relativistic regime ( i.e. a lorentz factor @xmath103 ) in a matter of @xmath104 s , according to the numerical simulation . the radius of the fireshell at transparency , @xmath105 cm , corresponds to more than a hundred light - seconds ; however relativistic motion in the direction of the observer squeezes the light curve by a factor @xmath106 , which makes the fireshell capable of traveling that distance under the observed timescale . the spectral analysis of this precursor is limited by the low number of counts . @xcite interpreted the spectrum with a blackbody plus power law model . this leads to a blackbody temperature of @xmath107 kev . the isotropic energy contained in the precursor amounts to @xmath108 erg . an interesting feature of the fireshell model is the possibility to infer a theoretical redshift from the observations of the p - grb and the prompt emission . in the case of grb 090510 , a comparison is therefore possible between the measured redshift @xmath88 and its theoretical derivation . an agreement between the two values would in particular strengthen the validity of our p - grb choice , which would in turn strengthen our results obtained with this p - grb . the feature of redshift estimate stems from the relations , engraved in the fireshell theory , between different quantities computed at the transparency point : the radius in the laboratory frame , the co - moving frame and blue - shifted temperatures of the plasma , the lorentz factor , and the fraction of energy radiated in the p - grb and in the prompt emission as functions of @xmath55 ( see figure 4 in @xcite ) . thus , the ratio @xmath109 implies a finite range for the coupled parameters @xmath43 and @xmath55 ( last panel of figure 4 in @xcite ) . assuming @xmath110 , this ratio is known since it is equal to the ratio between the observed fluences of the respective quantities : @xmath111 with the measured values @xmath112 erg @xmath113 and @xmath114 erg @xmath113 , we find @xmath115 . in addition , knowing the couple [ @xmath116 , @xmath55 ] gives the ( blue - shifted towards the observer ) temperature of the fireshell at transparency @xmath118 ( figure 4 in @xcite , second panel ) . but we also have the following relation between @xmath118 and the observed temperature at transparency @xmath119 , linking their ratio to the redshift : @xmath120 finally , since we assume that @xmath110 , we also have an expression of @xmath43 as a function of @xmath121 using the formula of the k - corrected isotropic energy : @xmath122 where @xmath123 is the photon spectrum of the grb and the fluence @xmath124 is obtained in the full gbm energy range 8 40000 kev . the use of all these relations allows a redshift to be determined by an iterative procedure , testing at every step the value of the parameters @xmath125 and @xmath118 . the procedure successfully ends when both values are consistent according to the relations described above . in the case of grb 090510 , we find @xmath126 , which provides a satisfactory agreement with the measured value @xmath88 . grb 090510 is associated with a high - energy emission , consistently with all other observed s - grbs , i.e. energetic events with @xmath21 erg . the only case of a s - grb without gev emission , namely grb 090227b , has been explained by the absence of alignment between the lat and the source at the time of the grb emission . nevertheless evidence of some gev emission in this source has been recently obtained ( ruffini et al . , in preparation ) . the gev light curve of grb 090510 is plotted in figure [ 090510gev ] together with other s - grb light curves and showing a common power - law behavior , which goes as @xmath127 , similar to the clustering of the gev light curves found by @xcite . these s - grbs are compared with that of the bdhne 130427a which shares a similar behavior . @xcite suggest and argue that the gev emission is related to the presence of a bh and its activity . this view is supported by the fact that the gev emission is delayed with respect to the @xmath38-ray emission : it starts only after the p - grb is over . the gev emission of grb 090510 is particularly intense , reaching @xmath128 erg . such a large value , one of the largest observed among s - grbs , is consistent with the large angular momentum of the newborn bh . this energetic can not be explained in terms of nss in view of the lower value of the gravitational binding energy . the absence of gev emission in s - grfs is also confirmed from the strong upper limit to the gev emission for s - grbs imposed by the fermi - lat sensitivity . we assume for a moment that the gev emission of a s - grf is similar to that of s - grb . we then compute the observed gev flux light curve of s - grb 090510 at different redshifts , e.g. , @xmath129 and @xmath130 , which correspond to the redshifts of the s - grb 081024b and of the s - grb 140402a , respectively ( aimuratov et al . , in preparation ) . the result is that if we compare these computed flux light curves with the _ fermi_-lat sensitivity of the pass 8 release 2 version 6 instrument response functions , which is approximately @xmath131 erg @xmath113s@xmath42 , all of them are always well above the lat broadband sensitivity by a factor @xmath132 ( see figure [ gevz ] ) . this result does not depend on the choice of the source . in their rest - frame all the s - grb gev light curves follow a similar behavior . therefore , the gev emission of s - grb 090510 is always above @xmath132 times to the lat sensitivity , even at higher redshifts . if we now assume that s - grfs do conform to the same behavior of s - grbs , the absence of detection of gev emission implies that the s - grfs have necessarily fluxes at least @xmath133@xmath134 times smaller than those of s - grbs . @xmath7 gev flux light curve of the s - grbs 090510 ( red squares ) , and the corresponding ones obtained by translating this s - grb at @xmath129 ( blue circles ) and at @xmath135 ( green diamonds ) . ] in order to estimate the energy requirement of the @xmath6@xmath7 gev emission of figure [ 090510gev ] we consider the accretion of mass @xmath136 onto a kerr - newman bh , dominated by its angular momentum and endowed with electromagnetic fields not influencing the geometry , which remains approximately that of a kerr bh . we recall that if the infalling accreted material is in an orbit co - rotating with the bh spin , up to @xmath137 of the initial mass is converted into radiation , for a maximally rotating kerr bh , while this efficiency drops to @xmath138 , when the infalling material is on a counter - rotating orbit ( see ruffini & wheeler 1969 , in problem 2 of @xmath139 104 in @xcite ) . therefore , the gev emission can be expressed as @xmath140 and depends not only on the efficiency @xmath141 in the accretion process of matter @xmath136 , but also on the geometry of the emission described by the beaming factor @xmath142 ( here @xmath143 is the half opening angle of jet - like emission ) . depending on the assumptions we introduce in equation [ accretion ] , we can give constraints on the amount of accreted matter or on the geometry of the system . for an isotropic emission , @xmath144 , the accretion of @xmath145 m@xmath71 , for the co - rotating case , and of @xmath146 m@xmath71 , for the counter - rotating case , is required . alternatively , we can assume that the accreted matter comes from the crustal material from an @xmath147 m@xmath71 ns ns binary progenitor . the crustal mass from the nl3 nuclear model for each of these nss is @xmath148 m@xmath71 ( see , e.g. , * ? ? ? * and figure [ nanda ] ) . assuming that crustal material accounts also for the baryon load mass , e.g. , @xmath149 m@xmath71 , the total available mass for accretion is @xmath150 m@xmath71 . then , the presence of a beaming is necessary : from equation [ accretion ] , a half opening beaming angle @xmath151 , for co - rotating case , and @xmath152 , for the counter - rotating case , would be required . the above considerations are clearly independent from the relativistic beaming angle @xmath153 , where the lower limit on the lorentz factor @xmath154 has been derived , in a different context , by @xcite to the gev luminosity light curve ( see figure [ 090510gev ] ) . further consequences on these results for the estimate of the rate of these s - grbs will be presented elsewhere ( ruffini et al . in preparation ) . it is interesting to recall some of the main novelties introduced in this paper with respect to previous works on grb 090510 . particularly noteworthy are the differences from the previous review of short bursts by @xcite , made possible by the discovery of the high energy emission by the fermi team in this specific source @xcite . a new family of short bursts characterized by the presence of a bh and associated high energy emission when lat data are now available , comprehends grbs 081024b , 090227b , 090510 , 140402a , and 140619b ( see , e.g. , figure [ 090510gev ] ) . the excellent data obtained by the fermi team and interpreted within the fireshell model has allowed to relate in this paper the starting point of the high energy emission with the birth of a bh . our fireshell analysis assumes that the @xmath38-ray and the gev components originate from different physical processes . first , the interpretation of the prompt emission differs from the standard synchrotron model : we model the collisions of the baryon accelerated by the grb outflow with the ambient medium following a fully relativistic approach ( see section 2 ) . second , we assume that the gev emission originates from the matter accretion onto the newly - born bh and we show that indeed the energy requirement is fulfilled . this approach explains also the delayed onset of the gev emission , i.e. , it is observable only after the transparency condition , namely after the p - grb emission . the joint utilization of the excellent data from the _ fermi_-gbm nai - n6 and n7 and the bgo - b1 detectors and from the mini - calorimeter on board agile @xcite has given strong observational support to our theoretical work . grb 090510 has been analyzed in light of the recent progress achieved in the fireshell theory and the resulting new classification of grbs . we show that grb 090510 is a s - grb , originating in a binary ns merger ( see figure [ rt_gw ] ) . such systems , by the absence of the associated sn events , are by far the simplest grbs to be analyzed . our analysis indicates the presence of three distinct episodes in s - grbs : the p - grb , the prompt emission , and the gev emission . by following the precise identification of successive events predicted by the fireshell theory , we evidence for the first indication of a kerr bh or , possibly , a kerr - newman bh formation : * the p - grb spectrum of grb 090510 , in the time interval from @xmath92 to @xmath94 s , is best - fitted by a comptonized component ( see figures [ pgrb ] and [ comp ] and table [ tab : fit ] ) , which is interpreted as a convolution of thermal spectra originating in a dyadotorus ( see @xcite and @xcite , figure [ dyadotorus ] , and section 2 ) . * the prompt emission follows at the end of the p - grb ( see figure [ spectotal ] ) . the analysis of the prompt emission within the fireshell model allows to determine the inhomogeneities in the cbm giving rise to the spiky structure of the prompt emission and to estimate as well an averaged cbm density of @xmath155 @xmath18 obtained from a few cbm clouds of mass @xmath48 g and typical dimensions of @xmath53 cm ( see figure [ 090510simlc ] ) . such a density is typical of galactic halos where binary ns are expected to migrate due to large natal kicks . * the late x - ray emission of grb 090510 does not follow the characteristic patterns expected in bdhn events ( see figure [ episode3 ] and @xcite ) . * the gev emission occurs at the end of the p - grb emission and is initially concurrent with the prompt emission . this sequence occurs in both s - grbs @xcite and bdhne @xcite . this delayed long lasting ( @xmath156 s ) gev emission in grb 090510 is one of the most intense ever observed in any grb ( see figure [ 090510gev ] and * ? ? ? * ; * ? ? ? * we then consider accretion on co - rotating and counter - rotating orbits ( see ruffini & wheeler 1969 , in problem 2 of @xmath139 104 in @xcite ) around an extreme kerr bh . assuming the accretion of the crustal mass @xmath157 m@xmath71 from a @xmath147 m@xmath71 ns ns binary , fulfilling global charge neutrality ( see figure [ nanda ] ) , geometrical beaming angles of @xmath152 , for co - rotating case , and @xmath151 , for the counter - rotating case , are inferred . in order to fulfill the transparency condition , the initial lorentz factor of the jetted material has to be @xmath158 ( see section 6.6 ) . * while there is evidence that the gev emission must be jetted , no beaming appears to be present in the p - grb and in the prompt emission , with important consequence for the estimate of the rate of such events @xcite . * the energetic and the possible beaming of the gev emission requires the presence of a kerr bh , or a kerr - newman bh dominated by its angular momentum and with electromagnetic fields not influencing the geometry ( see also section 6.5 ) . * the self - consistency of the entire procedure has been verified by estimating , on the ground of the fireshell theory , the cosmological redshift of the source . the theoretical redshift is @xmath159 ( see section 6.4 ) , close to and consistent with the spectroscopically measured value @xmath10 @xcite . * the values of @xmath160 and @xmath161 of grb 090510 fulfill with excellent agreement the muruwazha relation ( see section 5.2 , figure [ calderone ] and * ? ? ? the main result of this article is that the dyadotorus manifests itself by the p - grb emission and clearly preceeds the prompt emission phase , as well as the gev emission originating from the newly - formed bh . this contrasts with the usual assumption made in almost the totality of works relating bhs and grbs in which the bh preceeds the grb emission . in conclusion , in this article , we take grb 090510 as the prototype of s - grbs and perform a new time - resoved spectral analysis , in excellent agreement with that performed by the agile and the _ fermi _ teams . now this analysis , guided by a theoretical approach successfully tested in this new family of s - grbs , is directed to identify a precise sequence of different events made possible by the exceptional quality of the data of grb 090510 . this include a new structure in the thermal emission of the p - grb emission , followed by the onset of the gev emission linked to the bh formation , allowing , as well , to derive the strucutre of the circumburst medium from the spiky structure of the prompt emission . this sequence , for the first time , illustrates the formation process of a bh . it is expected that this very unique condition of generating a jetted gev emission in such a well defined scenario of a newly - born bh will possibly lead to a deeper understanding of the equally jetted gev emission observed , but not yet explained , in a variety of systems harboring a kerr bh . among these systems we recall binary x - ray sources ( see , e.g. , * ? ? ? * and references therein ) , microquasars ( see , e.g. , * ? ? ? * and references therein ) , as well as , at larger scale , active galactic nuclei . we thank the editor and the referee for their comments which helped to improve the presentation and the contextualization of our results . we are indebted to marco tavani for very interesting comments , as well as for giving us observational supporting evidences . this work made use of data supplied by the uk _ swift _ data center at the university of leicester . m. e. , m. k. , and y. a. are supported by the erasmus mundus joint doctorate program by grant numbers 2012 - 1710 , 2013 - 1471 , and 2014 - 0707 respectively , from the eacea of the european commission . c.c . acknowledges indam - gnfm for support . m.m . acknowledges the partial support of the project n 3101/gf4 ipc-11 , and the target program f.0679 of the ministry of education and science of the republic of kazakhstan . , j. 2003 , in american institute of physics conference series , vol . 662 , gamma - ray burst and afterglow astronomy 2001 : a workshop celebrating the first year of the hete mission , ed . g. r. ricker & r. k. vanderspek , 229236 , i. b. , klebesadel , r. w. , & evans , w. d. 1975 , in annals of the new york academy of sciences , vol . 262 , seventh texas symposium on relativistic astrophysics , ed . p. g. bergman , e. j. fenyves , & l. motz , 145158

in a new classification of merging binary neutron stars ( nss ) we separate short gamma - ray bursts ( grbs ) in two sub - classes . the ones with @xmath0 erg coalesce to form a massive ns and are indicated as short gamma - ray flashes ( s - grfs ) . the hardest , with @xmath1 erg , coalesce to form a black hole ( bh ) and are indicated as genuine short - grbs ( s - grbs ) . within the fireshell model , s - grbs exhibit three different components : the p - grb emission , observed at the transparency of a self - accelerating baryon-@xmath2 plasma ; the prompt emission , originating from the interaction of the accelerated baryons with the circumburst medium ; the high - energy ( gev ) emission , observed after the p - grb and indicating the formation of a bh . grb 090510 gives the first evidence for the formation of a kerr bh or , possibly , a kerr - newman bh . its p - grb spectrum can be fitted by a convolution of thermal spectra whose origin can be traced back to an axially symmetric dyadotorus . a large value of the angular momentum of the newborn bh is consistent with the large energetics of this s - grb , which reach in the @xmath3@xmath4 kev range @xmath5 erg and in the @xmath6@xmath7 gev range @xmath8 erg , the most energetic gev emission ever observed in s - grbs . the theoretical redshift @xmath9 that we derive from the fireshell theory is consistent with the spectroscopic measurement @xmath10 , showing the self - consistency of the theoretical approach . all s - grbs exhibit gev emission , when inside the _ fermi_-lat field of view , unlike s - grfs , which never evidence it . the gev emission appears to be the discriminant for the formation of a bh in grbs , confirmed by their observed overall energetics .

1607.02400

measurements at lep , sld , and the tevatron have been used extensively to limit models with physics beyond that of the standard model ( sm)@xcite . by performing global fits to a series of precision measurements , information about the parameters of new models can be inferred@xcite . the simplest example of this approach is the prediction of the @xmath3 boson mass . in the standard model , the @xmath3- boson mass , @xmath1 , can be predicted in terms of other parameters of the theory . the predicted @xmath3 boson mass is strongly correlated with the experimentally measured value of the top quark mass , @xmath4 , and increases quadratically as the top quark mass is increased . this strong correlation between @xmath1 and @xmath4 in the standard model can be used to limit the allowed region for the higgs boson mass@xcite . in a model with higgs particles in representations other than @xmath5 doublets and singlets , there are more parameters in the gauge / higgs sector than in the standard model . the sm tree level relation , @xmath6 no longer holds and when the theory is renormalized at one loop , models of this type will require extra input parameters@xcite . models with new physics are often written in terms of the sm lagrangian , @xmath7 plus an extra contribution , @xmath8 where @xmath9 represents contributions from new physics beyond the sm . phenomenological studies have then considered the contributions of @xmath7 at one - loop , plus the tree level contributions of @xmath9 . in this note , we give two specific examples with @xmath0 at tree level , where we demonstrate that this procedure is incorrect . we discuss in detail what happens in these models when the scale of the new physics becomes much larger than the electroweak scale and demonstrate explicitly that the sm is not recovered . the possibility of a heavy higgs boson which is consistent with precision electroweak data has been considered by chivukula , hoelbling and evans@xcite and by peskin and wells@xcite in the context of oblique corrections . in terms of the @xmath10 , @xmath11 and @xmath12 parameters@xcite , a large contribution to isospin violation , @xmath13 , can offset the contribution of a heavy higgs boson to electroweak observables such as the @xmath3 boson mass . the triplet model considered in this paper provides an explicit realization of this mechanism . the oblique parameter formulation neglects contributions to observables from vertex and box diagrams , which are numerically important in the example discussed here . in section [ renorm ] , we review the important features of the sm for our analysis . we discuss two examples in sections [ higgstrip ] and appendix [ lrmodel ] where the new physics does not decouple from the sm at one - loop . for simplicity , we consider only the dependence of the @xmath3 boson mass on the top quark mass and demonstrate that a correct renormalization scheme gives very different results from the sm result in these models . section [ higgstrip ] contains a discussion of the sm augmented by a real scalar triplet , and appendix [ lrmodel ] contains a discussion of a left - right @xmath14 symmetric model . in section [ nondecoupling ] , we show that the dependence on scalar masses in the w - boson mass is quadratic and demonstrate that the triplet is non - decoupling . our major results are summarized in eq . [ cc1]-[cc3 ] . these results are novel and have not been discussed in the literature before . section [ results ] contains our numerical results and section [ conc ] concludes this paper . similar results in the context of the littlest higgs model have previously been found in ref . . the one - loop renormalization of the sm has been extensively studied@xcite and we present only a brief summary here , in order to set the stage for sections [ higgstrip ] and appendix [ lrmodel ] . in the electroweak sector of the sm , the gauge sector has three fundamental parameters , the @xmath15 gauge coupling constants , @xmath16 and @xmath17 , as well as the vacuum expectation ( vev ) of the higgs boson , @xmath18 . once these three parameters are fixed , all other physical quantities in the gauge sector can be derived in terms of these three parameters and their counter terms . we can equivalently choose the muon decay constant , @xmath19 , the z - boson mass , @xmath20 , and the fine structure constant evaluated at zero momentum , @xmath21 , as our input parameters . experimentally , the measured values for these input parameters are@xcite , @xmath22 the w - boson mass then can be defined through muon decay@xcite , @xmath23\ ] ] where @xmath24 summarizes the radiative corrections , @xmath25 where @xmath26 , @xmath27 and @xmath28 is the weak mixing angle . the sm satisfies @xmath29 at tree level , @xmath30 in eq . ( [ rhodef ] ) , @xmath1 and @xmath31 are the physical gauge boson masses , and so our definition of the weak mixing angle , @xmath32 , corresponds to the on - shell scheme@xcite . it is important to note that in the sm , @xmath32 is not a free parameter , but is derived from @xmath33 the counterterms of eq . ( [ drdef ] ) are given by@xcite , @xmath34 where @xmath35 , for @xmath36 , are the gauge boson 2-point functions ; @xmath37 is defined as @xmath38 . the term @xmath39 contains the box and vertex contributions to the renormalization of @xmath40@xcite . the counterterm for @xmath41 can be derived from eq . ( [ rhodef ] ) , @xmath42 = \frac{\overline{c}_{\theta}^{2}}{\overline{s}_{\theta}^{2 } } \biggl [ \frac{\pi_{zz}(m_{z}^{2})}{m_{z}^{2 } } - \frac{\pi_{ww}(m_{w}^{2})}{m_{w}^{2 } } \biggr]\quad . \label{stdef}\ ] ] putting these contributions together we obtain , @xmath43\quad .\nonumber\end{aligned}\ ] ] these gauge boson self - energies can be found in ref . and and we note that the fermion and scalar contributions to the two - point function @xmath44 vanish . the dominant contributions to @xmath24 is from the top quark , and the contributions of the top and bottom quarks to the gauge boson self - energies are given in [ loop ] . the @xmath45 dependence in @xmath46 and @xmath47 exactly cancel , thus the difference , @xmath48 , depends on @xmath49 only logarithmically . the second term , @xmath50 , also depends on @xmath49 logarithmically . however , the quadratic @xmath45 dependence in @xmath51 and @xmath52 do not cancel . thus @xmath24 depends on @xmath49 quadratically , and is given by the well known result , keeping only the two - point functions that contain a quadratic dependence on @xmath49@xcite , @xmath53 where @xmath54 is the number of colors and the superscript @xmath55 denotes that we have included only the top quark contributions , in which the dominant contribution is quadratic . the complete contribution to @xmath24 can be approximated , @xmath56 the first term in eq . ( [ drsm ] ) results from the scaling of @xmath57 from zero momentum to @xmath20@xcite . in the numerical results , the complete contributions to @xmath24 from top and bottom quarks , the higgs boson as well as the gauge bosons are included , as given in eq . ( [ drsm1 ] ) . in this section , we consider the sm with an additional higgs boson transforming as a real triplet ( y=0 ) under the @xmath59 gauge symmetry . hereafter we will call this the triplet model ( tm)@xcite . this model has been considered at one - loop by blank and hollik@xcite and we have checked that our numerical codes are correct by reproducing their results . in addition , we derive the scalar mass dependence in this model and show that the triplet is non - decoupling by investigating various scalar mass limits . we also find the conditions under which the lightest neutral higgs can be as heavy as a tev , which has new important implications on higgs searches . these results concerning the scalar fields are presented in the next section . the @xmath58 higgs doublet in terms of its component fields is given by , @xmath60 with @xmath61 being the goldstone boson corresponding to the longitudinal component of the @xmath62 gauge boson . a real @xmath59 triplet , @xmath63 , can be written as @xmath64 , @xmath65 there are thus four physical higgs fields in the spectrum : there are two neutral higgs bosons , @xmath66 and @xmath67 , @xmath68 and the mixing between the two neutral higgses is described by the angle @xmath69 . the charged higgses @xmath70 are linear combinations of the charged components in the doublet and the triplet , with a mixing angle @xmath71 , @xmath72 where @xmath73 are the goldstone bosons corresponding to the longitudinal components of @xmath74 . the masses of these four physical scalar fields , @xmath75 , @xmath76 and @xmath77 , respectively , are free parameters in the model . the @xmath3 boson mass is given by , @xmath78 where @xmath79 is the vev of the neutral component of the @xmath59 higgs boson and @xmath80 is the vacuum expectation value of the additional scalar , leading to the relationship @xmath81 . a real triplet does not contribute to @xmath31 , leading to @xmath82 the main result of this section is to show that the renormalization of a theory with @xmath83 at tree level is fundamentally different from that of the sm . due to the presence of the @xmath58 triplet higgs , the gauge sector now has four fundamental parameters , the additional parameter being the vev of the @xmath58 triplet higgs , @xmath84 . a consistent renormalization scheme thus requires a fourth input parameter@xcite . we choose the fourth input parameter to be the effective leptonic mixing angle , @xmath85 , which is defined as the ratio of the vector to axial vector parts of the @xmath86 coupling , @xmath87 with @xmath88 and @xmath89 . this leads to the definition of @xmath85 , @xmath90 the measured value from lep is given by @xmath91@xcite . as usual the @xmath3 boson mass is defined through muon decay@xcite , @xmath92 where we have chosen @xmath93 instead of @xmath94 as an input parameter . the contribution to @xmath95 is similar to that of the sm , @xmath96 where the counter term @xmath97 is defined through @xmath98 , @xmath20 and @xmath85 as , @xmath99 unlike in the sm case where @xmath100 is defined through @xmath98 and @xmath20 as given in eq . ( [ stdef ] ) , now @xmath85 is an independent parameter , and its counter term is given by@xcite , @xmath101\ ; \biggr ] \ ; , \nonumber \\ & \equiv & \biggl(\frac{\hat{c}_{\theta}}{\hat{s}_{\theta } } \biggr ) \frac{re\biggl(\pi^{\gamma z}(m_{z}^{2})\biggr)}{m_{z}^{2 } } + \delta_{v - b}^{\prime } \ ; , \end{aligned}\ ] ] where @xmath102 is the axial part of the electron self - energy , @xmath103 and @xmath104 are the vector and axial - vector form factors of the vertex corrections to the @xmath105 coupling . these effects have been included in our numerical results@xcite . the total correction to @xmath95 in this case is then given by , @xmath106 where @xmath39 summarizes the vertex and box corrections in the tm model , and it is given by@xcite , @xmath107 , \qquad r \equiv m_{w}^{2 } / m_{z}^{2 } \ ; , \ ] ] where we show only the finite contributions in the above equation . keeping only the top quark contribution , @xmath108 where @xmath109 is the momentum cutoff in dimensional regularization and the definition of the function @xmath110 can be found in appendix a. as @xmath111 is logarithmic , the @xmath49 dependence of @xmath98 is now logarithmic . we note that this much softer relation between @xmath98 and @xmath49 is independent of the choice of the fourth input parameter . this will become clear in our second example , the left - right symmetric model , given in [ lrmodel ] . in our numerical results , we have included in @xmath95 the complete contributions , which are summarized in [ scalar ] , from the top and bottom quarks and the four scalar fields , as well as the gauge bosons , and the complete set of vertex and box corrections . as shown in [ scalar ] , @xmath95 depends on scalar masses quadratically . this has important implications for models with triplets , such as the littlest higgs model@xcite . the two point function @xmath44 does not have any scalar dependence , while @xmath112 and @xmath113 depend on scalar masses logarithmically . the quadratic dependence thus comes solely from the function @xmath114 . when there is a large hierarchy among the three scalar masses ( case ( c ) in [ scalar ] and its generalization ) , all contributions are of the same sign , and are proportional to the scalar mass squared , @xmath115 \nonumber\\ & & -s_{\delta}^{2}\frac{m_{h^{0}}^{2 } m_{h^{\pm}}^{2}}{2m_{w}^{4 } } % \biggl ( 1 + \ln \biggl(\frac{m_{h^\pm}^{2}}{m_{h^{0}}^{2}}\biggr ) % \biggr ) - c_{\delta}^{2 } \frac{2m_{k^{0}}^{2}m_{h^{\pm}}^{2}}{m_{w}^{4 } } % \biggl(1 + \ln \biggl(\frac{m_{h^\pm}^{2}}{m_{k^{0}}^{2}}\biggr)%\biggr ) \biggr\ } \ ; , \nonumber\end{aligned}\ ] ] for @xmath116 . thus the scalar contribution to @xmath95 in this case is very large , and it grows with the scalar masses . on the other hand , when the mass splitting between either pair of the three scalar masses is small ( case ( a ) and ( b ) and their generalization ) , the scalar contributions grow with the mass splitting@xcite , @xmath117 \nonumber\\ & & \qquad + \frac{5}{72}\biggl [ s_{\delta}^{2}\frac { \bigl(m_{h^{\pm}}^{2}-m_{h^{0}}^{2}\bigr)}{m_{h^{0}}^{2 } } + 4c_{\delta}^{2 } \frac { \bigl(m_{h^{\pm}}^{2 } - m_{k^{0}}^{2}\bigr)}{m_{k^{0}}^{2 } } \biggr ] \biggr\ } \nonumber \ ; , \end{aligned}\ ] ] for @xmath118 , and , @xmath119 \nonumber\\ & & -s_{\delta}^{2}\frac{m_{h^{0}}^{2}m_{h^{\pm}}^{2}}{2m_{w}^{4 } } % \biggl ( 1 + \ln \biggl(\frac{m_{h^\pm}^{2}}{m_{h^{0}}^{2}}\biggr ) % \biggr ) + \frac{5}{18}c_{\delta}^{2 } \frac{\bigl(m_{h^{\pm}}^{2 } - m_{k^{0}}^{2}\bigr)}{m_{k^{0}}^{2 } } \biggr\ } \nonumber \ ; , \end{aligned}\ ] ] for @xmath120 . cancellations can occur in this case among contributions from different scalar fields , leading to the viability of a heavier neutral higgs boson than is allowed in the sm@xcite . the non - decoupling property of the triplet is seen in eq . ( [ cc1 ] ) , ( [ cc2 ] ) and ( [ cc3 ] ) . because @xmath95 depends quadratically on the scalar masses@xcite , the scalars must be included in any effective field theory analysis of low energy physics . the scalar potential of the model with a @xmath58 triplet and an @xmath58 doublet is given by the following@xcite : @xmath121 where @xmath122 denotes the pauli matrices , and @xmath123 from the minimization conditions ( see [ minimize ] ) , @xmath124 we obtain the following conditions , @xmath125 the two mixing angles , @xmath69 and @xmath71 , in the neutral and charged higgs sectors defined in eqs . ( [ neutralmix ] ) and ( [ chargedmix ] ) , are solutions to the following two equations@xcite , @xmath126 \nonumber \label{cond1}\\ 0 & = & -\lambda_{4}v + \lambda_{3}vv^{\prime}+\tan\gamma \biggl [ \mu_{1}^{2 } - \mu_{2}^{2 } + 3 \lambda_{1 } v^{2 } - \frac{1}{2 } \lambda_{3 } v^{2 } -\lambda_{4}v^{\prime } -3\lambda_{2 } v^{\prime 2 } \\ & & \qquad + \frac{1}{2 } \lambda_{3 } v^{\prime 2 } + \lambda_{4 } v \tan\gamma - \lambda_{3 } vv^{\prime}\tan\gamma \biggr ] \ ; , \label{cond2 } \nonumber\end{aligned}\ ] ] which are obtained by minimizing the scalar potential . in terms of the parameters in the scalar potential , the masses of the four scalar fields are given by@xcite , @xmath127 this model has six parameters in the scalar sector , @xmath128 . equivalently , we can choose @xmath129 as the independent parameters . two of these six parameters , @xmath18 and @xmath130 , contribute to the gauge boson masses . when turning off the couplings between the doublet and the triplet in the scalar potential , @xmath131 , the triplet could still acquire a vev , @xmath132 , provided that @xmath133 is negative . since we have not observed any light scalar experimentally up to the ew scale , the triplet mass which is roughly of order @xmath134 has to be at least of the ew scale , @xmath135 . this is problematic because the vev of a real triplet only contributes to @xmath98 but not to @xmath31 , which then results in a contribution of order @xmath136 to the @xmath137 parameter , due to the relation , @xmath138 . for @xmath134 greater than @xmath18 , the ew symmetry is broken at a high scale . in order to avoid these problems , the parameter @xmath133 thus has to be positive so that the triplet does not acquire a vev via this mass term when @xmath139 is turned off . once the coupling @xmath140 is turned on while keeping @xmath141 , the term @xmath142 effectively plays the role of the mass term for @xmath63 and for @xmath143 . thus , similar to the reasoning given above , for @xmath144 , the coupling @xmath140 has to be positive so that it does not induce a large triplet vev . for simplicity , consider the case when there is no mixing in the neutral higgs sector , @xmath145 . in this case , when the mixing in the charged sector approaches zero , @xmath146 , the masses @xmath76 and @xmath77 approach infinity , and their difference @xmath147 approaches zero . the contribution due to the new scalars thus vanishes , and only the lightest neutral higgs contributes to @xmath95 . even though the contribution due to the new scalars vanishes , @xmath95 does not approach @xmath24 . this is because in the tm case , four input parameters are needed , while in the sm case three inputs are needed . there is no continuous limit that takes one case to the other@xcite . one way to achieve the @xmath146 limit is to take the mass parameter @xmath148 while keeping the parameter @xmath139 finite . ( [ cond1 ] ) then dictates that @xmath149 . however , satisfying eq . ( [ cond2 ] ) requires that @xmath150 . as @xmath151 , this condition implies that the dimensionless coupling constant @xmath140 has to scale as @xmath152 , which approaches infinity as @xmath146 . this can also be seen from eq . ( [ neutralmix1 ] ) . as there is no mixing in the neutral sector , @xmath153 the condition @xmath154 then follows . so , in the absence of the neutral mixing , @xmath145 , in order to take the charged mixing angle @xmath71 to zero while holding @xmath139 fixed , one has to take @xmath140 to infinity . in other words , for the triplet to decouple requires a dimensionless coupling constant @xmath140 to become strong , leading to the breakdown of the perturbation theory . alternatively , the neutral mixing angle @xmath69 can approach zero by taking @xmath155 while keeping @xmath140 and @xmath139 fixed . in this case , the minimization condition , @xmath156 where @xmath157 , implies that the charged mixing angle @xmath71 has to approach zero . this again corresponds to the case where the custodial symmetry is restored , by which we mean that the triplet vev vanishes , @xmath158 . in this case , severe fine - tuning is needed in order to satisfied the condition given in eq . ( [ cond2 ] ) . another way to get @xmath146 is to have @xmath159 , which trivially satisfies eq . ( [ cond1 ] ) . this can also be seen from eq . [ chargemix2 ] , @xmath160 eq . ( [ cond2 ] ) then gives @xmath161 . so for small @xmath140 , the masses of these additional scalar fields are of the weak scale , @xmath162 . this corresponds to a case when the custodial symmetry is restored . so unless one imposes by hand such symmetry to forbid @xmath139 , four input parameters are always needed in the renormalization . if there is a symmetry which makes @xmath163 ( to all orders ) , say , @xmath164 , then there are only three input parameters needed . so the existence of such a symmetry is crucial when one - loop radiative corrections are concerned . mass as a function of the top quark mass in the sm , tm and the lr model . the data point represents the experimental values with @xmath165 error bars@xcite . for the sm , we include the complete contributions from top and bottom quarks , the sm higgs boson with @xmath166 gev , and the gauge bosons . for the tm and the lr model , we include only the top quark contribution and the absolute normalization is fixed so that the curves intersect the data point . the @xmath167 boson mass is chosen to be @xmath168 tev in the lr model . ] mass in the tm as a function of the top quark mass for scalar masses @xmath169 gev , with @xmath93 and @xmath85 varying within their @xmath165 limits@xcite @xmath170 and @xmath171 . the solid curve indicates the prediction with @xmath93 and @xmath85 taking the experimental central values , @xmath172 and @xmath173 . the area bounded by the short dashed ( dotted ) curves indicates the prediction with @xmath174 ( @xmath175 ) and @xmath176 ( @xmath177 ) varying with its @xmath165 limits . the data point represents the experimental values with @xmath165 error bars@xcite . ] mass in the tm as a function of the top quark mass for scalar masses , @xmath75 , @xmath76 and @xmath77 , varying independently between ( a ) @xmath178 tev , ( b ) @xmath179 gev , and ( c ) @xmath180 gev . the data point represents the experimental values with @xmath165 error bars@xcite.,title="fig : " ] mass in the tm as a function of the top quark mass for scalar masses , @xmath75 , @xmath76 and @xmath77 , varying independently between ( a ) @xmath178 tev , ( b ) @xmath179 gev , and ( c ) @xmath180 gev . the data point represents the experimental values with @xmath165 error bars@xcite.,title="fig : " ] mass in the tm as a function of the top quark mass for scalar masses , @xmath75 , @xmath76 and @xmath77 , varying independently between ( a ) @xmath178 tev , ( b ) @xmath179 gev , and ( c ) @xmath180 gev . the data point represents the experimental values with @xmath165 error bars@xcite.,title="fig : " ] mass in the tm as a function of the lightest neutral higgs boson mass , @xmath75 , for various values of @xmath76 and @xmath77 . the area bounded by the two horizontal lines is the @xmath165 allowed region for @xmath98@xcite.,title="fig : " ] mass in the tm as a function of the lightest neutral higgs boson mass , @xmath75 , for various values of @xmath76 and @xmath77 . the area bounded by the two horizontal lines is the @xmath165 allowed region for @xmath98@xcite.,title="fig : " ] mass in the tm as a function of the lightest neutral higgs boson mass , @xmath75 , for various values of @xmath76 and @xmath77 . the area bounded by the two horizontal lines is the @xmath165 allowed region for @xmath98@xcite.,title="fig : " ] the previous section has presented analytic results for the triplet model , demonstrating that the dependence of the @xmath3 mass on the top quark mass is logarithmic , while the dependence on the scalar masses is quadratic . a dramatic change in the behavior of the @xmath3 mass is also observed in the @xmath181 model@xcite . for comparison with the triplet model , we summarize the results of the left - right model in appendix [ lrmodel ] . in this case , the dependence of the @xmath3 mass on the top quark mass is weakened from that of the sm since it depends on @xmath182 , where @xmath183 is the heavy charged gauge boson mass of the left - right model . the dependence of the @xmath3 mass on the top quark mass , @xmath49 , in the case of the sm , the model with a triplet higgs , and the minimal left - right model , are shown in fig . [ fg : mwmt ] . for the sm , we include the complete contributions from top and bottom quarks , the sm higgs boson with @xmath166 gev , and the gauge bosons . in this case , the @xmath49 dependence in the prediction for @xmath1 is quadratic . the range of values for the input parameter @xmath49 that give a prediction for @xmath98 consistent with the experimental @xmath165 limits@xcite , @xmath184 gev , is very narrow . it coincides with the current experimental limits@xcite , @xmath185 gev . for the triplet model and the lr model , we include only the top quark contribution . as we have shown in sec . [ higgstrip ] , the prediction for @xmath1 in the triplet model depends on @xmath49 only logarithmically . in our numerical result for the left - right model , we have used @xmath186 in the gauge sector , in addition to @xmath49 in the fermion sector , to predict @xmath98 . here we have identified @xmath187 and @xmath188 as the @xmath3 and @xmath62 bosons in the sm and consequently @xmath189 and @xmath190 . in this case , the @xmath49 dependence in the prediction for @xmath98 is similarly softer because the top quark contributions are suppressed by a heavy scale , @xmath183 . in the triplet model and the left - right model , the range of @xmath49 that gives a prediction for @xmath98 consistent with the experimental value is thus much larger , ranging from @xmath191 to @xmath192 gev . the presence of the triplet higgs thus dramatically changes the @xmath49 dependence in @xmath98 . this is clearly demonstrated in fig . [ fg : mwmt ] by the almost flat curves of the triplet and left - right symmetric models , contrary to that of the sm , which is very sensitive to @xmath49 . in fig . [ fg : mwmterr ] , we show the prediction for @xmath98 as a function of @xmath49 in the triplet model , with @xmath93 and @xmath85 varying within the @xmath165 limits@xcite , @xmath170 and @xmath171 . we find that the prediction for @xmath98 is very sensitive to the input parameters @xmath177 and @xmath85 . the complete contributions from the top and bottom quarks and the sm gauge bosons , as well as all four scalar fields in the triplet model are included in fig . [ fg : mwmt1 ] and [ fg : mwmh ] . we have also included the box and vertex corrections . in fig . [ fg : mwmt1 ] , we show the prediction in the triplet model for @xmath98 as a function of @xmath49 , allowing @xmath75 , @xmath77 and @xmath76 to vary independently between @xmath178 tev , @xmath179 gev and @xmath180 gev . interestingly , for all scalar masses in the range of @xmath178 tev , the prediction for @xmath98 in the tm model still agrees with the experimental @xmath165 limits . [ fg : mwmh ] shows the prediction for @xmath98 as a function of @xmath75 for various values of @xmath76 and @xmath77 . for small @xmath147 , the lightest neutral higgs boson mass can range from @xmath193 gev to a tev and still satisfy the experimental prediction for @xmath98 this agrees with our conclusion in sec . [ higgstrip ] that to minimize the scalar contribution to @xmath95 , the mass splitting @xmath147 has to be small and that when the mass splitting is small , cancellations can occur between the contributions of the lightest neutral higgs and those of the additional scalar fields . this has new important implications for the higgs searches . we have considered the top quark contribution to muon decay at one loop in the sm and in two models with @xmath0 at tree level : the sm with an addition real scalar triplet and the minimal left - right model . in these new models , because the @xmath137 parameter is no longer equal to one at the tree level , a fourth input parameter is required in a consistent renormalization scheme . these models illustrate a general feature that the @xmath49 dependence in the radiative corrections @xmath95 becomes logarithmic , contrary to the case of the sm where @xmath24 depends on @xmath4 quadratically . one therefore loses the prediction for @xmath49 from radiative corrections . on the other hand , due to cancellations between the contributions to the radiative corrections from the sm higgs and the triplet , a higgs mass @xmath75 as large as a few tev is allowed by the @xmath3 mass measurement . we emphasize that by taking the triplet mass to infinity , one does not recover the sm . this is due to the fact that the triplet scalar field is non - decoupling , and it implies that the one - loop electroweak results can not be split into a sm contribution plus a piece which vanishes as the scale of new physics becomes much larger than the weak scale . this fact has been overlooked by most analyses in the littlest higgs model@xcite , and correctly including the effects of the triplet can dramatically change the conclusion on the viability of the model@xcite . such non - decoupling effect has been pointed out in the two higgs doublet model@xcite , left - right symmetric model@xcite , and the littlest higgs model@xcite . it has _ not _ been discussed before in the model with a triplet . we comment that the non - decoupling observed in these examples do not contradict with the common knowledge that in gut models heavy scalars decouple . these two cases are fundamentally different because in gut models , heavy scalar fields do not acquire vev that break the ew symmetry , while in cases where non - decoupling is observed , heavy scalar fields _ do _ acquire vev that breaks the ew symmetry . the quadratic dependence in scalar masses in the triplet model can be easily understood physically . in sm with only the higgs doublet present , the quadratic scalar mass contribution is protected by the tree level custodial symmetry , and thus the higgs mass contribution is logarithmic at one - loop . this is the well - known screening theorem by veltman@xcite . as the custodial symmetry is broken in the sm at one - loop due to the mass splitting between the top and bottom quarks , the two - loop higgs contribution is quadratic . in models with a triplet higgs , as the custodial symmetry is broken already at the tree level , there is no screening theorem that protects the quadratic scalar mass dependence from appearing . our results demonstrate the importance of performing the renormalization correctly according to the ew structure of the new models . this manuscript has been authored by brookhaven science associates , llc under contract no . de - ac02 - 76ch1 - 886 with the u.s . department of energy . the united states government retains , and the publisher , by accepting the article for publication , acknowledges , a world - wide license to publish or reproduce the published form of this manuscript , or allow others to do so , for the united states government purpose . thanks w. marciano for useful discussions . we summarize below the leading contributions due to the sm top loop to the self - energies of the gauge bosons@xcite , where the definitions of the passarino - veltman functions utilized below are given in@xcite . @xmath194 \nonumber\\ \pi^{ww } ( 0 ) & = & -\frac{3 \alpha}{16\pi s_{\theta}^{2 } } \ , \cdot \ , m_{t}^{2 } \biggl [ 1 + 2 \ln \biggl ( \frac{q^{2}}{m_{t}^{2 } } \biggr ) \biggr ] \\ \pi^{zz } ( m_{z}^{2 } ) & = & -\frac{3 \alpha}{8\pi s_{\theta}^{2 } c_{\theta}^{2 } } \biggl [ \biggl(\biggl ( \frac{1}{2 } - \frac{4}{3}s_{\theta}^{2 } \biggr)^{2 } + \frac{1}{4 } \biggr ) h_{1}(m_{t}^{2})\\ & & -\frac{8}{3 } s_{\theta}^{2 } \biggl ( 1-\frac{4}{3}s_{\theta}^{2 } \biggr ) h_{2}(m_{t}^{2 } ) \biggr]\nonumber \\ \pi_{\gamma\gamma}^{\prime}(0 ) & = & \frac{4\alpha}{9\pi } \ln \biggl(\frac{q^{2}}{m_{t}^{2}}\biggr ) \\ \pi^{\gamma z } ( m_{z}^{2 } ) & = & -\frac{\alpha}{\pi s_{\theta}c_{\theta } } \biggl ( \frac{1}{2 } - \frac{4}{3 } s_{\theta}^{2 } \biggr ) m_{z}^{2 } \biggl [ \frac{1}{3 } \ln \biggl(\frac{q^{2}}{m_{t}^{2}}\biggr ) -2 i_{3}\biggl(\frac{m_{z}^{2}}{m_{t}^{2}}\biggr)\biggr]\end{aligned}\ ] ] where @xmath195 \\ h_{2}(m_{t}^{2 } ) & = & m_{t}^{2 } \biggl [ i_{1}\biggl(\frac{m_{z}^{2}}{m_{t}^{2}}\biggr ) - \ln \biggl(\frac{q^{2}}{m_{t}^{2}}\biggr ) \biggr]\quad .\end{aligned}\ ] ] the integrals are defined as , @xmath196 here @xmath197 is defined in the on - shell scheme ( eq . ( [ swos ] ) ) for the sm and as the effective weak mixing angle ( eq . ( [ swdef ] ) ) for the tm and lr model . the complete contributions to various two - point functions that appear in @xmath95 are given below , where the scalar and fermion contributions are given in ref . , and we have taken the sm gauge boson contributions from ref . . @xmath198 \\ & & + \frac{3 \alpha}{4\pi \hat{s}_{\theta}^{2 } } \biggl [ a_{0}(m_{t}^{2 } ) + m_{b}^{2 } b_{0}(m_{w}^{2},m_{b}^{2},m_{t}^{2 } ) \nonumber\\ & & \qquad -m_{w}^{2 } b_{1}(m_{w}^{2},m_{b}^{2},m_{t}^{2 } ) - 2 b_{22}(m_{w}^{2},m_{b}^{2},m_{t}^{2 } ) \biggr ] \nonumber\\ & & + { \alpha\over 4\pi \hat{s}_{\theta}^{2}}\biggl\ { s_\delta^2h(m_{h^0 } , m_{h^\pm})+c_\delta^2h(m_{h^0 } , m_w)\nonumber \\ & & \qquad + 4c_\delta^2h(m_{k^0 } , m_{h^\pm})+4s_\delta^2h(m_{k^0 } , m_w)\nonumber \\ & & \qquad + s_\delta^2h(m_z , m_{h^\pm})+c_\delta^2h(m_z , m_w)\biggr\ } \nonumber\\ & & + { \alpha\over 4 \pi \hat{s}_{\theta}^{2}}m_w^2 \biggl [ { s_\delta^2 c_\delta^2\over \hat{c}_{\theta}^{2 } } \biggl(b_0(0,m_z , m_{h^\pm})- b_0(m_{w},m_z , m_{h^\pm } ) \biggr ) \nonumber\\ & & \qquad + { ( s_\delta^2-\hat{s}_\theta^2)^2\over \hat{c}_{\theta}^{2 } } \biggl ( b_0(0,m_z , m_w ) - b_0(m_{w},m_z , m_w ) \biggr ) \nonumber \\ & & \qquad + \hat{s}_\theta^2 \biggl(b_0(0,0,m_w ) - b_0(m_w,0,m_w ) \biggr ) \nonumber\\ & & \qquad + c_\delta^2 \biggl ( b_0(0,m_{h^0 } , m_w ) - b_0(m_w , m_{h^0 } , m_w ) \biggr ) \nonumber\\ & & \qquad + 4 s_\delta^2 \biggl(b_0(0,m_{k^0},m_w ) - b_0(m_w , m_{k^0},m_w ) \biggr ) \biggr ] \nonumber \\ & & + \frac{\alpha}{4\pi \hat{s}_{\theta}^{2 } } \biggl [ \hat{c}_{\theta}^{2 } \biggl ( a_{1}(0,m_{z},m_{w } ) - a_{1}(m_w , m_z , m_w ) \biggr ) \nonumber\\ & & \qquad + \hat{s}_{\theta}^{2 } \biggl ( a_{1}(0,0,m_w ) - a_{1}(m_w,0,m_w ) \biggr ) \nonumber\\ & & \qquad - 2\hat{c}_{\theta}^{2 } h(m_z , m_w ) - 2\hat{s}_{\theta}^{2 } h(0,m_w ) \biggr]\nonumber \ ; , \\ & & \nonumber \\&&\nonumber \\&&\nonumber \\&&\nonumber \\&&\nonumber \\&&\nonumber \\&&\nonumber \\&&\nonumber \\&&\nonumber \\&&\nonumber\end{aligned}\ ] ] @xmath199 \\ & & + \frac{\alpha}{4\pi \hat{s}_{\theta}\hat{c}_{\theta } } \biggl [ 2(c_{\delta}^{2}-\hat{s}_{\theta}^{2 } + \hat{c}_{\theta}^{2})b_{22}(m_z , m_{h^\pm},m_{h^\pm } ) \nonumber\\ & & \qquad + 2 ( s_{\delta}^2-\hat{s}_{\theta}^2+\hat{c}_{\theta}^2 ) b_{22}(m_z , m_w , m_w ) \nonumber\\ & & \qquad + ( \hat{s}_{\theta}^2-\hat{c}_{\theta}^2-c_{\delta}^2)a(m_{h^\pm } ) + ( \hat{s}_{\theta}^2-\hat{c}_{\theta}^2-s_{\delta}^2)a(m_{w } ) \biggr ] \nonumber\\ & & + \frac{\alpha}{4\pi}\bigl(2m_{w}^{2 } ) \frac{\hat{s}_{\theta}^2-s_{\delta}^2 } { \hat{s}_{\theta}\hat{c}_{\theta } } b_{0}(m_z , m_w , m_w ) \nonumber\\ & & -\frac{\alpha}{4\pi \hat{s}_{\theta}^{2 } } \biggl [ \hat{s}_{\theta}\hat{c}_{\theta } a_{1}(m_z , m_w , m_w ) + 2 \hat{c}_{\theta}\hat{s}_{\theta } a_{2}(m_w ) \nonumber\\ & & \qquad + 2 \hat{s}_{\theta}\hat{c}_{\theta } b_{22}(m_z , m_w , m_w ) \biggr ] \ ; , \nonumber % % % \end{aligned}\ ] ] @xmath200 \ ; , % % % \end{aligned}\ ] ] @xmath201 \ ; , \nonumber\end{aligned}\ ] ] where @xmath202 and @xmath100 is defined in eq . ( [ swdef ] ) . to extract the dependence on the masses of the lightest neutral higgs , @xmath75 , and the extra scalar fields , @xmath203 and @xmath204 , we first note that , in the limit @xmath205 , @xmath206 -\frac{1}{12 } p^{2 } \ln \biggl ( \frac{q^{2}}{m_{1}^{2 } } \biggr ) -\frac{1}{72}\frac{p^{4}}{m_{1}^{2 } } \\ & & + \biggl [ \frac{1}{4 } \ln \biggl(\frac{q^{2}}{m_{1}^{2 } } \biggr ) + \frac{5}{72}\frac{p^{2}}{m_{1}^{2}}\biggr ] \delta m^{2 } + \mathcal{o}\biggl ( ( \delta m^{2})^{2 } , \ ; \biggl(\frac{p^{2}}{m_{1}^{2}}\biggr)^{2 } \biggr)\nonumber \\ & & + ( \mbox{terms with no scalar dependence})\nonumber \nonumber \ ; , \end{aligned}\ ] ] where we have defined @xmath207 and assumed that @xmath208 . using these relations , we then have , @xmath209 @xmath210 on the other hand , in the limit @xmath211 , we get , @xmath212 which gives , @xmath213 % -\frac{1}{12}p^{2 } \ln q^{2 } + \mathcal{o } \biggl ( \biggl(\frac{m_{2}^{2}}{m_{1}^{2}}\biggr ) , \ ; \biggl(\frac{p^{2}}{m_{1}^{2}}\biggr ) \biggr ) \\ & & + \ ; ( \mbox{terms with no scalar dependence } ) \nonumber\end{aligned}\ ] ] @xmath214 the two - point function @xmath44 does not have any scalar dependence , and the function @xmath215 depends on the scalar mass only logarithmically , @xmath216 the scalar dependence in the function @xmath113 is , @xmath217 \\ & = & \frac{\alpha}{4\pi \hat{s}_{\theta } \hat{c}_{\theta } } ( \hat{s}_{\theta}^{2 } -\hat{c}_{\theta}^{2}-c_{\delta}^{2 } ) m_{z}^{2 } \biggl [ \frac{1}{12 } \ln \biggl ( \frac{m_{h^{\pm}}^{2}}{q^{2}}\biggr ) - \frac{1}{72 } \frac{m_{z}^{2}}{m_{h^{\pm}}^{2}}\biggr ] \ ; , \nonumber\end{aligned}\ ] ] thus the dependence is also logarithmic . on the other hand , in the function @xmath114 , the scalar dependence is given by , @xmath218 \ ; . \nonumber\end{aligned}\ ] ] from eqs . ( [ b : eq ] ) and ( [ b0:eq2 ] ) , we know that the contributions from the terms in the square brackets of eq . ( [ square ] ) are logarithmic . thus the only possible quadratic dependence comes from terms in the curly brackets . we consider the following three limits , assuming that all scalar masses are much larger than @xmath98 and @xmath20 : * @xmath118 : in this case , the leading order scalar dependence is given by , @xmath219 \nonumber\\ & & + \frac{5}{72}\biggl [ s_{\delta}^{2}\frac{m_{w}^{2}}{m_{h^{0}}^{2 } } \bigl(m_{h^{\pm}}^{2}-m_{h^{0}}^{2}\bigr ) + 4c_{\delta}^{2 } \frac{m_{w}^{2}}{m_{k^{0}}^{2 } } \bigl(m_{h^{\pm}}^{2 } - m_{k^{0}}^{2}\bigr ) \biggr ] \biggr\ } \nonumber \ ; .\end{aligned}\ ] ] so the dominant scalar contribution to @xmath220 in this case is given by , @xmath221 \nonumber\\ & & \qquad + \frac{5}{72}\biggl [ s_{\delta}^{2}\frac { \bigl(m_{h^{\pm}}^{2}-m_{h^{0}}^{2}\bigr)}{m_{h^{0}}^{2 } } + 4c_{\delta}^{2 } \frac { \bigl(m_{h^{\pm}}^{2 } - m_{k^{0}}^{2}\bigr)}{m_{k^{0}}^{2 } } \biggr ] \biggr\ } \nonumber \ ; .\end{aligned}\ ] ] * @xmath222 : in this limit , the leading scalar dependence becomes , @xmath223 \nonumber\\ & & -s_{\delta}^{2}\frac{m_{h^{0}}^{2}}{2m_{w}^{2 } } % \biggl ( 1 + \ln \biggl(\frac{m_{h^\pm}^{2}}{m_{h^{0}}^{2}}\biggr ) % \biggr ) m_{h^{\pm}}^{2 } + \frac{5}{18}c_{\delta}^{2 } \frac{m_{w}^{2}}{m_{k^{0}}^{2 } } \bigl(m_{h^{\pm}}^{2 } - m_{k^{0}}^{2}\bigr ) \biggr\ } \nonumber \ ; .\end{aligned}\ ] ] the leading scalar contribution to @xmath220 is , @xmath224 \nonumber\\ & & -s_{\delta}^{2}\frac{m_{h^{0}}^{2}m_{h^{\pm}}^{2}}{2m_{w}^{4 } } % \biggl ( 1 + \ln \biggl(\frac{m_{h^\pm}^{2}}{m_{h^{0}}^{2}}\biggr ) % \biggr ) + \frac{5}{18}c_{\delta}^{2 } \frac{\bigl(m_{h^{\pm}}^{2 } - m_{k^{0}}^{2}\bigr)}{m_{k^{0}}^{2 } } \biggr\ } \nonumber \ ; .\end{aligned}\ ] ] * @xmath225 : in this limit , the leading scalar dependence becomes , @xmath226 \nonumber\\ & & -s_{\delta}^{2}\frac{m_{h^{0}}^{2}}{2m_{w}^{2 } } % \biggl ( 1 + \ln \biggl(\frac{m_{h^\pm}^{2}}{m_{h^{0}}^{2}}\biggr ) % \biggr ) m_{h^{\pm}}^{2 } - c_{\delta}^{2 } \frac{2m_{k^{0}}^{2}}{m_{w}^{2 } } % \biggl(1 + \ln \biggl(\frac{m_{h^\pm}^{2}}{m_{k^{0}}^{2}}\biggr)%\biggr ) m_{h^{\pm}}^{2 } \biggr\ } \ ; , \nonumber\end{aligned}\ ] ] the leading scalar contribution to @xmath220 is thus given by , @xmath227 \nonumber\\ & & -s_{\delta}^{2}\frac{m_{h^{0}}^{2 } m_{h^{\pm}}^{2}}{2m_{w}^{4 } } % \biggl ( 1 + \ln \biggl(\frac{m_{h^\pm}^{2}}{m_{h^{0}}^{2}}\biggr ) % \biggr ) - c_{\delta}^{2 } \frac{2m_{k^{0}}^{2}m_{h^{\pm}}^{2}}{m_{w}^{4 } } % \biggl(1 + \ln \biggl(\frac{m_{h^\pm}^{2}}{m_{k^{0}}^{2}}\biggr)%\biggr ) \biggr\ } \ ; . \nonumber\end{aligned}\ ] ] for the case @xmath228 , make the replacement , @xmath229 . as our second example to show that new physics does not decouple from the sm at one - loop , we consider the left - right ( lr ) symmetric model@xcite which is defined by the gauge group , @xmath230 the minimal left - right symmetric model contains a scalar bi - doublet , @xmath63 , and two @xmath5 triplets , @xmath231 and @xmath232 . we assume that the scalar potential is arranged such that the higgs fields obtain the following vev s : @xmath233 where the quantum numbers of these higgs fields under @xmath58 , @xmath234 and @xmath235 are given inside the parentheses . the vev @xmath236 breaks the @xmath237 symmetry down to @xmath238 of the sm , while the bi - doublet vev s @xmath239 and @xmath240 break the electroweak symmetry ; the vev @xmath241 may be relevant for generating neutrino masses@xcite . after the symmetry breaking , there are two charged gauge bosons , @xmath242 and @xmath243 , two heavy neutral gauge bosons , @xmath244 and @xmath245 , and the massless photon . we will assume that @xmath242 and @xmath244 are the lighter gauge bosons and obtain roughly their sm values after the symmetry breaking . turning off the @xmath58 triplet vev , @xmath246 , and assuming for simplicity that the @xmath247 gauge coupling constants satisfy @xmath248 , there are five fundamental parameters in the gauge / higgs sector , @xmath249 we can equivalently choose @xmath250 as input parameters . the counter term for the weak mixing angle is then defined through these parameters and their counter terms . assuming that the heavy gauge bosons are much heavier than the sm gauge bosons , @xmath251 , then to leading order @xmath252 , the counterterm @xmath253 is given as follows @xcite , @xmath254 where the effective weak mixing angle , @xmath85 , is defined as in eq . ( [ swdef ] ) . to go from the first to the second step in the above equation , we have used the following relation , @xmath255 . in the third line of eq . ( [ ds ] ) , we include only the leading top quark mass dependence . when the limit @xmath256 is taken , @xmath257 approaches zero , and thus the sm result , @xmath258 is _ not _ recovered , which is not what one would naively expect . one way to understand this is that in the left - right model , four input parameters are held fixed , while in the sm three input parameters are fixed . there is thus no continuous limit which takes one from one case ( @xmath83 at tree level ) to the other ( @xmath259 at tree level ) . this discontinuity , which has been pointed out previously@xcite , is closely tied to the fact that the triplet higgs boson is non - decoupling@xcite . due to this non - decoupling effect , even if the triplet vev is extremely small , as long as it is non - vanishing , there is the need for the fourth input parameter . the only exception to this is if there is a custodial symmetry which forces @xmath260 : in this case only the usual three input parameters are necessary . we also note that the contribution of the lightest neutral higgs in this case is given by@xcite , @xmath261 which depends on @xmath262 quadratically , and is suppressed by the heavy gauge boson masses , @xmath263 and @xmath264 . the contributions of the remaining scalar fields also have a similar structure . in this section , we summarize our results on minimization of the scalar potential in the model with a triplet higgs . from the minimization conditions , @xmath265 , we obtain the following conditions , @xmath266 we ues the short hand notaion , @xmath267 . the second derivatives are , @xmath268 if @xmath269 , then there is no mixing between the doublet and the triplet . this requires @xmath270 . a. sirlin , _ phys . rev . _ * d22 * , 971 ( 1980 ) . w. j. marciano and a. sirlin , _ phys . rev . _ * d22 * , 2695 ( 1980 ) [ erratum - ibid . * d31 * , 213 ( 1985 ) ] . a. sirlin and w. j. marciano , _ nucl . _ * b189 * , 442 ( 1981 ) . f. jegerlehner , _ renormalizaing the standard model _ , lecture given at theoretical advanced study institute in elementary particle physics ( tasi 90 ) , boulder , co , june 3 - 29 , 1990 . published in _ boulder tasi 90:476 - 590_. d.m . renormalization of supersymmetric theories _ , lecture given at the theoretical advanced study institute in elementary particle physics ( tasi 97 ) , boulder , co , june 1 - 7 , 1997 . published in _ boulder 1997 : supersymmetry , supergravity and supercolliders _ , 343 - 389 [ hep - ph/9805497 ] . j. r. forshaw , a. sabio vera and b. e. white , _ jhep _ * 0306 * , 059 ( 2003 ) [ arxiv : hep - ph/0302256 ] . g. passarino , _ phys . _ * b247 * , 587 ( 1990 ) . j. f. gunion , r. vega and j. wudka , _ phys . _ * d43 * , 2322 ( 1991 ) . j. c. pati and a. salam , _ phys . rev . _ * d10 * , 275 ( 1974 ) . r. n. mohapatra and j. c. pati , _ phys . _ * d11 * , 566 ( 1975 ) . r. n. mohapatra and j. c. pati , _ phys . rev . _ * d11 * , 2558 ( 1975 ) . g. senjanovic and r. n. mohapatra , _ phys . rev . _ * d12 * , 1502 ( 1975 ) . chen and k. t. mahanthappa , _ phys . rev . _ * d71 * , 035001 ( 2005 ) . chen and k. t. mahanthappa , _ int . j. mod * a18 * , 5819 ( 2003 ) . chen and k. t. mahanthappa , _ aip conf . _ * 721 * , 269 ( 2004 ) . .- c . chen and k. t. mahanthappa , _ phys . rev . _ * d62 * , 113007 ( 2000 ) . chen and k. t. mahanthappa , _ phys . rev . _ * d65 * , 053010 ( 2002 ) . chen and k. t. mahanthappa , _ phys . rev . _ * d68 * , 017301 ( 2003 ) . chen and k. t. mahanthappa , _ phys . _ * d70 * , 113013 ( 2004 ) .

electroweak precision data have been extensively used to constrain models containing physics beyond that of the standard model . when the model contains higgs scalars in representations other than singlets or doublets , and hence @xmath0 at tree level , a correct renormalization scheme requires more inputs than the three commonly used for the standard model case . in such cases , the one loop electroweak results can not be split into a standard model contribution plus a piece which vanishes as the scale of new physics becomes much larger than @xmath1 . we illustrate our results by presenting the dependence of @xmath1 on the top quark mass in a model with a higgs triplet and in the @xmath2 left - right symmetric model . in these models , the allowed range for the lightest neutral higgs mass can be as large as a few tev .

hep-ph0504286

atoms in highly excited rydberg states ( principal quantum number @xmath3 ) have large radii and electric - dipole transition matrix elements ( @xmath4 ) , large polarizabilities ( @xmath5 ) and strong van - der - waals interactions ( @xmath6 ) @xcite.these properties have led to a variety of interesting investigations and applications , including quantum information and logic gates @xcite , single - photon sources @xcite enabled by the rydberg excitation blockade effect @xcite , and many - body physics with strong long - range interactions @xcite . the large polarizability makes rydberg atoms sensitive to external fields , giving rise to applications in field measurement @xcite , quantum control @xcite and studies involving collisions @xcite and novel molecules @xcite . ( circles ) are prepared at an electric field @xmath7 = 3.14 v / cm . the field is then linearly ramped to @xmath8 = 3.99 v / cm with a rise time @xmath9 across a selected avoided crossing . the rydberg atoms undergo adiabatic / diabatic passage through the avoided crossing . atoms passing adiabatically are transformed into an @xmath10 high-@xmath11 elongated stark state ( ovals ) . during a hold time @xmath12 , they undergo efficient _ m_-mixing into stark states with high @xmath13 ( ovals of different shades ) . ( b ) timing diagram . after the hold time @xmath12 , a ramp ionization field ( except specified otherwise ) is applied that only ionizes the atoms in low-@xmath13 levels but not the ones in high-@xmath13 levels.,scaledwidth=50.0% ] we investigate how a controlled passage of a dense cloud of rydberg atoms through an avoided crossing alters the collision - induced dynamics of the sample . as a result of adiabatic state transformation , rydberg atoms passing adiabatically acquire large permanent electric dipole moments , which lead to enhanced dipole - dipole interactions . the accelerated state mixing is probed via time - delayed state - selective field ionization @xcite . previously , the ( single - atom ) dynamics of lithium rydberg atoms passing through an avoided crossing has been studied via a measurable difference in the ionization electric fields of the atoms exhibiting diabatic and adiabatic passage @xcite . l_-state redistribution has been controlled by application of a large - amplitude rectangular electric field pulse @xcite . in collision studies , _ l_-changing interactions of na @xmath3d rydberg atoms with slow na@xmath14 ions have been investigated using field - ionization templates for _ l _ =3 , 4 , and 5 @xcite . the effect of _ l _ and _ m_-mixing by weak , homogeneous dc electric fields and static ions has been predicted to lengthen the lifetimes of rydberg states @xcite . the mixing - induced long lifetimes of high-@xmath3 rydberg states play a central role in zero electron kinetic energy ( `` zeke '' ) spectroscopy @xcite . in cold rydberg - atom gases , plasma formation in a cloud of initially low-_l _ rydberg atoms and subsequent recombination processes can generate high-_l _ rydberg atoms @xcite . long - lived high-_l _ rydberg states have been created by _ l_- and _ m_-mixing collisions in rubidium @xcite and by state transfer induced with weak electric fields in cesium @xcite . here , we employ the adiabatic / diabatic passage of cesium rydberg atoms in a well - defined initial state , prepared within an applied electric field , through a selected multi - level avoided crossing . in alkali - metal atoms , low-_l _ ( _ _ l__@xmath152 ) rydberg states typically have low electric - dipole moments while the outermost linear stark levels have large ones , resulting in sequences of avoided crossings between low-_l _ states and linear , high-@xmath11 stark states @xcite . time - dependent studies of avoided crossings@xcite are of general relevance , since avoided crossings are a universal phenomenon in atoms and molecules . here , we are interested in the dynamics of cold , relatively dense rydberg atom samples after transformation of the rydberg - atom wavefunction in the avoided crossing . in our case , adiabatic transformation induces large permanent electric dipole moments , which have a profound effect on the subsequent collision - induced dynamics of the samples . we vary the speed of the electric - field ramp that drives the atoms through the avoided crossing as well as the atom interaction time after completion of the ramp . the final atom sample is analyzed via state - selective field - ionization . we interpret our results based on simulations of the passage behavior and the collision properties of the sample after the passage . we trap cesium atoms in a standard magneto - optical trap ( mot ) with temperature of @xmath16100 @xmath17k and peak density of @xmath18 @xmath19 , and use stepwise two - photon excitation to prepare rydberg states . the trapping laser is turned off during rydberg - atom excitation , manipulation and detection , whereas the repumping laser is left on to avoid optical pumping into 6s@xmath0(f=3 ) by off - resonant transitions . the lower - transition laser resonantly drives the 6s@xmath0 ( f=4 ) @xmath20 6p@xmath21 ( f@xmath22=5 ) transition and has a power of 660 @xmath17w and a gaussian beam waist of @xmath23 mm . the upper - transition laser drives a transition from 6p@xmath21 ( f@xmath22=5 ) into a rydberg state and has a waist diameter of @xmath24 m , yielding a cylindrical excitation volume with a length of @xmath16 800@xmath25 m and a diameter of @xmath16 50@xmath25 m . the excitation - pulse duration is 500 ns . in order to reproducibly excite selected , well - defined rydberg levels under presence of an applied electric field , where the density of states is large , we use a wavelength meter with an uncertainty of 30 mhz . the rydberg atoms are ionized using state - selective electric - field ionization @xcite . the released ions are detected with a calibrated micro - channel plate ( mcp ) . since the mcp ion detection efficiency is @xmath26 , the actual rydberg atom numbers and densities are about a factor of three higher than the numbers and densities of detected rydberg atoms . electric fields are applied using a pair of parallel , non - magnetic grids centered at the mot location ( spacing 15 mm ) . the electric fields are generated using an arbitrary - waveform generator ( voltage precision @xmath271 mv ) . the field is calibrated and zeroed via stark spectroscopy of 60d@xmath21 and 60d@xmath28 . the experiment and the time sequence are sketched in fig . 1 . and @xmath29 ( c ) of avoided crossings between the 49s@xmath0 state and the @xmath30 manifold of states in cesium . each avoided crossing involves three adiabatic states , labeled @xmath31 , @xmath29 , @xmath32 etc .. the points a and b in the stark map identify stark states that have primarily 49s@xmath0 character and are located between two adjacent avoided crossings . the point c identifies the high-@xmath11 stark state that becomes populated via adiabatic passage in our experiments . the line in ( c ) shows the result of the calculation ( top and right axes).,scaledwidth=40.0% ] in preparation for our time - dependent studies , it is important to locate a suitable avoided crossing , to verify that the field inhomogeneity is low enough that the crossing can be well resolved , and to measure its gap size @xmath33 . for our work we have calculated @xcite and measured cesium stark spectra in the vicinity of the @xmath30 hydrogen - like manifold of states ; respective results are shown in figs . 2 ( a ) and ( b ) . the stark map in ( a ) includes two three - level avoided crossings with adiabatic states labeled @xmath31 , @xmath29 , @xmath32 and @xmath34 , @xmath35 , @xmath36 ; the respective energies are e@xmath37 , e@xmath38 , e@xmath39 and e@xmath40 , e@xmath41 , e@xmath42 . the measured spectrum in fig . 2 ( b ) shows atom counts as a function of wavenumber for a selection of electric - field values , corresponding to the shaded field and energy range in fig . 2 ( a ) . both measurement and calculation demonstrate that the avoided crossings involve three atomic levels , whereby the line strength of the middle adiabatic level @xmath32 is much smaller than that of the @xmath31 and @xmath29-levels . the avoided crossings are characterized by a center electric field , @xmath43 , and the energy gap at @xmath43 . for each avoided crossing , @xmath43 is given by the electric - field value at which the energy gap , @xmath44 , between the adiabatic levels @xmath31 and @xmath29 exhibits a ( local ) minimum as a function of electric field @xmath45 , _ i.e. _ @xmath46 the gap minimum is realized when @xmath47 . as an example , in fig . 2 ( c ) we show the experimental energy - level difference @xmath44 ( symbols ) and calculation ( solid line ) between states @xmath31 and @xmath29 as a function of electric field for the anti - crossing in fig . 2 ( a ) . the measured center field is @xmath48 v / cm and the gap minimum @xmath49 mhz . the measured energy gap agrees with the calculation within about @xmath50 , and the field within @xmath51 . the intermediate level @xmath52 exhibits no significant curvature , leading us to infer that its coupling to the other levels is small and that its effect on @xmath33 is minor . this is confirmed below in sec . [ sec : lztheory ] . in our time - dependent studies we employ the avoided crossing that involves the adiabatic states @xmath34 , @xmath35 and @xmath36 defined in fig . 2 ( a ) ; this avoided crossing has a gap size , @xmath53 mhz . we initially prepare rydberg atoms in adiabatic state @xmath34 at point a in fig . 2 ( a ) , at an electric field of 3.14 v / cm . as the electric field is linearly ramped through the avoided crossing centered at @xmath54 v / cm to a final field of 3.99 v / cm , the atoms undergo diabatic or adiabatic passage into adiabatic states @xmath35 and @xmath34 at points b and c , respectively . the process strongly depends on the ramp time , @xmath55 , of the electric field . when the electric field varies fast , the atoms preferentially pass diabatically into level @xmath35 at point b , while for slow ramps the population mostly passes adiabatically into level @xmath34 at point c [ black arrow shown in fig . 2 ( a ) ] . the 49s@xmath0-character of the adiabatic quantum states at points a and b in fig . 2 ( a ) exceeds @xmath56 , while the adiabatic state at point c primarily contains hydrogenic states with @xmath57 . we therefore refer to the state at c as high-@xmath11 ( note , however , that the quantum number _ l _ is not conserved in non - zero fields ) . ignoring level @xmath36 and noting that the anti - crossings are sufficiently well separated that interactions between them @xcite are insignificant , the passage approximately follows landau - zener dynamics ( see ref . @xcite and sec . [ sec : lztheory ] ) . . the atoms are initially prepared in point a in fig . 2 , and the electric field is ramped from 3.14 to 3.99 v / cm ( through one of the avoided crossings in fig . 2 ) . the pulsed ionization electric field is set high enough that 49s@xmath0-like atoms ( point a in fig . 2 ) are detected , but it is too low to detect high-@xmath13 atoms ( @xmath58 ) . the rydberg - atom density is @xmath59@xmath19 and the holding time @xmath60 = 0 @xmath17s . ( b ) normalized atom number for a holding time of @xmath60 = 1 @xmath17s ( other conditions same as in ( a ) ) . the solid line shows an exponential fit to the data . the inset in ( b ) shows the decay behavior of pure 49s@xmath0-atoms at zero electric field ( black squares ) and of 49s@xmath0-like atoms at a fixed field of 3.14 v / cm ( point a in fig . 2 ; red circles ) . ( c ) detected number of penning - ionized atoms as a function of hold time @xmath61 for fixed @xmath55 = 80 ns and same density as in ( a ) and ( b).,scaledwidth=40.0% ] atoms adiabatically transferred into point c in fig . 2 ( a ) still have the same @xmath2 as the initially prepared 49s@xmath0-like state at point a , and are energetically very close to the 49s@xmath0-like state at point b ( which is populated via diabatic passage ) . therefore , atoms in points b and c can not be directly distinguished by state - selective field ionization ( in contrast to the case studied in ref . @xcite , where such a distinction was possible ) . this is shown in fig . 3 ( a ) , where we plot the counted atom number as a function of ramp time @xmath55 for a field ionization voltage that is set just high enough to ionize the initially populated 49s@xmath0-like atoms points a and b in fig . 2 ( a ) ) . based on our calculation of the passage probabilities in sec . [ sec : lztheory ] , in fig . 3 ( a ) the adiabatic - passage probability changes from near zero to near 100@xmath62 over the investigated range of @xmath9 , with no resultant discernable trend in the atom counts . we conclude that field ionization immediately after the ramp through the avoided crossing is not suitable to state - selectively detect atoms in points b and c. adiabatic passage transforms the atoms from weak into strong dipolar character . atoms in points a and b have electric dipole moments from about 130 e@xmath63 to 160 e@xmath63 , while atoms in point c have dipole moments of about 900 e@xmath63 . this is interesting in several respects . direct optical excitation of such a sample would be difficult to accomplish because of optical selection rules ( atoms in point c have only about @xmath64 49s@xmath0-character ) and because the rydberg excitation blockade @xcite suppresses optical or microwave excitation of high - density rydberg - atom samples at point c ( due to interaction - induced electric - dipole energy shifts ) . hence , atom samples prepared by adiabatic passage from a to c enable us to study collision - induced dynamics in dipolar rydberg atom gases under rydberg - atom density conditions that would likely not be attainable using alternate means . enhanced dipolar interactions change the sample dynamics after passaging through the avoided crossing . in fig . 3 ( b ) we hold the field for a time @xmath65s at its final value and show the normalized number of detected atoms as a function of the ramp time @xmath55 at a rydberg - atom density of @xmath66 @xmath19 ( blue triangles ) . as in fig . 3 ( a ) , we use a pulsed ionization field with voltage just high enough to ionize the initially populated 49s@xmath0-like atoms . over the investigated range of @xmath55 the landau - zener adiabatic - passage probability changes from near zero to near 100@xmath62 ( see sec . [ sec : lztheory ] below ) . in contrast to fig . 3 ( a ) , where the detected atom fraction is fairly constant at 100@xmath62 , in fig . 3 ( b ) the fraction of detected atoms drops from 100@xmath62 to near zero , as the adiabatic transition probability increases . it is concluded that the waiting time @xmath65s enables state - selective detection of adiabatic and diabatic passage of atoms through the avoided crossing . the adiabatically transformed atoms become undetectable over a wait time of @xmath67s . the signal decay observed in fig . 3 ( b ) is fitted quite well by a decaying exponential with a decay time constant of 103 ns @xmath27 5 ns . to explain the observed behavior , we first note that the atoms in points @xmath68 and @xmath69 in fig . 2 ( a ) are nearest - neighbor quantum states of the same @xmath10 stark map . such states can usually not be distinguished by using state - selective field ionization such as ours ; therefore the result in fig . 3 ( a ) is not surprising . however , if a sufficiently long hold time @xmath61 is introduced , the atoms in highly dipolar , elongated stark states ( point c in fig . 2 ( a ) ) may selectively undergo @xmath2-mixing collisions into energetically nearby stark states with higher @xmath13 . as @xmath13 exceeds a value of about 2.5 , atoms tend to ionize diabatically because the gaps of the avoided crossings in the stark maps trend towards zero @xcite . the ionization electric fields then increase by factors between 1.8 and 4 ( for a calculation of hydrogenic ionization rates see @xcite ) . hence , state - selective , efficient @xmath2-mixing and a corresponding increase in ionization field are the most likely reasons of why the adiabatically transformed atoms become undetectable . the results in fig . 3 ( b ) for short @xmath55 , where the passage is diabatic , further indicate that atoms in @xmath70-like states ( point b in fig . 2 ( a ) ) do not exhibit sufficient state mixing to alter the field ionization threshold . a state - mixing model is investigated in detail in sec . [ sec : mmixing ] below . to support the above interpretation of the data , in the inset of fig . 3 ( b ) we excite 49s@xmath0-like atoms at a fixed electric field of 3.14 v / cm ( point a in fig . 2 ( a ) ) and rydberg atom density @xmath71 , and we show the normalized detected atom number as a function of time @xmath9 over a very long range up to 300 @xmath17s ( red circles ) . for comparison , we also show the decay of 49s@xmath0 atoms in zero field ( black squares ) . the decays of the pure 49s@xmath0-atoms at zero field and the 49s@xmath0-like atoms at point a in fig . 2 ( a ) both occur on time scales that exceed the time scale found in the main panel of fig . 3 ( b ) by about two orders of magnitude ( to visualize this stark contrast , the data from the main panel in fig . 3 ( b ) are also shown in the inset ) . hence , the adiabatic passage through the avoided crossing and the dynamics that follows during the subsequent @xmath72s greatly accelerate the signal decay . less important but noteworthy are the following observations . the decay of the pure 49s@xmath0-atoms is exponential ( see fitted curve in the inset of fig . 3 ( b ) ) and has a fitted decay time of @xmath73s @xmath27 @xmath74s at zero field . this value is between the level lifetimes calculated for near - zero ( 0.1 k ) radiation temperature , where it is @xmath75s , and for 300 k radiation temperature , where it is @xmath76s . the chamber is at room temperature . the calculated lifetime at 300 k includes black - body - driven microwave and thz transitions into other rydberg levels . such transitions are not registered as decay events in the long - time data in the inset of fig . 3 ( b ) , which are obtained with a higher ionization electric field than the other data in fig . 3 . hence , the experimentally observed decay time is indeed expected to be between the calculated level lifetimes at 0.1 k and at 300 k. further , 49s@xmath0 atoms in zero field exhibit only weak van - der - waals forces and vanishing @xmath2-mixing ; hence any collision - induced effects are expected to be small even at long times . the observed exponential decay corroborates this expectation . the calculated level decay time for the 49s@xmath0-like atoms at point a in fig . 2 ( a ) also is about @xmath76s . in the experiment , the decay of the 49s@xmath0-like atoms at point a is non - exponential , with an accelerated initial decay time of @xmath77s and a longer - lived fraction of atoms remaining at times past @xmath78s . this indicates that initially the decay of the 49s@xmath0-like atoms at point a is enhanced by collisions . the 49s@xmath0-like atoms at point a have an electric - dipole moment of 130 e@xmath63 , which results in significant interatomic forces . over the long time scale covered in the inset in fig . 3 ( b ) , these forces are likely to cause @xmath3-mixing collisions and penning ionization @xcite . these collisions may cause the initial accelerated decay of the 49s@xmath0-like atoms at point a , in comparison with pure 49s@xmath0 atoms in zero field . a fraction of the product rydberg atoms are in long - lived high-@xmath13 states , which survive for much longer times @xcite . finally , to support our interpretation further , in fig . 3 ( c ) we show the number of free ions generated by penning ionization as a function of @xmath12 for a fixed ramp time @xmath79ns , which is sufficiently long for the passage to be @xmath80 adiabatic ( see sec . [ sec : lztheory ] ) . while for the wait time @xmath81s used in fig . 3 ( b ) the ion fraction is only about 2 @xmath62 , it rises to about 20 @xmath62 at @xmath82s . the fractions of atoms participating in penning - ionizing collisions are twice the ion fractions . thermal ionization due to black - body radiation is negligible over the time range in fig . 3 ( c ) . the fact that penning - ionizing collisions are seen in fig . 3 ( c ) makes it very plausible that @xmath2-mixing collisions , which are near - elastic and have larger rates than penning ionization , are highly probable for a wait time @xmath81s . we have established that two conditions are necessary for the 49s@xmath0-like atoms to become undetectable . firstly , the passage through the avoided crossing needs to be adiabatic . from fig . 3 ( b ) and sec . [ sec : lztheory ] it follows that adiabaticity requires @xmath83ns . secondly , our experiments show that a wait time @xmath84s is needed for the 49s@xmath0-like atoms to undergo @xmath2-mixing and to acquire an ionization electric field that is larger than the applied ionization field . the requirement on the wait time @xmath12 is supported by the @xmath2-mixing calculation presented in sec . [ sec : mmixing ] . in sec . [ sec : data ] we have attributed the signal decay observed in fig . 3 ( b ) to adiabatic passage and subsequent _ m_-mixing of atoms in point c of fig . 2 ( a ) during the hold time of @xmath61 = 1 @xmath17s . in order to test whether _ m_-mixing collisions are essential for atoms in point c to become undetectable , we have taken data equivalent to those in fig . 3 ( b ) and ( c ) at lower densities . the results shown in fig . 4 ( a ) clearly demonstrate that the loss in signal becomes less significant with decreasing atom density . also , the time scale over which the signal loss develops increases with decreasing density . both observations are consistent with collisions playing a central role for atoms in point c of fig . 2 ( a ) to become undetectable . as a function of ramp time @xmath55 at different densities , @xmath71 ( 3@xmath85@xmath19 ) , 0.5 @xmath71 and 0.3 @xmath71 , of rydberg atoms . the electric field is ramped from 3.14 to 3.99 v / cm through one of the avoided crossings in fig . 2 ( a ) for holding time @xmath61 = 1 @xmath17s . ( b ) detected number of penning - ionized atoms as a function of @xmath61 at @xmath71 , 0.5 @xmath71 and 0.3 @xmath71 with ramp time @xmath55 = 80 ns for @xmath71 and @xmath55 = 500 ns for 0.5 @xmath71 and 0.3 @xmath71 , respectively . in all cases , the @xmath55 are long enough to ensure primarily adiabatic passage.,scaledwidth=40.0% ] as an additional consistency test of our interpretation , in fig . 4 ( b ) we show the number of ions generated by penning ionization as a function of @xmath61 for fixed @xmath55 and for different rydberg atom densities . as the atom number and the atom density are lowered by about a factor of 3.3 , the number of ions generated drops by about a factor of 12 , while the curves remain linear ( within our experimental precision ) . the drop factor and the linear time dependence of the ion signal are consistent with binary penning - ionizing collisions generating the free ions . the time scales on which the signals observed in fig . 3 ( b ) and 4 ( a ) decay depend on both the passage behavior in the avoided crossing and the time required for atoms at point c in fig . 2 ( a ) to become undetectable due to @xmath2-mixing . atoms passing diabatically into point b in fig . 2 ( a ) have wavefunctions very similar to those of atoms in point a. these do not efficiently mix and remain near-@xmath86 detectable for several @xmath17s , as shown in the inset of fig . 3 ( b ) . for the interpretation we have presented to hold , the ramp time @xmath9 for the passage to be primarily adiabatic must be less than the signal decay time seen in fig . 3 ( b ) ( which is 103 ns ) . in the following we show that this condition is met . into adiabatic state @xmath87 in fig . 2 ( a ) as a function of the ramp time @xmath9 , according to a simplified landau - zener model explained in the text ( dashed ) and a complete simulation ( solid ) . ( b ) the squares show the probability of exiting the passage in the 49s@xmath0-state and the circles the probability of exiting in any state with angular momentum _ _ l__@xmath88.,scaledwidth=40.0% ] the avoided crossings under investigation involve three levels ( see fig . 2 ( a ) ) and may therefore differ from a simple two - level landau - zener case . in fig . 5 ( a ) we show the diabatic transition probability as a function of @xmath55 , obtained by solving the time - dependent schrdinger equation for passage through the relevant adiabatic crossing in fig . 2 ( a ) . the exact diabatic transition probability is compared with an approximate diabatic transition probability derived from a simplified landau - zener model that only involves levels @xmath89 and @xmath87 in fig . 2 ( a ) . the close agreement between the exact and the approximate landau - zener transition probabilities shows that the intermediate adiabatic levels ( @xmath90 and @xmath91 in fig . 2 ( a ) ) are indeed only of minor importance . the close agreement also accords with the facts that the intermediate adiabatic levels do not noticeably repel from other levels in fig . 2 ( a ) , and that the calculated probabilities for the atoms to be in the intermediate adiabatic levels are @xmath92 . according to the landau - zener model , the diabatic transition probability is given by @xmath93 where @xmath33 is the gap size and @xmath94 are the electric dipole moments of the coupled diabatic levels ( all in atomic units ) . while the latter are not a priori known , the @xmath94 are approximately given by the electric - dipole moments of the adiabatic states @xmath89 and @xmath87 at fields away from the crossing . equation ( [ eq : lz ] ) shows that the diabatic transition probability can be written as @xmath95 , with a characteristic time @xmath96 where @xmath97 is the field range of the linear electric field ramp . the values of @xmath98 depend on the gap size of the avoided crossing , the differential energy slope of the crossing levels and the ramp parameters . the landau - zener curve in fig . 5 ( a ) and eq . [ eq : lz2 ] yield a characteristic time @xmath99ns , which is close to the exponential - fit result for the complete simulation , @xmath100ns . these values are about a factor of four less than the 103 ns time scale observed for the signal decay in fig . 3 ( b ) ( which is for the highest - density condition we studied ) . this finding is consistent with our interpretation of the data . at higher rydberg - atom densities , the observed signal decay time should approach the critical value @xmath101 ns . we expect minor deviations of the actual passage dynamics from the landau - zener case , as there are some deviations of our system from the assumptions made in the landau - zener model . firstly , the diabatic states are a weak function of electric field , as evidenced by the fact that the electric dipole moments are slightly different before and after the crossings . secondly , the weakly coupled third level will play a role in the case of very slow ramps . finally , we have also calculated the probabilities of exiting the avoided crossing in the state @xmath70 or in any state with _ l _ @xmath88 . these probabilities , displayed in fig . 5 ( b ) , exhibit modulations at a period of 6 ns . the modulations are due to quantum interference between the 49s@xmath0 components present in both adiabatic levels populated after the passage . for a duration of several microseconds , the rydberg atoms primarily interact via the binary electric - dipole interaction , @xmath102 where the electric - dipole operators @xmath103 and @xmath104 act on the first and the second component of two - atom rydberg states @xmath105 , respectively . there , the states @xmath106 and @xmath107 are single - atom rydberg states within the ( non - zero ) electric field @xmath45 . the unit vector @xmath108 points from the center of mass of the first to that of the second atom , and @xmath109 is the magnitude of the interatomic separation . the matrix elements are calculated as described in ref . @xcite . to estimate the effect of @xmath2-mixing , we have integrated the time - dependent schrdinger equation for atom pairs picked at initial nearest - neighbor separations for a density of @xmath110@xmath19 , corresponding to our experimental conditions . for randomly positioned atom pairs , the initial atom separation follows a probability distribution @xmath111\ ] ] where @xmath112 is the wigner - seitz radius . in the simulation the range of initial values of @xmath109 is restricted to @xmath113 m , because lesser separations are unlikely due to the excitation blockade effect and penning ionization . the initial angle @xmath114 of the internuclear separation vector relative to the electric field is chosen at random ( with weighting @xmath115 ) . the initial atom velocities are randomly selected from a maxwell distribution ( temperature @xmath116 ) . the internuclear separation @xmath109 and its unit vector @xmath108 are a function of time , given by the randomly chosen initial values for positions and velocities . due to the short interaction time , the effect of interatomic dipole - dipole forces on the ( classical ) internuclear trajectories are neglected . since the internuclear separation vectors @xmath117 depend on time , the matrix elements of the electric - dipole interaction operator in eq . [ eq : dipoleint ] also depend on time . -quantum number due to binary electric - dipole interactions for atom pairs in the 49s@xmath0-like state at point b in fig . 2 ( a ) ( solid squares ; @xmath118k ) , and for atom pairs in the high-_l _ state at point c ( open squares ; large symbols : @xmath118k , small symbols : @xmath119k ) . all atoms in the simulation initially have @xmath120 = + 1/2 . for the high-_l _ state , the probability of @xmath2-change is much higher than for the 49s@xmath0-like state.,scaledwidth=40.0% ] the time - dependent schrdinger equation is integrated over a wait time @xmath121s , according to the experiment . the quantization axis is parallel to the electric field . the utilized internal - state space is restricted to two - body states for which both single - atom @xmath2-quantum numbers are within the range @xmath122 , where @xmath123 is the initial - state @xmath2-quantum number ( which is @xmath124 ) . also , the effective quantum numbers of both atoms are within the range @xmath125 . we run the simulation for two cases of the initial internal two - body state , namely @xmath126 or @xmath127 , according to the points b and c in fig . 2 ( a ) . finally , the two - body state space is restricted to states whose total energy differs by less than 40 mhz from the initial two - body state . this range of two - body states has been found large enough to yield convergent @xmath2-mixing probabilities . typically , there are several tens of thousands of two - body states in this range . from the final two - body state we extract the probabilities of the ( single - atom ) @xmath2 quantum numbers changing by amounts @xmath128 . according to our basis restriction , @xmath129 . the simulation is repeated and the results are averaged over 1000 random choices of initial positions and velocities . the selection rules of the electric - dipole interaction operator @xmath130 in eq . [ eq : dipoleint ] are , in first order , @xmath131 , for both of the atoms involved , and @xmath132 for the sum of the two single - atom @xmath2-quantum numbers . the interaction time of 1 @xmath17s is long enough that @xmath130 acts in higher order . as a result , large changes @xmath128 and @xmath133 are possible . a typical simulation result is shown in fig . it is seen that the high-@xmath134 elongated stark states @xmath2-mix much more readily than the 49s@xmath0-like ones . this may be expected because in the vicinity of point c in fig . 2 ( a ) the density of higher-@xmath13 `` background '' states is larger than it is in the vicinity of point b. also , the @xmath130-couplings of the 49s@xmath0-like atoms to other states are generally smaller , because at the small fields used in this work the @xmath135-character within the manifold of hydrogenic states is very small ( due to the large quantum defect of cs @xmath135-levels ) and the admixture of high-@xmath136 character in the 49s@xmath0-like levels also is small . as a result of @xmath2-mixing , the high-@xmath134 atoms selectively acquire higher field ionization thresholds . the mixing percentages in fig . 6 likely are underestimates because many - body effects beyond two atoms are not included ; such effects have been found earlier to enhance near - resonant many - body mixing effects @xcite . further , the effects of interatomic forces will lead to particle acceleration . we note the large permanent electric dipole moment of the high-@xmath134 elongated stark state , given by the large negative slope of the energy level at point c in fig . 2 ( a ) . the resultant strong interatomic permanent - electric - dipole forces may also enhance the @xmath2-mixing . figure 6 shows that at higher temperatures there is slightly more @xmath2-mixing . this indicates that the mixing may accelerate once the sample heats up due to the interatomic forces . adiabatic / diabatic processes are ubiquitous in natural science ; they are , for instance , manifested in atomic and molecular spectra , in collisions and in chemical reactions . in this work , we have prepared high - density gases of strongly interacting rydberg atoms based on avoided crossings formed by the 49s@xmath0-state and @xmath30 hydrogenic states of cesium in an electric field . electric - field ramps through the avoided crossings induce mixed adiabatic - diabatic passage behavior . in the adiabatic case , the atoms acquire large permanent electric dipole moments . the adiabatically transformed atoms are embedded in a background of other high-@xmath137 stark states . these conditions are conducive to selective @xmath2-mixing of the adiabatically transformed atoms , making adiabatic passage experimentally detectable . our measurements are in agreement with a model we have presented . in future work one may employ state transformation via adiabatic passage as a method to generate highly dipolar matter , against obstacles that may otherwise arise from inhomogeneous broadening , excitation blockade effects , and small rates for direct optical excitation of highly dipolar states . the oscillations seen in the calculations in fig . 5 ( b ) are a general interferometric characteristic of systems involving two coupled quantum states ; they may , in future work , enable measurements of dipole moments and other atomic properties . we thank prof . jrn manz for useful discussions . the work was supported by the 973 program ( grant no . 2012cb921603 ) , nsfc project for excellent research team ( grants no.61121064 ) , nnsf of china ( grants no . 11274209 , 61475090 , 61378039 and 61378013 ) and nsf grant no . phy-1205559 199 gallagher t f _ rydberg atoms _ ( cambridge university press , cambridge , 1994 ) . galindo a and martin - delgado m a 2002 _ rev . phys . _ * 74 * 347 garca - ripoll j j , zoller p and cirac j i 2005 _ j. phys . b _ * 38 * s567 jaksch d , cirac j i , zoller p , rolston s l , ct r and lukin m d 2000 _ phys . rev . lett . _ * 85 * 2208 lukin m d , fleischhauer m , cote r , duan l m , jaksch d , cirac j i and zoller p 2001 _ phys . lett . _ * 87 * 037901 isenhower l , urban e , zhang x l , gill a t , henage t , johnson t a , walker t g and saffman m 2010 _ phys . rev . lett . _ * 104 * 010503 dudin y o and kuzmich a 2012 _ science _ * 336 * 887 comparat d and pillet p 2010 _ j. opt . soc . b _ * 27 * a208 tong d , farooqi s m , stanojevic j , krishnan s , zhang y p , ct r , eyler e e and gould p l 2004 _ phys . lett . _ * 93 * 063001 vogt t , viteaut m , zhao j , chotia a , comparat d and pillet p 2006 _ phys . lett . _ * 97 * 083003 vogt t , viteaut m , chotia a , zhao j , comparat d and pillet p 2007 _ phys . lett . _ * 99 * 073002 saffman m and walker t g 2005 _ phys . a _ * 72 * 042302 mller m , lesanovsky i , weimer h , bchler h p and zoller p 2009 _ phys . rev . lett . _ * 102 * 170502 olmos b , gonzlez - frez r and lesanovsky i 2009 _ phys . rev . lett . _ * 103 * 185302 saffman m and walker t g 2010 _ rev . _ * 82 * 2313 bariani f , dudin y o , kennedy t a b and kuzmich a 2012 _ phys . _ * 108 * 030501 mohapatra a k , bason m g , butscher b , weatherill k j and adams c s 2008 _ nature physics _ * 4 * 890 osterwalder a and merkt f 1999 _ phys . rev . lett . _ * 82 * 1831 nipper j , balewski j b , krupp a t , hofferberth s , lw r and pfau t 2012 _ phys . x _ * 2 * 031011 marcassa l g and shaffer j p 2014 _ adv . at . phy . _ * 63 * 47 dietrich h j , mller - dethlefs k and baranov l y 1996 _ phys . rev . lett . _ * 76 * 3530 grtler a and van der zande w j 2004 _ phys . a _ * 324 * 315 rubbmark j r , kash m m , littman m g and kleppner d 1981 _ phys . * 23 * 3107 tannian b e , stokely c l , dunning f b , reinhold c o and burgdrfer j 1999 _ j. phys . b : at . mol . phys . _ * 32 * l517 sun x and macadam k b 1993 _ phys . rev . a _ * 47 * 3913 merkt f and zare r n 1994 _ j. chem . _ * 101 * 3495 merkt f 1994 _ j. chem . phys . _ * 100 * 2623 mller - dethlefs k , schlag e w , grant e r , wang k and mckoy b v 1995 _ adv . chem . phys . _ * 90 * 1 robinson m p , laburthe tolra b , noel m w , gallagher t f and pillet p 2000 _ phys . rev . lett . _ * 85 * 4466 dutta s k , feldbaum d , walz - flannigan a , guest j r and raithel g 2001 _ phys . rev . lett . _ * 86 * 3993 zhang h , wang l , zhang l , li c , xiao l , zhao j , and jia s , cheinet p , comparat d and pillet p 2013 _ phys . a _ * 87 * 033405 zimmerman m l , littman m g , kash m m and kleppner d 1979 _ phys . a _ * 20 * 2251 harmin d a 1997 _ phys . rev . a _ * 56 * 232 kleppner d , 1980 _ proc . les houches summer school session xxxiv _ , course 7 ( amsterdam : north - holland ) damburg r j and kolosov v v 1979 _ j. phys . b : atom . molec . * 12 * 2637 nakamura h 1969 _ j. phys . * 26 * 1473 reinhard a , cubel liebisch t , younge k c , berman p r and raithel g 2008 _ phys . lett . _ * 100 * 123007 younge k c , reinhard a , pohl t , berman p r and raithel g 2009 _ phys . a _ * 79 * 043420 - 5

the passage of cold cesium 49s@xmath0 rydberg atoms through an electric - field - induced multi - level avoided crossing with nearby hydrogen - like rydberg levels is employed to prepare a cold , dipolar rydberg atom gas . when the electric field is ramped through the avoided crossing on time scales on the order of 100 ns or slower , the 49s@xmath0 population adiabatically transitions into high-_l _ rydberg stark states . the adiabatic state transformation results in a cold gas of rydberg atoms with large electric dipole moments . after a waiting time of about @xmath1s and at sufficient atom density , the adiabatically transformed highly dipolar atoms become undetectable , enabling us to discern adiabatic from diabatic passage behavior through the avoided crossing . we attribute the state - selectivity to @xmath2-mixing collisions between the dipolar atoms . the data interpretation is supported by numerical simulations of the passage dynamics and of binary @xmath2-mixing collisions .

1504.00518

granular crystals are material systems based on the assembly of particles in one- , two- and three - dimensions inside a matrix ( or a holder ) in ordered closely packed configurations in which the grains are in contact with each other @xcite . the fundamental building blocks constituting such systems are macroscopic particles of spherical , toroidal , elliptical or cylindrical shapes @xcite , arranged in different geometries . the mechanical , and more specifically dynamic , properties of these systems are governed by the stress propagation at the contact between neighboring particles . this confers to the overall system a highly nonlinear response dictated , in the case of particles with an elliptical or spherical contact , by the discrete hertzian law of contact interaction @xcite . geometry and/or material anisotropy between particles composing the systems allows for the observation of interesting dynamic phenomena deriving from the interplay of discreteness and nonlinearity of the problem ( i.e. anomalous reflections , breathers , energy trapping and impulse fragmentation ) @xcite . these findings open up a large parameter space for new materials design with unique properties sharply departing from classical engineering systems . one of the prototypical excitations that have been found to arise in the granular chains are traveling solitary waves , which have been extensively studied both in the absence @xcite ( see also @xcite for a number of recent developments ) , as well as in the presence @xcite of the so - called precompression . the precompression is an external strain a priori imposed on the ends of the chain , resulting in a displacement of the particles from their equilibrium position . as has been detailed in these works , the profile of these traveling waves is fundamentally different in the former , in comparison to the latter case . without precompression , waves exist for any speed , featuring a doubly exponential ( but not genuinely compact ) decay law , while in the case with precompression , waves are purely supersonic ( i.e. , exist for speeds beyond the speed of sound in the medium ) and decay exponentially in space . in fact , the fpu type lattices such as the one arising also from the hertzian chain in the presence of precompression have been studied extensively ( see @xcite and references cited therein for an overview of the history of the fpu model ) . it is known , both formally @xcite and rigorously @xcite ( on long but finite time scales ) that kdv approximates fpu @xmath0-type lattices for small - amplitude , long - wave , low - energy initial data . this fact has been used in the mathematical literature to determine the shape @xcite and dynamical stability @xcite of solitary waves and even of their interactions @xcite . we remark that the above referenced remarks in the mathematical literature are valid `` for @xmath1 sufficiently small '' , where @xmath1 is a parameter characterizing the amplitude and inverse width , as well as speed of the waves above the medium s sound speed . one of the aims of the present work is to determine the range of the parameter @xmath1 for which this theory can be numerically validated , an observation that , in turn , would be of considerable use to ongoing granular chain experiments . it is that general vein of connecting the non - integrable traveling solitary wave interactions of the granular chain ( that can be monitored experimentally ) with the underlying integrable ( and hence analytically tractable ) approximations , that the present work will be following . in particular , our aim is to quantify approximations of the hertzian contact model to two other models , one continuum and one discrete in which soliton and multi - soliton solutions are analytically available . these are , respectively , the kdv equation and the toda lattice . the former possesses only uni - directional waves . since hamiltonian lattices are time - reversible , a single kdv equation can not capture the evolution of general initial data . it is typical to use a pair of uncoupled kdv equations , one moving rightward and one moving leftward to capture the evolution of general initial data @xcite . on the other hand , the toda lattice has several benefits as an approximation of the granular problem . firstly , it is inherently discrete , hence it is not necessary to use a long wavelength type approximation that is relevant for the applicability of the kdv reduction @xcite . secondly , the toda lattice admits two - way wave propagation , hence a single equation can capture the evolution of all ( small amplitude ) initial data . once these approximations are established , we will `` translate '' two - soliton solutions , as well as superpositions of 1-soliton solutions of the integrable models into initial conditions of the granular lattice and will dynamically evolve and monitor their interactions in comparison to what the analytically tractable approximations ( kdv and toda ) yield for these interactions . we will explore how the error in the approximations grows , as a function of the amplitude of the interacting waves , so as to appreciate the parametric regime where these approximations can be deemed suitable for understanding the inter - soliton interaction . we believe that such findings will be of value to theorists and experimentalists alike . on the mathematical / theoretical side , they are relevant for appreciating the limits of applicability of the theory and the sharpness of its error bounds . on the experimental side , these explicit analytical expressions provide a yardstick for quantifying solitary wave collisions ( at least within an appropriate regime ) in connection to the well - characterized by now direct observations . our presentation will be structured as follows . in section ii , we will present the analysis and comparisons for the kdv reduction . in section iii , we will do the same for the toda lattice , examining in this case both co - propagating and counter - propagating soliton collisions . finally , in section iv , we will summarize our findings and present some conclusions , as well as some directions for future study . in the appendix , we will present some rigorous technical aspects of the approximation of the fpu solution by the toda lattice one . our starting point here will be an adimensional , rescaled form of the granular lattice problem , with precompression @xmath2 @xcite that reads : @xmath3^p_+ - [ \delta_0+y_{n}-y_{n+1}]^p_+\ ] ] where @xmath4 is the displacement of the @xmath5-th particle from equilibrium , and @xmath6_+ = max\{0 , x\}$ ] . defining the strain variables as @xmath7 , we obtain the symmetrized strain equation : @xmath8^p_+ - 2[\delta_0+u_{n}]^p_+ + [ \delta_0+u_{n+1}]^p_+ . \label{eqn2}\ ] ] in the context of the kdv approximation @xcite ( see also more recently and more specifically to the granular problem @xcite ) , we seek traveling waves at the long wavelength limit , which is suitable for the consideration of a continuum limit . we thus use the following spatial and temporal scales @xmath9 , @xmath10 . assuming then a strain pattern depending on these scales @xmath11 , we get @xmath12 + \frac{\epsilon^2}{12}\partial_x^4[(\delta_0+a)^p]+\frac{\epsilon^4}{360}\partial_x^6[(\delta_0+a)^p]+\cdots,\ ] ] while by consideration of the variable @xmath13 measuring the strain as a fraction of the precompression , we can also use the expansion of the nonlinear term as : @xmath14.\ ] ] this finally yields : @xmath15 + \frac{\epsilon^2}{12}\partial_x^4[pb + \frac{1}{2}p(p-1)b^2+\cdots]+\frac{\epsilon^4}{360}\partial_x^6[pb + \frac{1}{2}p(p-1)b^2+\cdots]+\cdots.\ ] ] now consider @xmath16 , with @xmath17 , @xmath18 , @xmath19 , with @xmath0 a small parameter , we get @xmath20 + \frac{\epsilon^2}{12}\partial_\xi^4[b + \frac{1}{2}(p-1)b^2+\cdots]+\frac{\epsilon^4}{360}\partial_\xi^6[b + \frac{1}{2}(p-1)b^2+\cdots]+\cdots.\ ] ] we now proceed to drop lower order terms such as @xmath21 , @xmath22 , @xmath23 , @xmath24 , and thus obtain the kdv approximation of the form : @xmath25 eq . ( [ kdv0 ] ) after the transformations @xmath26 , @xmath27 , @xmath28 , can be converted to the standard form : @xmath29 which has one soliton solutions as : @xmath30,\ ] ] as well as two soliton solutions given by : @xmath31 here , @xmath32 , and @xmath33 ^ 2 $ ] ; see e.g. @xcite , as well as the more recent work of @xcite , for more details on multi - soliton solutions of the kdv . if the initial positions of the two solitons satisfy @xmath34 , we need @xmath35 for the two solitons to collide . a typical example of the approximation of collisional dynamics of the solitons in the granular chain through the kdv is shown in figs . [ twosoliton ] and [ twosoliton_image ] . the first figure shows select snapshots of the profile of the two waves in the strain variable @xmath36 presenting the comparison of the analytical kdv approximation shown as a dashed ( blue ) line with the actual numerical granular chain evolution , of eq . ( [ eqn2 ] ) [ shown by solid ( red ) line ] . in this , as well as in all the cases that follow , we use the rescalings developed above ( and also for the toda lattice below ) to transform the integrable model solution into an approximate solution for the granular chain and initialize in our granular crystal numerics that solution at @xmath37 . i.e. , the analytical and numerical results share the same initial condition and their observed / measured differences are solely generated by the dynamics . it is clear that the kdv limit properly captures the individual propagation of the waves and is proximal not only qualitatively but even semi - quantitatively to the details of the inter - soliton interaction , as is illustrated from the middle and especially the bottom panels of the figure . nevertheless , there is a quantative discrepancy in tracking the positions of the solitary waves , especially so after the collision . the second figure shows a space - time plot of the very long scale of the observed time evolution . it is clear from the latter figure that small amplitude radiation ( linear ) waves are present in the actual granular chain , while such waves are absent in the kdv limit , due to its integrable , radiationless soliton dynamics . in fact , these linear radiation waves are also clearly discernible as small amplitude `` blips '' in fig . [ twosoliton ] . we believe that the very long time scales of the interaction of the waves enable numerous `` collisions '' also with these small amplitude waves thereby apparently reducing the speed of the larger waves in comparison to their kdv counterparts , as is observed in the bottom panels of fig . [ twosoliton ] . in that light , this is a natural consequence of the non - integrability of our physical system in comparison to the idealized kdv limit . nevertheless , we believe that the latter offers a very efficient means for monitoring the solitary wave collisions even semi - quantitatively . as a final comment on this comparison , we would like to point out that because of the very slow ( long time ) nature of the interaction , we are monitoring the dynamics in a periodic domain , merely for computational convenience . the position shifts of the kdv two - soliton solution after the collision are given by @xmath38 and @xmath39 for the faster and slower solitons respectively @xcite . if we use the position shift in kdv to predict the relevant position shifts in the granular lattice , for the parameters used in fig . ( [ twosoliton ] ) , the shift should be @xmath40/\epsilon = 7.88 $ ] and @xmath41/\epsilon = -11.15 $ ] for the fast and slow soliton respectively . numerically , we compare the soliton position with and without the collision , by tracing the peak of the soliton , and accordingly obtain a position shift of @xmath42 for the fast soliton and @xmath43 for the slower soliton , in line with our comments above about a semi - quantitative agreement between theory and numerics . , @xmath44 , @xmath45 , @xmath46 . the initial condition consists of a two - soliton solution containing waves of amplitude of @xmath47 and @xmath48 centered at @xmath49 and @xmath50 , respectively . from top to bottom , left to right snapshots at different times of the collisional evolution are shown , namely : @xmath51 . the solid ( red ) line represents the actual ( non - integrable ) granular lattice numerical evolution dynamics , while the dashed ( blue ) line stems from the qualitatively ( and even semi - quantitatively ) accurate integrable kdv two - soliton approximation.[twosoliton ] , title="fig : " ] , @xmath44 , @xmath45 , @xmath46 . the initial condition consists of a two - soliton solution containing waves of amplitude of @xmath47 and @xmath48 centered at @xmath49 and @xmath50 , respectively . from top to bottom , left to right snapshots at different times of the collisional evolution are shown , namely : @xmath51 . the solid ( red ) line represents the actual ( non - integrable ) granular lattice numerical evolution dynamics , while the dashed ( blue ) line stems from the qualitatively ( and even semi - quantitatively ) accurate integrable kdv two - soliton approximation.[twosoliton ] , title="fig : " ] , @xmath44 , @xmath45 , @xmath46 . the initial condition consists of a two - soliton solution containing waves of amplitude of @xmath47 and @xmath48 centered at @xmath49 and @xmath50 , respectively . from top to bottom , left to right snapshots at different times of the collisional evolution are shown , namely : @xmath51 . the solid ( red ) line represents the actual ( non - integrable ) granular lattice numerical evolution dynamics , while the dashed ( blue ) line stems from the qualitatively ( and even semi - quantitatively ) accurate integrable kdv two - soliton approximation.[twosoliton ] , title="fig : " ] , @xmath44 , @xmath45 , @xmath46 . the initial condition consists of a two - soliton solution containing waves of amplitude of @xmath47 and @xmath48 centered at @xmath49 and @xmath50 , respectively . from top to bottom , left to right snapshots at different times of the collisional evolution are shown , namely : @xmath51 . the solid ( red ) line represents the actual ( non - integrable ) granular lattice numerical evolution dynamics , while the dashed ( blue ) line stems from the qualitatively ( and even semi - quantitatively ) accurate integrable kdv two - soliton approximation.[twosoliton ] , title="fig : " ] , @xmath44 , @xmath45 , @xmath46 . the initial condition consists of a two - soliton solution containing waves of amplitude of @xmath47 and @xmath48 centered at @xmath49 and @xmath50 , respectively . from top to bottom , left to right snapshots at different times of the collisional evolution are shown , namely : @xmath51 . the solid ( red ) line represents the actual ( non - integrable ) granular lattice numerical evolution dynamics , while the dashed ( blue ) line stems from the qualitatively ( and even semi - quantitatively ) accurate integrable kdv two - soliton approximation.[twosoliton ] , title="fig : " ] , @xmath44 , @xmath45 , @xmath46 . the initial condition consists of a two - soliton solution containing waves of amplitude of @xmath47 and @xmath48 centered at @xmath49 and @xmath50 , respectively . from top to bottom , left to right snapshots at different times of the collisional evolution are shown , namely : @xmath51 . the solid ( red ) line represents the actual ( non - integrable ) granular lattice numerical evolution dynamics , while the dashed ( blue ) line stems from the qualitatively ( and even semi - quantitatively ) accurate integrable kdv two - soliton approximation.[twosoliton ] , title="fig : " ] , but the space - time contour plot of the strains is shown for the granular lattice evolution . notice the long time scale of the interaction , the exchange of the relative positions of the solitary waves and the nontrivial presence ( and thus impact on the interaction ) of small amplitude linear radiation waves stemming from the non - integrability of the model . periodic boundary conditions have been employed.,scaledwidth=100.0% ] the toda lattice model has the well - known form @xcite @xmath52}- e^{[x_{n}-x_{n+1}]}\nonumber\\ & = & [ 1+(x_{n-1}-x_n)+ \frac{1}{2 } ( x_{n-1}-x_n)^2 + \frac{1}{6 } ( x_{n-1}-x_n)^3 + \cdots ] -[1+(x_{n}-x_{n+1 } ) + \frac{1}{2 } ( x_{n}-x_{n+1})^2 + \frac{1}{6 } ( x_{n}-x_{n+1})^3 + \cdots]\nonumber\\ & = & ( x_{n-1}-2x_n+x_{n+1 } ) + \frac{1}{2}[(x_{n-1}-x_n)^2-(x_{n}-x_{n+1})^2 ] + \cdots \label{toda}\end{aligned}\ ] ] in the 2nd and 3rd lines above , we have expanded the lattice into an fpu-@xmath0 type form ( i.e. , maintaining the leading order nonlinear term ) . on the other hand , a similar expansion ( notice that now no long wavelength assumptions are needed ) of our granular chain model reads : @xmath53^p - [ \delta_0+y_{n}-y_{n+1}]^p\nonumber\\ & = & \delta_0^p[1+p\frac{y_{n-1}-y_n}{\delta_0}+\frac{1}{2}p(p-1)(\frac{y_{n-1}-y_n}{\delta_0})^2+\cdots]-\delta_0^p[1+p\frac{y_{n}-y_{n+1}}{\delta_0}+\frac{1}{2}p(p-1)(\frac{y_{n}-y_{n+1}}{\delta_0})^2+\cdots]\nonumber\\ & = & p\delta_0^{p-1}\{(y_{n-1}-2y_n+y_{n+1 } ) + \frac{1}{2}\frac{(p-1)}{\delta_0 } [ ( y_{n-1}- y_n)^2- ( y_{n}- y_{n+1})^2 ] + \cdots\ } \label{gl}\end{aligned}\ ] ] then , rescaling time and displacements according to @xmath54 and @xmath55 , the relevant eqn . ( [ gl ] ) becomes @xmath56 + \cdots \label{sgl}\end{aligned}\ ] ] where @xmath57 is the derivative with respect to @xmath58 . hence , eqs . ( [ sgl ] ) and eqn . ( [ toda ] ) agree up to second order , and thus the leading order error in our granular chain approximation by the toda lattice will stem from the cubic term ( for which it is straightforward to show that it can not be matched between the two models i.e. , we have expended all the scaling freedom available within the discrete granular lattice model ) . to see the closeness of the two models , we define the error term @xmath59 by the relation @xmath60 . here @xmath59 will remain of order one or smaller and @xmath1 controls the size of the error term . we proceed by using the evolution for @xmath59 to control how small we can choose @xmath1 while keeping @xmath59 of order one over timescales of interest . we compute @xmath61 here @xmath62 is a linear operator with a norm that scales roughly like @xmath63 , @xmath64 is quadratic and the residual given by the disparity between the interaction potential for toda and that for the granular chain is : @xmath65^p + \frac{p-1}{p}\left [ 1 + \frac{x_n - x_{n+1}}{p-1}\right]^p \\ \\ & = & \frac{1}{6}(1-\frac{p-2}{p-1})((x_{n-1}-x_n)^3 - ( x_n - x_{n+1})^3 ) + \mathcal{o}\left((x_{n-1}-x_n)^4 + ( x_n - x_{n+1})^4\right ) . { \end{array}}\ ] ] since the discrete wave equation conserves the @xmath66 norm exactly , the @xmath66 norm of @xmath59 , for time scales on which @xmath67 , will be bounded above by a constant times @xmath68 times the @xmath66 norm of @xmath69 . in other words @xmath59 remains of order one on timescale @xmath70 so long as @xmath71 in the sequel we will consider solutions for which @xmath72 over timescales @xmath73 . thus @xmath74 and we obtain an upper bound on the approximation error of @xmath75 . we note that this improves on the estimate of @xmath76 which appears e.g. in @xcite . [ a number of details towards making this argument rigorous are presented in the appendix ] . after describing the single and multiple solitary wave solutions of the toda lattice , we will return to the numerical examination of the validity of this concrete prediction . in starting our comparison of the evolution of toda lattice analytical solutions with the granular crystal dynamical evolution , we consider the single soliton solution of the toda lattice of form @xmath77}{1+\exp[-2k(n-1 ) \pm 2(\sinh k)t]}\right\}. \label{todaone}\end{aligned}\ ] ] by composing two counter propagating solitons we get a typical dynamical evolution such as the one presented in figs . [ toda1 ] and [ toda2 ] . once again ( as in the kdv case ) , the former represents the snapshots at specific times , while the latter the contour plot of the strain variable evolution ( as will be the case in all the numerical experiments presented herein ) . the figure contains the comparison of 3 waveforms . the solid ( red ) one is from the time integration of the granular chain dynamics . the dashed ( blue ) line is a plain superposition of two one - solitons of the toda lattice , while the dash - dotted ( green ) line shows the evolution of the toda lattice . detailed examination of the latter two suggests that the dashed and the dash - dotted curves do not perfectly coincide ( although such differences are not straightforwardly discernible in the scale of fig . [ toda1 ] ) . this is the well - known feature of the presence of _ phase shifts _ as a result of the solitonic collisions in the integrable dynamics . it is however relevant to add here that admittedly not only qualitatively but even quantitatively the toda lattice appears to be capturing the counter - propagating soliton dynamics of our granular chain , both before , during and after the collision . , @xmath44 , @xmath78 are used . the initial condition consists of two solitons of the same amplitude at @xmath79 and @xmath80 . here , the granular crystal ( non - integrable ) dynamics is also compared to the mere addition of two one - soliton solutions of the toda lattice . from top to bottom , left to right the snapshots shown are at @xmath81 . the ( red ) solid line is for the actual ( numerical ) granular lattice dynamics , the ( blue ) dashed line is the plain superposition of two toda one - soliton solutions of eqn . ( [ todaone ] ) , and the green dash - dotted line represents the numerical evolution of the toda chain . [ toda1],title="fig : " ] , @xmath44 , @xmath78 are used . the initial condition consists of two solitons of the same amplitude at @xmath79 and @xmath80 . here , the granular crystal ( non - integrable ) dynamics is also compared to the mere addition of two one - soliton solutions of the toda lattice . from top to bottom , left to right the snapshots shown are at @xmath81 . the ( red ) solid line is for the actual ( numerical ) granular lattice dynamics , the ( blue ) dashed line is the plain superposition of two toda one - soliton solutions of eqn . ( [ todaone ] ) , and the green dash - dotted line represents the numerical evolution of the toda chain . [ toda1],title="fig : " ] , @xmath44 , @xmath78 are used . the initial condition consists of two solitons of the same amplitude at @xmath79 and @xmath80 . here , the granular crystal ( non - integrable ) dynamics is also compared to the mere addition of two one - soliton solutions of the toda lattice . from top to bottom , left to right the snapshots shown are at @xmath81 . the ( red ) solid line is for the actual ( numerical ) granular lattice dynamics , the ( blue ) dashed line is the plain superposition of two toda one - soliton solutions of eqn . ( [ todaone ] ) , and the green dash - dotted line represents the numerical evolution of the toda chain . [ toda1],title="fig : " ] , @xmath44 , @xmath78 are used . the initial condition consists of two solitons of the same amplitude at @xmath79 and @xmath80 . here , the granular crystal ( non - integrable ) dynamics is also compared to the mere addition of two one - soliton solutions of the toda lattice . from top to bottom , left to right the snapshots shown are at @xmath81 . the ( red ) solid line is for the actual ( numerical ) granular lattice dynamics , the ( blue ) dashed line is the plain superposition of two toda one - soliton solutions of eqn . ( [ todaone ] ) , and the green dash - dotted line represents the numerical evolution of the toda chain . [ toda1],title="fig : " ] , @xmath44 , @xmath78 are used . the initial condition consists of two solitons of the same amplitude at @xmath79 and @xmath80 . here , the granular crystal ( non - integrable ) dynamics is also compared to the mere addition of two one - soliton solutions of the toda lattice . from top to bottom , left to right the snapshots shown are at @xmath81 . the ( red ) solid line is for the actual ( numerical ) granular lattice dynamics , the ( blue ) dashed line is the plain superposition of two toda one - soliton solutions of eqn . ( [ todaone ] ) , and the green dash - dotted line represents the numerical evolution of the toda chain . [ toda1],title="fig : " ] , @xmath44 , @xmath78 are used . the initial condition consists of two solitons of the same amplitude at @xmath79 and @xmath80 . here , the granular crystal ( non - integrable ) dynamics is also compared to the mere addition of two one - soliton solutions of the toda lattice . from top to bottom , left to right the snapshots shown are at @xmath81 . the ( red ) solid line is for the actual ( numerical ) granular lattice dynamics , the ( blue ) dashed line is the plain superposition of two toda one - soliton solutions of eqn . ( [ todaone ] ) , and the green dash - dotted line represents the numerical evolution of the toda chain . [ toda1],title="fig : " ] . the left panel represents the dynamical evolution of two colliding solitary waves of the granular lattice . the right panel shows the _ difference _ between the granular lattice and the superposition of two one - soliton solutions of the toda lattice . the very small magnitude of the difference ( to be quantified further below ) in the colorbar in comparison to the left panel illustrates the relevance of our approximation.,title="fig : " ] . the left panel represents the dynamical evolution of two colliding solitary waves of the granular lattice . the right panel shows the _ difference _ between the granular lattice and the superposition of two one - soliton solutions of the toda lattice . the very small magnitude of the difference ( to be quantified further below ) in the colorbar in comparison to the left panel illustrates the relevance of our approximation.,title="fig : " ] the two - soliton solution of the toda lattice is of the form @xcite @xmath82 with @xmath83+a_2\exp[2(k_2 n -\beta_2 t ) ] + \exp[2(k_1+k_2 ) n -2(\beta_1+\beta_2)t ] \right\ } \label{toda_aux}\end{aligned}\ ] ] and @xmath84 the results of this evolution are very similar to the ones illustrated above and hence are not shown here . in order to appreciate the role of the wave amplitude ( and thus of the speed in this mono - parametric family of soliton solutions ) in the outcome of the interaction , we have also explored higher amplitude collisions , as shown in figs . [ todatwosnap2]-[todatwo2 ] . in these cases , the small amplitude wakes of radiation traveling ( at the speed of sound ) behind the supersonic wave are more clearly discernible . nevertheless , once again the toda lattice approximation appears to capture accurately the result of such a collision occurring at strain amplitudes of about half the precompression . [ todatwo2 ] again captures not only the granular chain evolution but also the relative error between that and the corresponding toda lattice evolution . here , it is more evident that the eventual mismatch of speeds of the waves between the approximation and the actual evolution yields a progressively larger difference between the two fields . as a systematic diagnostic of the `` distance '' of the numerical granular crystal and approximate toda - lattice - based solutions ( and as a check of our theoretical prediction presented above ) , we have measured the @xmath85 norm ( maximum absolute value in space and time ) and the maximum of the @xmath66 norm in space of @xmath86 till the two counter - propagating solitons are well separated . we measured this quantity as a function of the parameter @xmath87 ( with @xmath88 , @xmath89 ) and report it as a function of the amplitude of @xmath90 in fig . ( [ error_glmtoda ] ) . as shown in fig . ( [ error_glmtoda ] ) , both graphs indicate a power law growth of the relevant error , with an exponent of @xmath91 and @xmath92 for the @xmath85 and @xmath66 norm of the error respectively . these results can be connected with the theoretical expectations for this power law . in particular , as we saw above the theoretical prediction for @xmath1 scales as @xmath93 , while the amplitude , @xmath94 , of the solution is proportional to @xmath95 , hence the scaling of the quantity measured in our numerics is theoretically predicted as @xmath96 . the close agreement with our numerics suggests that the theoretical estimate is tight i.e. , there is no normal form transformation which could push the residual between fpu and toda to higher order . , @xmath44 , @xmath97 , @xmath98 . the initial conditions consisted of a two - soliton solution with waves of the same amplitude centered at @xmath99 and @xmath100 at the toda lattice level . from left to right , top to bottom , the snapshots at times @xmath101 are shown . once again , the solid ( red ) line denotes the numerical granular chain evolution dynamics while the dashed ( blue ) line stems from the exact toda lattice two - soliton solution of eq . ( [ todatwosoliton ] ) . the three curves are nearly coincident for all the times considered . [ todatwosnap2 ] , title="fig : " ] , @xmath44 , @xmath97 , @xmath98 . the initial conditions consisted of a two - soliton solution with waves of the same amplitude centered at @xmath99 and @xmath100 at the toda lattice level . from left to right , top to bottom , the snapshots at times @xmath101 are shown . once again , the solid ( red ) line denotes the numerical granular chain evolution dynamics while the dashed ( blue ) line stems from the exact toda lattice two - soliton solution of eq . ( [ todatwosoliton ] ) . the three curves are nearly coincident for all the times considered . [ todatwosnap2 ] , title="fig : " ] , @xmath44 , @xmath97 , @xmath98 . the initial conditions consisted of a two - soliton solution with waves of the same amplitude centered at @xmath99 and @xmath100 at the toda lattice level . from left to right , top to bottom , the snapshots at times @xmath101 are shown . once again , the solid ( red ) line denotes the numerical granular chain evolution dynamics while the dashed ( blue ) line stems from the exact toda lattice two - soliton solution of eq . ( [ todatwosoliton ] ) . the three curves are nearly coincident for all the times considered . [ todatwosnap2 ] , title="fig : " ] , @xmath44 , @xmath97 , @xmath98 . the initial conditions consisted of a two - soliton solution with waves of the same amplitude centered at @xmath99 and @xmath100 at the toda lattice level . from left to right , top to bottom , the snapshots at times @xmath101 are shown . once again , the solid ( red ) line denotes the numerical granular chain evolution dynamics while the dashed ( blue ) line stems from the exact toda lattice two - soliton solution of eq . ( [ todatwosoliton ] ) . the three curves are nearly coincident for all the times considered . [ todatwosnap2 ] , title="fig : " ] , @xmath44 , @xmath97 , @xmath98 . the initial conditions consisted of a two - soliton solution with waves of the same amplitude centered at @xmath99 and @xmath100 at the toda lattice level . from left to right , top to bottom , the snapshots at times @xmath101 are shown . once again , the solid ( red ) line denotes the numerical granular chain evolution dynamics while the dashed ( blue ) line stems from the exact toda lattice two - soliton solution of eq . ( [ todatwosoliton ] ) . the three curves are nearly coincident for all the times considered . [ todatwosnap2 ] , title="fig : " ] , @xmath44 , @xmath97 , @xmath98 . the initial conditions consisted of a two - soliton solution with waves of the same amplitude centered at @xmath99 and @xmath100 at the toda lattice level . from left to right , top to bottom , the snapshots at times @xmath101 are shown . once again , the solid ( red ) line denotes the numerical granular chain evolution dynamics while the dashed ( blue ) line stems from the exact toda lattice two - soliton solution of eq . ( [ todatwosoliton ] ) . the three curves are nearly coincident for all the times considered . [ todatwosnap2 ] , title="fig : " ] , and the space - time contour plot of the granular lattice evolution , as well as the difference of that from the two - soliton solution of toda lattice are shown . [ todatwo2],title="fig : " ] , and the space - time contour plot of the granular lattice evolution , as well as the difference of that from the two - soliton solution of toda lattice are shown . [ todatwo2],title="fig : " ] norm of the error ( i.e. , difference of granular evolution from the toda lattice 2-soliton solution ) until the two solitons are well separated after the collision , versus the amplitude of the initial data . the right panel is the @xmath66 norm of the same quantity . both clearly represent a power law with a best fit exponent of @xmath91 and @xmath92 respectively ( shown in red dash line ) . [ error_glmtoda],title="fig : " ] norm of the error ( i.e. , difference of granular evolution from the toda lattice 2-soliton solution ) until the two solitons are well separated after the collision , versus the amplitude of the initial data . the right panel is the @xmath66 norm of the same quantity . both clearly represent a power law with a best fit exponent of @xmath91 and @xmath92 respectively ( shown in red dash line ) . [ error_glmtoda],title="fig : " ] lastly , we explore the case of the toda lattice approximation for the case of two co - propagating solitary waves . i.e. , recalling that one of the advantages of the toda lattice approximation is not only its discrete nature , but also its ability to capture both co - propagating and counter - propagating solutions , we use the 2-soliton solution of toda lattice of the form @xcite @xmath102 with @xmath103 + b \cosh[k_2(n - n_2)-\beta_2 t ] \right\}.\end{aligned}\ ] ] for the waves propagating in the same direction @xmath104 and the result of a typical example of the dynamical evolution is shown in figs . [ todatworrsnap]-[todatworr ] . it can be clearly observed here that the co - propagating case yields a far less accurate description than the counter - propagating one . this is presumably because of the shorter ( non - integrable ) interaction time in the latter in comparison to the former . furthermore notice that again the disparity between the two evolutions is far more pronounced for large amplitude waves , as the small amplitude one is accurately captured throughout the collision process . nevertheless , once again our integrable approximation is quite useful in providing at least a qualitative , essentially analytical handle on the interaction dynamics observed herein . , @xmath44 , @xmath105 , @xmath106 , with an initial condition of two solitons at @xmath99 and @xmath107 propagating in the same direction ( to the right ) is shown . from left to right , top to bottom , the snapshots of @xmath108 are shown . the ( red ) solid line represents the granular lattice evolution and the ( blue ) dashed line the toda 2-soliton solution of eq . ( [ todatwosoliton2 ] ) . [ todatworrsnap ] , title="fig : " ] , @xmath44 , @xmath105 , @xmath106 , with an initial condition of two solitons at @xmath99 and @xmath107 propagating in the same direction ( to the right ) is shown . from left to right , top to bottom , the snapshots of @xmath108 are shown . the ( red ) solid line represents the granular lattice evolution and the ( blue ) dashed line the toda 2-soliton solution of eq . ( [ todatwosoliton2 ] ) . [ todatworrsnap ] , title="fig : " ] , @xmath44 , @xmath105 , @xmath106 , with an initial condition of two solitons at @xmath99 and @xmath107 propagating in the same direction ( to the right ) is shown . from left to right , top to bottom , the snapshots of @xmath108 are shown . the ( red ) solid line represents the granular lattice evolution and the ( blue ) dashed line the toda 2-soliton solution of eq . ( [ todatwosoliton2 ] ) . [ todatworrsnap ] , title="fig : " ] , @xmath44 , @xmath105 , @xmath106 , with an initial condition of two solitons at @xmath99 and @xmath107 propagating in the same direction ( to the right ) is shown . from left to right , top to bottom , the snapshots of @xmath108 are shown . the ( red ) solid line represents the granular lattice evolution and the ( blue ) dashed line the toda 2-soliton solution of eq . ( [ todatwosoliton2 ] ) . [ todatworrsnap ] , title="fig : " ] , @xmath44 , @xmath105 , @xmath106 , with an initial condition of two solitons at @xmath99 and @xmath107 propagating in the same direction ( to the right ) is shown . from left to right , top to bottom , the snapshots of @xmath108 are shown . the ( red ) solid line represents the granular lattice evolution and the ( blue ) dashed line the toda 2-soliton solution of eq . ( [ todatwosoliton2 ] ) . [ todatworrsnap ] , title="fig : " ] , @xmath44 , @xmath105 , @xmath106 , with an initial condition of two solitons at @xmath99 and @xmath107 propagating in the same direction ( to the right ) is shown . from left to right , top to bottom , the snapshots of @xmath108 are shown . the ( red ) solid line represents the granular lattice evolution and the ( blue ) dashed line the toda 2-soliton solution of eq . ( [ todatwosoliton2 ] ) . [ todatworrsnap ] , title="fig : " ] ] we believe that the present work provides an insightful and meaningful ( not only qualitative but even quantitatively , where appropriate ) way for considering the interactions of solitary waves in the realm of granular crystal dynamics . what makes this work particularly timely and relevant is that the granular chain problem is currently both theoretically interesting and experimentally , as well as computationally tractable . two types of approximations were proposed herein for developing a qualitative and even semi - quantitative understanding of such collisions . the first was based on the well known kdv equation . while this is a useful approximation , some of its limitations were discussed , the most notable being the continuum , long - wavelength nature of the approximation , as well as the uni - directional character of the interaction ( i.e. , co - propagating waves ) . to avoid these constraints , a second approximation , based on the toda lattice was also presented . the latter provided a high - quality description especially of counter - propagating wave collisions , while its 2-soliton solutions can also be used for capturing co - propagating cases , at least qualitatively . it is relevant to note here that at the level of @xmath109 ( i.e. , when the precompression is absent ) such collisions have _ already _ been studied in @xcite . it thus seems that an extension of that work to experimentally consider collisions in the more theoretically and analytically tractable case of finite precompression would be possible , as much as it would be desirable . we believe that this line of thinking , and especially the approximation of using a discrete model such as the toda lattice could provide a useful tool for understanding different forms of solitary wave interactions in hertzian systems @xcite . a more ambitious generalization would involve the consideration of two - dimensional lattices and the potential reduction thereof to kadomtsev - petviashvilli continuum models ( i.e. , 2d generalizations of the kdv ) or perhaps to other lattice models in order to understand the dynamics of higher dimensional such chains . another challenging problem would be to obtain some analytical understanding of the collisions without precompression ; the difficulty in that case stems from the absence of a well - established , yet analytically tractable ( discrete or continuum ) description for capturing multi - soliton interactions . such directions are currently under consideration and will be presented in future publications . to obtain rigorous estimates it is first useful to write the general fpu chain as a first order system . observe that the chain of oscillators @xmath110 can be rewritten as the system @xmath111 upon making the change of variables @xmath112 , @xmath113 . both the granular chain and the toda lattice are special cases with @xmath114 and @xmath115 respectively . writing the hamiltonian @xmath116 and the operator @xmath117 defined by @xmath118_n = ( p_{n-1}-p_n , r_n - r_{n+1})$ ] the equation is rewritten as the system of hamiltonian odes @xmath119 notice that in the above expression @xmath120_n = \left({\begin{array}}{c } v'(r_n ) \\ p_n { \end{array}}\right)$ ] . having provided this general setup for our toda and granular hamiltonian chains , we now proceed to present the proposition that estimates the proximity between the 2-soliton solutions of these two models for small amplitude initial data ( quantified by @xmath121 below ) and long times ( quantified by @xmath73 below ) . let @xmath122 denote the toda hamiltonian and let @xmath123 denote its four - parameter family of two - soliton solutions . let @xmath124 be a general smooth interaction potential satisfying @xmath125 as well as @xmath126 and @xmath127 and let @xmath128 denote the solution of the corresponding fpu lattice with initial condition @xmath129 . in the case of counterpropagating solitary waves , the time scale @xmath73 is sufficiently long for the waves to pass through each other . the content of the theorem is that the amount of energy that is transferred from coherent modes to radiative modes is very small compared to the energy in the coherent modes . the proposition can be regarded as a corrolary of three lemmas . before we embark into their technical description , let us give a brief outline of the physical significance of each one . lemma 3 below shows that in the context of the toda 2-soliton solution , the interaction of two broad shallow waves does not produce high ( i.e. order greater than @xmath87 ) frequency ripples i.e. , `` radiation '' corresponding to such wavenumbers . lemma 1 makes use of lemma 3 to quantify the `` local truncation error '' for the scheme given by evolving a toda 2-soliton in lieu of fpu . i.e. , when we evolve with toda 2-soliton initial conditions in our fpu ( non - integrable ) lattice instead of the integrable toda one , there is a local truncation error stemming from the difference between the two lattice dynamics . this lemma quantifies this difference as a function of the solution amplitude ( represented by @xmath121 ) . finally , lemma 2 estimates how the difference of our fpu - type lattice and the toda lattice evolves over time on the basis of the above local truncation error and how the latter `` accumulates '' over a long interval of time @xmath70 ( characterized by @xmath73 ) . let @xmath136 denote a toda 2-soliton solution . let @xmath117 , @xmath137 and @xmath122 be defined as above . define @xmath138 and define @xmath139 . let @xmath140 and @xmath141 be fixed numbers and let the amplitude parameters for @xmath136 be given by @xmath142 and @xmath143 . let the following be given : a hilbert space @xmath152 with inner product @xmath153 and associated norm @xmath154 , an open subset @xmath155 , @xmath156 functions @xmath157 and @xmath158 from @xmath159 to @xmath152 , with identical and symplectic linear part , i.e. @xmath160 satisfying @xmath161 for all @xmath162 . there exist positive constants @xmath2 and @xmath163 such that if the estimates @xmath164 } \|f'(x_*)-j\|_{h \to h } < \frac{\delta_0}{t},\ ] ] @xmath165 and @xmath164 } \|f(x_*)-f_*(x_*)\| < \frac{1}{c_0 \delta_0 ^ 2 t^2}\ ] ] hold for some solution @xmath166 on some time interval @xmath167 $ ] , then the following hold : introduce the new variable @xmath59 by the equation @xmath172 . the proof will proceed by deriving first an evolution equation for @xmath59 , and then an evolution equation for @xmath173 , using a bootstrapping argument to show that if @xmath174 is not too large , then @xmath175 remains not too large for @xmath176 $ ] . we begin by computing @xmath177 the first term is the linearization about zero . the second term is the linear part owing to the fact that the linearization about @xmath136 is not equal to the linearization about zero . the third term incorporates all of the quadratic and higher order terms in @xmath157 and the fourth term owes to the fact that @xmath136 and @xmath168 satisfy different des . thus @xmath181 and hence @xmath182 } \|\dot{y}-jy\| t}\ ] ] for @xmath183 . in light of we see that @xmath184 now let @xmath58 be the largest time for which @xmath185 } e(t ) \le 2 \lceil e^4 \rceil$ ] . the hypotheses of the lemma guarantee that each of the terms @xmath186 , @xmath187 , @xmath186 and @xmath188 are bounded above by one , hence the exponential of the sum is bounded above by @xmath189 . in particular @xmath190 for as long as @xmath191 and also @xmath176 $ ] . thus the inequality @xmath190 holds for all @xmath176 $ ] . sinkovits and s. sen , phys . 74 * , 2686 ( 1995 ) ; d.p . visco , s. swaminathan , t.r.krishna mohan , a. sokolow and s. sen , phys . e * 70 * , 051306 ( 2004 ) ; s. sen , t.r.krishna mohan , d.p . visco , s. swaminathan , a. sokolow , e. avalos and m. nakagawa , int . j. mod b * 19 * , 2951 ( 2005 ) . g. schneider and c.e . counter - progagating waves on fluid surfaces and the continuum limit of the fermi - pasta - ulam model _ , in k. fiedler , b. grger and j. sprekels ( eds . ) , equadiff99 ; proceedings of the international conference on differential equations , world scientific , ( singapore , 2000 ) .

our aim in the present work is to develop approximations for the collisional dynamics of traveling waves in the context of granular chains in the presence of precompression . to that effect , we aim to quantify approximations of the relevant hertzian fpu - type lattice through both the korteweg - de vries ( kdv ) equation and the toda lattice . using the availability in such settings of both 1-soliton and 2-soliton solutions in explicit analytical form , we initialize such coherent structures in the granular chain and observe the proximity of the resulting evolution to the underlying integrable ( kdv or toda ) model . while the kdv offers the possibility to accurately capture collisions of solitary waves propagating in the same direction , the toda lattice enables capturing both co - propagating and counter - propagating soliton collisions . the error in the approximation is quantified numerically and connections to bounds established in the mathematical literature are also given .

1405.1768

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we show that the centrality and system - size dependence of elliptic flow measured at rhic are fully described by a simple model based on eccentricity scaling and incomplete thermalization . we argue that the elliptic flow is at least 25% below the ( ideal ) `` hydrodynamic limit '' , even for the most central au - au collisions . this lack of perfect equilibration allows for estimates of the effective parton cross section in the quark - gluon plasma and of its viscosity to entropy density ratio . we also show how the initial conditions affect the transport coefficients and thermodynamic quantities extracted from the data , in particular the viscosity and the speed of sound . when two ultrarelativistic nuclei collide at non - zero impact parameter , their overlap area in the transverse plane has a short axis , parallel to the impact parameter , and a long axis perpendicular to it . this almond shape of the initial profile is converted by the pressure gradient into a momentum asymmetry , so that more particles are emitted along the short axis @xcite . the magnitude of this effect is characterized by elliptic flow , defined as @xmath0 where @xmath1 is the azimuthal angle of an outgoing particle , @xmath2 is the azimuthal angle of the impact parameter , and angular brackets denote an average over many particles and many events . the unexpected large magnitude of elliptic flow at rhic @xcite has generated a lot of activity in recent years . elliptic flow results from the interactions between the produced particles , and can be used to probe local thermodynamic equilibrium . if the produced matter equilibrates , it behaves as an ideal fluid . hydrodynamics predicts that at a given energy , @xmath3 scales like the eccentricity @xmath4 of the almond @xcite . it is independent of its transverse size @xmath5 , as a consequence of the scale invariance of ideal - fluid dynamics . if , on the other hand , equilibration is incomplete , then eccentricity scaling is broken and @xmath6 also depends on the knudsen number @xmath7 , where @xmath8 is the length scale over which a parton is deflected by a large angle . here , we show that the centrality dependence of @xmath6 , for both au+au and cu+cu collisions , can be described by the following simple formula @xcite : @xmath9 @xmath6 is largest in the hydrodynamic limit @xmath10 . the first order corrections to this limit , corresponding to viscous effects , are linear in @xmath11 . for large mean - free path , far from the hydrodynamic limit , @xmath12 vanishes like the number of collisions per particle . one expects the transition between these two regimes to occur when @xmath13 , hence that @xmath14 . a recent transport calculation @xcite in two spatial dimensions indeed obtained @xmath15 . elliptic flow develops gradually during the early stages of the collision . due to the strong longitudinal expansion , the thermodynamic properties of the medium depend on the time @xmath16 , of course . the average particle density , for instance , decreases like @xmath17 ( if their number is approximately conserved , see recent discussion in @xcite ) : @xmath18 where @xmath19 denotes the total ( charged + neutral ) multiplicity per unit rapidity , and @xmath20 is the transverse overlap area between the two nuclei . the quantities that we shall extract from @xmath3 should be intepreted as averages over the transverse area @xmath20 , and over some time interval around @xmath21 , which is the typical time scale for the build - up of @xmath3 in hydrodynamics @xcite . @xmath22 denotes the velocity of sound . the knudsen number @xmath11 is defined by evaluating the mean free path @xmath23 ( @xmath24 is a partonic cross section ) at @xmath25 . thus , @xmath26 the purpose of this letter is to show that the centrality and system - size dependence of the data for @xmath3 at rhic is described very well by eqs . ( [ v2k ] ) and ( [ knud ] ) . this provides three important pieces of information . first , such a fit allows us to `` measure '' the knudsen number corresponding to a given centrality , which quantifies how close the dense matter produced in heavy - ion collisions at rhic is to perfect fluidity . second , the extrapolation to @xmath27 allows us to read off the limiting value for @xmath28 extracted from the _ data _ ; this is useful for constraining the equation of state ( eos ) of qcd via hydrodynamic simulations , and we shall also see that it exhibits a rather surprising dependence on the initial conditions . finally , using eq . ( [ knud ] ) , we can convert the knudsen number into the typical partonic cross section @xmath24 ( and viscosity ) in the quark - gluon plasma ( qgp ) . since only the combination @xmath29 actually appears in eq . ( [ v2k ] ) , uncertainties in @xmath30 or @xmath22 then translate into corresponding uncertainties of @xmath24 . unless mentioned otherwise , our standard choice is @xmath31 ( ideal quark - gluon plasma ) and @xmath32 . letting @xmath33 and @xmath34 instead reduces the estimated @xmath24 by a factor of two ; on the other hand , taking @xmath35 and @xmath36 increases @xmath24 by the same factor . for the elliptic flow , @xmath3 , we use phobos data for au - au @xcite and cu - cu @xcite collisions . the same analysis could be carried out using data from phenix @xcite or star @xcite . the initial eccentricity @xmath4 and the transverse density @xmath37 are evaluated using a model of the collision . two such models will be compared . the remaining parameters @xmath38 and @xmath24 are fit to the data . the first step is to plot @xmath6 versus @xmath37 @xcite . such plots have already been obtained at sps and rhic @xcite , and they are puzzling : while @xmath6 increases with centrality , it shows no hint of the _ saturation _ predicted by eq . ( [ v2k ] ) for @xmath39 , suggesting that the system is far from equilibrium @xcite . on the other hand , the value of @xmath3 for central au - au collisions at rhic is about as high as predicted by hydrodynamics , which is widely considered as key evidence that a `` perfect liquid '' has been created at rhic @xcite . it was understood only recently that the eccentricity of the overlap zone has so far been underestimated , as the result of two effects . the first effect is fluctuations in initial conditions @xcite : the time scale of the nucleus - nucleus collision at rhic is so short that each nucleus remains in a frozen configuration , with its nucleons distributed according to the nuclear wave function . fluctuations in the nucleon positions result in fluctuations of the overlap area . their effect on elliptic flow was first pointed out in ref . @xcite . it was later realized by the phobos collaboration @xcite that the orientation of the almond may also fluctuate , so that @xmath2 in eq . ( [ defv2 ] ) is no longer the direction of impact parameter , but the minor axis of the ellipse defined by the positions of the nucleons . these fluctuations explain both the large magnitude of @xmath3 for small systems , such as cu - cu collisions , as well as the non - zero magnitude of @xmath3 in central collisions , where the eccentricity would otherwise vanish . they have to be taken into account in order to observe the expected saturation of @xmath6 at high density mentioned above . the eccentricity is usually estimated from the distribution of participant nucleons in the transverse plane ( glauber model ) . more precisely , we assume here that the density distribution of produced particles is given by a fixed 80%:20% superposition of participant and binary - collision scaling , respectively @xcite . for au - au collisions , this simple model reproduces the centrality dependence of the multiplicity reasonably well ( we assume that charged particles are 2/3 of the total multiplicity , and that @xmath40 at midrapidity ) , while it underestimates it for central cu - cu collisions by about 10% . at high energies a second effect which increases the eccentricity is perturbative gluon saturation , which determines the @xmath41-integrated multiplicity from weak - coupling qcd without additional models for soft particle production . high - density qcd ( the `` color - glass condensate '' ) predicts a different distribution of produced gluons , @xmath42 , which gives a similar centrality dependence of the multiplicity @xcite but a larger eccentricity @xcite . when particle production is dominated by transverse momenta below the saturation scale of the denser nucleus , then @xmath43 traces the participant density of the more dilute collision partner , rather than the average as in the glauber model @xcite . precise figures depend on how the saturation scale is defined @xcite . naively , the larger initial eccentricity predicted by the gluon saturation approach is expected to require more dissipation in order to reproduce the same experimentally measured @xmath3 . somewhat surprisingly , we shall find that this expectation is incorrect , which underscores the non - trivial role played by the initial conditions . both effects , fluctuations and gluon saturation , were recently combined by drescher and nara @xcite . in their approach , the saturation momenta and the unintegrated gluon distribution functions of the colliding nuclei are determined for each configuration individually . the finite interaction range of the nucleons is also taken into account . upon convolution of the projectile and target unintegrated gluon distribution functions and averaging over configurations , the model leads to a very good description of the multiplicity for both au - au as well as cu - cu collisions over the entire available range of centralities . having determined the density distributions of produced particles from either model as described above , we obtain the eccentricity via @xcite @xmath44 @xmath45 , @xmath46 are the respective root - mean - square widths of the density distributions , and @xmath47 ( a bar denotes a convolution with the density distribution for a given configuration while brackets stand for averages over configurations ) . the overlap area @xmath20 is defined by @xmath48 @xcite . we find it more appropriate to define these moments via the number density distribution @xmath42 rather than the energy density distribution @xmath49 . the reason is twofold : first , @xmath3 is extracted experimentally from the azimuthal distribution of particle number , not transverse energy ; second , our cgc approach describes the centrality dependence of the _ measured _ final - state multiplicity very well , which indicates that the ratio of final - state particles to initial - state gluons ( including possible gluon multiplication processes @xcite ) is essentially constant . ) and ( [ knud ] ) . [ fig : glauber ] ] , using cgc initial conditions . [ fig : cgc ] ] figs . [ fig : glauber ] and [ fig : cgc ] display @xmath6 as a function of @xmath37 for au - au and cu - cu collisions at various centralities , within the glauber and cgc approaches , respectively . for both types of initial conditions , cu - cu and au - au collisions at the same @xmath37 give the same @xmath6 within error bars . eccentricity fluctuations are crucial for this agreement @xcite . the figures also show that eqs . ( [ v2k ] ) and ( [ knud ] ) provide a good fit to the data , for both sets of initial conditions . on the other hand , the values of the fit parameters clearly depend on the initial conditions , which has important consequences for the physics . the first physical quantity extracted from the fit is the hydrodynamic limit , @xmath50 , obtained by extrapolating to @xmath51 . the values are @xmath52 with the glauber parameterization , and @xmath53 with cgc initial conditions . comparing these numbers to the experimental data points one observes that deviations from ideal hydrodynamics are as large as 30% , even for central au - au collisions . this is our first important result . so far , a quantitative extraction of the qcd eos from rhic data via hydrodynamic analysis was hampered by the fact that @xmath6 had not been factorized into the perfect - fluid part @xmath50 and the dissipative correction @xmath54 . for example , huovinen found @xcite that an eos with a rapid cross - over rather than a strong first - order phase transition , as favored by lattice qcd @xcite , overpredicted the flow data . this finding was rather puzzling , too , as it was widely believed that the rhic data fully saturates the hydrodynamic limit . our results suggest that ideal hydrodynamics _ should _ in fact overpredict the measured flow . that is , that one should not choose an eos in perfect - fluid simulations that fits the data rather , the eos could be extracted by comparing ideal hydrodynamics to @xmath50 . the next result is that cgc initial conditions , which predict a higher initial eccentricity @xmath4 , naturally lead to a lower hydrodynamic limit @xmath50 . now , close to the ideal - gas limit ( @xmath55 ) , @xmath50 scales approximately like the sound velocity @xmath22 @xcite . this means that cgc initial conditions imply a lower average speed of sound ( softer equation of state ) than glauber initial conditions , by a factor of @xmath56 . the second fit parameter is the partonic cross section @xmath24 . the larger @xmath24 , the faster the saturation of @xmath6 as a function of @xmath37 . for our standard values of @xmath30 and @xmath22 we obtain @xmath57 mb for glauber initial conditions and @xmath58 mb for cgc initial conditions . these values are significantly smaller than those found in previous transport calculations @xcite , but match the findings of ref . @xcite . cgc initial conditions imply a larger value of @xmath24 than glauber initial conditions , that is , a _ lower _ viscosity . this can be easily understood . as already mentioned above , the cgc predicts a larger eccentricity @xmath4 than the glauber model for semi - central collisions of large nuclei ( when there is a large asymmetry in the local saturation scales of the collision partners , along a path in impact - parameter direction away from the origin @xcite ) . however , for very peripheral collisions or small nuclei , there is of course very little asymmetry in the saturation scales , and the eccentricity approaches the same value as in the glauber model . this has been checked numerically in fig . 7 of ref . @xcite , and can also be clearly seen by comparing our figures : while in fig . [ fig : cgc ] @xmath6 for semi - central au+au collisions is lower than in fig . [ fig : glauber ] , there is no visible difference for peripheral cu+cu collisions . in all , with cgc initial conditions the scaled flow grows less rapidly with the transverse density , which is the reason for the larger elementary cross - section . the dependence of @xmath24 on the initial conditions is probably even stronger than the numerical values above suggest , for the following reason . as alluded to above , our fit to the data really determines the product @xmath59 , rather than @xmath24 alone . it appears reasonable to assume that @xmath30 does not depend on the initial conditions . however , for consistency , the speed of sound @xmath22 entering the knudsen number should match the one underlying the hydrodynamic limit @xmath60 ; hence , if cgc initial conditions require a smaller @xmath22 by a factor @xmath61 , the elementary cross - section obtained above should be rescaled accordingly . this leads to our final estimate @xmath62 mb . our numerical results for @xmath24 should be taken as rough estimates rather than precise figures , because of the uncertainties related to the precise values of @xmath30 and @xmath22 . it is , however , tempting to convert them into estimates of the shear viscosity @xmath63 , which has been of great interest lately . a universal lower bound @xmath64 ( where @xmath65 is the entropy density ) has been conjectured using a correspondence with black - hole physics @xcite , and it has been argued that the viscosity of qcd might be close to the lower bound . extrapolations of perturbative estimates to temperatures @xmath66 mev , on the other hand , suggest that the viscosity of qcd could be much larger @xcite . on the microscopic side , @xmath63 is related to the scattering cross - section @xmath24 . following teaney @xcite , the relation for a classical gas of massless particles with isotropic differential cross sections ( which applies , for example , to a boltzmann - transport model ) is @xmath67 @xcite . on the other hand , the entropy density of a classical ultrarelativistic cas is @xmath68 , with @xmath69 the particle density , so that @xmath70 the relevant particle density in au - au collisions at rhic , which is estimated at the time when @xmath3 develops @xcite , is 3.9 @xmath71 for both glauber and cgc initial conditions , and @xmath72 mev . our two estimates @xmath73 mb ( glauber initial conditions ) and @xmath74 mb ( cgc initial conditions ) thus translate into @xmath75 fm , @xmath76 and @xmath77 fm , @xmath78 , respectively . these values for @xmath79 agree with those from ref . @xcite if the mean - free path is scaled to our result , and also with estimates of @xmath79 based on the observed energy loss and elliptic flow of heavy quarks @xcite , on transverse momentum correlations @xcite , or bounds on entropy production @xcite . hence , for our best fit(s ) @xmath79 is slightly larger than the conjectured lower bound , but significantly smaller than extrapolations from perturbative estimates . on the other hand , our lower value is close to a recent lattice estimate @xcite for su(3 ) gluodynamics , which gives @xmath80 at @xmath81 . a complementary approach to incorporate corrections from the ideal - fluid limit is viscous relativistic hydrodynamics . a formulation that is suitable for applications to high - energy heavy - ion collisions has been developped in recent years @xcite . a first calculation of elliptic flow @xcite shows that for glauber initial conditions and @xmath82 , @xmath3 reaches about @xmath83 of the ideal - fluid value for semi - central au - au collisions . it is interesting to note that our simple estimates are in good agreement with this finding . using eq . ( [ eta ] ) , @xmath82 corresponds to @xmath84 mb , for which eqs . ( [ v2k ] ) and ( [ knud ] ) give @xmath85 . the comparison to experimental data in ref . @xcite , however , appears to favor lower values of @xmath79 because the eos used there underpredicts @xmath86 required for glauber initial conditions . alternatively , simulations could be performed with cgc initial conditions which require only @xmath87 . in summary , we have shown that the centrality and system - size dependence of the _ measured _ @xmath3 can be understood as follows : @xmath3 scales like the initial eccentricity @xmath4 ( as predicted by hydrodynamics ) , multiplied by a correction factor due to off - equilibrium ( i.e. , viscous ) effects . this correction involves the multiplicity density in the overlap area , @xmath37 . two types of initial conditions have been compared : a glauber - type model , and a color - glass condensate approach . phobos data can be described with both . in particular , there is good agreement between cu - cu and au - au data . the resulting estimates for thermodynamic quantities and transport coefficients , on the other hand , depend significantly on the initial conditions . color glass condensate - type initial conditions require _ lower _ viscosity and a _ softer _ equation of state ( smaller speed of sound ) . the scaled flow extrapolated to vanishing mean - free path is lower than for glauber initial conditions by a factor of @xmath88 ; the effective speed of sound should also be lower by about the same factor . our estimates for the viscosity are @xmath89 for glauber initial conditions , and @xmath90 for cgc initial conditions , but these numbers should be taken only as rough estimates . we have also shown that the data for the scaled flow indeed _ saturate _ at high densities to a hydrodynamic limit . in central au - au collisions at rhic , @xmath3 reaches 70% ( resp . 75% ) of the hydrodynamic limit for glauber ( cgc ) initial conditions . the corrections to ideal hydrodynamics are therefore significant , but reasonably small compared to unity , implying that ( viscous ) hydrodynamics should be a valid approach for understanding flow at rhic . also , the asymptotic limit of @xmath6 has been isolated and could now be used to test realistic equations of state from lattice - qcd with hydrodynamic simulations of heavy - ion collisions .

0704.3553

the rotor mechanism , firstly proposed in the theory of self - organized criticality @xcite under name `` eulerian walk '' @xcite , was rediscovered independently as a tool for a derandomization of the random walk @xcite . the subsequent studies were concerned with collective properties of the medium `` organized '' by the walk and with statistical properties of the walk itself @xcite . the dynamics of the rotor - router walk can be described as follows . consider a square lattice with arrows attached to the lattice sites . arrows attached to the lattice sites are directed toward one of their neighbors on the lattice . a particle called usually _ chip _ , performs a walk jumping from a site to a neighboring site . arriving to a given site , the chip changes direction of the arrow at that site in a prescribed order and moves toward the neighbor pointed by new position of the arrow . thus , given an initial orientation of arrows on the whole lattice , the rotor - router walk is deterministic . the walk started from uniformly distributed random initial configurations can be called uniform rotor walk . three steps of the rotor walk on the square lattice are shown in fig.[steps ] . if the lattice is finite , the walk starting from an arbitrary site settles into an eulerian circuit where each edge of the lattice is visited exactly once in each direction @xcite . when the walker is in the eulerian circuit , configurations of rotors @xmath4 associated to each site are recurrent . a graphic representation of the recurrent configuration is _ unicycle _ which is a specific state where the arrows form a spanning set of directed edges containing a unique directed cycle which the chip belongs to @xcite . if the position of the chip on the cycle is @xmath5 , we denote the unicycle as @xmath6 . along with the unicycle , we can define the _ multicycle _ @xcite as a graph containing exactly @xmath7 cycles together with @xmath7 chips at vertices @xmath8 belonging to the cycles . for multicycles , we use the notation @xmath9 . for the infinite lattice , both questions on trajectories of the walker and on the configurations of arrows become more complicated . a basic problem here is to find the range of the rotor walk , i.e. the number of distinct sites visited in @xmath10 steps and , given the lattice symmetry and the rotor mechanism , to find a shape of the area visited by the walker . one conjecture and one theorem shed light on this problem . kapri and dhar @xcite conjectured that the set of sites visited by the clockwise uniform rotor walk in @xmath10 steps on the infinite square lattice is asymptotically a disk of average radius @xmath11 where @xmath12 is a constant . florescu , levine and peres @xcite proved that for an infinite @xmath13-dimensional lattice , regardless of a rotor mechanism or an initial rotor configuration , the rotor walk in @xmath10 steps visits at least on the order of @xmath14 distinct sites . monte carlo simulations in @xcite showed that the average number of visits of a site inside the disk is a linear decreasing function of its distance from the origin . the authors of @xcite give the following explanation of this characteristic behavior . after a moment when two sites at different distances from the origin are visited by the rotor walk , both sites are visited equally often because of the local euler - like organization of arrows . then , the difference between the numbers of visits of these sites remains bounded for an arbitrary number of subsequent steps . the key point in this explanation is the local eulerian organization which is proven rigorously only for finite graphs as a part of the total organization . for the infinite lattice , any bounded domain tends to the entirely organized state only asymptotically being repeatedly visited by the rotor walk . a question , however , is in the periodicity of returns . the mean number of returns and the mean - square displacement should be in a definite proportion to provide the sub - diffusive behavior of the rotor walk . so , it is desirable to find in the system of rotors some structure which provides sufficiently often returns of the walker to the origin and , as a consequence , to any previously visited site . the construction of such a structure is the main goal of the present paper . in the recent work @xcite , we have considered the motion of the clockwise rotor - router walk inside closed contours emerged in random rotor configurations on the infinite square lattice . we proved a property called the _ weak reversibility _ : even though the configuration of rotors inside the contour is random , the rotor - router walk inside the contour demonstrates some regularity , namely , the chip entering the clockwise contour @xmath15 in a vertex @xmath16 leaves the contour at the same vertex @xmath17 , and then the clockwise orientation of rotors on @xmath15 becomes anti - clockwise . we referred to the sites where rotors complete clockwise contours as _ labels _ , and noticed that the sequence of labels forms a spiral structure . after averaging over initial random configurations of rotors , the sequence approaches asymptotically the _ archimedean _ spiral . however , the spiral structure as such does not explain the obligatory periodic visits of the origin by the rotor walk . in section iii , we consider particular labels called _ nodes_. the set of nodes being a subset of that of labels has also the spiral structure . the difference between labels and nodes lies in the disposition of contours corresponding to them . in the case of labels , a contour completed at given site is not necessarily adjacent to the contour associated with the previous label . in case of nodes , each new contour associated with a node either has common sites with that corresponding to the previous node , or contains this contour inside . in section iv , we analyze the structure of contours associated with nodes . according to the week reversibility , each contour after visiting its interior becomes anti - clockwise and left by the chip . at the moment of exit from the node located at vertex @xmath17 , there is a directed path formed by arrows and connecting one of neighboring sites of @xmath17 with the previous node . a collection of paths obtained at the moment @xmath10 is a tree rooted at the current location of the chip . the tree structure together with spiral - like motion of the chip provides the obligatory visits to the origin for each turn of the spiral . depending on the location of the origin with respect to the first clockwise contour , the chip returns to the vicinity of origin by different ways . once the spiral structure is formed , the number of visits to the origin is 4 for each rotation around the origin . then , the total average number of visits @xmath0 for @xmath3 rotations performed by the chip starting from the uniform random initial configurations of arrows is @xmath2 , when @xmath18 . since the archimedean spiral has a constant interval between coils , we obtain the linear dependence between the radius of the spiral and the number of returns to the origin . in the separate section v , we analyze the convergence of the set of labels ( nodes ) to the archimedean spiral . as it was noticed in @xcite , this convergence is extremely slow . the existence of the limit for the averaged ratio of radius @xmath19 to angle @xmath20 which is a constant @xmath21 for the purely archimedean case , is not proven yet . a prospective value of @xmath21 can be obtained from the scaling law for the average number of visits conjectured by kapri and dhar @xcite . extensive simulations show that the deviation from the constant @xmath21 remains considerable for very large number of nodes @xmath22 and huge number of steps @xmath23 . we discuss a possible reason for this deviation . we consider the infinite square lattice , and fix the clockwise rotor mechanism at each site . in the initial rotor state , the arrows at each lattice site are directed randomly to one of four directions with equal probabilities , and the chip is in the origin . at each step of discrete time , the chip arriving at a site rotates the arrow at that site 90 degrees clockwise , and moves to the neighboring site pointed by the new position of arrows . the motion of the chip is determined by the current rotor state . given a rotor state , we say that a group of arrows forms a directed path if the arrows are attached to sites @xmath24 such that @xmath25 and @xmath26 are neighbors , and the arrow at @xmath25 is directed toward @xmath26 for @xmath27 . the directed path of arrows becomes a cycle if @xmath28 . a shortest possible cycle consists of two adjacent sites @xmath29 , @xmath30 , which are connected by a pair of edges from @xmath29 to @xmath30 and back . we call such cycles _ dimers _ by analogy with lattice dimers covering two neighboring sites . a cycle formed by more than two edges is called _ contour_. the configuration of arrows inside a contour is either free of cycles or contains a number of cycles . in the first case , the arrows inside the contour form a spanning forest rooted at the contour . in the second case , the arrows form a spanning forest where each tree is rooted either at the external contour , or at one of the internal cycles . correspondingly , two theorems describe the behavior of the chip approaching contours . _ theorem 1 _ on reversibility ( @xcite and @xcite , corollary 4.11 ) . let @xmath31 be a planar graph containing a unicycle @xmath6 with the contour @xmath15 oriented clockwise and @xmath32 . after the rotor - router walk makes some number of steps , each rotor internal to @xmath15 has performed a full rotation , each rotor external to @xmath15 has not moved , each rotor on @xmath15 has performed a partial rotation so that @xmath15 is now oriented anti - clockwise and the chip has returned to @xmath5 . to describe the motion of the chip in the second case , we consider a more general situation , where several chips are involved into the evolution of arrows inside the contour . _ theorem 2 _ , @xcite . let @xmath31 be a connected bidirected planar graph and @xmath9 be a multicycle with the external contour @xmath33 oriented clockwise together with @xmath34 internal cycles @xmath35 oriented anti - clockwise . the rotor - router operation is sequentially applied to the chip at @xmath36 until the moment @xmath37 when the chip returns to @xmath38 , and the rotor at @xmath38 is made oriented anticlockwise . then , the same is applied to chips at @xmath39 until the moments @xmath40 when chips starting from @xmath41 return to @xmath42 and the rotors at @xmath42 are made oriented clockwise . then , all rotors on @xmath33 are becoming oriented anticlockwise , all rotors on @xmath35 become oriented clockwise , and all vertices internal to @xmath33 and external to @xmath35 perform a full rotation . the description of motion of the single chip inside a contour is given by a reduced version of theorem 2 : _ theorem 3 _ on weak reversibility . let @xmath31 be a planar graph containing the external contour @xmath33 oriented clockwise and some number of internal cycles inside @xmath33 . the rotor - router walk starting at site @xmath17 , @xmath43 moves until the moment when the chip returns to @xmath17 . as a result , all rotors on @xmath33 become oriented anti - clockwise , and the chip leaves contour @xmath33 at the next time step . the rotors internal to @xmath15 perform either a full rotation or a partial rotation or do not move at all . the theorem 2 allows us to specify internal sites of contour @xmath33 which perform the full rotation . _ corollary . _ let @xmath44 be a set of sites inside the clockwise contour @xmath33 which belong to the forest rooted at @xmath33 . assume that there is the site @xmath45 and @xmath46 . the chip starting at site @xmath43 moves until the moment when the chip returns to @xmath17 . then , the rotor at @xmath47 performs the full rotation . _ let @xmath48 be the total number of steps in the process described in theorem 2 . the process can be divided into two stages , i and ii : stage i for steps @xmath49 and stage ii for steps @xmath50 . consider the site @xmath51 belonging to the forest @xmath44 rooted at @xmath33 . according to theorem 2 , the rotor at @xmath47 performs the full rotation to the moment @xmath52 . the number of rotations is @xmath53 . assume that a part of rotations is performed during stage i and the rest of them during stage ii . since stage ii follows stage i , a sequence of arrows resulting after @xmath52 steps is directed from @xmath47 to one of sites @xmath54 . but @xmath47 belongs to a tree rooted at @xmath33 by the condition . therefore , all @xmath53 rotations are performed during stage i , which coincides with the process described in theorem 3 . below , we apply the theorems 1,3 and corollary to investigate the uniform rotor walk on the infinite square lattice . the rotor walk during the time evolution creates contours of arrows @xmath55 sequentially . in @xcite , we considered the set of sites @xmath56 where each contour becomes closed at time steps @xmath57 , and called these sites labels . according to theorems 1,3 , the rotor walk leaves each contour @xmath58 at step @xmath59 at the same site @xmath60 . we skip details of evolution in the time intervals between @xmath61 and @xmath62 and ignore possible new contours appearing inside @xmath58 during these intervals . the only fact of the evolution between @xmath61 and @xmath62 we take into account is reversing the clockwise orientation of the contour @xmath58 . it was found in @xcite that the labels @xmath56 are not simply situated in the cluster of visited sites but form a spiral structure . an example of the spiral of labels is shown in fig.[fig2](a ) . whereas each particular spiral has an irregular form , their average over uniform random initial rotor states tends to the spiral obeying the archimedean property @xmath63 in planar coordinates @xmath64 , with constant @xmath5 and @xmath21 . a convergence of spirals to the archimedean law ( [ archimed ] ) for a large number of steps is discussed in section v. despite the surprising property , the spiral of labels does not say anything about periodicity of the chip returns to the vicinity of the origin needed for organization of the cluster of visited sites . to answer this question , we consider the sequence of contours @xmath55 corresponding to labels in more detail . let @xmath58 and @xmath65 be two successive contours in the sequence @xmath55 . there are three possibilities for the disposition of @xmath65 with respect to @xmath66 : a ) the set of sites @xmath67 where arrows of @xmath65 are attached has no common sites with @xmath68 and contour @xmath65 is outside @xmath66 ; b)the set @xmath67 has no common sites with @xmath68 and contour @xmath66 is inside @xmath65 ; c ) the set @xmath67 has at least one common site with @xmath68 . to provide for the condition b ) , the contour @xmath65 should contain inside at the moment @xmath69 all sites visited at moments @xmath70 . otherwise , there are lattice sites outside @xmath65 which do not connected with @xmath58 at the moment @xmath61 by any path of arrows , what is impossible for a single walk . when the cluster of visited sites grows , the probability of a contour enveloping the cluster of previously visited sites dramatically decreases and we can exclude the case b ) from the consideration . then , we select from the set of labels @xmath56 a subset of labels @xmath71 whose contours obey criterion c ) and assume that @xmath72 coincides with @xmath29 . we call the selected labels _ nodes_. fig.[fig2](b ) shows the spiral consisting of nodes selected from the labels of fig.[fig2](a ) . let @xmath73 be the sequence of nodes generated by the rotor walk . by the construction , at the moment of exit from the node located at vertex @xmath74 , there is a directed path formed by arrows and connecting one of neighboring sites of @xmath74 with the previous node @xmath75 . thus , the system of directed paths and remaining parts of contours associated with nodes forms a connected network . in what follows , we will see that the obtained network is a tree constructed from topologically uniform elements . the chip moving from a fixed site @xmath76 to site @xmath77 traces a path of arrows directed from @xmath76 to @xmath77 . fig.[labstar](a ) illustrates the path traced by the chip moving between two successive nodes @xmath78 and @xmath79 selected from the sequence of nodes @xmath73 . the first node is at the site @xmath78 marked by 1 . the directed path between sites 6 and 1 is a part of the contour @xmath80 corresponding to the first node . the chip creates a clockwise contour at site 2 , reverses it to the anti - clockwise one and leaves it at the same site 2 . assume , that this contour has no common sites with line @xmath81 . if it also has no intersections with either of previous contours , the site 2 is a label but not a node . the situation is repeated at site 3 and the chip continues motion creating an arbitrary number of labels until reaching site 5 . in general , site 5 does not belong to line @xmath81 because of a possible sequence of arrows directed from site 5 to site 6 existing before the chip could reach it . as a result , the clockwise contour appears @xmath82 at site 5 which is a node @xmath74 according to criterium c ) . after reversing @xmath83 to the anti - clockwise contour @xmath84 , the chip leaves @xmath85 at site 5 . the parts of reversed contours corresponding to the labels 2 and 3 and possible others are just branches attached to the path from 4 to 5 which do not affect connectivity of this path and therefore can be ignored . the resulting path shown in fig.[labstar](b ) is a building block of the tree we are going to construct . the tree is constructed by consecutive adding the building blocks , one by one for each new node . in fig.[tree ] we show schematically how the tree grows if one neglects a possible difference in sizes of the blocks and takes into account their topological structure only . consider the first clockwise contour containing the origin . in fig.[tree ] , this contour is formed by the path starting at site @xmath86 and reaching site 2 . due to the possible directed sequence from site 2 to 3 , the contour is closed at site 2 which is the first node . at site 2 , the chip leaves the contour after reversing its direction and reaches site 5 which is connected with site 6 by the directed sequence of arrows , if exists ( otherwise site 5 coincides with 6 ) . then the clockwise contour 2,4,5,6,3,2 appears with the next node at site 5 . the interior of this contour covers the sector of space bounded by two radial branches 3,2,4 and 3,6,5 . reversing this contour to the anti - clockwise one , the chip continues motion from node 5 to node 8 and so on . by the construction of the building blocks , the resulting graph @xmath87 is a tree . at each moment of time , all directed paths of arrows constituting the tree are oriented towards the current position of the chip . consider the chip at the last node situated at site 20 . its further path can trace a directed path of arrows to any reachable site of the tree @xmath87 . assume this site is on the interval between sites 10 and 8 . then the next clockwise contour will contain the sites 8,9,6,3,2,21,20 and cover the sector bounded by radial branches 3,2,21,20 and 3,6,9,8 . the continuation of the spiral up to full rotation creates subsequent contours covering one by one _ all _ sectors between branches of the tree , including the sector bounded by line @xmath88 and containing the origin . if the initial rotor configuration contains one , two or three sequences of arrows flowing into the origin , the structure of the tree @xmath87 admits two , three or four sectors having the origin on the boundary between sectors . the described scenario being empirical is nevertheless typical for any random initial configuration of rotors . the obtained tree is stable after each rotation of the spiral . indeed , by theorem 2 , the chip visiting a contour changes its orientation but not its form . branches of the tree situated inside contours consist of sites which perform the full rotation according to the corollary in section ii and therefore remain stable as well . the arrows inside the contours not belonging to these branches can not create new cycles due to the rotor mechanism and can only add new branches to the existing tree . the main conclusion we can draw from the existence of the tree structure and the spiral ordering of nodes is that every turn of the spiral generates necessarily either a contour containing the origin inside it or generates contours containing the origin on their boundaries . now we are ready to answer the question about the number of visits to the origin for each rotation . consider the first clockwise contour @xmath89 containing the origin inside . by the construction of the tree @xmath87 , the origin @xmath86 belongs to the forest rooted at sites of @xmath89 . it follows from the corollary , that the arrow at the origin performs the full rotation when @xmath89 is reversed . as the tree @xmath87 grows , the subsequent contours containing the origin become larger and the chip visiting them not necessarily visits all sites of their interiors . however , the forest arisen in the first contour remains rooted at all subsequent contours containing the origin . then by the corollary , the rotor at the origin performs the full rotation each time when such a contour appears . since it happens for each turn of the spiral , the number of visits of the origin @xmath90 depends of the number of turns @xmath91 as @xmath92 where @xmath93 is due to possible visits to the origin before the moment when the first loop of the spiral is formed . the result of simulation for a single spiral is shown in fig.[s1 ] . vs number of rotations @xmath91.,width=264 ] if the initial configuration contains one or more sequences of arrows directed to @xmath86,the origin can belong to the boundary of adjacent sectors of the tree . since all sectors are covered by contours appearing during one rotation of the spiral , the total number of visits to the origin remains 4 . for instance , let @xmath94 and @xmath95 be two adjacent contours and the origin 0 is at the boundary between them . reversing contour @xmath94 at some stage of rotation , the chip rotates the arrow at 0 by angle @xmath96 or @xmath97 depending on the form of the boundary . contour @xmath95 is also reversed during the same turn with the arrow rotation by the additional angles @xmath98 or @xmath99 giving the full rotation per one turn . the emergence of a spiral structure in the clockwise rotor - router walk comes from a rather simple reason . indeed , consider a label @xmath100 situated near the boundary of the cluster of visited sites . a preferable position for the next label @xmath101 is on the right side from the branch @xmath102 to provide the clockwise orientation of the contour @xmath103 if it has common edges with @xmath102 . then , the preferable direction of successive positions of labels @xmath104 is clockwise with respect to the origin of the cluster . since the size of the cluster grows with time , the positions @xmath104 form a spiral - like structure . the set of nodes , being a subset of labels has the spiral form as well . moreover , the condition for contours corresponding to two successive nodes to be adjacent imposes an additional restriction on the positions of nodes . the comparison of typical spirals in fig.[fig2](a ) and ( b ) shows that this restriction makes the spiral of nodes more regular than that of labels . introducing the polar coordinates , we denote by @xmath105 the distance from the origin and by @xmath106 the winding angle of @xmath7-th node . then , we say that the spiral of nodes is asymptotically archimedean in average if @xmath107 where the average is taken over the uniformly distributed states of the spiral . a numerical verification of the archimedean property for labels in @xcite showed a very slow converge to the asymptotic law ( [ definition ] ) . here , we use the more pronounced spiral structure of nodes to determine the asymptotic behavior of @xmath108 with greater accuracy . first , we compare the conjectured archimedean property ( [ definition ] ) with the conjecture by kapri and dhar @xcite , who supposed that the average number of visits @xmath109 to the site separated from the origin by distance @xmath110 for @xmath111 steps satisfies the scaling form @xmath112 where @xmath113 is the scaling function @xmath114 , @xmath115 . the data obtained in @xcite for @xmath116 and @xmath117 collapse with @xmath118 and @xmath119 . then , the limiting radius of the circle of visited sites depends on the number of visits to the origin @xmath90 as @xmath120 the spiral of nodes , by the construction , is contained inside the circle of visited sites . according to the archimedean law eq.([archimed ] ) with @xmath121 , for the large number of rotations @xmath91 , the spiral curve tends to the circle of radius @xmath122 . then , due to eq.([number ] ) , we obtain again the linear dependence of the radius on the number of visits to the origin @xmath123 thus , the conjectured archimedean property is consistent with the linear scaling law eq.([scaling ] ) . in the scaling limit , we can expect that @xmath124 . then , the constant @xmath21 in eq.([definition ] ) is related to the constants of the scaling law as @xmath125 . taking @xmath118 and @xmath119 , we obtain @xmath126 . to study the asymptotical behavior of the spiral , we extended our simulations to number of time steps @xmath127 and number of samples @xmath128 . the numbers of nodes and spiral rotations we use in our analysis are shown in fig.[fig5 ] . the result of averaging of the ratio @xmath129 over uniformly distributed initial conditions as a function of the node number is shown in fig.[fig4 ] . the horizontal line in fig.[fig4 ] corresponds to @xmath126 obtained from the scaling conjecture @xcite . we can see that the deviation from the constant @xmath21 remains considerable up to very large node numbers . vs number of node @xmath7.,width=415 ] despite the apparent decrease of the slope of the curve in fig.[fig4 ] , we can not guarantee the coincidence of the limiting value of @xmath130 with the scaling value @xmath126 and even the existence of the limit as such . the monte - carlo simulations show that the ratio of radius to angle grows with @xmath7 not faster than @xmath131 with @xmath132 . the logarithmic deviation obtained for finite @xmath7 suggests to try an asymptotic expansion in powers of @xmath133 . the obtained data yield the following lower bound : @xmath134 a reason for so slow convergence is a geometrical non - equivalence of averaging spirals . indeed , if the random spirals differ one from another only by the distances between coils , the averaging over large number of samples would give a well defined mean distance even for a relatively broad distance distribution . instead , we observe some number of meanders in the spiral structure at different regions of the spiral . this leads to an effective broadening of the intervals between coils , but since the meanders are rare events , the determination of parameters of the averaged spiral needs an enormously large statistics . vbp thanks the rfbr for support by grant 16 - 02 - 00252 . vsp thanks the jinr program `` ter - antonyan - smorodinsky '' .

we study the rotor - router walk on the infinite square lattice with the outgoing edges at each lattice site ordered clockwise . in the previous paper [ j.phys.a : math . theor . 48 , 285203 ( 2015 ) ] , we have considered the loops created by rotors and labeled sites where the loops become closed . the sequence of labels in the rotor - router walk was conjectured to form a spiral structure obeying asymptotically an archimedean property . in the present paper , we select a subset of labels called `` nodes '' and consider spirals formed by nodes . the new spirals are directly related to tree - like structures which represent the evolution of the cluster of vertices visited by the walk . we show that the average number of visits to the origin @xmath0 by the moment @xmath1 is @xmath2 where @xmath3 is the average number of rotations of the spiral . _ keywords _ : rotor - router walk , archimedean spiral , sub - diffusion .

1512.00280

from the time of hawking s discovery that black holes radiate with the black - body radiation , the problem of information stored in a black hole @xcite attracted much attention . different ideas were discussed , in particular those of remnants @xcite , `` fuzziness '' of the black hole @xcite and refs . therein , quantum hair @xcite and refs.therein . , and smearing of horizon by quantum fluctuations @xcite . the underlying idea of the last approach is that small fluctuations of the background geometry lead to corrections to the form of the density matrix of radiation . these corrections are supposed to account for correlations between the black hole and radiation and contain the imprint of information thrown into the black hole with the collapsing matter . the idea that horizon of the black hole is not located at the rigid position naturally follows from the observation that a black hole as a quantum object is described by the wave functional over geometries @xcite . in particular , the sum over horizon areas yields the black hole entropy . in papers @xcite the density matrix of black hole radiation was calculated in a model with fluctuating horizon . horizon fluctuations modify the hawking density matrix producing off - diagonal elements . horizon fluctuations were taken into account by convolution the density matrix calculated with the instantaneous horizon radius @xmath0 with the black hole wave function which was taken in the gaussian form @xmath1 . effectively the wave function introduces the smearing of the classical horizon radius @xmath2 . the width of the distribution , @xmath3 , was taken of order the plank lengths @xmath4 @xcite . in paper @xcite it was stated that the `` horizon fluctuations do not invalidate the semiclassical derivation of the hawking effect until the black hole mass approaches the planck mass '' . in this note we reconsider calculation the density matrix of radiation emitted from the black hole formed by the collapsing shell . the shell is supposed to follow the infalling trajectory which is the exact solution to the matching equations connecting the interior ( minkowski ) and exterior ( schwarzschild ) geometries of the space - time @xcite . in this setting one can trace propagation of a ray ( we consider only s - modes ) through the shell from the past to the future infinity . for the rays propagating in the vicinity of the horizon we obtain an exact formula connecting @xmath5 at the past infinity and @xmath6 at the future infinity . we obtain the expression for the `` smeared '' density matrix of hawking radiation of the black hole with the horizon smeared by fluctuations . in the limit @xmath7 the smeared density matrix turns to the hawking density matrix . the smeared density matrix is not diagonal and can be expressed as a sum of the `` classical part '' and off - diagonal correction which is roughly of order @xmath8 of the classical part . as a function of of frequencies @xmath9 of emitted quanta the distribution is concentrated around @xmath10 with the width of order @xmath11 . the paper is constituted as follows . in sect . 2 we review the geometry of the thin collapsing shell which follows a trajectory consisting of two phases . the trajectory is a solution of the matching equations connecting the internal and external geometries of the shell . we trace propagation of a light ray from the past to future infinity . in sect.3 we introduce the wave function of the shell which saturates the uncertainty relations . in sect.4 , we calculate the density matrix of black hole radiation smeared by horizon fluctuations . following the approach of paper @xcite calculation is performed by two methods : by the `` @xmath12 '' prescription and by using the normal - ordered two - point function . in sect.5 , using the exact expressions for the smeared radiation density matrix , we study the diagonal `` classical '' part of the density matrix and the off - diagonal elements . in this section we introduce notations and review the geometry of space with collapsing thin spherical shell @xcite . outside of the shell the exterior geometry is schwarzschild space - time , the interior geometry is minkowsky space - time . in the eddington - finkelstein coordinates the metric of the exterior space - time is 1.1 ds^2_(ext)=-(1-r / r ) dv^2 + 2dv dr + r^2d^2,r > r where @xmath13 @xmath14 and @xmath15 the metric of the interior space - time is 1.2 ds^2_(int ) = -dv^2 + 2dvdr + r^2 d^2 , where @xmath16 the light rays propagate along the cones @xmath17 in the exterior and along @xmath18 in the interior regions . trajectory of the shell is @xmath19 , where @xmath20 is proper time on the shell . the matching conditions of geometries on the shell , at @xmath21 , are 1.3 dv - du=2dr_s , dv - du= , dudv= ( 1-r / r_s ) dudv , , where the differentials are taken along the trajectory . from the matching conditions follow the equations 1.4 2r_s ( 1-u ) = u^2 - ( 1-r / r_s ) , + 1.5 2_s ( 1- ) = -^2 + ( 1-r / r_s ) .here prime and dot denote derivatives over @xmath22 and @xmath23 along the trajectory . the shell is in the phase i , for @xmath24 in the phase ii . @xmath25 is the point of horizon formation . ] the trajectory of the shell consists of two phases @xcite @xmath26 @xmath27 from the equations ( [ 1.4 ] ) , ( [ 1.5 ] ) are obtained the following expressions for the trajectory ; in the phase i 1.6 u(u)=l_0 u -2r_0 + 2r , v(v)=l_0 ( v-2x(r_0 ) ) + 2r , where @xmath28 . in the phase ii 1.7 v=2r , u= 2r-2r_s , + v=2x(r_0 ) u=2x(r_0 ) -2x(r_s ) . horizon is formed at @xmath29 and @xmath30 . we consider the modes propagating backwards in time . at @xmath31 the ray is in phase i , after crossing the shell it reaches @xmath32 in the phase ii . let the in - falling ray be at @xmath31 at @xmath33 , where @xmath34 is the point at which @xmath35 between the points 1 - 2 the ray propagates outside the shell in the phase i with @xmath36 . at the point 2 the ray crosses the shell and we have @xmath37 and @xmath38 . the ray propagates in the interior of the shell , and at the point 3 , at @xmath39 , we have @xmath40 . reflection condition at the point 3 is @xmath41 . at the crossing point 4 we have @xmath42 , where @xmath43 here @xmath44 stands for the radial position of the shell trajectory at the point 4 . the equation for @xmath45 can be written as @xmath46 in the region @xmath47 , where @xmath48 , neglecting in the first term @xmath49 as compared with @xmath50 , we obtain the approximate equation for @xmath45 1.8 = + thus , we have v_1 -v_0 = l_0 ^ -1v_2 = l_0 ^ -1v_3=l_0 ^ -1u_4 ( u_4 ) + = -l_0 ^ -12(r_0 -r)e^-(u_4 -2r_0)/r -1 1.8a removing the indices , we obtain our final result as 1.10 v = v_0 -2(el_0 ) ^-1(r_0 -r)e^-(u -2r_0)/r the above formulas are purely classical , modifications due to back reaction of hawking radiation are neglected . quantum nature of horizons of the black holes was discussed in the work of carlip and teitelboim @xcite , where it was shown that the area of horizon @xmath51 and the opening angle , @xmath52 or , equivalently , the deficit angle @xmath53 form the canonical pair . in paper @xcite it was shown that canonical pair is formed by the opening angle and the wald entropy @xmath54 @xcite 4.1 \ { , } = 1 . when the black hole is quantized , the poisson bracket is promoted to the commutation relation 4.2 [ , _ w ] = i. the wave function of the black hole satisfies the relation 4.3 -i=2 . the minimal uncertainty @xmath55 wave function is ( ) ~e^c(-2)^2 e^ < s_w > , where @xmath56 . for the spherically symmetric configurations which we consider the wave function written through the instantaneous horizon radius @xmath0 is 4.41 |(r)| = n^-1 e^- . the scale of horizon fluctuations is @xmath57 @xcite , where @xmath58 is the planck length and @xmath59 is the classical horizon radius of the black hole of the mass @xmath60 . the normalization factor @xmath61 is 4.5 n^-2=^_0 4dr r^2 e^ |r^2 . let us turn the calculation of hawking radiation of the massless real scalar field in the background of the black hole formed by the shell . to perform quantization of the field , we restrict ourselves to the @xmath62-wave modes . expanding the scalar field in the orthonormal set of solutions @xmath63 of the klein - gordon equation which at the past null infinity @xmath31 have only positive frequency modes we have 2.1 = _ i ( a_i u^(-)_i + a^+_i u^(-)*_i ) . the scalar product of the fields is 2.1a ( _ 1 , _ 2 ) = i_d^_2^ * _ _ 1 .alternatively the field @xmath64 can be expanded at the hypersurface @xmath65 where @xmath32 is the future null infinity and @xmath66 is the event horizon 2.2 = _ i ( b_i u^(+)_i + b^+_i u^(+)*_i + c_i q_i + c^+_i q^*_i ) . here @xmath67 is the orthonormal set of modes which contain at the @xmath32 only positive frequencies and @xmath68 is the orthonormal set of solutions of the wave equation which contains no outgoing components @xcite . the operators @xmath69 and @xmath70 are quantized with respect to the vacua @xmath71 and @xmath72 correspondingly . the modes @xmath73 can be expanded in terms of the modes @xmath74 2.3 u^(+)_i = _ j ( _ ij u^(-)_j + _ ij u^(-)*_j ) , where @xmath75 and @xmath76 are given by the scalar products @xmath77 for the spherically - symmetric collapse , the basis for the in- and outgoing modes is @xmath78 omitting the angular parts , the modes @xmath79 and @xmath80 are 2.4 u^(-)_(v)|_i^- ~,u^(+)_(u)|_i^+ ~ . from ( [ 1.10 ] ) we find @xmath81=f(r)-2r\ln \f{r_0 -r}{r},\ ] ] where @xmath82,\,\ , c=-\ln ( el_0 ) $ ] . to simplify formulas , we consider the case @xmath83 , so @xmath84 , and @xmath85 note that both @xmath86 and @xmath87 have explicit dependence on @xmath0 . the bogolubov coefficient @xmath88 smeared by horizon fluctuations is obtained by convoluting it with the function @xmath89 5.1 & & |__1_2= _ 0^dr r^2 e^-(r -|r ) ^2 /2 ^ 2n^2 _ _1_2~ + & & ~(_2 /_1 ) ^1/2_-^v_0 dv e^-i_1 ( f(r ) -2r ( ( v_0 -v)/2r ) -i_2 vdr n^2 r^2 e^-(r -|r ) ^2/2 ^ 2 .direct evaluation of the smeared bogolubov coefficient ( [ 5.1 ] ) yields ( cf.@xcite ) 5.1n |__1_2~dr r^2 n^2 e^-(r -|r ) ^2 /2 ^ 2r , following the paper @xcite we consider two ways of calculating the density matrix _ _1_2 = i_i^- dv u^(+)__1 ( v)_v u^(+)*__2 ( v)= i_i^- dvd d _ _1^*__2 u^(-)_ ( v ) _ v u^(-)*_ ( v ) + = d _ _1^*__2 3.1 , where in the last equality we used that the modes @xmath90 form the orthonormal set of functions on @xmath31 . the density matrix smeared by horizon fluctuations is m1 |__1_2=d _ _1 ^*__2 _ 0^dr_1 r_1 ^ 2 e^-(r_1 -|r ) ^2 /2 ^ 2n^2 _ 0^dr_2 r_2 ^ 2 e^-(r_2 -|r ) ^2 /2 ^ 2n^2 . substituting ( [ 5.1n ] ) , we have m2 |__1_2~_0^dr_1 r_2 ( -2ir_1 + ) ^-2ir_1 _1 ( 2ir_2+)^2ir_2_2 ( -2ir_1_1 ) ( 2ir_2_2 ) + e^-i_1 f(r_1 ) + i_2 f(r_2 ) dr_1 dr_2 r_1 ^ 2 r_2 ^ 2 n^4 e^-(r_1 -|r ) ^2 /2 ^ 2 e^-(r_2 -|r ) ^2 /2 ^ 2 the terms with @xmath86 have cancelled . integrating over @xmath91 , we obtain m3 |__1_2 ~dr_1 dr_2 r^2_1 r^2_2(r_1_1 -r_2_2 ) ( -)^-2ir_1_1 + e^-i_1 f(r_1 ) + i_2 f(r_2 ) n^4 e^-(r_1 -|r ) ^2 /2 ^ 2e^-(r_2 -|r ) ^2 /2 ^ 2 + = dr_1 dr_2 r_1 ^ 3 r_2 ^ 3 ( r_2 -r_1 ) + ( e^4r_1 _1 - 1 ) ^-1 e^-2ir_0 ( _1 -_2 ) n^4 e^-(r_1 -|r)^2/2 ^ 2 e^-(r_2 -|r ) ^2 /2 ^ 2 , where , taking into account the @xmath92-function , we substituted & & ( -2ir_1 ) ^-2ir_1_1(2ir_2 ) ^2ir_2_2(-r_1/r_2 ) ^-2ir_1_1= e^2r_1_1(r_1/r_2)^-2ir_1_1 + & & ( -2ir_1_1 ) ( 2ir_2_2 ) and e^-i_1 f(r_1 ) + i_2 f(r_2 ) e^2ir_1_1r_1 -2ir_2_2r_2=(r_1/r_2 ) ^2ir_1_1 . integration over @xmath93 yields m4 & & |__1_2~dr_1 r_1 ^ 5 ( ) ^3 ( e^4r_1_1 -1 ) ^-1 e^ -2ir_0 ( _1 -_2 ) + & & because @xmath94 both exponents have sharp extrema . integrating over @xmath95 , we arrive to the density matrix of the form m5 & & |__1_2~()^3 ( ) ^5 ( e^4|r ( _1 + _2 ) _1_2/(_1 ^ 2 + _2 ^ 2 ) -1)^-1 + & & e^- 2ir_0 ( _1 -_2 ) alternatively , the density matrix can be presented in the following form 3.2 |_=d_1 _ d_1^u^(+)*_lm(x_1 ) _ u^(-)__1 l_1 m_1 ( x_1 ) _ d_2^u^(+)_ l m ( x_2 ) _ u^(-)*__1 l_1 m_1 ( x_2 ) where for the initial value hypersurface can be taken either @xmath31 or @xmath32 . expanding @xmath64 in the basis @xmath96 @xmath97 where @xmath71 and @xmath72 are vacuum states at @xmath31 and @xmath32 , and using the relation @xmath98 we obtain 3.3 _ _1_2=_d_1^_d_2^<in|:(x_1 ) ( x_2 ) : |in > to perform calculation of ( [ 3.3 ] ) one can use the expansion of the two - point function @xmath99 on @xmath32 to obtain @xcite 3.4 _ _1_2~(_1 _2 ) ^-1/2_i^+du_1 du_2 e^-iu_1_1 + iu_2_2 ( - ) .where for @xmath100 we take the function ( [ 1.10 ] ) . for the density matrix modified by horizon fluctuations we obtain 5.3 & & |__1_2~(_1 _2 ) ^-1/2 _ i^+du_1_i^+ du_2 + & & e^-(r_1 -|r)^2/2 ^ 2 - ( r_2 -|r ) ^2 /2 ^ 2 n^4 r_1 ^ 2 r_2 ^ 2d r_1dr_2 , where for @xmath100 is taken the function ( [ 1.10 ] ) . extracting in the denominator the factor @xmath101 , shifting @xmath102 and changing variables @xmath103 , we obtain 5.4 |__1_2~(_1 _2 ) ^-1/2 ^_- du_1^_- du_2 e^-4i_1 u_1 r_1 + 4i_2 u_2 r_2 -2ir_0 ( _1 -_2 ) ^-2(u_1 -u_2 -i ) + e^-(r_1 -|r)^2/2 ^ 2 - ( r_2 -|r ) ^2/2 ^ 2 n^4r_1 ^ 2 r_2 ^ 2 d r_1 dr_2 performing the contour integration over @xmath104 around the pole in the upper half plane using the formula @xmath105 we have 5.5 |__1_2~(_1 _2 ) ^-1/2dr_1 dr_2 r_1 ^ 2 r_2 ^ 2 n^4 _1 r_1 du_2 e^-4i_1 r_1 u_2 + 4i_2 r_2 u_2 + e^-2ir_0 ( _1 -_2 ) e^-(r_1 -|r)^2/2 ^ 2 - ( r_2 -|r ) ^2 /2 ^ 2 . integration over @xmath106 yields 5.6 |__1_2~(_1_2 ) ^-1/2 dr_1 dr_2 r_1 ^ 2 r_2 ^ 2 n^4 _1 r_1 ( _1 r_1 -_2 r_2 ) + e^-2ir_0 ( _1 -_2 ) e^-(r_1 -|r)^2/2 ^ 2 - ( r_2 -|r ) ^ 2 . integrating over @xmath93 and removing the @xmath92-function , we obtain 5.7 |__1_2~ dr_1 r_1 ^ 5()^3 ( e^4r_1_1 - 1)^-1 e^-2ir_0 ( _1 -_2 ) + .expression ( [ 5.7 ] ) is identical to that obtained by method 1 in ( [ m4 ] ) . in the limit @xmath107 the expression 5.71 becomes the delta function @xmath108 . the density matrix @xmath109 eq . ( [ 5.7 ] ) turns into the formula for the hawking spectrum 5.8 _ _1_2 ~(_1 -_2 ) ( e^2|r ( _1 + _2 ) -1)^-1 . the smeared density matrix contains the off - diagonal elements . because the density matrix has the sharp maximum at @xmath10 , it is natural to divide it into the `` classical '' contribution 5.9 _ _1_2^cl ~(2 ^ 1/2- |1- | ) \{- } ( e^2|r(_1 + _2 ) -1 ) ^-1 and the off - diagonal correction . as mentioned above , in the classical contribution the factor multiplying @xmath110 in the limit @xmath111 turns into the @xmath92-function . at @xmath112 the expression ( [ 5.71 ] ) equals @xmath113 . at @xmath114 ( [ 5.71 ] ) is of order unity . stated differently , at the distance @xmath115 from the extremum , the off - diagonal part is of order @xmath116 of the classical expression at the point of extremum . to make this difference explicit we extract the factor @xmath117 : 5.11 _ _1 _2 = _ _1_2^cl+__1_2 , where 5.10 _ _1_2~__1_2 ( |1-|-2 ^ 1/2 ) . it is of interest to evaluate the contribution of small distances to the smeared density matrix ( cf . it is convenient to use the method 2 . starting from ( [ 5.3 ] ) and making the change of variables @xmath118 , we have 5.12 |__1_2~(_1_2 ) ^-1/2 du_1du_2 e^-4i_1 u_1 r_1/|r + 4i_2 u_2 r_2/|r ^-2(-i ) + n^4e^-(r_1 -|r)^2/2 ^ 2 - ( r_2 -|r ) ^2/2 ^ 2 r_1 ^ 2 r_2 ^ 2 d r_1 dr_2 , where we omitted the irrelevant for the estimate terms . introducing @xmath119 , we integrate over @xmath120 in the interval @xmath121 and over @xmath122 in the interval @xmath123 : 5.13 |__1_2~(_1 _2 ) ^-1/2 _ -^dz e^-iz ( r_1 _1 + r_2 _2 ) /|r |r ( r_1_1 -r_2 _2 ) + n^4e^-(r_1 -|r)^2/2 ^ 2 - ( r_2 -|r ) ^2/2 ^ 2 r_1 ^ 2 r_2 ^ 2 d r_1 dr_2 integrating over @xmath93 , we obtain the expression structurally similar to ( [ m4 ] ) and ( [ 5.7 ] ) . because this expression has sharp extremum at @xmath124 and @xmath10 , for our estimates we can set in the integrand @xmath95 and @xmath125 equal to the extremal values . the resulting density matrix is 5.14 i~(_1_2 ) ^-1/2_-^dz e^-iz ( -|1 -| ) . the integral in ( [ 5.14 ] ) was estimated in @xcite for @xmath126 and it was shown that that the ratio of ( [ 5.14 ] ) to the hawking spectrum is 5.15 ~/|r . taking @xmath127 and assuming for an estimate that the mass of the black hole is of order of several solar masses , we obtain that @xmath128 . in this paper we discussed modifications of the density matrix of radiation of the black hole formed by the collapsing shell resulting from horizon fluctuations of black hole . horizon fluctuations are inherent to the black hole considered as a quantum object . in distinction with the original hawking calculation based on the rigid horizon , horizon fluctuations provide the off - diagonal matrix elements of the density matrix . qualitatively , the off - diagonal matrix elements account for correlations between the particles in radiation and for information stored in these correlations . the construction of the density matrix discussed in the present note is parallel to that of papers @xcite , where the density matrix with the off - diagonal corrections was obtained in the form @xmath129 , where @xmath130 is the original hawking matrix and the off - diagonal correction is of order @xmath131 , where @xmath132 , where @xmath60 is the mass of the shell . the fact that the expansion parameter in both approaches is the same is rather obvious because @xmath133 is the only dimensionless parameter connecting the horizon radius and the scale of fluctuations . because the structural form ( but not the explicit form ) of the smeared density matrix obtained in the present paper is similar to that in papers @xcite , we arrive at the same qualitative conclusions concerning the information problem as in these papers . it is possible to construct the @xmath61-particle density matrix @xmath134 having dimensionality @xmath135 and to calculate the entropy of radiation @xmath136 . calculating the information contained in radiation , which is defined as the difference between the thermal bekenstein - hawking entropy @xmath137 of radiation , @xmath138 , one obtains the qualitatively correct page purification curve @xcite . however , the above results pose a question . in @xcite it was shown that the schwarzschield metric admits construction of `` nice slices''.the nice slices are at @xmath139 inside the horizon , and one can take @xmath140 . for @xmath141 horizon fluctuations which are on the scale @xmath4 are insignificant for particle production on the nice slices . if , however , the horizon fluctuations are somehow connected with hair ( in spirit of @xcite and refs . therein ) , then the niceness is broken and horizon fluctuations can be connected with the release of information from the black hole . the expressions for the density matrix discussed in the present paper refer to ethernal black holes . because of the outgoing flux of particles , the mass of collapsing shell is not constant , but decreases with time @xmath142 here @xmath143 is the @xmath144 component of the radiation stress tensor . in papers @xcite it was found that for the mass of black hole @xmath145 , where @xmath146 is the planck mass , the backreaction of black hole radiation does not prevent formation of the event horizon . when the outgoing flux is small and slowly varying , the calculation is self - consistent . the metric of the exterior geometry of the shell at large distances @xmath147 becomes @xmath148 where @xmath149 , and @xmath150 . for the case considered in sect.2 at the leading order @xmath151 where @xmath60 is the mass of the shell . substituting @xmath152 , we have @xmath153 in the near - horizon region @xmath154 , and we obtain @xmath155 . this shows that our semiclassical treatment is valid . hawking , _ breakdown of predictability in gravitational collapse _ , phys . rev . * d14 * ( 1976 ) 2460 . s.b . giddings , _ comments on information loss and remnants _ , phys . rev . * d49 * ( 1994 ) 4078 [ hep - th/9310101 ] j. preskill , _ do black holes destroy information ? _ [ hep - th/9209058 ] s.b . giddings , _ nonviolent nonlocality _ , phys . rev . * d88 * ( 2013 ) 064023 [ hep - th 1211.7070 ] s.d . mathur , _ the information paradox . a pedagogical introduction _ * 26 * ( 2009 ) 224001 [ gr - qc/0909.1038 ] s.d . mathur , _ what exactly is the information paradox ? notes phys . * 769 * ( 2009 ) 3 [ hep - th/0803.2030 ] s. coleman , j. preskill and f. wilczek , _ quantum hair on black holes _ , * b378 * ( 1992 ) 175 [ hep - th/9201059 ] s. w. hawking , m. j. perry and a. strominger , _ soft hair on black holes _ , [ hep - th/1601.00921 ] . g. compere and j. long , _ classical static final state of collapse with supertranslation memory _ , [ gr - qc/1602.05197 ] ford , n.f . svaiter , _ cosmological and black hole horizon fluctuations _ , phys.rev . * d56 * ( 1997 ) 2226 [ gr - qc/9704050 ] r. brustein , _ origin of the blackhole information paradox _ , [ hep - th/1209.2686 ] r. brustein and a.j.m . medved , _ restoring predictability in semiclassical gravitational collapse _ , jhep * 09 * ( 2013 ) 015 [ hep - th/1305.3139 ] r. brustein and a.j.m . medved , _ phases of information release during black hole evaporation _ , jhep * 02 * ( 2014 ) 116 [ hep - th/1310.5861 ] s. carlip , c. teitelboim , _ the off - shell black hole _ , * 12 * ( 1995 ) 1699 [ gr - qc/9312002 ] r. brustein and m. hadad , _ wave function of the quantum black hole _ , phys . lett . * b718 * ( 2012 ) 653 [ hep - th/1202.5273 ] . medved , _ on the `` universal '' quantum area spectrum _ , * a24 * ( 2009 ) 2601 [ hep - th/0906.2641 ] r. brout , s. massar , r. parentani , p. spindel , _ a primer for black hole quantum physics _ , * 260 * ( 1995 ) 329 [ gr - qc/0710.4345 ] a. paranjape , t. padmanabhan , _ radiation from collapsing shells , semiclassical backreaction and black hole formation _ phys.rev . * d80 * ( 2009 ) 044011 [ gr - qc/0906.1768 ] robert m. wald , _ black hole entropy is noether charge _ , phys . * d48 * ( 1993 ) r3427 [ gr - qc/9307038 ] i. agullo , j. navarro - salas , g.j . olmo and l. parker , _ short - distance contribution to the spectrum of hawking radiation _ , phys . rev . * d76 * ( 2007 ) 044018 [ hep - th/0611355 ] d.n . page , _ black hole information _ [ hep - th/9305040 ] , _ information in black hole radiation _ , phys.rev.lett . 71 ( 1993 ) 3743 [ hep - th/9306083 ] d. n. page , _ time dependence of hawking radiation entropy _ , [ hep - th/1301.4995 ] d. harlow , _ jerusalem lectures on black holes and quantum information _ , rev . * 88 * ( 2016 ) 15002 [ hep - th/1409.1231 ]

the density matrix of hawking radiation is calculated in the model of black hole with fluctuating horizon . quantum fluctuations smear the classical horizon of black hole and modify the density matrix of radiation producing the off - diagonal elements . the off - diagonal elements may store information of correlations between radiation and black hole . the smeared density matrix was constructed by convolution of the density matrix calculated with the instantaneous horizon with the gaussian distribution over the instantaneous horizons . the distribution has the extremum at the classical radius of the black hole and the width of order of the planck length . calculations were performed in the model of black hole formed by the thin collapsing shell which follows a trajectory which is a solution of the matching equations connecting the interior and exterior geometries . * density matrix of radiation of black hole with fluctuating horizon * * mikhail z. iofa * skobeltsyn institute of nuclear physics moscow state university moscow 119991 , russia

1603.07480

a choice of nonconstant meromorphic function @xmath28 on a compact riemann surface @xmath0 realizes @xmath0 as a finite sheeted branched covering of the riemann sphere @xmath29 . _ log - riemann surfaces of finite type _ are certain branched coverings , in a generalized sense , of @xmath30 by a punctured compact riemann surface , namely , which are given by certain transcendental functions of infinite degree . formally a log - riemann surface consists of a riemann surface together with a local holomorphic diffeomorphism @xmath31 from the surface to @xmath30 such that the set of points @xmath19 added to the surface , in the completion @xmath18 with respect to the path - metric induced by the flat metric @xmath32 , is discrete . log - riemann surfaces were defined and studied in @xcite ( see also @xcite ) , where it was shown that the map @xmath31 restricted to any small enough punctured metric neighbourhood of a point @xmath33 in @xmath19 gives a covering of a punctured disc in @xmath30 , and is thus equivalent to either @xmath34 restricted to a punctured disc @xmath35 ( in which case we say @xmath33 is a ramification point of order @xmath1 ) or to @xmath36 restricted to a half - plane @xmath37 ( in which case we say @xmath33 is a ramification point of infinite order ) . a log - riemann surface is said to be of finite type if it has finitely many ramification points and finitely generated fundamental group . we will only consider those for which the set of infinite order ramification points is nonempty ( otherwise the map @xmath31 has finite degree and is given by a meromorphic function on a compact riemann surface ) . in @xcite , @xcite , uniformization theorems were proved for log - riemann surfaces of finite type , which imply that a log - riemann surface of finite type is given by a pair @xmath38 , where @xmath0 is a compact riemann surface , and @xmath31 is a meromorphic function on the punctured surface @xmath16 such that the differential @xmath39 has essential singularities at the punctures of a specific type , namely _ exponential singularities_. given a germ of meromorphic function @xmath40 at a point @xmath41 of a riemann surface , a function @xmath42 with an isolated singularity at @xmath41 is said to have an exponential singularity of type @xmath40 at @xmath41 if locally @xmath43 for some germ of meromorphic function @xmath13 at @xmath41 , while a 1-form @xmath44 is said to have an exponential singularity of type @xmath40 at @xmath41 if locally @xmath45 for some germ of meromorphic 1-form @xmath46 at @xmath41 . note that the spaces of germs of functions and 1-forms with exponential singularity of type @xmath40 at @xmath41 only depend on the equivalence class @xmath47 $ ] in the space @xmath48 of germs of meromorphic functions at @xmath41 modulo germs of holomorphic functions at @xmath41 . thus the uniformization theorems of @xcite , @xcite give us @xmath1 germs of meromorphic functions @xmath49 at the punctures @xmath50 , with poles of orders @xmath3 say , such that near a puncture @xmath51 the map @xmath31 is of the form @xmath52 , where @xmath53 is a germ of meromorphic function near @xmath51 and @xmath28 a local coordinate near @xmath51 . the punctures correspond to ends of the log - riemann surface , where at each puncture @xmath51 , @xmath54 infinite order ramification points are added in the metric completion , so that the total number of infinite order ramification points is @xmath55 . the @xmath54 infinite order ramification points added at a puncture @xmath51 correspond to the @xmath54 directions of approach to the puncture along which @xmath56 so that @xmath57 decays exponentially and @xmath58 converges . in the case of genus zero with one puncture for example , which is considered in @xcite , @xmath31 must have the form @xmath59 where @xmath60 is a rational function and @xmath61 is a polynomial of degree equal to the number of infinite order ramification points . in @xcite , certain spaces of functions and @xmath6-forms on a log - riemann surface @xmath62 of finite type were defined , giving rise to a derham cohomology group @xmath63 . the integrals of the @xmath6-forms considered along curves in @xmath62 joining the infinite ramification points converge , giving rise to a pairing between @xmath63 and @xmath21 , which was shown to be nondegenerate ( @xcite ) . the spaces of functions and @xmath6-forms defined were observed to depend only on the types @xmath49 of the exponential singularities , and so a notion less rigid than that of a log - riemann surface was defined , namely the notion of an _ exp - algebraic curve _ , which consists of a compact riemann surface @xmath0 together with @xmath1 equivalence classes of germs of meromorphic functions modulo germs of holomorphic functions , @xmath2 , \cdots , [ h_n ] \}$ ] , with poles of orders @xmath3 at points @xmath4 . the relevant spaces of functions and @xmath6-forms with exponential singularities at @xmath4 of types @xmath9 , \cdots , [ h_n]$ ] can then be defined as follows : @xmath64 , \cdots , [ h_n ] \ } \\ { { \mathcal o}}_{{{\mathcal h } } } : = & \ { f \in { { \mathcal m}}_{{{\mathcal h } } } \,\mid\ , f \hbox { holomorphic on } s ' \ } \\ \omega_{{{\mathcal h } } } : = & \ { \omega \,\mid\ , \omega \hbox { meromorphic 1-form on } s ' \hbox { with exponential singularities}\\ & \hbox { of types } [ h_1 ] , \cdots , [ h_n ] \ } \\ \omega^0_{{{\mathcal h } } } : = & \ { \omega \in \omega_{{{\mathcal h } } } \,\mid\ , \omega \hbox { holomorphic on } s ' \}.\end{aligned}\ ] ] for @xmath65 ( respectively , @xmath66 ) we can define a divisor @xmath67 ( respectively , @xmath68 ) by @xmath69 if @xmath70 and @xmath71 if @xmath72 , where @xmath13 is a germ of meromorphic function at @xmath10 such that @xmath12 ( respectively , @xmath73 if @xmath70 and @xmath74 if @xmath72 , where @xmath46 is a germ of meromorphic @xmath6-form at @xmath10 such that @xmath15 ) . in @xcite it is shown how to naturally associate to an exp - algebraic curve @xmath75 a degree zero line bundle @xmath76 together with a meromorphic connection @xmath77 with poles at @xmath4 . the connection @xmath6-form of @xmath78 near @xmath10 is given ( with respect to an appropriate local trivialization ) by @xmath79 , so that the pair @xmath80 determines the exp - algebraic curve @xmath81 . there are naturally defined isomorphisms between the space of meromorphic sections of @xmath23 ( respectively , the space of meromorphic @xmath23-valued @xmath6-forms ) and @xmath82 ( respectively , @xmath83 ) , such that a meromorphic section @xmath84 of @xmath23 ( respectively , a meromorphic @xmath23-valued @xmath6-form @xmath46 ) maps to an @xmath65 with the same divisor as @xmath84 ( respectively , an @xmath66 with the same divisor as @xmath46 ) . in particular the space @xmath24 of meromorphic @xmath23-valued @xmath6-forms which are holomorphic on @xmath16 is naturally isomorphic to the space @xmath7 . fixing an @xmath11 inducing a log - riemann surface structure on @xmath0 , with completion @xmath18 , the @xmath6-forms in @xmath7 can be integrated along curves in @xmath21 , giving a map @xmath85 let @xmath25 denote the image of @xmath21 in @xmath86 . then our torelli - type theorem for exp - algebraic curves states that the pair @xmath87 determines the exp - algebraic curve @xmath88 : [ mainthm ] let @xmath89 be two exp - algebraic curves with the same underlying riemann surface @xmath0 , and the same set of punctures @xmath4 . suppose that @xmath90 is nontrivial , that the line bundles @xmath91 are isomorphic and that the induced isomorphism @xmath92 maps @xmath93 to @xmath94 . then @xmath95 . finally , we mention briefly some appearances of functions with exponential singularities in the literature . certain functions with exponential singularities , namely the @xmath1-point _ baker - akhiezer functions _ ( @xcite , @xcite ) , have been used in the algebro - geometric integration of integrable systems ( see , for example , @xcite , @xcite and the surveys @xcite , @xcite , @xcite , @xcite ) . given a divisor @xmath96 on @xmath16 , an @xmath1-point baker - akhiezer function ( with respect to the data @xmath97 ) is a function @xmath42 in the space @xmath98 satisfying the additional properties that the divisor @xmath99 of zeroes and poles of @xmath42 on @xmath16 satisfies @xmath100 , and that @xmath101 is holomorphic at @xmath51 for all @xmath102 . for @xmath96 a non - special divisor of degree at least @xmath13 , the space of such baker - akhiezer functions is known to have dimension @xmath103 . functions and differentials with exponential singularities on compact riemann surfaces have also been studied by cutillas ripoll ( @xcite , @xcite , @xcite ) , where they arise naturally in the solution of the _ weierstrass problem _ of realizing arbitrary divisors on compact riemann surfaces , and by taniguchi ( @xcite , @xcite ) , where entire functions satisfying certain topological conditions ( called structural finiteness " ) are shown to be precisely those entire functions whose derivatives have an exponential singularity at @xmath104 , namely functions of the form @xmath105 , where @xmath106 are polynomials . * acknowledgements . * this work grew out of a visit of the second author to tifr , mumbai . he would like to thank tifr for its hospitality . we recall some basic definitions and facts from @xcite , @xcite , @xcite . a log - riemann surface is a pair @xmath107 , where @xmath0 is a riemann surface and @xmath108 is a local holomorphic diffeomorphism such that the set of points @xmath19 added to @xmath0 in the completion @xmath109 with respect to the path metric induced by the flat metric @xmath32 is discrete . the map @xmath31 extends to the metric completion @xmath62 as a @xmath6-lipschitz map . in @xcite it is shown that the map @xmath31 restricted to a sufficiently small punctured metric neighbourhood @xmath110 of a ramification point is a covering of a punctured disc @xmath111 in @xmath30 , and so has a well - defined degree @xmath112 , called the order of the ramification point ( we assume that the order is always at least @xmath113 , since order one points can always be added to @xmath0 and @xmath31 extended to these points in order to obtain a log - riemann surface ) . a log - riemann surface is of finite type if it has finitely many ramification points and finitely generated fundamental group . for example , the log - riemann surface given by @xmath114 is of finite type ( with the metric @xmath32 it is isometric to the riemann surface of the logarithm , which is simply connected , with a single ramification point of infinite order ) , as is the log - riemann surface given by the gaussian integral @xmath115 , which has two ramification points , both of infinite order , as in the figure below : log - riemann surface of the gaussian integral in @xcite , it is shown that a log - riemann surface of finite type ( which has at least one infinite order ramification point ) is of the form @xmath116 , where @xmath16 is a punctured compact riemann surface @xmath8 and @xmath31 is meromorphic on @xmath16 and @xmath39 has exponential singularities at the punctures @xmath4 . let @xmath49 be the types of the exponential singularities of @xmath39 at the punctures @xmath4 . as described in @xcite , each puncture @xmath51 corresponds to an end of the log - riemann surface where @xmath54 infinite order ramification points are added , @xmath54 being the order of the pole of @xmath117 at @xmath51 . let @xmath33 be an infinite order ramification point associated to a puncture @xmath51 . an @xmath118-ball @xmath119 around @xmath33 is isometric to the @xmath118-ball around the infinite order ramification point of the riemann surface of the logarithm ( given by cutting and pasting infinitely many discs together ) , and there is an argument function @xmath120 defined on the punctured ball @xmath121 . while the function @xmath31 , which is of the form @xmath122 in a punctured neighbourhood of @xmath51 for some meromorphic @xmath6-form @xmath123 , extends continuously to @xmath33 for the metric topology on @xmath62 , in general functions of the form @xmath124 ( where @xmath46 is a @xmath6-form meromorphic near @xmath51 ) do * not * extend continuously to @xmath33 for the metric topology ( @xcite ) . limits of these functions in sectors @xmath125 do exist however and are independent of the sector ; we say that the function is _ stolz continuous _ at points of @xmath19 . define spaces of functions and @xmath6-forms on @xmath62 : @xmath126 we remark that these are simply the spaces @xmath127 defined in the introduction , where @xmath2 , \cdots , [ h_n ] \}$ ] . functions in @xmath98 are stolz continuous at points of @xmath19 taking the value @xmath128 there . the integrals of @xmath6-forms @xmath44 in @xmath129 over curves @xmath130 \to s^*$ ] joining points @xmath131 of @xmath19 converge if @xmath132 is disjoint from the poles of @xmath44 and tends to these points through sectors @xmath133 ( since any primitive of @xmath44 on a sector is stolz continuous ) . the definitions of the above spaces only depend on the types @xmath134 \in { { \mathcal m}}_{p_i}/{{\mathcal o}}_{p_i } \}$ ] of the exponential singularities of the @xmath6-form @xmath39 , which do not change if @xmath39 is multiplied by a meromorphic function . it is natural to define then a structure less rigid than that of a log - riemann surface of finite type . given a punctured compact riemann surface @xmath8 , two meromorphic functions @xmath135 on @xmath16 inducing log - riemann surface structures of finite type are considered equivalent if @xmath136 is meromorphic on the compact surface @xmath0 . an exp - algebraic curve is an equivalence class of such log - riemann surface structures of finite type . it follows from the uniformization theorem of @xcite that an exp - algebraic curve is given by the data of a punctured compact riemann surface and @xmath1 ( equivalence classes of ) germs of meromorphic functions @xmath137 \in { { \mathcal m}}_{p_i}/{{\mathcal o}}_{p_i } \}$ ] with poles at the punctures . we can associate a topological space @xmath138 to an exp - algebraic curve , given as a set by @xmath139 , where @xmath19 is the set of infinite ramification points added with respect to any map @xmath31 in the equivalence class of log - riemann surfaces of finite type , and the topology is the weakest topology such that all maps @xmath140 in the equivalence class extend continuously to @xmath138 . finally , for a meromorphic function @xmath42 on @xmath16 ( respectively , meromorphic @xmath6-form @xmath44 on @xmath16 ) with exponential singularities of types @xmath49 at points @xmath4 we can define a divisor @xmath67 ( respectively , @xmath68 ) by @xmath69 if @xmath70 and @xmath71 if @xmath72 , where @xmath13 is a germ of meromorphic function at @xmath10 such that @xmath12 ( respectively , @xmath73 if @xmath70 and @xmath74 if @xmath72 , where @xmath46 is a germ of meromorphic @xmath6-form at @xmath10 such that @xmath15 ) . note that the divisor @xmath99 can also be defined by @xmath141 , so it follows from the residue theorem applied to the meromorphic @xmath6-form @xmath142 that the divisor @xmath99 has degree zero . let @xmath143 be an exp - algebraic curve , where @xmath0 is a compact riemann surface of genus @xmath13 and @xmath49 are germs of meromorphic functions at points @xmath4 . let @xmath144 be the space of holomorphic @xmath6-forms on @xmath0 . the data @xmath145 defines a degree zero line bundle @xmath76 together with a transcendental section @xmath146 of this line bundle which is non - zero on the punctured surface @xmath16 as follows : solving the mittag - leffler problem locally for the distribution @xmath147 gives meromorphic functions on an open cover such that the differences are holomorphic on intersections , and hence gives an element of @xmath148 . under the exponential this gives a degree zero line bundle as an element of @xmath149 . explicitly this is constructed as follows : let @xmath150 be pairwise disjoint coordinate disks around the punctures @xmath4 and let @xmath151 be an open subset of @xmath16 intersecting each disk @xmath152 in an annulus @xmath153 around @xmath10 such that @xmath154 is an open cover of @xmath0 . define a line bundle @xmath76 by taking the functions @xmath155 to be the transition functions for the line bundle on the intersections @xmath156 . define a holomorphic non - vanishing section of @xmath76 on @xmath16 by : @xmath157 define a connection @xmath77 on @xmath76 by declaring that @xmath158 . then for any holomorphic section @xmath84 on @xmath151 , @xmath159 for some holomorphic function @xmath42 , and @xmath160 , so @xmath77 is holomorphic on @xmath151 . on each disk @xmath152 , letting @xmath161 be the section which is constant equal to @xmath6 on @xmath152 ( with respect to the trivialization on @xmath152 ) , for any holomorphic section @xmath84 on @xmath152 , @xmath162 for some holomorphic function @xmath42 , and @xmath163 , so @xmath164 thus the connection @xmath6-form of @xmath77 with respect to @xmath161 is given by @xmath79 , so @xmath77 is meromorphic on @xmath152 with a single pole at @xmath10 of order @xmath165 . let @xmath166 be the unique section of the dual bundle @xmath167 on @xmath16 such that @xmath168 on @xmath16 . then for any non - zero meromorphic section @xmath84 of @xmath23 , the function @xmath169 is meromorphic on @xmath16 with exponential singularities at @xmath4 of types @xmath49 , and the divisors of @xmath84 and @xmath42 coincide . thus the line bundle @xmath76 has degree zero . in summary we have : [ existence ] for any log - riemann surface of finite type @xmath62 , the line bundle @xmath76 has degree zero and the maps @xmath170 ( respectively , @xmath171 ) are mutually inverse isomorphisms between the spaces of meromorphic sections of @xmath23 and @xmath98 ( respectively , the spaces of meromorphic @xmath23-valued @xmath6-forms and @xmath172 ) preserving divisors . in particular the vector spaces @xmath173 are non - zero . since the isomorphisms above preserve divisors , the spaces @xmath174 correspond to the spaces of meromorphic sections of @xmath23 and meromorphic @xmath23-valued @xmath6-forms which are holomorphic on @xmath16 , both of which are non - empty . the correspondence @xmath175 gives a one - to - one correspondence between exp - algebraic structures on @xmath0 and degree zero line bundles on @xmath0 with meromorphic connections with all poles of order at least two , zero residues , and trivial monodromy . since the connection @xmath6-form of @xmath78 is given by @xmath79 on @xmath152 , all residues of @xmath78 are equal to zero , while the monodromy of @xmath78 is trivial since @xmath176 is a single - valued horizontal section . conversely , given such a meromorphic connection @xmath177 on a degree zero line bundle @xmath178 , if @xmath4 are the poles of @xmath177 and @xmath179 are the connection @xmath6-forms of @xmath177 with respect to trivializations near @xmath4 , then each @xmath180 has zero residue at @xmath10 and pole order at least two , hence there exist meromorphic germs @xmath49 near @xmath4 such that @xmath181 . we obtain an exp - algebraic curve @xmath182 . it is clear for an exp - algebraic curve @xmath26 that @xmath183 , so the correspondences are inverses of each other . finally we remark that by serre duality , the degree zero line bundle @xmath76 , given as an element of @xmath148 , can also be described as an element of @xmath184 using residues , as the linear functional @xmath185 we proceed to the proof of theorem [ mainthm ] . we will need the following theorems from @xcite and @xcite : [ approxn ] let @xmath0 be a compact riemann surface and @xmath188 a closed subset such that @xmath189 has finitely many connected components @xmath190 , and for each @xmath191 let @xmath192 be a point of @xmath193 . then any continuous function @xmath42 on @xmath194 which is holomorphic in the interior of @xmath194 can be uniformly approximated on @xmath194 by functions meromorphic on @xmath0 with poles only in the set @xmath195 . let @xmath196 be two exp - algebraic curves with the same underlying riemann surface @xmath0 and the same set of punctures @xmath4 , and suppose the hypothesis of theorem 1.1 holds , namely the line bundles @xmath91 are isomorphic and the induced isomorphism @xmath197 maps @xmath93 to @xmath94 . since the spaces @xmath198 are isomorphic to the spaces of meromorphic sections of @xmath199 and @xmath200 respectively , there is an induced isomorphism @xmath201 which preserves divisors . we fix non - zero functions @xmath202 which correspond to each other under this isomorphism , and let @xmath203 denote the completions induced by the corresponding log - riemann surface structures . we also fix a meromorphic @xmath6-form @xmath204 on @xmath0 . then the divisor preserving isomorphisms @xmath201 and @xmath205 induced by the isomorphism @xmath206 can be expressed as @xmath207 respectively , where @xmath13 varies over all meromorphic functions on @xmath0 . the hypothesis of theorem [ mainthm ] implies that for any @xmath212 , there is a @xmath213 such that @xmath214 for all meromorphic functions @xmath13 on @xmath0 such that @xmath215 is holomorphic on @xmath16 for @xmath216 . if @xmath208 is a meromorphic function on @xmath0 such that @xmath217 is holomorphic on @xmath16 , then @xmath218 for some meromorphic function @xmath13 on @xmath0 such that @xmath219 is holomorphic on @xmath16 . since the isomorphism @xmath220 is divisor preserving , we have that @xmath221 is also holomorphic on @xmath16 , so for any @xmath212 there is a @xmath222 such that @xmath223 since this is true for all @xmath212 , it follows from theorem [ pairing ] that @xmath224 , so there exists a meromorphic function @xmath225 on @xmath0 such that @xmath226 is holomorphic on @xmath16 and @xmath227 . in this case there exists a closed curve @xmath132 disjoint from the punctures @xmath4 and the poles and zeroes of @xmath235 such that @xmath236 is connected . fix a non - zero meromorphic function @xmath237 on @xmath0 such that @xmath238 ( and hence also @xmath239 ) is holomorphic on @xmath16 . if the meromorphic @xmath6-form @xmath240 ( which is holomorphic outside the punctures and the zeroes and poles of @xmath235 ) is not identically zero , then we can choose a continuous function @xmath241 on @xmath132 such that @xmath242 ( since the @xmath6-form @xmath243 is holomorphic and not identically zero on @xmath132 ) . by theorem [ approxn ] , since @xmath236 is connected and contains @xmath244 , we can choose a meromorphic function @xmath245 on @xmath0 which is holomorphic on @xmath246 and uniformly close enough to @xmath241 on @xmath132 such that @xmath247 . letting @xmath248 , we have that @xmath217 is holomorphic on @xmath16 and @xmath249 , contradicting lemma [ exact ] . it follows that @xmath250 , from which it follows easily that @xmath95 . in this case @xmath251 and we may assume @xmath252 . fix a non - zero polynomial @xmath61 such that @xmath253 are holomorphic on @xmath16 . then by lemma [ exact ] , for all @xmath254 , taking @xmath255 we have @xmath256 from which it follows that the laurent series of @xmath257 around @xmath258 vanishes identically , hence @xmath259 and @xmath95 . in this case @xmath251 and we may assume the single puncture @xmath260 , and that the functions @xmath235 are of the form @xmath261 for some polynomials @xmath262 . in this case it follows from the main theorem of @xcite that the dimension of @xmath263 equals @xmath264 . since @xmath93 and @xmath94 are isomorphic by hypothesis , it follows that @xmath265 say , where @xmath266 since @xmath267 is non - trivial . let @xmath268 . let @xmath269 be a basis for @xmath267 as described in section 4 of @xcite , each @xmath270 being a curve joining a pair of ramification points @xmath271 , where @xmath272 . by hypothesis , for each curve @xmath273 there is a @xmath274 such that @xmath275 for all polynomials @xmath276 . consider the @xmath277 matrix @xmath278 it follows from theorem iii.1.5.1 of @xcite that the @xmath279 @xmath6-forms @xmath280 span @xmath281 , and hence form a basis for @xmath281 . since by theorem [ pairing ] the pairing with @xmath267 is nondegenerate , it follows that the @xmath282 submatrix formed by the first @xmath279 columns of @xmath283 is nonsingular , thus @xmath283 has rank @xmath279 . on the other hand , since @xmath284 , it follows that @xmath285 hence there is a scalar @xmath286 such that @xmath287 , so @xmath288 . it follows from lemma [ exact ] that for any polynomial @xmath276 the @xmath6-form @xmath289 lies in @xmath211 . thus if @xmath290 , then @xmath291 , hence @xmath292 for all polynomials @xmath276 . since all @xmath6-forms in @xmath293 are of the form @xmath294 for some polynomial @xmath61 and any @xmath295 for some polynomial @xmath276 , it follows that @xmath296 is trivial , a contradiction . thus @xmath297 , so @xmath298 and hence @xmath95 .

an exp - algebraic curve consists of a compact riemann surface @xmath0 together with @xmath1 equivalence classes of germs of meromorphic functions modulo germs of holomorphic functions , @xmath2 , \cdots , [ h_n ] \}$ ] , with poles of orders @xmath3 at points @xmath4 . this data determines a space of functions @xmath5 ( respectively , a space of @xmath6-forms @xmath7 ) holomorphic on the punctured surface @xmath8 with exponential singularities at the points @xmath4 of types @xmath9 , \cdots , [ h_n]$ ] , i.e. , near @xmath10 any @xmath11 is of the form @xmath12 for some germ of meromorphic function @xmath13 ( respectively , any @xmath14 is of the form @xmath15 for some germ of meromorphic @xmath6-form ) . for any @xmath14 the completion of @xmath16 with respect to the flat metric @xmath17 gives a space @xmath18 obtained by adding a finite set @xmath19 of @xmath20 points , and it is known that integration along curves produces a nondegenerate pairing of the relative homology @xmath21 with the derham cohomology group defined by @xmath22 . there is a degree zero line bundle @xmath23 associated to an exp - algebraic curve , with a natural isomorphism between @xmath7 and the space @xmath24 of meromorphic @xmath23-valued @xmath6-forms which are holomorphic on @xmath16 , so that @xmath21 maps to a subspace @xmath25 . we show that the exp - algebraic curve @xmath26 is determined uniquely by the pair @xmath27 . [ section ] [ theorem]proposition [ theorem]lemma [ theorem]corollary [ theorem]definition

1606.06449

in 1988 , a. barkai and c.d mcquaid reported a novel observation in population ecology while studying benthic fauna in south african shores @xcite : a predator - prey role reversal between a decapod crustacean and a marine snail . specifically , in malgas island , the rock lobster _ jasus lalandii _ preys on a type of whelk , _ burnupena papyracea_. as could be easily expected , the population density of whelks soared upon extinction of the lobsters in a nearby island ( marcus island , just four kilometers away from malgas ) . however , in a series of very interesting controlled ecological experiments , barkai and mcquaid reintroduced a number of _ jasus lalandii _ in marcus island , to investigate whether the equilibrium observed in the neighboring malgas island could be restored . the results were simply astounding : _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ `` the result was immediate . the apparently healthy rock lobsters were quickly overwhelmed by large number of whelks . several hundreds were observed being attacked immediately after release and a week later no live rock lobsters could be found at marcus island . '' _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ surprisingly , and despite observations such as the report in @xcite , theoretical population biology has largely ignored the possibility of predators and preys switching their roles . of importance , the paper of barkai and mcquaid suggests the existence of a threshold control parameter responsible for switching the dynamics between ( a ) a classical predator - prey system with sustained or decaying oscillations , and ( b ) a predator ( the former prey ) driving its present - day prey to local extinction . it is worth noting there are some papers in the literature describing ratio - dependent predation ( see , for example @xcite and @xcite ) , but they are not related to the possibility of role - reversals . on the other hand , the likelihood of changing ecological roles as a result of density dependence has already been documented for the case of mutualism by breton @xcite and , in 1998 , hernndez made an interesting effort to build a mathematical scheme capable of taking into account the possible switches among different possible ecological interactions @xcite . so , to the best of our knowledge , there are no theoretical studies supported by field evidence specifically addressing predator - prey role - reversals yet . predator - prey systems are generally modeled by adopting one of the many variations of the classical lotka - volterra model : @xmath0 where @xmath1 denotes the intrinsic preys rate of growth , @xmath2 corresponds to the rate of predation upon preys , @xmath3 stands for the predators death rate in absence of preys , and @xmath4 represents the benefit of predators due to the encounters with preys . our goal is to assess whether modeling the role - reversal behavior observed by barkai & mcquaid @xcite is possible , when adopting appropriate parameters and assumptions . for instance , if one considers quadratic density dependence in the preys as well as in the predators , non - constant rates of consumption of preys by the predators , and the profiting of predators by the existence of preys , then it is possible to suggest the following system : @xmath5 where @xmath6 represents the intrinsic growth rate of the prey in the absence of predators , @xmath7 the carrying capacity of the prey s habitat , @xmath8 the rate of preys consumption by the population of predators , @xmath9 the predators decay rate in the absence of preys , @xmath10 the intraspecific rate of competition among predators and , finally , @xmath11 the factor of predator s profiting from preys . the ratio @xmath12 is then the fraction of prey biomass that is actually converted into predator biomass . the latter should remain constant , since the fraction of preys biomass converted to predators biomass is a physiological parameter , rather than a magnitude depending on demographical variables . thus , a particular case of system ( [ e1 ] ) in the appropriate rescaled variables is : @xmath13 where all the parameters are positive and @xmath14 . in fact , all of the parameters have a relevant ecological interpretation : @xmath15 is the normalized intrinsic growth rate of the species with density @xmath16 , @xmath17 is a measure of the damage intensity of the second species on the first one , @xmath18 is the normalized rate of predators decay and @xmath19 is the benefit ( damage ) the second population gets from the first one . note the crucial role played by the interaction term @xmath20 , where @xmath21 stands for the first population threshold to switch from being prey to predator . the horizontal nullcline of the system of equations ( [ e2 ] ) , that is @xmath22x=0 $ ] , has two branches : the vertical axis and the nontrivial branch @xmath23 which is a symmetric hyperbola with asymptotes : @xmath24 and @xmath25 ( see figure 1 ) . [ fig:1 ] ) . the dotted line and the vertical axis are the horizontal nullcline . the continuous downward facing parabola and the horizontal axis are the vertical nullcline . @xmath26 , @xmath27 and @xmath28 are the non - trivial equilibria . the origin of coordinates is a trivial equilibrium.,title="fig:",width=453,height=359 ] the vertical nullcline , @xmath29y=0 $ ] , also has two branches : the horizontal axis and @xmath30 which is a parabola with @xmath31 , attaining its maximum at @xmath32 , the value of which is @xmath33 this term is positive if and only if @xmath34 . the zeros , @xmath35 and @xmath36 of equation ( [ e4 ] ) are given by @xmath37 the latter are real numbers if and only if @xmath38 . the rate of change of @xmath39 is then @xmath40 to analyze the system while keeping in mind the ecological interpretation of the variables and parameters , we will now consider the left branch of the horizontal nullcline ( [ e3 ] ) , @xmath34 with @xmath14 and the region of the phase plane of the system of equations ( [ e2 ] ) defined as @xmath41 the system of equations ( [ e2 ] ) has the equilibria : @xmath42 , @xmath43 plus those states of the system stemming from the intersection of the nullclines @xmath44 and @xmath39 in the region @xmath45 . such equilibria are defined by the @xmath16 in the interval @xmath46 satisfying the identity @xmath47 or , equivalently , the @xmath16 that are roots of the third order polynomial @xmath48 where @xmath49 , @xmath50 , @xmath51 and @xmath52 . the calculation of the nontrivial equilibria of ( [ e2 ] ) follows from the determination of the roots of ( [ e6 ] ) . consequently , due to the qualitative behavior of the functions @xmath44 and @xmath39 on @xmath45 , we are faced with the following possibilities : 1 . the nontrivial branches of the nullclines do not intersect each other in the region of interest . in such a case , the system ( [ e2 ] ) has just two equilibria : @xmath53 and @xmath26 in @xmath54 . figure 2a shows the relative position of the nullclines in this case , and figure 3a the phase portrait of the system . + for fixed positive values of @xmath55 and @xmath19 , and @xmath56 such that @xmath57 one can see that both nullclines become closer with increasing values of @xmath21 . + , both nullclines touch tangentially . further changes in the parameter lead to a saddle - node bifurcation , and to the two transversal intersections depicted in figure 1.,title="fig:",width=230 ] [ fig:1a ] + , both nullclines touch tangentially . further changes in the parameter lead to a saddle - node bifurcation , and to the two transversal intersections depicted in figure 1.,title="fig:",width=230 ] [ fig:1b ] + + 2 . the nullclines @xmath44 and @xmath39 touch each other tangentially at the point @xmath58 in the region @xmath54 . again , figure 2b shows the relative position of the nullclines in this case , and figure 3b the phase portrait of the system . in such a case @xmath59 , in addition to satisfy ( [ e5 ] ) , must also satisfy the condition @xmath60 _ i.e. _ , + @xmath61 + if one assumes the existence of @xmath59 satisfying ( [ e5 ] ) , the required extra condition ( [ e7 ] ) imposes the restriction @xmath62 on @xmath59 , due to the positiveness of its left hand side . moreover , from a geometrical interpretation of ( [ e7 ] ) it follows that : * \(i ) if + @xmath63 there is not any @xmath64 such that @xmath65 . * \(ii ) if @xmath66 the condition ( [ e7 ] ) is satisfied just at @xmath67 . * \(iii ) if @xmath68 there exists exactly one value , @xmath69 , of @xmath70 such that the equality ( [ e7 ] ) holds . + in any case , the point @xmath71 is a _ non - hyperbolic _ equilibrium of the system of equations ( [ e2 ] ) . in fact , the proof that a tangential contact of the nullclines results in a point where the determinant of the jacobian matrix of the system vanishes follows immediately , implying that at least one of its eigenvalues is zero . the nullclines intersect each other transversally at two points , @xmath27 and @xmath28 , belonging the region @xmath54 . for reference , please refer to figure 1 . in this case the system of equations ( [ e2 ] ) has two extra equilibria which arise from the _ bifurcation _ of @xmath71 . + here , if in addition to choosing the parameters @xmath19 , @xmath21 and @xmath18 such that @xmath72 , we select the rest of them such that : * \(i ) @xmath73 , i.e. , + @xmath74>\frac{b}{c}(2-k),\ ] ] + guaranteeing the existence of the equilibria @xmath75 and @xmath76 above mentioned . moreover , the coordinates of these points satisfy @xmath77 , @xmath78 , @xmath79 with @xmath80 . here * \(ii ) @xmath82 , i.e. , @xmath74=\frac{b}{c}(2-k).\ ] ] + here we have @xmath75 with @xmath77 and @xmath83 . meanwhile , @xmath84 . part of the local analysis of the system of equations ( [ e2 ] ) is based on the linear approximation around its equilibria . thus , we calculate the jacobian matrix of the system ( [ e2 ] ) : @xmath85_{(x , y ) } = \left[\begin{array}{cc } b(1 - 2x)-cy(k-2x ) & -cx(k - x)\\ fy(k-2x ) & -e-2ey+fx(k - x)\end{array } \right].\ ] ] by a straightforward calculation , we obtain the eigenvalues of the jacobian matrix ( [ e9 ] ) at the point @xmath53 . these are : @xmath86 and @xmath87 . hence , @xmath53 is saddle point of the system ( [ e2 ] ) , for all positive parameter values . by carrying out similar calculations we obtain the corresponding eigenvalues of matrix ( [ e9 ] ) at @xmath26 , which are : @xmath88 and @xmath89 . the restriction @xmath14 on @xmath21 implies that @xmath90 . therefore , @xmath26 is an asymptotically stable node for all the positive parameter values appearing in system ( [ e2 ] ) . now we carry out the local analysis of ( [ e2 ] ) . we notice two cases , depending on the relative position of the nullclines : * case 1 . * the main branches ( [ e3 ] ) and ( [ e4 ] ) of the nullclines do not intersect on @xmath54 . here , any trajectory of system ( [ e2 ] ) starting at the initial condition @xmath91 with positive @xmath92 and @xmath93 tends to the equilibria @xmath53 as time goes to infinity . thus , the region @xmath94 is the basin of attraction of @xmath26 . invariably , the species with density @xmath95 vanishes , implying non coexistence among the interacting species . meanwhile , the other species approach the associated carrying capacity . * the nullclines intersect each other at the points @xmath75 and @xmath96 , where none is tangential . here @xmath97 and @xmath98 satisfy @xmath77 and @xmath78 . in a neighborhood of @xmath27 and @xmath28 , the functions @xmath99 and @xmath100 satisfy the implicit function theorem . in particular , each one of the identities @xmath101 and @xmath102 define a function there . actually , these are @xmath103 and @xmath104 given in ( [ e3 ] ) and ( [ e4 ] ) , respectively . their derivative at @xmath105 with @xmath81 is calculated as follows @xmath106 by using these equalities , we can state the following proposition . * proposition 1 . * _ the equilibrium @xmath27 is not a saddle point . meanwhile , the equilibrium @xmath28 is a saddle point for all the parameter values . _ a proof of this proposition and some remarks can be found in appendix a. as we have already shown , system ( [ e2 ] ) has four equilibrium points . these are illustrated in figure 4 . the origin is a saddle point with the horizontal and vertical axis as its unstable and stable manifolds . @xmath26 and @xmath27 are , respectively , a node and a saddle for parameter values after the bifurcation , and @xmath28 is a stable node . the stable manifold of the saddle point is a separatrix dividing the phase space in two disjoint regions : the set of initial conditions going to @xmath26 , and the complement with points going to @xmath28 . moreover , our numerical solution shows the existence of an homoclinic trajectory starting and ending in the saddle point . thus , we have a bistable system . ) . from left to right : @xmath27 is a stable node , @xmath28 is a saddle and @xmath26 is another stable node . the heteroclinic trajectory joining the saddle point to the stable node is easily identified . the stable manifold is a separatrix between the basin of attraction of @xmath26 and @xmath27.,width=453,height=359 ] the bistability of system ( [ e2 ] ) has an interesting ecological interpretation : the coexistence of the interacting species occurs whenever the initial population densities @xmath91 are located in the region above the saddle point unstable manifold . in this case , both populations evolve towards the attractor @xmath27 . on the other hand , if the initial population densities @xmath91 are below the separatrix , the population densities @xmath107 evolve towards the equilibria @xmath26 implying the non - coexistence of the species and , invariably , the species with population density @xmath95 vanishes . the heteroclinic trajectory of system ( [ e2 ] ) connecting the saddle ( @xmath28 ) with the node ( @xmath27 or focus , depending on the set of parameters ) , in addition to the coexistence of the species , also tells us that this occurs by the transition from one equilibrium to another as time increases . to describe more accurately our role - reversal system , we extended our model of system ( [ e2 ] ) to incorporate the spatial variation of the population densities . here , if we denote by @xmath108 and @xmath109 the population density of the whelks and lobsters at the point @xmath110 at time @xmath111 , the resulting model is : @xmath112 where the subscript in @xmath113 and @xmath114 denotes the partial derivative with respect the time , and @xmath115 is the laplacian operator . here , @xmath116 , @xmath117 correspond to the diffusivity of the species with density @xmath113 and @xmath114 , i.e. that of whelks and lobsters , respectively . it is worth noting that the original variables have been rescaled , but still denote population densities . we then proceeded to construct numerical solutions of the system ( 10 ) in three different domains : a circle with radius 2.2 length units ( lu ) , an annulus defined by concentric circles of radii 2.2 lu and 1 lu , and a square with side length of 4.6 lu . all domains were constructed to depict similar distances between malgas island and marcus island ( roughly 4 km ) . in the first one , the annular domain , we try to mimic the island habitat of whelks and lobsters as a concentric domain . the other two domains are used to confirm the pattern formation characteristic of excitable media , and to reject any biases from the shape of the boundaries . to obtain numerical solutions of all spatial cases , we used the finite element method with adaptive time - stepping , and assumed zero - flux boundary conditions . accordingly , we discretized all spatial domains by means of delaunay triangulations , until a maximal side length of 0.17 was obtained . the latter defines the approximation error of the numerical scheme . we attempted to describe two entirely different situations by using a single set of kinetic parameters : that of malgas island , where both species co - exist , and marcus island , where whelks soar and lobsters become extinct . the only difference between these two cases was the initial conditions used . aside , one could intuitively assume whelks motion to be very slow , or even negligible in comparison to that of lobsters . however , it is worth considering how slow , and whether fluid motion could aftect this speed . while there is no data specific to _ jasus lalandii _ and _ burnupena papyracea _ in islands of the saldanha bay , data of similar species can be found in the literature . for instance , a related rock - lobster species , _ jasus edwardii _ has been found to move at a rate of 5 - 7 km / day @xcite . in contrast , whelks within the superfamily _ buccinoidea _ have been found to move towards food at rates between 50 and 220 meters / day ( see @xcite and @xcite ) . importantly , predation by whelks remains seemingly unaffected by variations in water flow @xcite . by putting these findings together , we argue a reasonable model need not incorporate influences from shallow water currents , and would assume whelks to move toward ` bait ' at a speed roughly one order of magnitude smaller than that of lobsters . thus , we opted for a two - dimensional habitat , and one order of magnitude difference between the non - dimensional isotropic diffusion rates ( @xmath118 and @xmath119 ) . aside , our choice of reaction parameters was : @xmath120 , @xmath121 , @xmath122 , @xmath123 , and @xmath124 . regarding initial conditions , we adopted the following scenarios , representing the different scenarios of weighted biomass : 1 . malgas island : the initial density of whelks at each element was drawn from a uniform distribution 0.1 * u(0.25 , 0.05 ) , and that of lobsters from u(0.25 , 0.05 ) . marcus island : the initial density of whelks at each element was drawn from a uniform distribution u(0.25 , 0.05 ) , and that of lobsters from 0.1 * u(0.25 , 0.05 ) . results are shown in figure 5 , corresponding to averaged densities of whelks and lobsters in the three different spatial domains , respectively . simulations in an annular domain can be found in the supplementary material . interestingly , changes in density are usually accompanied with wave - like spatial transitions in each species density . examples of this spatial transient patterns can be found in figures 6 and 7 , for annular and rectangular domains in malgas island and marcus island , respectively . we have modeled a well documented case of role - reversal in a predator - prey interaction . our model pretends to capture the essential ecological factors within the study of barkai and mcquaid @xcite , who did an extraordinary field work and meticulously reported this striking role - reversal phenomenon happening between whelks and lobsters in the saldanha bay . the analysis of our model and corresponding numerical solutions clearly predict the coexistence of both populations and the switching of roles between the once denoted predators and preys . here , the coexistence scenario corresponds to the case when lobsters predate upon whelks , and role - reversal corresponds to the case when whelks drive the population of lobsters to extinction , as observed by barkai and mcquaid in the field . moreover , by introducing spatial variables and letting both populations diffuse within a spatial domain , we obtain patterns that are characteristic of excitable media @xcite . of particular interest is the upper row of figure 6 , where self - sustained waves travel in the annular region . the latter is not entirely surprising , as the ordinary differential equation model in which the spatial case was based shows bistability . nevertheless , our findings are novel in that , to the best of our knowledge , there are no reports of ecological interactions behaving as excitable media . pm was supported by unam - in107414 funding , and wishes to thank oist hospitality during last stages of this work . tml was supported by oist funding . * proof . * first we prove the second part of our proposition . at @xmath28 we have @xmath131 implying that @xmath132 and @xmath133 have the same sign at @xmath134 but @xmath135 with @xmath78 and @xmath136 , then @xmath137 and @xmath138 . by using the above calculations we obtain @xmath139 and @xmath140 then , the determinant of the jacobian matrix of the system ( [ e2 ] ) at @xmath28 for the proof of the first part of the proposition we follow a similar sign analysis as we did previously , by considering that @xmath77 , @xmath142 and that at @xmath97 the inequality @xmath143 holds where both derivatives are positive . thus , given that @xmath144 and @xmath145 the inequalities given that its discriminant[multiblock footnote omitted ] @xmath151 , ( [ e11 ] ) is a quadratic equation of hyperbolic type . in order the see more details of such quadratic , we calculate its gradient . this is the zero vector at the point @xmath159\left[\frac{\partial^2 trj}{\partial y^2}\right]-\left\{\frac{\partial^2 trj}{\partial y\partial x}\right\}^2=-4c^2<0.\ ] ] therefore @xmath155 is a saddle point of the surface ( [ e10 ] ) . the value of @xmath153 $ ] at @xmath155 is s. s. powers , j. n. kittinger , ( 2002 ) , hydrodynamic mediation of predator - prey interactions : differential patterns of prey susceptibility and predator success explained by variation in water flow . journal of experimental marine biology and ecology , 85 : 245 - 257 .

predator - prey relationships are one of the most studied interactions in population ecology . however , little attention has been paid to the possibility of role exchange between species once determined as predators and preys , despite firm field evidence of such phenomena in the nature . in this paper , we build a model capable of reproducing the main phenomenological features of one reported predator - prey role - reversal system , and present results for both the homogeneous and the space explicit cases . we find that , depending on the choice of parameters , our role - reversal dynamical system exhibits excitable - like behaviour , generating waves of species concentrations that propagate through space .

1404.2685

starting from the first telescopic sunspot observations by david and johannes fabricius , galileo galilei , thomas harriot and christoph scheiner , the 400-year sunspot record is one of the longest directly recorded scientific data series , and forms the basis for numerous studies in a wide range of research such as , e.g. , solar and stellar physics , solar - terrestrial relations , geophysics , and climatology . during the 400-year interval , sunspots depict a great deal of variability from the extremely quiet period of the maunder minimum @xcite to the very active modern time @xcite . the sunspot numbers also form a benchmark data series , upon which virtually all modern models of long - term solar dynamo evolution , either theoretical or ( semi)empirical , are based . accordingly it is important to review the reliability of this series , especially since it contains essential uncertainties in the earlier part . the first sunspot number series was introduced by rudolf wolf who observed sunspots from 1848 until 1893 , and constructed the monthly sunspot numbers since 1749 using archival records and proxy data @xcite . sunspot activity is dominated by the 11-year cyclicity , and the cycles are numbered in wolf s series to start with cycle # 1 in 1755 . when constructing his sunspot series wolf interpolated over periods of sparse or missing sunspot observations using geomagnetic proxy data , thus losing the actual detailed temporal evolution of sunspots @xcite . sunspot observations were particularly sparse in the 1790 s , during solar cycle # 4 which became the longest solar cycle in wolf s reconstruction with an abnormally long declining phase ( see fig . [ fig : wsn]a ) . the quality of wolf s sunspot series during that period has been questioned since long . based on independent auroral observations , it was proposed by elias loomis already in 1870 that one small solar cycle may have been completely lost in wolf s sunspot reconstruction in the 1790 s @xcite , being hidden inside the interpolated , exceptionally long declining phase of solar cycle # 4 . this extraordinary idea was not accepted at that time . a century later , possible errors in wolf s compilation for the late 18th century have been emphasized again based on detailed studies of wolf s sunspot series @xcite . recently , a more extensive and consistent sunspot number series ( fig . [ fig : wsn]b ) , the group sunspot numbers ( gsn ) , was introduced by @xcite , which increases temporal resolution and allows to evaluate the statistical uncertainty of sunspot numbers . we note that the gsn series is based on a more extensive database than wolf s series and explicitly includes all the data collected by wolf . however , it still depicts large data gaps in 17921794 ( this interval was interpolated in wolf s series ) . based on a detailed study of the gsn series , @xcite revived loomis idea by showing that the lost cycle ( a new small cycle started in 1793 , which was lost in the conventional wolf sunspot series ) agrees with both the gsn data ( fig . [ fig : wsn]b ) and indirect solar proxies ( aurorae ) and does not contradict with the cosmogenic isotope data . the existence of the lost cycle has been disputed by @xcite based on data of cosmogenic isotope and sunspot numbers . however , as argued by @xcite , the lost cycle hypothesis does not contradict with sunspots or cosmogenic isotopes and is supported by aurorae observations . using time series analysis of sparse sunspot counts or sunspot proxies , it is hardly possible to finally verify the existence of the lost solar cycle . therefore , the presence of the lost cycle has so far remained as an unresolved issue . here we analyze newly restored original solar drawings of the late 18th century to ultimately resolve the old mystery and to finally confirm the existence of the lost cycle . most of wolf s sunspot numbers in 1749 - 1796 were constructed from observations by the german amateur astronomer johann staudacher who not only counted sunspots but also drew solar images in the second half of the 18th century ( see an example in fig . [ fig : staud ] ) . however , only sunspot counts have so far been used in the sunspot series , but the spatial distribution of spots in these drawings has not been analyzed earlier . the first analysis of this data , which covers the lost cycle period in 1790 s , has been made only recently @xcite using staudacher s original drawings . additionally , a few original solar disc drawings made by the irish astronomer james archibald hamilton and his assistant since 1795 have been recently found in the archive of the armagh observatory @xcite . after the digitization and processing of these two sets of original drawings @xcite , the location of individual sunspots on the solar disc in the late 18th century has been determined . this makes it possible to construct the sunspot butterfly diagram for solar cycles # 3 and 4 ( fig . [ fig : wsn]c ) , which allows us to study the existence of the lost cycle more reliably than based on sunspot counts only . despite the good quality of original drawings , there is an uncertainty in determining the actual latitude for some sunspots ( see @xcite for details ) . this is related to the limited information on the solar equator in these drawings . the drawings which are mirrored images of the actual solar disc as observed from earth , can not be analyzed by an automatic prodecure adding the heliographic grid . therefore , special efforts have been made to determine the solar equator and to place the grid of true solar coordinates for each drawing ( see fig . [ fig : staud ] ) . depending on the information available for each drawing , the uncertainty in defining the solar equator , @xmath0 , ranges from almost 0@xmath1 up to a maximum of 15@xmath1 @xcite . the latitude error of a sunspot , identified to appear at latitude @xmath2 , can be defined as @xmath3 where @xmath0 is the angular uncertainty of the solar equator in the respective drawing , and @xmath4 is the angular distance between the spot and the solar disc center . accordingly , the final uncertainty @xmath5 can range from 0@xmath1 ( precise definition of the equator or central location of the spot ) up to 15@xmath1 . we take the uncertain spot location into account when constructing the semiannually averaged butterfly diagram as follows . let us illustrate the diagram construction for the second half - year ( jul - dec ) of 1793 ( fig . [ fig : dist ] ) . during this period there were only two daily drawings by staudacher with the total of 8 sunspots : two spots on august 6th , which were located close to the limb near the equator , and six spots on november 3rd , located near the disc center at higher latitudes . the uncertainty in definition of the equator was large ( @xmath6 ) for both drawings . because of the near - limb location ( large @xmath4 ) of the first two spots , the error @xmath5 of latitude definition ( eq . [ eq : delta ] ) is quite large . the high - latitude spots of the second drawing are more precisely determined because of the central location of the spots . the latitudinal occurrence of these eight spots and their uncertainties are shown in fig . [ fig : dist ] as stars with error bars . the true position of a spot is within the latitudinal band @xmath7 , where @xmath5 is regarded as an observational error and @xmath2 as the formal center of the latitudinal band . accordingly , when constructing the butterfly diagram , we spread the occurrence of each spot within this latitudinal band with equal probability ( the use of other distribution does not affect the result ) . finally , the density of the latitudinal distribution of spots during the analyzed period is computed as shown by the histogram in fig . [ fig : dist ] . this density is the average number of sunspots occurring per half - year per 2@xmath1 latitudinal bin . each vertical column in the final butterfly diagram shown in fig . [ fig : wsn]c is in fact such a histogram for the corresponding half - year . typically , the sunspots of a new cycle appear at rather high latitudes of about 2030@xmath1 . this takes place around the solar cycle minimum . later , as the new cycle evolves , the sunspot emergence zone slowly moves towards the solar equator . this recurrent `` butterfly''-like pattern of sunspot occurrence is known as the _ sprer law _ @xcite and is related to the action of the solar dynamo ( see , e.g. * ? ? ? it is important that the systematic appearance of sunspots at high latitudes unambiguously indicates the beginning of a new cycle @xcite and thus may clearly distinguish between the cycles . one can see from the reconstructed butterfly diagram ( fig . [ fig : wsn]c ) that the sunspots in 17931796 appeared dominantly at high latitudes , clearly higher than the previous sunspots that belong to the late declining phase of the ending solar cycle # 4 . thus , a new `` butterfly '' wing starts in late 1793 , indicating the beginning of the lost cycle . since sunspot observations are quite sparse during that period , we have performed a thorough statistical test as follows . the location information of sunspot occurrence on the original drawings during 17931796 ( summarized in table [ tab1 ] ) allows us to test the existence of the lost cycle . the observed sunspot latitudes were binned into three categories : low ( @xmath8 ) , mid- ( @xmath9@xmath10 ) and high latitudes ( @xmath11 ) , as summarized in column 2 of table [ tab2 ] . we use all available data on latitude distribution of sunspots since 1874 covering solar cycles 12 through 23 ( the combined royal greenwich observatory ( 18741981 ) and usaf / noaa ( 19812007 ) sunspot data set : http://solarscience.msfc.nasa.gov/greenwch.shtml ) as the reference data set . we tested first if the observed latitude distribution of sunspots ( three daily observations with low - latitude spots , one with mid - latitude and three with only high - latitude spots , see table [ tab2 ] ) is consistent with a late declining phase ( d - scenario , i.e. the period 17931796 corresponds to the extended declining phase of cycle # 4 ) or with the early ascending phase ( a - scenario , i.e. , the period 17931796 corresponds to the ascending phase of the lost cycle ) . we have selected two subsets from the reference data set : d - subset corresponding to the declining phase which covers three last years of solar cycles 12 through 23 and includes in total 11235 days when 33803 sunspot regions were observed ; and a - subset corresponding to the early ascending phase which covers 3 first years of solar cycles 13 through 23 and includes 10433 days when 47096 regions were observed . first we analyzed the probability to observe sunspot activity of each category on a randomly chosen day . for example , we found in the d - subset 4290 days when sunspots were observed at low latitudes below 8@xmath1 . this gives the probability @xmath12 ( see first line , column 3 in table [ tab2 ] ) to observe such a pattern on a random day in the late decline phase of a cycle . similar probabilities for the other categories in table [ tab2 ] have been computed in the same way . next we tested whether the observed low - latitude spot occurrence ( three out of seven daily observations ) corresponds to declining / ascending phase scenario . the corresponding probability to observe @xmath13 events ( low - latitude spots ) during @xmath14 trials ( observational days ) is given as @xmath15 where @xmath16 is the probability to observe the event at a single trial , and @xmath17 is the number of possible combinations . we assume here that the results of individual trials are independent on each other , which is justified by the long separation between observational days . thus , the probability to observed three low - latitude spots during seven random days is @xmath18 and 0.07 for d- and a - hypotheses , respectively . the corresponding probabilities are given in the first row , columns 56 of table [ tab2 ] . the occurrence of three days with low latitude activity is quite probable for both declining and ascending phases . thus , this criterion can not distinguish between the two cases . the observed mid - latitude spot occurrence ( one out of seven daily observations ) is also consistent with both d- and a - scenarios . the corresponding confidence levels ( 0.06 and 0.22 , respectively , see the second row , columns 56 of table [ tab2 ] ) do not allow to select between the two hypotheses . next we tested the observed high - latitude spot occurrence ( three out of seven observations ) in the d / a - scenarios ( the corresponding probabilities are given in the third row of table [ tab2 ] ) . the occurrence of three days with high - latitude activity is highly improbable during a late declining phase ( d - scenario ) . thus , the hypothesis of the extended cycle # 4 is rejected at the level of @xmath19 . the a - scenario is well consistent ( confidence 0.26 ) with the data . thus , the observed high - latitude sunspot occurrence clearly confirms the existence of the lost cycle . we also noticed that sunspots tend to appear in northern hemisphere ( 13 out of 16 observed sunspots appeared in the northern hemisphere ) . despite the rather small number of observations , the statistical significance of asymmetry is quite good ( confidence level 99% ) , i.e. it can be obtained by chance with the probability of only 0.01 , in a purely symmetric distribution . nevertheless , more data are needed to clearly evaluate the asymmetry . thus , a statistical test of the sunspot occurrence during 17931796 confirms that : * the sunspot occurrence in 17931796 contradicts with a typical latitudinal pattern in the late declining phase of a normal solar cycle ( at the significance level of @xmath19 ) . * the sunspot occurrence in 17931796 is consistent with a typical ascending phase of the solar cycle , confirming the start of the lost solar cycle . we note that it has been shown earlier @xcite , using the group sunspot number , that the sunspot number distribution during 17921793 was statistically similar to that in the minimum years of a normal solar cycle , but significantly different from that in the declining phase . * the observed asymmetric occurrence of sunspots during the lost cycle is statistically significant ( at the significance level of 0.01 ) . therefore , the sunspot butterfly diagram ( fig . [ fig : wsn]c ) unambiguously proves the existence of the lost cycle in the late 18th century , verifying the earlier evidence based on sunspot numbers @xcite and aurorae borealis @xcite . an additional cycle in the 1790 s changes cycle numbering before the dalton minimum , thus verifying the validity of the _ gnevyshev - ohl _ rule of sunspot cycle pairing @xcite and the related 22-year periodicity @xcite in sunspot activity throughout the whole 400-year interval . another important consequence of the lost cycle is that , instead of one abnormally long cycle # 4 ( min - to - min length @xmath2015.5 years according to gsn @xcite ) there are two shorter cycles of about 9 and 7 years ( see fig . [ fig : wsn]d ) . note also that some physical dynamo models even predict the existence of cycles of small amplitude and short duration near a grand minimum @xcite . the cycle # 4 ( 17841799 in gsn ) with its abnormally long duration dominates empirical studies of relations , e.g. , between cycle length and amplitude . replacing an abnormally long cycle # 4 by one fairly typical and one small short cycle changes empirical relations based on cycle length statistics . this will affect , e.g. , predictions of future solar activity by statistical or dynamo - based models @xcite , and some important solar - terrestrial relations @xcite . the lost cycle starting in 1793 depicts notable hemispheric asymmetry with most sunspots of the new cycle occurring in the northern solar hemisphere ( fig . [ fig : wsn]c ) . this asymmetry is statistically significant at the confidence level of 99% . a similar , highly asymmetric sunspot distribution existed during the maunder minimum of sunspot activity in the second half of the 17th century @xcite . however , the sunspots during the maunder minimum occurred preferably in the southern solar hemisphere @xcite , i.e. , opposite to the asymmetry of the lost cycle . this shows that the asymmetry is not constant , contrary to some earlier models involving the fossil solar magnetic field @xcite . interestingly , this change in hemispheric asymmetry between the maunder and dalton minimum is in agreement with an earlier , independent observation , based on long - term geomagnetic activity , that the north - south asymmetry oscillates at the period of about 200 - 250 years @xcite . concluding , the newly recovered spatial distribution of sunspots of the late 18th century conclusively confirms the existence of a new solar cycle in 17931800 , which has been lost under the preceding , abnormally long cycle compiled by rudolf wolf when interpolating over the sparse sunspot observations of the late 1790 s . this letter brings the attention of the scientific community to the need of revising the sunspot series in the 18th century and the solar cycle statistics . this emphasizes the need to search for new , yet unrecovered , solar data to restore details of solar activity evolution in the past ( e.g. , * ? ? ? the new cycle revises the long - held sunspot number series , restoring its cyclic evolution in the 18th century and modifying the statistics of all solar cycle related parameters . the northern dominance of sunspot activity during the lost cycle suggests that hemispheric asymmetry is typical during grand minima of solar activity , and gives independent support for a systematic , century - scale oscillating pattern of solar hemispheric asymmetry . these results have immediate practical and theoretical consequences , e.g. , to predicting future solar activity and understanding the action of the solar dynamo . we are grateful to dr . john butler from armagh observatory for his help with finding the old notes of hamilton s data . support from the academy of finland and finnish academy of sciences and letters ( visl foundation ) are acknowledged .

because of the lack of reliable sunspot observation , the quality of sunspot number series is poor in the late 18th century , leading to the abnormally long solar cycle ( 17841799 ) before the dalton minimum . using the newly recovered solar drawings by the 1819th century observers staudacher and hamilton , we construct the solar butterfly diagram , i.e. the latitudinal distribution of sunspots in the 1790 s . the sudden , systematic occurrence of sunspots at high solar latitudes in 17931796 unambiguously shows that a new cycle started in 1793 , which was lost in traditional wolf s sunspot series . this finally confirms the existence of the lost cycle that has been proposed earlier , thus resolving an old mystery . this letter brings the attention of the scientific community to the need of revising the sunspot series in the 18th century . the presence of a new short , asymmetric cycle implies changes and constraints to sunspot cycle statistics , solar activity predictions , solar dynamo theories as well as for solar - terrestrial relations .

0907.0063

this work was supported by gesellschaft fr schwerionenforschung , darmstadt ( gsi ) and by the loewe initiative of the state of hessen through the helmholtz international center for fair . s. a. bass , m. gyulassy , h. stoecker and w. greiner , j. phys . g * 25 * , r1 ( 1999 ) [ arxiv : hep - ph/9810281 ] . k. adcox _ et al . _ [ phenix collaboration ] , nucl . phys . a * 757 * , 184 ( 2005 ) [ arxiv : nucl - ex/0410003 ] . j. adams _ et al . _ [ star collaboration ] , nucl . phys . a * 757 * , 102 ( 2005 ) [ arxiv : nucl - ex/0501009 ] . x. n. wang , acta phys . hung . a * 24 * , 307 ( 2005 ) . i. vitev , phys . b * 639 * , 38 ( 2006 ) [ arxiv : hep - ph/0603010 ] . n. armesto , a. dainese , c. a. salgado and u. a. wiedemann , phys . d * 71 * , 054027 ( 2005 ) [ arxiv : hep - ph/0501225 ] . d. molnar and m. gyulassy , nucl . a * 697 * , 495 ( 2002 ) [ erratum - ibid . a * 703 * , 893 ( 2002 ) ] [ arxiv : nucl - th/0104073 ] . z. xu and c. greiner , phys . rev . c * 76 * , 024911 ( 2007 ) [ arxiv : hep - ph/0703233 ] . z. xu and c. greiner , arxiv:0710.5719 [ nucl - th ] . z. xu , c. greiner and h. stocker , arxiv:0711.0961 [ nucl - th ] . a. el , z. xu and c. greiner , arxiv:0712.3734 [ hep - ph ] . j. zimanyi , t. s. biro , t. csorgo and p. levai , phys . b * 472 * , 243 ( 2000 ) [ arxiv : hep - ph/9904501 ] . r. j. fries , b. muller , c. nonaka and s. a. bass , phys . c * 68 * , 044902 ( 2003 ) [ arxiv : nucl - th/0306027 ] . d. krieg and m. bleicher , phys . rev . c ( 2008 ) in print [ arxiv:0708.3015 [ nucl - th ] ] . s. sakai [ phenix collaboration ] , j. phys . g * 34 * , s753 ( 2007 ) . v. greco , c. m. ko and r. rapp , phys . b * 595 * , 202 ( 2004 ) [ arxiv : nucl - th/0312100 ] . a. adare _ et al . _ [ phenix collaboration ] , phys . rev . lett . * 98 * , 172301 ( 2007 ) [ arxiv : nucl - ex/0611018 ] . l. yan , p. zhuang and n. xu , phys . * 97 * , 232301 ( 2006 ) [ arxiv : nucl - th/0608010 ] . l. ravagli and r. rapp , phys . b * 655 * , 126 ( 2007 ) [ arxiv:0705.0021 [ hep - ph ] ] . o. linnyk , e. l. bratkovskaya and w. cassing , arxiv:0801.4282 [ nucl - th ] . s. a. voloshin , phys . c * 55 * , 1630 ( 1997 ) [ arxiv : nucl - th/9611038 ] . p. huovinen , p. f. kolb , u. w. heinz , p. v. ruuskanen and s. a. voloshin , phys . lett . b * 503 * , 58 ( 2001 ) [ arxiv : hep - ph/0101136 ] . s. a. voloshin , arxiv : nucl - th/0202072 . f. retiere and m. a. lisa , phys . c * 70 * , 044907 ( 2004 ) [ arxiv : nucl - th/0312024 ] . s. pratt and s. pal , nucl . phys . a * 749 * , 268 ( 2005 ) [ phys . c * 71 * , 014905 ( 2005 ) ] [ arxiv : nucl - th/0409038 ] . m. bleicher and h. stoecker , phys . b * 526 * , 309 ( 2002 ) [ arxiv : hep - ph/0006147 ] .

we discuss one of the most prominent features of the very recent preliminary elliptic flow data of @xmath0 meson from the phenix collaboration @xcite . even within the the rather large error bars of the measured data a negative elliptic flow parameter ( @xmath1 ) for @xmath0 in the range of @xmath2 is visible . we argue that this negative elliptic flow at intermediate @xmath3 is a clear and qualitative signature for the collectivity of charm quarks produced in nucleus - nucleus reactions at rhic . within a parton recombination approach we show that a negative elliptic flow puts a lower limit on the collective transverse velocity of heavy quarks . the numerical value of the transverse flow velocity @xmath4 for charm quarks that is necessary to reproduce the data is @xmath5 and therefore compatible with the flow of light quarks . the main goal of the current and past heavy ion programs is the search for a new state of matter called the quark - gluon - plasma ( qgp ) @xcite . major breakthroughs for the potential discovery @xcite of this new state of matter were the observation of constituent quark number scaling of the elliptic flow @xmath6 , with @xmath7 being the number of constituent quarks in the respective hadron as well as the observation of jet quenching at intermediate transverse momenta @xcite . together with the standard hydrodynamical interpretation this implies a rapid thermalization and a strong collective flow of the qcd matter created at rhic . however , open questions remain : how can one obtain a consistent description of the high @xmath3 suppression and the elliptic flow of heavy flavour quarks and hadrons . i.e. is the collectivity at rhic restricted to light quarks ( up , down , strange ) or do even charm ( bottom ) quarks participate in the collective expansion of the partonic system and reach local kinetic equilibrium ? previously , it was assumed that local equilibrium of ( heavy ) quarks could not be achieved within pqcd transport simulations . in fact , older studies @xcite based on a parton cascade dynamics restricted to @xmath8 parton interactions seemed to indicate that the opacity needed to achieve local equilibrium would be at least an order of magnitude higher than pqcd estimates . however , recent state - of - the - art parton cascade calculations ( including @xmath9 parton interactions ) have clearly shown that pqcd cross sections are sufficient to reach local ( gluon ) equilibrium and allow to describe the measured elliptic flow data @xcite . the aim of the present letter is to investigate whether also the charm quark does locally equilibrate and therefore follows the flow of the light quarks . here we will focus on the @xmath0 because it reflects the momentum distribution of the charm quarks directly , in addition first experimental data on the @xmath0 elliptic flow just became available . we will show that the recently measured negative elliptic flow of @xmath0 s provides a unique _ lower _ bound on the charm quark s collective velocity . under the assumption of local equilibration of light quarks a hydrodynamic parametrization of the freeze - out hyper - surface to parametrize the quark emission function , namely the blast - wave model , can be employed . for the charm quarks , the same emission function is used , however , with the transverse collective velocity as a free parameter to be determined by the preliminary phenix data . to calculate @xmath0 s from the charm quark emission function , we apply the well known parton recombination approach @xcite . details ( like the exact form of the freeze - out hyper - surface ) of the specific approach employed here can be found in @xcite . different from there we used a linear increasing transverse flow rapidity instead of a constant one , but the mean value has been preserved . here we summarize the most important features : in a coalescence process the quarks contribute equally to the hadrons momentum , so it inherits its azimuthal asymmetry directly from its constituents . therefore in recombination the elliptic flow of @xmath0 s emerges directly from a negative @xmath1 of the charm quark . to incorporate the asymmetry , the transverse expansion rapidity @xmath10 depend on the azimuthal angle @xmath11 and the radial coordinate @xmath12 as @xmath13 with the eccentricity @xmath14 and @xmath15 to model the damping at high @xmath3 . with the factor @xmath16 we recover @xmath17 as the mean transverse rapidity after integrating over @xmath18 . by applying the definition of the elliptic flow one obtains @xcite @xmath19 k_1 \left[b(\phi,\rho)\right ] \ , d\phi\ , \rho\ , d\rho } { \int i_0 \left[a(\phi,\rho)\right ] k_1\left[b(\phi,\rho)\right ] \ , d\phi\ , \rho\ , d\rho}\ ] ] with @xmath20 , @xmath21 and the modified bessel functions @xmath22 and @xmath23 . for a more general hydrodynamical hypersurface one could assume a dependence of freeze - out time @xmath24 on the radial coordinate @xmath18 . this would lead to additional terms involving @xmath25 and bessel functions of other order . we have checked that the modifications are only minor and therefore neglect the contributions in this letter for brevity . let us investigate the elliptic flow of the @xmath0 at midrapidity as a function of the transverse momentum for various transverse flow velocities as shown in fig . [ plt : jpsi_flow ] . the lines from top to bottom indicate calculations with a charm quark mass @xmath26 for different mean expansion velocities @xmath27 , the data by the phenix collaboration are shown as symbols with error bars indicating a negative elliptic flow for @xmath0 s at intermediate transverse momenta . the calculation shows that with increasing transverse flow a negative @xmath1 at low @xmath3 ( above @xmath28 gev , the elliptic flow values turn positive again ) develops for the @xmath0 , posing a lower bound of @xmath29 for the charm quarks flow . the best fit to the data is obtained with a mean charm flow velocity of @xmath30 equal to the light quark flow velocity extracted from previous fits within the same model . in fig . [ plt : compare_flow ] we use @xmath30 and compare the elliptic flow to other heavy mesons and fig . [ plt : quark_flow ] shows the same for the quarks . the value for @xmath31 , with a bottom quark mass of @xmath32 , is negativ in the whole range of applicability . in contrast to @xmath0 , the @xmath1 of @xmath33 stays positiv . this is due to the positive light quark @xmath1 , which competes with the negative one for the charm quark , and results in nearly zero elliptic flow at low @xmath3 . while the @xmath34 meson follows the @xmath33 flow for @xmath35 , it is much more suppressed at higher @xmath3 due to the strong negative flow of the bottom quark and approximately zero up to @xmath36 . data on @xmath37-meson elliptic flow is not yet available . when comparing it to the non - photonic electron @xmath1 , our calculations fail to predict the data @xcite . these two observables have been predicted to be similiar @xcite , since the non - photonic electrons are mainly from , @xmath37-meson decays , but with a small contribution of @xmath38-meson decays . but the electron elliptic flow is no straightforward probe for the @xmath33 @xmath1 . since the electron is not the only decay product , the decay kinematics might smear out the resulting elliptic flow of the electrons . at low @xmath3 , the increase of the @xmath33 flow is similiar to the electron data , but shifted to higher transverse momenta . above @xmath39 the electron @xmath1 starts to decrease which might be due to contributions from the @xmath38-mesons or an early onset of the fragmentation regime . direct measurements on the elliptic flow of heavy mesons will be available in the near future with the heavy - flavor tracker for star , which will allow a better analysis . therefore the presented results are based only the @xmath0 elliptic flow data . ) of @xmath0 s for @xmath40 for different mean transverse expansion velocities ( lines ) compared to preliminary data from phenix collaboration @xcite . while the @xmath1 of @xmath0 s is smaller than for light hadrons , the mean transverse velocity for the best - fit case ( @xmath41 for charm quarks ) is the same as for light quarks . ] ) for @xmath0 , @xmath33 , @xmath31 and @xmath34 at @xmath40 with @xmath41 to data of non - photonic electrons from phenix collaboration @xcite . ] ) of light , charm and bottom quarks at @xmath40 with @xmath41 . ] these results provide strong evidence for a substantial collectivity and transverse expansion of the charm quarks in nucleus - nucleus reactions at rhic . due to the large error bars this has to be verified when more precise data is available . note that our present findings are different from previous approaches that assume incomplete thermalization of the charm @xcite . we also verified our findings within a boltzmann approach to coalescence @xcite using our parametrizations and received similar results . one should also note that the observation of negative elliptic flow of heavy particles is well known in the literature ( even if not conclusively observed experimentally up to now ) . it appears due to an interplay between transverse expansion and particle mass , the more flow and the heavier the particle the more negative values does the elliptic flow reach . e.g. , negative values of the elliptic flow parameter for heavy hadrons has also been found in previous exploratory studies and seem to be a general feature of the blast - wave like flow profile at high transverse velocities @xcite . it reflects the depletion of the low @xmath3 particle abundance , when the source elements are highly boosted in the transverse direction . the difference to the present study is that here , @xmath1 is already negative on the quark level . negative elliptic flow values will even be encountered for light quarks at asymptotically high bombarding energies as discussed in @xcite . one might argue that this is an artefact of the blast - wave peak and will not survive in more realistic calculations , however also transport model calculations show slightly negative @xmath1 values for heavy particles at low transverse momenta @xcite . in conclusion , we have shown that the recent preliminary phenix data exhibiting a negative elliptic flow at low @xmath3 can be explained within a parton recombination approach using a blast - wave like parametrization . we point out that studying @xmath42 from @xmath0 offers the possibility to put a lower limit on the charm quark transverse velocity . from the present quantitative analysis we expect the transverse velocity of charm quarks to be above @xmath43 . within the limits of the present model the best description of the data is obtain for a charm transverse velocity equal to the light quark velocity of @xmath44 . so if more precise data will still support the negative @xmath1 , we conclude from this observation that charm quarks reach a substantial amount of local kinetic equilibration .

0806.0736