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You are an expert at summarizing long articles. Proceed to summarize the following text: motion camouflage is a stealth strategy employed by various visual insects and animals to achieve prey capture , mating or territorial combat . in one type of motion camouflage , the predator camouflages itself against a fixed background object so that the prey observes no relative motion between the predator and the fixed object . in the other type of motion camouflage , the predator approaches the prey such that from the point of view of the prey , the predator always appears to be at the same bearing . ( in this case , we say that the object against which the predator is camouflaged is the point at infinity . ) assuming that the prey can readily observe optical flow , but only poorly sense looming , this type of motion by the predator is then difficult to detect by the prey . for example , insects with compound eyes are quite sensitive to optical flow ( which arises from the transverse component of the relative velocity between the predator and the prey ) , but are far less sensitive to slight changes in the size of images ( which arise from the component of the relative velocity between the predator and prey along the line between them ) . more broadly such interactions may also apply in settings of mating activity or territorial maneuvers as well . in the work , @xcite of srinivasan and davey , it was suggested that the data on visually mediated interactions between two hoverflies , _ syritta pipiens _ obtained earlier by collett and land @xcite , supports a motion camouflage hypothesis . later , mizutani , chahl and srinivasan @xcite , observing territorial aerial maneuvers of dragonflies _ hemianax papuensis _ , concluded that the flight pattern is motivated by motion camouflage ( see figure 1 in their paper ) . see also @xcite for a review of related themes in insect vision and flight control . motion camouflage can be used by a predator to stealthily pursue prey , but a motion camouflage strategy can also be used by the prey to evade a predator . the only difference between the strategy of the predator and the strategy of the evader is that the predator seeks to approach the prey while maintaining motion camouflage , whereas the evader seeks to move away from the predator while maintaining motion camouflage . besides explaining certain biological pursuit strategies , motion camouflage may also be quite useful in certain military scenarios ( although the `` predator '' and `` prey '' labels may not be descriptive ) . in some settings , as is the case in @xcite , @xcite , @xcite it is more appropriate to substitute the labels `` shadower '' and `` shadowee '' for the predator - prey terminology . in this work , we take a structured approach to deriving feedback laws for motion camouflage , which incorporate biologically plausible ( vision ) sensor measurements . we model the predator and prey as point particles moving at constant ( but different ) speeds , and subject to steering ( curvature ) control . for an appropriate choice of feedback control law for one of the particles ( as the other follows a prescribed trajectory ) , a state of motion camouflage is then approached as the system evolves . ( in the situation where the predator follows a motion - camouflage law , and the speed of the predator exceeds the speed of the prey , the predator is able to pass `` close '' to the prey in finite time . in practice , once the predator is sufficiently close to the prey , it would change its strategy from a pursuit strategy to an intercept strategy . ) what distinguishes this work from earlier study of motion - camouflage trajectories in @xcite is that we present _ biologically plausible feedback laws _ leading to motion camouflage . furthermore , unlike the neural - network approach used in @xcite to achieve motion camouflage using biologically - plausible sensor data , our approach gives an explicit form for the feedback law which has a straightforward physical interpretation . the study of motion camouflage problems also naturally extends earlier work on interacting systems of particles , using the language of curves and moving frames @xcite-@xcite . for concreteness , we consider the problem of motion camouflage in which the predator ( which we refer to as the `` pursuer '' ) attempts to intercept the prey ( which we refer to as the `` evader '' ) while appearing to the prey as though it is always at the same bearing ( i.e. , motion camouflaged against the point at infinity ) . in the model we consider , the pursuer moves at unit speed in the plane , while the evader moves at a constant speed @xmath0 . the dynamics of the pursuer are given by @xmath1 where @xmath2 is the position of the pursuer , @xmath3 is the unit tangent vector to the trajectory of the pursuer , @xmath4 is the corresponding unit normal vector ( which completes a right - handed orthonormal basis with @xmath3 ) , and the plane curvature @xmath5 is the steering control for the pursuer . similarly , the dynamics of the evader are @xmath6 where @xmath7 is the position of the evader , @xmath8 is the unit tangent vector to the trajectory of the evader , @xmath9 is the corresponding unit normal vector , and @xmath10 is the steering control for the evader . figure [ framefig2d ] illustrates equations ( [ pursuer2d ] ) and ( [ evader2d ] ) . note that @xmath11 and @xmath12 are planar natural frenet frames for the trajectories of the pursuer and evader , respectively . we model the pursuer and evader as point particles ( confined to the plane ) , and use natural frames and curvature controls to describe their motion , because this is a simple model for which we can derive both physical intuition and concrete control laws . ( furthermore , although we save the details for a future paper , this approach generalizes nicely for three - dimensional motion . ) flying insects and animals ( also unmanned aerial vehicles ) have limited maneuverability and must maintain sufficient airspeed to stay aloft , so treating their motion as constant - speed with steering control is physically reasonable , at least for some range of flight conditions . ( note that the steering control directly drives the angular velocity of the particle , and hence is actually an acceleration input . however , this acceleration is constrained to be perpendicular to the instantaneous direction of motion , and therefore the speed remains unchanged . ) we refer to ( [ pursuer2d ] ) and ( [ evader2d ] ) as the `` pursuit - evader system . '' in what follows , we assume that the pursuer follows a _ feedback _ strategy to drive the system toward a state of motion camouflage , and close in on the evader . the evader , on the other hand , follows an open - loop strategy . the analysis we present for the pursuer feedback strategy also suggests ( with a sign change in the control law ) how the evader could use feedback and a motion - camouflage strategy to conceal its flight from the pursuer . ultimately , it would be interesting to address the game - theoretic problem in which both the pursuer and evader follow feedback strategies , so that the system would truly be a pursuit - evader system . ( what we address in this work would be more properly described as a pursuer - pursuee system . however , we keep the pursuer - evader terminology , because it sets the stage for analyzing the true pursuer - evader system , which we plan to address in a future paper . ) motion camouflage with respect to the point at infinity is given by @xmath13 where @xmath14 is a fixed unit vector and @xmath15 is a time - dependent scalar ( see also section 5 of @xcite ) . let @xmath16 be the vector from the evader to the pursuer . we refer to @xmath17 as the `` baseline vector , '' and @xmath18 as the `` baseline length . '' we restrict attention to non - collision states , i.e. , @xmath19 . in that case , the component of the pursuer velocity @xmath20 transverse to the base line is @xmath21 and similarly , that of the evader is @xmath22 the _ relative _ transverse component is @xmath23 * lemma * ( infinitesimal characterization of motion camouflage ) : the pursuit - evasion system ( [ pursuer2d ] ) , ( [ evader2d ] ) is in a state of motion camouflage without collision on an interval iff @xmath24 on that interval . * proof * : @xmath25 suppose motion camouflage holds . thus @xmath26.\ ] ] differentiating , @xmath27 . hence , @xmath28.\end{aligned}\ ] ] @xmath29 suppose @xmath30 on @xmath31 $ ] . thus @xmath32 so that @xmath33 where @xmath34 and @xmath35 . @xmath36 it follows from the * lemma * that the set of all motion camouflage states constitutes a 5-dimensional smooth manifold with two connected components , each diffeomorphic to @xmath37 in the 6-dimensional state space @xmath38 of the problem . in practice we are interested in how far the pursuit - evasion system is from a state of motion camouflage . in what follows , we offer a measure of this . consider the ratio @xmath39 which compares the rate of change of the baseline length to the absolute rate of change of the baseline vector . if the baseline experiences pure lengthening , then the ratio assumes its maximum value , @xmath40 . if the baseline experiences pure shortening , then the ratio assumes its minimum value , @xmath41 . if the baseline experiences pure rotation , but remains the same length , then @xmath42 . noting that @xmath43 we see that @xmath44 may alternatively be written as @xmath45 thus , @xmath44 is the dot product of two unit vectors : one in the direction of @xmath17 , and the other in the direction of @xmath46 note that @xmath47 is well - defined except at @xmath48 , since @xmath49 for convenience , we define the notation @xmath50 to represent the vector @xmath51 rotated counter - clockwise in the plane by an angle @xmath52 . thus , for example , @xmath53 . the transverse component @xmath54 of relative velocity , expression ( [ wdefn ] ) , then becomes @xmath55 \left(\frac{\bf r}{|{\bf r}|}\right)^{\perp } \nonumber \\ \hspace{-.2 cm } & = & \hspace{-.2 cm } - \left(\frac{\bf r}{|{\bf r}| } \cdot \dot{\bf r}^{\perp}\right ) \left(\frac{\bf r}{|{\bf r}|}\right)^{\perp}.\end{aligned}\ ] ] for convenience , we define @xmath56 to be the ( signed ) magnitude of @xmath54 , i.e. , @xmath57 and refer also to @xmath56 as the transverse component of the relative velocity . from the orthogonal decomposition @xmath58 \left(\frac{\dot{\bf r}}{|\dot{\bf r}|}\right)^\perp,\ ] ] it follows that @xmath59 ^ 2 = \gamma^2 + \frac{|w|^2}{|\dot{\bf r}|^2}.\ ] ] thus @xmath60 is a measure of the distance from motion camouflage . differentiating @xmath47 along trajectories of ( [ pursuer2d ] ) and ( [ evader2d ] ) gives @xmath61 \nonumber \\ & & + \frac{1}{|\dot{\bf r}|}\left [ \frac{\bf r}{|{\bf r}| } - \left(\frac{\bf r}{|{\bf r}|}\cdot \frac{\dot{\bf r}}{|\dot{\bf r}|}\right ) \frac{\dot{\bf r}}{|\dot{\bf r}| } \right ] \cdot \ddot{\bf r}.\end{aligned}\ ] ] from ( [ rdefnplanar ] ) we obtain @xmath62 and @xmath63 also , @xmath64 \hspace{-.2 cm } & = & \hspace{-.2 cm } \left[\frac{\bf r}{|{\bf r}| } \cdot \left(\frac{\dot{\bf r}}{|\dot{\bf r}|}\right)^{\perp } \right ] \left(\frac{\dot{\bf r}}{|\dot{\bf r}|}\right)^{\perp } \nonumber \\ \hspace{-.2 cm } & = & \hspace{-.2 cm } \frac{1}{|\dot{\bf r}|^2 } \left(\frac{\bf r}{|{\bf r}| } \cdot \dot{\bf r}^{\perp } \right ) \dot{\bf r}^{\perp}.\end{aligned}\ ] ] then from ( [ dotgamma ] ) we obtain @xmath65 \nonumber \\ & & + \frac{1}{|\dot{\bf r}|}\left [ \frac{\bf r}{|{\bf r}| } - \left(\frac{\bf r}{|{\bf r}|}\cdot \frac{\dot{\bf r}}{|\dot{\bf r}|}\right ) \frac{\dot{\bf r}}{|\dot{\bf r}| } \right ] \cdot \big ( { \bf y}_p u_p - \nu^2 { \bf y}_e u_e \big ) \nonumber \\ \hspace{-.2 cm } & = & \hspace{-.2 cm } \frac{|\dot{\bf r}|}{|{\bf r}|}\left [ \frac{1}{|\dot{\bf r}|^2 } \left(\frac{\bf r}{|{\bf r}| } \cdot \dot{\bf r}^{\perp } \right)^2 \right ] \nonumber \\ & & + \frac{1}{|\dot{\bf r}|}\left[\frac{1}{|\dot{\bf r}|^2 } \left(\frac{\bf r}{|{\bf r}| } \cdot \dot{\bf r}^{\perp } \right ) \dot{\bf r}^{\perp}\right ] \cdot \big ( { \bf y}_p u_p - \nu^2 { \bf y}_e u_e \big ) . \nonumber \\\end{aligned}\ ] ] noting that @xmath66 and @xmath67 we obtain @xmath68 \nonumber \\ & & + \frac{1}{|\dot{\bf r}|}\left[\frac{1}{|\dot{\bf r}|^2 } \left(\frac{\bf r}{|{\bf r}| } \cdot \dot{\bf r}^{\perp } \right ) \right ] \big(1 - \nu ( { \bf x}_p \cdot { \bf x}_e ) \big ) u_p \nonumber \\ & & + \frac{1}{|\dot{\bf r}|}\left[\frac{1}{|\dot{\bf r}|^2 } \left(\frac{\bf r}{|{\bf r}| } \cdot \dot{\bf r}^{\perp } \right ) \right ] \big(\nu - ( { \bf x}_p \cdot { \bf x}_e ) \big ) \nu^2 u_e.\end{aligned}\ ] ] suppose that we take @xmath69 \nu^2 u_e,\ ] ] where @xmath70 , so that the steering control for the pursuer consists of two terms : one involving the motion of the evader , and one involving the transverse component of the relative velocity . then @xmath71 \left [ \frac{1}{|\dot{\bf r}| } \left(\frac{\bf r}{|{\bf r}| } \cdot \dot{\bf r}^{\perp } \right ) \right]^2,\ ] ] and for any choice of @xmath70 , there exists @xmath72 such that @xmath73 for all @xmath17 such that @xmath74 . thus , for control law ( [ planarup ] ) , @xmath75 control law ( [ planarup ] ) has the nice property that for any value of the gain @xmath76 , there is a disc of radius @xmath77 ( depending on @xmath78 ) such that @xmath79 outside the disc . however , the problem with ( [ planarup ] ) is that the pursuer needs to know ( i.e. , sense and estimate ) the evader s steering program @xmath10 . here we show that by taking @xmath78 sufficiently large , motion camouflage can be achieved ( in a sense we will make precise ) using a control law depending only on the transverse relative velocity : @xmath80 in place of ( [ planarup ] ) , provided @xmath81 is bounded . comparing ( [ planarupgain ] ) to ( [ transrelvel3 ] ) , we see that , indeed , @xmath5 is proportional to the signed length of the relative transverse velocity vector . we will designate this as the _ motion camouflage proportional guidance _ ( mcpg ) law for future reference ( see section v below ) . as is further discussed in section v , ( [ planarupgain ] ) requires range information as well as pure optical flow sensing . however , the range information can be coarse , since range errors ( within appropriate bounds ) have the same effect in ( [ planarupgain ] ) as gain variations . we say that ( [ planarupgain ] ) is _ biologically plausible _ because the only critical sensor measurement required is optical flow sensing . optical flow sensing does not yield the relative transverse velocity directly , but rather the angular speed of the image of the evader across the pursuer s eye . in fact , it is the sign of the optical flow that is most critical to measure correctly , since errors in the magnitude of the optical flow , like range errors , only serve to modulate the gain in ( [ planarupgain ] ) . for biological systems , the capabilities of the sensors _ vis - a - vis _ the sensing requirements for implementing ( [ planarupgain ] ) constrain the range of conditions for which ( [ planarupgain ] ) represents a feasible control strategy . in the high - gain limit we focus on below , sensor noise ( which is amplified by the high gain ) would be expected to have significant impact . however , to illustrate the essential behavior , here we neglect both sensor limitations and noise . let us consider control law ( [ planarupgain ] ) , and the resulting behavior of @xmath47 as a function of time . from ( [ dotgammaupue ] ) , we obtain the inequality @xmath82 \left [ \frac{1}{|\dot{\bf r}| } \left(\frac{\bf r}{|{\bf r}| } \cdot \dot{\bf r}^{\perp } \right ) \right]^2 \nonumber \\ & & + \frac{1}{|\dot{\bf r}|}\left[\frac{1}{|\dot{\bf r}|^2 } \left(\frac{\bf r}{|{\bf r}| } \cdot \dot{\bf r}^{\perp } \right ) \right ] \big(\nu - ( { \bf x}_p \cdot { \bf x}_e ) \big ) \nu^2 u_e \nonumber \\ \hspace{-.2 cm } & \le & \hspace{-.2 cm } - \left(1-\gamma^2 \right ) \left[\frac{\mu}{|\dot{\bf r}|}\big(1-\nu ( { \bf x}_p \cdot { \bf x}_e)\big ) - \frac{|\dot{\bf r}|}{|{\bf r}| } \right ] \nonumber \\ & & + \frac{1}{|\dot{\bf r}|^2}\sqrt{1-\gamma^2 } \big| \big(\nu - ( { \bf x}_p \cdot { \bf x}_e ) \big ) \nu^2 u_e \big| \nonumber \\ \hspace{-.2 cm } & \le & \hspace{-.2 cm } - \left(1-\gamma^2 \right ) \left[\frac{\mu}{|\dot{\bf r}|}(1-\nu ) - \frac{|\dot{\bf r}|}{|{\bf r}| } \right ] \nonumber \\ & & + \left(\sqrt{1-\gamma^2}\right)\frac{\nu^2(1+\nu)(\max |u_e| ) } { |\dot{\bf r}|^2 } \nonumber \\ \hspace{-.2 cm } & \le & \hspace{-.2 cm } - \left(1-\gamma^2 \right ) \left[\mu\left(\frac{1-\nu}{1+\nu}\right ) - \frac{1+\nu}{|{\bf r}| } \right ] \nonumber \\ & & + \left(\sqrt{1-\gamma^2}\right)\left[\frac{\nu^2(1+\nu)(\max |u_e| ) } { ( 1-\nu)^2}\right],\end{aligned}\ ] ] where we have used ( [ dotrbound ] ) . for convenience , we define the constant @xmath83 as @xmath84 for any @xmath70 , we can define @xmath72 and @xmath85 such that @xmath86 ( and it is clear that many such choices of @xmath77 and @xmath87 exist ) . note that ( [ mudecomp ] ) implies @xmath88 thus , for @xmath89 , ( [ dotgammabound ] ) becomes @xmath90 \nonumber \\ & & + \left(\sqrt{1-\gamma^2}\right)c_1 \nonumber \\ \hspace{-.2 cm } & = & \hspace{-.2 cm } - \left(1-\gamma^2 \right ) c_o + \left(\sqrt{1-\gamma^2 } \right)c_1.\end{aligned}\ ] ] suppose that given @xmath91 , we take @xmath92 . then for @xmath93 , @xmath94 where @xmath95 * remark * : there are two possibilities for @xmath96 the state we seek to drive the system toward has @xmath97 ; however , ( [ oneminusgammasqreps ] ) can also be satisfied for @xmath98 . ( recall that @xmath99 . ) there is always a set of initial conditions such that ( [ oneminusgammasqreps ] ) is satisfied with @xmath98 . we can address this issue as follows : let @xmath100 denote how close to @xmath101 we wish to drive @xmath47 , and let @xmath102 denote the initial value of @xmath47 . take @xmath103 so that ( [ dotgammabound2 ] ) with ( [ c2defn ] ) applies from time @xmath104 . @xmath36 from ( [ dotgammabound2 ] ) , we can write @xmath105 which , integrating both sides , leads to @xmath106 where @xmath107 . noting that @xmath108 we see that for @xmath89 , ( [ dotgammabound2 ] ) implies @xmath109 where we have used the fact that @xmath110 is a monotone increasing function . now we consider estimating how long @xmath89 , which in turn determines how large @xmath111 can become in inequality ( [ gammabound ] ) , and hence how close to @xmath112 will @xmath44 be driven . from ( [ gammadotprod ] ) we have @xmath113 which from ( [ dotrbound ] ) and @xmath114 , @xmath115 , implies @xmath116 from ( [ dotnormrbound ] ) , we conclude that @xmath117 and , more to the point , @xmath118 ) to be meaningful for the problem at hand , we assume that @xmath119 . then defining @xmath120 to be the minimum interval of time over which we can guarantee that @xmath79 , we conclude that @xmath121 from ( [ gammafinal ] ) , we see that by choosing @xmath122 sufficiently large ( which can be accomplished by choosing @xmath92 sufficiently large ) , we can force @xmath123 . noting that @xmath124 for @xmath125 , we see that @xmath126 thus , if @xmath92 is taken to be sufficiently large that @xmath127 then we are guaranteed ( under the conditions mentioned in the above calculations ) to achieve @xmath128 at some finite time @xmath129 . * definition * : given the system ( [ pursuer2d ] ) , ( [ evader2d ] ) with @xmath47 defined by ( [ gammadotprod ] ) , we say that `` motion camouflage is accessible in finite time '' if for any @xmath130 there exists a time @xmath131 such that @xmath128 . @xmath36 * proposition * : consider the system ( [ pursuer2d ] ) , ( [ evader2d ] ) with @xmath47 defined by ( [ gammadotprod ] ) , and control law ( [ planarupgain ] ) , with the following hypotheses : * @xmath132 ( and @xmath133 is constant ) , * @xmath10 is continuous and @xmath81 is bounded , * @xmath134 , and * @xmath135 . motion camouflage is accessible in finite time using high - gain feedback ( i.e. , by choosing @xmath70 sufficiently large ) . * proof * : choose @xmath72 such that @xmath136 . choose @xmath137 sufficiently large so as to satisfy ( [ c2bound ] ) , and choose @xmath87 accordingly to ensure that ( [ dotgammabound2 ] ) holds for @xmath138 . then defining @xmath78 according to ( [ mudecomp ] ) ensures that @xmath139 , where @xmath140 is defined by ( [ bigtdefn ] ) . @xmath36 * remark * : assumption @xmath141 above can be generalized to @xmath142 . ( the @xmath143 case corresponds to a stationary `` evader , '' so that the natural frenet frame ( [ evader2d ] ) and steering control @xmath10 for the evader are not defined . ) the following simulation results illustrate the behavior of the pursuit - evasion system ( [ pursuer2d ] ) , ( [ evader2d ] ) , under the control law ( [ planarupgain ] ) for the pursuer and various open - loop controls for the evader . the simulations also confirm the analytical results presented above . figure [ straight_traj ] shows the behavior of the system for the simplest evader behavior , @xmath144 , which corresponds to straight - line motion . because control law ( [ planarupgain ] ) is the same as ( [ planarup ] ) when @xmath144 , @xmath47 tends monotonically toward @xmath101 ( for the initial conditions and choice of gain @xmath78 used in the simulation shown ) . in figure [ straight_traj ] , as in the subsequent figures showing pursuer and evader trajectories , the solid light lines connect the pursuer and evader positions at evenly - spaced time instants . for a pursuit - evasion system in a state of motion camouflage , these lines would all be parallel to one another . also , each simulation is run for finite time , at the end of which the pursuer and evader are in close proximity . ( the ratio of speeds is @xmath145 in all of the simulations shown . ) figure [ sine_traj ] illustrates the behavior of the pursuer for a sinsusoidally - varying steering control @xmath10 of the evader , and figure [ sine_gamma ] shows the corresponding behavior of @xmath44 . in figure [ sine_gamma ] , increasing the value of the feedback gain @xmath78 by a factor of three is observed to decrease the peak difference between @xmath47 and @xmath101 by a factor of about @xmath146 . this is consistent with the calculations in the proof of the * proposition*. figure [ random_traj ] illustrates the behavior of the pursuer for a randomly - varying steering control @xmath10 of the evader , and figures [ random_gamma ] and [ random_gamma_initial ] show the corresponding behavior of @xmath44 . similarly to figure [ sine_gamma ] , figure [ random_gamma ] shows that increasing the feedback gain @xmath78 by a factor of three decreases the peak difference between @xmath47 and @xmath101 by a factor of about @xmath146 . figure [ random_gamma_initial ] shows the initial transient in @xmath44 for @xmath111 small . as would be expected , increasing the gain @xmath78 increases the convergence rate . ( the time axes for figures [ random_gamma ] and [ random_gamma_initial ] differ by a factor of 200 , which is why the initial transient can not be seen in figure [ random_gamma ] . ) finally , figure [ circle_traj ] illustrates the behavior of the pursuer for a constant steering control @xmath10 , resulting in circling motion by the evader . there is a vast literature on the subject of missile guidance in which the problem of pursuit of an ( evasively ) maneuvering target by a tactical missile is of central interest . a particular class of feedback laws , known as _ pure proportional navigation guidance _ ( ppng ) occupies a prominent place @xcite . for planar missile - target engagements , the ppng law determining the steering control for the missile / pursuer is @xmath147 where @xmath148 denotes the rate of rotation ( in the plane ) of the line - of - sight ( los ) vector from the pursuer to the evader . here the gain @xmath149 is a dimensionless positive constant known as the navigation constant . notice that our motion camouflage guidance law ( mcpg ) given by ( [ planarupgain ] ) has a gain @xmath78 which has the dimensions of @xmath150^{-1 } $ ] . also , it is easy to see that @xmath151 so , to make a proper comparison we let @xmath77 as in section iii be a length scale for the problem and define the dimensionless gain @xmath152 thus , our mpcg law takes the form @xmath153 it follows that motion camouflage uses range information to support a high gain in the initial phase of the engagement , ramping down to a lower value in the terminal phase @xmath154 . in nature this extra freedom of gain control is particularly relevant for echolocating bats ( see @xcite ) , which have remarkable ranging ability . analysis of the performance of the ppng law is carried out in @xcite , using arguments similar to ours ( although our sufficient conditions appear to be weaker ) . while motion camouflage as a strategy is discussed in @xcite , under `` parallel navigation , '' to the best of our knowledge , the current work is the first to present and analyze a feedback law for motion camouflage . in work under preparation , we have generalized the analysis to the three - dimensional setting , and to planar motion camouflage with respect to a finite point . the three - dimensional analysis is made possible by the use of natural frenet frames , analogously to the three - dimensional unit - speed particle interaction laws described in @xcite . because we are able to treat the motion camouflage problem within the same framework as our earlier formation control and obstacle - avoidance work @xcite-@xcite , we would like to understand how teams of vehicles can make use of motion camouflage , and whether we can determine the convergence behavior of such systems . various biologically - inspired scenarios for motion camouflage with teams have been described in @xcite . considering additional military applications without biological analogs , there are thus a variety of team motion camouflage problems to study . the authors would like to thank m.v . srinivasan of the research school of biological sciences at the australian national university for valuable discussions and helpful comments on an earlier draft of this paper . this research was supported in part by the naval research laboratory under grants no . n00173 - 02 - 1g002 , n00173 - 03 - 1g001 , n00173 - 03 - 1g019 , and n00173 - 04 - 1g014 ; by the air force office of scientific research under afosr grants no . f49620 - 01 - 0415 and fa95500410130 ; by the army research office under oddr&e muri01 program grant no . daad19 - 01 - 1 - 0465 to the center for communicating networked control systems ( through boston university ) ; and by nih - nibib grant 1 r01 eb004750 - 01 , as part of the nsf / nih collaborative research in computational neuroscience program .

motion camouflage is a stealth strategy observed in nature . we formulate the problem as a feedback system for particles moving at constant speed , and define what it means for the system to be in a state of motion camouflage . ( here we focus on the planar setting , although the results can be generalized to three - dimensional motion . ) we propose a biologically plausible feedback law , and use a high - gain limit to prove accessibility of a motion camouflage state in finite time . we discuss connections to work in missile guidance . we also present simulation results to explore the performance of the motion camouflage feedback law for a variety of settings .

You are an expert at summarizing long articles. Proceed to summarize the following text: nova cephei 2014 was discovered as a transient by nishiyama and kabashima ( 2014 ) at a magnitude of 11.7 on unfiltered ccd frames ( limiting magnitude 13.7 ) taken around 2014 march 8.792 ut . the object was confirmed to be a classical nova by munari et al . ( 2014 ) who obtained a low - resolution spectrogram ( range 395 - 852 nm , 0.21 nm / pixel ) on 2014 march 9.792 ut . the spectrum showed a red continuum with strong emission lines from the balmer series , o i 777.4 and 844.6 nm , ca ii 849.8 nm , and fe ii multiplets 42 , 48 , and 49 . all emission lines showed strong p - cyg absorptions which were blue - shifted by 660 km s@xmath4 for the balmer lines , 780 km s@xmath4 for the fe ii lines , and 900 km s@xmath4 for the o i lines . the emission lines had a width of about 800 km s@xmath4 . the intensity of the o i 844.6 nm emission line was seen to be about twice that of o i 777.4 nm , indicating that there was fluorescent pumping from hydrogen lyman@xmath5 photons . photometry on 2014 march 10.094 ut showed a large value of the color b - v = + 1.27 which indicated significant reddening consistent with the red slope of the continuum observed in the spectrum . the object s spectrum showed it to be a highly reddened fe ii class nova observed close to maximum brightness . no detailed study of this nova , in any wavelength regime , has been presented till date . nova sco 2015 was discovered as a bright transient on 2015 february 11.8367 ut at an unfiltered ccd magnitude of 8.2 by tadashi kojima using a 150-mm f/2.8 lens @xmath6 a digital camera ( nakano 2015 ) . nothing was visible on a frame from the same camera on feb . 10.827 ut . ( vsnet - alert 18276 : ; aavso special notice no . 397 ) . the object was designated pnv j17032620 - 3504140 on the cbat transient object confirmation page ( tocp ) . an echelle spectrum on 2015 february 13 at 09:38ut ( walter 2015 ) confirmed that the object was a nova . h@xmath0 had an equivalent width of -14 nm and full width at half maximum ( fhwm ) @xmath72000 km s@xmath4 . there were symmetrically displaced emission features at about @xmath8 4500 kms@xmath4 which resembled those seen in fast he / n novae . h@xmath0 and h@xmath5 showed p cyg absorption features at about -4200,-3200 , and -2300 km s@xmath4 o i 777 nm and 845 nm were in emission . a strong emission line at 588 nm with a prominent p cyg absorption was either he i 587 nm or modestly blue shifted na i. broad ( 2000 km s@xmath4 fwzi ) he i 706 nm emission was possibly also present . similarly broad emission was seen in the prominent fe ii multiplet 42 lines at 492 , 502 , and 517 nm , though the first two may have had some he i contribution . the apparently rapid fading and bright possible near - ir counterpart suggested this was a system with an m giant donor ( walter 2015 ) , like v745 sco or nova sco 2014 . and @xmath9 light curves of nova cep 2014 and nova sco 2015 from aavso data in black and gray symbols respectively . the outburst dates are taken as 2014 march 8.792 ut ( jd 2456725.2920 ) and 2015 february 11.8367 ut ( jd 2457065.3333 ) respectively.,width=336,height=432 ] .log of the photometry@xmath10 from mount abu ir telescope [ cols="^,^,^,^,^,^,^ " , ] [ table_obsspec ] a , b : the spectroscopic observation of nova sco 2015 on 2015 march 23.63 ut was made from the irtf telescope . the rest of the spectra were obtained from mt . abu . early x - ray and radio observations of nova sco 2015 by nelson et al ( 2015 ) implicated strong shocks against a red giant wind . their observations of nova sco 2015 were carried out at x - ray , uv and radio wavelengths . the x - ray observations were carried out with the swift satellite between 2015 february 15.5 and 16.3 ut ( about 4 days after discovery ) . an x - ray source was clearly detected at the position of the nova . the spectrum was hard and could be modeled as a highly absorbed , hot thermal plasma ( n(h ) @xmath7 6@xmath81 x 10@xmath11 @xmath12 ; kt greater than 41 kev ) . however , a significant excess of counts over the model prediction was observed between 1 and 2 kev , possibly indicating the presence of a second , softer emission component . nelson et al . ( 2015 ) also observed nova sco 2015 at radio wavelengths with the karl g. jansky very large array ( vla ) on 2015 february 14.5 , approximately 3 days after optical discovery . the nova was detected at frequencies from 4.55 to 36.5 ghz with a spectrum typical of non - thermal synchrotron emission ( spectral index between -0.7 and -0.9 ) . the presence of hard , absorbed x - rays and synchrotron radio emission at this early stage of the outburst suggested that the nova - producing white dwarf was embedded within the wind of a red - giant companion , with collisions between the ejecta and this wind shock - heating plasma and accelerating particles ( as in , e.g. rs oph , v407 cyg and v745 sco ( banerjee et al . 2009 , munari et al . 2011 , banerjee et al . this interpretation is supported by our nir observations . in this paper we present our nir spectroscopic and photometric observations of nova cep 2014 and of nova sco 2015 , preliminary reports of which were made in ashok et al . ( 2014 ) and srivastava et al.(2015 ) . the observations of nova cep 2014 span 9 epochs covering 5 to 90 days after the outburst and the observations of nova sco 2015 span 11 epochs covering 7 to 47 days after the outburst . we present the observations in section [ sec_observations ] . the analysis and results for nova cep 2014 and nova sco 2015 are described in section [ sec_ncep_results ] and section [ sec_nsco_results ] respectively . near - ir spectroscopy in the 0.85 to 2.4 @xmath13 m region at resolution @xmath14 @xmath7 1000 was carried out with the 1.2 m telescope of the mount abu infrared observatory using the near - infrared camera / spectrograph ( nics ) equipped with a 1024x1024 hgcdte hawaii array . spectra were recorded with the star dithered to two positions along the slit with one or more spectra being recorded in both of these positions . the co - added spectra in the respective dithered positions were subtracted from each other to remove sky and dark contributions . the spectra from these sky - subtracted images were extracted using iraf tasks and wavelength calibrated using a combination of oh sky lines and telluric lines that register with the stellar spectra . to remove telluric lines from the target s spectra , it was ratioed with the spectra of a standard star ( sao 18998 spectral type a2iv in case of nova cep 2015 and sao 206599 , spectral type a0/a1v in the case of nova sco 2015 ) from whose spectra the hydrogen paschen and brackett absorption lines had been removed . the spectra were finally multiplied by a blackbody at the effective temperature of the standard star to yield the resultant spectra . all spectra were covered in three settings of the grating that cover the @xmath15 and @xmath16 regions separately . a spectrum of nova sco 2015 was obtained using the 3 m irtf telescope on 2015 march 23.625ut covering the 0.8 to 2.5 @xmath13 m region . this spectrum was obtained using spex ( rayner et al . 2003 ) in the cross - dispersed mode using the @xmath17 slit ( @xmath18 ) and a total integration time of 360s . the spex data were reduced and calibrated using the spextool software ( cushing et al . 2004 ) , and corrections for telluric absorption were performed using the idl tool xtellcor ( vacca et al . the log of the observations are given in tables [ table_obsphot ] and [ table_obsspec ] . fig [ fig_lightcurves ] shows the @xmath19 and @xmath9 band light curves of the nova cep 2014 using data from american association of variable star observers ( aavso ) . the nova showed a climb to maximum that lasted for 5 days before peaking at @xmath7 11.05 mag in @xmath19 on 2014 march 13.9198 ( jd2456730.4198 ) . from the light curve we determine t@xmath20 and t@xmath21 - the time for the brightness to decline by 2 and 3 magnitudes respectively from maxima - to be 22@xmath8 2d and 42 @xmath8 1d thereby putting it in the fast speed class . the observed @xmath22 values at maximum and at t@xmath20 equal 1.18 and 0.9 respectively in contrast to the expected values of 0.23 @xmath8 0.06 and -0.02 @xmath8 0.04 respectively at these epochs ( van den bergh & younger 1987 ) . the large values of @xmath22 imply considerable reddening ; the excess @xmath23 values are equal to 0.95 and 0.92 respectively . we adopt a mean value of 0.935 for the reddening and thus an extinction @xmath24 . for @xmath25 , the mmrd relation of della valle & livio ( 1995 ) gives m@xmath26 = -7.84 @xmath8 0.5 which implies a distance to the nova of 15.8 @xmath8 4 kpc . similarly , the mmrd relations for t@xmath20 and t@xmath21 by downes & duerbeck ( 2000 ) yield similar m@xmath26 magnitudes of -7.90 @xmath8 0.66 and -7.87 @xmath8 0.92 respectively . these translate to a mean distance estimate of 17.2 @xmath8 7 kpc . clearly , a large distance to the nova is suggested . the possibility is unlikely that the distance estimate is being boosted up because of a low choice of extinction @xmath27 in the distance - modulus relation @xmath28 . the extinction of 2.9 that we have used is close to the total galactic extinction of 2.995 mag in the direction of the nova as estimated by schlafly & finkbeiner ( 2011 ) from dust extinction maps . our choice of a@xmath26 is also consistent with the extinction modeling of marshall et al ( 2006 ) who find that the extinction a@xmath26 rapidly rises , in the direction of the nova , to 3.36 @xmath8 0.33 by a distance of 1.23 kpc and remains at this value for larger distances . the j , h and k band spectra for nova cep 2014 are shown in fig [ ncep14_spec_jhk ] . these show that the outburst and evolution of nova cep 2014 was that of a conventional fe ii class nova . the spectra of such novae , in the near - ir , are characterized by strong hi lines of the paschen and brackett series but what differentiates them from the he / n class is the presence of several strong ci lines seen around maximum and during the early decline ( banerjee & ashok , 2012 ) . these ci lines are all prominently seen in the spectrum of nova cep 2014 ( examples being ci 1.0685 , 1.165 , 1.176 @xmath13 m and a strong blend of ci lines in the region 1.73 to 1.78 @xmath13 m ) . the detailed identification of the lines is presented in fig [ fig_ncep_lineid ] and is discussed in the appendix and table [ table_linelist ] . p - cygni absorption components are seen in many of the lines in the spectra taken during 2014 march . the line profile widths do not vary much over time ; for e.g. the fwhm of the paschen @xmath5 1.2818 @xmath13 m line changes from @xmath71200 km@xmath29 to @xmath71500 km@xmath29 between the epochs 2014 march 14 ( 5.22d ) to 2014 april 7 ( 29.20d ) . @xmath30 ( small dash - dotted lines ) , 10@xmath31 @xmath30 ( solid lines ) , 10@xmath32 @xmath30 ( big dash lines ) , 10@xmath33 @xmath30 ( small dash lines ) , 10@xmath34 @xmath30 ( dotted lines ) , and 10@xmath35 @xmath30 ( dash - dotted lines ) . , scaledwidth=50.0% ] we do not find any evidence of dust formation in this nova , as manifested by an ir excess , during the early decline stage . to check whether dust formation may have occurred later , photometry was done recently ( 2015 april 28 , jd 2457140.5 ) . however the nova was not detected in any of the j , h or k bands . the limiting magnitudes of our observations in j , h and k band are @xmath715.0 . this , taken in conjunction with the latest v magnitudes of 18.943 on 2015 april 6.87 ut ( jd 2457119.36553 ) from the aavso database , yields @xmath36 @xmath37 3.9 . the small value of the @xmath36 color indicates that dust formation is very unlikely to have occurred . a recombination case b analysis was done , but only for selected dates of 2014 march 15.02 , april 05.00 and april 07.99 ( i.e. 6 , 27 and 30 days after the outburst ) when contemporaneous photometric observations were available for flux calibrating the spectra . the measured line fluxes for the h i brackett lines are given in table [ table_lineluminosity_ncep2014 ] and fig [ ncep14_recombanalysis ] shows the brackett line strengths with respect to br 12 set to unity . we find that the line fluxes do not match predicted case b recombination values . in particular , it is seen from fig [ ncep14_recombanalysis ] that the br7 ( br@xmath38 ) line strength is significantly lower than the predicted values of storey & hummer ( 1995 ) . though expected to be stronger than the other br series of lines , it is found to be significantly weaker than for e.g br10 and br11 . such behavior is expected in the early phase of outbursts signifying that br@xmath38 is optically thick and so possibly are the other br lines . such optical depth effects in the brackett lines are also seen in several other novae systems e.g. nova oph 1998 ( lynch et al . 2000 ) , v2491 cyg and v597 pup ( naik at al . 2009 ) , rs oph ( banerjee et al . 2009 ) and t pyx ( joshi et al . 2014 ) . lccc + emission line & & integrated line flux & + and wavelength & & at days after outburst & + ( @xmath13 m ) & & ( @xmath39 watt/@xmath40 ) & + & 6.23d & 27.21d & 30.19d + br17 1.5439 & 35.0 & - & 7.44 + br16 1.5556 & 81.4 & 28.1 & 33.5 + br15 1.5701 & 50.1 & 9.04 & + br14 1.5881 & 72.5 & 27.9 & 19.0 + br13 1.6109 & - & 25.5 & 30.8 + br12 1.6407 & 92.8 & 46.0 & 41.5 + br11 1.6807 & 88.5 & 137.0 & 118.0 + br10 1.7362 & 94.1 & 147.0 & 158.0 + br7 2.1655 & 82.4 & 85.5 & 108.0 + [ table_lineluminosity_ncep2014 ] although the lines are optically thick , we can estimate the emission measure @xmath2 of the ejecta following the opacity data given by hummer & storey ( 1987 ) and storey & hummer ( 1995 ) and using the fact that br@xmath38 line is found to be optically thick . the optical depth at line - center @xmath41 is given by @xmath42 , where @xmath3 , @xmath43 , @xmath44 and @xmath45 are the electron number density , ion number density , path length and opacity corresponding to the transition from upper level @xmath46 to lower level @xmath47 , respectively . further , the opacity factor @xmath45 does not vary significantly within the density or temperature range that is expected to prevail in the ejecta . for e.g. , from storey & hummer 1995 , the opacity @xmath45 for br@xmath38 line for the temperature @xmath48 k , and number densities @xmath49 to @xmath50 @xmath51 vary only between 1.3 @xmath52 10@xmath53 to 7.46 @xmath52 10@xmath53 . we will assume that the densities in the early stage of the nova outburst are high and lie in the in the above range of @xmath49 to @xmath50 @xmath51 . as @xmath54 the emission measure @xmath2 for above values is estimated to be in the range of @xmath55 to @xmath56 @xmath57 . we constrain the electron density by taking @xmath44 as the kinematical distance @xmath58 traveled by the ejecta where @xmath59 is the velocity of ejecta and @xmath60 is the time after outburst . we consider a typical ejecta velocity of @xmath59 @xmath7 1000 kms@xmath4 as measured from half the fwzi of the pa@xmath5 1.2818 @xmath13 m line and @xmath60 to range from 6 to 30 days . with the constraints that @xmath61 , the lower limit on electron density @xmath3 is found to be in the range @xmath62 @xmath51 to @xmath63 @xmath51 ( assuming @xmath64 ) . it should be noted that these derived lower limits are likely to be smaller than the actual @xmath3 values because @xmath65 ( br@xmath38 ) can be considerably @xmath66 . the density in the nova ejecta remains significantly high over the entire duration of our observations . lynch et al . ( 2000 ) showed that high densities of @xmath67 @xmath51 or more tend to thermalize the level populations through collisions and thereby bring about deviations from case b predictions . the same has been observed here in hi lines . the gas mass of the ejecta may be estimated by @xmath68 = @xmath69@xmath19@xmath3@xmath70 where @xmath19 is the volume ( = 4/3@xmath71@xmath72 ) , @xmath69 is the volume filling factor and @xmath70 is the proton mass . for @xmath44 varying between the distance traversed in 6d to 30d and the corresponding lower limits on @xmath3 as estimated above , the mass @xmath68 varies between ( @xmath73 @xmath74)@xmath69 m@xmath75 . this is a wide range and the mass is poorly constrained but nevertheless the mass range is consistent with the typical ejecta masses estimated in novae of @xmath76 to @xmath77 m@xmath75 . the v and b band light curves are shown in the lower panel of fig [ fig_lightcurves ] using the data from aavso . the nova showed a monotonic decline and we determine t@xmath20 and t@xmath21 values of 14@xmath82 d and 19@xmath81 d which puts the nova sco 2015 in the fast speed class similar to nova cep 2014 discussed earlier in subsection [ subsec_ncep_lightcurve ] . the observed ( b - v ) value near the optical maximum and t@xmath20 are 0.87 and 0.79 respectively . comparing these values with the expected values of 0.23 @xmath8 0.06 and -0.02 @xmath8 0.04 respectively at these epochs from van den bergh & younger ( 1987 ) , we get an average value of 0.72 for the color excess e(b - v ) and interstellar extinction @xmath78 . by using the mmrd relation of della valle & livio ( 1995 ) we get @xmath79 for @xmath80 d which implies a distance of @xmath81 kpc for @xmath78 . similarly by using the mmrd relations for t@xmath20 and t@xmath21 of downes & duerbeck ( 2000 ) we get m@xmath26 values of @xmath82 and @xmath83 and these translate to a mean distance of @xmath84 kpc , which is adopted as the distance to nova sco 2015 . the extinction value of @xmath78 used in these calculations is slightly larger than the total galactic extinction of 1.99 in the direction of the nova as estimated by schlafly & finkneiner ( 2011 ) from the dust extinction maps . the near - infrared spectra of nova sco 2015 at different epochs are shown in fig [ nsco15_spec_jhk ] and fig [ fig_nsco_irtf ] . the prominent spectral features in these spectra are the brackett and paschen recombination lines of h i and he i lines at 1.0831 , 1.7002 and 2.0581 @xmath13 m , with the 1.0831 @xmath13 m line being overwhelmingly strong . the n i blend at 1.2461 and 1.2469 @xmath13 m and the lyman @xmath5 fluoresced o i lines at 0.8446 & 1.1287 @xmath13 m are also present . these spectra are typical of of the he / n class of nova with strong lines of he i seen starting from the first set of observations on 7.16 d after the outburst . the absence of c i lines all through the span of present observations is also consistent with the he / n class ( banerjee & ashok , 2012 ) . the p cygni absorption features are clearly seen in the higher resolution irtf spectra obtained on 2015 march 23.625 ut . another notable feature seen in the irtf spectra is the presence of blue emission components in the profiles of pa @xmath38 , pa @xmath5 and br @xmath38 hi emission lines . the magnified sections of the selected lines from the irtf spectra are shown in figs [ fig_nsco_irtf ] and [ fig_nsco_irtf_magnified ] to highlight the p cygni features and the weak blue components . a detailed list of emission lines observed in the spectra is given in the appendix and table [ table_linelist ] . with the help of the present near - ir observations , we establish that the secondary component of nova sco 2015 is a late type cool giant star ( see section [ subsec_secondarynature ] ) . the evolution of the velocity profiles seen in the emission lines of nova sco 2015 spectra suggests and supports this possibility . the t crb sub class of recurrent novae ( rne ) with a giant cool red companion typically show a significant decrease in the width of the emission line profiles with time after outburst ( banerjee et al . this behavior is expected as the high velocity ejecta thrown out during the eruption moves through the wind of the companion and thereby undergoes a deceleration . such a deceleration causes a fast temporal decrease of the expansion velocity resulting in the narrowing of the emission line widths . this behavior has been well documented in the nir in the case of 4 other similar symbiotic systems viz . in the 2006 outburst of rs oph ( das , banerjee & ashok 2006 ) , in v407 cyg ( munari et al . 2011 ) where the donor is a high mass losing mira variable , in the recurrent nova v745 sco ( banerjee et al . 2014 ) and in nova sco 2014 ( joshi et al . 2015 ) . our near ir observations of nova sco 2015 show a similar behavior . fig [ pabeta_timeevol ] shows the evolution of the pa@xmath5 1.2818 @xmath13 m line profile during our observations . the narrowing of the line profile is clearly observed here . fig [ fig_nsco_fwhmvstime ] shows the time evolution of the observed line widths ( fwhm ) of the pa@xmath5 1.2818 @xmath13 m line . the intrinsic fwhm of the profiles have been obtained by deconvolving the observed profiles from instrumental broadening by assuming a gaussian profile for both the observed and instrumental profiles ( a reasonable assumption ) from which it follows that the fwhms will combine in quadrature ( @xmath85 + @xmath86 = @xmath87 ) . the fwhm of the instrumental profile for 2015 february 19 to 2015 march 8 data from nics on mt . abu telescope is measured to be 560 kms@xmath4 . for the 2015 march 23 data from the irtf telescope , the same is measured to be 150 kms@xmath4 from an argon lamp arc spectrum which is equivalent to the resolution of 2000 cited for spex . a power law fit to the evolving intrinsic line widths , of the form @xmath88 , is shown in fig [ fig_nsco_fwhmvstime ] which is seen to give a reasonable fit for a value of @xmath89 . the impact of the high - velocity nova ejecta with the wind of the giant companion is known to produce a strong shock which can heat the gas to high temperatures . this hot , shocked gas can be the site of hard x - rays ( sokolski et al . 2006 ; bode et al . 2006 ) and also @xmath38-ray production created by diffusive acceleration of particles across the shock to tev energies . the accelerated protons can subsequently either inverse - comptonize ambient low - energy radiation to the @xmath38 ray regime or participate in production of neutral pions which decay with the emission of gamma rays . the early x - ray and radio observations of nova sco 2015 by nelson et al ( 2015 ) implicate the presence of such a strong shock forming in the red giant wind . bode & kahn ( 1985 ) , for e.g. , have discussed the propagation of such a shock wave into the dense ambient medium surrounding the white dwarf . it may be described as a three stage process viz . a free expansion or ejecta dominated stage where the ejecta expands freely into the red giant wind . this phase lasts till the mass of the swept - up material from the donor wind is smaller than the mass of the the nova ejecta . a constant velocity of the shock is seen during this time . 2 . an adiabatic phase or sedov - taylor stage where the majority of the ejecta kinetic energy has been transferred to the swept - up ambient gas and there is negligible cooling by radiation losses . this phase is characterized by the temporal evolution of shock velocity @xmath59 as @xmath90 , assuming a @xmath91 dependence for the decrease in density of the wind . 3 . in phase 3 , the shocked material has cooled by radiation , and here the expected dependence of the shock velocity is @xmath92 . in the case of nova sco 2015 the free expansion stage , if it occurred in the first instance , is clearly missed . this is possibly because our observations began late viz . our earliest spectrum being recorded 7 days after the outburst . the deceleration that accompanies phases 2 or 3 is seen but the decay is too fast and the index @xmath93 that we get deviates substantially from that expected in either phase 2 or 3 . such deviations were also noticed in other recurrent novae as well e.g. rs oph ( das et al . 2006 ) , v745 sco ( banerjee et al . 2014 ) . this likely happens due to the propagation of ejecta into a non - symmetrical wind . in such cases , the ejecta would be slowed down more effectively in the parts moving in the direction of the giant due to the increasing density in that direction . in addition there could be anisotropic distribution of the material over the equatorial plane . thus as a combination , the mass distribution of the ejecta around the white dwarf would be anisotropic and the shock front would then propagate as an aspherical one ( chomiuk et al . 2012 , fig 6 therein ) . this nova possibly has a bipolar flow associated with it based on the description of the early optical spectrum by walter ( 2015 ) where , apart from the main central feature , symmetrically displaced emission features at about @xmath8 4500 kms@xmath4 were seen in the h@xmath0 profile . such a profile structure is typical of a bipolar flow and has been seen in quite a few novae viz . rs oph ( banerjee et al . 2009 ) , kt eri 2009 ( ribeiro et al , 2013 ; raj et al . 2013 ) , t pyx ( joshi et al . ( 2014 ) . from our own data , indication for an asymmetrical ejecta flow also comes from the velocity profiles of the pa@xmath38 @xmath94 m , pa@xmath5 @xmath95 m and br@xmath38 @xmath96 m lines shown in fig [ fig_nsco_velplot ] . here a weak blue component is seen in each of the profiles , separated from the principal profile , by -650 , -765 , -690 @xmath97 for pa@xmath38 , pa@xmath5 and the br @xmath38 lines respectively . this could be the blue symmetrically displaced component found by walter ( 2015 ) which has undergone considerable deceleration from its original value of -4000 km / s down to the present value of @xmath7 -600 to -700 km / s . the intensity of its counterpart red component could have dropped below detection limits . in short , indications are clearly present for deviations from spherical symmetry in the velocity kinematics seen in nova sco 2015 . this could be one of the reasons for the deviation of the deceleration index @xmath0 from model values . we have performed the recombination case b analysis following the same lines as done earlier for nova cep 2014 . we analyze observed spectra spanning 6 epochs covering the first 33 d of our observations . the measured line strengths from the flux calibrated spectra of n sco 2015 are given in table [ table_lineluminosity ] . flux calibrations of the spectra are done using near ir photometric observations from smarts consortium . fig [ nsco15_recombanalysis ] shows the observed relative line strengths for the brackett lines which have been normalized with respect to br12 set to unity . br22 and br23 line values ( as given in table [ table_lineluminosity ] ) are not considered for this analysis as they are not resolved properly . the predicted case b values are shown in fig [ nsco15_recombanalysis ] for a representative temperature 10000 k and for electron number densities of @xmath3 = 10@xmath98 , 10@xmath99 , 10@xmath100 , 10@xmath101,10@xmath102 and 10@xmath103 @xmath104 . as can be seen , in this nova also , the br@xmath38 line is weaker than expected implying it is optically thick . thus , using the same formalism described for nova cep 2014 , the emission measure @xmath2 is estimated to be in the range of @xmath55 to @xmath56 @xmath57 , the corresponding lower limit of the electron density @xmath3 is in the range @xmath105 @xmath51 to @xmath106 @xmath51 and the mass @xmath68 is between ( @xmath107 @xmath108 ) @xmath69 m@xmath75 where @xmath69 is the filling factor . the electron density and mass estimates are again reasonably consistent with expected values . @xmath30 ( small dash - dotted lines ) , 10@xmath31 @xmath30 ( solid lines ) , 10@xmath32 @xmath30 ( big dash lines ) , 10@xmath33 @xmath30 ( small dash lines ) , 10@xmath34 @xmath30 ( dotted lines ) , and 10@xmath35 @xmath30 ( dash - dotted lines ) . , scaledwidth=50.0% ] lllclll + emission line & & & line flux on & & & + and wavelength & & & days after outburst & & & + ( @xmath13 m ) & & & ( @xmath39 w/@xmath40 ) & & & + + & 7.16d & 8.16d & 9.17d & 14.18d & 15.18d & 40.0d + + + pa9 0.9226 & 414.0 & 396.0 & 301.0 & 102.0 & 171.0 & 2.07 + pa8 0.9546 & 158.0 & 182.0 & 145.0 & 41.1 & 22.4 & 1.36 + pa7 1.0049@xmath10 & 349.0 & 427.0 & 391.0 & 84.8 & 71.2 & 1.44 + hei + pa6 @xmath109 & 2890.0 & 2970.0 & 2380.0 & 624.0 & 568.0 & 9.79@xmath110 + oi 1.1287 & 246.0 & 194.0 & 185.0 & 49.5 & 23.8 & 3.31 + pa5 1.2818 & 680.0 & 862.0 & 635.0 & 161.0 & 158.0 & 3.95 + br22 + br23 @xmath111 & - & & 9.87 & - & 1.65 & 0.37 + br21 1.5133 & 5.62 & - & 7.71 & 5.42 & 2.04 & 0.20 + br20 1.5192 & 5.83 & 6.11 & 4.83 & 2.73 & - & 0.22 + br19 1.5261 & 10.9 & 11.5 & 7.19 & 4.85 & 1.88 & 0.22 + br18 1.5342 & 17.9 & 15.6 & 12.1 & 5.85 & 3.69 & 0.26 + br17 1.5439 & 32.9 & 25.9 & 19.3 & 11.2 & 4.81 & 0.23 + br16 1.5556 & 41.8 & 39.3 & 28.0 & 7.59 & 4.07 & 0.26 + br15 1.5701 & 46.6 & 58.5 & 36.7 & 8.15 & 6.76 & 0.31 + br14 1.5881 & 77.0 & 62.9 & 43.6 & 8.00 & 10.2 & 0.30 + br13 1.6109 & 89.3 & 58.2 & 55.3 & 10.8 & 5.59 & 0.30 + br12 1.6407 & 109.0 & 79.1 & 76.3 & 9.94 & 9.01 & 0.33 + br11 1.6807 & 123.0 & 93.2 & 81.9 & 12.0 & 6.56 & 0.40 + hei 1.7002 & 54.5 & 31.3 & 31.0 & 4.70 & 3.92 & 0.19 + br10 1.7362 & 201.0 & 167.0 & 110.0 & 16.1 & 13.4 & 0.44 + hei 2.0581 & 79.7 & 71.2 & 56.8 & 7.10 & - & 0.19 + hei 2.112 & 20.3 & 17.5 & 13.2 & 1.62 & - & 0.08 + br7 2.1655 & 125.0 & 162.0 & 87.5 & 22.5 & 8.59 & 0.63 + [ table_lineluminosity ] a : blended with other lines b : hei 1.0831 is blended with hi pa6 1.0938 c : the integrated line flux of hei and pa6 d : integrated line flux of br22 and br 23 our near - ir observations along with the archival data from 2mass survey indicate that nova sco 2015 presents a strong case to belong to the class of symbiotic system consisting of a wd and and a late type giant companion . as symbiotic systems are rare among novae , identification of a new object likely to belong to this group is of much significance . the bright near ir counterpart of nova sco 2015 from the 2mass archival database ( 2mass j17032617 - 3504178 ) is likely to be a symbiotic system based on its 2mass magnitudes of @xmath112 , @xmath113 & @xmath114 . these magnitudes are transformed to bessel & brett ( 1988 ) homogenized system using transformation equations given by carpenter ( 2001 ) and corrected for interstellar extinction using the relations given by rieke & lebofsky ( 1985 ) . the ir color indices are thus obtained as @xmath115 , and @xmath116 . these colors are consistent with the values of 0.68 and 0.14 expected for k3 iii / k4 iii respectively ( bessel & brett 1988 ) . on the other hand the sed in quiescence suggests a slightly different class for the secondary . for constructing the sed , the wavelength coverage was extended on the either size of the 2mass coverage by using wise w1 and w2 bands data and denis i band data . it may be noted that the denis magnitudes of @xmath117 = 13.45 @xmath8 0.07 and @xmath118 = 12.20 @xmath8 0.10 are in good agreement with the corresponding 2mass values of 13.40 @xmath8 0.03 and 12.22 @xmath8 0.03 respectively . the wise w3 and w4 magnitudes were not used for the sed because the star is not seen as a point source in the wise images in these bands , only the background diffuse ir cirrus is being picked up . the extremely low snr ( between 3 to 6 ) and the poor quality flags of the w3 and w4 data show that they should not be used . all magnitudes were corrected for interstellar extinction using a@xmath119 = 2.23 . the sed shown in fig [ nsco2015_sed ] , using i(0.79 @xmath13 m ) = 15.25 @xmath8 0.04 , w1(3.3 @xmath13 m ) = 11.24 @xmath8 0.03 and w2(4.6@xmath13 m ) = 11.46 @xmath8 0.03 , is well fit by a black body of temperature @xmath120 k , suggesting a spectral class of m4 - 5 iii . considering the spectral class of k3 iii / k4 iii suggested earlier from the near - infrared @xmath121 colors it appears that the determination of the spectral type of the companion is uncertain to few sub - classes @xmath122 at best we can say that range of the likely spectral class is either k or m. to have a more definitive classification , the spectral lines of the secondary must be recorded in a good snr spectrum in quiescence . further support for the secondary to be in the giant class comes from its absolute magnitude estimation . assuming that the quiescent k band brightness is dominated by the secondary ( i.e m@xmath123 of the secondary = 12.22 from 2mass ) and using the mmrd distance estimate to the nova of @xmath7 14.7 kpc , the k band absolute magnitude of the secondary , is calculated as @xmath124 . whereas using the intricsic color @xmath125 ( bessell & brett 1988 ) and absolute magnitude @xmath126 ( lang 1990 ) for k5 iii spectral class , the corresponding @xmath127 is determined as @xmath128 . as the two are in good agreement , it further strengthens the classification of the secondary as a red giant . on the other hand , the possibility of the companion being a dwarf instead of a giant can be ruled out from the following consideration . for a mid - k to mid - m spectral class dwarf as suggested by our previous analyses , the k absolute magnitudes is in the range of @xmath129 respectively ( pecaut & mamajek 2013 ; see online version of table 5 therein ) . using the distance modulus relation , the corresponding distance range comes out to be @xmath130 pc . this is severely in disagreement with the distance estimated earlier using mmrd relations . further for such a close distance range of @xmath130 pc , the extinction versus distance models of marshall et al . ( 2006 ) suggest that we should get an extinction value of 0.45 for a@xmath119 which is again inconsistent with the observed estimate of 2.23 . an additional confirmation for the symbiotic nature of nova sco 2015 comes from the comparison of its h@xmath0 and r band images . the supercosmos archive/ ] h@xmath0 image is considerably brighter than the r band counter part indicating the presence of strong h@xmath0 emission . the supercosmos values of h@xmath0 and short - r band magnitudes are 15.48 and 16.33 respectively , whereas the mean value of the ( h@xmath0 - short r ) magnitude for about 275 listed sources in a 3 arc - minute square field around the object is found to be @xmath131 . that is , the source is considerably bright in h@xmath0 . in fact , pronounced h i emission is used as one of the principal criteria for the classification of symbiotic stars ( belczynski et al . , 2000 ) . this can be seen from the spectra of symbiotic stars , for e.g. in the catalogue of munari & zwitter ( 2002 ) , wherein they are seen to exhibit strong h@xmath0 emission . however in addition to the presence of hi lines , the other criteria for a definitive symbiotic star classification requires the presence of higher excitation lines ( e.g. [ oiii ] , heii ) . after the system has returned to quiescence , and the nova ejecta faded , it may be checked whether such lines are seen . we estimate the outburst amplitude of nova sco 2015 by associating the optical counter part nomad-1 0549 - 0492872 with @xmath19 = 17.0 as the progenitor suggested by guido & howes ( 2015 ) . the @xmath19 = 9.492 near the maximum from the smarts database gives an outburst amplitude ( a ) of @xmath77.5 in the @xmath19 band which is relatively small . it lies @xmath7 4.5 magnitude below the outburst amplitude ( a ) vs log ( @xmath132 ) plot for classical novae ( warner 1995 ; fig 5.4 ) . in an extensive study of a large sample of classical novae and the recurrent novae pagnotta & schaefer ( 2014 ) have identified the characteristics common to recurrent novae to identify the potential recurrent novae among the known classical novae . the data discussed earlier shows that most of these characteristics , namely , small outburst amplitude , near ir colors resembling the colors of late type giant , expansion velocity exceeding 2000 kms@xmath4 and the presence of high excitation lines are fulfilled by nova sco 2015 , thus it presents a strong case for to be a potential recurrent nova . it is worth noting the similar case of nova sco 2014 , wherein joshi et al ( 2015 ) have shown that the outburst occurred in a symbiotic binary system and also suggested that it could be a recurrent nova . no @xmath38-ray emission was detected by the fermi observatory from nova sco 2015 or nova sco 2014 ( joshi et al . 2015 ) which are both similar systems . on the other hand , two other similar symbiotic systems were detected in @xmath38 rays viz . v407 cyg and v745 sco . the non - detections in nova sco 2014 and nova sco 2015 could be a consequence of both objects being sufficiently distant that any emission from them falls below the detection threshold of fermi . but it is desirable to check the fermi data from both these novae carefully for any weak or suggested signs of detection . this has bearings on the origin of @xmath38-ray emission from novae where the latest paradigm suggests that @xmath38-ray emission could be a generic property of all novae and not intrinsic to just symbiotic systems ( ackermann et al . novae from which @xmath38-ray emission has been detected , but which are not symbiotic systems , are nova sco 2012 , nova mon 2012 , nova del 2013 and nova cen 2013 . we present near - infrared photometric and spectroscopic observations of nova cep 2014 and nova sco 2015 which were discovered in outburst on 2014 march 8.79 ut and 2015 february 11.84 ut respectively . our observations for nova cep 2014 cover 9 epochs from 5 to 90 days after outburst and for nova sco 2015 cover 11 epochs covering 7 to 47 days after outburst . nova cep 2014 shows the conventional characteristics of a fe ii class characterized by strong ci lines together with hi and o i lines , whereas nova sco 2015 is classified as he / n class , shows strong he i emission lines together with hi and oi emission features . using mmrd relations for the novae , we estimate the distances for nova cep 2014 and nova sco 2015 as @xmath133 kpc and @xmath134 kpc respectively . for nova sco 2015 , the presence of a decelerative shock seen through a narrowing of the line profiles , presents a strong case for it to be a symbiotic system . we discuss the evolution of the strength and shape of the emission line profiles . the ejecta velocity shows a power law decay with time ( @xmath1 ) and case is presented for asymmetric ejecta flow in the winds of a cool giant companion star . the sed of the secondary in quiescence shows it to be a late cool type giant and the h@xmath0 excess seen from the system in quiescence is also indicative of the symbiotic nature of the system . constraints are put on the spectral type of the companion star . a case b recombination analysis shows the brackett lines to be optically thick . this in turn helps us to estimate , for nova sco 2015 , that the emission measure @xmath2 is in the range of @xmath55 to @xmath56 @xmath57 , the corresponding lower limit of the electron density @xmath3 is in the range @xmath105 @xmath51 to @xmath106 @xmath51 and the mass @xmath68 of the ejecta is between ( @xmath135 @xmath136)@xmath69 m@xmath75 where @xmath69 is the filling factor . for nova cep 2014 , the corresponding estimates are @xmath55 to @xmath56 @xmath57 for the emission measure , @xmath62 @xmath51 to @xmath63 @xmath51 for the electron density and ( @xmath73 @xmath74)@xmath69 m@xmath75 for the mass of the ejecta where @xmath69 is the filling factor . the research work at the physical research laboratory is funded by the department of space , government of india 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of presolar materials . , new york , p. 203 storey p. j. , hummer d. g. , 1995 , mnras , 272 , 41 vacca , w. d. , cushing , m. c. , rayner , j. t. , 2003 , pasp , 115 , 389 van den berg s. , younger p.s . , 1987 , a&as , 70 , 125 walter f. , astron . telegram , 7060,1 warner b. , 1995 , cataclysmic variable stars . cambridge astrophysics series , cambridge univ . press , cambridge , new york ; williams , r.e . , 2012 , aj , 144 , 98 most of the nir lines that appear in the spectra of novae have been identified in das et al ( 2008 ) . however , the spectra presented there were in the 1.08 to 2.4 @xmath13 m region , whereas the present spectra are taken with a newer instrument extend up to 0.85 @xmath13 m . a robust identification of the numerous lines that appear in the 0.85 to 1.08 @xmath13 m ( ij band region ) is thus desirable . to identify the lines that contribute to a nova s spectrum , we use an lte model to build synthetic spectra as in das et al ( 2008 ) and ashok & banerjee ( 2003 ) . assumptions of lte may not strictly prevail in an nova environment although , around maximum and the early decline stage , when the particle density can be high ( up to even @xmath138 ) collisions will be a dominant mechanism and will tend to drive the gas towards a boltzmann distribution and lte . yet , in spite of the limitations of the lte assumption we find that the model - generated spectra , greatly aid in a more secure identification of the lines observed . briefly ( more details in das et al , 2008 ) the model spectra are generated by considering only those elements whose lines can be expected at discernible strength . since nucleosynthesis calculations of elemental abundances in novae ( starrfield et al . 1997 ; jose @xmath137 hernanz , 1998 ) show that h , he , c , o , n , ne , mg , na , al , si , p , s are the elements with significant yields in novae ejecta , only these elements have been considered . the saha ionization equation was applied to calculate the fractional percentage of the species in different ionization stages and subsequently the boltzman equation was applied to calculate level populations . by switching off or greatly increasing the abundance of an element , it is easy to identify the positions where the lines of that element disappear or build up . fig [ fig_ncep_lineid ] shows the ij band spectrum of nova cep 2014 of 2014 march 20 in black and a typical synthetic lte spectrum in gray below . the lte spectrum has been computed for @xmath3 = 10@xmath139 @xmath30 , @xmath140 = 8000k and abundances typically found in co novae as given in starrfield et al . ( 1997 ) and jose @xmath137 hernanz ( 1998 ) . a total of @xmath7 2500 of the strongest lines were considered for these elements compiled from the kurucz atomic line list and national institute of standards and technology ( nist ) line list database . based on the line identifications done here , and lines in novae spectra known from earlier studies ( williams , 2012 ; das et al . 2008 ) , the observed line list is given in the table [ table_linelist ] . lrrrrlrrrr + wavelength & species & other con- & nova cep & nova sco & wavelength & species & other con- & nova cep & nova sco + ( @xmath141 m ) & & tributors/ & & & ( @xmath141 m ) & & tributors/ & & + & & remarks & & & & & remarks & & + 0.8359 & hi pa22 & & & x & 1.2527 & he i & & & x + 0.8374 & hi pa 21 & & & x & 1.2562,1.2569 & c i & & x & + 0.8392 & hi pa20 & & & x & 1.2620 & c i & & x & + 0.8413 & hi pa 19 & & & x & 1.2755 & u.i & & & x + 0.8446 & o i & pa18 0.8438 & & x & 1.2763 & u.i & & & x + 0.8467 & pa17 & & & x & 1.2818 & hi pa5 & & x & x + 0.8498 & ca ii & pa16 0.8502 & & x & 1.2963 & u.i & & & x + 0.8542 & ca ii & pa15 0.8545 & & x & 1.3164 & o i & & x & x + 0.8598 & hi pa14 & & & x & 1.34 - 1.38 & n i & blend of many & x & + & & & & & & & ni lines & & + 0.8665 & hi pa13 & ca ii 0.8662 & & x & 1.4420 & c i & & x & + 0.8680 & ni & & x & & 1.4543 & c i & & x & + 0.8750 & hi pa12 & & & x & 1.4539 & u.i & ci 1.4543 ? & & x + 0.8802 & u.i & 0.8807 mg i ? & & x & 1.4757 & n i & & x & + 0.8863 & hi pa11 & & & x & 1.4906 & hi br27 & & & x + 0.8909 & u.i & & & x & 1.4938 & hi br26 & & & x + 0.8923 & u.i & & & x & 1.4967 & hi br25 & & & x + 0.9015 & hi pa10 & & & x & 1.5000 & hi br24 & & & x + 0.9021 & ni & & x & & 1.5039 & hi br23 & & & x + 0.9089 & ci & & x & & 1.5083 & hi br22 & & & x + 0.9174 & u.i & & & x & 1.5133 & h1 br21 & & & x + 0.9226 & hi pa9 & & & x & 1.5192 & hi br20 & & x & x + 0.9264 & oi & & x & & 1.5261 & hi br19 & & x & x + 0.9396 & ni & & x & & 1.5342 & hi br18 & & x & x + 0.9406 & ci & & x & & 1.5439 & hi br17 & & x & x + 0.9402 & u.i & ci 0.9406 ? & & x & 1.5556 & hi br16 & & x & x + 0.9546 & hi pa8 & & & x & 1.5701 & hi br15 & & x & x + 0.9863 , 0.9872 & ni & & x & & 1.5881 & hi br14 & & x & x + 0.9993 & u.i & & & x & 1.6005 & c i & & x & + 1.0049 & hi pa7 & & & x & 1.6109 & hi br13 & & x & x + 1.0112 & ni & ci 1.0119 & x & & 1.6407 & hi br12 & & x & x + 1.0124 & he ii & & & x & 1.6807 & hi br11 & & x & x + 1.0308 & u.i & & & x & 1.6872 & fe ii & & & x + 1.0399 & u.i & & x & & 1.6890 & c i & & x & x + 1.0457 & u.i & & & x & 1.7002 & he i & & & x + 1.0497 & u.i & & & x & 1.7362 & hi br10 & & x & x + 1.0534 & ni & & x & & 1.7200 - 1.7900 & c i & blend of many & x & + & & & & & & & ci lines & & + 1.0685 & c i & & x & x & 1.7413 & feii & & & x + 1.0831 & he i & & x & x & 1.8174 & hi br9 & & & x + 1.0938 & hi pa6 & & x & x & 1.9446 & hi br8 & & & x + 1.1287 & o i & & x & x & 1.9722 & c i & & x & + 1.1330 & c i & & x & & 2.0581 & he i & & x & x + 1.1659 & c i & & x & & 2.0703 & u.i & & & x + 1.1753 & c i & & x & x & 2.1023 & c i & & x & + 1.1828 & mg i & & & x & 2.1120 , 21132 & he i & & & x + 1.1880,1.1896 & c i & & x & x & 2.1156 - 2.1295 & ci & blend of & x & + & & & & & & & ci lines & & + 1.1969 & he i & & & x & 2.1361 & u.i & & & x + 1.2028 & u.i & & & x & 2.1425 & u.i & & & x + 1.2249,1.2264 & c i & & x & & 2.1655 & hi br7 & & x & x + 1.2281 & u.i & & & x & 2.2906 & c i & & x & + 1.2461 , 1.2469 & n i & & x & x & 2.3438 - 2.4945 & hi pf30 to 17 & & & x +

we present multi - epoch near - infrared photo - spectroscopic observations of nova cephei 2014 and nova scorpii 2015 , discovered in outburst on 2014 march 8.79 ut and 2015 february 11.84 ut respectively . nova cep 2014 shows the conventional nir characteristics of a fe ii class nova characterized by strong ci , hi and o i lines , whereas nova sco 2015 is shown to belong to the he / n class with strong he i , hi and oi emission lines . the highlight of the results consists in demonstrating that nova sco 2015 is a symbiotic system containing a giant secondary . leaving aside the t crb class of recurrent novae , all of which have giant donors , nova sco 2015 is shown to be only the third classical nova to be found with a giant secondary . the evidence for the symbiotic nature is three - fold ; first is the presence of a strong decelerative shock accompanying the passage of the nova s ejecta through the giant s wind , second is the h@xmath0 excess seen from the system and third is the spectral energy distribution of the secondary in quiescence typical of a cool late type giant . the evolution of the strength and shape of the emission line profiles shows that the ejecta velocity follows a power law decay with time ( @xmath1 ) . a case b recombination analysis of the h i brackett lines shows that these lines are affected by optical depth effects for both the novae . using this analysis we make estimates for both the novae of the emission measure @xmath2 , the electron density @xmath3 and the mass of the ejecta . [ firstpage ] infrared : spectra - line : identification - stars : novae , cataclysmic variables - stars : individual nova cephei 2014 , nova scorpii 2015 - techniques : spectroscopic , photometric .

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Proceed to summarize the following text: most astrophysical systems , e.g. accretion disks , stellar winds , the interstellar medium ( ism ) and intercluster medium are turbulent with an embedded magnetic field that influences almost all of their properties . this turbulence which spans from km to many kpc ( see discussion in armstrong , rickett , & spangler 1995 ; scalo 1987 ; lazarian , pogosyan , & esquivel 2002 ) holds the key to many astrophysical processes . for instance , propagation of cosmic rays and their acceleration is strongly affected by mhd turbulence . recent research has shown that a substantial part of the earlier results in the field require revision . earlier research used ad hoc models of mhd turbulence and this entailed erroneous conclusions . before we start a discussion of mhd turbulence let us recall some basic properties of the hydrodynamic turbulence . all turbulent systems have one thing in common : they have a large reynolds number " ( @xmath0 ; l= the characteristic scale or driving scale of the system , v = the velocity difference over this scale , and @xmath1=viscosity ) , the ratio of the viscous drag time on the largest scales ( @xmath2 ) to the eddy turnover time of a parcel of gas ( @xmath3 ) . a similar parameter , the magnetic reynolds number " , @xmath4 ( @xmath5 ; @xmath6=magnetic diffusion ) , is the ratio of the magnetic field decay time ( @xmath7 ) to the eddy turnover time ( @xmath3 ) . the properties of the flows on all scales depend on @xmath8 and @xmath4 . flows with @xmath9 are laminar ; chaotic structures develop gradually as @xmath8 increases , and those with @xmath10 are appreciably less chaotic than those with @xmath11 . observed features such as star forming clouds and accretion disks are very chaotic with @xmath12 and @xmath13 . let us start by considering incompressible hydrodynamic turbulence , which can be described by the kolmogorov theory ( kolmogorov 1941 ) . suppose that we excite fluid motions at a scale @xmath14 . we call this scale the _ energy injection scale _ or the _ largest energy containing eddy scale_. for instance , an obstacle in a flow excites motions on scales of the order of its size . then the energy injected at the scale @xmath14 cascades to progressively smaller and smaller scales at the eddy turnover rate , i.e. @xmath15 , with negligible energy losses along the cascade .. ] ultimately , the energy reaches the molecular dissipation scale @xmath16 , i.e. the scale where the local @xmath17 , and is dissipated there . the scales between @xmath14 and @xmath16 are called the _ inertial range _ and it typically covers many decades . the motions over the inertial range are _ self - similar _ and this provides tremendous advantages for theoretical description . the beauty of the kolmogorov theory is that it does provide a simple scaling for hydrodynamic motions . if the velocity at a scale @xmath18 from the inertial range is @xmath19 , the kolmogorov theory states that the kinetic energy ( @xmath20 as the density is constant ) is transferred to next scale within one eddy turnover time ( @xmath21 ) . thus within the kolmogorov theory the energy transfer rate ( @xmath22 ) is scale - independent , @xmath23 and we get the famous kolmogorov scaling @xmath24 the one - dimensional , which , for isotropic turbulence , is given by @xmath25 . ] energy spectrum @xmath26 is the amount of energy between the wavenumber @xmath27 and @xmath28 divided by @xmath29 . when @xmath26 is a power law , @xmath30 is the energy _ near _ the wavenumber @xmath31 . since @xmath32 , kolmogorov scaling implies @xmath33 kolmogorov scalings were the first major advance in the theory of incompressible turbulence . they have led to numerous applications in different branches of science ( see monin & yaglom 1975 ) . however , astrophysical fluids are magnetized and the a dynamically important magnetic field should interfere with eddy motions . paradoxically , astrophysical measurements are consistent with kolmogorov spectra ( see lpe02 ) . for instance , interstellar scintillation observations indicate an electron density spectrum is very close to @xmath34 for @xmath35 - @xmath36 ( see armstrong et al . 1995 ) . at larger scales lpe02 summarizes the evidence of -@xmath37 velocity power spectrum over pc - scales in hi . solar - wind observations provide _ in - situ _ measurements of the power spectrum of magnetic fluctuations and leamon et al . ( 1998 ) also obtained a slope of @xmath38 -@xmath37 . is this a coincidence ? what properties is the magnetized compressible ism expected to have ? we will deal with these questions , and some related issues , below . here we discuss a focused approach which aims at obtaining a clear understanding on the fundamental level , and considering physically relevant complications later . the creative synthesis of both approaches is the way , we think , that studies of astrophysical turbulence should proceed . certainly an understanding of mhd turbulence in the most ideal terms is a necessary precursor to understanding the complications posed by more realistic physics and numerical effects . for review of general properties of mhd , see a recent book by biskamp ( 1993 ) . in what follows , we first consider observational data that motivate our study ( 2 ) , then discuss theoretical approaches to incompressible mhd turbulence ( 3 ) . we move to the effects of compressibility in 4 and discuss implications of our new understanding of mhd turbulence for cosmic ray dynamics in 5 . we present the summary in 6 . kolmogorov turbulence is the simplest possible model of turbulence . since it is incompressible and not magnetized , it is completely specified by its velocity spectrum . if a passive scalar field , like `` dye particles '' or temperature inhomogeneities , is subjected to kolmogorov turbulence , the resulting spectrum of the passive scalar density is also kolmogorov ( see lesieur 1990 ; warhaft 2000 ) . in compressible and magnetized turbulence this is no longer true , and a complete characterization of the turbulence requires not only a study of the velocity statistics but also the statistics of density and magnetic fluctuations . direct studies of turbulence have been done mostly for interstellar medium and for the solar wind . while for the solar wind _ in - situ _ measurements are possible , studies of interstellar turbulence require inverse techniques to interpret the observational data . attempts to study interstellar turbulence with statistical tools date as far back as the 1950s ( von horner 1951 ; kamp de friet 1955 ; munch 1958 ; wilson et al . 1959 ) and various directions of research achieved various degree of success ( see reviews by kaplan & pickelner 1970 ; dickman 1985 ; armstrong et al . 1995 ; lazarian 1999a , 1999b ; lpe02 ) . solar wind ( see review goldstein & roberts 1995 ) studies allow pointwise statistics to be measured directly using spacecrafts . these studies are the closest counterpart of laboratory measurements . the solar wind flows nearly radially away from the sun , at up to about 700 km / s . this is much faster than both spacecraft motions and the alfvn speed . therefore , the turbulence is `` frozen '' and the fluctuations at frequency @xmath39 are directly related to fluctuations at the scale @xmath27 in the direction of the wind , as @xmath40 , where @xmath41 is the solar wind velocity ( horbury 1999 ) . usually two types of solar wind are distinguished , one being the fast wind which originates in coronal holes , and the slower bursty wind . both of them show , however , @xmath42 scaling on small scales . the turbulence is strongly anisotropic ( see klein et al . 1993 ) with the ratio of power in motions perpendicular to the magnetic field to those parallel to the magnetic field being around 30 . the intermittency of the solar wind turbulence is very similar to the intermittency observed in hydrodynamic flows ( horbury & balogh 1997 ) . studies of turbulence statistics of ionized media ( see spangler & gwinn 1990 ) have provided information on the statistics of plasma density at scales @xmath43-@xmath44 cm . this was based on a clear understanding of processes of scintillations and scattering achieved by theorists ( see narayan & goodman 1989 ; goodman & narayan 1985 ) . a peculiar feature of the measured spectrum ( see armstrong et al . 1995 ) is the absence of the slope change at the scale at which the viscosity by neutrals becomes important . scintillation measurements are the most reliable data in the `` big power law '' plot in armstrong et al . however there are intrinsic limitations to the scintillations technique due to the limited number of sampling directions , its relevance only to ionized gas at extremely small scales , and the impossibility of getting velocity ( the most important ! ) statistics directly . therefore with the data one faces the problem of distinguishing actual turbulence from static density structures . moreover , the scintillation data does not provide the index of turbulence directly , but only shows that the data are consistent with kolmogorov turbulence . whether the ( 3d ) index can be -4 instead of -11/3 is still a subject of intense debate ( higdon 1984 ; narayan & goodman 1989 ) . in physical terms the former corresponds to the superposition of random shocks rather than eddies . additional information on the electron density is contained in the faraday rotation measures of extragalactic radio sources ( see simonetti & cordes 1988 ; simonetti 1992 ) . however , there is so far no reliable way to disentangle contributions of the magnetic field and the density to the signal . we feel that those measurements may give us the magnetic field statistics when we know the statistics of electron density better . spectral line data cubes are unique sources of information on interstellar turbulence . doppler shifts due to supersonic motions contain information on the turbulent velocity field which is otherwise difficult to obtain . moreover , the statistical samples are extremely rich and not limited to discrete directions . in addition , line emission allows us to study turbulence at large scales , comparable to the scales of star formation and energy injection . however , the problem of separating velocity and density fluctuations within hi data cubes is far from trivial ( lazarian 1995 , 1999b ; lazarian & pogosyan 2000 ; lpe02 ) . the analytical description of the emissivity statistics of channel maps ( velocity slices ) in lazarian & pogosyan ( 2000 ) ( see also lazarian 1999b ; lpe02 for reviews ) shows that the relative contribution of the density and velocity fluctuations depends on the thickness of the velocity slice . in particular , the power - law asymptote of the emissivity fluctuations changes when the dispersion of the velocity at the scale under study becomes of the order of the velocity slice thickness ( the integrated width of the channel map ) . these results are the foundation of the velocity - channel analysis ( vca ) technique which provides velocity and density statistics using spectral line data cubes . the vca has been successfully tested using data cubes obtained via compressible magnetohydrodynamic simulations and has been applied to galactic and small magellanic cloud atomic hydrogen ( hi ) data ( lazarian et al . 2001 ; lazarian & pogosyan 2000 ; stanimirovic & lazarian 2001 ; deshpande , dwarakanath , & goss 2000 ) . furthermore , the inclusion of absorption effects ( lazarian & pogosyan 2002 ) has increased the power of this technique . finally , the vca can be applied to different species ( co , h@xmath45 etc . ) which should further increase its utility in the future . within the present discussion a number of results obtained with the vca are important . first of all , the small magellanic cloud ( smc ) hi data exhibit a kolmogorov - type spectrum for velocity and hi density from the smallest resolvable scale of 40 pc to the scale of the smc itself , i.e. 4 kpc . similar conclusions can be inferred from the galactic data ( green 1993 ) for scales of dozens of parsecs , although the analysis has not been done systematically . deshpande et al . ( 2000 ) studied absorption of hi on small scales toward cas a and cygnus a. within the vca their results can be interpreted as implying that on scales less than 1 pc the hi velocity is suppressed by ambipolar drag and the spectrum of density fluctuations is shallow @xmath46 . such a spectrum ( deshpande 2000 ) can account for the small scale structure of hi observed in absorption . magnetic field statistics are the most poorly constrained aspect of ism turbulence . the polarization of starlight and of the far - infrared radiation ( fir ) from aligned dust grains is affected by the ambient magnetic fields . assuming that dust grains are always aligned with their longer axes perpendicular to magnetic field ( see the review lazarian 2000 ) , one gets the 2d distribution of the magnetic field directions in the sky . note that the alignment is a highly non - linear process in terms of the magnetic field and therefore the magnetic field strength is not available . the statistics of starlight polarization ( see fosalba et al . 2002 ) is rather rich for the galactic plane and it allows to establish the spectrum @xmath47 , where @xmath48 is a two dimensional wave vector describing the fluctuations over sky patch .. for sufficiently small areas of the sky analyzed the multipole analysis results coincide with the fourier analysis . ] for uniformly sampled turbulence it follows from lazarian & shutenkov ( 1990 ) that @xmath49 for @xmath50 and @xmath51 for @xmath52 , where @xmath53 is the critical angular size of fluctuations which is proportional to the ratio of the injection energy scale to the size of the turbulent system along the line of sight . for kolmogorov turbulence @xmath54 . however , the real observations do not uniformly sample turbulence . many more close stars are present compared to the distant ones . thus the intermediate slops are expected . indeed , cho & lazarian ( 2002b ) showed through direct simulations that the slope obtained in fosalba et al . ( 2002 ) is compatible with the underlying kolmogorov turbulence . at the moment fir polarimetry does not provide maps that are really suitable to study turbulence statistics . this should change soon when polarimetry becomes possible using the airborne sofia observatory . a better understanding of grain alignment ( see lazarian 2000 ) is required to interpret the molecular cloud magnetic data where some of the dust is known not to be aligned ( see lazarian , goodman , & myers 1997 and references therein ) . another way to get magnetic field statistics is to use synchrotron emission . both polarization and intensity data can be used . the angular correlation of polarization data ( baccigalupi et al . 2001 ) shows the power - law spectrum @xmath55 and we believe that the interpretation of it is similar to that of starlight polarization . indeed , faraday depolarization limits the depth of the sampled region . the intensity fluctuations were studied in lazarian & shutenkov ( 1990 ) with rather poor initial data and the results were inconclusive . cho & lazarian ( 2002b ) interpreted the fluctuations of synchrotron emissivity ( giardino et al . 2001 , 2002 ) in terms of turbulence with kolmogorov spectrum . attempts to describe magnetic turbulence statistics were made by iroshnikov ( 1963 ) and kraichnan ( 1965 ) . their model of turbulence ( ik theory ) is isotropic in spite of the presence of the magnetic field . for simplicity , let us suppose that a uniform external magnetic field ( @xmath56 ) is present . in the incompressible limit , any magnetic perturbation propagates _ along _ the magnetic field line . since wave packets are moving along the magnetic field line , there are two possible directions for propagation . if all the wave packets are moving in one direction , then they are stable to nonlinear order ( parker 1979 ) . therefore , in order to initiate turbulence , there must be opposite - traveling wave packets and the energy cascade occurs only when they collide . the ik theory starts from this observation , one of the consequences of which is the modification of the energy cascade timescale : @xmath57 , where @xmath58 is alfven velocity of the mean field . here , the ik theory assumes that opposite - traveling isotropic wave packets of similar size interact . from this and the scale - invariance of energy cascade rate , they obtained @xmath59 however , the presence of the uniform magnetic component has non - trivial dynamical effects on the turbulence fluctuations . one obvious effect is that it is easy to mix field lines in directions perpendicular to the local mean magnetic field and much more difficult to bend them . the ik theory assumes isotropy of the energy cascade in fourier space , an assumption which has attracted severe criticism ( montgomery & turner 1981 ; shebalin , matthaeus , & montgomery 1983 ; montgomery & matthaeus 1995 ; sridhar & goldreich 1994 ; matthaeus et al . mathematically , anisotropy manifests itself in the resonant conditions for 3-wave interactions : @xmath60 where @xmath61 s are wavevectors and @xmath62 s are wave frequencies . the first condition is a statement of wave momentum conservation and the second is a statement of energy conservation . alfvn waves satisfy the dispersion relation : @xmath63 , where @xmath64 is the component of wavevector parallel to the background magnetic field . since only opposite - traveling wave packets interact , @xmath65 and @xmath66 must have opposite signs . then from equations ( [ k123 ] ) and ( [ w123 ] ) , either @xmath67 or @xmath68 must be equal to 0 and @xmath69 must be equal to the nonzero initial parallel wavenumber . that is , zero frequency modes are essential for energy transfer ( shebalin et al . therefore , in the wavevector space , 3-wave interactions produce an energy cascade which is strictly perpendicular to the mean magnetic field . however , in real turbulence , equation ( [ w123 ] ) does not need to be satisfied exactly , but only to within an an error of order @xmath70 ( goldreich & sridhar 1995 ) . this implies that the energy cascade is not strictly perpendicular to @xmath56 , although clearly very anisotropic . we assume throughout this discussion that the rms turbulent velocity at the energy injection scale is comparable to the alfvn speed of the mean field and consider only scales below the energy injection scale . this is called _ strong _ turbulence regime . note that , as a consequence , the regime of @xmath71 is not considered in this review . however , the regime of @xmath72 is still relevant to the strong turbulence regime because scales below the energy equipartition scale is expected to fall in the strong turbulence regime ( cho & vishniac 2000a ) . an ingenious model very similar in its beauty and simplicity to the kolmogorov model has been proposed by goldreich & sridhar ( 1995 ; hereinafter gs95 ) for incompressible strong mhd turbulence . they pointed out that motions perpendicular to the magnetic field lines mix them on a hydrodynamic time scale , i.e. at a rate @xmath73 , where @xmath74 is the wavevector component perpendicular to the local mean magnetic field and @xmath75 . these mixing motions couple to the wave - like motions parallel to magnetic field giving a _ critical balance _ condition @xmath76 where @xmath64 is the component of the wavevector parallel to the local magnetic field . when the typical @xmath64 on a scale @xmath74 falls below this limit , the magnetic field tension is too weak to affect the dynamics and the turbulence evolves hydrodynamically , in the direction of increasing isotropy in phase space . this quickly raises the value of @xmath64 . in the opposite limit , when @xmath64 is large , the magnetic field tension dominates , the error @xmath77 in the matching conditions is reduced , and the nonlinear cascade is largely in the @xmath74 direction , which restores the critical balance . if conservation of energy in the turbulent cascade applies locally in phase space then the energy cascade rate ( @xmath78 ) is constant : @xmath79 combining this with the critical balance condition we obtain an anisotropy that increases with decreasing scale @xmath80 and a kolmogorov - like spectrum for perpendicular motions @xmath81 which is not surprising since the magnetic field does not influence motions that do not bend it . at the same time , the scale - dependent anisotropy reflects the fact that it is more difficult for the weaker , smaller eddies to bend the magnetic field . gs95 shows the duality of motions in mhd turbulence . those perpendicular to the mean magnetic field are essentially eddies , while those parallel to magnetic field are waves . the critical balance condition couples these two types of motions . numerical simulations ( cho & vishniac 2000b ; maron & goldreich 2001 ; cho , lazarian , & vishniac 2002 ) show reasonable agreements with the gs95 model . .notations for compressible turbulence [ cols= " < , < " , ] [ cho_table1 ] for the rest of the review , we consider mhd turbulence of a single conducting fluid . while the gs95 model describes incompressible mhd turbulence well , no universally accepted theory exists for compressible mhd turbulence despite various attempts ( e.g. , higdon 1984 ) . earlier numerical simulations of compressible mhd turbulence covered a broad range of astrophysical problems , such as the decay of turbulence ( e.g. mac low 1998 ; stone , ostriker , & gammie 1998 ) or turbulent modeling of the ism ( see recent review vazquez - semadeni 2002 ; see also passot , pouquet , & woodward 1988 ; vazquez - semadeni , passot , & pouquet 1995 ; passot , vazquez - semadeni , & pouquet 1995 ; vazquez - semadeni , passot , & pouquet 1996 for earlier pioneering 2d simulations and ostriker , gammie , & stone 1999 ; ostriker , stone , & gammie 2001 ; padoan et al . 2001 ; klessen 2001 ; boldyrev 2002 for recent 3d simulations ) . in what follows , we concentrate on the fundamental properties of compressible mhd . and @xmath82 represent the directions of displacement of fast and slow modes , respectively . in the fast basis ( @xmath83 ) is always between @xmath84 and @xmath85 . in the slow basis ( @xmath82 ) lies between @xmath86 and @xmath87 . here , @xmath86 is perpendicular to @xmath84 and parallel to the wave front . all vectors lie in the same plane formed by @xmath88 and @xmath89 . on the other hand , the displacement vector for alfvn waves ( not shown ) is perpendicular to the plane . ( b ) directions of basis vectors for a very small @xmath90 drawn in the same plane as in ( a ) . the fast bases are almost parallel to @xmath85 . ( c ) directions of basis vectors for a very high @xmath90 . the fast basis vectors are almost parallel to @xmath89 . the slow waves become pseudo - alfvn waves . ] lies between @xmath86 and @xmath91 and @xmath83 between @xmath84 and @xmath85 . again , @xmath86 is perpendicular to @xmath84 and parallel to the wave front . note also that , for the fast wave , for example , density ( inferred by the directions of the displacement vectors ) becomes higher where field lines are closer , resulting in a strong restoring force , which is why fast waves are faster than slow waves . ] let us start by reviewing different mhd waves . in particular , we describe the fourier space representation of these waves . the real space representation can be found in papers on modern shock - capturing mhd codes ( e.g. brio & wu 1988 ; ryu & jones 1995 ) . for the sake of simplicity , we consider an isothermal plasma . figure [ fig_modes ] and figure [ fig_modes - real ] give schematics of slow and fast waves . for slow and fast waves , @xmath88 , @xmath92 ( @xmath93 ) , and @xmath89 are in the same plane . on the other hand , for alfvn waves , the velocity of the fluid element @xmath94 is orthogonal to the @xmath95 plane . as before , the alfvn speed is @xmath96 where @xmath97 is the average density . fast and slow speeds are @xmath98^{1/2},\ ] ] where @xmath99 is the angle between @xmath88 and @xmath89 . see table [ cho_table1 ] for the definition of other variables . when @xmath90 ( @xmath100=@xmath101 ; @xmath102= gas pressure , @xmath103= magnetic pressure ; hereinafter @xmath104 average @xmath105 ) goes to zero , we have @xmath106 figure [ fig_modes ] shows directions of displacement ( or , directions of velocity ) vectors for these three modes . we will call them the basis vectors for these modes . the alfvn basis is perpendicular to both @xmath84 and @xmath87 , and coincides with the azimuthal vector @xmath107 in a spherical - polar coordinate system . here hatted vectors are unit vectors . the fast basis @xmath108 lies _ between _ @xmath84 and @xmath85 : @xmath109 ^ 2 k_{\| } \hat{\bf k}_{\| } + k_{\perp } \hat{\bf k}_{\perp},\ ] ] where @xmath110 , and @xmath111 is the averaged @xmath90 ( = @xmath112 ) . the slow basis @xmath113 lies _ between _ @xmath114 and @xmath87 ( = @xmath115 ) : @xmath116 ^ 2 k_{\perp } \hat{\bf k}_{\perp}.\ ] ] the two vectors @xmath108 and @xmath113 are mutually orthogonal . proper normalizations are required for both bases to make them unit - length . when @xmath90 goes to zero ( i.e. the magnetically dominated regime ) , @xmath108 becomes parallel to @xmath85 and @xmath113 becomes parallel to @xmath87 ( fig . [ fig_modes]b ) . the sine of the angle between @xmath87 and @xmath113 is @xmath117 . when @xmath90 goes to infinity ( i.e. gas pressure dominated regime ) ) but finite , so that @xmath118 means the gas pressure @xmath119 . ] , @xmath108 becomes parallel to @xmath84 and @xmath113 becomes parallel to @xmath114 ( fig . [ fig_modes]c ) . this is the incompressible limit . in this limit , the slow mode is sometimes called the pseudo - alfvn mode ( goldreich & sridhar 1995 ) . here we address the issue of mode coupling in a low @xmath90 plasma . it is reasonable to suppose that in the limit where @xmath120 turbulence for mach numbers ( @xmath121 ) less than unity should be largely similar to the exactly incompressible regime . thus , lithwick & goldreich ( 2001 ) conjectured that the gs95 relations are applicable to slow and alfvn modes with the fast modes decoupled . they also mentioned that this relation can carry on for low @xmath90 plasmas . for @xmath122 and @xmath123 , we are in the regime of hydrodynamic compressible turbulence for which no theory exists , as far as we know . in the diffuse interstellar medium @xmath90 is typically less than unity . in addition , it is @xmath124 or less for molecular clouds . there are a few simple arguments suggesting that mhd theory can be formulated in the regime where the alfvn mach number ( @xmath125 ) is less than unity , although this is not a universally accepted assumption . alfvn modes describe incompressible motions . arguments in gs95 are suggestive that the coupling of alfvn to fast and slow modes will be weak . consequently , we expect that in this regime the alfvn cascade should follow the gs95 scaling . moreover the slow waves are likely to evolve passively ( lithwick & goldreich 2001 ) . for @xmath126 their nonlinear evolution should be governed by alfvn modes so that we expect the gs95 scaling for them as well . the phase velocity of alfvn waves and slow waves depend on a factor of @xmath127 and this enables modulation of the slow waves by the alfvn ones . however , fast waves do not have this factor and therefore can not be modulated by the changes of the magnetic field direction associated with alfvn waves . the coupling between the modes is through the modulation of the local alfvn velocity and therefore is weak . for alfvn mach number ( @xmath128 ) larger than unity a shock - type regime is expected . however , generation of magnetic field by turbulence ( cho & vishniac 2000a ) is expected for such a regime . it will make the steady state turbulence approach @xmath129 . therefore in cho & lazarian ( 2002a ) we consider turbulence in the limit @xmath123 , @xmath130 , and @xmath131 . for these simulations , we mostly used @xmath132 , @xmath133 , and @xmath134 . the alfvn speed of the mean external field is similar to the rms velocity ( @xmath135 ) , and we used an isothermal equation of state . although the scaling relations presented below are applicable to sub - alfvnic turbulence , we cautiously speculate that small scales of super - alfvnic turbulence might follow similar scalings . boldyrev , nordlund , & padoan ( 2001 ) obtained energy spectra close to @xmath136 in solenoidally driven super - alfvnic supersonic turbulence simulations . the spectra are close to the kolmogorov spectrum ( @xmath137 ) , rather than shock - dominated spectrum ( @xmath138 ) . this result might imply that small scales of super - alfvnic mhd turbulence can be described by our sub - alfvnic model presented below , which predicts kolmogorov - type spectra for alfvn and slow modes . alfvn modes are not susceptible to collisionless damping ( see spangler 1991 ; minter & spangler 1997 and references therein ) that damps slow and fast modes . therefore , we mainly consider the transfer of energy from alfvn waves to compressible mhd waves ( i.e. to the slow and fast modes ) . , @xmath139 . ( _ right _ ) the ratio of @xmath140 to @xmath141 . the stronger the external field ( @xmath142 ) is , the more suppressed the coupling is . the ratio is not sensitive to @xmath90 . from cho & lazarian ( 2002a ) , title="fig : " ] , @xmath139 . ( _ right _ ) the ratio of @xmath140 to @xmath141 . the stronger the external field ( @xmath142 ) is , the more suppressed the coupling is . the ratio is not sensitive to @xmath90 . from cho & lazarian ( 2002a ) , title="fig : " ] in cho & lazarian ( 2002a ) , we carry out simulations to check the strength of the mode - mode coupling . we first obtain a data cube from a driven compressible numerical simulation with @xmath143 . then , after turning off the driving force , we let the turbulence decay . we go through the following procedures before we let the turbulence decay . we first remove slow and fast modes in fourier space and retain only alfvn modes . we also change the value of @xmath88 preserving its original direction . we use the same constant initial density @xmath97 for all simulations . we assign a new constant initial gas pressure @xmath102 . and @xmath102 preserve the alfvn character of perturbations . in fourier space , the mean magnetic field ( @xmath88 ) is the amplitude of @xmath144 component . alfvn components in fourier space are for @xmath145 and their directions are parallel / anti - parallel to @xmath146 (= @xmath147 ) . the direction of @xmath146 does not depend on the magnitude of @xmath142 or @xmath102 . ] after doing all these procedures , we let the turbulence decay . we repeat the above procedures for different values of @xmath142 and @xmath102 . [ fig_coupling]a shows the evolution of the kinetic energy of a simulation . the solid line represents the kinetic energy of alfvn modes . it is clear that alfvn waves are poorly coupled to the compressible modes , and do not generate them efficiently therefore , we expect that alfvn modes will follow the same scaling relation as in the incompressible case . [ fig_coupling]b shows that the coupling gets weaker as @xmath142 increases : @xmath148 the ratio of @xmath149 to @xmath150 is proportional to @xmath151 . ) for alfven velocity shows anisotropy similar to the gs95 . conturs represent eddy shapes . from cho & lazarian ( 2002a ) . , title="fig : " ] ) for alfven velocity shows anisotropy similar to the gs95 . conturs represent eddy shapes . from cho & lazarian ( 2002a ) . , title="fig : " ] this marginal coupling is in good agreement with a claim in gs95 , as well as earlier numerical studies where the velocity was decomposed into a compressible component @xmath152 and a solenoidal component @xmath153 . the compressible component is curl - free and parallel to the wave vector @xmath89 in fourier space . the solenoidal component is divergence - free and perpendicular to @xmath89 . the ratio @xmath154 is an important parameter that determines the strength of any shock ( passot et al . 1988 ; pouquet 1999 ) . porter , woodward , & pouquet ( 1998 ) performed a hydrodynamic simulation of decaying turbulence with an initial sonic mach number of unity and found that @xmath155 evolves toward @xmath156 . matthaeus et al . ( 1996 ) carried out simulations of decaying weakly compressible mhd turbulence ( zank & matthaeus 1993 ) and found that @xmath157 , where @xmath158 is the sonic mach number . in boldyrev ( 2001 ) a weak generation of compressible components in solenoidally driven super - alfvnic supersonic turbulence simulations was obtained . [ fig_alf ] shows that the spectrum and the anisotropy of alfvn waves in this limit are compatible with the gs95 model : @xmath159 and scale - dependent anisotropy @xmath160 that is compatible with the gs95 theory . slow waves are somewhat similar to pseudo - alfvn waves ( in the incompressible limit ) . first , the directions of displacement ( i.e. @xmath161 ) of both waves are similar when anisotropy is present . the vector @xmath162 is always between @xmath114 and @xmath115 . in figure [ fig_modes ] , we can see that the angle between @xmath114 and @xmath115 gets smaller when @xmath163 . therefore , when there is anisotropy ( i.e. @xmath163 ) , @xmath113 of a low @xmath90 plasma becomes similar to that of a high @xmath90 plasma . second , the angular dependence in the dispersion relation @xmath164 is identical to that of pseudo - alfvn waves ( the only difference is that , in slow waves , the sound speed @xmath165 is present instead of the alfvn speed @xmath166 ) . goldreich & sridhar ( 1997 ) argued that the pseudo - alfvn waves are slaved to the shear - alfvn ( i.e. ordinary alfvn ) waves in the presence of a strong @xmath88 , meaning that the energy cascade of pseudo - alfvn modes is primarily mediated by the shear - alfven waves . this is because pseudo - alfvn waves do not provide efficient shearing motions . similar arguments are applicable to slow waves in a low @xmath90 plasma ( cho & lazarian 2002a ) ( see also lithwick & goldreich 2001 for high-@xmath90 plasmas ) . as a result , we conjecture that slow modes follow a scaling similar to the gs95 model ( cho & lazarian 2002a ) : @xmath167 fig . [ fig_slow]a shows the spectra of slow modes . for velocity , the slope is close to @xmath168 . [ fig_slow]b shows the contours of equal second - order structure function ( @xmath169 ) of slow velocity , which are compatible with @xmath160 scaling . in low @xmath90 plasmas , density fluctuations are dominated by slow waves ( cho & lazarian 2002a ) . from the continuity equation @xmath170 @xmath171 we have , for slow modes , @xmath172 hence , this simple argument implies @xmath173 where we assume that @xmath174 and @xmath158 is the mach number . on the other hand , only a small amount of magnetic field is produced by the slow waves . similarly , using the induction equation ( @xmath175 ) , we have @xmath176 which means that equipartition between kinetic and magnetic energy is not guaranteed in low @xmath90 plasmas . in fact , in fig . [ fig_slow]a , the power spectrum for density fluctuations has a much larger amplitude than the magnetic field power spectrum . since density fluctuations are caused mostly by the slow waves and magnetic fluctuation is caused mostly by alfvn and fast modes , we _ do not _ expect a strong correlation between density and magnetic field , which agrees with the ism simulations ( padoan & nordlund 1999 ; ostriker et al . 2001 ; vazquez - semadeni 2002 ) . figure [ fig_fast ] shows fast modes are isotropic . the resonance conditions for interacting fast waves are : @xmath177 since @xmath178 for the fast modes , the resonance conditions can be met only when all three @xmath89 vectors are collinear . this means that the direction of energy cascade is _ radial _ in fourier space , and we expect an isotropic distribution of energy in fourier space . using the constancy of energy cascade and uncertainty principle , we can derive an ik - like energy spectrum for fast waves . the constancy of cascade rate reads @xmath179 on the other hand , @xmath180 can be estimated as @xmath181 if contributions are random , the denominator can be written by the square root of the number of interactions ( @xmath182 ) times strength of individual interactions ( @xmath183 ) . , where @xmath99 is the angle between @xmath89 and @xmath88 . thus marginal anisotropy is expected . it will be investigated elsewhere . ] here we assume locality of interactions : @xmath184 . due to the uncertainly principle , the number of interactions becomes @xmath185 , where @xmath186 is the typical transversal ( i.e. not radial ) separation between two wave vectors @xmath187 and @xmath188 ( with @xmath189 ) . therefore , the denominator of equation ( [ eq14_fast ] ) is @xmath190 . we obtain an independent expression for @xmath180 from the uncertainty principle ( @xmath191 with @xmath192 ) . from this and equation ( [ eq14_fast ] ) , we get @xmath193 which yields @xmath194 combining equations ( [ cas_rate_fast ] ) and ( [ t_cascade ] ) , we obtain @xmath195 or @xmath196 . this is very similar to acoustic turbulence , turbulence caused by interacting sound waves ( zakharov 1967 ; zakharov & sagdeev 1970 ; lvov , lvov , & pomyalov 2000 ) . zakharov & sagdeev ( 1970 ) found @xmath197 . however , there is debate about the exact scaling of acoustic turbulence . here we cautiously claim that our numerical results are compatible with the zakharov & sagdeev scaling : @xmath198 magnetic field perturbations are mostly affected by fast modes ( cho & lazarian 2002a ) when @xmath90 is small : @xmath199 if @xmath200 . the turbulent cascade of fast modes is expected to be slow and in the absence of collisionless damping they are expected to propagate in turbulent media over distances considerably larger than alfvn or slow modes . this effect is difficult to observe in numerical simulations with @xmath201 . a modification of the spectrum in the presence of the collisionless damping is presented in yan & lazarian ( 2002 ) . many astrophysical problems require some knowledge of the scaling properties of turbulence . therefore we expect a wide range of applications of the established scaling relations . here we show how recent progress in understanding mhd turbulence affects cosmic ray propagation . the propagation of cosmic rays is mainly determined by their interactions with electromagnetic fluctuations in interstellar medium . the resonant interaction of cosmic ray particles with mhd turbulence has been repeatedly suggested as the main mechanism for scattering and isotropizing cosmic rays . in these analysis it is usually assumed that the turbulence is _ isotropic _ with a kolmogorov spectrum ( see schlickeiser & miller 1998 ) . how should these calculations be modified ? consider resonance interaction first . particles moving with velocity @xmath41 get into resonance with mhd perturbations propagating along the magnetic field if the resonant condition is fulfilled , namely , @xmath202 ( @xmath203 ) , where @xmath204 is the wave frequency , @xmath205 is the gyrofrequency of relativistic particle , @xmath206 , where @xmath207 is the pitch angle of particles . in other words , resonant interaction between a particle and the transverse electric field of a wave occurs when the doppler shifted frequency of the wave in the particle s guiding center rest frame @xmath208 is a multiple of the particle gyrofrequency . for cosmic rays , @xmath209 , so the slow variation of the magnetic field with time can be neglected . thus the resonant condition is simply @xmath210 . from this resonance condition , we know that the most important interaction occurs at @xmath211 . it is intuitively clear that resonant interaction of particles in isotropic and anisotropic turbulence should be different . chandran ( 2001 ) obtained strong suppression of scattering by alfvenic turbulence when he treated turbulence anisotropies in the spirit of goldreich - sridhar model of incompressible turbulence . his treatment was improved in yan & lazarian ( 2002 , henceforth yl02 ) who used a more rigorous description of magnetic field statistics . moreover , they took into account cr scattering by compressible mhd modes and found that fast modes absolutely dominate cosmic ray scattering . in our description we shall follow yl02 treatment of the problem . we employ quasi - linear theory ( qlt ) to obtain our estimates . qlt has been proved to be a useful tool in spite of its intrinsic limitations ( chandran 2000 ; schlickeiser & miller 1998 ; miller 1997 ) . for moderate energy cosmic rays , the corresponding resonant scales are much smaller than the injection scale . therefore the fluctuation on the resonant scale @xmath212 even if they are comparable at the injection scale . qlt disregards diffusion of cosmic rays that follow wandering magnetic field lines ( jokipii 1966 ) and this diffusion should be accounted separately . obtained by applying the qlt to the collisionless boltzmann - vlasov equation , the fokker - planck equation is generally used to describe the evolvement of the gyrophase - average distribution function @xmath213 , @xmath214,\ ] ] where @xmath215 is particle momentum . the fokker - planck coefficients @xmath216 are the fundamental physical parameter for measuring the stochastic interactions , which are determined by the electromagnetic fluctuations ( schlickeiser & achatz 1993 ) . from ohm s law @xmath217 we can get the electromagnetic fluctuations from correlation tensors of magnetic and velocity fluctuations @xmath218 @xmath219 . here , @xmath220 for alfven modes , cho , lazarian and vishniac ( 2002 ) obtained @xmath221 where @xmath222 is a 2d matrix in x - y plane , @xmath223 is the wave vector along the local mean magnetic field , @xmath224 is the wave vector perpendicular to the magnetic field and the normalization constant @xmath225 . we assume that for the alfven modes @xmath226 @xmath227 where the fractional helicity @xmath228 is independent of @xmath229 ( chandran 2000 ) . according to cho & lazarian ( 2002a ) , fast modes are isotropic and have one dimensional spectrum @xmath197 . in low @xmath230 medium , the velocity fluctuations are always perpendicular to @xmath231 for all @xmath229 , while the magnetic fluctuations are perpendicular to @xmath229 . thus @xmath232 @xmath233 of fast modes are not equal , @xmath234={\frac{l^{-1/2}}{8\pi } } j_{ij}k^{-7/2}\left[\begin{array}{c } \cos ^{2}\theta \\ \sigma \cos ^{2}\theta \\ 1\end{array } \right],\label{fast_tensor_lowb}\ ] ] where @xmath235 is also a 2d tensor in @xmath236 plane . @xmath237 are 3d matrixes . however , the third dimension is not needed for our calculations . @xmath233 is different from that in schlickeiser & miller ( 1998 ) . the fact that the fluctuations @xmath238 in fast modes are in the @xmath229-@xmath239 plane place another constrain on the tensor so that the term @xmath240 does nt exist . ] in high @xmath230 medium , the velocity fluctuations are radial , i.e. , along the direction of @xmath241 . fast modes in this regime are essentially sound waves compressing magnetic field ( gs95 ; lithwick & goldreich 2001 , cho & lazarian , in preparation ) . the compression of magnetic field depends on plasma @xmath230 . the corresponding x - y components of the tensors are @xmath234={\frac{l^{-1/2}}{8\pi } } \sin ^{2}\theta j_{ij}k^{-7/2}\left[\begin{array}{c } \cos ^{2}\theta /\beta \\ \sigma \cos \theta /\beta ^{1/2}\\ 1\end{array } \right].\label{fast_tensor_highb}\ ] ] adopting the approach in schlickeiser & achatz ( 1993 ) , we can obtain the fokker - planck coefficients in the lowest order approximation of @xmath242 , @xmath243={\frac{\omega ^{2}(1-\mu ^{2})}{2b_{0}^{2}}}\left[\begin{array}{c } 1\\ mc\\ m^{2}c^{2}\end{array } \right]{\mathcal{r}}e\sum _ { n=-\infty } ^{n=\infty } \int _ { k_{min}}^{k_{max}}dk^{3 } & & \nonumber \\ \int _ { 0}^{\infty } dte^{-i(k_{\parallel } v_{\parallel } -\omega + n\omega ) t}\left\ { j_{n+1}^{2}({\frac{k_{\perp } v_{\perp } } { \omega } } ) \left[\begin{array}{c } p_{{\mathcal{rr}}}({\mathbf{k}})\\ t_{{\mathcal{rr}}}({\mathbf{k}})\\ r_{{\mathcal{rr}}}({\mathbf{k}})\end{array } \right]\right . & & \nonumber \\ + j_{n-1}^{2}({\frac{k_{\perp } v_{\perp } } { \omega } } ) \left[\begin{array}{c } p_{{\mathcal{ll}}}({\mathbf{k}})\\ -t_{{\mathcal{ll}}}({\mathbf{k}})\\ r_{{\mathcal{ll}}}({\mathbf{k}})\end{array } \right]+j_{n+1}({\frac{k_{\perp } v_{\perp } } { \omega } } ) j_{n-1}({\frac{k_{\perp } v_{\perp } } { \omega } } ) & & \nonumber \\ \left.\left[e^{i2\phi } \left[\begin{array}{c } -p_{{\mathcal{rl}}}({\mathbf{k}})\\ t_{{\mathcal{rl}}}({\mathbf{k}})\\ r_{{\mathcal{rl}}}({\mathbf{k}})\end{array } \right]+e^{-i2\phi } \left[\begin{array}{c } -p_{{\mathcal{lr}}}({\mathbf{k}})\\ -t_{{\mathcal{lr}}}({\mathbf{k}})\\ r_{{\mathcal{lr}}}({\mathbf{k}})\end{array } \right]\right]\right\ } & & \label{genmu}\end{aligned}\ ] ] where @xmath244 , @xmath245 corresponds to the dissipation scale , @xmath246 is the relativistic mass of the proton , @xmath247 is the particle s velocity component perpendicular to @xmath231 , @xmath248 @xmath249 represent left and right hand polarization , the expression is only true for alfv\en modes . there are additional compressional terms for compressable modes . ] . noticing that the integrand for small @xmath224 is substantially suppressed by the exponent in the anisotropic tensor ( see eq . ( [ anisotropic ] ) ) so that the large scale contribution is not important , we can simply use the asymptotic form of bessel function for large argument . then if the pitch angle @xmath207 is not close to 0 , we can derive the analytical result for anisotropic turbulence ( yl02 ) , @xmath250=\frac{v^{2.5}\cos \alpha ^{5.5}}{2\omega ^{1.5}l^{2.5}\sin \alpha } \gamma [ 6.5,k_{max}^{-2/3}k_{res}l^{1/3}]\left[\begin{array}{c } 1\\ \sigma mv_{a}\\ m^{2}v_{a}^{2}\end{array } \right],\label{ana}\ ] ] where @xmath251 is the injection sale , @xmath245 corresponds to the dissipation scale , @xmath252 $ ] represents the incomplete gamma function . the scattering frequency @xmath253 is plotted for different models in fig.([fig : incom]a ) . it is clear that anisotropy suppresses scattering . although our results are larger than those obtained in chandran ( 2001 ) using an _ ad hoc _ tensor with a step function , they are still much smaller than the estimates for isotropic model . unless we consider very high energy crs ( @xmath254 ) with corresponding larmor radius comparable to the injection scale , we can neglect scattering by the alfvnic turbulence . what is the alternative way to scatter cosmic rays ? for compressible modes we discuss two types of resonant interaction : gyroresonance and transit - time damping ; the latter requires longitudinal motions . the contribution from slow modes is not larger than that by alfvn modes since the slow modes have the similar anisotropies and scalings . more promising are fast modes , which are isotropic ( cho & lazarian 2002a ) . however , fast modes are subject to collisionless damping if the wavelength is smaller than the proton mean free path or by viscous damping if the wavelength is larger than the mean free path . according to cl02 , fast modes cascade over time scales @xmath255 where @xmath256 is the eddy turn - over time , @xmath257 is the turbulence velocity at the injection scale . consider gyroresonance scattering in the presence of collisionless damping . the cutoff of fast modes corresponds to the scale where @xmath258 and this defines the cutoff scale @xmath259 . using the tensors given in eq . ( [ fast_tensor_lowb ] ) we obtain the corresponding fokker - planck coefficients for the crs interacting with fast modes by integrating eq.([genmu ] ) from @xmath260 to @xmath261 ( see fig.([fig : incom]b ) ) . when @xmath259 is less than @xmath262 , the results of integration for damped and undamped turbulence coincides . since the damping increases with @xmath230 , the scattering frequency decreases with @xmath230 . adopting the tensors given in eq . ( [ fast_tensor_highb ] ) , it is possible to calculate the scattering frequency of crs in high @xmath230 medium . for instance , for density @xmath263@xmath264 temperature @xmath265k , magnetic field @xmath266 g , the mean free path is smaller than the resonant wavelength for the particles with energy larger than @xmath267 , therefore collisional damping rather than landau damping should be taken into account . nevertheless , our results show that the fast modes still dominate the crs scattering in spite of the viscous damping . apart from the gyroresonance , fast modes potentially can scatter crs by transit - time damping ( ttd ) ( schlickeiser & miller 1998 ) . ttd happens due to the resonant interaction with parallel magnetic mirror force @xmath268 . for small amplitude waves , particles should be in phase with the wave so as to have a secular interaction with wave . this gives the cherenkov resonant condition @xmath269 , corresponding to the @xmath270 term in eq.([genmu ] ) . from the condition , we see that the contribution is mostly from nearly perpendicular propagating waves ( @xmath271 ) . according to eq . ( [ fast_tensor_lowb]),we see that the corresponding correlation tensor for the magnetic fluctuations @xmath233 are very small , so the contribution from ttd to scattering is not important . self - confinement due to the streaming instability has been discussed by different authors(see cesarsky 1980 , longair 1997 ) as an effective alternative to scatter crs and essential for cr acceleration by shocks . however , we will discuss in our next paper that in the presence of the turbulence the streaming instability will be partially suppressed owing to the nonlinear interaction with the background turbulence . thus the gyroresonance with the fast modes is the principle mechanism for scattering cosmic rays . this demands a substantial revision of cosmic ray acceleration / propagation theories , and many related problems may need to be revisited . for instance , our results may be relevant to the problems of the boron to carbon abundances ratio . we shall discuss the implications of the new emerging picture elsewhere . recently there have been significant advances in the field of compressible mhd turbulence and its implications to cosmic ray transport : 1 . simulations of compressible mhd turbulence show that there is a weak coupling between alfvn waves and compressible mhd waves and that the alfvn modes follow the goldreich - sridhar scaling . fast modes , however , decouple and exhibit isotropy . 2 . scattering of cosmic rays by alfvenic modes is suppressed and therefore the scattering by fast modes is the dominant process provided that turbulent energy is injected at large scales . 3 . the scattering frequency by fast modes depends on collisionless damping or viscous damping and therefore on plasma @xmath230 . * acknowledgments : * we acknowledge the support of the nsf through grant ast-0125544 . this work was partially supported by national computational science alliance under ast000010n and utilized the ncsa sgi / cray origin2000 . lazarian , a. 1999b , in plasma turbulence and energetic particles in astrophysics , ed . m. ostrowski & r. schlickeiser ( cracow , poland : obserwatorium astronomiczne , uniwersytet jagiellonski ) , 28 ( astro - ph/0001001 ) stone , j. m. , ostriker , e. c. , & gammie , c. f. 1998 , apj , 508 , l99 swordy , s. p. , 2001 , space science rev . , 99 , 85 vazquez - semadeni , e. 2002 , in seeing through the dust , ed . r. taylor , t. landecker , & a. willis ( san francisco : asp ) ( astro - ph/0201072 )

turbulence is the most common state of astrophysical flows . in typical astrophysical fluids , turbulence is accompanied by strong magnetic fields , which has a large impact on the dynamics of the turbulent cascade . recently , there has been a significant breakthrough on the theory of magnetohydrodynamic ( mhd ) turbulence . for the first time we have a scaling model that is supported by both observations and numerical simulations . we review recent progress in studies of both incompressible and compressible turbulence . we compare iroshnikov - kraichnan and goldreich - sridhar models , and discuss scalings of alfvn , slow , and fast waves . we discuss the implications of this new insight into mhd turbulence for cosmic ray transport .

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Proceed to summarize the following text: a general theory of games first introduced in @xcite has found several applications in the field of economics and engineering . a solution concept or a notion of equilibrium was proposed by nash ( known as nash equilibrium ) in @xcite and was shown to exist in every finite normal - form game . further generalizations of nash equilibrium such as correlated equilibrium and coarse correlated equilibrium were also introduced and studied . it is well known that for every game the set correlated and coarse - correlated equilibria are convex subsets of the strategy space . but in general the set of nash equilibria is not convex . a number of methods have been proposed to compute a nash equilibium strategy . lemke - howson s algorithm for bi - matrix games@xcite , global newton method@xcite , homotopy based methods@xcite are some of the few methods to compute a nash equilibrium strategy . for a general n - player game , the associated optimization problem is non - linear and non - convex and hence is difficult to solve . it is known that the problem of computing nash equilibria in bi - matrix games is a linear complementarity problem and for the general n - player scenario it is a non - linear complementarity problem . linear complimentarity problems ( the ones arising from games ) can be solved using lemke - howson s method , while non - linear complimentarity problems are in general hard to solve and require some sufficient conditions to be imposed on the problem to solve them which is not satisfied by every game . in this paper we present optimization problems with biconvex objective function and linear constraints such that the set of global minima of the optimization problems is the same as the set of nash eqilibria of a n - player general - sum normal form game . global optimization algorithms exist that can compute the global minima of such optimization problems@xcite . the main idea in the formulation of these optimization problems is the fact that correlated or coarse - correlated equilibrium which are product of individual player s strategy is a nash equilibrium . we further show that the objective function is an invex function i.e. the set of stationary points is the same as the set of global minima . we also consider a projected gradient descent scheme and prove that is converges to a partial optimum of the objective function . the remainder of this paper is organised as follows : in section 2 , necessary definitions and notations are stated . in section 3 , functions with required properties are defined . in section 4 , properties of the functions defined in section 2 are proved . in section 5 , optimization problems are presented . in section 6 , the projected gradient descent algorithm is stated and convergence analysis is performed . in section 7 , simulation results of the projected gradient descent algorithm on certain test cases are presented . in section 8 , we summarize and present directions for future research . in this section we shall state definitions , introduce variables and notations used later in this paper . a normal form game ( or simply a game ) ( @xmath0 ) is defined by tuple @xmath1 where , @xmath2 denotes the set of players ( @xmath3 ) , @xmath4 denotes the set of actions of player @xmath5 ( @xmath6 ) . let @xmath7 and @xmath8 denotes the utility function of player @xmath5 . for every @xmath9 , @xmath10 denotes the set of probability distributions on @xmath11 . @xmath10 is identified by the probability simplex @xmath12 . @xmath13 denotes a generic element of @xmath10 . let @xmath14 which is identified as a vector in @xmath15 where @xmath16 . let @xmath17 . let @xmath18 denote the set of probability distributions on @xmath19 . @xmath18 is identified by the probability simplex @xmath20 where @xmath21 . @xmath22 denotes a generic element in @xmath18 . for every @xmath9 , @xmath23 and @xmath24 denotes a generic element in @xmath25 . similarly , this can be extended to more than one player . @xmath26 . similarly define @xmath27 and @xmath28 denote a generic element in @xmath29 . @xmath30 . for every @xmath9 , @xmath31 where @xmath32 . for every @xmath9 , @xmath33 where @xmath34 . for every @xmath9 , @xmath35 and @xmath36 . similarly define @xmath37 . @xmath38 is said to be a * nash equilibrium strategy of the game @xmath0 ( or just n.e . ) if @xmath39 . let @xmath40 denote the set of nash equilibria strategies of game @xmath0 . * @xmath41 is said to be a * correlated equilibrium strategy of the game @xmath0 ( or just c.e . ) if @xmath42 . let @xmath43 denote the set of correlated equilibria of the game @xmath0 . * @xmath41 is said to be a * coarse correlated equilibrium strategy of the game @xmath0 ( or just c.c.e . ) if @xmath44 . let @xmath45 denote the set of coarse correlated equilibria of the game @xmath0 . * define @xmath46 , s.t . , @xmath47 where @xmath48 . let the graph of the function @xmath49 be @xmath50 . in the following lemma we summarize the relationship between the various equilibrium concepts defined.*lemma 2.1 : given a game @xmath0 . the following hold . * * @xmath51 . * @xmath52 . * @xmath53 . the results in lemma follow directly from definitions . @xmath54 is a * nash equilibrium profile of game @xmath0 if @xmath55 is a nash equilibrium strategy of game @xmath0 and @xmath56 . * let @xmath57 and @xmath58 be two convex subsets of @xmath59 and @xmath60 respectively . a function @xmath61 is said to be a * biconvex function if @xmath62 is a convex function and @xmath63 is a convex function . @xmath64 is a * partial optimum of a biconvex function @xmath65 if @xmath66 and @xmath67 . for a detailed study of biconvex functions see @xcite . * * let @xmath68 be a subset of @xmath69 and @xmath70 . @xmath71 is said to the global optimum of the optimization problem @xmath72 , if , @xmath73 . in this section we shall define functions whose set of zeros is the same as the set of nash equilibria of the game @xmath0 . the following theorem gives a necessary and sufficient condition for @xmath74 to be in @xmath75 . * theorem 3.1 : given @xmath74 . then , @xmath76 iff @xmath77 . * proof : [ @xmath78 assume @xmath76 . fix @xmath79 . then@xmath80 and @xmath81 . therefore @xmath82 . since @xmath83 are arbitrary , @xmath84 , @xmath85 . [ @xmath86 fix @xmath87 . from data , we know that @xmath88 . using the above , we get , @xmath89 . from data , we also know that @xmath90 . therefore by substituting for the sum , we get , @xmath91 . similarly repeating the above procedure for actions of the third player we get , @xmath92 . proceeding all the way upto player @xmath93 we get , @xmath94 . since @xmath41 , we know that @xmath95 . therefore , @xmath96 . since @xmath97 is arbitrary , @xmath98.@xmath99 * * using the above theorem we now define a non - negative function on @xmath100 such that the function takes the value zero on @xmath75 and is positive on @xmath101 . let @xmath102 such that , @xmath103 . * corollary 3.1 : given @xmath74 . then , @xmath104 iff @xmath76 . * from the definitions of coarse - correlated equilibrium and correlated equilibrium we now define the following non - negative functions on @xmath18 such that they take the value zero on the set of coarse - correlated equilibria ( @xmath45 ) and correlated equilibria ( @xmath43 ) respectively . let @xmath105 , such that , @xmath106 and @xmath107 , such that , @xmath108 . * lemma 3.1 : given @xmath109 . * * @xmath110 iff @xmath111 . * @xmath112 iff @xmath113 . * proof : follows directly from the definitions of correlated equilibrium and coarse correlated equilibrium in section 2.@xmath99 * let @xmath114 s.t . the idea is that when @xmath116 and @xmath117 , then , @xmath118 is a best response to @xmath28 . * lemma 3.2 : given @xmath116 . @xmath117 iff @xmath55 is a nash equilibrium.*proof : [ @xmath78since @xmath117 , we have , @xmath119 . hence @xmath120 . since @xmath121 , @xmath122 and @xmath123 . therefore , @xmath124 , which by definition of a nash equilibrium strategy in section 2 , implies @xmath55 is nash equilibrium . * * [ @xmath86 since @xmath55 is a nash equilibrium , we have , @xmath124 . since @xmath121 , @xmath122 and @xmath123 . therefore , @xmath120 , which further implies , @xmath119 . thus @xmath117.@xmath99 we now characterise the set of nash equilibria of a game ( @xmath0 ) using the functions @xmath125 and @xmath126 . * theorem 3.2 : given @xmath54 . * * @xmath127 is a nash equilibrium profile iff @xmath128 . * @xmath127 is a nash equilibrium profile iff @xmath129 . * @xmath127 is a nash equilibrium profile iff @xmath130 . * proof : first we shall prove ( 1).[@xmath78 assume @xmath127 is a nash equilibrium . then , by definition of nash equilibrium profile in section 2 , @xmath55 is a n.e . and @xmath56 . by lemma 2.1 , since @xmath55 is a n.e . @xmath131 and since @xmath56 , @xmath121 . thus @xmath104 and @xmath110 by theorem 3.1 and lemma 3.1 respectively . therefore @xmath128.assume @xmath128 . since both @xmath132 and @xmath133 are non - negative , @xmath104 and @xmath110 . by theorem 3.1 , @xmath104 will imply @xmath121 and by lemma 3.1 @xmath110 will imply @xmath111 . since @xmath134 and @xmath56 , from lemma 2.1 , we have that @xmath55 is a n.e . thus @xmath127 is a nash equilibrium . * proof of ( 2 ) is similar to that of ( 1 ) and the proof of ( 3 ) follows from lemma 3.2 and corollary 3.1.@xmath99 in this section we shall prove certain properties of the functions constructed in section * no . first , we shall prove that @xmath132 is biconvex and that @xmath133 and @xmath126 are convex . * lemma 4.1 : @xmath132 is a biconvex function i.e. @xmath135 is convex and @xmath136 is convex.*proof : @xmath137 where @xmath48 , @xmath138 is a linear function of @xmath41 and an affine function of @xmath139 . by proposition 1.1.4 in @xcite , @xmath140 is convex in @xmath41 and @xmath38 with the other fixed . since sum of convex functions is convex , @xmath141 is convex in @xmath142 for every fixed @xmath38 and is convex in @xmath55 for every fixed @xmath143.@xmath99 * lemma 4.2 : @xmath144 and @xmath145 are convex functions of @xmath41.*proof : first we shall show @xmath133 is convex . @xmath146 , @xmath147 is linear in @xmath41 . since supremum of convex functions is convex , we have , @xmath146 , @xmath148 . since composition of nondecreasing function and convex function is convex , @xmath146 , @xmath149 , is convex . therefore , @xmath150 is a convex function . * * * * * similarly we can show that @xmath145 is also a convex function.@xmath99 it is easy to show @xmath151 and @xmath145 are continuously differentiable on an open set containing their respective domains ( for a similar proof refer @xcite ) . let @xmath152^t$ ] , where @xmath153 and @xmath154 . for every @xmath155 , @xmath156\end{aligned}\ ] ] so as to compute @xmath157 , we shall write @xmath158 , where @xmath159 s.t . @xmath160 ( which is possible since @xmath138 is linear in @xmath142 ) . therefore , @xmath161 the following lemma says that set of partial optima of @xmath132 , the set of stationary points of @xmath132 and the set of global minima of @xmath132 are all the same.*lemma 4.3 : given @xmath162 . then the following are equivalent . * * @xmath163 is a partial optimum of @xmath132 . * @xmath163 is s.t . @xmath104 . * @xmath163 is s.t . @xmath164 . * proof : [ @xmath165 . since @xmath163 is a partial optimum of @xmath132 , @xmath166 . hence , @xmath167 . therefore , @xmath168 . * [ @xmath169 . since @xmath104 , @xmath77 . substituting the above in the expression of @xmath170 and @xmath157 we get , @xmath164 . [ @xmath171 . since @xmath132 is biconvex ( from lemma 4.1 ) , @xmath172 and @xmath173 are convex functions . from proposition 1.1.7 in @xcite , we get , @xmath174 and @xmath175 . substituting @xmath176^t=0 $ ] , will give , @xmath177 and @xmath178 . thus , @xmath163 is a partial optimum of @xmath132.@xmath99 so as to compute @xmath179 , we shall write @xmath180 where @xmath181 ( which is possible since @xmath147 is linear in @xmath142 ) . then @xmath182 . the following lemma says that the set of global minima of @xmath133 and the set of stationary points of @xmath133 are the same . * lemma 4.4 : given @xmath183 . @xmath184 iff @xmath185 . * proof : follows directly from the expression of the gradient and the convexity of @xmath133.@xmath99 * * a similar result can be derived for @xmath126 . in what follows in this paper results proved for @xmath133 can be extended to @xmath126 as well . in theorem 3.2 we showed that the set of zeros of @xmath186 is the same as the set of nash equilibrium profiles of the game @xmath0 . in the following lemma we show that the set of zeros of @xmath186 is the same as the set of stationary points of the function @xmath186 . * lemma 4.5 : given @xmath187 . @xmath188 iff @xmath189 . * proof : [ @xmath78 since @xmath188 and that @xmath132 and @xmath133 are non - negative , will imply that @xmath168 and @xmath184 . thus , @xmath176^t=0 $ ] and @xmath185 by lemma 4.3 and 4.4 respectively . therefore , @xmath190^t=0 $ ] . * * [ @xmath86since @xmath190^t=0 $ ] , we have @xmath191 . @xmath192 . by substituting the expressions for @xmath193 and @xmath194 we get , @xmath195 and @xmath196 . therefore , @xmath197.@xmath99 lemma 4.5 shows that the function @xmath186 is invex . similarly it can shown that @xmath198 is also invex . in following lemma we show that @xmath199 is a biconvex function . as a consequence of this lemma , lemma 4.1 and lemma 3.3 in @xcite , we get , @xmath200 is a biconvex function . * lemma 4.6 : @xmath199 is a biconvex function i.e. @xmath201 is a convex function and @xmath202 is a convex function . * proof : proof is similar to that of lemma 4.1.@xmath99 * * in this section we shall state the optimization problems obtained using the functions constructed in the previous sections such that the global minima of the optimization problem correspond to nash equilibria of the game @xmath0 . first optimization problem ( @xmath203 ) is stated below : @xmath204 the constraints in the above optimization problem ensure that the feasible set is @xmath205 . the second optimization problem ( @xmath206 ) is stated below : @xmath207 the following theorem says that the set of global minima of the optimization problem ( @xmath203 ) is the same as the set of nash equilibria profiles of the game @xmath0.*theorem 5.1 : for every game @xmath0 , there exists @xmath162 s.t . @xmath188 . further given @xmath208 , @xmath188 iff @xmath163 is a nash equilibrium profile.*proof : since for every game there exists @xmath209 , s.t . , @xmath210 is a n.e . ( see @xcite ) . thus by theorem 3.2 , @xmath163 with @xmath211 satisfies @xmath188 . the other part follows directly from theorem 3.2.@xmath99 * * a similar claim can be proved for @xmath206 . the above two optimization problems have a biconvex objective function with convex ( linear ) constraints . global optimization algorithm exists that solves the above two optimization problems ( see @xcite ) . in this section we shall consider a projected gradient descent algorithm to solve @xmath203 . the algorithm is stated below : * input : * * @xmath212 : initial point for the algorithm , * @xmath0 : the underlying game , * @xmath213 : step size sequences chosen as follows : * * @xmath214 , * * @xmath215 , * * @xmath216 , * @xmath217 : projection operator ensuring that @xmath127 remains in @xmath205 . * output : after sufficiently large number of iterations(@xmath218 ) the algorithm outputs the terminal strategy @xmath219.@xmath220 * in what follows we shall present the convergence analysis of the above projected gradient descent algorithm . we shall analyse the behaviour of the above algorithm using the o.d.e . method presented in @xcite . in order to use the results from @xcite , we need the gradient function to be lipschitz continuous on @xmath205 , which is proved in the following lemma.*lemma 6.1 : there exists @xmath221 , s.t . , @xmath222 , * @xmath223 * proof : it is easy to see that the function @xmath224 is twice continuously differentiable on an open set containing @xmath205 . thus @xmath225 is continuously diffrentiable on @xmath205 . hence @xmath226 for some @xmath227 . by mean value theorem , we have , @xmath225 is lipschitz continous with lipschitz constant @xmath228 . let @xmath229 . fix @xmath230 . clearly , @xmath231 . therefore , we have , @xmath232 where @xmath233}|$ ] . since @xmath234 , we have , @xmath235 , where @xmath236 . since sum of two lipschitz continuous functions is lipschitz continuous , we have , @xmath237 is lipschitz continous with lipschitz constant @xmath238.@xmath99 * in order to study the asymptotic behaviour of the recursion presented in the algorithm , by results in section 3.4 of @xcite , it is enough to study the asymptotic behaviour of the o.d.e . , @xmath239 where @xmath240 i.e. the directional derivative of @xmath217 at @xmath241 along the direction @xmath242 . the above o.d.e . is well posed i.e. has a unique solution for every initial point in @xmath205 ( for a proof see @xcite ) . @xmath205 , is a cartesian product of simplices and hence the projection of @xmath243 on to @xmath205 is the same as projection of @xmath244 on to @xmath245 and @xmath246 on to @xmath18 i.e. @xmath247^t$ ] where @xmath248 denotes the projection operator which projects every vector in @xmath69 on to @xmath249 . thus , in order to compute the directional derivative of @xmath217 , it is enough to consider the directional derivative of the projection operator on to individual simplices and then juxtaposing them would give us the directional derivative of @xmath217 . the computation of the directional derivative of a projection operation on to a simplex can be found in @xcite which we shall state here . let @xmath250 and @xmath251 . then , @xmath252 where @xmath253 , s.t . , @xmath254 . let @xmath255 . fix @xmath256 be a initial point of the o.d.e . [ o.d.e . ] and the corresponding unique solution be @xmath257 . then , @xmath258 by substituing [ dd ] and the fact that @xmath259 in the above equation we get , @xmath260 where the last inequality follows from the application of cauchy schwartz and the fact that @xmath261 . therefore along every solution of the o.d.e . ] , the value of the potential function @xmath262 reduces and hence the above o.d.e . converges to an internally chain transitive invariant set contained in @xmath263 . in the following lemma we shall prove that @xmath264 is an equilibrium point of o.d.e . [ o.d.e.].*lemma 6.2 : if @xmath264 , then , @xmath265.*proof : if @xmath264 is such that @xmath266 , then @xmath267 . assume @xmath268 . since @xmath264 , @xmath269 . by cauchy schwartz inequality , @xmath270 and @xmath271 . since their sum is zero , we get , @xmath272 and @xmath273 . hence , @xmath274 and @xmath275 . by , definition of @xmath276 in equation [ dd ] , we get , @xmath277 and @xmath278 . substituing for @xmath279 and @xmath280 in the expression for @xmath281 and @xmath282 and using the fact that @xmath283^t$ ] we get the desired result.@xmath99 * * in fact the converse is also true and the proof is similar to that of the previous lemma . therefore @xmath284 where @xmath285 denotes the set of equilibrium points of o.d.e.[o.d.e . ] . the following lemma says that every point in the set @xmath286 is a partial optimum of the biconvex function @xmath186.*lemma 6.3 : @xmath287 , then , @xmath288 and @xmath289.*proof : if @xmath287 is such that @xmath266 , then by lemma 4.5 the result follows . assume @xmath268 . then by lemma 6.2 we have , @xmath277 and @xmath278 . * * by equation [ dd ] , @xmath290 and hence @xmath291 . therefore @xmath292 . by convexity of @xmath293 and proposition 1.1.8 in @xcite , we get @xmath289 . by equation [ dd ] , @xmath294 and hence @xmath295 . therefore @xmath296 . since @xmath297 , we get , @xmath298 . thus by convexity of @xmath299 and by proposition 1.1.8 in @xcite , we have , @xmath300.@xmath99 even though the proof guarantees convergence to the set of partial optimum of the biconvex function in simulation on various test cases it was observed that the iterates converge to the set of nash equilibria of the game @xmath0 . in the simulations carried out , in order to perform the projection operation in every iteration we use the procedure in @xcite . we consider the following version of the standard rock - paper - scissor game . @xmath301 in the above game , @xmath302 is the only nash equilibrium strategy . having started the algorithm from a random initial point , variation of the objective function value and the strategies are shown in the plots below . 0.47 0.495 the plots in fig:[fig : rps_ap ] show that the action probabilities converge to the nash equilibrium of the game . as the action probabilities converge to nash equilibrium strategy the objective function value approaches zero as seen in fig:[fig : rps_obj ] . the general form of jordan s game can be found in @xcite . we consider the following version . * player 3 action @xmath303 : @xmath304 * player 3 action @xmath305 : @xmath306 in the above game , @xmath307 is the only nash equilibrium strategy . having started the algorithm from a random initial point , variation of the objective function value and the strategies are shown in the plots in fig:[fig : jg_ap ] and fig:[fig : jg_ap_obj ] . 0.435 0.485 0.435 0.45 simulations were also carried out on other versions of this game obtained from the general form in @xcite and convergence to nash equilibrium was observed . the following game was introduced in @xcite in order to show non - convergence of certain class of algorithms . the game is stated below . @xmath308 in the above game , @xmath309 and @xmath310 are the two nash equilibrium strategies . having started the algorithm from a random initial point , variation of the objective function value and the strategies are shown in the plots in fig:[fig : hm_ap ] and fig:[fig : hm_obj ] . 0.428 0.428 @xmath311 in the above game , @xmath312 is the set of nash equilbria . having started the algorithm from a random initial point , variation of the objective function value and the strategies are shown in the plots in fig:[fig : ie_ap ] and fig:[fig : ie_obj ] . 0.46 0.4 we have presented optimization problems ( @xmath203 and @xmath206 ) such that the global minima of these optimization problems are nash equilibria of the game @xmath0 . the objective functions were shown to be bi - convex and in case of @xmath203 the objective function was also shown to be an invex function . we also considered a projected gradient descent scheme and proved that it converges to a partial optimum of the objective function . even though the proof gaurantees convergence to the set of partial optimum in various test cases considered we have seen convergence to a nash equilibrium strategy . in future we wish to extend the above optimization problem formulation to discounted stochastic games and prove convergence to nash equilibrium or construct a counter example where the algorithm converges to a partial optimum which is not a nash equilibrium strategy . 99 von neumann j.and o. morgenstern . theory of games and economic behaviour , princeton university press . equilibrium points in n - person games . proceedings of national academy of sciences , vol 44 , pp 48 - 49 , 1950 . lemke c. e. and j. t. howson . equilibrium points of bimatrix games . siam journal on applied mathematics , vol 12 , pp 413 - 423 , 1964 . s. govindan and r. wilson . a global newton method to compute nash equilibria . journal of economic theory , vol 110,issue 1 , pp 65 - 86 , 2003 . p c. a. floudas and v. vishweswaran . a global optimization algorithm for certain classes of nonconvex nlps - i . computers chem . engng , vol . 1397 - 1417 , 1990 . j. gorski , f. pfeuffer and k. klamroth . biconvex sets and optimization with biconvex functions - a survey and extensions . methods of operations res , vol 66 , issue 3 , pp 373 - 407 , 2007 . v. s. borkar . stochastic approximations : a dynamical systems viewpoint . dimitri p. bertsekas . convex optimization theory . r. d. mckelvey . a liapunov function for nash equilibria . social science working paper , california institute of technology , 1998 . yunmei chen and xiojing ye . projection onto a simplex . p. dupuis and a. nagurney . dynamical systems and variational inequalities . annals of operations research , vol 44 , pp 7 - 42 , 1993 . sergiu hart and andreu mas - colell . uncoupled dynamics do not lead to nash equilibrium . , vol 93 , pp 1830 - 1836 , 2003 . sergiu hart and andreu mas - colell . stochastic uncoupled dynamics and nash equilibrium . games and economic behaviour , vol 57 , pp 286 - 303 , 2006 .

in this paper we present optimization problems with biconvex objective function and linear constraints such that the set of global minima of the optimization problems is the same as the set of nash eqilibria of a n - player general - sum normal form game . we further show that the objective function is an invex function and consider a projected gradient descent algorithm . we prove that the projected gradient descent scheme converges to a partial optimum of the objective function . we also present simulation results on certain test cases showing convergence to a nash equilibrium strategy .

You are an expert at summarizing long articles. Proceed to summarize the following text: a classical example of self - assembly is crystallization . at low temperatures the crystalline phase is typically stable , and thus grows spontaneously from solution through homogeneous nucleation . if several competing crystalline phases are allowed by microscopic interactions , the efficient production of a desired phase often requires heterogeneous nucleation from a seed of this phase , with precise annealing protocols @xcite . more complex microscopic interactions may lead to a glassy phase with many competing structures ; however , it is generally impossible to control local compositions or microscopic interactions to obtain a particular structure . recently , there has been a dramatic change in macromolecular and colloidal assembly techniques , made possible by the use of biopolymers , such as dna , to create a large variety of inter - component interactions . indeed , biomolecules offer exquisite control of microscopic interactions that allows self - assembly of diverse large structures . examples range from nanoparticle assemblies @xcite , which can also form macroscopic crystals @xcite , to structures using dna itself as a building material . in the latter case , dna origami uses short dna strands to controllably fold a long backbone strand into different well controlled structures @xcite , while short strands of dna by themselves can also build up complex three dimensional objects @xcite . similar efforts are underway using rationally - designed proteins by creating complementary binding sites on their surfaces @xcite . up until now , however , experimental and theoretical studies have been limited to devising interactions for the assembling a single structure . this is to be contrasted with biological systems , where many different self - assembled structures can be formed within the cell cytoplasm . these assembled structures can in fact share some of their components and can be dynamically induced independently from one another @xcite . here , we propose a new mechanism for the self - assembly of many different structures from one large set of shared components . each structure is multifarious , i.e. , is made out of many different types of components . such self - assembling systems , which we propose calling `` multifarious assembly mixtures '' , are stable and yet responsive . this means that the mixtures do not form structures spontaneously , but can be controllably induced to assemble a specific structure . different structures are encoded " through the choice of molecular interactions and thus stored " in the mixture , to then be retrieved " by changing only a small number of parameters . the theoretical framework introduced below allows calculation of the capacity of these systems , i.e. , how many different independent structures can be stored and retrieved in a mixture of @xmath0 species of components . in the traditional approach of self assembly without shared components , if each structure s is composed of the same number @xmath1 of different species , only @xmath2 different structures can be self - assembled . in contrast , in multifarious assembly mixtures , many more distinct structures can be stored . any stored structure can be retrieved with a ( super)critical nucleation seed . multiple seeds can induce the simultaneous assembly of multiple corresponding structures . moreover , we show that each different structure can be retrieved by changing only a small number of chemical potentials or interspecies interactions , where the number of tuned components is the number of components in a ( super)critical nucleation seed of the desired structure . classical nucleation theory implies that the size of this seed is only weakly dependent on the size of the structure that is built . consider @xmath0 species of interacting components in a solution kept at a constant temperature @xmath3 . in principle , each species can have a different chemical potential @xmath4 , @xmath5 , but for simplicity we assume for now that the chemical potentials have the same value @xmath6 . we would like the components to be able to self - assemble into one of @xmath7 distinct , multifarious structures , @xmath8 , on demand ( fig . [ fig : generalidea]a ) . a typical multifarious structure @xmath9 is built of @xmath1 component species . in general , each species @xmath10 in the structure @xmath9 has its own multiplicity @xmath11 . in contrast to traditional studies of self - assembly , e.g. of crystals , where the same component species appears in many copies in the assembled structure , for multifarious assembly mixtures we are interested in the case of small values of @xmath11 . indeed , for simplicity , we assume here that all component species have a single copy in every stored structures , @xmath12 , so that the number of species @xmath1 used in the structure equals the size of the structure @xmath13 . additionally , we make a simplifying assumption that all the structure sizes @xmath14 have the same value @xmath15 , so @xmath16 . both cellular systems and recent dna - mediated assembly experiments show that a single structure @xmath9 can be robustly assembled if each pair of neighboring components , of species @xmath10 and @xmath17 ( @xmath18 ) interact through a specific binding interaction . our next simplifying assumption is that all these interaction energies are equal : @xmath19 and we will also set all non - specific interactions to zero . the binding interactions between different components are mediated through a discrete number of binding sites " , with a species @xmath10 having a valence @xmath20 . for simplicity we assume that all components have the same valence @xmath21 . how might we choose an interaction energy matrix @xmath22 so that the components are capable of assembling different desired structures @xmath8 ( fig . [ fig : generalidea]a ) ? the simplest general prescription that can work for arbitrary structures is to assume that two species @xmath10 and @xmath17 bind specifically with energy @xmath23 if and only if at least one of the desired structures @xmath9 requires this binding . such a matrix @xmath22 then has the potential for `` storing '' each structure @xmath9 as a local free energy minimum ( fig . [ fig : generalidea]b ) . this matrix can be written as , @xmath24 this form of energy matrix implies that component species can be promiscuous in their interactions . indeed , since a given species @xmath10 binds specifically to its partners in each of the stored structures , the total number of specific binding partners for species @xmath10 can be large . in addition to the free energy minima corresponding to the desired structures , other undesired local minima might emerge . these correspond to chimeric structures , or chimeras " , made of chunks of different desired structures that can bind together due to the promiscuity implied by eqn . the stability of the stored structures is determined by the size of the free energy barriers between the different minima . for instance , if the barriers are low , chimeras will form spontaneously , even if their local free energy minima lie higher than those of desired structures ( fig . [ fig : generalidea]a ) . similarly , the free energy barriers between the solution of unbound components and other minima determines the solution s characteristic time @xmath25 , beyond which stored structures nucleate spontaneously and the process of the controlled retrieval of stored structures is compromised . thus , @xmath25 is the functional `` lifetime '' of the multifarious assembly mixture . how many different multifarious structures , each of size @xmath15 , can one store by using @xmath0 different species of components with well - chosen interspecies interactions defined by eqn ? if each species contributed to only a single structure , the maximum capacity would be @xmath26 . by sharing species between structures , however , a much larger number of structures can be stored before chimeras start to dominate . to find this increased capacity , consider components attaching to the boundary of a growing seed . the promiscuous interactions implied by eqn . might allow the seed to bind different sets of components , resulting in chimeras . therefore , let us compute the number of species that can specifically bind to a given boundary site of the seed . since each component in the bulk of a stored structure has @xmath21 nearest neighbors , for an incoming component to bind stably , it must form specific bonds with @xmath27 components on the seed s boundary . due to the promiscuous nature of eqn . , each of these @xmath27 boundary components can bind specifically to a set of @xmath28 other species . is randomly constituted from the @xmath0 species , a given species will occur in @xmath28 of the @xmath7 stored structures and typically have a different partner in each of them . hence , a typical species will have @xmath28 specifically binding partners . ] for randomly constituted @xmath7 structures , each set contains a fraction @xmath29 of all the @xmath0 component species . the intersection of these @xmath27 sets , of the size @xmath30 , determines the species that can specifically bind to all the @xmath27 boundary components . when this number is larger than @xmath31 , many different species can attach to a given boundary site on a growing seed , resulting in a proliferation of chimeras . hence , the largest number @xmath7 of structures that can be stored is @xmath32 for @xmath33 , the exponent @xmath34 is positive and this equation implies that the capacity @xmath35 can be much larger than the traditional estimate of the capacity @xmath26 . it is instructive to understand why @xmath36 structures i.e. , linear chains , can not share components . binding to an end of a growing chain requires forming a bond with just one component . if that component is promiscuous , the seed can always grow in a non - unique chimeric manner . hence the promiscuity of individual species , implied by eqn , must be countered by the requirement on incoming particles to form multiple ( i.e. , @xmath27 @xmath37 ) bonds . the above argument shows that the number of structures that can be stored and stabilized with @xmath0 components is large . for this to be useful , we need to be able to retrieve each of them easily . the retrieval can be done in three different ways . one can introduce a nucleation seed , i.e. , a part of a stored structure , into the solution . alternatively , one can enhance the formation of such a seed by increasing the chemical potential of its components by an appropriate amount @xmath38 , or by strengthening the interactions @xmath39 by @xmath40 for bonds found in such a seed . these methods enhance the nucleation of one stored structure without nucleating others , despite all stored structures being made of the same set of components . such selective nucleation is possible only for multifarious structures ; it relies on the fact that small contiguous subsets of distinct structures typically have distinct compositions . such subsets can be used as selective nucleating seeds , or to selectively lower the nucleation barrier for one structure using the other two methods described above . the critical question is how many different species have to be tuned in this way to successfully retrieve a particular stored structure . the answer follows directly from general nucleation theory , which specifies a critical nucleation radius @xmath41 in terms of the chemical potential @xmath6 and bond energy @xmath42 @xcite . the minimal seed size @xmath43 needed to recover a structure is set by @xmath41 ; smaller seeds dissolve back into components while larger seeds are supercritical and grow into stored structures . we can make the multifarious assembly mixture responsive to smaller seeds by lowering the critical nucleation radius , for example , by lowering the ratio @xmath44 of chemical potential to bond energy ( see si text and si figs . s2 , s5 and s6 ) . however , lowering the critical nucleation radius also lowers the barrier to spontaneous homogenous nucleation . as noted above , critical seeds can spontaneously assemble on a characteristic timescale @xmath25 and grow into random stored structures , without any external input . thus , at a minimum , we need @xmath25 to be much longer than the retrieval time , i.e. , time necessary for a supercritical seed to grow into a full structure . nucleation theory , adapted to multifarious @xmath45-dimensional structures , determines @xmath25 as @xmath46 where @xmath47 is the free energy barrier , @xmath48 is the free energy per area required for creating the critical seed and @xmath49 is a time scale connected with microscopic processes . the second term on the right hand side arises because we must account for the multiplicity @xmath50 of distinct nucleation paths leading to the @xmath7 different stored structures . for small @xmath7 , we can estimate @xmath51 to account for critical seeds from different parts of the @xmath7 stored structures of size @xmath15 each ( see si text ) . for a fixed @xmath52 , eqn . can be solved for the nucleation radius @xmath41 , and hence the minimal number of components @xmath43 that must be tuned to retrieve a structure . if all the components have a typical size @xmath53 , this number @xmath43 is of order @xmath54 where @xmath55 depends only weakly ( logarithmically ) on @xmath7 , @xmath15 and @xmath52 . we thus conclude , that since @xmath43 is determined by the nucleation barrier , it is essentially independent of the size of the structure , @xmath15 , that is being retrieved . note that the above equations show an unavoidable trade - off : increasing the lifetime of the multifarious assembly mixture @xmath25 necessarily increases the nucleation radius @xmath41 and hence increases @xmath43 . thus a more stable multifarious assembly mixture requires a larger seed for recovering stored structures . in order to study different regimes of self - assembly of multifarious structures , we have considered assembly based on eqn . , on a simple @xmath56 square lattice . individual components are square tiles that can be one of @xmath57 species . all @xmath7 stored structures consist of @xmath58 tiles , each tile being of different species positioned inside a @xmath59 square block . more precisely , we assume that all the species are present in all the structures , and each species appears only once in each structure , @xmath60 for all @xmath10 and all @xmath9 , so that @xmath61 for all structures @xmath8 . in other words , each stored structure is simply a different random permutation of the tiles inside the square block . we assume that each tile component can bind up to @xmath62 neighbors through specific binding interactions given by eqn , and that all species of tiles have the same chemical potential @xmath6 . we run grand canonical monte carlo simulations with different numbers @xmath7 of stored structures on a square lattice of total size @xmath63 , for different values of temperature @xmath3 and chemical potential @xmath6 ( see si text and si fig . s3 ) . starting from a particular supercritical seed ( of linear size @xmath64 ) of one of the @xmath7 stored structures , fig . [ fig : pd ] shows a diagram of the different outcomes of our simulations , as a function of the number of stored structures , @xmath7 , and the temperature @xmath3 ( or more precisely , @xmath65 , where @xmath42 is the specific binding energy ) , for a fixed @xmath6 . we visualize the different stored structures with different colors , with the desired structure colored in dark red . for low @xmath7 and @xmath3 , the supercritical seed indeed grows into the desired structure . in this regime of parameter space ( regime i ) , the solution behaves as a useful multifarious assembly mixture : the mixture is stable for a long time @xmath25 and stored structures can be retrieved through heterogeneous nucleation . , is much longer than the timescale for recovery , @xmath66 . thus , even though the monte carlo dynamics do not reflect the dynamics of a realistic self - assembly system ( see e.g.@xcite ) , they do substantiate the predictions of nucleation theory , and expose different regimes of self - assembly of multifarious structures . ] for higher number of stored structures @xmath7 ( and at higher temperatures @xmath3 ) another behavior appears ( regime ii ) . it is characterized by the spontaneous homogeneous nucleation of all stored structures from the solution : in this regime , the multifarious assembly mixture is too short lived to allow the structure retrieval , i.e. , @xmath25 becomes comparable to the time taken for a supercritical seed to grow into a full desired structure , @xmath66 . at even higher values of @xmath7 we find yet another regime of behavior ( regime iii ) , where chimeric structures dominate . finally , at high temperatures @xmath3 , and for all values of @xmath7 , we encounter regime iv , where any initial seed disintegrates into small clusters of individual components . the extent of different regimes depends of course on the chosen model parameters . in particular , the chemical potential @xmath6 influences the extent of regimes i and ii ( see si text and si figs . s7 and s9 ) . simulations presented in fig . [ fig : diffsa ] confirm that , in regime i , the assembly of a structure can be triggered not only with supercritical nucleating seeds , but also by enhancing the chemical potential of a small set of tile species , or by increasing the bond energies between the tile species from such a set . numerical simulations are also a way to gauge the capacity of an multifarious assembly mixture to store structures , and to compare it with the theoretical predictions presented above . to do this , we have introduced the entire target structure as a supercritical seed , and have examined it after a fixed simulation time chosen to be shorter than the mixture s lifetime @xmath25 . we have assessed the quality of retrieval by measuring the error , i.e. , the fraction of the final assembled structure that differs from the initial target structure ( see si text and si fig . [ fig : scaling]a depicts the error as a function of the number of stored structures @xmath7 , for different number of particle species @xmath0 ( structure sizes being @xmath58 ) , at fixed temperature @xmath3 and chemical potential @xmath6 . there is a transition at critical value @xmath67 , above which the error rises rapidly . we show that the error curves for different @xmath0 collapse onto each other when plotted against @xmath68 , fig . [ fig : scaling]b , where @xmath35 increases with increasing @xmath0 as @xmath69 with @xmath70 . this is in a good agreement with the prediction of eq . that the memory capacity scales as @xmath71 , for the square lattice model with @xmath62 nearest neighbors ( see si text ) . finally , we have also assessed the trade - off , expressed in eqn , between the stability of the multifarious assembly mixture , i.e. , its lifetime @xmath25 , and the minimal size @xmath43 of a seed needed for retrieval ( fig [ fig : tplots ] ) . the minimal seed size @xmath43 increases slowly with increasing @xmath25 , and remains a small fraction of the total number of components ( @xmath72 , in this case ) in a stored structure . the number of stored structures @xmath7 has only a modest effect on @xmath43 , in agreement with eqn ( see si text and si fig . s7 ) . to conclude , we have demonstrated that it is possible to store multiple structures in a solution of components with designed interactions between them . using @xmath0 different component species , we can store as many as @xmath73 different multifarious structures of size @xmath15 and of average coordination number @xmath21 . in an extended region of parameter values ( e.g. , temperature , chemical potentials , binding energies ) , such a multifarious assembly mixture " with many stored structures is both stable and responsive ; each of the multifarious structures can be selectively grown ( retrieved ) by modifying chemical potentials or binding energies of only a small fraction of the @xmath0 component types , or by introducing an appropriate seed . the model that we have explored is very similar to the way associative neural networks , such as hopfield s classical networks @xcite store multiple memories in a distributed way . in these models , a neural network is programmed to have multiple stable states , i.e. , memories , using a prescription for neuronal connections that is very similar in spirit to the pooled energy matrix in eqn . it has been shown @xcite that if the number of programmed memories is sufficiently small , each memory is indeed a stable state and can be recovered through initial conditions in a robust manner . however , if the number of stored memories exceeds the capacity of the network , recovery is spoiled by the presence of many `` spurious memories '' undesired stable states resulting in regimes @xcite similar to those shown in fig . [ fig : pd ] . a distinctive feature of multifarious assembly mixtures , however , is that we require the stability of the unassembled mixture itself for a long time @xmath25 , in addition to the stability of the stored structures ( see si text ) . in our simulated lattice model , different stored structures have identical components rearranged in random permutations . thus stored structures are assumed to be independent , or orthogonal " , as in the case of stored memories in the original hopfield model @xcite . an important extension of the present model would be to study stored structures with built - in correlations , such as the presence of shared modules . after all , the controlled assembly of chimeric structures could be useful . it is also important to stress that designing specific binding interactions between different components based on superposition ( eqn ) , is not the only way to create functional multifarious assembly mixtures . although it is arguably the simplest prescription that works for generic structures , other non - linear prescriptions can be tailored for particular structures by exploiting structural motifs ( e.g. , creating multifarious assembly mixtures with higher capacity or longer lifetimes ) . such tailored interactions have been used to store and retrieve a particular set of structures composed of a small number of component species in recent work on dna programmed assembly @xcite . in similar vein , the ability of a protein sequence to code for multiple stored internal structures has been studied in the context of protein folding @xcite . beyond immediate applications to artificial systems with controllable binding specificity , the present model proposes a new paradigm to understand molecular aggregates in biology . for instance , our calculations show that instead of creating new proteins for every individual structure , it is more efficient if individual proteins are used in a multiplicity of structures , as is the case in many cellular assemblies , ranging from transcription factors @xcite to ribonucleoproteins such as spliceosomes @xcite . our calculations also indicate that such versatility can be quite high , increasing rapidly with the number of different component species in the pool . nonetheless , different structures can be selectively assembled by reprogramming molecular interactions , e.g. by a simple modulation of the expression levels , or of the specific binding energies via post - translation modifications , of a small number of selected components . this is indeed what seems to happen often in cellular assembly . we hope that the theoretical framework presented here , properly generalized to far - from - equilibrium situations , will form a basis for quantitative studies of functioning , regulation and evolution of biological assembly . we would like to thank our colleagues for discussions and their comments on the manuscript , in particular john hopfield , david huse and olivier rivoire . z.z . acknowledges support from the george f. carrier fellowship . m.p.b . acknowledges funding by the national science foundation through the harvard materials research science and engineering center ( dmr-0820484 ) , the division of mathematical sciences ( dms-0907985 ) , and by grant rfp-12 - 04 from the foundational questions in evolutionary biology fund . m.p.b . is an investigator of the simons foundation . 10 lifshitz , i & slyozov , v. 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( 2009 ) the spliceosome : design principles of a dynamic \{rnp } machine . 136:701 718 . in our general arguments , we assumed that species @xmath10 and @xmath17 interact specifically if they are bound together in any one of the @xmath7 stored structures : @xmath74 this superposition form of @xmath75 implies that each species has multiple specifically binding partners . we define the promiscuity of a binding site on a component of a given species as the number of species that can specifically bind to it . in our model with @xmath0 total species of components and @xmath7 stored structures of size @xmath15 each , a given species will occur in about @xmath76 stored structures . the component will typically be bound to a different species in each of these structures . thus the promiscuity of a typical binding site is @xmath28 . when the number of stored structures @xmath7 is large , the species interactions are highly promiscuous . as a result , a given seed may be able to grow non - uniquely by binding distinct combinations of components and thus form chimeric structures . here we show that there is a sudden onset of chimeras at some @xmath77 which defines the capacity . to study this , we can specify arbitrary components along the boundary of a seed ( black tiles in fig . s[fig : sifig_boundary_conditions]a ) and ask if there are multiple choices of species that can fill in the positions @xmath78 shown in fig . s[fig : sifig_boundary_conditions]a , such that @xmath79 form specific bonds with each other and with all the seed components . for example , the number of species that can form a specific bond with the component @xmath80 from the right ( i.e. , number of choices for site @xmath81 in fig . s[fig : sifig_boundary_conditions]b ) is given by the promiscuity @xmath28 of @xmath80 s right - side binding site . similarly , we have another set of @xmath82 choices of species for site @xmath83 that bind specifically to the component @xmath84 from below . the choices for site @xmath85 , i.e. species that can bind specifically to both @xmath80 and @xmath84 , is given by the intersection of these two sets . if the @xmath7 stored structures are randomly and independently constituted from the @xmath0 species , we can assume that these sets of choices for @xmath83 and @xmath81 are two random uncorrelated sets of size @xmath76 contained in the set of all @xmath0 species . we then estimate the probability of having at least one element in the intersection of the two sets to be , @xmath86 to estimate the probability that at least one extended chimeric structure of length @xmath87 ( like that shown in red in fig . s[fig : sifig_boundary_conditions]a ) exists , we assume a choice of species for site @xmath85 that is stably bound and compute the probability of stably - bound chimeric choices for @xmath88 , fix a choice for @xmath88 and compute choices for @xmath89 and so on . such an estimate of the probability of existence of an extended chimeric structure of length @xmath87 is given by : @xmath90 taylor expanding for small @xmath91 gives @xmath92 for large @xmath87 , this probability sharply grows at @xmath93 , where @xmath94 a similar argument applies for a general lattice structure with coordination number @xmath21 . in this more general case , a component occupying site @xmath79 must form specific bonds with @xmath27 boundary components . hence the choice of species for each @xmath79 is given by the intersection of @xmath27 sets of size @xmath91 each , so that the taylor expansion for @xmath95 is modified to @xmath96 . the formula for capacity with general coordination number @xmath21 is therefore : @xmath97 we can also explicitly count the number of stably - bound chimeric structures @xmath98 of length @xmath87 , i.e. , the number of choices for the set @xmath99 shown in fig . s[fig : sifig_boundary_conditions ] . thus , @xmath100 estimated above is the probability that @xmath101 . we can numerically compute @xmath102 for a square lattice through explicit enumeration using a transfer - matrix - like method @xcite . let @xmath103 be the set of species that can occupy site @xmath104 in fig . s[fig : sifig_boundary_conditions]a by forming a specific bond with the boundary component from the right . ( for instance , @xmath105 is the set of species that specifically bind to the component species @xmath106 from the right . ) we form a `` transfer matrix '' @xmath107 between sites @xmath53 and @xmath108 by restricting the matrix @xmath109 to rows @xmath10 which are species found in @xmath103 and to columns @xmath17 which are species found in @xmath110 . hence @xmath107 is a @xmath111 matrix of boltzmann factors for binding between species @xmath103 and @xmath110 that can stably occupy sites @xmath53 and @xmath108 . ( note that we use the interaction matrix @xmath75 between top and bottom faces , i.e. , vertical direction in fig . s[fig : sifig_boundary_conditions]a , in this construction of @xmath112 . ) then , the sum @xmath113 of all entries of the matrix product , @xmath114 gives the partition function summed over all chimeric structures made of components @xmath115 which , by construction , form specific bonds with the ( black ) boundary components to the left in fig . s[fig : sifig_boundary_conditions]a . the most stable of these chimeric structures will also contain specific ( vertical ) bonds between every pair @xmath116 , i.e. , @xmath117 specific vertical bonds of energy @xmath23 each . hence , if we multiply the partition function @xmath113 by @xmath118 and take the limit @xmath119 , only terms corresponding to chimeric structures with @xmath87 specific vertical bonds will survive . in fact , in the large @xmath17 limit , @xmath120 precisely gives us the number @xmath102 of such stably - bound chimeric structures . using this method , we can compute @xmath102 explicitly for given boundary conditions . averaging this count over @xmath121 realizations of @xmath7 random structures of size @xmath58 , we obtained @xmath122 shown in fig . s[fig : sifig_boundary_conditions]c . here we chose @xmath123 to only count chimeric structures of length comparable to the side length of the @xmath124 structure itself . ( however , any @xmath87 that grows with @xmath0 gives similar results , as supported by the probabilistic argument for @xmath100 above . ) by varying @xmath0 between @xmath125 and @xmath126 , the inset of fig . s[fig : sifig_boundary_conditions]c shows that @xmath102 rises rapidly at @xmath127 . both our probabilistic arguments , using @xmath100 and explicitly counting @xmath102 of chimeric structures , agree on the scaling of capacity @xmath128 . these results also agree with the scaling extracted from monte - carlo simulations of the lattice model , presented in fig . 4 and in section [ sec : scalingsim ] below . we require two distinct behaviors of the multifarious assembly mixture : * * responsive : * the multifarious assembly mixture must produce structures in response to an externally introduced seed ( or equivalent perturbation ) of small size , * * stable : * the multifarious assembly mixture must be stable in the absence of external signals and must not produce any structures spontaneously . nucleation theory , adapted to multifarious structures , dictates whether such competing requirements can be implemented . we begin with the question of how large a seed is needed to recover a stored structure . the change in free energy , with respect to the multifarious assembly mixture , due to the presence of an @xmath129 square seed taken from one of the @xmath7 stored structures is : @xmath130 since such a structure has @xmath131 components and @xmath132 strong bonds . here @xmath133 is the energy of specific bonds and @xmath134 is the chemical potential of each species . ( note that we neglect the change of entropy in eqn . . ) the general shape of @xmath135 is shown in fig . s[fig : sifig_nucleationbarriers ] ; as in conventional nucleation theory , @xmath135 has a maximum for some critical size @xmath41 . a sub - critical seed of size @xmath136 will dissolve back into its components while supercritical seeds ( i.e. , size @xmath137 ) will grow in size and into the full stored structure . hence the minimal size of the seed we must introduce to recover structures is simply given by the size @xmath41 of the critical seed . we can calculate @xmath41 by setting @xmath138 . we find @xmath139 with @xmath140 . note that this relationship is independent of the number of stored structures @xmath7 ; the minimal seed size is determined by a local condition @xmath141 on the free energy landscape and is not affected by the presence of other minima . the excess free energy potential @xmath135 shown in fig . s[fig : sifig_nucleationbarriers ] implies that the multifarious assembly mixture is intrinsically unstable even without the external introduction of any seed , a critical seed of size @xmath41 could spontaneously emerge on some timescale @xmath25 , leading to the nucleation of random stored structures . hence the timescale of such spontaneous nucleation @xmath25 sets the useful lifetime of the multifarious assembly mixture . in conventional nucleation theory , the timescale for spontaneous nucleation @xmath25 is given by arrhenius s formula for barrier crossing @xmath142 . however , with @xmath7 multifarious structures , we need to modify this formula to account for multiple inequivalent seeds that can spontaneously nucleate distinct stored structures . if there are @xmath143 inequivalent barriers that can be crossed , the timescale for spontaneously crossing any one of the barriers and assembling a stable stored structure is given by , @xmath144 where @xmath49 is a timescale associated with microscopic processes . in conventional nucleation theory , there is only one ( or @xmath145 ) stable phase , while seeds can vary only in shape and not in composition . in contrast , our multifarious assembly mixture can form at least @xmath7 stable structures ( i.e. , the @xmath7 stored structures ) in addition to any stable chimeric structures that might exist . further , seeds from different parts of these multifarious structures are inequivalent in composition . for small @xmath146 , we can ignore chimeric structures and estimate @xmath147 since the @xmath7 stored structures can each be nucleated with @xmath148 inequivalent seeds . we do not pursue the detailed form of this correction any further here ; the correction to eqn . [ eqn : tstarfstar ] is logarithmic and numerical simulations discussed below confirm that the @xmath149 correction is modest . using the expression for @xmath150 , we find , @xmath151 note that we have an unavoidable trade - off : increasing the lifetime of the multifarious assembly mixture @xmath25 would necessarily increase the critical seed size @xmath41 . thus a more stable multifarious assembly mixture requires a larger seed for recovering stored structures . we can also rewrite the lifetime @xmath25 in terms of the parameters @xmath6 and @xmath42 as @xmath152 and we tested this relationship in our monte carlo simulations . we carry out monte carlo simulations of this system , with @xmath0 different species of square tiles on a square grid , see fig . [ fig : simsetup ] . each stored structure is a @xmath153 sized square composed as a random permutation of @xmath154 tiles , with each tile being of different species . since we work in the grand canonical ensemble , the total system was larger , with a side length @xmath155 , when @xmath156 was even , and @xmath157 otherwise . the energy of a state of the system is specified by @xmath158 , where @xmath159 labels the stored structures , while @xmath160 labels the bond directionality of the nearest neighbor tile positions , with @xmath62 being the coordination number ( or valance ) . for our square grid , @xmath161 labels up , right , down and left neighbor positions , respectively . for a given structure @xmath9 we consider all nearest neighbor pairs of tiles . when a pair of tiles of species @xmath162 is found , @xmath163 is determined as the directionality of the distance vector @xmath164 , and @xmath165 is set to @xmath23 . the elements of the total interaction matrix for the system are : @xmath166 the maximum function caps the matrix elements , since we consider a model with all bonds of the same strength . finally , we define the energy of states . the square grid has @xmath167 sites , each being either empty or occupied by a tile which can be one of @xmath0 different species . an empty site can be simply treated as a tile of zeroth species , which has no binding energy , so @xmath168 when any of @xmath162 is zero , and with chemical potential @xmath169 . a state of the system is then described by a vector array @xmath170 , where @xmath171 if @xmath172 is the tile species at grid site @xmath173 , and @xmath174 otherwise , with @xmath175 labeling site @xmath176 void of tile . the energy of such a state is : @xmath177 where @xmath178 is the directionality defined by nearest neighbor pair @xmath179 , and @xmath180 the vector of chemical potentials for species @xmath10 . the monte carlo algorithm we used in the simulations chooses a random grid position and changes its species with a probability @xmath181 , where @xmath40 is the total energy cost of the change calculated using eq . typical simulations were run for @xmath182 , where @xmath183 is one lattice sweep , i.e. , @xmath167 monte carlo moves . 4a shows the error in an assembled structure , observed at the end of simulation , as a function of the number of stored structures @xmath7 , for different structure sizes @xmath15 ( and correspondingly different system sizes @xmath0 and numbers of tile species @xmath58 ) . each simulation starts with one selected complete structure of a given size @xmath15 ( in shape of a square ) and runs for a fixed amount of time @xmath184 . we define the error using a three - step procedure , see fig . s[fig : errordef ] : first , in the final state of the simulation we identify the largest contiguous area of tiles that are bonded , i.e. , the largest connected structure , which we call the `` final structure '' ; next , the union of the area of the final structure with the area of the initial structure ( which is a square ) gives a total area @xmath185 , and the number of tiles that match between the initial and final structures inside @xmath185 is divided by the total number of tiles in @xmath185 to give the overlap of structures . by definition , this overlap is between zero and one . however , since the initial structure in the simulations never dissolves in the considered regimes ( i and iii ) , the overlap is at least @xmath186 ( for the system sizes we studied @xmath187 ) . finally , the error is defined as one minus the overlap . this definition of error , which uses @xmath185 , consistently takes into account the multifarious assembly mistakes that occur by changing or loosing tiles in the initial structure , as well as by attaching tiles to the boundary of the initial structure . for any given structure size @xmath15 , the error in the simulations sharply rises from zero when a certain number of stored structures was reached , and then quickly saturates at @xmath188 . the capacity @xmath35 is the number of structures that can be stored without having a large error , and we extract its value using a finite size scaling analysis . our ansatz for the scaling function is @xmath189 , with @xmath190 , where @xmath7 is the number of stored structures and @xmath191 is the only fit parameter in function @xmath192 . we find it robust for analysis to consider only the datapoints for which @xmath193 , which is consistent with our interest in the sharp rise of error from zero . we rescale the datapoints in each error curve using @xmath194 , with @xmath191 fixed , and we fit the complete set of datapoints using a polynomial of fifth degree which represents the unknown function @xmath195 . we quantify the quality of the fit for the given @xmath191 with @xmath196 , where @xmath197 is total of squared fit residuals , @xmath198 is total of squared mean prediction errors , and @xmath199 is total of squared datapoint errors @xcite . we minimize @xmath200 with respect to @xmath191 . the @xmath200 varied smoothly with @xmath191 and its minimum is easily found , giving the optimal value of @xmath191 . 4b shows a very good collapse of error curves for different number of species @xmath0 when the optimal @xmath191 is used ( the parts of the curves with @xmath201 are also plotted ) , which validates the scaling ansatz . finally , we describe the procedure for estimating the error of the calculated optimal value of @xmath191 . we use a simple bootstrap method @xcite . consider the error curve for some system size , and let there be @xmath202 datapoints in this curve . we form a new dataset by randomly choosing a datapoint from the curve @xmath202 times . the new curve therefore has the same number of points as the original , but some of the original points might be missing and some might occur multiple times . this sampling is applied to every error curve . using the new datasets , the new optimal value of @xmath191 is calculated . we repeat the entire procedure a hundred times ( always starting from the original datasets ) , giving a distribution of optimal @xmath191 values . the mean of the distribution is the @xmath191 quoted in the main text , while the standard deviation is its error . we check that using higher order polynomials in the fit procedure does not significantly improve the quality of fits . in addition , instead of taking the error cutoff equal to @xmath203 , we also vary it between @xmath204 and @xmath205 , but the variation in obtained optimal @xmath191 was of the same order of magnitude as the error obtained by the bootstrapping procedure . to test the relationship between @xmath41 , @xmath6 and @xmath7 ( eqns . and ) , we ran a series of simulations starting with a square - shaped seed of different sizes and various values of @xmath6 , and measure the probability that the seed dissolves . we consider @xmath206 and @xmath207 stored structures at fixed temperature @xmath208 , well inside the retrieval regime . s[fig : diss ] shows the dissolving probability as a function of the number of tiles in a seed , @xmath209 , each given seed approximating a disc in shape . each probability averages @xmath210 simulation runs . from this data we extract the critical radius of the seed , @xmath41 , at which the probability to dissolve drops to zero . we show the extracted @xmath41 as a function of @xmath6 in fig . s[fig : rstarmu ] . note that in this section we redefine the quantity @xmath41 as dimensionless and equals the critical seed radius measured in the units of tile length . we investigate different time scales in the monte carlo simulations of the lattice model . s[fig : tstarmum]a plots both the spontaneous nucleation time @xmath25 and the recovery time @xmath211 measured as functions of the chemical potential @xmath6 . we define the spontaneous nucleation time @xmath25 as the time when any structure , being spontaneously assembled in the homogeneous solution , reached an area of @xmath212 tiles , which is above the critical nucleation size for the largest value of the chemical potential we considered ( see fig . s[fig : diss ] ) . recovery time @xmath211 identifies the moment in the simulation when we first observed the seeded structure completely assemble ( seed had @xmath213 tiles , which is supercritical for all chemical potentials considered ) . in some simulations , at the moment of completed assembly , there are a few erroneous tiles attached to the structure , which we neglect here . as @xmath214 , the @xmath25 diverges , while the recovery time is much smaller and essentially independent of @xmath6 . this demonstrates that there is a parameter range of the model where structure retrieval occurs much more quickly than spontaneous nucleation from the solution , fig . s[fig : tstarmum]a . in fig . s[fig : tstarmum]b shows how the nucleation time @xmath25 depends on the number of stored structures . the obtained result is consistent with eqns . [ eq : q ] and [ eq : tstar2 ] of the main text . here we present details of the finite temperature transition between regimes iii and iv presented in fig . 2 . as presented in fig . s[fig : transitiont ] , we pick the simulations with @xmath215 and @xmath216 stored structures as examples , and look at the size of largest structure as function of temperature . at the end of each simulation we identify the `` biggest structure '' as the largest contiguous area of tiles that are bonded specific interactions , i.e. , the largest connected structure . at each temperature we average the size ( number of tiles ) of the biggest structure over @xmath106 independent runs . s[fig : transitiont ] reveals that the biggest structure covered most of the entire system at low temperatures ( regime iii ) , but its size sharply dropped at a certain temperature . at temperatures above this transition ( regime iv ) , the system is mostly filled with tiles , however , there are hardly any specific interactions between them , i.e. , the state was a solution of fluctuating components . as expected , the transition temperature increases slowly with increasing number of stored structures to @xmath217 , with @xmath23 the binding energy . in fig . s[fig : regimesmu]a - b we show regimes observed in monte carlo simulations as a function of the number of stored structures @xmath7 and temperature @xmath65 , starting from a particular supercritical seed ( shown in bottom panels ) . as in fig . 2 , we use different colors to visualize different stored structures , with the seeded structure colored in dark red . bottom panels distinguish the four regimes identified in the diagram . in regime i the desired structure is retrieved through heterogeneous nucleation since the solution remains stable in the time required for assembly . the solution in this regime is a functional multifarious assembly mixture . regime ii is characterized by homogeneous nucleation of all structures due to reduced stability of the solution . in regime iii , formation of structures is dominated by chimeras . finally , in regime iv , any initial seed is disintegrated into the solution . these figures differ from fig . 2 in the value of the chemical potential @xmath6 , which is the same for all the species . in fig . s[fig : regimesmu]a the lifetime @xmath25 of the multifarious assembly mixture is suppressed due to the lower value of @xmath218 . hence , regime ii suppresses the retrieval regime i compared to the result in fig . 2 , where @xmath219 . in fig . s[fig : regimesmu]b the chemical potential is higher @xmath220 . consequently the characteristic lifetime of the multifarious assembly mixture is @xmath221 ( see fig . s[fig : tstarmum ] ) , resulting in complete suppression of regime ii . our model of multifarious assembly mixtures is closely related to models of associative memory @xcite . in these models @xcite , multiple `` memories '' are stored as stable states of a neural network by choosing the connections between @xmath0 neurons appropriately . just as with multifarious assembly mixtures , neural networks have a finite capacity ; with @xmath0 neurons , a limited number of memories @xmath222 can be reliably stored and retrieved . if the capacity @xmath223 is exceeded , many spurious memories undesired stable states appear and interfere with retrieval . neural networks have also been studied in the thermodynamic limit of a large number of neurons @xmath0 @xcite . in this limit @xcite , energy barriers separating the memories grow large and each memory becomes a stable thermodynamic phase of the system , provided the number of memories @xmath224 is less than a critical value @xmath225 . however , there is a phase transition at @xmath226 to a spin - glass phase with an exponential number of other spurious but stable states . the properties of such phases have been worked out for different neural networks with different models of interactions between neurons , resulting in phase diagrams that resemble fig . 2 of our paper . in fact , a recently studied @xcite @xmath56-lattice neural network , albeit with long - ranged interactions , is closely related to the large coordination number @xmath21 limit of our model . hence , it is natural to ask about the thermodynamic properties of multifarious assembly mixtures . for example , we can take the size of programmed structures @xmath227 to be large , with the ratio @xmath26 held finite . such a limit might allow for a growing number of stored structures @xmath228 to become stable phases of the system , provided @xmath229 is less than a critical @xmath230 . however , a crucial intrinsically - kinetic feature of multifarious assembly mixtures , not found in neural networks , is the stability of the unassembled mixture itself . as we showed in the paper , the unassembled mixture has a finite lifetime @xmath25 after which random stored structures are spontaneously nucleated . this lifetime @xmath25 is set by the ratio @xmath44 of chemical potential to bond energy and hence , in principle , is independent of structure size @xmath15 and can stay finite in the thermodynamic limit . as a result , for multifarious assembly mixtures , the question of practical interest is a finite - time kinetic question can one recover specific structures using seeds in a time shorter than @xmath25 ? to focus on this question , we define capacity and the transition to chimeric phase through a practical finite - time notion of error we introduce an initial seed and measured error in recovery after a finite simulation time chosen to be much smaller than the lifetime of the mixture . a thermodynamic analysis of our model would require a modified set of quantities and parameter limits . for example , the unassembled mixture can itself be stabilized as a phase only if its lifetime @xmath25 diverges in the thermodynamic limit , which requires tuning the chemical potential @xmath231 with growing structure size @xmath227 . in such a limit , our finite - time error can be replaced by order parameters analogous to those used in @xcite to study the transition to the chimeric regime . we leave a detailed study of the thermodynamic limit of multifarious assembly mixtures to future work .

self - assembly materials are traditionally designed so that molecular or meso - scale components form a single kind of large structure . here , we propose a scheme to create multifarious assembly mixtures " , which self - assemble many different large structures from a set of shared components . we show that the number of multifarious structures stored in the solution of components increases rapidly with the number of different types of components . yet , each stored structure can be retrieved by tuning only a few parameters , the number of which is only weakly dependent on the size of the assembled structure . implications for artificial and biological self - assembly are discussed .

You are an expert at summarizing long articles. Proceed to summarize the following text: for many years now , spin physics has played a very prominent role in qcd . the field has been carried by the hugely successful experimental program of polarized deeply - inelastic lepton - nucleon scattering ( dis ) , and by a simultaneous tremendous progress in theory . this talk summarizes some of the interesting new developments in spin physics in the past roughly two years . as we will see , there have yet again been exciting new data from polarized lepton - nucleon scattering , but also from the world s first polarized @xmath0 collider , rhic . there have been very significant advances in theory as well . it will not be possible to cover all developments . i will select those topics that may be of particular interest to the attendees of a conference in the `` dis '' series . until a few years ago , polarized inclusive dis played the dominant role in qcd spin physics @xcite . at the center of attention was the nucleon s spin structure function @xmath1 . [ fig1 ] shows a recent compilation @xcite of the world data on @xmath1 . these data have provided much interesting information about the nucleon and qcd . for example , they have given direct access to the helicity - dependent parton distribution functions of the nucleon , @xmath2 polarized dis actually measures the combinations @xmath3 . from @xmath4 extrapolation of the structure functions for proton and neutron targets it has been possible to test and confirm the bjorken sum rule @xcite . polarized dis data , when combined with input from hadronic @xmath5 decays , have allowed to extract the unexpectedly small nucleon s axial charge @xmath6 , which to lowest order unambiguously coincides with the quark spin contribution to the nucleon spin @xcite . the results from polarized inclusive dis have also led us to identify the next important goals in our quest for understanding the spin structure of the nucleon . the measurement of gluon polarization @xmath7 rightly is a main emphasis at several experiments in spin physics today , since @xmath8 could be a major contributor to the nucleon spin . also , more detailed understanding of polarized quark distributions is clearly needed ; for example , we would like to know about flavor symmetry breakings in the polarized nucleon sea , details about strange quark polarization , and also about the small-@xmath9 and large-@xmath9 behavior of the densities . again , these questions are being addressed by current experiments . finally , we would like to find out how much orbital angular momentum quarks and gluons contribute to the nucleon spin . ji showed @xcite that their total angular momenta may be extracted from deeply - virtual compton scattering , which has sparked much experimental activity also in this area . there are several lepton - nucleon scattering fixed - target experiments around the world with dedicated spin physics programs . this will not be a complete list ; i will mention only those that play a role in this talk . hermes at desy uses hera s 27.5 gev polarized electron beam on polarized targets . they have just completed a run with a transversely polarized target . semi - inclusive dis ( sidis ) measurements are one particular strength of hermes . compass at cern uses a 160 gev polarized muon beam . their main emphasis is measuring gluon polarization ; they have completed their first run . there is also a very large spin program at jefferson lab , involving several experiments . large-@xmath9 structure functions and the dvcs reaction are just two of many objectives there . finally , an experiment e161 at slac aims at measuring @xmath8 in photoproduction , but has unfortunately been put on hold awaiting funding . for the more distant future , there are plans to develop a polarized electron - proton _ collider _ at bnl , erhic @xcite . a new milestone has been reached in spin physics by the advent of the first polarized proton - proton collider , rhic at bnl . by now , two physics runs with polarized protons colliding at @xmath10 gev have been completed , and exciting first results are emerging . we will see one example toward the end of this talk . all components crucial for the initial phase of the spin program with beam polarization up to 50% are in place @xcite . this is true for the accelerator ( polarized source , siberian snakes , polarimetry by proton - carbon elastic scattering ) as well as for the detectors . rhic brings to collision 55 bunches with a polarization pattern , for example , @xmath11 in one ring and @xmath12 in the other , which amounts to collisions with different spin combinations every 106 nsec . it has been possible to maintain polarization for about 10 hours . there is still need for improvements in polarization and luminosity for future runs . the two larger rhic experiments , phenix and star , have dedicated spin programs focusing on precise measurements of @xmath8 , quark polarizations by flavor , phenomena with transverse spin , and many others . as mentioned above , the measurement of @xmath8 is a main goal of several experiments . the gluon density affects the @xmath13-evolution of the structure function @xmath1 , but the limited lever arm in @xmath13 available so far has left @xmath8 virtually unconstrained . one way to access @xmath8 in lepton - nucleon scattering is therefore to look at a less inclusive final state that is particularly sensitive to gluons in the initial state . one channel , to be investigated by compass in particular , is heavy - flavor production via the photon - gluon fusion process @xcite . an alternative reaction is @xmath14 , where the two hadrons in the final state have large transverse momentum @xcite . rhic will likely dominate the measurements of @xmath8 . several different processes will be investigated @xcite that are sensitive to gluon polarization : high-@xmath15 prompt photons @xmath16 , jet or hadron production @xmath17 , @xmath18 , and heavy - flavor production @xmath19 . in addition , besides the current @xmath10 gev , also @xmath20 gev will be available at a later stage . all this will allow to determine @xmath21 in various regions of @xmath9 , and at different scales . one can compare the @xmath8 extracted in the various channels , and hence check its universality implied by factorization theorems . in this way , we will also likely learn a lot more about high-@xmath15 reactions in qcd . we emphasize that for all the reactions relevant at rhic we now know the next - to - leading order ( nlo ) qcd corrections to the underlying hard scatterings of polarized partons @xcite . this significantly improves the theoretical framework , since it is known from experience with the unpolarized case that the corrections are indispensable in order to arrive at quantitative predictions for hadronic cross sections . for instance , the dependence on factorization and renormalization scales in the calculation is much reduced when going to nlo . therefore , only with knowledge of the nlo corrections will one be able to extract @xmath8 reliably . figure [ fig2 ] shows nlo predictions @xcite for the double - spin asymmetry @xmath22 for the reaction @xmath23 at rhic , using various different currently allowed parameterizations @xcite of @xmath21 . it also shows the statistical errors bars expected for a measurement by phenix in @xmath23 with lower polarization and luminosity were reported by phenix @xcite . ] under the assumption of 50% beam polarizations and 7/pb integrated luminosity . it is evident that the prospects for determining @xmath8 in this reaction , and in related ones , are excellent . we stress that phenix has recently presented a measurement of the unpolarized high-@xmath15 @xmath24 cross section @xcite that agrees well with an nlo perturbative - qcd calculation over the whole range of @xmath15 accessed . this provides confidence that the theoretical hard scattering framework used for fig . [ fig2 ] is indeed adequate . as mentioned earlier , inclusive dis via photon exchange only gives access to the combinations @xmath25 . there are at least two ways to distinguish between quark and antiquark polarizations , and also to achieve a flavor separation . semi - inclusive measurements in dis are one possibility , explored by smc @xcite and , more recently and with higher precision , by hermes @xcite . one detects a hadron in the final state , so that instead of @xmath26 the polarized dis cross section becomes sensitive to @xmath27 , for a given quark flavor . here , the @xmath28 are fragmentation functions , with @xmath29 . [ fig3 ] shows the latest results on the flavor separation by hermes @xcite , obtained from their lo monte - carlo code based `` purity '' analysis . within the still fairly large uncertainties , they are not inconsistent with the large negative polarization of @xmath30 in the sea that has been implemented in many determinations of polarized parton distributions from inclusive dis data @xcite ( see curves in fig . [ fig3 ] ) . on the other hand , there is no evidence either for a large negative strange quark polarization . for the region @xmath31 , the extracted @xmath32 integrates @xcite to the value @xmath33 , while analyses of inclusive dis prefer an integral of about -0.025 . there is much theory activity currently on sidis , focusing also on possible systematic improvements to the analysis method employed in @xcite , among them nlo corrections , target fragmentation , and higher twist contributions @xcite . we note that at rhic @xcite one will use @xmath34 production to determine @xmath35 with good precision , making use of parity - violation . comparisons of such data taken at much higher scales with those from sidis will be extremely interesting . new interesting information on the polarized quark densities has also recently been obtained at high @xmath9 . the hall a collaboration at jlab has published their data for the neutron asymmetry @xmath36 @xcite , shown in fig . [ fig4 ] ( left ) . the new data points show a clear trend for @xmath36 to turn positive at large @xmath9 . such data are valuable because the valence region is a particularly useful testing ground for models of nucleon structure . the right panel of fig . [ fig4 ] shows the extracted polarization asymmetry for @xmath37 . the data are consistent with constituent quark models @xcite predicting @xmath38 at large @xmath9 , while `` hadron helicity conservation '' predictions based on perturbative qcd and the neglect of quark orbital angular momentum @xcite give @xmath39 and tend to deviate from the data , unless the convergence to 1 sets in very late . besides the unpolarized and the helicity - dependent densities , there is a third set of twist-2 parton distributions , transversity @xcite . in analogy with eq . ( [ eq1 ] ) they measure the net number ( parallel minus antiparallel ) of partons with transverse polarization in a transversely polarized nucleon : @xmath40 in a helicity basis , one finds @xcite that transversity corresponds to a helicity - flip structure , as shown in fig . this precludes a gluon transversity distribution at leading twist . it also makes transversity a probe of chiral symmetry breaking in qcd @xcite : perturbative - qcd interactions preserve chirality , and so the helicity flip required to make transversity non - zero must primarily come from soft non - perturbative interactions for which chiral symmetry is broken . measurements of transversity are not straightforward . again the fact that perturbative interactions in the standard model do not change chirality ( or , for massless quarks , helicity ) means that inclusive dis is not useful . collins , however , showed @xcite that properties of fragmentation might be exploited to obtain a `` transversity polarimeter '' : a pion produced in fragmentation will have some transverse momentum with respect to the fragmenting parent quark . there may then be a correlation of the form @xmath41 . the fragmentation function associated with this correlation is the collins function . the phase is required by time - reversal invariance . the situation is depicted in fig . the collins function would make a _ leading - power _ @xcite contribution to the single - spin asymmetry @xmath42 in the reaction @xmath43 : @xmath44 where @xmath45 ( @xmath46 ) is the angle between the lepton plane and the @xmath47 plane ( and the transverse target spin ) . as is evident from eq . ( [ eq3 ] ) , this asymmetry would allow access to transversity if the collins functions are non - vanishing . a few years ago , hermes measured the asymmetry for a longitudinally polarized target @xcite . for finite @xmath48 , the target spin then has a transverse component @xmath49 relative to the direction of the virtual photon , and the effect may still be there , even though it is now only one of several `` higher twist '' contributions @xcite . if `` intrinsic '' transverse momentum in the fragmentation process plays a crucial role in the asymmetry for @xmath50 , a natural question is whether @xmath51 in the initial state can be relevant as well . sivers suggested @xcite that the @xmath51 distribution of a quark in a transversely polarized hadron could have an azimuthal asymmetry , @xmath52 , as shown in fig . [ fig7 ] . there is a qualitative difference between the collins and sivers functions , however . while phases will always arise in strong interaction final - state fragmentation , one does not expect them from initial ( stable ) hadrons , and the sivers function appears to be ruled out by time - reversal invariance of qcd @xcite . until recently , it was therefore widely believed that origins of single - spin asymmetries as in @xmath53 and other reactions were more likely to be found in final - state fragmentation effects than in initial state parton distributions . however , then came a model calculation @xcite that found a leading - power asymmetry in @xmath54 not associated with the collins effect . it was subsequently realized @xcite that the calculation of @xcite could be regarded as a model for the sivers effect . it turned out that the original time - reversal argument against the sivers function is invalidated by the presence of the wilson lines in the operators defining the parton density . these are required by gauge invariance and had been neglected in @xcite . under time reversal , however , future - pointing wilson lines turn into past - pointing ones , which changes the time reversal properties of the sivers function and allows it to be non - vanishing . now , for a `` standard '' , @xmath51-integrated , parton density the gauge link contribution is unity in the @xmath55 gauge , so one may wonder how it can be relevant for the sivers function . the point , however , is that for the case of @xmath51-dependent parton densities , a gauge link survives even in the light - cone gauge , in a transverse direction at light - cone component @xmath56 @xcite . thus , time reversal indeed does not imply that the sivers function vanishes . the same is true for a function describing transversity in an unpolarized hadron @xcite . it is intriguing that these new results are based entirely on the wilson lines in qcd . if the sivers function is non - vanishing , it will for example make a leading - power contribution to @xmath53 , of the form @xmath57 this is in competition with the collins function contribution , eq . ( [ eq3 ] ) ; however , the azimuthal angular dependence is discernibly different . hermes has just completed a run with transverse polarization , and preliminary results are expected soon . we note that the collins function may also be determined separately from an azimuthal asymmetry in @xmath58 annihilation @xcite . it was pointed out @xcite that comparisons of dis and the drell - yan process will be particularly interesting : from the properties of the wilson lines it follows that the sivers functions relevant in dis and in the drell - yan process have opposite sign , violating universality of the distribution functions . this is a striking prediction awaiting experimental testing . for work on the process ( in)dependence of the collins function , see @xcite ; recent model calculations of the function in the context of the gauge links may be found in @xcite . originally , the sivers function was proposed @xcite as a means to understand and describe the significant single - spin asymmetries @xmath59 observed @xcite in @xmath60 , with the pion at high @xmath15 . these are inclusive `` left - right '' asymmetries and may be generated by the sivers function from the effects of the quark intrinsic transverse momentum @xmath51 on the partonic hard - scattering which has a steep @xmath15 dependence . the resulting asymmetry @xmath59 is then power - suppressed as @xmath61 in qcd , where @xmath62 is an average intrinsic transverse momentum . similar effects may arise also from the collins function . fits to the available @xmath59 data have been performed recently @xcite , assuming variously dominance of the collins or the sivers mechanisms . an exciting new development in the field is that the star collaboration has presented the first data on @xmath63 from rhic @xcite . the results are shown in fig . as one can see , a large @xmath59 persists to these much higher energies . [ fig9 ] also shows predictions based on the collins and the sivers effects @xcite , and on a formalism @xcite that systematically treats the power - suppression of @xmath59 in terms of higher - twist parton correlation functions ( for a connection of the latter with the sivers effect , see @xcite ) . the star data clearly give valuable information already now . for the future , it will be important to extend the measurements to higher @xmath15 where the perturbative - qcd framework underlying all calculations will become more reliable . it was recognized some time ago that certain fourier transforms of generalized parton densities with respect to momentum transfer give information on the position space distributions of partons in the nucleon @xcite . for a transversely polarized nucleon , one then expects @xcite a distortion of the parton distributions in the transverse plane , which could provide an intuitive physical picture for the origins of single - spin asymmetries . we finally note that _ double_-transverse spin asymmetries @xmath64 in @xmath0 scattering offer another possibility to access transversity . candidate processes are drell - yan , prompt photon , and jet production . recently , the nlo corrections to @xmath65 have been calculated @xcite . the results show that @xmath64 is expected rather small at rhic . i am grateful to the organizers of dis 2003 for their invitation . i thank d. boer , g. bunce , m. grosse - perdekamp , s. kretzer , z. meziani , g. rakness , m. stratmann for very useful discussions and help , and riken , brookhaven national laboratory and the u.s . department of energy ( contract number de - ac02 - 98ch10886 ) for providing the facilities essential for the completion of this work . 0 for recent reviews , see : b. lampe , e. reya , phys . * 332 * ( 2000 ) 1 ; + e.w . hughes , r. voss , annu . nucl . part . * 49 * ( 1999 ) 303 ; + b.w . filippone , x .- d . ji , adv . nucl * 26 * ( 2001 ) 1 . u. stsslein , acta phys . * b33 * ( 2002 ) 2813 . bjorken , phys . rev . * 148 * ( 1966 ) 1467 ; _ ibid . _ * d1 * ( 1970 ) 1376 . x .- ji , phys . * 78 * ( 1997 ) 610 . for information on the erhic project , see : http://www.bnl.gov/eic l.c . bland , hep - ex/0212013 . c. marchand , these proceedings . a. bravar , d. von harrach , a.m. kotzinian , phys . * b421 * ( 1998 ) 349 ; a. airapetian et al . , hermes collab . lett . * 84 * ( 2000 ) 4047 . see , for example : g. bunce , n. saito , j. soffer , and w. vogelsang , annu . nucl . part . sci . * 50 * , 525 ( 2000 ) , and references therein . see : b. jger , a. schfer , m. stratmann , w. vogelsang , phys . * d67 * ( 2003 ) 054005 , and references therein . m. glck et al . , e. reya , m. stratmann , w. vogelsang , phys . rev . * d63 * ( 2001 ) 094005 . a. bazilevsky , talk presented at the `` x@xmath66 workshop on high energy spin physics ( spin-03 ) '' , dubna , russia , sep . 16 - 20 , 2003 . adler et al . , phenix collab . , hep - ex/0304038 . b. adeva et al . , smc , phys . lett . * b420 * ( 1998 ) 180 . a. airapetian et al . , hermes collab . , hep - ex/0307064 . j. blmlein , h. bttcher , nucl . * b636 * ( 2002 ) 225 . m. stratmann , w. vogelsang , phys . * d64 * ( 2001 ) 114007 ; m. glck , e. reya , hep - ph/0203063 ; a. kotzinian , phys . * b552 * ( 2003 ) 172 ; g. navarro , r. sassot , eur . j. * c28 * ( 2003 ) 321 ; e. christova , s. kretzer , e. leader , eur . j. * c22 * ( 2001 ) 269 ; e. leader , d.b . stamenov , phys . * d67 * ( 2003 ) 037503 ; s.d . bass , phys . * d67 * ( 2003 ) 097502 . x. zheng et al . , jefferson lab hall a collab . , nucl - ex/0308011 . n. isgur , phys . rev . * d59 * ( 1999 ) 034013 . g. farrar , d.r . jackson , phys . * 35 * ( 1975 ) 1416 ; s.j . brodsky , m. burkardt , i. schmidt , nucl . * b441 * ( 1995 ) 197 ; see also : e. leader , a.v . sidorov , d.b . stamenov , int . j. mod a13 * ( 1998 ) 5573 . ralston , d.e . soper , nucl . * b152 * ( 1979 ) 109 ; x. artru , m. mekhfi , z. phys . * c45 * ( 1990 ) 669 ; r.l . jaffe , x. ji , phys . * 67 * ( 1991 ) 552 ; nucl . phys . * b375 * ( 1992 ) 527 . collins , nucl . b394 * ( 1993 ) 169 . collins , nucl . * b396 * ( 1993 ) 161 . a. airapetian et al . , hermes collab . , * 84 * ( 2000 ) 4047 . oganessyan , h.r . avakian , n. bianchi , a.m. kotzinian , hep - ph/9808368 ; p.j . mulders , r.d tangerman , nucl . b461 * ( 1996 ) 197 . sivers , phys . rev . * d41 * ( 1990 ) 83 ; _ ibid . _ * d43 * ( 1991 ) 261 . brodsky , d.s . hwang , i. schmidt , phys . * b530 * ( 2002 ) 99 . collins , phys . * b536 * ( 2002 ) 43 . belitsky , x. ji , f. yuan , nucl . 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we briefly review some of the recent developments in qcd spin physics . bnl - nt-03/28 + rbrc-338 +

You are an expert at summarizing long articles. Proceed to summarize the following text: in the past decade , a series of experiments , in particular the wilkinson microwave anisotropy probe ( wmap ) @xcite , have measured the anisotropy in the cosmic microwave background ( cmb ) radiation with extraordinary precision . the data from these experiments have helped to usher in the era of precision cosmology . most of the cosmological results that have been obtained from these experiments have been derived from the microwave anisotropy at relatively large angular scales . the remarkable agreement between the angular power spectrum of these data and the predictions of the adiabatic inflationary scenario have established the empirical success of inflationary cosmology . cosmic strings , though ruled out as the origin of cosmological structure , have recently enjoyed a renaissance . this renewed popularity has been brought about by the recognition that a variety of string theory - motivated and hybrid models for inflation generically predict the formation of cosmic string networks @xcite . strings are limited to producing less than about 10% of the primordial cmb anisotropy @xcite , though it was shown in @xcite that cmb data can actually favor a contribution from strings if the inflationary spectrum is exactly harrison - zeldovich ( @xmath3 ) ; this corresponds to a string tension ( @xmath4 ) between @xmath5 and @xmath6 for a standard set of string network parameters . the b ( i.e. , curl ) mode polarization in the cmb caused by the active perturbations of a cosmic string network has a spectrum distinct from those expected either from inflationary gravity waves or the lensing of e ( i.e. , gradient ) mode polarization into b mode by large scale structure . when the above bound is marginally saturated , this b mode polarization should be measurable @xcite , providing a powerful test of the presence of cosmic strings . another consequence is the power spectrum of perturbations that strings source at large @xmath7 ( small angular size ) . cosmic string networks continually generate cmb anisotropies , both primordially through active density perturbations and subsequent to recombination through the lensing of the primary cmb light the kaiser - stebbins effect ( ks ) @xcite . the cmb anisotropy due to the ks effect alone at large @xmath7 is expected to decrease only as @xmath8 @xcite . this rate of decrease is much slower than that expected for inflationary perturbations ( which fall off exponentially as a function of @xmath7 due to silk damping , which is due to radiative diffusion ) . if @xmath9 is not too small , this large @xmath7 power spectrum may be measurable . it should appear as an excess above the prediction from inflationary perturbations . in particular , for @xmath10 , the power created on small angular scales , @xmath2 , by cosmic strings will actually dominate over that created by the primary inflationary perturbations . this range of angular scales in the cmb is presently being measured by a number of experiments , so that this prediction of cosmic string networks will be tested soon . in this note , we present the large @xmath7 power spectrum due to cosmic strings . we use cmbact @xcite , a modified version of cmbfast @xcite , to produce the string sourced anisotropy spectra . the model , described in refs . @xcite , is based on representing the cosmic string network as a collection of moving straight string segments . in brief , there are two important length scales in this model : @xmath11 , the length of a string segment , which represents the typical length of roughly straight segments in a full network ; and @xmath12 , the typical length between two string segments , which sets the number density of strings in a given volume ( @xmath13 ) . we use the velocity - dependent one - scale model ( vos ) @xcite to evolve the network length - scales and density . in the simplest and most relevant cases , @xmath14 . with these parameters set , we average over a set of randomized realizations of the network approximated as a set of straight string segments . this model was introduced in @xcite , based on the approach suggested in @xcite , and was developed into its present form in @xcite . the overall normalization of the spectrum has a simple dependence on the string tension and number density : c_^strings n_s ( g)^2 ( ) ^2 . lower reconnection probability @xcite for cosmic strings will rescale the amplitude of this spectrum , but will not change its shape . although this model assumes a single tension cosmic string network , the results described here should generally apply for more complex multi - tension string network models of the sort that may be produced in the aftermath of brane inflation @xcite . [ f1 ] shows our result . the dominance of vector modes ( the blue dashed line in the left panel ) for @xmath15 is an effect that is a distinctive signature of cosmic strings . as we show with the magenta , dash - dotted line , the string spectrum at @xmath16 where vector - sourced perturbations start to dominate is well approximated by a @xmath17 fall off . this can be captured by a simple fitting formula : ( ( + 1)/ 2 ) c_^tt[strings ] 500 ( ) ^1.5 ( ) ^2 for 1000 . we find that towards the very high @xmath7 end of the considered range , i.e. at @xmath18 , the fall off is better described by @xmath19 , in agreement with the pure ks contribution analytically predicted in @xcite . at smaller @xmath7 , the residual fluctuations from the last - scattering surface are non - negligible leading to a @xmath20 fall off in the @xmath21 range . the right panel in fig . [ f1 ] shows the high-@xmath7 power sourced by strings relative to the inflationary contribution . the string spectrum s amplitude is set by saturating the observational bound : it accounts for @xmath22% of power for @xmath0 . ; average coherence length @xmath23 ( measured in units of the horizon size ) ; rms velocity near @xmath24 ; and average wiggliness parameter @xmath25 . the solid black line is the total power . the red dotted line is the power due to scalar perturbations , the blue dashed line represents the vector mode perturbations . the ( green ) long dashed line shows the ( negligible ) contribution from tensor modes . note that vector modes dominate above @xmath26 , a distinctively stringy effect . the magenta dash - dotted line shows the @xmath17 fitting formula . _ right : _ contributions from strings ( solid red ) , inflation ( blue dot ) ( including lensing , from camb @xcite ) , and their combination ( black dash ) at high-@xmath7 . the string contribution satisfies the @xmath22% bound on total power imposed on large angular scales.,title="fig:",width=288 ] ; average coherence length @xmath23 ( measured in units of the horizon size ) ; rms velocity near @xmath24 ; and average wiggliness parameter @xmath25 . the solid black line is the total power . the red dotted line is the power due to scalar perturbations , the blue dashed line represents the vector mode perturbations . the ( green ) long dashed line shows the ( negligible ) contribution from tensor modes . note that vector modes dominate above @xmath26 , a distinctively stringy effect . the magenta dash - dotted line shows the @xmath17 fitting formula . _ right : _ contributions from strings ( solid red ) , inflation ( blue dot ) ( including lensing , from camb @xcite ) , and their combination ( black dash ) at high-@xmath7 . the string contribution satisfies the @xmath22% bound on total power imposed on large angular scales.,title="fig:",width=288 ] the spectrum in fig . [ f1 ] should not be taken as the unique prediction of the @xmath27 spectrum from strings , but as a representative example of what one can get for a reasonably motivated string network . it corresponds to particular values of string model parameters , such as wiggliness , coherence length , and rms velocity . it also relies on the moving segments " approximation used in cmbact . this model is designed to describe statistical properties of scaling string networks . for instance , it can be used to calculate cmb power spectra , but can not make a sky map of string network effects . real string simulations that capture more of the physics of networks such as their curvature and loops are too computationally costly to be used over many expansion times . the segments model has been shown to match cmb spectra from full simulations reasonably well over the scales where they can be compared ( for a fuller discussion , see refs . @xcite ) . while varying the string tension , @xmath4 , simply renormalizes the spectrum , changing the other parameters can re - distribute the power between the large and small scales . generally , smaller velocities and smaller coherence lengths enhance the power in vector modes at high @xmath7 @xcite . more wiggliness tends to make strings move more slowly . this suppresses all contributions to the anisotropic stress , including the vector modes @xcite . for this plot , we used a model with @xmath28 with an average string coherence length of @xmath29 of the horizon size , . ] a rms velocity of @xmath30 , and an average wiggliness parameter @xmath31 of @xmath32 . these values are fairly close to those seen in numerical simulations ( e.g. @xcite ) , but are slightly different from those used in the default version of cmbact ; the values chosen here are those which slightly enhance the string - sourced power at high @xmath7 . fractionally greater ( or attenuated ) power at small angular scales can be achieved with different model parameters , but the overall behavior at high @xmath7 is a generic feature of string networks . all cosmological parameters , _ i.e. _ @xmath33 , @xmath34 , etc , are those of the latest wmap best fit @xcite . cosmic strings produce power on small angular scales because they are _ active _ sources that continue contributing to the anisotropy after the last scattering . for that reason they evade silk damping , _ i.e. _ the erasure of anisotropies on small scales due to the finite thickness of the last scattering surface . a network of cosmic strings with a tension near the present observational bound of @xmath35 can dominate the power spectrum of cmb fluctuations in the strongly silk - damped regime ( @xmath36 ) of the microwave background anisotropy , creating an apparent excess of power over what is expected from an inflationary adiabatic perturbation spectrum . this high @xmath7 regime is accessible to existing fine scale resolution experiments like the cosmic background imager ( cbi ) @xcite and the arcminute cosmology bolometer array receiver ( acbar ) @xcite now , and will soon be measured very accurately by experiments like the south pole telescope @xcite and the atacama cosmology telescope @xcite . it is interesting to note that these experiments published data already show some hints of excess power in the high - multipole range ; see fig . [ withacbar ] for a rough comparison with the data from acbar . cmb anisotropy in the acbar range , plotted against the latest data from the acbar experiment . the dotted blue line is the inflationary prediction alone ( including lensing , from the software package camb @xcite ) ; the dashed black line includes the contribution from a cosmic string network . the amplitude fitting has been done roughly , by eye , to match the two theoretical curves with the second acbar data point ; the cosmological parameters are all those of the wmap best fit . , width=288 ] although we have not done a statistical analysis , the preliminary results presented here suggest that cosmic strings with @xmath37 could contribute enough power to account for the excess over inflation suggested by the acbar data at @xmath38 without exceeding observational bounds on string contributions at @xmath39 . excess power in the high-@xmath7 cmb can also be generated by other physical phenomena , including the sunyaev - zeldovich effect @xcite and tangled primordial magnetic fields @xcite . using the sz effect to account for any substantial excess measurable by today s experiments is problematic , however , because the amount of small - scale gravitational clustering ( measured by the parameter @xmath40 ) required to generate a large excess over the inflationary prediction via the sz effect is in some conflict with values determined by other experiments . tangled magnetic fields , on the other hand , are not meaningfully constrained by competing experiments , but their existence at the necessary epoch is by no means accounted for . discovery of significant high @xmath7 excess power in cmb that can not be explained by more conventional means could be taken as evidence for existence of cosmic strings with tensions near the observational bound . on the other hand , if no excess is seen at large @xmath7 , non - observation of this effect will provide a useful bound on the properties of any cosmic string network . strings with tensions in this range can be searched for by some other means , such as gravitational lensing and microlensing @xcite , gravitational radiation bursts @xcite , pulsar timing or non - gaussian step - like fluctuations in cmb temperature @xcite . an especially promising signature of such strings would be a substantial b mode polarization @xcite ; for @xmath4 around @xmath41 , the b mode polarization fluctuations from strings could exceed expected power from e to b conversion by gravitational lensing by factors of a few . we thank j. richard bond , anthony readhead , and mark hindmarsh for discussions leading to the exploration of this question ; we also thank david chernoff for conversations and jonathan sievers for comments on the draft . lp s work is supported by a discovery grant from the national sciences and engineering research council of canada . shht s work is supported by the national science foundation ( nsf ) under grant phy-0355005 . iw s work is supported by the nsf under grant phy-0555216 . the work of m. w. at the perimeter institute is supported by the government of canada through industry canada and by the province of ontario through the ministry of research & innovation . j. dunkley _ et al . _ [ wmap collaboration ] , arxiv:0803.0586 [ astro - ph ] . s .- h. tye , arxiv : hep - th/0610221 ; + j. polchinski , arxiv:0707.0888 [ astro - ph ] ; 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cosmic string networks generate cosmological perturbations actively throughout the history of the universe . thus , the string sourced anisotropy of the cosmic microwave background is not affected by silk damping as much as the anisotropy seeded by inflation . the spectrum of perturbations generated by strings does not match the observed cmb spectrum on large angular scales ( @xmath0 ) and is bounded to contribute no more than @xmath1 of the total power on those scales . however , when this bound is marginally saturated , the anisotropy created by cosmic strings on small angular scales @xmath2 will dominate over that created by the primary inflationary perturbations . this range of angular scales in the cmb is presently being measured by a number of experiments ; their results will test this prediction of cosmic string networks soon .

You are an expert at summarizing long articles. Proceed to summarize the following text: effects of physical dimensionality on crystallization are among the most important problems in condensed matter physics . as was established by landau and pierls @xcite , the long - range crystalline order in 2d is universally destroyed by thermal fluctuations . the problem has been revisited in 1970s by kosterlitz and thouless @xcite-@xcite , who have shown that crystals do exist in 2d , within a new topological definition . nevertheless , their melting temperature is believed to be always lower than in 3d systems ( assuming the same interparticle potential ) . one of the manifestations of this effect is the phenomenon of surface melting : a microscopic liquid layer normally appear at the interface of a crystalline solid well below its bulk melting temperature . in this paper , we describe a remarkable example in which the thermal fluctuation _ stabilize _ a 2d crystalline solid , embedded in 3d physical space , with respect to transition to an alternative 3d structure . the model system discussed below has been introduced in the recent work by one of us @xcite , in order to describe dna assisted self assembly of colloids . its essential ingredients are cohesive interparticle interactions and medium range soft core repulsion . the binary system of same size spheres ( a and b ) discussed in ref . @xcite , combines the repulsive potential @xmath0 acting between same type particles , with a b attraction . as was shown at that work , both interactions may be induced by properly designed dna . it was found that this colloid dna mixture may exhibit an unusually diverse phase diagram as a function of two control parameter : the relative strength of attraction and repulsion , and aspect ratio @xmath1 ( @xmath2 is the range of repulsive potential @xmath3 , and @xmath4 is the particle diameter ) . among various self assembled phases expected for that system , it was especially striking to find quasi-2d membrane with the in plane square order ( sq ) . in other words , according to our calculations , this 3d system may prefer to self assemble into a lower dimensional structure . we will refer to this phenomenon as spontaneous self compactification . of course , there are other known examples of self compactified structures in condensed matter , such as lipid membranes @xcite . however , our case is quite unique because it is based on _ isotropic pair potentials _ ( in contrast to anisotropic interactions between lipids , or covalent bonding of carbon atoms in graphite ) . note also that at the found sq phase is not a stacking of weakly coupled layers ( lamella like ) , but rather an isolated membrane like structure . in our early calculations , we have only accounted for the interplay of repulsive and attractive energies , while the thermal fluctuations were totally ignored . even though this approximation is applicable when the characteristic energies considerably exceed @xmath5 , the entropic effects are expected to be significant in any realistic case . given that fluctuations are known to strongly affect 2d crystals , one might wonder whether the phenomenon of self compactification will still be present if the fluctuations are introduced . below we present the detailed study of this question . before going into a specific example , we describe our model and its generic features . we start with particles packed into ideal crystalline lattices , whose fluctuation free energies are to be compared . the interparticle potentials will be replaced with linear springs , whose spring constants correspond to the second derivatives of the corresponding potentials . the first derivatives of the potentials will give rise to a pre existing stresses in those springs . we consider only very short range interactions , so that any connections beyond second nearest neighbors will be neglected . let @xmath6 and @xmath7 be spring constants for the first and second nearest neighbor bonds , respectively . repulsion between the second nearest neighbors induces tension @xmath8 in @xmath7springs , which should be balanced by an appropriate compressive force @xmath9 @xmath10 in @xmath6springs . if the interaction range @xmath2 is much shorter than nearest neighbors distance @xmath4 , one expects @xmath11 , and @xmath12 . this gives rise to a hierarchy of elastic constants in this system : @xmath13 @xmath14 . below , we will use harmonic analysis to diagonalize the phonon hamiltonian and find the fluctational contributions to free energies of the competing phases . as an example , we compare 2d square lattice ( sq ) ( embedded in 3d physical space ) to an alternative three dimensional phase with a very similar local structure . figure [ sq ] shows this 3d counterpart of sq ( referred below as `` dual phase '' ) , which also has four nearest neighbors lying in one plane around each particle . the difference from sq is that now there are eight , rather than four second nearest neighbors for each site . because of this difference in the number of repulsive bonds , sq phase is generally preferred energetically over its 3d dual , at the zero temperature limit . below , we will see whether the free energy balance between the two phases may be reversed by the thermal fluctuations . before we present the exact results for sq , and its 3d - dual phase , we discuss certain general features of this problem . given the hierarchy of the spring constants , one can distinguish between several kinds of phonon modes . namely , there are _ hard _ modes , which involve deformations of strong @xmath15springs , and _ soft _ modes which only depend on spring constant @xmath7 . there are @xmath16 degrees of freedom per particle , and @xmath17 of them correspond to hard modes , both for 2d lattice and its 3d dual phase . in 3d structure , the third mode is the soft one ( @xmath7-mode ) . however , if there were no stresses @xmath18 and @xmath8 , within harmonic approximation there would be no restoring force for the out - of - plane fluctuations in the 2d structure . this means that in 2d lattice the effective spring constant for the transverse modes should be of the order of @xmath19 ( since @xmath20 , we may justifiably call these modes of 2d lattice @xmath21 ) . because of the replacement of the soft modes with the supersoft ones , we do expect the entropy of the quasi2d structure to be higher , which will further shift the free energy balance to its favor . in addition , in small@xmath22 limit , the out - of - plane modes in 2d structure will universally have dispersion @xmath23 ( associated with bending rigidity ) . this will give another negative correction to free energy compared to dispersion @xmath24 of regular acoustic phonons . note that these arguments are quite universal , and can be applied beyond the particular case of sq lattice . we now proceed with the discussion of the specific example , 2d sq vs. its 3d - dual phase . the latter 3d structure may be obtained from a simple cubic lattice , by removing the particles occupying sites . @xmath25 and @xmath26 . in this geometry , the equilibrium condition requires @xmath27 , compared to @xmath28 for sq . note that it is tension @xmath8 , not @xmath18 which is expected to be nearly identical for the two competing structures , since it is given by derivative of the potential , @xmath29 ( taken at the distance of the second nearest neighbor ) . in contrast , @xmath18 can be considerably varied by relatively small deformation of the strong @xmath6spring . an obvious choice of unit cell for 2d lattice is a square containing one atom . for the 3d structure , consider a cube containing eight smaller cubes , as shown on figure [ sq](b ) . the particles are located at @xmath30 , where @xmath31 is @xmath32 , @xmath32 , @xmath33 , @xmath34 , @xmath35 , @xmath36 . the translational symmetry in this structure is generated by vectors @xmath37 , @xmath38 and @xmath39 , i. e. the cube on the picture corresponds to two unit cells . given the translational symmetry , the particle at @xmath39 is equivalent to @xmath40 ( we label them as type a particles ) , @xmath41 to @xmath32 ( type b ) and @xmath34 to @xmath42 ( type c ) . we can now obtain hamiltonian @xmath43 and calculate the phonon contribution to free energy . the number of particles per a unit cell is , @xmath44 for sq and @xmath45 for its 3d - dual . for the 2d phase , displacement of a particle at ( m , n ) is @xmath46 . for the 3d phase , the displacements there are three families of modes , corresponding to the three types of non - equivalent particles , a , b , and c : @xmath47 the phonon free energy has general form @xmath48 \exp \left ( -\frac{h}{k_{b}t}\right ) \label{free - en}\ ] ] here index @xmath49 parameterizes all the modes for a given wave vector @xmath50 , and hamiltonian @xmath43 is given by@xmath51by performing the gaussian integration over @xmath52 , one can transform eq . ( [ free - en ] ) into : @xmath53 \right ) \label{dets}\ ] ] the integration here is performed over a single brilluen zone , i.e. each component of the wave vector runs from @xmath54 to @xmath55 . note that in our regime ( @xmath56 and @xmath57 ) , one can neglect the coupling between hard and soft modes , which allows one represent the determinants entering eq . ( [ dets ] ) in factorized form : @xmath58 \simeq \det \left [ \hat{\gamma}^{\left ( hard\right ) } \left ( \mathbf{q}% \right ) \right ] \det \left [ \hat{\gamma}^{\left ( soft\right ) } \left ( \mathbf{% q}\right ) \right ] $ ] . furthermore , the hard modes have essentially identical spectra in the both structures ( corresponding to one - dimensional chain of @xmath15-springs ) . therefore , the difference of the fluctuational free energies is mainly due to the soft mode contributions . in the case of sq phase , there is only one soft mode : @xmath59 = \hat{\gamma}_{zz}\left ( \mathbf{q}\right ) = \frac{4\sqrt{2}\tilde{% \tau}}{a}\sin ^{2}\left ( \frac{q_{1}}{2}\right ) \sin ^{2}\left ( \frac{q_{2}}{% 2}\right)\]]for the 3d dual phase , 3 out of 9 modes are soft , namely @xmath60 : @xmath61\]]here @xmath62,@xmath63 $ ] , and @xmath64 . thus , @xmath65 \simeq 8\tilde{\kappa}^{3}\left ( 1 + 3\alpha \right ) ^{3}\left [ 1-\varpi_{1}\left ( \frac{1+\alpha } { 8\left ( 1 + 3\alpha \right ) } \right ) ^{2}+\varpi_{2}\left ( \frac{1+\alpha } { 8\left ( 1 + 3\alpha \right ) } \right ) ^{3}\right]\ ] ] here @xmath66 , and @xmath67 . now we can calculate the fluctuational contribution into the free energy difference between the two structures : @xmath68 \label{result}\ ] ] consistent with the above general arguments , we obtain @xmath69 , i.e. the entropic effects enhance the stability of the self compactified phase . one can identify the physical origin of each of the three contributions at the above result . the first logarithmic term is due to an entropic gain of a single fluctuation particle with all of its neighbors fixed . one can see that these fluctuations are enhanced in 2d case ( for an arbitrary value of parameter @xmath70 ) , mainly due to the smaller number of second nearest neighbors there . the second term , @xmath71 represent an additional gain due to @xmath72 dispersion of the soft acoustic phonon in sq phase . in contrast , the soft modes in 3d phase correspond to optical phonons , whose dispersion gives rise to the negligible third term in eq . ( [ result ] ) . since the effects responsible for the dominant contributions to our result , eq . ( [ result ] ) , are not specific for the above example , we expect our conclusions to be rather universal , and applicable to other self compactified phases . in fact , the very same binary system introduced in ref . @xcite , provides other examples of such phases , e. g. _ honeycomb _ lattice . this structure has several 3d counterparts @xcite , some of which are shown on fig.([honey ] ) . we expect the thermal fluctuations to enhance the stability of honeycomb with respect to transition to alternative 3d structures , as in the case of sq . another aspect of our observation is that the region of the phase diagram corresponding to quasi-2d structures ( like sq or honeycomb ) , should expand when thermal fluctuations are introduced . however , in this study we have not discuss the transition between solid and liquid phases . in fact , the strong fluctuations in 2d structures indicate that they should melt earlier than 3d solid phases , which is consistent with the classical picture . in other words , the thermal fluctuations enhance the stability of quasi-2d latices with respect to transition to 3d crystalline structures , but not to the disordered liquid phase . note also that the self - compactified structure need not to be an ideal crystal , even topologically . introduction of defects such as disclination or dislocation , normally associated with 2d melting , is not expected affect our arguments , as long as the systems remains effectively two - dimensional . that is because the dominant contribution to the free energy is associated with length scales of the order of the interparticle distance . hence , the macroscopic properties and conformation of these lattices are not relevant for the discussed phenomenon . as long at the large - scale behavior is concern , one might expect their properties to be similar to those of tethered ( solid - like ) membranes @xcite . it should be noted that the non - zero bending rigidity is known to stabilize such a membrane with respect to crumpling transition . one may give a simple qualitative interpretation to the obtained results . the self compactification in zero - fluctuation limit corresponds to the regime of the interparticle repulsion which is too weak to destabilize the 2d structure , yet strong enough to cause an overall repulsion between two such layers . introduction of thermal fluctuations results in an additional effective repulsion between these 2d layers , an effect very similar to helfrich interaction between conventional membranes @xcite , helfrich .

we discuss the phenomenon of spontaneous self compactification in a model colloidal system , proposed in a recent work on dna mediated self assembly . we focus on the effect of thermal fluctuations on the stability of membrane - like self assembled phase with in - plane square order . surprisingly , the fluctuations are shown to enhance the stability of this quasi2d phase with respect to transition to alternative 3d structures . * pacs numbers : 64.70.kb , 64.70.nd *

You are an expert at summarizing long articles. Proceed to summarize the following text: different regression models have been proposed for lifetime data such as those based on the gamma , lognormal and weibull distributions . these models typically provide a satisfactory fit in the middle portion of the data , but very often fail to deliver a good fit at the tails , where only a few observations are generally available . the family of distributions proposed by birnbaum and saunders ( 1969 ) can also be used to model lifetime data and it is widely applicable to model failure times of fatiguing materials . this family has the appealing feature of providing satisfactory tail fitting . this family of distributions was originally obtained from a model for which failure follows from the development and growth of a dominant crack . it was later derived by desmond ( 1985 ) using a biological model which followed from relaxing some of the assumptions originally made by birnbaum and saunders ( 1969 ) . the random variable @xmath0 is said to be birnbaum saunders distributed with parameters @xmath1 , say @xmath2-@xmath3 , if its cumulative distribution function ( cdf ) is given by @xmath4,\quad t > 0,\ ] ] where @xmath5 is the standard normal distribution function and @xmath6 and @xmath7 are shape and scale parameters , respectively . it is easy to show that @xmath7 is the median of the distribution : @xmath8 . for any @xmath9 , then @xmath10-@xmath11 . mccarter ( 1999 ) considered parameter estimation under type ii data censoring for the @xmath2-@xmath12 distribution . lemonte et al . ( 2007 ) derived the second - order biases of the maximum likelihood estimates ( mles ) of @xmath6 and @xmath7 , and obtained a corrected likelihood ratio statistic for testing the parameter @xmath6 . lemonte et al . ( 2008 ) proposed several bootstrap bias corrected estimates of @xmath6 and @xmath7 . further details on the birnbaum saunders distribution can be found in johnson et al . ( 1995 ) . rieck and nedelman ( 1991 ) proposed a log - linear regression model based on the birnbaum saunders distribution . they showed that if @xmath13-@xmath12 , then @xmath14 is sinh - normal distributed , say @xmath15 , with shape , location and scale parameters given by @xmath6 , @xmath16 and @xmath17 , respectively . their model has been widely used as an alternative model to the gamma , lognormal and weibull regression models ; see rieck and nedelman ( 1991 , 7 ) . diagnostic tools for the birnbaum saunders regression model were developed by galea et al . ( 2004 ) , leiva et al . ( 2007 ) and xie and wei ( 2007 ) , and the bayesian inference was considered by tisionas ( 2001 ) . in this paper we propose a class of birnbaum saunders nonlinear regression models which generalizes the regression model introduced by rieck and nedelman ( 1991 ) . we discuss maximum likelihood estimation of the regression parameters and obtain the fisher information matrix . as is well known , however , the mles , although consistent , are typically biased in finite samples . in order to overcome this shortcoming , we derive a closed - form expression for the bias of the mle in these models which is used to define a bias corrected estimate . bias adjustment has been extensively studied in the statistical literature . in fact , cook et al . ( 1986 ) proposed bias correction in normal nonlinear models . young and bakir ( 1987 ) obtained bias corrected estimates for a generalized log - gamma regression model . cordeiro and mccullagh ( 1991 ) gave general matrix formulae for bias correction in generalized linear models , whereas paula ( 1992 ) derive the second - order biases in exponential family nonlinear models . cordeiro et al . ( 2000 ) obtained bias correction for symmetric nonlinear regression models . more recently , vasconcellos and cribari neto ( 2005 ) calculate the biases of the mles in a new class of beta regression . cordeiro and demtrio ( 2008 ) propose formulae for the second - order biases of the maximum quasi - likelihood estimates , whereas cordeiro and toyama ( 2008 ) derive the second - order biases in generalized nonlinear models with dispersion covariates . the rest of the paper is as follows . section [ reg_nonlinear ] introduces the class of birnbaum - saunders nonlinear regression models and discusses maximum likelihood estimation . using general results from cox and snell ( 1968 ) , we derive in section [ bias ] the second - order biases of the mles of the nonlinear parameters in our class of models and define bias corrected estimates some special models are considered in section 4 . simulation results are presented and discussed in section [ simulation ] for two nonlinear regression models . we show that the bias corrected estimates are nearly unbiased with mean squared errors very close to the corresponding ones of the uncorrected estimates . section [ application ] gives an application of the proposed regression model to a real fatigue data set , which provides a better fit at the tail of the data . finally , section [ conclusions ] concludes the paper . let @xmath13-@xmath3 . the density function of @xmath18 has the form ( rieck and nedelman , 1991 ) @xmath19 this distribution has a number of interesting properties ( rieck , 1989 ) : ( i ) it is symmetric around the location parameter @xmath20 ; ( ii ) it is unimodal for @xmath21 and bimodal for @xmath22 ; ( iii ) the mean and variance of @xmath23 are @xmath24 and var@xmath25 , respectively . there is no closed - form expression for @xmath26 , but rieck ( 1989 ) obtained asymptotic approximations for both small and large values of @xmath6 ; ( iv ) if @xmath27 , then @xmath28 converges in distribution to the standard normal distribution when @xmath29 . we define the nonlinear regression model @xmath30 where @xmath31 is the logarithm of the @xmath32th observed lifetime , @xmath33 is an @xmath34 vector of known explanatory variables associated with the @xmath32th observable response @xmath31 , @xmath35 is a vector of unknown nonlinear parameters , and @xmath36 . we assume a nonlinear structure for the location parameter @xmath37 in model ( [ eq1 ] ) , say @xmath38 , where @xmath39 is assumed to be a known and twice continuously differentiable function with respect to @xmath40 . for the linear regression @xmath41 , the model ( [ eq1 ] ) reduces to rieck and nedelman s ( 1991 ) model . the log - likelihood function for the vector parameter @xmath42 from a random sample @xmath43 obtained from ( [ eq1 ] ) , except for constants , can be expressed as @xmath44 where @xmath45/2)$ ] , @xmath46/2)$ ] for @xmath47 . the function @xmath48 is assumed to be regular ( cox and hinkley , 1974 , ch . 9 ) with respect to all @xmath40 and @xmath6 derivatives up to third order . further , the @xmath49 local matrix @xmath50 of partial derivatives of @xmath51 with respect to @xmath40 is assumed to be of full rank , i.e. , rank(@xmath52 for all @xmath40 . the nonlinear predictors @xmath53 are embedded in an infinite sequence of @xmath54 vectors that must satisfy these regularity conditions for the asymptotics to be valid . under these assumptions , the mles have good asymptotic properties such as consistency , sufficiency and normality . the derivatives with respect to the components of @xmath40 and @xmath6 are denoted by : @xmath55 , @xmath56 , @xmath57 , @xmath58 , @xmath59 , @xmath60 , etc . further , we use the following notation for joint cumulants of log - likelihood derivatives : @xmath61 , @xmath62 , @xmath63 , etc . let @xmath64 , etc . all @xmath65 s and their derivatives are assumed to be of order @xmath66 . also , we adopt the notation @xmath67 and @xmath68 for the first and second partial derivatives of @xmath37 with respect to the elements of @xmath40 . it is easy to see by differentiating ( [ eq2 ] ) that @xmath69 @xmath70 @xmath71 the score function for @xmath40 is @xmath72 , where @xmath73 is an @xmath74-vector whose @xmath32th element is equal to @xmath75 . it is well - known that , under general regularity conditions ( cox and hinkley , 1974 , ch . 9 ) , the mles are consistent , asymptotically efficient and asymptotically normal . let @xmath76 be the mle of @xmath42 . we can write @xmath77 for @xmath74 large , where @xmath78 denotes approximately distributed , @xmath79 is the block - diagonal fisher information matrix given by @xmath80 , @xmath81 is its inverse , @xmath82 is the information matrix for @xmath40 and @xmath83 is the information for @xmath6 . also , @xmath84 where @xmath85 is the error function given by @xmath86 details on @xmath87 can be found in gradshteyn and ryzhik ( 2007 ) . since @xmath79 is block - diagonal , the vector @xmath40 and the scalar @xmath6 are globally orthogonal ( cox and reid , 1987 ) and @xmath88 and @xmath89 are asymptotically independent . it can be shown ( rieck , 1989 ) that @xmath90 for @xmath6 small and @xmath91 for @xmath6 large . the mle @xmath92 satisfies @xmath93 equations @xmath94 for the components of @xmath40 and @xmath6 . the fisher scoring method can be used to estimate @xmath40 and @xmath6 simultaneously by iteratively solving the equations @xmath95 where @xmath96 and @xmath97 for @xmath98 . the above equations show that any software with a weighted linear regression routine can be used to calculate the mles of @xmath40 and @xmath6 iteratively . initial approximations @xmath99 and @xmath100 for the iterative algorithm are used to evaluate @xmath101 and @xmath102 from which these equations can be used to obtain the next estimates @xmath103 and @xmath104 . these new values can update @xmath105 and @xmath106 and so the iterations continue until convergence is achieved . we now obtain some joint cumulants of log - likelihood derivatives and their derivatives : @xmath107 @xmath108 @xmath109 let @xmath110 and @xmath111 be the @xmath112 biases of @xmath113 ( @xmath114 ) and @xmath89 , respectively . the use of cox and snell s ( 1968 ) formula to obtain these biases is greatly simplified , since @xmath40 and @xmath6 are globally orthogonal and the cumulants corresponding to the parameters in @xmath40 are invariant under permutation of these parameters . from now on we use einstein summation convention with the indices varying over the corresponding parameters . we have @xmath115 and @xmath116 where @xmath117 is the @xmath118th element of the inverse @xmath119 of the information matrix for @xmath40 , @xmath120 and @xmath121 denotes the summation over all combinations of parameters @xmath122 . first , we consider equation ( [ vies - beta ] ) from which we readily have that the second sum is zero since @xmath123 . it follows that @xmath124 by rearranging the summation terms we obtain @xmath125 let @xmath126 @xmath127 and @xmath128 @xmath129 be vectors containing the first and second partial derivatives of the mean @xmath37 with respect to the @xmath130 s . we can write the above equation in matrix notation as @xmath131 where @xmath132 is the @xmath133th row of the @xmath134 identity matrix and vec@xmath135 is the operator which transforms a matrix into a vector by stacking the columns of the matrix one underneath the other . it is straightforward to check that @xmath136 where @xmath137 and @xmath138 are @xmath49 and @xmath139 matrices of the first and second partial derivatives of the mean vector @xmath140 with respect to @xmath40 , respectively . the @xmath112 bias vector @xmath141 of @xmath88 can then be written as @xmath142 where @xmath143 is an @xmath144 vector defined as @xmath145 . we now calculate the @xmath112 bias of @xmath89 . using ( [ vies - alpha ] ) , we obtain @xmath146 where @xmath147 denotes the trace operator . now , making use of the fact that @xmath148 , we can rewrite the @xmath112 bias of @xmath89 as @xmath149 equations ( [ bias - beta ] ) and ( [ bias - alpha ] ) represent the main results of the paper . the bias vector @xmath141 can be obtained from a simple ordinary least - squares regression of @xmath143 on the columns of @xmath150 . it depends on the nonlinearity of the regression function @xmath151 and the parameter @xmath6 . the bias vector @xmath141 will be small when @xmath143 is orthogonal to the columns of @xmath150 . also , it can be large when @xmath152 and @xmath74 are both small . equation ( [ bias - beta ] ) is easily handled algebraically for any type of nonlinear regression , since it involves simple operations on matrices and vectors . for special models with closed - form information matrix for @xmath40 , it is possible to obtain closed - form expressions for @xmath141 . for linear models , the matrix @xmath153 and the vector @xmath143 vanish and hence @xmath154 , which is in agreement with the result due to rieck and nedelman ( 1991 , p. 54 ) that the mles are unbiased to order @xmath112 . expression ( 6 ) depends directly on the nonlinear structure of the regression model only through the rank @xmath155 of @xmath150 . it shows that the bias is always a linear function of the dimension @xmath155 of @xmath40 . in the right - hand sides of expressions ( [ bias - beta ] ) and ( [ bias - alpha ] ) , which are both of order @xmath112 , consistent estimates of the parameters @xmath40 and @xmath6 can be inserted to define bias corrected estimates @xmath156 and @xmath157 , where @xmath158 and @xmath159 are the values of @xmath141 and @xmath111 , respectively , at @xmath160 . the bias corrected estimates @xmath161 and @xmath162 are expected to have better sampling properties than the classical mles @xmath88 and @xmath89 . in fact , we present some simulations in section [ simulation ] to show that @xmath161 and @xmath162 have smaller biases than their corresponding uncorrected estimates , thus suggesting that these bias corrections have the effect of shrinking the adjusted estimates toward to the true parameter values . however , we can not say that the bias corrected estimates offer always some improvement over the mles , since they can have mean squared errors larger . it is worth emphasizing that there are other methods to obtain bias corrected estimates . in regular parametric problems , firth ( 1993 ) developed the so - called `` preventive '' method , which also allows for the removal of the second - order bias . his method consists of modifying the original score function to remove the first - order term from the asymptotic bias of these estimates . in exponential families with canonical parameterizations , his correction scheme consists in penalizing the likelihood by the jeffreys invariant priors . this is a preventive approach to bias adjustment which has its merits , but the connections between our results and his work are not pursued in this paper since they could be developed in future research . additionally , it should be mentioned that it is possible to avoid cumbersome and tedious algebra on cumulant calculations by using efron s bootstrap ( efron and tibshirani , 1993 ) . we use the analytical approach here since this leads to a nice formula . moreover , the application of the analytical bias approximation seems to generally be the most feasible procedure to use and it continues to receive attention in the literature . we now calculate the second - order bias @xmath163 of the mle @xmath164 of the @xmath32th mean @xmath38 . we can easily show by taylor series expansion that @xmath165 where @xmath166 is a @xmath134 matrix of second partial derivatives @xmath167 ( for @xmath168 ) , @xmath169 is the asymptotic covariance matrix of @xmath88 and the vectors @xmath170 and @xmath141 were mentioned previously . all quantities in the above equation should be evaluated at @xmath88 . the asymptotic variance of @xmath171 can also be expressed explicitly in terms of the covariance of @xmath88 by @xmath172 equation ( [ bias - beta ] ) is easily handled algebraically for any type of nonlinear model , since it involves simple operations on matrices and vectors . this equation , in conjunction with a computer algebra system such as maple ( abell and braselton , 1994 ) will compute @xmath141 algebraically with minimal effort . in particular , ( [ bias - beta ] ) may simplify considerably if the number of nonlinear parameters is small . moreover , for any nonlinear special model , we can calculate the bias @xmath141 numerically via a software with numerical linear algebra facilities such as ox ( doornik , 2001 ) and r ( r development core team , 2008 ) . first , we consider a nonlinear regression model which depends on a single nonlinear parameter . equation ( [ bias - beta ] ) gives @xmath173 where @xmath174 and @xmath175 . the constants @xmath176 and @xmath177 are evaluated at @xmath178 and @xmath89 to yield @xmath179 and the corrected estimate @xmath180 . for example , the simple exponential model @xmath181 leads to @xmath182 and @xmath183 . as a second example , we consider a partially nonlinear regression model defined by @xmath184 where @xmath185 is a known @xmath186 matrix of full rank , @xmath187 is an @xmath144 vector , @xmath188 , @xmath189 and @xmath7 and @xmath190 are scalar parameters . this class of models occurs very often in statistical modeling ; see cook et al . ( 1986 ) and cordeiro et al . ( 2000 ) . for example , @xmath191 ( gallant , 1975 ) , @xmath192 ( darby and ellis , 1976 ) and @xmath193 ( stone , 1980 ) . ratkowsky ( 1983 , ch . 5 ) discusses several models of the form ( [ mspecial ] ) which include the asymptotic regression and weibull - type models given by @xmath194 and @xmath195 , respectively . the @xmath49 local model matrix @xmath150 takes the form @xmath196 $ ] and , after some algebra , we can obtain from ( [ bias - beta ] ) @xmath197,\ ] ] where @xmath198 is a @xmath199 vector with a one in the last position and zeros elsewhere , @xmath200 ) is simply the set of coefficients from the ordinary regression of the vector @xmath201 on the matrix @xmath150 , and @xmath202 and @xmath203 are the large - sample second moments obtained from the appropriate elements of the asymptotic covariance matrix @xmath204 . it is clear from ( [ b1 ] ) that @xmath141 does not depend explicitly on the linear parameters in @xmath205 and it is proportional to @xmath206 . further , the covariance term @xmath203 contributes only to the bias of @xmath207 . we now use monte carlo simulation to evaluate the finite - sample performance of the mles of the parameters and of their corrected versions in two nonlinear regression models . the mles of the parameters were obtained by maximizing the log - likelihood function using the bfgs quasi - newton method with analytical derivatives . this method is generally regarded as the best - performing nonlinear optimization method ( mittelhammer et al . , 2000 , p. 199 ) . the covariate values were selected as random draws from the uniform @xmath208 distribution and for fixed @xmath74 those values were kept constant throughout the experiment . also , the number of monte carlo replications was 10,000 . all simulations were performed using the ox matrix programming language ( doornik , 2001 ) . in order to analyze the performance of the estimates , we computed , for each sample size and for each estimate : the relative bias ( the relative bias of an estimate @xmath209 , defined as @xmath210 , is obtained by estimating @xmath211 by monte carlo ) and the root mean square error ( @xmath212 ) , where mse is the estimated mean square error from the 10,000 monte carlo replications . first , we consider the nonlinear regression model @xmath213 where @xmath36 for @xmath47 . the sample sizes were @xmath214 and 45 . without loss of generality , the true values of the regression parameters were taken as @xmath215 , @xmath216 , @xmath217 , @xmath218 and @xmath219 and @xmath220 . table [ tab1 ] gives the relative biases of both uncorrected and corrected estimates to show that the bias corrected estimates are much closer to the true parameters than the unadjusted estimates . for instance , when @xmath221 and @xmath222 , the average of the estimated relative biases for the estimates of the model parameters is @xmath223 , whereas the average of the estimated relative biases for the corrected estimates is @xmath224 . hence , the average bias ( in absolute value ) of the mles is almost four times greater than the average bias of the corrected estimates . this fact suggests that the second - order bias of the mles should not be ignored in samples of small to moderate size since they can be non - negligible . the figures in table 2 show that the root mean squared errors of the uncorrected and corrected estimates are very close . hence , the figures in both tables suggest that the corrected estimates have good properties . ccl rrrrr@xmath6 & @xmath74 & & @xmath225 & @xmath226 & @xmath227 & @xmath228 & @xmath229 + 0.5 & 15 & mle & 0.0006 & @xmath230 & 0.0011 & 0.0020 & @xmath231 + & & bce & 0.0007 & @xmath232 & 0.0001 & 0.0008 & @xmath233 + & 30 & mle & 0.0001 & @xmath230 & 0.0013 & 0.0009 & @xmath234 + & & bce & 0.0002 & @xmath235 & 0.0007 & @xmath236 & @xmath237 + & 45 & mle & 0.0003 & @xmath235 & 0.0007 & 0.0008 & @xmath238 + & & bce & 0.0003 & @xmath232 & 0.0003 & 0.0001 & @xmath239 + 1.5 & 15 & mle & @xmath240 & @xmath224 & 0.0248 & 0.0197 & @xmath241 + & & bce & @xmath242 & @xmath243 & 0.0113 & 0.0056 & @xmath244 + & 30 & mle & @xmath245 & @xmath246 & 0.0079 & 0.0078 & @xmath247 + & & bce & @xmath232 & @xmath248 & 0.0027 & 0.0012 & @xmath249 + & 45 & mle & @xmath250 & @xmath251 & 0.0052 & 0.0026 & @xmath252 + & & bce & @xmath253 & @xmath248 & 0.0023 & @xmath254 & @xmath255 + ccl rrrrr@xmath6 & @xmath74 & & @xmath225 & @xmath226 & @xmath227 & @xmath228 & @xmath229 + 0.5 & 15 & mle & 0.4093 & 0.4920 & 0.2707 & 0.0924 & 0.1234 + & & bce & 0.4093 & 0.4921 & 0.2709 & 0.0922 & 0.1067 + & 30 & mle & 0.3006 & 0.3806 & 0.2113 & 0.0688 & 0.0763 + & & bce & 0.3006 & 0.3806 & 0.2114 & 0.0686 & 0.0702 + & 45 & mle & 0.2434 & 0.2874 & 0.1768 & 0.0567 & 0.0590 + & & bce & 0.2434 & 0.2874 & 0.1769 & 0.0566 & 0.0555 + 1.5 & 15 & mle & 1.6302 & 1.1230 & 0.9756 & 0.3235 & 0.3938 + & & bce & 1.6333 & 1.1274 & 0.9819 & 0.3152 & 0.3315 + & 30 & mle & 0.9684 & 0.7003 & 0.5785 & 0.1931 & 0.2399 + & & bce & 0.9693 & 0.7011 & 0.5807 & 0.1908 & 0.2155 + & 45 & mle & 0.6505 & 0.5575 & 0.3895 & 0.1318 & 0.1837 + & & bce & 0.6507 & 0.5577 & 0.3901 & 0.1311 & 0.1700 + when the parameter @xmath6 increases , the finite - sample performance of the mles deteriorates ( see tables [ tab1 ] and [ tab2 ] ) . for instance , when @xmath221 , the relative biases of @xmath207 ( mle ) and @xmath256 ( bce ) were 0.0020 and 0.0008 ( for @xmath219 ) and 0.0197 and 0.0056 ( for @xmath222 ) , which indicate an increase in the relative biases of nearly 10 and 7 times , respectively . also , the root mean squared errors in the same order were 0.0924 and 0.0922 ( for @xmath219 ) and 0.3235 and 0.3152 ( for @xmath222 ) . next , we consider the very known michaelis menton model , which is very useful for estimating growth curves , where it is common for the response to approach an asymptote as the stimulus increases . the michaelis menton model ( ratkowsky , 1983 ) provides an hyperbolic form for @xmath37 against @xmath257 given by @xmath258 where the curve has an asymptote at @xmath259 . here , the sample sizes were @xmath260 and 50 . also , the true values of the regression parameters were taken as @xmath261 and @xmath262 , with @xmath219 . table [ tab3 ] gives the relative biases and root mean squared errors of the uncorrected and corrected estimates . the figures in this table reveal that the mles of the parameters can be substantially biased , even when @xmath263 , and that the bias correction is very effective . in terms of mse , the adjusted estimates are slightly better than the ordinary mles . cl rrr|rrr & & & + @xmath74 & & @xmath7 & @xmath190 & @xmath6 & @xmath7 & @xmath190 & @xmath6 + 20 & mle & 0.0476 & 0.1718 & @xmath264 & 0.6984 & 0.3947 & 0.0859 + & bce & @xmath245 & @xmath265 & @xmath266 & 0.5264 & 0.2783 & 0.0847 + 30 & mle & 0.0313 & 0.1077 & @xmath267 & 0.5245 & 0.2750 & 0.0684 + & bce & @xmath268 & @xmath269 & @xmath270 & 0.4478 & 0.2252 & 0.0678 + 40 & mle & 0.0215 & 0.0754 & @xmath271 & 0.4222 & 0.2207 & 0.0582 + & bce & @xmath236 & @xmath272 & @xmath273 & 0.3835 & 0.1954 & 0.0578 + 50 & mle & 0.0160 & 0.0558 & @xmath274 & 0.3609 & 0.1862 & 0.0516 + & bce & @xmath275 & @xmath236 & @xmath254 & 0.3380 & 0.1710 & 0.0514 + obviously , due to the genesis of the birnbaum saunders distribution , the fatigue processes are by excellence ideally modeled by this model . we now consider an application to a biaxial fatigue data set reported by rieck and nedelman ( 1991 ) on the life of a metal piece in cycles to failure . the response @xmath276 is the number of cycles to failure and the explanatory variable @xmath277 is the work per cycle ( mj / m@xmath278 ) . the data of forty six observations were taken from table 1 of galea et al . ( 2004 ) . rieck and nedelman ( 1991 ) proposed the following model for the biaxial fatigue data : @xmath279 where @xmath280 and @xmath281 , for @xmath282 . the mles ( the corresponding standard errors in parentheses ) are : @xmath283(0.3942 ) , @xmath284(0.1096 ) and @xmath285(0.0428 ) . we take the logarithm of @xmath277 to ensure a linear relationship between the response variable ( @xmath286 ) and the covariate in ( [ rnmodel ] ) ; see galea et al . ( 2004 , figure 1 ) . however , figure [ fig1 ] suggests a nonlinear relationship between the response variable and the covariate @xmath277 . here , we proposed the nonlinear regression model @xmath287 where @xmath36 . the mles ( the standard errors in parentheses ) are : @xmath288(0.7454 ) , @xmath289(0.5075 ) , @xmath290(7.3778 ) and @xmath291(0.0417 ) . the bias corrected estimates are : @xmath292(0.7734 ) , @xmath293(0.5266 ) , @xmath294(7.6548 ) and @xmath295(0.0433 ) . hence , the uncorrected estimates are slightly different from the bias corrected estimates even for large samples ( @xmath296 observations ) . figure [ fig2 ] gives the scatter - plot of the data , the fitted model ( [ nonlinear ] ) and the fitted straight line , say @xmath297 , where the mles are : @xmath298(0.1622 ) , @xmath299(0.0036 ) and @xmath300(0.0542 ) . figure [ fig2 ] shows that the nonlinear model ( 10 ) ( unlike the linear model ) fits satisfactorily to the fatigue data . the @xmath301th observation ( the one with work per cycle near 100 ) can be an influential data . however , it is not possible to say whether this observation is influential or not without using an efficient way to detect influential observations in the new class of models . influence diagnostic analysis for this class of models will be developed in future research . following xie and wei ( 2007 ) , we obtain the residuals @xmath302 and @xmath303 . figure [ fig3 ] gives the scatter - plot of @xmath304 versus the predicted values @xmath171 for both fitted models : ( i ) @xmath305 ; and ( ii ) @xmath306 . figure [ fig3 ] shows that the distribution of @xmath304 is approximately normal for model ( ii ) but this is not true for model ( i ) . based upon the fact that @xmath307 if @xmath308 , then the residual @xmath309 should follow approximately a sinh - normal distribution . versus @xmath171.,title="fig:",width=453,height=340 ] versus @xmath171.,title="fig:",width=453,height=340 ] the birnbaum saunders distribution is widely used to model times to failure for materials subject to fatigue . the purpose of the paper was two fold . first , we propose a new class of birnbaum saunders nonlinear regression models which generalizes the regression model described in rieck and nedelman ( 1991 ) . second , we give simple formulae for calculating bias corrected maximum likelihood estimates of the parameters of these models . the simulation results presented show that the bias correction derived is very effective , even when the sample size is large . indeed , the bias correction mechanism adopted yields adjusted maximum likelihood estimates which are nearly unbiased . we also present an application to a real fatigue data set that illustrates the usefulness of the proposed model . future research will be devoted to a study of diagnostics and influence analysis in the new class of nonlinear models . we gratefully acknowledge grants from fapesp and cnpq ( brazil ) . the authors are also grateful to an associate editor and two referees for helpful comments and suggestions .

we introduce , for the first time , a new class of birnbaum saunders nonlinear regression models potentially useful in lifetime data analysis . the class generalizes the regression model described by rieck and nedelman [ 1991 , a log - linear model for the birnbaum saunders distribution , _ technometrics _ , * 33 * , 5160 ] . we discuss maximum likelihood estimation for the parameters of the model , and derive closed - form expressions for the second - order biases of these estimates . our formulae are easily computed as ordinary linear regressions and are then used to define bias corrected maximum likelihood estimates . some simulation results show that the bias correction scheme yields nearly unbiased estimates without increasing the mean squared errors . we also give an application to a real fatigue data set . bias correction , birnbaum saunders distribution , maximum likelihood estimation , nonlinear regression .

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Proceed to summarize the following text: the phenomenon of color superconductivity @xcite is of general interest , in particular , in studies of the qcd phase structure @xcite and applications in the astrophysics of compact stars @xcite . observable consequences are expected for , e. g. , the cooling behavior @xcite . different aspects have been investigated so far , whereby models of the njl type have been widely employed @xcite in studies of the phase structure in the vicinity of the hadronization transition . recently , it has been shown in these investigations that for low temperatures ( @xmath2 ) and not too large chemical potentials ( @xmath3 ) the two - flavor color superconductivity ( 2sc ) phase is favored over alternative color superconducting phases @xcite . according to @xcite , the color - flavor - locked ( cfl ) phase occurs only at @xmath4 mev . it is generally agreed that at low temperatures the transition of the matter from the phase with broken chiral symmetry to the color superconducting phase is of the first order ( see e. g. @xcite ) . from the point of view of phenomenological applications , as e.g. in compact star physics , the order of the phase transition to quark superconducting matter plays an important role . the conclusion about the first order phase transition was drawn within models without vector interaction channels taken into account ; the vector interaction has been considered in few papers @xcite . it was found that the presence of quark interaction in the vector channel moves the critical line in the in @xmath5 plain to larger @xmath3 @xcite . recently it has been demonstrated @xcite that the critical line of first order phase transition in the @xmath5 plane can have a second end - point at low temperatures , besides the well known one at high temperatures . the latter one could even be subject to experimental verification in heavy - ion collisions @xcite whereas the former could be of relevance for neutron stars . while in ref . @xcite this feature of the phase diagram was a consequence of the presence of interaction in the vector channel , we would like to investigate in the present work the sensitivity of the phase diagram to the choice of model parameters without interaction in the vector channel . we will demonstrate that in the absence of the vector channel interaction the phase transition is not necessarily of the first order , thus revising statements in refs . @xcite . it is worth noting that some progress has recently been done in lattice calculations . there are methods being developed that allow to extend lattice results to the case of finite chemical potentials @xcite . however , these methods are valid only for small chemical potentials ( see e. g. @xcite ) , below the conditions at which the color superconductivity phase is expected to form . the structure of our paper is as follows . in sect . 2 , a chiral quark model is introduced , its lagrangian is given and the model parameters are fixed from the vacuum state in two different schemes . temperature and chemical potential are introduced into the quark model in sect . 3 , using the matsubara formalism . the conclusions and a discussion of the obtained results are given in sect . in order to study the quark matter phase diagram including color superconductivity , one should generalize the concept of the single order parameter related to the quark - antiquark condensate in the case of chiral symmetry breaking to a set of order parameters when condensation can occur in other interaction channels too . the simplest extension is the scalar diquark condensate @xmath6 for @xmath7 and @xmath8 quarks @xmath9 which is an order parameter characterizing the domain where the color symmetry is spontaneously broken and the quark matter finds itself in the ( two - flavor ) color superconducting ( 2sc ) state . this quantity is the most important one among other possible condensates that can be constructed in accordance with the pauli principle @xcite . in ( [ delta ] ) the matrix @xmath10 is the charge conjugation matrix operator for fermions @xmath11 the matrices @xmath12 and @xmath13 are pauli and gell - mann matrices , respectively . the first one acts on the flavor indices of spinors while the second one acts in the color space . if the electroweak interaction is discarded and only the strong coupling is in focus , the resulting quark matter phase diagram is essentially determined by nonperturbative features of the qcd vacuum state . one therefore has to resort to nonperturbative approaches to describe the behavior of particles at various conditions , ranging from cold and dilute matter up to the hot and dense one . a reliable and widely tested model to nonperturbative strong coupling qcd is provided by the dyson - schwinger equations @xcite , however , for qualitative studies like the one we attempt here it proves to be too complex . therefore , we will use here a simple and tractable nonperturbative model of quark interaction , the nambu jona - lasinio ( njl ) model @xcite , which has been extensively exploited for the description of the properties of the light meson sector of qcd ( also to describe the color superconductivity phase @xcite ) and proved to be a model respecting the low - energy theorems . before we proceed to the case of finite temperature and density , the model parameters that determine the quark interaction should be fixed . this shall be done for the vacuum state where hadronic properties are known . we will assume , according to common wisdom that , once fixed , these parameters ( originating from the nonperturbative gluon sector of qcd ) will not change , even in the vicinity of the transition to the quark matter . this transition is thus caused by medium effects in the quark sector only . in the present paper we restrict ourselves to the two - flavor case , leaving the strange quark and effects related to it beyond our consideration . as we constrain ourselves to only two order parameters , the quark and scalar diquark condensates , during our investigation , the interaction of quarks will be represented in the lagrangian by @xmath14 symmetric scalar , pseudoscalar quark - antiquark , and scalar diquark vertices : @xmath15,\\ { { \mathcal l } } _ { qq}&=&\frac{h}{2}(\bar\psi i\gamma_5\tau_2\lambda_2c\bar\psi^t)(\psi^{t } c i\gamma_5\tau_2\lambda_2\psi),\end{aligned}\ ] ] where @xmath16 is the diagonal current quark mass matrix @xmath17 , @xmath18 and @xmath19 are constants describing the interaction of quarks in the scalar , pseudoscalar , and scalar diquark channels , respectively . we work in the isospin symmetric case @xmath20 , thus @xmath21 . by the standard hubbard - stratonovich procedure , we introduce auxiliary scalar ( @xmath22 ) , pseudoscalar triplet ( @xmath23 ) , and diquark ( @xmath24 ) fields together with yukawa - like terms in the lagrangian density instead of the four - quark vertices : @xmath25 in order to integrate out the quark degrees of freedom by gaussian path integration , it is appropriate to represent the quark fields by the bispinor @xmath26 and to introduce the matrix propagator @xmath27 : @xmath28 integrating over @xmath29 and @xmath30 , we then obtain an effective lagrangian in terms of collective scalar and pseudoscalar quark - antiquark and scalar diquark excitations . here , we restrict ourselves to the mean - field approximation , leaving the next - to - leading order corrections in the @xmath31 expansion beyond our model . finally , the effective lagrangian density reads @xmath32 the trace in ( [ invpropagator ] ) is taken in the dirac , color , and flavor space . the matrix @xmath33 contains @xmath22 and @xmath23 fields : @xmath34 the sum over @xmath35 is assumed , and @xmath36 . as it was mentioned above , we are working in the mean - field approximation and the quark condensates are of interest . therefore , further study can be performed in terms of the effective potential @xmath37 where @xmath38 is 4-dimensional volume . the vacuum expectation values of the collective variables @xmath22 , @xmath23 , @xmath39 , and @xmath40 determine the absolute minimum of @xmath41 . they are given by the equation @xmath42 _ a priori _ it is known that in the vacuum only the @xmath22 field acquires a nonvanishing expectation value . the diquark fields @xmath39 , @xmath40 are expected to have nonzero mean values only in dense matter . the mean value of the pseudoscalar isotriplet field @xmath23 is always equal to zero , therefore we omit it hereafter . having solved eq . ( [ vmin ] ) for the field @xmath22 , one can work in terms of the constituent quark mass @xmath43 , connected with the current quark mass by the gap equation @xmath44 in the chiral limit ( @xmath45 ) , the constituent quark mass is proportional to the quark condensate and thus can be treated as the order parameter . in the njl model the quark condensate is @xmath46 where @xmath47 the divergence in @xmath48 is eliminated by means of a sharp 3d cut - off at the scale @xmath49 . in our model we have four parameters : the four - quark interaction constants @xmath18 and @xmath19 , cut - off @xmath49 , and the current quark mass @xmath50 . without diquarks , there are only three : @xmath18 , @xmath49 , and @xmath50 . they are fixed by the following relations : 1 . the goldberger - treiman relation ( gtr ) : @xmath51 where @xmath52 mev is the pion weak coupling constant and @xmath53 describes the coupling of a pion with quarks @xmath54 @xmath55 2 . the quark condensate ( qc ) from qcd sum rules @xmath56 3 . the decay constant @xmath57 for the @xmath58 ( r2pd ) process @xmath59 the @xmath60 transitions are omitted here . the current quark mass @xmath50 is fixed from the gmor relation : @xmath61 in the chiral limit @xmath62 , @xmath45 . 5 . with the diquark channel included , there is an additional parameter @xmath19 which can be fixed as @xmath63 from the fierz transformation ( as e. g. in@xcite ) .. it turned out that within our model the resulting phase diagram is not much affected if one makes the choice in favor of @xmath64 . however , it would be preferable to fix the constant @xmath19 from some observable , e. g. from the nucleon mass . ] in the item 2 , we have given two alternatives : one can either use the value of the quark condensate taken from qcd sum rule estimates or demand from the model that it should describe the @xmath58 decay . the latter is well observable in experiment contrary to the quark condensate . for simplicity , we perform all calculations in the chiral limit @xmath45 . in this case , when investigating the hot and dense quark matter , the borders between phases turn out to be sharp and the critical temperature and chemical potential are well defined . with the finite current quark mass , the transitions from one phase to the other become smooth . as a result , one obtains two different parameter sets shown in table [ paramset ] . in the type i parameter set the interaction of quarks is stronger , the uv cut - off is smaller , and the constituent quark mass is greater . one can calculate the dimensionless constant @xmath65 . it equals 4.6 for the type i and 3.72 for the type ii , respectively . as we will see further , these two parametrizations result in qualitatively different phase diagrams we extend the njl model to the case of finite temperatures @xmath2 and chemical potentials @xmath3 , applying the matsubara formalism , and restrict ourselves to the isospin symmetric case where up and down quark chemical potentials coincide . the thermodynamical potential per volume is @xmath66 where @xmath67 are matsubara frequencies for fermions , and the chemical potential is included into the definition of inverse quark propagator @xmath68 the expression in ( [ thpot ] ) can be simplified using the equations @xmath69\nonumber\\ & + & 2\left[\ln\left(\frac{(\omega_n^2+{\epsilon^+}^2)(\omega_n^2 + { \epsilon^-}^2)}{t^4}\right)\right]\end{aligned}\ ] ] and @xmath70.\ ] ] @xmath71\right.\right.\nonumber\\ & + & \left.2 t \ln\left[1+\exp\left(-\frac{\epsilon^-}{t}\right)\right]\right)\nonumber\\ & + & 4\left(\left(e^++e^-\right)\theta(\lambda^2-\vec{p}^2)+2 t \ln\left[1+\exp\left(-\frac{e^+}{t}\right)\right]\right.\nonumber\\ & + & \left.\left.2 t \ln\left[1+\exp\left(-\frac{e^-}{t}\right)\right]\right)\right\ } + \frac{m^2}{2g}+\frac{|\delta|^2}{2 h}~,\end{aligned}\ ] ] where @xmath72 @xmath73 the cold matter limit @xmath74 looks as follows : @xmath75 \nonumber\\ & \times&\theta(\lambda^2-\vec{p}^2)+\frac{m^2}{2g}+\frac{|\delta|^2}{2 h}~.\end{aligned}\ ] ] the thermodynamical potential can not be calculated in closed form for arbitrary @xmath2 and @xmath3 . however , in the cold matter limit one can easily obtain analytic expressions for the thermodynamical potential or its derivatives if only one of the collective variables @xmath22 or @xmath76 has a nonvanishing average value . this allows to find what kind of phase transition is to be expected for different parameter choices . we evaluate the remaining 3d momentum integrals numerically and calculate the value of thermodynamical potential at different @xmath2 and @xmath3 for the two types of model parameter sets . the equilibrium state for each @xmath2 and @xmath3 is determined by @xmath77 and @xmath78 corresponding to the minimum of @xmath79 . it is quite illustrative to look at the contour plots of the thermodynamical potential . for several values of @xmath3 at @xmath74 they are shown in figs . [ thdp_i_1][thdp_i_4 ] where one can follow the appearance and disappearance of local minima , maxima , and saddle points of the thermodynamical potential with increasing chemical potential . for zero temperature and chemical potential we have , as expected , a nonzero constituent quark mass ( quark condensate ) corresponding to the absolute minimum of the thermodynamical potential at @xmath80 mev and @xmath81 in fig . [ thdp_i_1 ] . at a certain chemical potential , a new local minimum related to the diquark condensate near @xmath82110 mev and @xmath83 ( fig . [ thdp_i_2 ] ) , but it does not yet give the absolute minimum . there is also a local maximum around @xmath84 mev and @xmath81 . as the matter becomes more dense , the second minimum lowers until it becomes degenerate with the first minimum while the average value of @xmath22 ( or @xmath85 ) remains almost unchanged ( see fig . [ thdp_i_3 ] ) . above the corresponding ( critical ) chemical potential @xmath86 mev , the second minimum becomes the absolute one and a first order transition occurs , during which @xmath87 discontinuously changes to zero while the diquark condensate acquires nonzero value breaking the color symmetry of the strong interaction . this characterizes the color superconducting phase transition in quark matter . furthermore , the local minimum on the @xmath43 axis merges the saddle point ( see fig . [ thdp_i_4 ] ) and , at still higher @xmath3 , only the local minimum on the @xmath39 axis near @xmath88 130 mev and @xmath83 remains . at a fixed chemical potential above @xmath89 , with the temperature rising , the average value of @xmath76 decreases until it reaches zero at the critical temperature @xmath90 which can be roughly estimated using the bcs theory formula @xmath91 above this temperature quark matter is in the symmetric phase where the chiral and color symmetries are restored . finally , we obtain the phase diagram shown on fig . [ phdi ] with three phases : the hadron phase , 2sc phase , and symmetric phase . all three phases coexist at the triple point : @xmath92 mev and @xmath93 mev . as for the type i parameter set , at zero @xmath2 and @xmath3 only the constituent quark mass @xmath43 , being the order parameter for the chiral condensate , is nonzero , whereas the diquark gap @xmath39 vanishes . however , the vacuum value of @xmath43 is lower than that for the type i and , with the chemical potential increasing , @xmath3 becomes equal to the vacuum value of @xmath43 before the second local minimum , corresponding to the diquark condensate , appears . at further increase of @xmath3 the constituent quark mass decreases , and it would vanish at @xmath94 , @xmath95 if the diquark condensate did not appear . actually , at the critical value @xmath96 both the quark condensate and the diquark condensate are small but nonzero . the changes of the local extrema for increasing chemical potential are similar to those shown in figs . [ thdp_i_1][thdp_i_4 ] . the cases of dilute ( @xmath97 ) and very dense matter ( @xmath98 mev ) are qualitatively analogous , only the absolute values of @xmath43 and @xmath76 at which the local minima are found are different . at intermediate densities , however , there is a qualitative difference . within a very narrow range of values of the chemical potential , there exists a new phase of massive superconducting matter . one can see this in fig . [ thdp_ii ] for @xmath99 mev . at higher @xmath3 the chiral symmetry is restored and the quark matter is in the pure superconducting phase . a possibility of the chiral diquark condensates to coexist at certain condition has been already noticed in ref . @xcite thus , for the type ii parameter set , the transition from the hadronic to the superconducting phase is of the second order . in this case there are no degenerate local minima in the thermodynamical potential separated by a barrier . this behavior is unlike to what is commonly expected for a cold and dense matter but it parallels the findings of ref . @xcite where vector interactions are responsible for this behavior . the average value of @xmath76 is much smaller than for the type i parameter set . as a consequence , the border between 2sc and the symmetric phases of quark matter lies at noticeably lower temperatures . the phase diagram obtained in our model for the type ii parameter set is shown in fig . [ phdii ] . in the framework of the simple njl model for two flavors , a phase diagram is obtained for @xmath74200 mev and @xmath97450 mev . three phases are found for the type i parameter set and four phases for the type ii parameter set . the critical temperature and chemical potential obtained in the type i scheme differ from those obtained with the type ii parameter set . at @xmath74 , @xmath100 mev for the type i parameter set and @xmath101 mev for the type ii . the corresponding quark densities differ by a factor 1.5 1.7 . the critical temperature for the type ii parameter set is as low as @xmath102 mev and thus much closer to critical temperatures for the paring instability in nuclear matter systems ( see @xcite ) whereas for the type i parameter set the critical temperatures are an order of magnitude larger . this striking difference in the critical parameters obtained within the same model calls for a more detailed investigation of the question of model parametrization . in our work , the constant @xmath19 was not obtained from a fit to observable data . instead , fierz transformation arguments have been used to fix the ratio @xmath103 . a parameterization would be favourable where ( in the spirit of the type ii model ) experimentally measured quantities , like the @xmath104 meson width , are used rather than non - observable ones ( quark condensate etc . ) . it would therefore be more consistent to fit the constant @xmath19 from baryon properties , see @xcite and also to go beyond the mean field level of description . these investigations shall be performed in future work where it remains to be clarified which critical parameters for the color superconducting phase transition can be considered more realistic and of which order the phase transition is . the authors are grateful to m. buballa , t. kunihiro , and m. kitazawa for useful discussions . this work has been supported by daad and heisenberg - landau programs . v.y . and m.k.v . acknowledge support by rfbr grant no . 02 - 02 - 16194 . 99 b. barrois , nucl . b * 129 * , 390 ( 1977 ) . d. bailin and a. love , phys . rep . * 107 * , 325 ( 1984 ) . m. iwasaki and t. iwado , phys . b * 350 * , 163 ( 1995 ) . m. iwasaki and t. iwado , prog . phys . * 94 * , 1073 ( 1995 ) . m. alford , k. rajagopal , and f. wilczek , phys . b * 422 * , 247 ( 1998 ) . r. rapp , t. schfer , e. v. shuryak , and m. velkovsky , phys . lett . * 81 * , 53 ( 1998 ) . m. a. halasz , a. d. jackson , r. e. shrock , m. a. stephanov , and j. j. verbaarschot , phys . d * 58 * , 096007 ( 1998 ) ; m. a. stephanov , k. rajagopal , and e. v. shuryak , phys . lett . * 81 * , 1998 ( 4816 ) . j. c. r. bloch , c. d. roberts , and s. m. schmidt , phys . rev . c * 60 * , 065208 ( 1999 ) . m. alford , k. rajagopal , and f. wilczek , nucl . b * 573 * , 443 ( 1999 ) . r. d. pisarski and d. h. rischke , phys . rev . lett . * 83 * , 37 ( 1999 ) . d. t. son , phys . rev . d * 59 * , 094019 ( 1999 ) . t. schfer and f. wilczek , phys . d * 60 * , 114033 ( 1999 ) . j. berges and k. rajagopal , nucl . b * 538 * , 215 ( 1999 ) . r. rapp , t. schfer , e. v. shuryak , and m. velkovsky , ann . phys . * 280 * , 35 ( 2000 ) . r. d. pisarski and d. h. rischke , phys . rev . d * 61 * , 051501 ( 2000 ) . k. rajagopal and f. wilczek , arxiv : hep - ph/0011333 and references therein , in : shifman , m. 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[ cols="<,^,^,^,^,^,^ " , ] and @xmath39 at zero temperature and the chemical potential @xmath97 mev . ]

the phase diagram for quark matter is investigated within a simple nambu jona - lasinio model without vector correlations . it is found that the phase structure in the temperature density plane depends sensitively on the parametrization of the model . we present two schemes of parametrization of the model where within the first one a first order phase transition from a phase with broken chiral symmetry to a color superconducting phase for temperatures below the triple point at @xmath0 mev occurs whereas for the second one a second order phase transition for temperatures below @xmath1 mev is found . in the latter case , there is also a coexistence phase of broken chiral symmetry with color superconductivity , which is a new finding within this class of models . possible consequences for the phenomenology of the qcd phase transition at high baryon densities are discussed . rostock university + preprint no . mpg - vt - ur 235/02

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Proceed to summarize the following text: the demand for new materials with interesting and useful physical properties has led to a fast development in material science . properties such as superconductivity , magnetic ordering , nearly ferromagnetic fermi - liquid and heavy fermion behavior have been observed in many materials , including the family of complex intermetallic compounds @xmath15zn@xmath1 ( @xmath16 = rare earth , @xmath2 = transition metal).@xcite this family , first reported two decades ago by nasch @xmath17.@xcite , has been extensively used as a model system due to its rather unique structure which features a complex but well ordered crystal structure . the @xmath15zn@xmath1 compounds have a cubic structure with @xmath18 ( no . 227 ) space group in which the @xmath16 and @xmath2 ions occupy the crystallographic sites 8@xmath19 and 16@xmath14 , respectively . moreover , these ions are each encapsulated in quasi - spherical cages formed exclusively by zn ions , which occupy three inequivalent wyckoff positions given by 16@xmath20 , 48@xmath21 , and 96@xmath22 as can be viewed in fig . [ fig : fig0 ] . two particular properties observed in these materials have attracted great attention : a remarkably high magnetic ordering temperature observed in the @xmath16fe@xmath3zn@xmath1 series ( although it contains less than @xmath23 of @xmath16 ion ) and a nearly ferromagnetic fermi - liquid behavior in yfe@xmath3zn@xmath1.@xcite several works have argued , based on macroscopic measurements and band structure calculations , that the elevated magnetic ordering temperatures and the type of magnetic ordering are attributed to a high density of states ( dos ) at the fermi level.@xcite for instance , the co based compounds gdco@xmath3zn@xmath1 and tbco@xmath3zn@xmath1 present an antiferromagnetic ( afm ) ordering below @xmath24 5.7 k and 2.5 k , respectively , in which the larger @xmath16-@xmath16 distance ( @xmath24 6 ) supports an indirect interaction and a low transition temperature . when the co ions are replaced by fe ions ( gdfe@xmath3zn@xmath1 and tbfe@xmath3zn@xmath1 ) the compounds exhibit ferromagnetic ( fm ) ordering with the transition temperatures drastically raised to 86 k and 66 k , respectively . the relatively long distance between rare earth ions in the structure weakens the ruderman - kittel - kasuya - yosida ( rkky ) exchange interaction , while the high density of fe 3@xmath14 bands at the fermi level directly affects the conduction electrons . recently , a detailed study of the magnetic structure of tbco@xmath3zn@xmath1 and tbfe@xmath3zn@xmath1 through magnetic neutron scattering at low temperature was reported.@xcite although the tb - based compounds present similar magnetic properties as compared to gdco@xmath3zn@xmath1 and gdfe@xmath3zn@xmath1 , the absence of crystalline electric field ( cef ) at first order and a strong rkky interaction in the gd - compounds affect the electronic and magnetic interactions between the rare earth ions and the surrounding matrix . in particular , the investigation of the compounds with half - filled 4@xmath25 shell ( gd - based materials ) at low temperature can provide information about the physical properties and are very important as reference compounds due to their lack of orbital momentum , i.e. , @xmath26 and @xmath27 , which leads to magnetic properties that are unaffected by spin - orbit coupling . in order to understand the implications of such interactions we have used spectroscopy and magnetic scattering techniques to probe in detail the electronic and magnetic properties of the gd@xmath0zn@xmath1 family . the large gd neutron absorption cross section leaves x - ray measurements as the ideal option to probe of the magnetic and electronic properties in these materials . furthermore , the incoming beam energy can be tuned to the absorption edge , thus providing chemical and atomic selectivity , i.e. , the magnetic response of each element can be probed separately . here we report the magnetic and electronic structure at low temperature of the gd@xmath0zn@xmath1 ( @xmath2 = fe , co ) compounds using x - ray resonant magnetic scattering ( xrms ) , x - ray absorption near - edge structure ( xanes ) and x - ray magnetic circular dichroism ( xmcd ) techniques . the xrms measurements performed on gdco@xmath3zn@xmath1 reveal a commensurate antiferromagnetic ordering with a magnetic propagation vector @xmath28 below @xmath29 k in which the gd magnetic moments are aligned following the magnetic representation @xmath7 . the xmcd measurements performed below the curie temperature ( @xmath30 k ) in gdfe@xmath3zn@xmath1 display a dichroic signal of 12.5 @xmath8 and 9.7 @xmath8 of the absorption jump for gd @xmath31 and @xmath11 edges , respectively . surprisingly , a magnetic signal of about 0.06 @xmath8 is detected at the zn @xmath12-edge which suggests that the zn ions are spin polarized . this magnetic signal might originate from the hybridization between the extended gd 5@xmath14 bands with the empty zn 4@xmath32 states . absorption measurements performed at the fe @xmath12-edge do not reveal any magnetic contribution coming from the iron ions above the background level . high quality single crystals of gdfe@xmath3zn@xmath1 and gdco@xmath3zn@xmath1 were grown at ufabc by zn self flux method@xcite similar to that reported in previous studies on the family.@xcite in order to perform the absorption measurements at the gd @xmath33 , fe and zn @xmath12 edge , selected single crystals of gdfe@xmath3zn@xmath1 were ground and sieved , resulting in fine powders with grain sizes around @xmath34-@xmath35 . the magnetic diffraction measurements were done on a high quality gdco@xmath3zn@xmath1 single crystal cut to dimensions of approximately 2 x 2 x 0.5 mm@xmath36 . the crystalline piece was carefully polished to achieve a flat surface perpendicular to the @xmath37 $ ] direction , yielding a mosaic width of approximately 0.02@xmath38 . the phase purity of the samples was confirmed by powder diffraction using conventional laboratory x - ray sources . temperature dependent magnetic susceptibility measurements ( not shown here ) were performed using a commercial superconducting quantum interference device ( quantum design mpms - squid ) to verify the magnetic ordering temperatures , the effective magnetic moments and the curie - weiss constants . the resonant diffraction measurements were performed at beamline 6-id - b at the advanced photon source ( aps ) , argonne national laboratory ( argonne , il / usa ) , whereas the absorption measurements were conducted at 4-id - d ( aps)@xcite and at beamline p09 at petra iii ( desy , hamburg / germany).@xcite xanes and xmcd spectra obtained at low temperature for the gd @xmath33 , fe and zn @xmath12-absorption edges were performed in transmission geometry on powdered gdfe@xmath3zn@xmath1 samples . the samples were cooled down by a displex cryostat with base temperature around 7 k. xmcd spectra were performed in helicity switching mode in which the left and right circular polarization was obtained by means of diamond phase plates.@xcite the degree of circularly polarized beam was higher than 95 @xmath8 for both beamlines ( p09 and 4-id - d).@xcite an external magnetic field of h = 2.0 t ( at aps ) and 0.8 t ( at desy ) was applied in the gdfe@xmath3zn@xmath1 samples along and opposite to the incident beam wave vector @xmath39 to align the ferromagnetic domains and to correct for non - magnetic artifacts in the xmcd data . those external magnetic fields were enough to reach the saturation magnetization according to the macroscopic measurements . xrms measurements were performed at t = 4.5 k on gdco@xmath3zn@xmath1 single crystal , mounted inside the closed - cycle displex cryostat in a six - circle diffractometer at the 6-id - b beamline . the single crystal was oriented with the @xmath37 $ ] direction parallel to the vertical diffraction plane . several magnetic superlattice reflections of the type @xmath40 with @xmath41 were measured and their integrated intensities were compared to the simulated intensities to determine the magnetic structure below @xmath42 . to enhance the magnetic bragg peak intensities , the energy of the incident beam was tuned near the gd @xmath10 or @xmath11 absorption edges . in addition , in order to investigate any magnetic contribution from co and zn ions , the energy of the incident beam was also tuned to the co ( 7709 ev ) and zn ( 9659 ev ) @xmath12-edges and a search for superlattice reflections was performed . the charge and magnetic contributions present in the scattered beam were separated by a pyrolytic graphite [ c@xmath43 analyzer crystal installed on the 2@xmath44 arm of the diffractometer . since the incident beam presents the polarization perpendicular to diffraction plane ( @xmath45 polarization ) , by rotation of the analyzer crystal around the scattered beam wave vector @xmath46 we were able to select the two polarization channels ( @xmath47 and @xmath48 ) in this experimental geometry.@xcite the experimental results are organized into two sections : the first part is dedicated to the absorption measurements on powdered gdfe@xmath3zn@xmath1 in its fm state . the subsequent section shows the results obtained by the xrms technique on the gdco@xmath3zn@xmath1 single crystal in its afm state . xanes and xmcd measurements performed at the gd @xmath33 edges in gdfe@xmath3zn@xmath1 are shown in fig . [ fig : fig1 ] . dipolar selection rules make the dichroic signal at the @xmath33 absorption edges sensitive to the spin polarization of the intermediate 5@xmath14 level . the gd xanes reported in fig . [ fig : fig1 ] are normalized to one at the @xmath11 and half at the @xmath10 edge to reflect the 2:1 ratio of the initial state at these edges ( 2@xmath49 and 2@xmath50 , respectively ) . figure [ fig : fig1 ] also shows the xmcd spectra at the gd @xmath33 edges in which each spectrum is normalized to the corresponding edge jump of the absorption spectrum . the xmcd signal obtained across the two edges show different intensities with a strong dichroic magnetic signal around 12.5 @xmath8 at the @xmath10 and 9.7 @xmath8 at the @xmath11 absorption edges , which is consistent with gd - based compounds.@xcite the size and the shape of the magnetic contribution obtained by fitting the xmcd signals with lorentzian function can describe additional properties of this system . the widths of the dipolar contributions ( e1 ) contributions observed at the @xmath10 and @xmath11 absorption edges are 4.3(2 ) ev and 4.5(2 ) ev , respectively , which reflects a short 2@xmath13 core hole lifetime . using the integrated intensities , the @xmath51 ratio ( or branching ratio value - br)@xcite obtained experimentally is @xmath52 . absorption measurements at the fe and zn @xmath12 edges were also carried out on powdered samples . the absorption measurement near the @xmath12 edge , in which the dipolar transition is probed ( 1@xmath53 4@xmath13 ) , is crucial towards understanding the magnetic and electronic properties due to the delocalized character of the @xmath13 states.@xcite . since the probed @xmath13 states are very delocalized , a strong influence of the surrounding matrix can be expected due to the hybridization between the rare earth and the transition metal ions . as shown in figure [ fig : fig1_n ] , the measurements performed near the fe @xmath12 edge do not reveal any magnetic contribution from the fe ions higher than the background level ( @xmath24 0.07 @xmath8 ) . the inset in fig . [ fig : fig1_n ] exhibits the xmcd measurements obtained for a 5 @xmath54 m fe - foil in the same experimental conditions : a clear dichroic signal can be observed near the edge . however , the spectroscopy measurement at the zn @xmath12 edge manifests an interesting behavior . figure [ fig : fig2 ] shows the absorption and dichroism results at the zn @xmath12 edge in which an induced magnetic signal around 0.06 @xmath8 is detected . this magnetic signal is due to a hybridization with the rare earth 5@xmath14 orbitals . the xmcd spectrum exhibits the main positive feature located at 9665 ev with a width around 2.5 ev , surrounded by two negative peaks 6.5 ev away . the broad feature localized around 9680 ev ( @xmath24 20 ev above the edge ) is likely due to magnetic exafs.@xcite a clear evidence of the zn 4@xmath13 states polarization due to the gd ions can be found in the temperature and field dependence reported in panels [ fig : fig2tdep](a ) and [ fig : fig2tdep](b ) , respectively . the magnetic intensities for both gd and zn ions follow the same temperature evolution and disappear around the critical temperature ( t@xmath55 85 k ) . in addition , the two hysteresis loops obtained at the maximum xmcd intensity shows clearly the zn magnetism dependence in relation to the gd ions and therefore it suggests a spin polarization of the zn 4@xmath13 bands by the gd sub - lattice . figure [ fig : fig3 ] shows the evolution of the integrated intensity for the magnetic bragg reflection @xmath56 as a function of temperature for the gdco@xmath3zn@xmath1 compound fitted by a lorentzian - squared function . the magnetic peak intensity decreases smoothly to zero as the temperature approaches @xmath42 , indicating a standard second order phase transition from an afm to a paramagnetic state . a dashed red line in fig . [ fig : fig3 ] displays a fitting using a critical power - law expression , @xmath57 , above 5.0 k. the fitting around the nel temperature yields a @xmath42 = 5.72(6 ) k and a critical exponent @xmath58 = 0.36(3 ) . the value of @xmath42 is in good agreement with bulk magnetic susceptibility measurements and previous works.@xcite the critical exponent @xmath58 close to 0.367 suggests a three - dimensional ( 3d ) heisenberg magnetic model . @xcite in blue symbol ( fig . [ fig : fig3 ] ) is also reported the full width at half maximum ( fwhm ) of the magnetic superlattice peak @xmath56 as a function of temperature . this figure clearly shows a peak broadening and a decrease in intensity near the phase transition temperature characteristic of a loss of long - range order . energy dependences across the gd-@xmath10 and @xmath11 edges performed at 4.5 k are displayed in fig . [ fig : fig4 ] . the top panels [ fig . [ fig : fig4](a ) and [ fig : fig4](c ) ] show the normalized absorption coefficients ( @xmath54 ) obtained from the fluorescence yield , while the bottom panels [ fig . [ fig : fig4](b ) and [ fig : fig4](d ) ] exhibit the energy profile obtained at the magnetic superlattice position @xmath59 with the analyzer crystal set to the @xmath48 polarization channel . a resonant enhancement of over two orders of magnitude at both absorption edges can be seen . in addition , the maximum intensities are observed about 2 - 3 ev above the absorption edge ( defined by the vertical dashed lines ) , which is a characteristic signature of a dipole electronic transition . the same energy dependence was performed in the @xmath47 polarization channel , and no significant contribution was observed . the strong resonant enhancement in the spectra [ fig . [ fig : fig4](b ) and [ fig : fig4](d ) ] indicates a significant overlap between the initial 2@xmath13 and 5@xmath14 states , and a strong exchange interaction between the 4@xmath21 - 5@xmath14 orbitals . this magnetic polarization of the 5@xmath14 bands via 4@xmath21 states helps shed light on the magnetic structures of these rare earth based materials using the @xmath60 absorption edge measurements , i.e. , 2@xmath61 5@xmath14 transitions . moreover , the asymmetric peak shape expressed as a long tail below the absorption edges arises from the interference between the resonant and non - resonant magnetic scattering contributions.@xcite the normalized energy line shape dependence after absorption correction for selected magnetic bragg peaks @xmath62 with @xmath41 performed at gd-@xmath10 and @xmath11 edges are displayed in fig . [ fig : fig5 ] . the magnetic reflections show a narrow resonant line shape at the @xmath10 ( @xmath63 4.9 ev ) and @xmath11 ( @xmath63 5.9 ev ) edges . the smaller energy broadening for the @xmath10 edge is associated with a relatively short lifetime compared with the @xmath11 edge . the magnetic reflections were only observed in the @xmath48 polarization channel ; in the @xmath47 channel , no intensity was detected . the integrated intensities of the magnetic reflections were used to extract the @xmath51 ratio and to determine the direction of magnetic moment . the br values show an unusual behavior in which the values varies between @xmath64 - @xmath65 for different magnetic reflections . for xrms , the branching ratio is expected to be equal to 1 . the exact value for the br ratio is difficult to obtain due to the various corrections that must be applied to the experimental data , such as self - absorption and angular corrections , and therefore , we can not affirm that @xmath66 for this afm compound . the magnetic structure of the gd spins are determined comparing the experimental integrated intensities in fig . [ fig : fig5 ] with simulated data from selected magnetic reflections . the program sara_h _ @xcite was run to determine the possible magnetic arrangements that the gd ion can adopt inside the unit cell , i.e. , the magnetic representation ( @xmath67 ) . in addition , we assumed that only the gd ions carry magnetic moments in this compound . for this material , whose magnetic propagation vector is @xmath5 , the space group is @xmath18 , and the gd ions occupy the 8@xmath19 crystallographic site , the magnetic representation ( mr ) can be decomposed into four non - zero irreducible representations ( irs ) : two one - dimensional ( 1d - @xmath68 ) and two two - dimensional ( 2d - @xmath69 ) irs . the four possible magnetic representations for the gdco@xmath3zn@xmath1 compound are summarized in , where the labeling of the propagation vector and the magnetic representation follows the kovalev notation.@xcite .basis vectors ( bv ) for the space group @xmath18:2 with @xmath70 . the decomposition of the magnetic representation ( mr ) for the gd site can be written like @xmath71 . the two rare earth atoms positions of the nonprimitive basis are defined according to 1 : @xmath72 and 2 : @xmath73 . [ cols="^,^,^,^,^,^,^,^,^,^,^,^,^,^,^,^,^,^,^ " , ] to determine the magnetic structure , the intensities were calculated assuming only dipole transition ( e1 ) and hence , the x - ray magnetic scattering cross section model can be written as:@xcite @xmath74 where @xmath75 and @xmath76 . f^{e1}_{n } & = [ ( \hat{\varepsilon}'\cdot \hat{\varepsilon})f^{(0 ) } - i(\hat{\varepsilon}'\times \hat{\varepsilon } ) \cdot \hat{z}_n f^{(1 ) } \\ & + ( \hat{\varepsilon}'\cdot \hat{z}_n ) ( \hat{\varepsilon } \cdot \hat{z}_n)f^{(2 ) } ] . \end{split}\end{aligned}\ ] ] the term @xmath77 [ eq . ( [ eq : eq3 ] ) ] contains the absorption correction and the lorentz factor . @xmath78 is the angle between the wave - vector transfer @xmath79 and the @xmath37 $ ] crystal direction , and @xmath44 is half of the 2@xmath44 scattering angle . ( [ eq : eq4 ] ) shows the resonant term . it carries information about the @xmath80 ( @xmath81 ) and @xmath46 ( @xmath82 ) , i.e. , the incident and scattered wave ( polarization ) vectors , respectively , and the magnetic moment direction at _ n__th site ( @xmath83 ) . the terms @xmath84 are related to the dipole matrix transition and by atomic properties.@xcite the exponential function in eq . [ eq : eq2 ] is a function of the wave - vector transfer @xmath85 , and the position @xmath86 of the _ _ n__th gd ion inside the unit cell . for the xrms technique probing afm materials , we assumed that the magnetic intensity at the first harmonic satellites are due to only dipole contribution comes from the linear term on magnetic moment direction [ @xmath87 displayed in eq . ( [ eq : eq4 ] ) . the simulated intensities obtained using eq . ( [ eq : eq2])-([eq : eq4 ] ) and the experimental intensities obtained at the gd @xmath10 edge [ fig . [ fig : fig4](g - l ) ] are displayed in fig . [ fig : fig6 ] . comparing the intensities , the best agreement between experimental ( symbol , @xmath88 ) and simulated data ( curves ) is obtained for the magnetic representation @xmath89 . to identify different magnetic propagation vectors , such as @xmath90 and @xmath91 , a systematic search for commensurate and incommensurate magnetic reflections in the reciprocal space was performed below @xmath42 . however , only magnetic bragg reflections of the type @xmath92 were observed . in addition , to probe a possible presence of afm magnetic moments in the co and zn ions , the beam energy was tuned to the co and zn @xmath12 edges . search for magnetic superlattice reflections were performed below @xmath42 , and no measurable magnetic reflection at 4.5 k could be observed . the nature of the electronic and magnetic properties of the intermetallic @xmath15zn@xmath1 systems depends strongly on the interactions between the rare earth and the transition metals ions . since the gd 5@xmath14 states participate on the conduction bands , the resonant absorption and diffraction measurements at the gd @xmath33 edges provide valuable information . as reported in susceptibility measurements and band structure calculations@xcite , the replacement of the transition metal affects the electronic density of states at the fermi level ( @xmath93 ) and the conduction band without changing significantly the lattice parameters.@xcite doping studies of gd(fe@xmath94co@xmath95zn@xmath1 compounds shows a monotonic increase of the magnetic ordering temperature when @xmath96 increases to 1 which indicates that there is an increase in the coupling between the rare earth magnetic moments.@xcite absorption measurements performed in powdered samples of fm gdfe@xmath3zn@xmath1 compound below @xmath97 reveal interesting behaviors . as can be seen in fig . [ fig : fig1]-[fig : fig2tdep ] , only at the gd @xmath60 and zn @xmath12-edges a dichroic signal is observed above the background level whilst an unexpected lack of magnetic intensity is observed at the fe @xmath12-edge . the intense magnetic signal at the rare earth @xmath60-edges occurs mainly due to the overlap between the gd 2@xmath13 and 5@xmath14 states and a strong energy splitting of the 5@xmath14 sub - bands as a result of a 4@xmath21 - 5@xmath14 exchange interaction.@xcite in addition , the splitting of the @xmath14 state into 5@xmath14 spin - up and spin - down wave function has considerable influence in the magnetism observed at the zn @xmath12-edge . the zn 3@xmath14 orbitals are completely filled ( 3@xmath98 ) and henceforth , a magnetic moment due to an overlap between the 3@xmath14 and 4@xmath32 orbitals in the zn ion is not expected to occur due to the filled 3@xmath14 orbitals being more contracted . therefore , the magnetic signal observed in the 4@xmath13 states is due to hybridization with the extended gd 5@xmath14 orbitals and not from the exchange interaction with the zn 3@xmath98 orbitals.@xcite following hund s rule , the gd ion has the 4@xmath21 state filled by seven spin - up electrons in which it pulls the 5@xmath14 sub band spin - up function towards the inner core due to a positive exchange interaction . the short distances between the first gd - zn ions ( @xmath24 3 ) drives a small hybridization between the broad zn 4@xmath13 and gd 5@xmath14 states , inducing a small amount of magnetic moment in 4@xmath13 states . theoretical works have suggested that the orbital moment should be almost zero for the 5@xmath14 band ( @xmath99 ) , i.e. , a quenching of the angular momentum , so the 5@xmath100 and 5@xmath101 sub - bands should display the same polarization and thereby the dichroism at the @xmath11 and @xmath10 edges should have equal magnetic intensity.@xcite for the gdco@xmath3zn@xmath1 compound , the br ratio vary between 0.9 - 1.2 , i.e. , close to the theoretical value and therefore we can not suggest any orbital moment for this compound . however , as showed in fig . [ fig : fig1 ] for the fm compound , the intensity recorder at the @xmath10 is higher than the @xmath11 edge , where we observe a @xmath51 ratio of approximately @xmath52.@xcite this slightly different value from the theoretical branching ratio expected for xmcd ( br @xmath102 ) suggests that the gd ions may carry a small orbital moment at the 5@xmath14 orbitals in the fm compound . mssbauer spectroscopy measurements@xcite and simulations@xcite for the dyfe@xmath3zn@xmath1 compound reported that the fe ions align afm with the dy magnetic moments in which the iron ions exhibit a very small magnetic moment @xmath24 @xmath103/fe . neutron diffraction measurements@xcite performed in tbfe@xmath3zn@xmath1 compound also reported that a small magnetic moment at the fe ions ( @xmath104 ) would improve the refinement . in addition , recently mssbauer measurement@xcite in gdfe@xmath3zn@xmath1 also reported the presence of a small magnetic contribution at the fe site . in order to verify the quality of our xmcd data at the fe @xmath12 edge , dichroic measurements at low temperature and under magnetic field in a 5 @xmath54 m fe - foil was also carried out . a magnetic signal around @xmath105 which , according to the literature@xcite , corresponds to a magnetic moment around @xmath106/fe was observed . assuming that fe spins order ferromagnetically in gdfe@xmath3zn@xmath1 with a magnetic signal below 0.07 @xmath8 ( noise level ) , it would result in a magnetic moment lower than @xmath107 , i.e. , the same order of magnitude as reported by band structure calculation@xcite and comparable with the dyfe@xmath3zn@xmath1 compound . therefore , if the fe ions carry magnetic contribution in the gdfe@xmath3zn@xmath1 materials , we can suggest that this magnetic moment must be lower than @xmath108 . one possible explanation for the difficulty in identifying the magnetic signal at the fe @xmath12-edge ( 7112 ev ) would be due to contamination coming from the gd @xmath11 edge ( 7243 ev ) . despite the fact that the two edges are apart by around 130 ev , the gd @xmath11 pre - edge increases the background around the fe @xmath12-edge which , as a consequence may hide the small magnetic signal . an afm ordering state for the fe ions can not be ruled out , but is less likely to occur . field dependent magnetization measurements@xcite show that the gdfe@xmath3zn@xmath1 compound reaches a saturation magnetization of 6.5 @xmath109/f.u . at low temperature . assuming that the gd ions contribute with 7.94 @xmath109/gd for the total magnetic moment ( theoretical value ) , and that the transition metals are coupling antiferromagnetically with the rare earth elements , the transition metals are found to carry a magnetic moment around 1.44 @xmath109/f.u . opposite to the gd ions . supposing that the fe ions have a total magnetic moment @xmath107 , consequently the zn ions present in this material would present an induced magnetic moment of approximately 0.05 @xmath109/zn . therefore , it strongly suggests that the interaction between the rare earth ions affects the environment around the atoms and consequently spin polarizes the transition metal ion . the rough estimate of magnetic moment for the zn ions based on spectroscopy and macroscopic measurements has to be further investigate by density functional theory calculation . to provide further information about these systems , the magnetic properties of the afm compound gdco@xmath3zn@xmath1 were also investigated , using the xrms technique rather than neutron scattering , due to the high neutron absorption cross - section of gd ions . the transition to the magnetically ordered phase driven by temperature is characterized by the appearance of superlattice magnetic reflections with a magnetic propagation vector @xmath110 . this @xmath111 magnetic vector indicates that the magnetic unit cell is represented by a doubled chemical unit cell in all three crystallographic directions . as displayed in fig . [ fig : fig5 ] , the gd spins follow the magnetic representation @xmath89 which is different when compared with the isostructural tbco@xmath3zn@xmath1 compound.@xcite since there is a large separation ( @xmath24 6 ) between the rare earth ions in this system , the magnetic properties will be mainly mediated via the conduction electrons . the large distance between the first @xmath16-@xmath16 ions explains quite well a weakening of the @xmath112 exchange interaction and thus a very low representative magnetic transition temperature , i.e. , approximately 5.7 k ( gdco@xmath3zn@xmath1 ) and 2.5 k ( tbco@xmath3zn@xmath1 ) . as a consequence of a poor @xmath112 coupling , the surrounding matrix or the zn cages around the @xmath16 ions are weakly affected and hence it is not possible to induce magnetic moment in the transition metal ions . the absence of magnetism in the transition metal ions are also observed in the tbco@xmath3zn@xmath1 compound , in which the best refinement of the magnetic structure was obtained with null magnetic moment in the co and zn ions . although gdco@xmath3zn@xmath1 and tbco@xmath3zn@xmath1 compounds display the same magnetic propagation vector @xmath5 , their magnetic structures are different , which is mainly related to the competition between the rkky and cef interactions . for the tb - based materials the crystalline electric field splitting can affect the @xmath112 constant and therefore influence the magnetic coupling and the temperature transition . it is well known that an energy level cef splitting induces magnetic anisotropies in the ground state and it may influence the total angular momentum . several tb - based intermetallics have exhibited distinct magnetic properties in relation to the gd counterparts as shown in the layered family @xmath113 ( @xmath16 = rare earth ; @xmath2 and @xmath114 = transition metal ; @xmath115 = 1,2 and @xmath116 = 0,1 ) . @xcite although gdco@xmath3zn@xmath1 and tbco@xmath3zn@xmath1 compounds display the same magnetic propagation vector @xmath5 , their magnetic structures are different , which is mainly related to the competition between the rkky and cef interactions . the modification in the magnetic structure for different rare earth ions but with the same propagation vector has already been extensively investigated and it is mainly related to the rkky and cef interactions . et al_. [ ] evaluated the cef parameters from the thermodynamic measurements for the entirely @xmath16co@xmath3zn@xmath1 series ( @xmath16 = tb - tm ) and they observe small energy scales ( small @xmath117 parameter ) and a large @xmath118 cef parameter for the complete series . this finding suggests a small energy level splitting and a strong influence of the zn cage on the rare earth ions , i.e. , guest - framework interaction . therefore , we can suggest that the rare earth ions located in this large polarized environment are strongly affected by the zn cages , which has direct influence in the electronic and magnetic properties . this can be seen in the different magnetic structures for the afm compounds and the spin polarization of the zn ions only for the gdfe@xmath3zn@xmath1 compound . hence , the cef effect has an important role in this class of compound . a detailed investigation for different rare earth elements would allow a better understanding of the @xmath16co@xmath3zn@xmath1 family . nevertheless macroscopic measurements up to 1.8 k only report magnetic ordering for the compounds with @xmath16 = gd and tb . we have investigated the intermetallic gd@xmath0zn@xmath1 system with @xmath119 co and fe at low temperature using the xrms and xanes / xmcd techniques , respectively . the xrms measurements performed in gdco@xmath3zn@xmath1 compound reveal a commensurate antiferromagnetic ordering with a magnetic propagation vector @xmath5 in which only the gd ions carry magnetic moments . selected magnetic reflections were measured in the polarization channel @xmath48 and we identified that the gd spins follow the magnetic representation @xmath7 , which is different from the isostructural compound tbco@xmath3zn@xmath1 , mainly due to the cef effects in the latter . the evolution of magnetic signal showed a magnetic phase transition below t@xmath120 = 5.72(6 ) k with a critical exponent @xmath58 = 0.36(3 ) , suggesting a three - dimensional ( 3d ) heisenberg magnetic model . the xanes and xmcd measurements performed at the gd @xmath33 edges in gdfe@xmath3zn@xmath1 compound reveal a strong magnetic signal ( @xmath24 12.5 @xmath8 - @xmath10 and 9.7 @xmath8 - @xmath11 ) indicating a splitting of the 5@xmath14 orbitals and a strong gd - gd exchange interaction as well as a non zero orbital moment . in addition , we observed a presence of a small magnetic dichroic signal at the zn @xmath12 edge due to the spin polarization of the gd 5@xmath14 orbitals . this indicates a large rkky exchange interaction between the gd - gd ions which polarizes the surrounding matrix . this work was supported by fapesp ( sp - brazil ) under contracts no . 2009/10264 - 0 , 2011/19924 - 2 , 2011/24166 - 0 , 2012/10675 - 2 and 2012/17562 - 9 . parts of this research were carried out at the light source petra iii at desy , a member of the helmholtz association ( hgf ) . work at argonne is supported by the u.s . department of energy , office of science , office of basic energy sciences , under contract no . de - ac-02 - 06ch11357 . we would like to thank r. d. reis and c. s.b . dias for their assistance in macroscopic measurements and j. w. kim for his assistance at beamline 6-id - b . we are indebted to d. reuther , r. kirchhof and h .- c . wille for their assistance with the electromagnet at p09/desy . s. nandi , y. su , y. xiao , s. price , x. f. wang , x. h. chen , j. herrero - martn , c. mazzoli , h. c. walker , l. paolasini , s. francoual , d. k. shukla , j. strempfer , t. chatterji , c. m. n. kumar , r. mittal , h. m. rnnow , ch . regg , d. f. mcmorrow , and th . brckel , phys . b * 84 * , 054419 ( 2011 ) . j.d . thompson , r. movshovich , z. fisk , f. bouquet , n.j . curro , r.a . fisher , p.c . hammel , h. hegger , m.f . hundley , m. jaime , p.g . pagliuso , c. petrovic , n.e . phillips , j.l . sarrao j. magn . magn . mater . * 226230 * , 5 ( 2001 ) .

we investigate the magnetic and electronic properties of the gd@xmath0zn@xmath1 ( @xmath2 = fe and co ) compounds using x - ray resonant magnetic scattering ( xrms ) , x - ray absorption near - edge structure ( xanes ) and x - ray magnetic circular dichroism ( xmcd ) techniques . the xrms measurements reveal that the gdco@xmath3zn@xmath1 compound has a commensurate antiferromagnetic spin structure with a magnetic propagation vector @xmath4 = @xmath5 below the nel temperature ( @xmath6 5.7 k ) . only the gd ions carry a magnetic moment forming an antiferromagnetic structure with magnetic representation @xmath7 . for the ferromagnetic gdfe@xmath3zn@xmath1 compound , an extensive investigation was performed at low temperature and under magnetic field using xanes and xmcd techniques . a strong xmcd signal of about 12.5 @xmath8 and 9.7 @xmath8 is observed below the curie temperature ( @xmath9 85 k ) at the gd-@xmath10 and @xmath11 edges , respectively . in addition , a small magnetic signal of about 0.06 @xmath8 of the jump is recorded at the zn @xmath12-edge suggesting that the zn 4@xmath13 states are spin polarized by the gd 5@xmath14 extended orbitals .

You are an expert at summarizing long articles. Proceed to summarize the following text: angle - resolved photo - emission spectroscopy ( arpes ) directly probes the dispersion of electrons ; identifying a kink associated with the active optical phonon in the cuprates @xcite . neutron scattering has shown an anomalous change in the phonon spectrum at the superconducting transition , indicating an interesting role for phonons in the cuprates @xcite . estimates of the magnitude of the electron - phonon coupling , and the energy of the phonon mode put the problem outside the limited region of applicability for bcs theory , so schemes to cope with a wider range of parameters need to be developed . moreover , since there a node in the superconducting gap consistent with d - wave symmetry @xcite , any theory implicating phonons as the mechanism must be able to explain the unconventional order . electron - phonon ( e - ph ) interactions can be described by the following generic model , @xmath3 here , @xmath4 and @xmath5 with @xmath6 and @xmath7 ( representing a quasi-2d system and taming the van - hove singularities ) . the holstein model is a special case where @xmath8 and @xmath9 , for which @xmath10 is the dimensionless e - ph coupling ( @xmath11 is the half - band width ) . the momentum - independent phonon dispersion approximates an optical phonon , and the momentum - independent e - ph coupling corresponds to a completely local interaction . a coulomb pseudo - potential , @xmath12 , may be added ( i.e. hubbard model at the hartree fock level ) . ( a ) overview of the diagrammatic representations of the self - energy used in this work . @xmath13 is the electron , and @xmath14 the phonon self energy . the row labeled me contains the vertex neglected and vc the corrected diagrams . ( b ) schematic of the dca ( quasi - local ) formalism for cluster size @xmath15 . within the squares @xmath16 where @xmath17 are represented by solid circles . equivalently , @xmath18 . a partial density of states ( dos ) is associated with each sub - zone . dca enables computations in the thermodynamic limit , via an equivalent cluster impurity problem.,scaledwidth=46.0% ] some treatments of the arpes kink used a standard eliashberg theory with anisotropic coupling and an assumed @xmath0-wave order parameter for fitting purposes @xcite . the extended eliashberg approach that i have developed differs significantly from previous work in the sense that it is fully self - consistent , with no prior assumptions about the form of the order parameter ( i.e. any @xmath0-wave order arises directly from the self - consistency of the theory ) @xcite . in this section , i discuss how quasi - local eliashberg equations can be constructed , and demonstrate how the lowest - order fock diagram does not contribute to the gap equations . the calculation is intended to be demonstrative , and shows that standard eliashberg equations fail for an optical phonon mediated @xmath0-wave state . the eliashberg equations are computed from the fock contribution to the self - energy : @xmath19 and equate @xmath20 with the approximate form , @xmath21 normally the eliashberg equations are computed using a local ( momentum - independent ) approximation . however , to examine @xmath0-wave states , the local approximation is not sufficient . substituting the green function @xmath22 in eq . [ eqn : se ] ( @xmath23 , @xmath24 are pauli matrices and @xmath25 is the chemical potential ) , and treating the gap function as constant in sub - zones of @xmath26-space in the manner of dca , one obtains a quasi - local approximation for the eliashberg equations , @xmath27 when gap functions have @xmath0-wave symmetry , @xmath28 and @xmath29 , and the partial dos @xmath30 has the symmetry of the lattice , it is immediately clear that the @xmath31 contribution to the quasi - local eliashberg equations vanishes , and 2nd order terms must be considered . when there is some weak momentum dependence in the e - ph coupling and phonon frequency , this cancellation will not be exact , but the contribution from 2nd - order terms can still be expected to be the largest in the perturbation series . thus a fully extended set of eliashberg equations including vertex corrections needs to be constructed . currently , it is not possible to construct a similar approximation to eq . [ eqn : qlapprox ] for the vertex corrected theory . however a numerical approach using the full dca equations is available to enlighten some aspects of the e - ph problem . the rest of this article concerns results from an extended eliashberg theory with momentum dependence and vertex corrections determined numerically using dca . no form for the order parameter is assumed in advance and full forms for the green functions and self - energies are used . full details of the translation of the diagrams in fig . [ fig : overview](a ) may be found in ref . figure [ fig : one ] shows image plots of the electron spectral functions @xcite . vertex corrections are most pronounced for @xmath32 , as expected from analysis of the parameter space @xcite . in particular , the vertex corrected theory shows a kink at the energy scale relating to the phonon frequency * measured * at the @xmath33 point ( i.e. renormalised zone edge phonon frequency ) in agreement with the arpes measurements in ref . @xcite . phonon spectral functions are shown in fig . [ fig : two ] . the phonon renormalisation is extremely small for @xmath34 . in contrast , modes at the @xmath35 point in the brillouin zone are strongly softened for @xmath36 when vertex corrections are not taken into account . the second order terms act against the softening and reinforce the theory . a small reduction in spectral weight is seen mid way along the @xmath37-x and @xmath37-w lines when @xmath32 . this might be a precursor to the strong softening seen in ref . @xcite on passing through the transition temperature , but further analysis of the superconducting state is necessary to be sure . @xmath0-wave superconductivity is one of the remarkable results from the theory extended with vertex corrections . figure [ fig : three ] shows the anomalous density as the chemical potential is varied first upwards from half - filling and then back from large filling showing that two solutions are stable . a small @xmath38 is applied . @xmath39 and @xmath40 . starting with a half - filled solution and increasing , then decreasing the chemical potential , first @xmath0-wave and then ( on decreasing from high electron density ) @xmath41-wave solutions to the self - consistent equations are found ( as detailed in the caption of fig . [ fig : three ] ) . highly anisotropic solutions form as the fermi - surface approaches the van - hove points . in the @xmath0-wave state , the first order terms are not the leading order of the perturbation theory in @xmath2 . it was argued in ref . @xcite that this was in part due to the nearly flat phonon spectrum , so that the phonon propagator @xmath42 and the 1st order contribution to the anomalous self - energy @xmath43 is small ( @xmath44 is the anomalous green function ) . the anomalous contribution to the pseudopotential term is @xmath45 . since @xmath44 is modulated in the @xmath0-wave state , then the contribution of the pseudopotential term is zero . in the @xmath41-wave state , @xmath44 is not modulated , and the resulting finite contribution destroys the @xmath41-wave order for large enough @xmath46 , thus selecting a dominant @xmath0-wave contribution . the phonon spectra in fig . [ fig : two ] demonstrate that so long as @xmath47 , then this condition is met , and 2nd order terms are essential . 2nd order terms remain the leading order correction to the anomalous self - energy , even for the larger @xmath2 used here . it is clear that the 2nd order terms are similar for both repulsive and attractive models , and thus it is perhaps not surprising that both attractive and repulsive models show a @xmath0-wave state . i have described spectroscopic results and order parameters from an extended eliashberg - style theory with vertex and momentum corrections . the extended theory is essential for the investigation of d - wave superconductivity mediated by optical phonons , since the vertex corrections are the lowest order terms in the perturbation expansion in electron - phonon coupling @xmath2 . indeed , d - wave superconductivity is immediately predicted with such an extension . there are two remaining areas which require further investigation . the first involves a better treatment of the coulomb repulsion , which is currently only treated as a simple hartree - fock pseudopotential . in particular , the ability to treat large repulsion and antiferromagnetism is essential for high-@xmath48 materials . treatment of momentum dependent e - ph coupling would also be of interest . however , the overriding challenge at this stage is to convert the numerical results into a coherent analytic theory with vertex corrections .

i present results from an approach that extends the eliashberg theory by systematic expansion in the vertex function ; an essential extension at large phonon frequencies , even for weak coupling . in order to deal with computationally expensive double sums over momenta , a dynamical cluster approximation ( dca ) approach is used to incorporate momentum dependence into the eliashberg equations . first , i consider the effects of introducing partial momentum dependence on the standard eliashberg theory using a quasi - local approximation ; which i use to demonstrate that it is essential to include corrections beyond the standard theory when investigating @xmath0-wave states . using the extended theory with vertex corrections , i compute electron and phonon spectral functions . a kink in the electronic dispersion is found in the normal state along the major symmetry directions , similar to that found in photo - emission from cuprates . the phonon spectral function shows that for weak coupling @xmath1 , the dispersion for phonons has weak momentum dependence , with consequences for the theory of optical phonon mediated d - wave superconductivity , which is shown to be 2nd order in @xmath2 . in particular , examination of the order parameter vs. filling shows that vertex corrections lead to @xmath0-wave superconductivity mediated via simple optical phonons . i map out the order parameters in detail , showing that there is significant induced anisotropy in the superconducting pairing in quasi-2d systems . * [ published as : journal of physics and chemistry of solids , vol . 69 , 2982 - 2985 ( 2008 ) ] * extended eliashberg theory , superconductivity , spectroscopy , unconventional pairing

You are an expert at summarizing long articles. Proceed to summarize the following text: the spontaneous breakdown of symmetries in the early universe that produce linear discontinuities in the field is called cosmic string ( kibble 1976 ) . the cosmic strings are also common in modern string cosmologies . in 2003 , general interest in cosmic strings was heightened by the discovery of what seemed at first to be a plausible candidate for lensing by a cosmic string . a pair of images of elliptical galaxies separated by 1.8 arc seconds was found to have the same redshift , @xmath1 , and the same spectra . sazhim et al ( 2003 ) confirmed that these images were not distorted in the way that would be expected for lensing of a single galaxy but are consistent with lensing by cosmic string . in the literature , the line like structure of cosmic strings with particle attached to them are considered as possible seeds for galaxy formation at the early stages of the evolution of universe . in the past time , stachel ( 1980 ) , letelier ( 1983 ) and vilenkin et al ( 1987 ) have studied different aspects of string cosmological models in general relativity . reddy ( 2003 ) , reddy and naidu ( 2007 ) and recently yadav ( 2013 ) have investigated anisotropic string cosmological models in scalar - tensor theory of gravitation . + harko et al ( 2011 ) proposed @xmath0 gravity theory by taking into account the gravitational lagrangian as the function of ricci scalar r and of the trace of energy - stress tensor t. they have obtained the equation of motion of test particle and the gravitational field equation in metric formalism both . myrzakulov ( 2011 ) presented point like lagrangian s for @xmath0 gravity . the @xmath0 gravity model that satisfy the local tests and transition of matter from dominated era to accelerated phase was considered by houndjo ( 2012 ) . recently chaubey and shukla ( 2013 ) , naidu et al ( 2013 ) and ahmad and pradhan ( 2013 ) have investigated anisotropic cosmological model in @xmath0 gravity . in this paper , our aim is to study the role of strings in @xmath0 gravity and bianchi - v space time . since bianchi - v models are natural generalization of frw models hence the bianchi - v cosmological models permit one to obtain more general cosmological model , in comparison to frw model . in the recent years several authors ( yadav 2011 , kumar and yadav 2011 , yadav 2012 , pradhan et al 2005 , singh et al 2008 ) have studied bianchi - v cosmological models in different physical context . + in this paper , we establish the existence of bianchi - v string cosmological model in @xmath0 gravity and examine a cosmological scenario by proposing power law expansion . we observe that strings do not survive for long time and eventually disappear from universe . the organization of the paper is as follows : the model and basic theory of @xmath0 gravity are presented in section 2 . the field equations are established in section 3 and their solution is presented in section 4 . section 5 deals with the cosmological parameters and classical potential of derived model . finally conclusions are presented in section 6 . we consider the bianchi - v metric in following form @xmath2 here , @xmath3 , @xmath4 & @xmath5 are scale factors in the x , y & z direction respectively and @xmath6 is constant . + the action of @xmath0 gravity is given by @xmath7 where @xmath8 , @xmath9 and @xmath10 are the ricci scalar , the trace of the stress - energy tensor of matter and the matter lagrangian density respectively . + the stress - energy tensor of matter is given by @xmath11 the gravitational field equation of @xmath0 gravity is given by @xmath12 where @xmath13 , @xmath14 , @xmath15 and @xmath16 denotes the covariant derivative . + the stress - energy tensor is given by @xmath17 in general , the field equations depend through the tensor @xmath18 , on the physical nature of the matter field . hence we obtain several theoretical models for different choice of f(r , t ) depending on the nature of the matter source . in the literature , chaubey @xmath19 shukla ( 2013 ) , reddy et al . ( 2012a , 2012b ) and naidu et al ( 2013 ) have been studied the cosmological models , assuming @xmath20 . recently ahmad @xmath19 pradhan studied consequence of bianchi - v cosmological model in f(r , t ) gravity by considering @xmath21 . they have assumed perfect fluid as source of matter while in this paper , we assumed the string fluid as source of matter to describe the physical consequences of early universe . thus our paper is all together different from the paper of ahmad and pradhan ( 2013 ) . assuming @xmath22 and @xmath23 where @xmath24 is arbitrary parameter . + now equation ( [ eq4 ] ) can be rewritten as @xmath25 throughout the paper , we use units @xmath26 . + the expansion scalar @xmath27 and shear scalar @xmath28 have the form @xmath29 @xmath30 - \frac{\theta^{2}}{3}\ ] ] the einstein s field equation ( [ eq6 ] ) for the line - element ( [ eq1 ] ) leads to the following system of equations @xmath32 @xmath33 @xmath34 @xmath35 @xmath36 we have system of five equation ( [ eq9])@xmath37([eq13 ] ) involving six unknown variables , namely @xmath38 , @xmath39 , @xmath40 , @xmath41 , @xmath42 & @xmath43 . thus , in order to completely solve the fields equations , we need at least one physical assumption among the unknown parameters . in the literature , it is common to use the law of variation of hubble s parameter which yields the constant value of deceleration parameter . this law deals with two type of cosmology ( i ) power law cosmology ( ii ) exponential law cosmology . it is well known that the exponential law projects the dynamics of future universe and such type of model does not have consistency with present day observations . since we are looking for a model , describing the late time acceleration of universe therefore we choose the average scale factor in following form @xmath44 where @xmath45 , is positive constant . ( [ eq14 ] ) can be easily obtain by law of variation of hubble s parameter . + we define average scale factor @xmath46 and mean hubble s parameter @xmath47 of the bianchi - v model as @xmath48 @xmath49 where @xmath50 , @xmath51 and @xmath52 are directional hubble s parameters in the direction of x , y , and z- direction respectively . + from equation ( [ eq10 ] ) and ( [ eq11 ] ) , we obtain the following relation @xmath53 where @xmath54 and @xmath55 are constant of integrations . + integrating eq . ( [ eq13 ] ) , we get @xmath56 from equations ( [ eq14 ] ) , ( [ eq15 ] ) , ( [ eq17 ] ) and ( [ eq18 ] ) , the metric functions can be explicitly written as @xmath57 @xmath58\ ] ] @xmath59\ ] ] provided @xmath60 . the average hubble s parameter @xmath47 , expansion scalar @xmath27 , spatial volume @xmath61 , deceleration parameter ( dp ) and shear scalar @xmath28 are given by @xmath62 @xmath63 @xmath64 @xmath65 @xmath66 for accelerating universe , we impose the restriction on the value of n @xmath67 . if we take @xmath68 then the value of dp is @xmath69 which exactly matches with the observed value of dp at present epoch ( cunha et al . hence we constrain @xmath68 in graphical representations and discussion of physical parameters of derived model . + the isotropic pressure @xmath70 , proper energy density @xmath71 , string tension density @xmath72 and particle energy density @xmath73 are found to be @xmath74\ ] ] @xmath75\ ] ] @xmath76\ ] ] @xmath77\ ] ] [ cols= " > , < " , ] from eq . ( [ eq17 ] ) , we obtain @xmath78 following saha and boyadjiev ( 2004 ) and yadav(2013 ) , we have the following equation of motion of a single particle with unit mass under force @xmath79 @xmath80 where @xmath81 and @xmath82 are the classical potential and amount of energy respectively . ( [ eq29 ] ) and ( [ eq30 ] ) lead to @xmath83 the classical potential in terms of hubble s parameter is given by @xmath84 the age of universe in connection with dp is given by @xmath85 from eq . ( [ eq33 ] ) , it is clear that the value of @xmath86 in the range @xmath87 increase the age of universe . + also we know that the speed of sound @xmath88 is less than the speed of light @xmath89 . in gravitational unit we take @xmath90 . therefore for a physical viable model , @xmath88 lies between 0 and 1 . + from eq . ( [ eq25 ] ) and ( [ eq26 ] ) , the speed of sound is given by @xmath91 where + @xmath92 + @xmath93 + @xmath94 + in figure panel 1 , we graphed the parameters of derived model against @xmath95 for @xmath68 . the behaviour is quite evident ; the scale factors along axial direction satisfies the anisotropic nature of universe ; @xmath96 shows that the particles dominate over the strings with the evolution of universe hence the strings are not observed today ; the classical potential @xmath97 is positive and speed of sound @xmath98 is less the speed of light throughout the expansion of universe from big bang to present epoch . in the present paper , we have considered @xmath0 gravity model with an arbitrary coupling between matter and geometry in bianchi - v space - time . we have derived the gravitational field equations for string fluid corresponding to @xmath0 gravity model . we observed that string tension density @xmath72 decreases with time and it approaches to zero at present epoch . therefore strings could not survive with the evolution process of universe . that is why the strings are not observed today but it play significant role in early universe . the classical potential @xmath97 is positive and it decreases with time . the scale factors vanish at @xmath99 . thus the model has point type singularity at @xmath99 . as @xmath100 the scale factors diverge and the physical parameters such as expansion , scalar @xmath27 , energy density @xmath71 and hubble s parameters @xmath47 tend to zero . therefore in the derived model , all matter and radiation are concentrated in the big bang . it is important to note that @xmath101 and @xmath102 for @xmath100 in the derived model which implies the fastest rate of expansion of universe . so , the derived model can be utilized to describe the dynamics of universe at present epoch . since @xmath103 , thus the model approaches isotropy at late times .

* abstract : * in this paper , we search the existence of bianchi - v string cosmological model in @xmath0 gravity with power law expansion . einstein s field equations have been solved by taking into account the law of variation of hubble s parameter that yields the constant value of deceleration parameter ( dp ) . we observe that the massive strings dominate the early universe but they do not survive for long time and finally disappear from the universe . we examine the nature of classical potential and also discuss the physical properties of universe . + * keywords : * early universe , @xmath0 gravity and cosmological parameters .

You are an expert at summarizing long articles. Proceed to summarize the following text: the formation and evolution of structure in the universe is a fundamental field of research in cosmology . clusters of galaxies represent the most extreme deviation from initial conditions , and are therefore good evolutionary probes for studying the formation of the large - scale structure . while clusters of galaxies have been studied extensively in the relatively nearby universe , their evolutionary history becomes obscure beyond roughly half the hubble time ( e.g. blakeslee et al . 2006 , mullis et al . 2005 , stanford et al . their progenitors are difficult to identify when the density contrast between the forming cluster and the field becomes small , and mass condensations on the scales of clusters are extremely rare at any epoch ( kaiser 1984 ) . while deep pencil beam surveys have allowed us to study many different kinds of galaxies and their relative contributions to the cosmic star formation rate out to @xmath3 , the study of the evolution of galaxy clusters has not progressed much beyond @xmath4 . however , the relatively advanced evolutionary state of some clusters even at @xmath5 suggests a formation epoch at @xmath2 , consistent with predictions for the ( early ) formation of large - scale structure in cosmological simulations ( springel et al . 2005 ) . finding and studying the progenitors of these clusters will give new clues to how the most massive structures in the universe came about . distant radio galaxies are important laboratories for studying the formation and evolution of massive galaxies in the early universe . they are among the most luminous and largest known galaxies at @xmath6 , having stellar masses in excess of @xmath7 @xmath8 ( e.g. de breuck et al . 2002 , villar - martin et al . there is evidence that protoclusters in the early universe ( @xmath9 ) can be found around ( but are not limited to ) radio galaxies ( e.g. kurk 2000 , 2004 ; miley et al . 2004 ; overzier et al . 2006a,2006b ; pentericci et al . 2000,2002 ; venemans et al . 2002 , 2004 , 2005 ) . these protoclusters have galaxy and mass overdensities that are sufficiently large for such structures to break away from the expanding universe , and form a virialized cluster before @xmath10 . note that although powerful 3c - type radio sources are very rare objects , their number density is comparable to that of abell clusters ( @xmath11@xmath12 mpc@xmath13 ) if estimated radio source life - times ( @xmath14 yr ) are taken into account ( bahcall & soneira 1983 , dunlop & peacock 1990 , west 1994 ) . it is believed that the fueling of a supermassive black hole due to cooling flows , mergers and interactions are responsible for triggering a powerful ( radio ) quasar ( e.g. heckman et al . 1986 , kauffmann & haehnelt 2000 , canalizo & stockton 2001 , croton et al . if this indeed provides the trigger for radio activity , then the cores of protoclusters are likely locations to find radio galaxies at high redshift . taken together with the highly special intrinsic properties of the host galaxies of distant luminous radio sources , it is probable that radio galaxies could be progenitors of the brightest cluster galaxies ( bcgs ) . the characteristic luminosity of radio sources increases with redshift in a flux limited sample , meaning that one could be observing an increasing fraction of proto - bcgs with increasing redshift , while at low redshift ( @xmath15 ) 3c sources span a factor of @xmath16 in radio luminosity and most of these will therefore not correspond to bcgs ( west et al . we obtained deep images of the radio galaxy mrc 1138262 with the _ hubble space telescope _ advanced camera for surveys ( acs , ford et al . 1998 ) , as part of an acs gto programme to map the central regions of radio galaxy protoclusters in order to study star formation and galaxy morphologies in overdense environments at @xmath18 . by coadding the images taken through the filters @xmath19 and @xmath20 ( both in the rest - frame uv ) we created a 44 ksec image of a @xmath21 kpc@xmath22 region surrounding the radio galaxy ( fig . 1 ) . initially studied in detail by pentericci et al . ( 1997,1998,2001 ) , the increased depth of the acs image reveals more fine details and complexity in this remarkable object ( miley et al . 2006 ) : image of the radio galaxy mrc 1138262 at @xmath0 . the system consists of a large ( @xmath23 kpc ) conglomeration of sub - galactic clumps embedded in a region of diffuse emission . the total star formation rate in the clumps and the diffuse component ( several hundred @xmath8 yr@xmath24 ) are about equal . comparison with the rest - frame optical magnitude ( pentericci et al . 1997 ) indicates that the mass of the star - forming component seen in this image is only a small fraction of the total mass of the system , suggesting that the mrc 1138262 system has elements of both early and late formation ( see miley et al . 2006 for details ) . contours show the radio emission at 4.5 ghz . the postage stamps to the right of the image show close - ups of objects having interesting morphologies ( marked with the letters a e in the image ) . ] the optical continuum emission of the galaxy consists of at least 10 distinct clumps , presumably satellite galaxies that are still merging with mrc 1138262 . several of the faint satellites have interesting morphologies . object ` a ' in fig . 1 shows four bright knots of emission in a line like a string of beads . these longitudinal galaxies that are barely resolved in their transverse dimension are known as ` chain ' galaxies , and could be distant , ancestral forms of local hubble type galaxies ( cowie et al . they are believed to form from star formation progressing along filaments of gas previously blown out from massive forming galaxies ( taniguchi & shioya 2001 ) . because the chain galaxies are believed to be inherently unstable and therefore short - lived , the chance of finding one could have been enhanced by the vigorous merging and gaseous outflows observed in the mrc 1138262 system . the objects marked ` c ' and ` d ' in fig . 1 can be classified as ` tadpole ' galaxies , which consist of a knot at one end and an extended tail at the other . these objects are believed to be early - stage mergers ( straughn et al . other objects also show evidence for disruption , multiple nuclei , and tidal tails . interestingly , faint diffuse emission is visible throughout the @xmath25 kpc inter - clump region . the extended emission is unlikely to be caused by scattered light of the obscured quasar nucleus , because its roughly spherical morphology and radial profile do not resemble that of a scattering cone . the mean colour of the diffuse emission is comparable to that of the star - forming satellites , consistent with the occurrence of ongoing star formation over a very extended region . the small - scale variations in the surface brightness of the extended emission seen to the south - west of the nucleus could be caused by gradients in the amount of dust , or by the subclustering of star - forming clumps too faint to be detected individually . the total extended luminosity ( comprising 45% of the total emission in @xmath19 ) corresponds to a star formation rate of @xmath26100 @xmath8 yr@xmath24 . this indicates that star formation over extended regions such as observed in mrc 1138262 could be important for building the most massive galaxies , besides merging and gas accretion . contours ( blue ) of the giant emission line nebula surrounding mrc 1138262 , and vla 8ghz radio contours ( red ) of the non - thermal radio structure superimposed on the composite ( @xmath19 + @xmath20 ) acs image . the gaseous nebula extends for @xmath27200 kpc . ] the galaxy is surrounded by a giant ( @xmath28 kpc ) emission line nebula ( fig . 2 ) , similar to the large low surface brightness halos of ionized gas seen around other high redshift radio galaxies . these halos could be remnant reservoirs of gas from which the massive galaxies are forming , or they could be due to gas swept up by previous episodes of starburst superwinds or large - scale agn outflows ( e.g. haiman et al . 2000 , taniguchi & shioya 2000 , haiman & rees 2001 , villar - martn et al . when a powerful source of ionizing radiation becomes active ( e.g. , agn or starburst ) the halo will begin to emit luminous line emission , and high velocities of gas can be observed in regions where it is perturbed by interaction with the radio structure . the size of the halos is comparable to that of the diffuse envelopes of cd galaxies in the local universe , but it is not clear whether these are related phenomena . we interpret the new hst image as showing hierarchical merging in a proto - bcg . the morphological complexity of the mrc 1138262 system agrees , at least qualitatively , with predictions of massive galaxy formation ( e.g. dubinsky 1998 , gao et al . the extensive merging provides a plausible mechanism for fueling the radio source , which may subsequently interact with its surrounding medium to regulate its own growth by preventing gas accretion from the larger halo when the source is ` on ' ( croton et al . evidence for this kind of coupling between the radio structure and the surrounding medium has been revealed through recent integral field spectroscopy of the gaseous halo surrounding mrc 1138262 ( fig . 2 ) . nesvadba et al . ( 2006 ) found very high gas velocities and velocity dispersions associated with the radio structure . the energy of the outflow is sufficiently high to expel a significant fraction of the mass in gas during the lifetime of the radio source . after the star formation has switched off , the galaxy will continue to grow through merging ( de lucia & blaizot 2006 ) . however , the total stellar mass contained in the population of satellites is believed to be small compared to the mass of the central galaxy ( @xmath29 @xmath8 ) , estimated from its @xmath30-band magnitude ( pentericci et al . this may indicate that mrc 1138262 is an exceptionally rare object that accumulated a large amount of its mass within a relatively short time at @xmath31 . the observations presented here provide a unique testbed for simulations of forming massive galaxies at the centers of clusters and protoclusters . bahcall , n. a. , & soneira , r. m. 1983 , apj , 270 , 20 blakeslee , j. p. , et al . 2006 , apj , 644 , 30 canalizo , g. , & stockton , a. 2001 , apj , 555 , 719 cowie , l. l. , hu , e. m. , & songaila , a. 1995 , aj , 110 , 1576 croton , d. j. , et al . 2006 , mnras , 365 , 11 de breuck , c. , van breugel , w. , stanford , s. a. , rttgering , h. , miley , g. , & stern , d. 2002 , aj , 123 , 637 de lucia , g. , & blaizot , j. , 2006 , mnras , submitted dubinski , j. 1998 , apj , 502 , 141 dunlop , j. s. , & peacock , j. a. 1990 , mnras , 247 , 19 ford , h. c. , et al . 1998 , proc . spie , 3356 , 234 gao , l. , loeb , a. , peebles , p. j. e. , white , s. d. m. , & jenkins , a. 2004 , apj , 614 , 17 haiman , z. , spaans , m. , & quataert , e. 2000 , apj , 537 , l5 haiman , z. , & rees , m. j. 2001 , apj , 556 , 87 heckman , t. m. , smith , e. p. , baum , s. a. , van breugel , w. j. m. , miley , g. k. , illingworth , g. d. , bothun , g. d. , & balick , b. 1986 , apj , 311 , 526 kaiser , n. 1984 , apj , 284 , l9 kauffmann , g. , & haehnelt , m. 2000 , mnras , 311 , 576 kurk , j. d. , et al . 2000 , a&a , 358 , l1 kurk , j. d. , pentericci , l. , overzier , r. a. , rttgering , h. j. a. , & miley , g. k. 2004 , a&a , 428 , 817 miley , g. k. , et al . 2004 , nature , 426 , 47 miley , g. k. , et al . 2006 , apj , 650 , l29 mullis , c. r. , rosati , p. , lamer , g. , bhringer , h. , schwope , a. , schuecker , p. , & fassbender , r. 2005 , apj , 623 , l85 pentericci , l. , rttgering , h. j. a. , miley , g. k. , carilli , c. l. , & mccarthy , p. 1997 , a&a , 326 , 580 nesvadba , n. , et al . 2006 , apj , in press overzier , r. a. , et al . 2006a , apj , submitted ( astro - ph/0601223 ) overzier , r. a. , et al . 2006b , apj , 637 , 58 pentericci , l. , rttgering , h. j. a. , miley , g. k. , spinrad , h. , mccarthy , p. j. , van breugel , w. j. m. , & macchetto , f. 1998 , apj , 504 , 139 pentericci , l. , et al . 2000 , a&a , 361 , l25 pentericci , l. , mccarthy , p. j. , rttgering , h. j. a. , miley , g. k. , van breugel , w. j. m. , & fosbury , r. 2001 , apj , 135 , 63 springel , v. , et al . 2005 , nature , 435 , 629 stanford , s. a. , et al . 2006 , apj , 646 , l13 straughn , a. n. , cohen , s. h. , ryan , r. e. , hathi , n. p. , windhorst , r. a. , & jansen , r. a. 2006 , apj , 639 , 724 taniguchi , y. , & shioya , y. 2000 , apj , 532 , l13 taniguchi , y. , & shioya , y. 2001 , apj , 547 , 146 venemans , b. p. , et al . 2002 , apj , 569 , l11 venemans , b. p. , et al . 2004 , a&a , 424 , l17 venemans , b. p. , et al . 2005 , a&a , 431 , 793 villar - martn , m. , et al . 2006 , mnras , 366 , l1 west , m. j. 1994 , mnras , 268 , 79

we present deep observations taken with the hst advanced camera for surveys of the central massive galaxy in a forming cluster at @xmath0 . the galaxy hosting the powerful radio source mrc 1138262 is associated with one of the most extensive merger systems known in the early universe . our hst / acs image shows many star - forming galaxies merging within a @xmath1200 kpc region that emits both diffuse line emission and continuum in the rest - frame uv . because this galaxy lives in an overdense environment , it represents a rare view of a brightest cluster galaxy in formation at @xmath2 which may serve as a testbed for predictions of massive cluster galaxy formation . early universe , galaxies : clusters : general

You are an expert at summarizing long articles. Proceed to summarize the following text: among the various approximations existing in the literature to describe a diluted bose condensed gas at finite temperature , the generalized random phase approximation ( grpa ) has been the subject of several studies @xcite . this approximation has attracted a special attention since it is the only one in the literature with two important properties : 1 ) in agreement with the hugenholtz - pines theorem @xcite , it predicts the observed gapless and phonon - like excitations ; 2 ) the mass , momentum and energy conservation laws are fulfilled in the gas dynamical description . an approximation that satisfies these properties is said to be _ gapless _ and _ conserving _ @xcite . besides these unique features , the grpa predicts also other phenomena , namely a second branch of excitations and the dynamical screening of the interaction potential . these phenomena appear also in the case of a gas of charged particles or plasma . the possibility of a second kind of excitation has been explained quite extensively in @xcite . there is a distinction between the single particle excitations and the collective excitations . in the case of a plasma , the first corresponds to the electrically charged excitations and its dispersion relation is obtained from the pole of the one particle green function . the second corresponds to the plasmon which is a chargeless excitation whose the dispersion relation is obtained from the pole of the susceptibility function . the plasmon mediates the interaction between two charged excitations . more precisely , during the interaction , one charged excitation emits a virtual plasmon which is subsequently reabsorbed by another charged excitation ( see fig.1 ) . remarkably , such a description holds also for a bose gas with single atom excitations carrying one unit of atom number and with gapless collective excitations with no atom number . the poles of the green functions have a similar structure above the critical point . but below this critical point , the existence of a macroscopic condensed fraction _ hybridizes _ the collective and single particle excitations so that the poles of the one particle green function and the susceptibility function mix to form common branches of collective excitations @xcite . thus , at the difference of a plasma , the presence of a condensed fraction prevents the direct observation of the atom - like excitation through the one particle green function . the dynamical screening effect predicted in the grpa appears much more spectacular in a bose gas . the screening effect of the coulombian interaction is well known to explain the dissociation of salt diluted in water into its ions ( see fig.2a ) . but it also provides an explanation to the superfluidity phenomenon i.e. the possibility of a metastable motion without any friction . most of the literature on superfluidity is usually devoted to the study of metastable motion in a toroidal geometry like , for example , an annular region between two concentric cylinders possibly in rotation @xcite . in this simply connected geometry , the angular momentum about the axis of the cylinder of the superfluid is quantized in unit of @xmath2 . the metastability of the motion is explained by the impossibility to go continuously from one quantized state to another due to the difficulty to surmount an enormous free - energy barrier . this is not the situation we want to address in this paper . we are rather focusing on the explanation of the superfluid ability to flow without any apparent friction with its surrounding . the landau criterion is a necessary but not sufficient condition for superfluidity . it tells about the kinematic conditions under which an external object can move relatively to a superfluid without damping its relative velocity by emitting a phonon - like collective excitation . for a dilute bose gas at low temperature , it amounts to saying that this relative velocity must be lower than the sound velocity @xcite . the external object is assumed to be macroscopic and can be an impurity @xcite , an obstacle like a lattice @xcite or even the normal fluid @xcite . in particular , this criterion does not taken into account the fact that the normal fluid is microscopically composed of thermal excitations . in a bose condensed gas , even though their relative velocity is on average lower than the critical one , many of these excitations are very energetic with a relative velocity high enough to allow the phonon emission . in the grpa where these excitations correspond to the thermal atoms and under the condition of the landau criterion , such a process is forbidden as shown qualitatively from fig.2b . the effect of an external perturbation of the condensed atoms caused for example by the thermal atoms is attenuated by the dynamical screening . this screening is total in the sense that no effective mutual binary interaction allows a collision process which would be essential for a dissipative relaxation of the superfluid motion . the purpose of this paper is to show that these peculiar phenomena could in principle be observed in a raman scattering process . this process induces a transition for a given frequency @xmath0 and a wavevector @xmath1 determined from the difference of the frequencies and the wavevectors of two laser beams @xcite . for each wavevector corresponding to the transferred momentum , one can arbitrarily tune the frequency in order to reach the resonance energy associated to the excitation . unlike the bragg scattering which allows the observation of the bogoliubov phonon - like collective excitation @xcite , the raman scattering is more selective . not only the gas is probed with a selected energy transition and transferred momentum , but the atoms are scattered into a selected second internal hyperfine level . through a zeeman splitter , they can be subsequently analyzed separately from unscattered atoms . according to the grpa , the scattered thermal atoms become distinguishable from the unscattered ones and thus release the gap energy due to the exchange interaction . in a previous study @xcite , we showed that this gap appears as a resonance in the frequency spectrum of the atom transition rate at @xmath3 . the possibility of momentum transfer allows to analyze the influence of the screening of the external perturbation induced by the raman light beams . the paper is divided as follows . in section 2 , we review the time - dependant hartree - fock ( tdhf ) equations for a spinor condensate and study the linear response function to an external potential which gives results equivalent to the grpa . sections 3 and 4 are devoted to the bragg and raman scatterings respectively . section 5 ends up with the conclusions and the perspectives . we start from the time - dependant hartree - fock equations for describing two component spinor bose gas @xcite labeled by @xmath4 . the atoms have a mass @xmath5 , feel the external potential @xmath6 and the hartree and fock mean field interaction potential characterized by the coupling constants @xmath7 expressed in terms of the scattering lengths @xmath8 between components @xmath9 and @xmath10 ( @xmath11 ) . note that no fock mean field ( or exchange ) interaction energy appears between condensed atoms . these equations describe the time evolution of a set of spinor wave function @xmath12 describing @xmath13 atoms labeled by @xmath14 and depending on the position @xmath15 and on the time @xmath16 . for the condensed mode ( @xmath17 ) , these are : @xmath18 for a non condensed mode ( @xmath19 ) , these are @xmath20 the non condensed spinors remain orthogonal during their time evolution in the thermodynamic limit . in general , the spinor associated to the condensed mode does not remain orthogonal with the others . but according to @xcite , the non orthogonality is not important in the thermodynamic limit for smooth external potential . another way of justifying the non orthogonality is to start from an ansatz where the condensed spinor mode is described in terms of a coherent state and the non condensed ones in terms of a complete set of orthogonal fock states i.e. @xmath21 where @xmath22 is the atom creation operator in the mode @xmath14 and @xmath23 . the theory remains _ conserving _ because the conservation laws are preserved on average but becomes non _ number conserving _ since the quantum state is not an eigenstate of the total particle number operator . this procedure is justified in the thermodynamic limit since the total particle number fluctuations are relatively small during the time evolution . in contrast , instead of using spinor wavefunctions , the alternative method based on the use of excitation operators is number conserving @xcite . the atom number @xmath13 for each mode is supposed time - independent in the tdhf . strictly speaking , a collision term must be added in order to allow population transfers between the various modes . these equations are valid in the collisionless regime i.e. on a time scale shorter than the average time between two collisions @xmath24 where @xmath25 is the average velocity and @xmath26 is the scattering cross section . in these conditions , the resulting frequency spectrum has a resolution limited by @xmath27 . the magnitude order of resolution of interest is given by the @xmath28 s so we require @xmath29 which is generally the case when @xmath30 . these conditions are fulfilled for the parameter values considered in this work . in the following , we will restrict our analysis to a bulk gas embedded in a volume @xmath31 . at @xmath32 , we assume all atoms in thermodynamic equilibrium in the level @xmath33 and that @xmath34 except for @xmath35 which is constant and fixes the energy shift between the two sub - levels . in that case , the solutions of the tdhf are orthogonal plane waves with @xmath14 corresponding to the momentum @xmath36 : @xmath37}{\sqrt{v } } \left(\begin{array}{c}1 \\ 0 \end{array } \right)\end{aligned}\ ] ] where we define the hartree - fock energy for atoms with momentum @xmath36 : @xmath38 where @xmath39 and where the condensed and total particle densities are @xmath40 and @xmath41 . eq.([drs ] ) corresponds to the dispersion relation of the single particle excitation . at equilibrium , @xmath42 - 1 ) \ ] ] is the bose - einstein distribution . below the condensation point , the chemical potential becomes @xmath43 and the macroscopic occupation @xmath44 is fixed to satisfy the total number conservation . for @xmath45 , we apply an external potential . for the bragg and raman scatterings , these are respectively : @xmath46\end{aligned}\ ] ] we solve the system through a perturbative expansion : @xmath47 the equations of motion for the first order corrections are for the case of bragg and raman scatterings respectively : @xmath48 \psi^{(1)}_{1,{\mathbf{k } } } = \nonumber \\ \left[v_{11}+\!\sum_{\mathbf{k ' } } g_{11 } ( 2-\delta_{{\mathbf{k'}},{\mathbf{0}}}\delta_{{\mathbf{k}},{\mathbf{0 } } } ) ( { \psi^{(0)*}_{1,{\mathbf{k ' } } } } \psi_{1,{\mathbf{k'}}}^{(1 ) } + c.c . ) n_{\mathbf{k ' } } \right]\ ! \psi^{(0)}_{1,{\mathbf{k } } } \\ \label{p21 } \left[i{\partial_t } + \frac{\nabla^2_{\mathbf{r}}}{2 m } - \omega_0 - g_{12}(n-\delta_{{\mathbf{k}},{\mathbf{0}}}n_{{\mathbf{0 } } } ) \right ] \psi^{(1)}_{2,{\mathbf{k } } } = \left[v_{12 } + g_{12}\sum_{\mathbf{k ' } } n_{\mathbf{k ' } } { \psi^{(0)*}_{1,{\mathbf{k ' } } } } \psi^{(1)}_{2,{\mathbf{k'}}}\right ] \psi^{(0)}_{1,{\mathbf{k}}}\end{aligned}\ ] ] these two set of integral equations can be solved exactly using the methods developed in @xcite . defining the fourier transforms : @xmath49 } v_{ab}({\mathbf{r}},t)\end{aligned}\ ] ] one obtains in the level 1 for the condensed mode : @xmath50 for the non condensed modes ( @xmath51 ) : @xmath52 and in the level 2 for all modes : @xmath53 these formulae resemble the one obtained from the non interacting bose gas excepted for the mean field term in ( [ psir ] ) and the extra factors representing the screening effect . for the bragg scattering , these factors can be written as @xcite : @xmath54 where @xmath55 -8g_{11}\chi_0({\mathbf{q}},\omega)g_{11 } n_{{\mathbf{0}}}\epsilon_{\mathbf{q}}\end{aligned}\ ] ] is the propagator for the collective excitations , @xmath56 is the bogoliubov excitation energy , @xmath57 is the sound velocity and @xmath58 is the susceptibility function describing the normal atoms . for the raman scattering , it is @xmath59 where @xmath60 knowing the fourier transform of the potential @xmath61 and @xmath62 , eqs.([psib0],[psib],[psir ] ) are calculated using the contour integration method over @xmath63 by analytic continuation in the lower half plane . as a consequence , the poles of the integrand tell about the excitation frequencies induced by the external perturbation . the pole of the propagator containing @xmath36 corresponds to atom excitation involving one mode only while the poles coming from the screening factors correspond to the excitations involving all modes @xmath36 collectively . thus , the tdhf approach predicts both single atom and collective excitations . note that the single mode excitation is not possible for the condensed atoms since the corresponding pole is compensated by a zero coming from the screening factor . the expressions ( [ psib0],[psib],[psir ] ) have an interpretation shown in fig.3 . an atom of momentum @xmath36 is scattered into a state of momentum @xmath64 by means of an external interaction mediated by a virtual collective excitation of momentum @xmath65 . [ vc ] let us first review the bragg scattering process . up to the second order in the bragg potential , the atoms number for any mode @xmath36 can be decomposed into an unscattered part : @xmath66\end{aligned}\ ] ] and a scattered part : @xmath67 is determined through the conservation relation @xmath68 . generally speaking within the sublevel 1 , the scattered atoms can not be distinguished from the unscattered ones . but in order to understand the underlying physics , we assume that distinction is possible . within the second order perturbation theory , the quantity of interest is the scattered atom rate per unit of time and is expected to reach a stationary value after a certain transition time . in the following , we shall analyze these transition rates for time long enough that transient effects disappear . in these conditions , a perturbative approach is still valid for very large time provided that the scattered atom number remains low compared to unscattered ones . this last requirement is always satisfied with a sufficiently weak external perturbation . at zero temperature , only the condensed wave function is modified and eq.([psib0 ] ) becomes after contour integration over @xmath63 : @xmath69\end{aligned}\ ] ] the response function is only resonant at the bogoliubov energy @xmath70 . also no transient response appears at zero temperature . using ( [ psib0 ] ) and ( [ psib ] ) , the total number of scattered atom can be obtained by determining the total momentum : @xmath71 through @xcite : @xmath72 using eq.([t0 ] ) , we recover that : @xmath73 where @xmath74 is the static structure factor . the delta function comes from the relation @xmath75 . the result ( [ suscb ] ) obtained in the grpa is identical to the one obtained from the bogoliubov approach where @xmath76 can be calculated equivalently from the four points correlation function @xcite . but in any case the generated phonon like excitation is still a part of the macroscopic wave function @xmath77 . at temperatures different from zero , the poles become imaginary which means that any bogoliubov excitation is absorbed by a thermal atom excitation @xcite . this phenomenon is known as the landau damping . so for long time , only the residues of ( [ psib0 ] ) with poles touching the real axis contribute whereas the others give rise to transient terms negligible for long time . thus the perturbative part becomes : @xmath78 using the property @xmath79 , the total number in the condensed mode reaches a constant value @xmath80 and the scattered thermal atom rate is given by : @xmath81 from ( [ p2 ] ) , we deduce for the imaginary susceptibility : @xmath82 the basic interpretation of these formulae is the following . at finite temperature , the collective excitation modes created by the external perturbation are damped over a time given by the inverse of the landau damping . so the number of collectively excited condensed atom reaches the constant value ( [ n0scat ] ) when the produced collective excitations rate compensates their absorption rate by thermal atoms . this constant value is higher for a transition frequency and a transferred momentum close to the resonance @xmath83 . the formula ( [ nscatt ] ) is a generalization of the fermi - golden rule when the screening effect is taken into account . the external potential perturbs the thermal atoms of momentum @xmath36 in two channels by transferring a momentum @xmath84 and a transition energy @xmath85 such that the resulting single atom excitation has a momentum @xmath86 and a kinetic energy @xmath87 . the presence of the screening factor amplifies or reduces the scattering rate . amplification ( or anti - screening ) occurs for a frequency close to the resonance energy @xmath88 of the collective excitations . on the contrary , dynamical screening occurs for a frequency close to the pole of the screening factor and is total for transition involving condensed atom at @xmath89 . thus , in grpa , attempt to generate incoherence through single condensed atom scattering is forbidden at finite temperature . only collective excitations affect the condensed mode but they are damped and therefore can not contribute to effectively transfer condensed atoms to a different mode @xcite . it is taught in standard textbooks @xcite that , in the impulse approximation used for large @xmath1 , the response of the system is sensitive to the momentum distribution of the gas , since the atoms behave like independent particle . in particular , a delta peak is expected to account for the presence of a condensate fraction . the difficulty of the observation of this peak could be explained by this impossibility of a single condensed atom excitation at finite temperature . for completeness , let us mention that interaction with thermal atoms can be also totally screened and inspection of the formulae ( [ k ] ) shows that this happens for @xmath90 @xcite . 4 shows these features in the frequency spectrum for the total momentum rate ( [ chit ] ) at fixed @xmath1 . we choose the typical density observed experimentally for @xmath91 at the trap center @xcite . these results can be put in direct relation with the analysis of impurity scattering @xcite . indeed , the dynamic response function is related to the dynamic structure factor through the fluctuation - dissipation theorem : @xmath92 . the dynamic structure factor is directly connected to the transition probability rate @xmath93 that an external particle or impurity changes its initial momentum @xmath94 and energy @xmath95 into @xmath96 and @xmath97 respectively : @xmath98 where @xmath99 is the fourier transform of the interaction potential between the impurity and the atom gas . the total rate of scattering @xmath100 results from a virtual process involving emission and absorption of the collective excitations : @xmath101 as a consequence , the impurity scattering is possible provided that the energy and momentum are conserved in a effective collision with a thermal atom of momentum @xmath36 mediated by a virtual collective excitation . note that total screening prevents impurity scattering involving ongoing and outgoing condensed atoms . in contrast , for temperature close to zero , the landau damping approaches zero since @xmath102 so that the application of eq.([suscb ] ) to ( [ gam ] ) leads to an on - energy shell process of absorption and emission of a collective excitation . we obtain : @xmath103 where @xmath104 . this limit case leads to the apparent interpretation of an impurity interacting with a thermal bath of phonon - like quasi - particle . this situation has been considered in @xcite in the study of the impurity dynamics . instead , eq.([gam2 ] ) provides a generalization for higher temperature emphasizing that any external particle can excite a single thermal atom alone but not a condensed one . the conclusions so far obtained in the bragg process can be extended straightforwardly to the case of raman scattering with the difference that only one channel of scattering is possible . for the purpose of simplicity , we choose the case @xmath105 . also this channel is easier to access experimentally . defining the detuning @xmath106 , explicit calculations of the spinor component ( [ psir ] ) in the second sublevel give : @xmath107 so we obtain for the atom number in the mode @xmath36 : @xmath108 by summing over all the modes , we obtain the density rate transferred in level 2 : @xmath109 where we define the imaginary part @xmath110 of the intercomponent susceptibility function : @xmath111 this last formulae is also the one obtained in the grpa @xcite . again we find a similar structure as the intracomponent case . in this process , thermal atoms with an initial momentum @xmath36 and energy @xmath112 are transferred into a second level with momentum @xmath113 and energy @xmath114 provided @xmath115 . in absence of screening , a resonance appears at the detuning @xmath116 . the first term corresponds to the usual recoil energy while the second is the gap energy @xmath117 that results from the exchange interaction . during the raman transition , the transferred atoms become distinguishable from the others and release this gap energy . the scattering rate is determined through the imaginary part of the susceptibility eq.([chirpa ] ) versus the transition frequency @xmath0 and at fixed @xmath1 . [ fig:1 ] and [ fig:2 ] show the corresponding resonance around this gap in absence of screening . the screening effect strongly reduces the raman scattering and , in particular , forbids it for atoms with momentum @xmath36 such that @xmath118 . this case corresponds to @xmath119 and includes also the condensed atoms ( @xmath120 ) . the graphs illustrate well the effect of the macroscopic wave function that deforms its shape in order to attenuate locally the external potential displayed by the raman light beams and to prevent incoherent scattering of the condensed atom . the experimental observation of this result would explain some of the reasons for which a superfluid condensate moves coherently without any friction with its surrounding . anti - screening occurs in the region close to the resonance frequency @xmath88 of the collective mode . at zero temperature , we recover @xmath121 @xcite while for non zero temperature the collective modes become damped for @xmath122 @xcite . these results can be compared to the one obtained from the bogoliubov non conserving approximation developed in @xcite and valid only for a weakly depleted condensate . this approach implicitly assumes that the only elementary excitations are the collective ones and form a basis of quantum orthogonal states that describe the thermal part of the gas . consequently , this formalism predicts no gap and no screening . instead , the intercomponent susceptibility describes transitions involving the two collective excitation modes of phonon @xmath123 and of rotation in spinor space @xmath124 : @xmath125 where @xmath126^{1/2}$ ] . this function does not preserve the f - sum rule associated to the @xmath127 symmetry . in contrast to the grpa , a delta peak describes a spinor rotation transition of the condensed fraction , and two other transitions involve the excitation transfer from a phonon mode into a rotation mode and the excitation creation in the two modes simultaneously . for small @xmath1 , these processes remain dispersive since the frequency transition depends on the momentum @xmath36 . as a consequence , the resulting spectrum shown in figs . 5 and 6 is broader . in particular , the process of creation in the two modes favors transition with positive frequency . note also the maximum of the curve separating the region involving a transition atom - atom like ( high @xmath36 ) and the one involving a transition phonon - atom like ( low @xmath36 ) . all these features established so far for the bulk case allow a clear comparison between the grpa and the bogoliubov approaches . in the real case of a parabolic trap , the inhomogeneity induces a supplementary broadening of the spectrum that prevents the direct observation of the screening . this effect as well as the finite time resolution and the difference between the scattering lengths will be discussed in a subsequent work . we have analyzed the many body properties that can be extracted from the raman scattering in the framework the grpa . the calculated spectrum allows to show the existence of a second branch of excitation but also the screening effect which prevents the excitation of the condensed mode alone . the observation of phenomena like the gap and the dynamical screening could have significant repercussions on our microscopic understanding of a finite temperature bose condensed gas and its superfluidity mechanism . on the contrary , the non - observation of these phenomena would imply that the _ gapless _ and _ conserving _ grpa is not valid . in that case , a different approximation has to be developed in order to explain what will be observed . as an alternative , the idea to use the bogoliubov approach has been also discussed . but unfortunately , the violation of the f - sum rule is a serious concern regarding this _ non conserving _ approach @xcite . all these aspects emphasize the importance of the experimental study of the raman scattering at finite temperature . pn thanks the referees for usefull comments and acknowledges support from the belgian fwo project g.0115.06 , from the junior fellowship f/05/011 of the kul research council , and from the german avh foundation . p. szpfalusy and i. kondor , ann . phys . * 82 * , 1 - 53 ( 1974 ) ; 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we explore the many body physics of a bose condensed atom gas at finite temperature through the raman transition between two hyperfine levels . unlike the bragg scattering where the phonon - like nature of the collective excitations has been observed , a different branch of thermal atom excitation is found theoretically in the raman scattering . this excitation is predicted in the generalized random phase approximation ( grpa ) and has a gapped and parabolic dispersion relation . the gap energy results from the exchange interaction and is released during the raman transition . the scattering rate is determined versus the transition frequency @xmath0 and the transferred momentum @xmath1 and shows the corresponding resonance around this gap . nevertheless , the raman scattering process is attenuated by the superfluid part of the gas . the macroscopic wave function of the condensate deforms its shape in order to screen locally the external potential displayed by the raman light beams . this screening is total for a condensed atom transition in order to prevent the condensate from incoherent scattering . the experimental observation of this result would explain some of the reasons why a superfluid condensate moves coherently without any friction with its surrounding .

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Proceed to summarize the following text: the presence of disorder and impurities in strongly correlated systems offers a good opportunity to better understand the role played by quantum fluctuations in such materials @xcite . either intrinsically present or explicitly added by doping , impurities in condensed matter systems can rarely be ignored , in particular when they induce new physics as compared to the disorder - free situation . prominent examples are the kondo effect @xcite , anderson localization @xcite , dirty bosons physics in disordered superconductors @xcite , impurities in magnetic semiconductors @xcite , spin glasses @xcite ... in the context of antiferromagnetic ( af ) mott insulators , parent compounds of high temperature superconducting cuprates for instance , spin ladder materials @xcite have been shown to display very interesting features , in particular when the number of legs is an even number . for example , a finite energy gap @xmath3 appears in the excitation spectrum of two - leg af spin-1/2 ladders @xcite , as seen for instance in srcu@xmath1o@xmath4 @xcite . furthermore , defects in gapped ladders induce very interesting effects @xcite , in particular the apparition of effective gapless modes below the bare spin gap @xmath3 . having in mind that the ground - state of a two - leg ladder displays short - range resonating valence bond like physics @xcite , a non - magnetic dopant is expected to break such a short - distance singlet , inducing a quasi - free spin-@xmath5 , strongly localized in the vicinity of the impurity . interesting questions arise then when a finite concentration of impurities is introduced in a spin ladder , as studied in a large number of theoretical works @xcite . similar physics is also at play in other spin - gapped materials : spin-@xmath6 ( haldane ) chains such as y@xmath1banio@xmath7 @xcite , or pbni1@xmath1v@xmath1o@xmath8 @xcite , spin - peierls chains such as cugeo@xmath4 @xcite for instance . indeed , a universal behavior can be observed for the impurity - induced three - dimensional ordering mechanism in such weakly coupled chains or ladder materials @xcite . nevertheless , several aspects of impurity effects in ladder and more generally spin gapped materials remain to be explored in order to better understand and interpret experimental results . regarding the effective pair - wise interaction between released moments , it is believed to remain non - frustrated even when the underlying spin systems is frustrated @xcite , but it is not clear to which extend such a result is robust when strong frustration leads to incommensurability @xcite , as expected for instance in the ladder material bicu@xmath1po@xmath2 @xcite . a natural question then arises regarding which effective model is able to quantitatively describe the low energy physics of randomly doped ladders . indeed , it was believed since the seminal work of sigrist and furusaki @xcite that a simple model of random ( in sign and magnitude , reflecting the random locations of impurities in a ladder ) nearest - neighbors couplings between effective spin-1/2 ( describing impurity degrees of freedom ) was able to correctly capture the low temperature physics of depleted ladders . this so - called random f - af chain model @xcite displays some universal behavior for various quantities such as uniform and staggered susceptibilities or the specific heat in the low temperature regime , with an interesting large spin phase occurring at very low temperature . however , in the context of depleted ladders , universality for such thermodynamic quantities has been first questioned using quantum monte carlo ( qmc ) simulations by miyazaki and co - workers @xcite where no clear signature of universal low temperature scalings were found , in agreement with a more recent qmc study @xcite . despite the large number of works devoted to such systems in the absence of external magnetic field , much less is known regarding finite field effects . indeed , as recently reviewed by giamarchi and co - workers @xcite , applying a finite external field on gapped af systems leads to the analog of a bose - einstein condensation ( bec ) of magnetic excitations @xcite ( hard - core bosons triplets ) when the field is sufficiently strong to close the spin gap @xmath3 . note that a true bec is only expected for dimension @xmath9 , which occurs at low enough temperature below some energy scale controlled by three dimensional couplings . nevertheless for low-@xmath10 a quasi - bec is expected , as observed in ultra - cold atom physics @xcite . in solid state physics , triplet bec has been observed in several quantum magnetic compounds , such as coupled dimers tlcucl@xmath4 @xcite , frustrated bilayers bacusi@xmath1o@xmath2 @xcite , coupled haldane chains in dtn @xcite , and also spin ladder materials like ( c@xmath7h@xmath11n)@xmath1cubr@xmath12 @xcite . however , when disorder is present in such spin gapped systems , a new phenomenology is expected with the interesting possibility to address bose - glass ( bg ) physics , as recently found in br - doped ipacucl@xmath4 @xcite or dtn @xcite ( see also ref . for a very recent review ) . while several issues remain unsolved regarding bg physics , _ e.g. _ for the excitation spectrum @xcite , the case where disorder comes from ligand substitution seems easier to understand from a microscopic point of view . indeed , such doping will essentially generate disorder in the af couplings without inducing local moments . on the other hand , doping on the magnetic sites is expected to be more complicated as gapless states will populate the clean gap . therefore the magnetic response will display non - trivial brillouin - like behaviors in most of the experimentally relevant situations . such cases have been studied theoretically by a few authors @xcite , showing a rich physics and various scenarios that demand further analysis . in this work , we focus on the two - leg ladder model to provide a systematic analysis of the physics of interacting impurities , building on both analytical and numerical arguments . in particular , we are interesting in the following issues : ( i ) the effective interaction between impurities for commensurate and incommensurate backgrounds ; ( ii ) the low energy emergence of large spins due to random signs in effective couplings in a realistic context including finite size effects due to chain breaking ; ( iii ) the temperature scaling of the curie constant of the uniform susceptibility , as obtained from both effective and realistic doped ladder models ; ( iv ) the deviations of the magnetic curve from the brillouin response as a probe of the magnitude of interactions . the ladder model used throughout this study is the one studied in refs . @xmath13\\ \nonumber + & j_{2 } \left[\mathbf{s}_{i,1}\cdot\mathbf{s}_{i+2,1 } + \mathbf{s}_{i,2}\cdot\mathbf{s}_{i+2,2}\right]\\ \label{eq : microh } + & j_{\perp } \,\mathbf{s}_{i,1}\cdot\mathbf{s}_{i,2 } \;,\end{aligned}\ ] ] where @xmath14 is the spin-@xmath0 operator acting at site @xmath15 of leg @xmath16 and the @xmath17s are the magnitude of the various couplings which are here taken to be antiferromagnetic ( @xmath18 ) . in the rest of the paper , the only parameter coming with the presence of impurities is their concentration @xmath19 . the doped microscopic model is numerically solved with two state - of - the - art methods : the density - matrix renormalization group ( dmrg ) technique @xcite and the stochastic series expansion ( sse ) quantum monte - carlo ( qmc ) technique @xcite . the paper is organized as follow : in a first part , we discuss in details the effective model describing two - body interactions between impurities . the resulting effective model is then compared to the solution of the microscopic model in the second part . the latter is dedicated to the study of the magnetization curve at field below the spin gap @xmath3 , i.e. in the region dominated by the impurity spins response . this region is itself divide in two regimes : ( i ) the small field regime @xmath20 , featuring a temperature - dependent curie constant @xmath21 , ( ii ) the intermediate field regime @xmath22 displaying again deviations from brillouin through an approximate power - law behavior . we do not investigate fields @xmath23 as the physics involves triplet bosons in a disordered medium which is exciting but beyond the scope of the present manuscript . in this first section , the emphasis is put on the quantitative analysis of the effective interaction between impurities from arguments similar to rkky theory . this provides an effective hamiltonian which couplings distribution is essential for understanding the magnetic responses . previous works along this direction are found in refs . . we start with the derivation of the low - energy effective hamiltonian accounting for effective interactions between impurities . for a generic heisenberg spin model with @xmath24 spins , the clean hamiltonian takes the general form @xmath25 where @xmath26 are the microscopic couplings , which depend only on the relative distance @xmath27 . we now consider that a few non - magnetic impurities occupy sites @xmath28 of the lattice . then , the hamiltonian reads @xmath29 or @xmath30 , where @xmath31 notice that effective spins operators are introduced at sites @xmath28 where the impurities live , while these sites are actually vacant . hamiltonian takes the form @xmath32 , in which @xmath33 is an effective magnetic field operator . assuming the perturbation @xmath34 can be treated using linear response theory , we may write the fourier transform of @xmath35 : @xmath36 with @xmath37 is the static susceptibility at wave - vector @xmath38 , and @xmath39 one can thus write the perturbation @xmath34 as @xmath40 in which @xmath41 is the effective two - body interaction between impurities . when the clean system possesses a spin gap associated to a spin correlation length @xmath42 , the susceptibility @xmath43 , and therefore the effective interaction @xmath44 , decreases exponentially with the distance @xmath45 . for a sufficiently small impurity concentration @xmath19 ( @xmath46 in one dimension ) , effective interactions remain much smaller than the spin gap . at temperatures smaller than this gap , the clean part of the doped system can be considered to be in the ground - state of @xmath47 while the impurities dynamics is governed by , in which one can take the zero - temperature behavior for the susceptibility @xmath37 . the static susceptibility of the ground - state of can be computed using the bond - order mean - field ( bomf ) approximation @xcite ( see appendix . [ app : bomf ] ) . in the strong - coupling limit @xmath48 , the spin gap is in the @xmath49 sector and the magnon branch is well separated from the two - magnons continuum . on can thus neglect the @xmath50 contribution and keep only the single magnon one . the details of the calculations are given in appendix [ app : bomf ] and show that the susceptibility displays the same singularities as the spin structure factor . in the large @xmath51 regime , the result reads @xmath52 before turning to the interaction between two impurities , it is first instructive to consider the magnetization pattern induced by a single impurity , which is also of interest for nuclear magnetic resonance ( nmr ) experiments . the impurity is located at site @xmath53 and the corresponding effective magnetic field defined by is simply given by @xmath54 . the expectation value of the spin operator @xmath35 is then given by linear response theory which , in fourier transform , reads @xmath55 this perturbative response is a priori valid far from the impurity . the magnetization profile in sector @xmath56 is then @xmath57 in which the general expression of the coupling of the frustrated ladder hamiltonian is @xmath58 after computing the integral limit of the sum over the brillouin zone , one obtains two different situations , depending on the behavior of the residues : in the commensurate regime @xmath59 , the profile is given by @xmath60 } + \frac{e^{-x/\xi_\text{spin}^-}}{\sinh\left(1/\xi_\text{spin}^-\right ) p'\left[-\cosh\left(1/\xi_\text{spin}^-\right)\right ] } \bigg ) \;,\ ] ] where @xmath61 are the spin correlation lengths defined in eqs . and @xmath62 is the derivative of the polynomial @xmath63 defined in eq . . in the incommensurate regime @xmath64 , one has @xmath65 } \bigg ] \;,\ ] ] where @xmath66 and @xmath42 are defined by eqs . and . notice that there is no unknown constant in these expressions . the key point of this result is that the transition from commensurate to incommensurate correlations induces a discontinuity in the features of the magnetization profile , which will show up in the effective interaction too . indeed , at each side of the transition , both residues diverge but their sum tends to zero . notice that exactly at the transition , the denominator factorizes , having a single pole of order 2 for which the residue is zero , corresponding to @xmath67 . on the contrary , the amplitude of the magnetization @xmath68 goes to @xmath69 at each side of the transition . but this does not mean that the magnetization profile diverges at short distance . of course , @xmath68 remains always bound by @xmath0 . however , the fact that the amplitude of the asymptotic behavior diverges which makes the perturbative analysis of the linear response fail . induced by a non - magnetic impurity at site @xmath70 , for an isotropic ladder ( @xmath71 ) with @xmath72 . the incommensurate regime displays oscillations at wave - vector @xmath66 . fit is done using . ] on fig . [ fig : aimantation_impurete ] , we compare the magnetization profile in the sector @xmath73 obtained by dmrg to the mean - field predictions . in practice , expressions and provide good estimates of the behaviors , but it is preferable to fit the magnetization profiles using the following ansatz : @xmath74 in the commensurate regime , and @xmath75 in the incommensurate one . remarkably , except near the onset of incommensurability where the amplitude diverges , these expressions remain correct at small distances , down to @xmath76 . in nmr experiments , an incommensurate @xmath66 would give a narrowing of the peak w.r.t . the commensurate case with the same @xmath77 since the magnetization will display smaller values even close to the impurity . and wave - vector @xmath66 extracted from the magnetization profile induced by a single impurity . dmrg results are compared to bomf predictions in the large @xmath51 regime . ] these profiles give a simple way to numerically access the fit parameters and compare them to bomf predictions . indeed , the values of @xmath42 and @xmath66 extracted from the magnetization profiles agree qualitatively well with the mean - field predictions , as shown on fig . [ fig : xi_q ] . in particular , we checked that the amplitude @xmath78 possesses a maximum close to the transition from commensurate to incommensurate . physically , these calculations provide an explicit illustration of the fact that the impurity generates a spinon that is confined close to it through an effective attractive potential acting over a typical length - scale @xmath42 . induced by a non - magnetic impurity at site @xmath70 , for an isotropic ladder ( @xmath71 ) for @xmath79 in the commensurate regime . the decay of the magnetization profile and effective interaction between two impurities are compared to the decay of spin correlations . these three quantities display the same length - scale @xmath42 in the exponential . yet , only correlations display a power - law correction ( see app . [ app : bomf ] ) . ] last , we stress that there is a qualitative difference between the magnetization profile and the spin correlation function ( see app . [ app : bomf ] for discussion of spin correlations in the model ) . one does not expect a power - law correction in the decay of the magnetization . this is clearly visible on fig [ fig : correction ] where fitting the envelope using @xmath80 gives an exponent @xmath81 while the exponent found for the fit of the correlations is rather @xmath82 , as expected from the usual arguments recalled in app . [ app : bomf ] . these quantitative results on the magnetization profiles and their sensitivity to the commensurate - incommensurate transition share similarities with those on the effective interaction between impurities which we now discuss . within the bomf approximation , valid in the strong - coupling limit , the effective interaction between impurities of eq . takes the following form in the thermodynamical limit : @xmath83 where @xmath84 is the polynom @xmath85 as for the magnetization profile , one can evaluate the integral using the residue theorem to obtain two cases : in the commensurate regime , one has @xmath86}{\sinh\left(1/\xi_\text{spin}^+\right ) p'\left[-\cosh\left(1/\xi_\text{spin}^+\right)\right]}e^{-x/\xi_\text{spin}^+ } + \frac{q^2\left[-\cosh\left(1/\xi_\text{spin}^-\right)\right]}{\sinh\left(1/\xi_\text{spin}^-\right ) p'\left[-\cosh\left(1/\xi_\text{spin}^-\right)\right ] } e^{-x/\xi_\text{spin}^- } \bigg ) \;,\end{gathered}\ ] ] while in the incommensurate regime , one has @xmath87}{\sin\left(q+i\xi_\text{spin}^{-1}\right)p'\left[\cos\left(q+i\xi_\text{spin}^{-1}\right)\right]}e^{iqx } \bigg ] \;.\ ] ] similarly to the magnetization profile , the amplitude of @xmath88 diverges close to the transition between the two regimes but is strictly zero at the transition point . this divergence does not mean that the effective interaction gets stronger but rather that the applicability of the long - distance result is limited to large distances . the effective interaction remains always bonded by the maximum of @xmath89 and @xmath90 ( see short distances behavior hereafter ) . further , no power - law corrections are expected in this quantity , contrarily to what is commonly proposed @xcite . yet , this result is valid in the strong - coupling limit and we observe that it remains correct down to the isotropic ladder regime . in the weak - coupling limit , it is possible to have power - law or logarithmic corrections but we have not studied this case quantitatively . between to impurities as a function of their relative distance for @xmath91 and @xmath92 . dmrg results ( symbols ) are fitted using the expression . ] numerically , we compute the effective interaction using dmrg by targeting the lowest energies in the singlet and triplet sectors . we assume that the lowest triplet excitation is due to the interaction between the two impurities ( the spin gap is large enough in this system ) so that one can use the relation @xmath93 only the total @xmath94 is fixed in dmrg calculations . thus , since the triplet sector has a contribution for @xmath95 , one accesses to the amplitude @xmath96 from the energy difference of the first two energies in this sector . in order to get the sign of @xmath97 , one compares the obtained energies in sector @xmath95 with the lowest in sector @xmath98 . on figure [ fig : jeff ] , we fit the curves using the function @xmath99 where @xmath100 and @xmath101 in the commensurate regime . as expected , the wave - vector @xmath66 and length - scale @xmath42 exactly correspond to the ones of the magnetization profile and correlations function . the behavior of the amplitude @xmath102 is not quantitatively predicted by the bomf theory as , for instance , the behavior of @xmath42 is not in perfect agreement , due to the approximations made in the bomf . still , as one sees on figure [ fig : j0 ] , that the amplitude @xmath102 displays a sharp increase in the vicinity of the commensurate - incommensurate transition , reminiscent from the divergence expected in eqs . - . of the effective interaction ( a ) evolution without frustration ( @xmath79 ) for increasing coupling @xmath103 . comparison with other energy scales is given : the mean antiferro magnetic couplings @xmath104 and the maximum value of the effective interaction couplings @xmath105 . ( b ) evolution in the strong - coupling regime @xmath91 for increasing frustration @xmath106 , showing the divergence at the transition to the incommensurate regime . ] the short distances effective interactions computed with dmrg , and displayed on fig . [ fig : jeff ] , do not follow the prescription . in fact , although linear response theory fails at these distances , one can guess the sign and magnitude of @xmath107 by looking at each configuration ( see fig [ fig : interaction_courte_distance ] ) . the first thing to notice is that a configuration with impurities on the same rung breaks the ladder and the elementary excitation is then a magnon in the largest piece which energy cost is slightly larger than the spin gap . such effect does not enter in the effective model since no spin-@xmath0 is located at the vicinity of an impurity in that case . without frustration ( @xmath79 ) , the effective interaction oscillates at @xmath108 , even at short distances , except when @xmath109 for which the effective coupling is almost zero within dmrg accuracy . indeed , we observe on figure [ fig : interaction_courte_distance](a ) that this configuration breaks the ladder . two spinons are then generated on disconnected fragments and behave independently , making the triplet and singlet states degenerate and the effective interaction equal to zero . the largest value of the effective interaction , which we write @xmath105 in the following , is obtained when two impurities are neighbour on the same chain . the magnitude is then controlled by @xmath89 and shown on fig . [ fig : j0](a ) . in the presence of frustration , the configuration with @xmath109 no longer breaks the ladder . figure [ fig : interaction_courte_distance](b ) shows that spinons freed by the impurities should anti - align due to @xmath90 , as for the configuration with @xmath110 sketched on [ fig : interaction_courte_distance](c ) . in both cases , the corresponding effective interaction is expected to be antiferromagnetic ( positive ) , in agreement with the dmrg result of figure [ fig : jeff ] . if @xmath90 is larger than @xmath89 , it typically sets the scale of the maximum coupling @xmath105 in the effective interaction . with @xmath79 , ( b ) @xmath111 with @xmath112 and ( c ) @xmath110 with @xmath112 . ] last , one can recall that two - body interactions is just an approximation and that terms involving more than two partners should be included to improve the comparison with ab - initio calculations involving many impurities . the validity of two - body interaction has been discussed in ref . to which we refer to for further details on this question . we here discuss the nature of the couplings distribution @xmath113 resulting from doping the ladder and which is a central quantity for the understanding of the magnetic responses . we use the following notation from now on : @xmath24 is the total number of sites , @xmath114 the length of the ladder , @xmath115 the number of impurities and @xmath116 the impurities concentration or doping . the latter corresponds to the probability for a site to be occupied by an impurity . the lattice spacing is taken to be one in both directions . the relative distance between two points on the ladder is written as @xmath117 with @xmath118 and @xmath119 . consider impurities that are randomly distributed on the ladder . the probability @xmath120 of having a distance @xmath121 between two impurities is given by a geometric law @xmath122 to understand this formula , one can scan all intermediate sites between the impurities following a zig - zag path . thus , within the ladder geometry which has the peculiarity to differ from a chain because of the possibility to put two impurities on the same rung ( case with @xmath123 and @xmath124 ) , we have the following results for the mean longitudinal and transverse distances : @xmath125 one recovers the intuitive behaviors @xmath126 and @xmath127 in the dilute limit @xmath128 . thus , in this limit , the typical average distance @xmath10 between impurities , as if they were on a chain , is given by the effective doping @xmath129 as @xmath130 and one has to keep in mind the presence of this factor two in qualitative reasoning . to obtain the distribution of couplings , we use for analytical calculations the simplified and generic relation @xmath131 with @xmath66 a dimensionless wave - vector which accounts for a possible incommensurability and @xmath132 the spin correlation length ( in a shortened notation ) , @xmath133 a phase - shift and @xmath102 an energy scale . their typical behavior with microscopic parameters was discussed in the previous subsections . formally , one obtains the distribution of couplings using the definition @xmath134 and use for the discrete case @xmath135 as we will see , the magnetic curve will be deeply connected to the coupling repartition function that we denote by @xmath136 we also introduce the repartition function of antiferromagnetic couplings only : @xmath137 indeed , negative @xmath17s corresponding to ferromagnetic couplings will yield polarized impurities as soon as the field is turn on . a correct way of defining an energy scale corresponding to a magnetic field in the problem is thus to average only the positive @xmath17s . then , we take the following definition @xmath138 for the typical energy scale of the antiferromagnetic couplings . in this case , the interaction is purely antiferromagnetic corresponding to @xmath100 . changing variables is done using @xmath139 with @xmath140 . [ [ continuous - distribution ] ] continuous distribution + + + + + + + + + + + + + + + + + + + + + + + + + we first consider the most elementary situation where @xmath141 is approximated by a continuous function , which requires @xmath128 and @xmath142 with fixed @xmath143 . then , what enters in the distance probability is the effective chain doping @xmath144 , giving the exponential law @xmath145 . the calculation yields a symmetric power - law distribution @xmath146 for @xmath147 $ ] , featuring the exponent @xmath148 . the corresponding repartition function reads @xmath149 \;. \label{eq : rep - com}\ ] ] the energy scale @xmath150 takes the simple form @xmath151 a similar expression has been used to interpret experiments with 3d effects @xcite . here , the effective volume of the interaction boils down to @xmath152 . [ [ exact - distribution - and - lattice - effects ] ] exact distribution and lattice effects + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + the continuous distribution ansatz is not justified for systems with very short correlation length such as the isotropic ladder . we here carry out the calculation in the discrete case to obtain the exact formula that can be compared to numerical histograms of the couplings used in numerical simulations . we have @xmath153 and @xmath154 for @xmath147 $ ] . one recovers the result of the continuous approximation under its assumption . in particular , one can see that lattice effects decouple the effect of the correlation length and of the impurity concentration , i.e. the exponent is not a function of @xmath143 only , but a function of both @xmath19 and @xmath132 , which makes the results not so universal . last , we notice that the distribution becomes flat for the particular value @xmath155 and we will see that this can have consequences on the shape of the magnetic curve . for a realistic value of the correlation length @xmath156 . the continuous approximation result is compared to the exact discrete result , showing lattice effects discussed in the text . ] for the typical energy scale @xmath104 , the exact result is computed by directly summing upon the @xmath157 and one obtains @xmath158 here again , one can check that is recovered provided @xmath142 and @xmath128 while keeping @xmath159 finite . otherwise , deviations from occur at all doping . in particular , the low doping regime @xmath128 at fixed @xmath132 is @xmath160 these lattice effects are illustrated in fig . [ fig : meanj ] for a realistic case with @xmath161 , which is characteristic of the isotropic ladder limit . in this case the @xmath162 function is not bijective which changes qualitatively the distribution of the @xmath17s . the presence of the cosine significantly lowers the weight of the largest @xmath17s while the smallest @xmath17s will see their weight increase . second , in the presence of fractional @xmath163 , commensurate effects happen while an irrational @xmath163 has qualitatively the behavior of a true quasi - periodic signal . in particular , for rational @xmath163 and @xmath101 , a fraction of couplings can be zero . yet this situation is unphysical for the model under consideration which generic case is non - zero phase shift and irrational @xmath163 . in order to illustrate the typical behavior of the repartition function in the commensurate and incommensurate regimes , we have sampled numerically the distribution of couplings . results are gathered in fig . [ fig : repartition ] . the essential features are the following : ( i ) up to discrete effect , the exponent @xmath164 in eq . captures well the power - law in the commensurate case ; ( ii ) for irrational @xmath163 , the repartition function is qualitatively very close to the commensurate case , with a similar exponent , and up to the weight redistribution towards lower @xmath17 which translates into a smaller energy scale @xmath150 . ( iii ) for rational @xmath163 and @xmath101 , plateaus appear at @xmath165 and @xmath166 in the figure , corresponding the many zero couplings , together with cusps in @xmath150 . yet , the latter situation being unphysical , the main conclusion is that incommensurability hardly affects the coupling distribution . the fact that frustration , throughout incommensurability , only lowers the energy scale @xmath150 but hardly affects the distribution is essential to understand that frustration will have only minor effects on the local magnetic responses studied in the next sections . for @xmath167 and @xmath156 with @xmath101 . energy scale @xmath150 vs the incommensurate wave - vector @xmath66 . ] in this section , we improve on the work of sigrist and furusaki@xcite for the calculation of the averaged total spin and zero temperature curie constant of a doped system on a bipartite lattice . the results that are obtained are more general than for the special case of a ladder , and can be useful in checking numerical simulations and understanding finite - size corrections . we assume a finite size sample containing @xmath24 sites and doped with @xmath168 impurities , where @xmath19 is the impurity concentration which is fixed . the impurities are assumed not to break the lattice into disconnected sublattices ( see discussion section [ sec : chainbreak ] ) . on a bipartite lattice , with two sublattices @xmath169 and @xmath170 which have the same number of sites @xmath171 , applying marshall s theorem yields that the total spin @xmath172 of a given impurity configuration reads : @xmath173 where @xmath174 ( resp . @xmath175 ) is the number of impurities on sublattice @xmath169 ( resp . @xmath170 ) . the probability of having a configuration with @xmath174 impurities on sublattice @xmath169 , is qualitatively similar to the result on a ferro - antiferromagnetic ( f - af ) chain @xcite , and given by @xmath176 then , the probability of having a total spin @xmath172 on a sample is @xmath177 where @xmath178 $ ] . this result is exact and can be used to compute numerically the mean total spin and the curie constant . for large @xmath115 and fixed @xmath19 , according to the central limit theorem , @xmath179 converges towards a gaussian . a saddle - point calculation gives the asymptotic behavior : @xmath180 of variance @xmath181 . one then obtains the average spin and the average square - spin in the @xmath182 limit as @xmath183 the total zero - temperature curie constant matches exactly @xmath184 from , we obtain the following asymptotic behavior @xmath185\;. \label{eq : c - exact - ladder}\ ] ] these asymptotic results are compared to numerical calculations using the exact distribution on fig . [ fig : finite - size - c ] . in the thermodynamical limit @xmath186 , we thus find that the curie constant _ per impurity _ is @xmath187 at @xmath188 , while the curie constant _ per spin _ is @xmath189 . notice that one can also compute exactly the average of the squared spin : @xmath190/\binom{n}{n_i}\;,\end{aligned}\ ] ] which is useful to cross - check the statistical convergence of averaging over samples , but we did not manage to compute exactly @xmath191 . , for four different doping @xmath19 , obtained from exact calculations ( points ) and compared to eq . the finite - size correction of the zero - temperature curie constant , compared to eq . [ eq : c - exact - ladder ] . the @xmath192 line shows the usually admitted result . the incommensurate case corresponds to an isotropic ladder with @xmath193 . ] these results remain correct in the commensurate regime because the effective model is still unfrustrated . but in the incommensurate regime , expressions are not valid anymore . the total spin and curie constant at zero - temperature can be computed from exact diagonalization on the effective model using the exact couplings extracted from dmrg data . the results for a frustrated isotropic ladder in the incommensurate regime are shown on fig . [ fig : finite - size - c ] . the frustration induced by the incommensurability yields a appreciable reduction of both the total spin and the curie constant . note that this reduction is essentially due to the short distance behavior of the effective interaction , hence the necessity to use the exact effective couplings computed in dmrg rather than the asymptotic law . the doping dependence of the prefactors and finite - size corrections were missed in previous work . they originate from the dilution of the lattice . they actually play a crucial role in the quantitative understanding of the numerics that usually work with a restricted number of impurities . last , the precise value at finite - size is essential in extracting the low - temperature exponent , as we will see in the next section . results in eqs . and are exact assuming that there is a fully connected cluster containing @xmath115 impurities over @xmath194 sites . of course , the network can be disconnected in many sub - clusters by the presence of impurities . then , each cluster , which is a finite size system , will eventually contribute to @xmath195 and @xmath78 in the thermodynamical limit , which was pointed out by sigrist and furusaki who computed an evaluation of the correction in the ladder case @xcite . thus , we expect that , in general , @xmath196 in the thermodynamical and that @xmath78 is not exactly given by . lastly , it has been proposed in ref . that these breaks provide an natural cut in the length scale ( and an energy scale ) which should affect the behavior of the correlations . this was confirmed numerically in ref . in which results on depleted two - leg ladders are consistent with an upper bound of the order of @xmath197 reached for very low temperatures . the probability and size distribution of these clusters are governed by percolation theory . the percolation transition distinguishes two main regimes : ( i ) there exist an infinite cluster of sites below a certain critical doping @xmath198 , ( ii ) for @xmath199 , only finite - size clusters exist and their size distribution is typically exponential , associated with a mean cluster size that will be denoted by @xmath200 in the following . at the critical doping @xmath201 , there is still an infinite cluster and scaling is expected for the mean cluster size . the value of @xmath198 is very sensitive to dimensionality and connectivity of the lattice . these percolation regimes induce important finite size effects and are essential for experiments and numerical simulations . on a chain , it is clear that @xmath202 , i.e. any finite doping will break the lattice and the mean cluster size is easily related to the doping @xmath203 . on a ladder , the situation is similar in the sense that any finite doping breaks the lattice into sub - clusters . this chain breaking effects have been shown to have quantitative results on the magnetic response of doped chains @xcite . yet , computing @xmath200 in the case of ladder is not trivial and the remaining clusters are themselves doped with various concentration of impurities which makes the predictions more involved . we give below an exact discussion of the cluster sizes in the ladder and apply the results to the chain breaking effects on ladders . we consider a ladder with nearest - neighbour only . connectivity of the network is broken if ( i ) two impurities fall on the same rung , or ( ii ) two impurities fall on diagonal positions on a plaquette ( see figure [ fig : interaction_courte_distance](a ) ) . if @xmath204 is the impurity position , there are three positions at which a second impurity can break the ladder : @xmath205 , @xmath206 et @xmath207 . occupying a site with an impurity has a probability @xmath19 . in the diluted limit @xmath128 , the density of cuts is then @xmath208 and the corresponding mean cluster size is given by @xmath209 ( the factor 3 was missing in ref . ) . notice that in the presence of frustration , chain - breaking requires at least four neighboring impurities which would make a different scaling @xmath210 . for large enough distances , breaks are uncorrelated so a fair description of the distribution law is that of a poissonian process @xmath211 with @xmath212 in the diluted and continuum limit . an exact calculation of the cluster sizes distribution is carried out in appendix [ app : breaks ] and supports this phenomenological approach . the exact distribution reaches very quickly an asymptotic behaviour given by a geometric law @xmath213 of parameter @xmath214 consequently , one recovers in the continuum limit and @xmath215 . in particular , this provides finite-@xmath19 corrections to the @xmath216 scaling which turn out to be quantitative even for a few percent doping as we see now . averaging the total spin and curie constant over clusters is not trivial since the doping of each cluster can now be distributed between zero and approximately @xmath0 . to handle a correct estimate , one would have to average using the joint distribution of cluster sizes and doping . we give below a rough estimate that consists in assuming a fix doping @xmath19 for all cluster and averaging only over cluster sizes @xmath217 using @xmath218 . neglecting the doping fluctuations should yield a good approximation in the diluted limit where cluster sizes diverge . averaging the equations is performed using @xmath219 using the approximate law for the size distribution and a continuous approximation for the average spin , we get @xmath220 to obtain the density of spin and curie constant , one has to multiply them by the clusters density @xmath221 . the mean spin density @xmath222 and curie constant density @xmath223 now read @xmath224 and @xmath225 which gives in the diluted limit @xmath226 @xmath227 in agreement with the results of sigrist and furusaki @xcite up to a prefactor . in the opposite limit of high - density for impurities @xmath228 , the system is equivalent to a few independent spin-@xmath0s which essentially behave as in a paramagnetic phase . therefore , we have that @xmath229 and @xmath230 . in particular , we infer that there is an optimal doping @xmath231 which maximizes the curie constant and a slightly different one which maximizes the total spin . the value of @xmath231 is non - trivial since it occurs at the crossing of the two asymptotes . and ( b ) curie constant density @xmath232 averaged . ] these predictions are a lower bound for @xmath233 and @xmath232 since very small clusters with a few sites have a total spin and susceptibility larger than the random - walk result . in order to show the quantitative role of chain breaking and the validity of the results , we plot on fig . [ fig : chainbreaking ] the limiting behaviors at low and high doping @xmath19 together with the formulas and . although these formula are not exact , they capture well the existence of a maximum at an optimal doping . we now turn to the effect of interactions on the magnetic curve of two - leg ladders doped with impurities . the magnetic excitation is denoted by @xmath234 . we choose to define the total magnetization density as @xmath235 so that the high field saturation density is @xmath236 . with this normalization , the contribution of non - interacting impurities carrying a spin-@xmath0 to the magnetization is the brillouin formula at temperature @xmath237 ( we set @xmath238 in the following ) @xmath239 \label{eq : brillouin}\ ] ] where @xmath19 thus corresponds to the _ impurity _ saturation magnetization , assuming that each impurity frees a spin-@xmath0 . the latter assumption is actually affected by chain breaking and will be discussed in more details in sec . [ sec : magnetization ] . in the low - field limit at finite - temperature , the curie constant density @xmath232 is defined as @xmath240 in the case of independent impurities , we would have the total curie constant @xmath241 corresponding to @xmath242 , as expected from . if one uses the exact result for correlated impurities , then the curie constant density reads @xmath243 without chain breaking or with it . we first consider a simple model , dubbed `` random dimer model '' in which impurities are assumed to build dimers with their neighbour . dimers are independent but have couplings randomly distributed according to @xmath113 . the magnetization of a single dimer of coupling @xmath17 is given by @xmath244}{1 + e^{j / t } + 2 \cosh[h / t]}\;.\ ] ] the total number of dimers is @xmath245 so that the total magnetization density averaged over the coupling distribution reads : @xmath246}{1 + e^{j / t } + 2 \cosh[h / t ] } \;. \label{eq : rdm - magnetization}\ ] ] the brillouin formula is recovered when @xmath247 or more physically , in the high - temperature limit when @xmath248 . taking the zero - field limit at finite temperature in yields a temperature - dependent curie constant @xmath21 : @xmath249 which reaches the free spins result @xmath250 in the high - temperature regime . the effective hamiltonian of interacting impurity spins is given by @xmath251 and is solved numerically using either ed for @xmath252 or qmc up to @xmath253 and provided there is no incommensurability , i.e. for @xmath100 . ed provides all energies @xmath254 in a sector of total spin @xmath255 so finite - temperature predictions are accessible . the couplings @xmath256 are obtained by sampling impurities configurations on a ladder and using either the approximate formula with chosen @xmath66 , @xmath102 and @xmath132 , or the exact couplings computed from dmrg . two `` ab - initio '' methods are also used to compute observables directly on the original microscopic hamiltonian : the dmrg technique , which gives accurate results for the zero - temperature magnetization curve , and quantum monte - carlo sse calculations well suited for finite - temperature dependence . in this section , we focus on the limit of vanishing magnetic excitation @xmath257 at finite @xmath237 . the order of limits matters and the situation @xmath258 for fixed @xmath234 will be studied in the next section . in this limit , a modified curie law is generically expected , written as @xmath259 where @xmath21 is a temperature - dependent curie constant , corresponding to a static susceptibility @xmath260 . the goal of this section is to investigate quantitatively the whole @xmath21 curve and analyze the effect of interactions , doping and frustration on its behavior . the various regimes of @xmath21 in depleted ladders were first sketched by sigrist and furusaki @xcite who gave the following picture : starting at high - temperatures , the spins are essentially independent because of thermal fluctuations so that @xmath261 . assuming that the spin gap @xmath3 is larger enough than the maximum coupling @xmath105 ( implicitly corresponding to the strong - coupling regime ) , lowering the temperature below the spin gap freezes all magnon excitations . only remain spin-@xmath0s freed by impurities which should behave independently for a range of temperatures @xmath262 . this yields a plateau around @xmath263 if one neglects chain breaking and @xmath264 if they are taken into account to first order corrections . lowering again temperature enables one to reach the zero - temperature plateau discussed above and which is approximately given by @xmath265 ( the @xmath189 plateau within ref . conventions ) . in the regime governed by impurity - spins interactions , real - space renormalization group ( rsrg ) arguments@xcite generally gives low - temperature corrections of the form @xmath266 with @xmath267 a non - universal constant and @xmath268 an exponent which generally depends on doping @xmath19 and that captures the interesting physics about impurities interactions . we now check and analyze this scenario using our various models and methods . the random dimer model formula for the curie constant already displays a non - trivial temperature dependence due to the coupling distribution . in the case of the power - law distribution for which the exponent @xmath269 is the one of the couplings distribution and @xmath270 ( see below eq . ) , the high - temperature expansion leads to @xmath271\end{aligned}\ ] ] while , at low-@xmath237 , a sommerfeld - like expansion , in which the constant three appearing the denominator of has to be carefully taken into account , yields a power - law : @xmath272 with the constant @xmath273 @xmath274 is of the order of @xmath275-@xmath276 and matches some simple numbers for specific values of @xmath268 : @xmath277 , @xmath278 . the curves for various @xmath268 are represented in fig . [ fig : dimermodel ] and show that the prediction works for a wide range of temperatures . in the limit of small @xmath268 , one has the expansion @xmath279 with @xmath280 . . _ inset _ : behavior of the constant @xmath281 from . ] it is clear that , even though there is a power - law , there is nothing universal in this result . the scaling originates only from the fact that the distribution is a power - law . last , the zero - temperature result @xmath282 , which already differs from free spins , is always expected in the case of a symmetric @xmath283 distribution . in ab - initio ed and qmc calculations , the temperature - dependent curie constant is computed exactly using the average over thermal states and disorder configurations : @xmath284 since @xmath285 when @xmath286 for both the microscopic and effective models due to su(2 ) symmetry . notice that , on the effective model , chain breaking effects discussed in sec . [ sec : chainbreak ] are not included . within the effective model description for @xmath287 impurities . ] we present on fig . [ fig : ed - coft ] the results obtained from exact diagonalization with @xmath287 impurities and averaged over 10000 samples . the first remarkable result is that the zero - temperature plateau is very well approximated on finite sizes by using or its exact numerical evaluation . in fact , the effective model is not a bipartite lattice model to which the theorem applies , but the fact that it originates from a bipartite model to which the theorem applies ( without frustration ) seems to make it hold even in the effective model . the reason for that is certainly that the sign of the couplings satisfy the bipartite nature of the original lattice . the low - temperature departure from the @xmath188 plateau is very sensitive to finite - size effects and disorder averaging . this makes it hard to capture the hypothetical thermodynamical behavior with this data . yet , for the intermediate temperatures regime up to the high - temperature saturation plateau , we obtain a very good fit of the @xmath288 data using a power - law , as one can see from fig . [ fig : ed - coft](b ) . collecting the fitted exponents fig . [ fig : ed - coft](c ) shows a very good agreement with the @xmath164 prediction of the random dimer model . however , the behavior in the thermodynamical limit within the effective model is difficult to address . to sketch a possible scenario , we refer to the works done on the f - af random chain @xcite . indeed , for reasonably short correlation lengths @xmath132 , the effective model should fall into the f - af universality class in the rsrg sense . this universality class has been dubbed as the large - spin phase , which is of the griffith s type , and for which it has been found that the total spin follows the random - walk scenario discussed above , and that a power - law correction to the zero - temperature curie constant is expected . as regards the possible universal exponents of this phase , it was found numerically @xcite that it is strongly dependent on the singular nature of the initial coupling distribution . by denoting @xmath289 the initial distribution of the couplings , the following scenario is proposed : ( i ) when @xmath290 with @xmath291 , the rsrg flows towards a non - universal fixed point with non - universal value of @xmath268 . this exponent should depend on @xmath19 and @xmath132 but is not necessarily equal to @xmath164 ; ( ii ) when @xmath292 ( initial distribution `` not too singular '' ) , the rsrg flows towards a universal fixed point with @xmath293 . qmc calculations @xcite have demonstrated the following typical behavior for the curie constant on the f - af random chain : at high - temperatures below the saturation plateau , @xmath21 strongly depends on the initial distribution coupling . yet , at low enough temperatures , the various @xmath21 curves collapse very close to the rsrg prediction with @xmath294 where @xmath295 and @xmath296 . coming back to the situation of doped ladders , we may propose the following scenario . provided the rsrg picture is applicable to the ladder , something certainly true for @xmath297 but hard to justify when @xmath298 , we first expect from refs . that the high - temperature regime is always dependent on the distribution . interestingly , in the doped ladder situation and within the random dimer picture , we found that this regime displays a power - law behavior with an exponent @xmath164 which is simply related to the coupling distribution exponent . then , one expects that the rsrg picture develops at low - temperatures with two possibles cases . the @xmath299 criteria translates on ladder to a critical doping @xmath300 such that ( i ) if @xmath301 , the low - temperature exponent is non - universal , dependent on @xmath19 and @xmath132 and could differ from the high - temperature exponent expected to be @xmath164 ; ( ii ) if @xmath302 , one can fall into the rsrg universality class and the low - temperature exponent should become independent of @xmath19 and @xmath132 and reaches @xmath303 . one must notice that the second situation can actually be realistic for doped ladders in the weakly coupled regime , since , for instance , @xmath304 for @xmath305 , giving @xmath306 . impurities and two correlation length @xmath161 ( a ) and @xmath307 ( b ) . disorder averaging has been done over a few thousands of random configurations . the curie constant per impurity ( from which the asymptotic value @xmath308 has been subtracted ) is shown _ vs. _ temperature for various concentrations ( symbols ) , together with power - law fits ( dashed lines ) of the form @xmath309 where @xmath310 is a varying exponent indicated on the plot . the lines in the high - temperature region are with exponent @xmath311 . the @xmath312 line is the universal regime found in the f - af chain by frischmuth _ et al . _ @xcite , including the same prefactor @xmath313 . ] in order to test this scenario which relies on several questionable assumptions , although it looks plausible within the usual pictures discussed in random one - dimensional magnets , we have carried out qmc simulations on the effective model up to @xmath253 impurities . the results for the curie constant are plotted in fig . [ fig : qmc - effective - model](a - b ) for two values of the correlation length @xmath161 and @xmath307 which respectively correspond to @xmath314 and @xmath315 . we observe on these data a crossover from a fast decaying high - temperature regime , roughly controlled by the exponent @xmath164 and a smaller doping - dependent exponent at lower temperatures . for large values of @xmath143 , the deviation is even clearer and the exponent does not seem to exceed the rsrg universal result of @xmath316 . we also show the f - af universal result on the same plot showing that data at large @xmath317 qualitatively saturates on this limit . these results give good confidence that the above scenario is plausible . to further test the scenario , we have extracted the low - temperature exponent and plot it against @xmath19 and @xmath317 on fig . [ fig : qmc - effective - model-2 ] . we observe that the low - doping regime is consistent with the @xmath311 limit while intermediate dopings display significant deviations and a tendency to saturate around the rsrg universal regime for @xmath318 . yet , the validity of the random f - af rsrg picture can be questioned for two main reasons : when @xmath143 becomes large , the dilute short range interaction limit fails and it is not guaranteed that the rsrg is still under control with longer range interaction . second , as we will see on the magnetic curve , the discretized nature of the distribution can play a quantitative role . the criteria for the initial distribution exponent @xmath319 is valid within a continuous description but the discretized nature of the coupling can make the distribution more singular . interpretation in that sense was proposed on the same model through the study on correlation lengths@xcite . still , we observe that the rsrg argument does capture a lowering of the @xmath320 curve w.r.t . the @xmath311 naive expectation . having an accurate quantitative description of this curve yet remains a challenging question . frustration makes the lattice non - bipartite so that the exact results do not apply . still , within the effective model , if frustration is not too strong , the system remains commensurate and the above results remain valid and shows that the behavior is the same . when frustration is large enough to induce incommensurability in the system , the effective model is affected and next - nearest neighbor couplings can become frustrating . in this situation , qmc calculations are not possible due to the sign problem and we carry out ed calculations , limited to a @xmath287 impurities . we do not show the data because the picture remains essentially and quantitatively the same as for the commensurate regime . this absence of strong qualitative differences is certainly due to the fact that the spin correlation length is small and prevents frustrating effects to develop on large scales . furthermore , rsrg arguments tell that the commensurate and incommensurate cases should fall into the same f - af random chain picture so that incommensurability does not actually plays a fundamental role in this model , at least on its one - dimensional version . we now discuss the overall behavior of @xmath21 computed on the original microscopic model doped with impurities in order to test the scenario by sigrist and furusaki discussed above . we take the situation of an isotropic ladder which has @xmath321 but in which the energy scales @xmath102 , @xmath105 and @xmath89 are very close to each other ( see fig . [ fig : j0 ] ) . the first significant effect as seen on fig . [ fig : qmc - coft - full](a ) is thus the absence of an intermediate plateau of independent impurity spins . thermal magnons are activated before impurity spins become uncorrelated by thermal excitations so that the contributions of both can never be separated . this absence of plateau in the isotropic ladder will have its counterpart in the magnetic curve while scanning the energy scales with the magnetic field rather than with the thermal energy ( see sec . [ sec : magnetization ] ) . in the large @xmath322 limit , the separation of energy scales suggests that the plateau could be visible but we have not checked it numerically . at temperatures slightly below the temperature corresponding to the spin gap @xmath323 , a power - law behavior is clearly visible showing the regime in which the effective model accounts for the physics . the exponent is found to depend on doping , with very small exponents at low dopings which could give the impression of the presence of a plateau , although this is not correct . a systematic extraction of the exponent ( see fig . [ fig : qmc - coft - full](b ) ) gives the results plotted on fig . [ fig : qmc - effective - model-2 ] against the effective model results . slightly larger exponents are found but the agreement can be viewed as correct considering the low values of the exponents and the difficulty to tackle this low - temperature regime numerically . consequently , the effective model seems to capture the physics at low - energy of the interacting impurity spins . although the convergence towards the universal rsrg regime is plausible at small @xmath322 and low doping from our results on the effective model , the numerical challenge it represents on the microscopic model is beyond the scope of this paper ( the spin gap becomes significantly smaller ) . another way to probe the effective interactions between impurities is to scan the energies using a magnetic field rather than temperature . as we are studying the part of the magnetic curve which typically lies below the spin gap @xmath3 , the results correspond to accessible magnetic fields , and are then particularly relevant to experimental measurements . we aim at proposing some possible relevant fits of this regime . above the spin gap , the elementary excitations involve magnons which can localize in the disordered environment . there , the physics becomes quite different and we do not address these questions related to bose - glass physics . we now turn to the generic behavior of the magnetic curve @xmath324 . in the previous section , the non - trivial behavior when @xmath257 at finite @xmath237 was discussed . physically , it corresponded to susceptibility measurements performed with @xmath325 . strictly speaking , we must have @xmath326 when @xmath286 due to the su(2 ) symmetry . if one now considers a finite - system with @xmath188 and a small but finite magnetic field , the degeneracy within a sector of total spin @xmath172 will be lifted to favor the state @xmath327 . then , there exists a disorder averaged magnetization jump @xmath328 which matches @xmath329 . this is typical of a partially ferromagnetic state . if one does not take into account chain breaking effects , as we do for the effective model , the scaling of @xmath191 yields a magnetization jump that vanishes in the thermodynamical limit as @xmath330 interestingly , we notice that , due to the random walk argument , the prefactor is actually related to the zero - temperature curie - constant @xmath232 by @xmath331 which makes a connection between the two non - commutating limits of the magnetic responses under study . if we take chain breaking effects into account , then there exists a jump _ even in the thermodynamical limit _ which reads @xmath332 in the diluted limit @xmath333 . lastly , one expects that , within a picture of impurities bringing each exactly one spin , the saturation plateau corresponding to the polarization of all these spins equals @xmath334 . yet , chain breaking effects should lower this value since configurations where two impurities are on the same rung do not bring any free spin . taking this effect into account gives an expected saturation at @xmath335 . this effect matters for dmrg or qmc data as well as experiments . using the random dimer model , the low - temperature magnetization curve ( for @xmath336 ) takes a fermi - dirac form to a good approximation @xmath337 where @xmath234 plays the role of the chemical potential . this is physically transparent as the system is equivalent in this limit to a collection of two - level systems with only the singlet and triplet @xmath338 states contributing to the low - energy physics which naturally maps onto fermionic statistics . in particular , the @xmath188 limit of this model gives that the magnetic curve is simply related to the repartition function @xmath339 of the couplings through @xmath340 . in the case of the continuous distribution , this yields a power - law behavior @xmath341 with @xmath269 and for @xmath342 , which already deviates significantly from the brillouin picture . it is important to notice that situation where @xmath343 from is physical in the case of systems with a large correlation length @xmath344 . then , the curvature of the magnetic curve is expected to change from concave to convex . and finite temperature . @xmath345 represents the magnetization jump . ] still , we see that the random dimer model fails to reproduce the correct @xmath257 limit and gives for the jump @xmath346 ( @xmath347 for eq . ) . one can incorporate the exact result in the rdm by stretching the repartition function of antiferromagnetic couplings @xmath348 . we thus define the phenomenological stretched random dimer ansatz as @xmath349 physically , the issue of the random dimer model is that it works with total spins @xmath350 and @xmath351 and can not capture the large - spin formation . these features and ansatz of the random dimer model are represented on fig . [ fig : magnetization - rdm](a ) . lastly , this rough understanding of the shape of the curve leads to the following simple power - law fit which could be useful for experiments or numerical calculations : @xmath352 in which one can leave free the three parameters @xmath345 , @xmath102 and @xmath268 . interestingly , rsrg arguments @xcite have also proposed a power - law behavior for the magnetic curve when @xmath353 based on energy scales phenomenology . we compute numerically the magnetization curves at zero temperature using dmrg on the microscopic model and average over many configurations . the results for the isotropic ladder @xmath354 are displayed on fig . [ fig : magnetization - t0](a - b ) for both a system without frustration and with frustration in the incommensurate regime . qualitatively , the two curve are essentially governed by the coupling distribution and frustration does not have a drastic qualitative effect . interestingly , the simple approximations described in the preceding section account rather well of the behavior of the curve . first , the ed on the effective model captures the power - law like behavior and even underlines the discrete nature of the coupling distribution . this discrete nature is transparent from the ansatz using the exact effective couplings . the dmrg does show faded steps corresponding to the larger couplings , and ed too . the envelope of the random dimer model is captured by the continuous version of the coupling distribution . yet , we see that one really needs the discretized version to be quantitative . last , we show that a fit of the form captures the mean power - law behavior of the curve in a satisfactory way . this is all the more relevant as we will see that temperature tends to fade the steps due to the discrete couplings . one can notice the slight difference between the ed and dmrg results . we attribute these to two main possible effects . first , as the systems are chosen to have the same total number of impurities , the limitation of the two - body interaction effective model can play a role . many - impurity interactions could become relevant even though these are subdominant effects . second , we have seen that chain breaking effects must make the saturation plateau occur at @xmath355 , but it also has the effect of averaging magnetic curves over various dopings . indeed , in the presence of chain breaking effects , each piece has a different doping which approaches @xmath19 on average but can be lower or higher . this should have significant effects compare to the fixed @xmath19 curve of the ed on the effective model . the last important remark is that no saturation plateau is reached in the isotropic ladder . as for the curie constant plateau , this is due to the fact that the typical energy scales are of the same order of magnitude @xmath356 ( see fig . [ fig : j0 ] ) . then , magnons become activated by the magnetic field before all impurities are truly polarized . the energy scales separation in the strong - coupling limit suggests that such a plateau could be possible at large @xmath322 but we have not investigated this situation in details . in particular , the microscopic model displays small couplings in this limit which are harder to capture with dmrg . the approximation of eq . , naturally yields low - temperature corrections from a sommerfeld expansion , valid provided @xmath357 , which reads @xmath358 , where @xmath359 is the derivative of the coupling distribution . for instance , in the case of the continuous distribution and taking the approximation @xmath360 , the temperature corrections depend on doping and magnetic field through @xmath361 here again , the response to a small temperature is expected to strongly depend on the side of the limiting case @xmath362 , displaying a change in sign on the corrections . in the particular situation where @xmath362 , for which @xmath113 is flat , the magnetization curve of the random dimer model can actually be computed exactly @xmath363}{1 + 2\cosh[h / t ] } \\ & \times \frac{t}{j_0}\ln\left\ { \frac{1+e^{j_0/t}(1 + 2\cosh[h / t])}{1+e^{-j_0/t}(1 + 2\cosh[h / t ] ) } \right\}\;. \end{split } \label{eq : m - exact - z1}\ ] ] following the previous remark on the inability of the random dimer model to account for the large - spin formation , we can devise an extension of the stretched dimer model at finite temperatures using the following ansatz : @xmath364 where the first part accounts for the contribution of ferromagnetic couplings , while the second accounts for the magnetization process of antiferromagnetic dimers . the first term should in principle correspond to a brillouin function of spin @xmath195 but this version already gives satisfactory results . of depleted ladders of size @xmath365 sites , averaged over @xmath366 disordered samples , at finite temperature @xmath367 . different impurity concentrations @xmath368 are shown , together with the clean case at the same temperature for comparison . horizontal dashed lines show the expected saturation value for the impurities @xmath369 , and the full lines are ed results obtained with pbc on 1000 random clusters of 10 impurities from which one clearly sees that the saturation value is only reached when @xmath370 . ] the effect of temperature is first discussed on fig . [ fig : magnetization - finitet ] by showing the comparison between the brillouin response to the ed and random dimer model predictions for four increasing temperatures . all curves should collapse at high temperatures @xmath248 . we see that the zero - temperature steps are rapidly faded as temperature is turn on . still , the deviation from the brillouin curve due to the interaction remains well visible for finite temperature and actually makes the random dimer model almost exact . in order to validate the above comparison , we have compared the ed on the effective model to qmc , which is the appropriate method for finite temperature calculations on the microscopic model . in fig . [ fig : magnetization ] , one observes a rather good agreement for several realistic dopings . the larger the doping , the larger the deviation from the brillouin curve is and the larger the distance from saturation is when the magnons set in . the slight difference between ed and qmc could here again be attributed to many - impurities interactions not taken into account in the effective model and also to the effective doping averaging induced by chain breaking effects , as for the zero - temperature curve . then , the following message is almost quantitatively correct from the comparison between all different approaches : the low - part of the magnetic curve probes the couplings distribution between the impurities . this is evident in the random dimer model and the picture survives to the microscopic model rather well . this simple analysis is certainly due to the fact that we are discussing a simple observable ( density of magnetization ) which is little affected by low - energy behavior or correlations in the system . therefore , it could be accessible and interesting to test such phenomenology in experiments working with quasi one - dimensional systems . our theoretical study could in principle apply to several realistic spin gapped materials . however , as seen above , a clear separation between different energy scales the spin gap @xmath3 below which free local moments are expected , and the largest effective coupling @xmath105 below which they start to correlate upon random f - af exchanges would be difficult to achieve in systems close to the isotropic ladder limit . the separation remains plausible in the strong - coupling limit , although we have not investigate this point quantitatively in this paper . in the isotropic case , a saturation regime of impurity spins will be hardly detectable . nevertheless , the regime of large spin could be detected in curie tails at low temperature , provided the three dimensional ordering of induced moments ( expected below temperatures set by three dimensional couplings ) occurs at low enough temperature . in such a respect , a new analysis of susceptibility data of zn ( @xmath350 ) or ni ( @xmath351 ) doped bicu@xmath1po@xmath2 form ref . may give interesting results , although the three dimensional ordering of induced moments occurs below a few kelvins @xcite . perhaps more promising is the doped haldane chain system y@xmath1banio@xmath7 @xcite where a very small inter - chain coupling @xmath371 k is expected from neutron scattering @xcite despite a very large spin gap @xmath372 k. more generally , our study clearly shows that curie tails , present in all af materials even for undoped ones , because of intrinsic defects or imry - ma domain formation with random couplings @xcite , have to be analyzed perhaps more carefully than what is usually done . in particular , the assumption of free impurities leading to the extraction of their concentration @xmath19 through the simple form @xmath373 is not expected to be valid in many experimental situations . regarding the magnetization curve , our work can potentially apply to many materials where brillouin - like responses are observed upon increasing the external field . for spin - gapped systems , the effective couplings between local moments can strongly renormalize downwards the brillouin - like magnetization , and pushes the saturation towards larger magnetic fields , possibly larger than the spin gap @xmath3 . this means that , at the critical field where magnon excitations start to appear , not all impurity - induced moments have been saturated . such a phenomenology is expected for bicu@xmath1po@xmath2 in a field @xcite . nevertheless , for this ladder material @xcite , and also for other systems such as the herbertsmithite kagom compound @xcite , the presence of non - negligible dzyaloshinskii - moriya ( dm ) anisotropies make the situation much more difficult to analyze since dm terms induce a finite magnetic response also below the spin gap . the modification of the brillouin - like response due to the competition between impurity physics and dm interactions in an external field at finite temperature is a very interesting subject , relevant for many realistic systems , that we leave for future studies . the physics of randomly depleted ladder , studied initially in the seminal work of sigrist and furusaki @xcite , offers a remarkable playground for studying the effect of impurity disorder in gapped systems without and with frustration . in this contribution to the field , we improved on several intuitive results of ref . to provide quantitative predictions and comparison to numerics and adressed the shape of the magnetization curve . based on a detailed analysis of the effective couplings between impurities and of the corresponding coupling distribution , we focussed the main two magnetic responses : the zero - field susceptibility , through the temperature - dependent curie constant , and the magnetization curve . the first one is shown to have a non - trivial power - law behavior at very - low temperature in qualitative agreement with a rsrg scenario . the high - temperature deviation from free impurity spins is well captured by a simple random dimer model . this model also accounts qualitatively well for the magnetization curve for which we give several phenomenological fits at zero and finite - temperature which are in good agreement with accurate numerical calculations . one of the key outcome of this study is that incommensurability ( induced by frustration ) plays little role in the local quantities we looked at . indeed , the main consequence of incommensurability is a mere reduction of the zero - temperature spontaneous magnetization and of the low - temperature limit of the curie constant . the situation might be different in higher dimensional system but the one - dimensional version seems to be in the same universality class , as expected from rsrg arguments . these predictions on the magnetic responses could motivate experiments in that direction since the required temperatures and magnetic fields are accessible for several compounds . al thanks ccile delaporte and antoine channarond for insightful discussions on markov chains calculations . al and gr acknowledge support from the french anr program anr-2011-bs04 - 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order mean - field theory developed for the frustrated ladder in ref . . we will use the same notations as in this reference . the dynamical and static structure factors of the model has also been addressed recently in ref . . within bomf theory in which the singlet operators on rungs are assumed to condense @xmath374 , the mean - field hamiltonian is solved through a bogoliubov transformation on the triplet operators in @xmath375-space @xmath376 @xmath377 in which @xmath378 and @xmath379 satisfy @xmath380 . this leads to the diagonal version of the hamiltonian @xmath381 where @xmath382 is the dispersion relation which depends on @xmath383 and the chemical potential @xmath384 used to enforce the hard - core nature of the triplets on rungs . these two parameters are usually computed self - consistently with numerical methods . in this paper , in order to have tractable analytical formulas , we use the following approximations , which turn out to be good in the strong - coupling limit , @xmath385 and @xmath386 the zeros of @xmath382 extended to the complex plane will control the singularities of most physical quantities . to this end , we introduce the polynomial @xmath387 such that @xmath388 in ref . , we obtained that the spin structure factor defined by @xmath389 where @xmath390 are real - space spin correlations , is given in the bomf approximation by @xmath391 we now give more details on the two commensurable and incommensurable regimes , limited to the strong - coupling regime . we recall that the transition occurs at @xmath392 . * for @xmath393 , @xmath394 has two real roots , lower than @xmath395 . consequently , @xmath396 has branch cuts and four branching points on the axis @xmath397=\pi$ ] , with imaginary parts denoted by @xmath398 that define two correlation lengths @xmath61 ( @xmath399 ) , such that @xmath400 in the strong - coupling limit . * for @xmath401 , @xmath394 factorizes exactly and the square root disappears in the denominator of @xmath396 . there is no longer branch cuts and the branching points merge to give two poles on the axis @xmath397=\pi$ ] , with imaginary part @xmath402 , where @xmath403 * for @xmath404 , the roots of @xmath394 have a non - zero imaginary part . consequently , the branching points leaves the axis @xmath397=\pi$ ] . there coordinates can be written as @xmath405 where @xmath66 is the incommensurate wave - vector associated to the real - space correlations and @xmath42 is the spin correlation length . in the large @xmath51 limit , we obtain @xmath406 real - space behavior of the correlation function is recovered after a fourier transform of the static structure factor : @xmath407 one can not easily compute this integral using the theorem of residues , because of the branch cuts , but one can argue that the behavior in @xmath408 is essentially controlled by @xmath409 et @xmath410 with @xmath411 and @xmath412 the singularities of @xmath396 in the upper half plane . furthermore , due to the presence of the square - root in the denominator , one may guess the following asymptotic behavior @xmath413 indeed , as for the @xmath89-@xmath90 chain @xcite , the @xmath414 correction yields better fits of the numerical results . we thus have the following scenario for the correlation functions : * for @xmath415 , in the commensurate regime : @xmath416 where @xmath61 are given by and @xmath169 and @xmath170 are two constants that depend on @xmath61 . * for the transition point @xmath401 , we remark that the factorization of the denominator makes the decay purely exponential . then , one expects @xmath417 with @xmath42 given by and @xmath78 a constant depending on@xmath42 . * for @xmath404 , in the incommensurate regime : @xmath418 where @xmath66 and @xmath42 are given by and , and @xmath419 and @xmath133 are constant depending on @xmath66 and @xmath42 . in order to compute the magnetic susceptibility at @xmath420 , one applies a magnetic field corresponding to the wave - vector @xmath421 @xmath422 in the bomf approximation , the hamiltonian then reads @xmath423 the energy correction is obtained from second order perturbation theory in @xmath234 as @xmath424 by definition , the susceptibility @xmath425 enters in the expression through linear response theory @xmath426 from which we deduce the following expression for the static susceptibility : @xmath427 where @xmath394 is the polynom defined in . using the result for the susceptibility , on gets for the magnetization profile the prediction @xmath428 where the position @xmath429 are relative to the impurity site . the integral can be computed with help of the residues theorem applied over a rectangle of base between @xmath430 and @xmath431 and infinite in the vertical direction . performing the change of variables requires the calculation of the derivative @xmath432 \\ & = -\frac{j(r)}{\xi}\left [ 1 + q\xi \sqrt{\left(\frac{j_0}{j(r)}\right)^2 e^{-2r/\xi } -1 } \right ] \label{eq : derivative}\end{aligned}\ ] ] the zeros of the derivative of denoted by @xmath433 and satisfy the equation @xmath434 with @xmath435 $ ] . consequently , we have the zeros @xmath436 one can take by convention @xmath437 to define the intervals in which the sign of the derivative is constant @xmath438 $ ] . it is clear that the intervals have the same size @xmath439 . formally , one can write @xmath440 the solutions @xmath441 of the equation @xmath442 are not analytically computable in general . for a given @xmath17 , there is at most one solution in each interval @xmath443 and there is a least one solution for @xmath444 . let us denote by @xmath445 the number of solutions at a given @xmath17 so that the index ranges @xmath446 . using , we have @xmath447 we restrict the discussion to the continuous distribution case since the analytical formula for the discrete version do not help with respect to a direct numerical sampling . in this case , the weighting by the continuous approximation for @xmath141 gives @xmath448 the reduction of probability of large @xmath17 is understood by studying the situation where @xmath449 so that there is only a single solution @xmath450 . then , one can write @xmath451 with @xmath452 since the effect of @xmath66 is to decrease the position of the solution w.r.t . the @xmath100 result . this gives @xmath453 although it is not obvious in the formula , one may convince one - self graphically that @xmath454 corresponding to a decrease of the weight at large @xmath17 . consequently , the weight of small @xmath17s increase since the signal can approach zero at any distance . in this section , we study the distribution of sizes of disconnected ladders @xmath455 for a given impurity doping @xmath19 . if we consider an impurity at position @xmath204 , there are three positions for a second impurity that break the ladder : @xmath205 , @xmath206 and @xmath207 . in the diluted limit @xmath333 , the density of cuts is then @xmath208 and the average length of disconnected ladders @xmath456 . as the cuts are not correlated ( at least at large enough distances ) , it is reasonable to assume that the number of cuts follows a geometric law of parameter @xmath457 : @xmath458 in fact , the distribution can be calculated exactly . for this , the ladder is described by a markov chain @xmath459 , where @xmath460 represents the configuration of the plaquette made of the two consecutive rungs @xmath461 and @xmath462 ( see fig . [ fig : markov ] ) . the markov property is verified : the configuration on plaquette @xmath462 only depends of the configuration on plaquette @xmath461 as they have a rung in common . the @xmath463 configurations on a plaquette are classified as follows : 1 . the plaquette dos not break the ladder and there is no impurity on the second rung , 2 . the plaquette dos not break the ladder and there is one impurity on the second rung , 3 . the plaquette breaks the ladder . the transition matrix @xmath84 , whose elements are the probabilities to go to configuration @xmath16 from configuration @xmath15 , writes @xmath464 where @xmath465 the transition probabilities from configuration @xmath466 are not needed and we can take this configuration as a trap state . we want to calculate the distribution of distances to reach configuration @xmath466 @xmath467 starting from an initial distribution for @xmath468 @xmath469 equation can be expanded as @xmath470 where@xmath471 one can easily show by mathematical induction that @xmath472 as a result , we have @xmath473 the distribution is not exactly a geometric law but @xmath474 converges really quickly to a constant @xmath475 independent of @xmath476 , that is , independent of the initial distribution @xmath477 . in the limit @xmath333 , one recovers @xmath457 .

the magnetic responses of a spin-@xmath0 ladder doped with non - magnetic impurities are studied combining both analytical and numerical methods . the regime where frustration induces incommensurability is taken into account . several improvements are made on the results of the seminal work by sigrist and furusaki [ sigrist and a. furusaki , j. phys . soc . jpn . , * 65 * , 2385 ( 1996 ) ] , and deviations from the brillouin magnetization curve due to interactions are also analyzed . we first discuss the magnetic profile around a single impurity and the effective interactions between impurities within the bond - operator mean - field theory . the results are compared to density - matrix renormalization group calculations . in particular , these quantities are shown to be sensitive to the transition to the incommensurate regime . we then focus on the behavior of the zero - field susceptibility through an effective curie constant . at zero - temperature , we give doping - dependent corrections to the results of sigrist and furusaki on general bipartite lattices , and compute exactly the distribution of ladder clusters due to chain breaking effects . solving the effective model with exact diagonalization and quantum monte - carlo gives the temperature dependence of the curie constant . its high - temperature limit is understood within a random dimer model , while the low - temperature tail is compatible with a real - space renormalization group scenario . interestingly , solving the full microscopic model does not show a plateau corresponding to the saturation of the impurities _ in isotropic ladders_. the second magnetic response which is analyzed is the magnetic curve . below fields of the order of the spin gap , the magnetization process is controlled by the physics of interacting impurity spins . the random dimer model is shown to capture the bulk of the curve , accounting for the deviation from a brillouin behavior due to interactions . the effective model calculations agree rather well with density - matrix renormalization group calculations at zero temperature , and with quantum monte - carlo at finite temperature . in all , the effect of incommensurability does not display a strong qualitative effect on both the magnetic susceptibility and the magnetic curve . consequences for experiments on the bicu@xmath1po@xmath2 compound and other spin - gapped materials are briefly mentioned .

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Proceed to summarize the following text: the heavy - ion physics has attracted much attention during the last three decades @xcite . the behavior of nuclear matter under the extreme conditions of temperature , density , angular momentum etc . , is a very important aspect of heavy - ion physics . one of the important quantity which has been used extensively to study this hot and dense nuclear matter is the collective transverse in - plane flow @xcite . this quantity has a beauty of vanishing at a certain incident energy . this energy is dubbed as balance energy ( @xmath1 ) or the energy of vanishing flow ( evf ) @xcite . this beauty is due to the counterbalancing of attractive mean - field at low incident energies and repulsive nucleon - nucleon ( nn ) collisions at higher incident energies . the balance energy of the masses ranging from @xmath2 to @xmath3 at different colliding geometries was studied experimentally and theoretically and found to be sensitive with the composite mass of the system @xcite as well as with the impact parameter of a reaction @xcite . + with the passage of time , isospin degree of freedom in terms of symmetry energy and nn cross section is found to affect the balance energy or energy of vanishing flow and related phenomenon in heavy - ion collisions @xcite . + experimentally , pak _ et al . _ , studied the isospin effects on the collective flow and balance energy at central and peripheral collision geometries @xcite . on the other hand , theoretically , this effect is studied by using the isospin - dependent boltzmann uehling - uhlenbeck model ( ibuu ) @xcite , and isospin - dependent quantum molecular dynamics ( iqmd ) model @xcite . + as noted , balance energy is due to the counterbalancing of the attractive mean - field and repulsive nucleon- nucleon collisions . the coulomb interaction in intermediate energy heavy - ion collisions is expected to play a dominant role in balance energy due to its repulsive nature . these effects are supposed to be more pronounced in the presence of isospin effects @xcite . the comparative study which will show the shift in balance energy due to coulomb interactions in the presence of isospin effects by taking into account asymmetry of reaction in a controlled fashion is still missing in the literature . the second point is the asymmetry of the reaction . in some of the studies , the asymmetry of a reaction is taken into care , but not in other , which is very important to study the isospin effects @xcite . the asymmetry of the reaction can be defined by the parameter @xmath4 ; where @xmath5 and @xmath6 are the masses of target and projectile . the @xmath7 = 0 corresponds to the symmetric reactions , whereas , non - zero value of @xmath7 define different asymmetry of the reaction . it is worth mentioning that the reaction dynamics in a symmetric reaction ( @xmath7 = 0 ) can be quite different compared to asymmetric reaction ( @xmath8 ) @xcite . this is due to the deposition of excitation energy in the form of compressional energy and thermal energy in symmetric and asymmetric reactions , respectively . the effect of the asymmetry of a reaction on the multifragmentation is studied many times in the literature @xcite . unfortunately , very little study is available for the asymmetry of the reaction in terms of transverse in - plane flow . + in this paper , we will perform the first ever study for the balance energy in terms of asymmetry of the reaction and then observe the effect of coulomb interactions , symmetry energy , equations of state as well as different frame of references . the iqmd model used for the present analysis is explained in the sec .- ii . the results are presented in sec .- iii , leading to the conclusions in sec .- iv . the isospin - dependent quantum molecular dynamics ( iqmd)@xcite model treats different charge states of nucleons , deltas and pions explicitly @xcite , as inherited from the vlasov - uehling - uhlenbeck ( vuu ) model @xcite . the iqmd model was used successfully in analyzing the large number of observables from low to relativistic energies @xcite . one of its version ( qmd ) , has been very successful in explaining the subthreshold particle production @xcite , multi - fragmentation @xcite , collective flow @xcite , disappearance of flow @xcite , and density temperature reached in a reaction @xcite . we shall not take relativistic effects into account , since in the energy domain we are interested , there is no relativistic effect @xcite . the isospin degree of freedom enters into the calculations via both cross sections , mean field and coulomb interactions @xcite . the details about the elastic and inelastic cross sections for proton - proton and neutron - neutron collisions can be found in refs.@xcite . + in this model , baryons are represented by gaussian - shaped density distributions + @xmath9 nucleons are initialized in a sphere with radius r = @xmath10 fm , in accordance with the liquid drop model . each nucleon occupies a volume of @xmath11 so that phase space is uniformly filled . the initial momenta are randomly chosen between 0 and fermi momentum @xmath12 . the nucleons of the target and projectile interact via two and three - body skyrme forces and yukawa potential . the isospin degrees of freedom is treated explicitly by employing a symmetry potential and explicit coulomb forces between protons of the colliding target and projectile . this helps in achieving the correct distribution of protons and neutrons within the nucleus . + the hadrons propagate using hamilton equations of motion : + @xmath13 with + @xmath14 is the hamiltonian . @xmath15 the baryon - baryon potential @xmath16 , in the above relation , reads as + @xmath17 where @xmath18 , @xmath19 and @xmath20 . the values of @xmath21 and @xmath22 depends on the values of @xmath23 , @xmath24 and @xmath25 @xcite . here @xmath26 and @xmath27 denote the charges of the @xmath28 and @xmath29 baryon , and @xmath30 , @xmath31 are their respective @xmath32 components ( i.e. 1/2 for protons and -1/2 for neutrons ) . the parameters @xmath33 and @xmath34 are adjusted to the real part of the nucleonic optical potential . for the density dependence of the nucleon optical potential , standard skyrme - type parameterizations is employed . the skyrme energy density have been shown to be very successful at low incident energies where fusion is dominant channel @xcite . the yukawa term is quite similar to the surface energy coefficient used in the calculations of nuclear potential for fusion @xcite . the choice of the equation of state ( or compressibility ) is still a controversial one . many studies advocates softer matter , whereas , much more believe the matter to be harder in nature @xcite . we shall use both hard ( h ) and soft ( s ) equations of state that have compressibilities of 380 and 200 mev , respectively . as discussed earlier , asymmetry of a reaction is found to affect the phenomena of muti - fragmentation in intermediate energy heavy - ion collisions @xcite . on the other hand , system mass dependence of balance energy is studied many times in the literature @xcite . to check the effect of coulomb interactions and asymmetry of a reaction on the balance energy , we have fixed ( @xmath0 = @xmath35 = 152 ) and varied the asymmetry of the reaction just like this : @xmath36 ( @xmath37 ) , @xmath38 ( @xmath39 ) , @xmath40 ( @xmath41 ) , @xmath42 ( @xmath43 ) , @xmath44 ( @xmath45 ) , @xmath46 ( @xmath47 ) . the asymmetry of a reaction with multi - fragmentation in this fashion is varied many times @xcite . the whole reaction dynamics is studied at semi - central geometry by varying the incident energy between 50 and 250 mev / nucleon with an increment of 50 mev / nucleon by employing hard as well as soft equations of state . we have checked the stability of the reacting nuclei in laboratory ( lab ) as well as in center of mass ( c.m . ) frame by taking into account the coulomb interactions . our main purpose here is to understand the effect of equations of state and coulomb interactions on the energy of vanishing flow or alternatively , on the balance energy by taking into account the asymmetry of a reaction . + the directed transverse flow is calculated using @xmath48 @xcite + @xmath49 where y(i ) and @xmath50 are , respectively , the rapidity distribution and transverse momentum of the @xmath28 particle . + asymmetry dependence of directed flow in lab as well as center of mass frame in l.h.s , while , r.h.s for the relative effect of coulomb interactions . the different lines in the figure are representing the effect of symmetry energy and coulomb interactions . ] to check the effect of frame of reference , we display in fig.1 , variation of the asymmetry @xmath7 on directed flow @xmath51 in lab . as well as center of mass frame at incident energy of e = 50 mev / nucleon . the top and bottom panels in the right hand side of the figure are representing the relative coulomb effect with respect to the asymmetry of the reaction . this relative effect is calculated as : + @xmath52 as is evident from the figure , directed flow is found to increase in a systematic manner in c.m . as well as in lab frame at e = 50 mev / nucleon with asymmetry of the reaction . the inclusion of coulomb interactions does not alter the conclusions . note that the increase in the asymmetry is related with the increase in the n / z ratio . because the symmetry potential for the neutron rich systems is stronger compared to the neutron poor systems due to large relative neutron strength . furthermore , the symmetry potential is repulsive for neutrons and attractive for protons . on the other hand , more negative value of directed flow ( dominating the mean field ) is observed in the absence of coulomb interactions in center of mass as well as in lab . this is due to the enhancement of the chemical and mechanical instability domains in the absence of coulomb interactions @xcite . similar type of study and conclusion was also performed for nuclear stopping in ref . + extensive study in the literature proves the stability of reactions in lab . frame , but , keep in mind that the reaction under consideration in these studies were symmetric in nature . as is clear from the figure , asymmetric systems are found to be more stable in the center of mass frame compared to the lab frame . moreover , if one consider lab frame , one is surely missing the effect of asymmetry . to further strengthen the stability of center of mass frame with asymmetry , the relative effect of coulomb interactions @xmath53 is studied in both frames . the relative effect of the coulomb interactions is found to decrease with increase in the asymmetry of a reaction . the systematic decrease can be seen in center of mass frame with asymmetry , while very weak dependence of coulomb interactions on the asymmetry is obtained in lab frame . for the further study , we have opted the center of mass frame because we want to see the shift in the balance energy due to coulomb interactions , whose effect is clearly visible in center of mass frame . + the time evolution of directed flow at different incident energies in center of mass frame in the absence of coulomb interactions . the left and right panels are for soft and hard equations of state , respectively . ] before we proceed further , let us check the time evolution of directed transverse flow . in fig . [ fig2 ] , we display the time evolution of directed flow from e = 50 to 200 mev / nucleon in center of mass frame for soft ( l.h.s ) and hard ( r.h.s ) equations of state . note that the compressibilities of soft and hard equations of state are 200 and 380 mev , respectively @xcite . the time evolution is plotted in the absence of coulomb interactions to see the maximum effect of asymmetry of a reaction on the directed in - plane flow . the figure reveals the following points : + a ) . the quantity @xmath48 is observed to be constant throughout the whole distribution of time , while , the large variation is observed in the value at initial and final time steps when observed in the lab frame @xcite + b ) . with the increase in the incident energy , the directed flow is approaching towards more positive value . this is due to the well known fact of increase in the frequent nn collisions with increase in the incident energy @xcite . the behavior of the directed flow with asymmetry of a reaction follows the opposite trend at e = 50 mev / nucleon as compared to other high incident energies . it has been discussed many times in literature and also clear from the present findings that attractive mean - field is dominating at e = 50 mev /nucleon compared to higher incident energies under consideration @xcite . this is due to the different mechanisms contributing at low and high incident energies within isospin - dependent quantum molecular dynamics . it was shown by us as well as by others @xcite that symmetry potential dominates the physics at low incident energy , while , nn cross sections one major driven force at higher incident energies . furthermore , at low incident energies , symmetry potential is repulsive for neutrons and attractive for protons . with the increase in the asymmetry of a reaction , the number of neutrons increases and hence comparative repulsion due to neutrons also increases leading to less attractive value of the flow . it is also clear from ref . @xcite , that with the increase in asymmetry of a reaction , the participant zone decreases and spectator zone increases . at higher incident energies , where the effect of symmetry energy is negligible , the contribution of nn collisions comes from the participant zone whereas mean - field contribution comes from the spectator zone . hence directed flow is found to be less positive with increase in the asymmetry of the reaction . the directed flow has less positive value with soft equation of state compared to hard equation of state . the less positive means the dominance of mean - field . this is due to the different compressibilities of hard ( 380 mev ) and soft ( 200 mev ) equations of state . naturally , more is the compressibility , more are the number of collisions and hence more positive is the directed flow . this is indicating that the directed flow is sensitive towards the equations of state . + excitation function of directed flow at different asymmetries with and without coulomb interactions at semi - central geometry . ] finally , in figs.[fig3 ] and [ fig4 ] , we display the excitation function of directed flow at different asymmetries from @xmath7 = 0.2 to 0.7 . the value of abcissa at zero value of @xmath48 corresponds to the energy of vanishing flow ( evf ) or alternatively , the balance energy ( @xmath1 ) . the fig . [ fig3 ] shows the shift in the balance energy due to coulomb interactions , while , fig . [ fig4 ] is representing the shift in the balance energy due to different equations of state . in fig . [ fig3 ] , one sees a linear enhancement in the nuclear flow with increase in the incident energy . this increase in the transverse flow is sharp at smaller incident energies ( upto 200 mev / nucleon ) . if one goes to higher incident energies , the value gets saturated as discussed in ref @xcite . we have displayed here the results upto 250 mev / nucleon , since we are interested in and around balance energy . in the presence of coulomb interactions , more positive value of the flow is obtained . this is due to the well known repulsive nature of coulomb interactions . at higher energies , the repulsion due to coulomb interactions is stronger during the early phase of the reaction and transverse momentum increases sharply . the overall effect depends on the asymmetry of the reaction . if one looks at the balance energy , the shift in the incident energy towards the higher value is obtained at @xmath54 with the asymmetry of the reaction . this is showing that with increase in asymmetry of the reaction and in the absence of coulomb interactions , attractive mean - field is dominating the large region of incident energy . the systematics of the balance energy with asymmetry of the reaction is discussed in fig . . + excitation function of directed flow with hard and soft equations of state in the absence of coulomb interactions for different asymmetries . ] as we have seen in fig.[fig2 ] , that different equations of state show sensitivity towards the directed flow with respect to the asymmetry of a reaction . the detailed analysis with soft ( s ) and hard ( h ) equations of state is displayed in fig . the excitation function of directed flow follows similar trend as explained in fig . [ fig3 ] . for nearly symmetric systems ( @xmath7 = 0.2 ) , the balance energy is found to be independent of the equations of state , however , resonable differences are observed at higher incident energies . more positive values of directed flow are obtained with hard equation of state compared to soft equation of state . this is true at all asymmetries from ( @xmath7 = 0.3 to 0.7 ) . this is due to different compressibilities of hard ( 380 mev ) and soft ( 200 mev ) equations of state . with an increase in the asymmetry of a reaction , the shift in the balance energy towards the higher value of incident energy takes place with soft equation of state compared to hard one . this is consistent with the findings in the literature @xcite . + power law dependence of balance energy with asymmetry of the reaction . the lower panel is representing the relative @xmath55 effect of coulomb interactions on the balance energy . ] to sum up , in fig.[fig5 ] , we have displayed the asymmetry dependence of balance energy . we displayed the results for hard and soft equations of state by switching off the coulomb interactions . in addition , for a comparative study , the results in the presence of coulomb interactions with soft equation of state are also shown . all the lines are fitted with power law of the form @xmath56 , where c and @xmath57 are the constants . the values of @xmath57 in the absence of coulomb interactions for soft and hard equations of state are 0.375 and 0.282 , respectively , while in the presence of coulomb interactions for soft equation of state is @xmath57 = 0.06067 . + if we compare the asymmetry dependence of balance energy with mass dependence , the trend is opposite @xcite . it clearly indicates that if one wants to study the isospin effect , then one has to consider the systems with @xmath58 , otherwise one is not able to get the exact information about the isospin effects . it is also clear from the figure , that shift in the balance energy is observed due to coulomb interactions as well as due to equations of state with asymmetry of the reaction . the shift is more due to coulomb interactions in comparison to equations of state , indicating the importance of coulomb interactions in intermediate energy heavy - ion collisions . the higher balance energy is obtained with coulomb - off + soft equation of state followed by coulomb - off + hard equation of state and finally coulomb - on + soft equation of state . + for the further understanding , the relative percentage difference in the balance energy is plotted in the lower panel denoted by the quantity @xmath59 given by + @xmath60 { \times}100\ ] ] the @xmath59 is found to increase with the increase in the asymmetry of a reaction . this indicates shift of the nuclear matter towards the attractive mean - field region in the absence of coulomb interactions with the asymmetry of the reaction . this difference @xmath61 = 30 mev / nucleon at @xmath37 , while it is 115 mev / nucleon at @xmath47 . the difference of 90 mev / nucleon in the shift of balance energy with asymmetry can not be ignored . this is the first ever study and experiments are called to verify the results . our present aim was to understand the influence of coulomb interactions as well as equations of state on the dynamics of large asymmetric reactions in semi - central heavy - ion collisions . at low incident energies , the contribution of mean - field is more for the nearly symmetric systems , while at higher incident energies , opposite scenario is observed . the balance energy is found to increase with the increase in the asymmetry of the reaction . the balance energy is affected by the coulomb interactions compared to different equations of state . this work has been supported by the grant from department of science and technology ( dst ) , government of india , new delhi , vide grant no.sr/wos-a/ps-10/2008 . 999 w. scheid _ et al . _ , phys . rev . lett . * 32 * , 741 ( 1974 ) ; 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we aim to understand the role of coulomb interactions as well as of different equations of state on the disappearance of transverse flow for various asymmetric reactions leading to same total mass . for the present study , the total mass of the system is kept constant ( @xmath0 = 152 ) and asymmetry of the reaction is varied between 0.2 and 0.7 . we find that the contribution of mean - field at low incident energies is more for nearly symmetric systems , while the trend is opposite at higher incident energies . the coulomb interactions as well as different equations of state are found to affect the balance energy significantly for larger asymmetric reactions .

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Proceed to summarize the following text: there may be a deep connection between the origin of matter in the universe and the observed neutrino oscillations . this speculation is inspired by the idea that the heavy right handed majorana neutrinos that are added to the standard model for understanding small neutrino masses via the seesaw mechanism@xcite can also explain the origin of matter via their decay . the mechanism goes as follows@xcite : cp violation in the same yukawa interaction of the right handed neutrinos , which go into giving nonzero neutrino masses after electroweak symmetry breaking , lead to a primordial lepton asymmetry via the out of equilibrium decay @xmath11 ( where @xmath12 are the known leptons and @xmath13 is the standard model higgs field ) . this asymmetry subsequently gets converted to baryon - anti - baryon asymmetry observed today via the the electroweak sphaleron interactions@xcite , above @xmath14 ( @xmath15 being the weak scale ) . since this mechanism involves no new interactions beyond those needed in the discussion of neutrino masses , one would expect that better understanding of neutrino mass physics would clarify one of the deepest mysteries of cosmology both qualitatively as well as quantitatively . this question has been the subject of many investigations in recent years@xcite in the context of different neutrino mass models and many interesting pieces of information about issues such as the spectrum of right handed neutrinos , upper limit on the neutrino masses etc have been obtained . in a recent paper , @xcite , two of the authors showed that if one assumes that the lepton sector of minimal seesaw models has a leptonic @xmath0 interchange symmetry@xcite , then one can under certain plausible assumptions indeed predict the magnitude of the matter - anti - matter asymmetry in terms of low energy oscillation parameter , @xmath16 and a high scale cp phase . the choice of @xmath0 symmetry was dictated by the fact that it is the simplest symmetry of neutrino mass matrix that explains the maximal atmospheric mixing as indicated by data . using present experimental value for @xmath16 , one obtains the right magnitude for the baryon asymmetry of the universe . the results of the paper @xcite were derived in the limit that @xmath0 interchange symmetry is exact . if however a nonzero value for the neutrino mixing angle @xmath8 is detected in future experiments , this would imply that this symmetry is only approximate . also , since in the standard model @xmath17 and @xmath18 are members of the @xmath19 doublets @xmath20 and @xmath21 , any symmetry between @xmath17 and @xmath18 must be a symmetry between @xmath22 and @xmath23 at the fundamental lagrangian level . the observed difference between the muon and tau masses would therefore also imply that the @xmath0 symmetry has to be an approximate symmetry . in view of this , it is important to examine to what extent the results of ref.@xcite carry over to the case when the symmetry is approximate . we find two interesting results under some very general assumtions : ( i ) a simple formula relating the lepton asymmetry and neutrino oscillation observables for the case of three right handed neutrinos , i.e. @xmath24 and ( ii ) a relation of the form @xmath25 for the case of two right handed neutrinos . measurement of @xmath8 will have important implications for both the models ; in particular we show that in a class of models with two right handed neutrinos with approximate @xmath0 symmetry breaking , there is a lower limit on @xmath8 , which is between 0.1 to 0.15 depending on the values of the cp phase . these values are in the range which will be probed in experiments in near future@xcite . the basic assumption under which the two results are derived are the following : \(a ) type i seesaw formula is responsible for neutrino masses : \(b ) @xmath0 symmetry for leptons is broken only at high scale in the mass matrix of the right handed neutrinos . the paper is organized as follows : in sec . ii , we outline the general framework for our discussion ; in sec . iii , we rederive the result of ref.@xcite for the case of exact @xmath0 symmetry ; in sec . iv , we derive the connection between @xmath1 and oscillation parameters for the case of approximate @xmath0 symmetry . iv is devoted to the case of two right handed neutrinos , where we present the allowed range of @xmath8 dictated by leptogenesis argument . in sec . v , we describe a class of simple gauge models where these conditions are satisfied . we start with an extension of the minimal supersymmetric standard model ( mssm ) for the generic the type i seesaw model for neutrino masses . the effective low energy superpotential for this model is given by @xmath26 here @xmath27 are leptonic superfields ; @xmath28 are the higgs fields of mssm . @xmath29 and @xmath30 are general matrices where we choose a basis where @xmath31 is diagonal . we do not display the quark part of the superpotential which is same as in the mssm . after electroweak symmetry breaking , this leads to the type i seesaw formula for neutrino masses given by @xmath32 the constraints of @xmath0 symmetry will manifest themselves in the form of the @xmath29 and @xmath30 . it has been pointed out that if we go to a basis where the right handed neutrino mass matrix is diagonal , we can solve for @xmath29 in terms of the neutrino masses and mixing angles as follows@xcite : @xmath33 where @xmath34 is a complex matrix with the property that @xmath35 . the unitary matrix @xmath36 is the lepton mixing matrix defined by @xmath37 the complex orthogonal matrices @xmath34 can be parameterized as : @xmath38 with @xmath39 and similarly for the other matrices . @xmath40 are complex angles . let us now turn to lepton asymmetry : the formula for primordial lepton asymmetry in this case , caused by right handed neutrino decay is @xmath41 ^ 2_{1j}}{(\tilde{y}_\nu \tilde{y}^{\dagger}_\nu)_{11 } } f(\frac{m_1}{m_j } ) \label{nl}\end{aligned}\ ] ] where @xmath42 is defined in a basis where righthanded neutrinos are mass eigenstates and their masses are denoted by @xmath43 where @xmath44$]@xcite . in the case where that the right handed neutrinos have a hierarchical mass pattern i.e. @xmath45 , we get @xmath46 . in this approximation , we can write the lepton asymmetry in a simple form@xcite @xmath47_{11}}{v^2(\tilde{y}_\nu \tilde{y}^{\dagger}_\nu)_{11 } } \label{buch}\end{aligned}\ ] ] where using the expression for @xmath29 given above , we can rewrite @xmath1 as : @xmath48_{11}}{v^2|r(z_{ij}){\cal m}_\nu r^{\dagger}(z_{ij})|^2_{11 } } \label{buch1}\end{aligned}\ ] ] we will now apply this discussion to calculate the lepton asymmetry in the general case without any symmetries . in the following sections , we follow it up with a discussion of two cases : ( i ) the cases of exact @xmath0 symmetry and ( ii ) the case where this symmetry is only approximate . since the formula in eq . ( 9 ) assumes that there are three right handed neutrinos , we will focus on this case in the next two sections . in a subsequent section , we consider the case of two right handed neutrinos @xmath49 , which transform into each other under the @xmath0 symmetry . both cases are in agreement with the observed neutrino mass differences and mixings . it follows from eq.[buch1 ] that @xmath50}{v^2|r(z_{ij}){\cal m}_\nu r^{\dagger}(z_{ij})|^2_{11 } } \label{for}\end{aligned}\ ] ] since the matrix @xmath34 is an orthogonal matrix , we have the relation @xmath51 using this equation in eq.[for ] , we get @xmath52}{v^2\sum_j(|r_{1j}|^2 m_j ) } \label{for1}\end{aligned}\ ] ] this relation connects the lepton asymmetry to both the solar and the atmospheric mass difference square@xcite . to make a prediction for the lepton asymmetry , we need to the lengths of the complex quantities @xmath53 . the out of equilibrium condition does provide a constraint on @xmath54 as follows : @xmath55 it is clear from eq . ( 13 ) that if neutrinos are quasidegenerate i.e. @xmath56 , then using eq . ( 11 ) , we find that the left hand side of eq . ( 13 ) has a lower bound of @xmath57 which is clearly much bigger than the right hand side of the inequality . defining @xmath58 , this means that @xmath59 . this implies that the right handed neutrinos decays are in equilibrium at @xmath60 . this will cause dilution of the lepton asymmetry generated with the dilution factor given by @xmath61 . using a parameterization for the dilution factor @xmath62@xcite , we get @xmath63 which will make the baryon to photon ratio much too small . based on this argument , we conclude that a degenerate mass spectrum with @xmath64 ev will most likely be in conflict with observations , if type i seesaw is responsible for neutrino masses . it must however be noted that a more appealing and natural scenario for degenerate neutrino masses is type ii seesaw formula@xcite , in which case the above considerations do not apply . therefore , it is not possible to conclude based on the leptogenesis argument alone that a quasi - degenerate neutrino spectrum is inconsistent . in a hierarchical neutrino mass picture , ( 13 ) implies that @xmath65 and @xmath66 . if we assume that the upper limit in the eq.[equ1 ] is saturated , then we get the atmospheric neutrino mass difference square in eq.[for1 ] to give the dominant contribution . we will see below that if one assumes an exact @xmath0 symmetry for the neutrino mass matrix , the situation becomes different and it is the solar mass difference square that dominates . in this section , we consider the case of three right handed neutrino with an exact @xmath0 symmetry in the dirac mass matrix as well as the right handed neutrino mass matrix . in this case , the right handed neutrino mass matrix @xmath30 and the dirac yukawa coupling @xmath29 can be written respectively as : @xmath67 where @xmath68 and @xmath69 are all complex . an important property of these two matrices is that they can be cast into a block diagonal form by the same transformation matrix @xmath70 on the @xmath71 s and @xmath72 s . let us denote the block diagonal forms by a tilde i.e. @xmath42 and @xmath73 . we then go to a basis where the @xmath73 is subsequently diagonalized by the most general @xmath74 unitary matrix as follows : @xmath75 where @xmath76 where @xmath77 is the most general @xmath74 unitary matrix given by @xmath78 . the @xmath79 case therefore reduces to a @xmath74 problem . the third mass eigenstate in both the light and the heavy sectors play no role in the leptogenesis as well as generation of solar mixing angle@xcite . note also that we have @xmath80 . the seesaw formula in the 1 - 2 subsector has exactly the same form except that all matrices in the left and right hand side of eq . ( 9 ) are @xmath74 matrices . the formula for the dirac yukawa coupling in this case can be inverted to the form : @xmath81 where @xmath82 . using this , we can cast @xmath1 in the form : @xmath83 this could also have been seen from eq.([for1 ] ) by realizing that for the case of exact @xmath0 symmetry , we have @xmath84 and @xmath85 . the above result reproduces the direct proportionality between @xmath1 and solar mass difference square found in ref.@xcite . to simplify this expression further , let us note that out of equilibrium condition for the decay of the lightest right handed neutrino leads to the condition : @xmath86 \leq 14 \frac{m^2_1}{m_{p\ell}}\end{aligned}\ ] ] which implies that @xmath87 since solar neutrino data require that in a hierarchical neutrino mass picture @xmath88 ev , in eq.([equ ] ) , we must have @xmath89 . if we parameterize @xmath90 , we recover the conclusions of ref.@xcite . this provides a different way to arrive at the conclusions of ref.@xcite . in this section , we consider the effect of breaking of @xmath0 symmetry on lepton asymmetry . within the seesaw framework , this breaking can arise either from the dirac mass matrix for the neutrinos or from the right handed neutrino sector or both . we focus on the case , when the symmetry is broken in the right handed sector only . such a situation is easy to realize in seesaw models where the theory obeys exact @xmath0 symmetry at high scale ( above the seesaw scale ) prior to b - l symmetry breaking as we show in a subsequent section . we will also show that in this case there is a simple generalization of the lepton asymmetry formula that we derived in the exact @xmath0 symmetric case @xcite symmetric model where the dirac yukawa coupling has the form @xmath91 has been discussed in ref.@xcite . our discussion applies more generally . ] . in this case the neutrino yukawa matrix is given in the mass eigenstates basis of the right handed neutrinos by @xmath92 where @xmath93 is the neutrino dirac matrix in the flavor basis ; the notation @xmath94 denotes a unitary @xmath74 matrix in the @xmath95 subspace . in the above equation , @xmath96 . now if we substitute for @xmath97 the expression in eq . [ ynu ] and use maximal mixing for the atmospheric neutrino we obtain @xmath98 = v_{1/3}m^{1/2}_r r_{1/2}r_{1/3}m^{1/2}_{\nu}u_{1/2}^{+}u_{1/3}^{+}\ ] ] since the @xmath0 symmetry breaking is assumed to be small and from reactor neutrino experiments @xmath99 we will expand the mixing matrices in the @xmath100 subspace to first order in mixing parameter : @xmath101 where @xmath102\ ] ] to first order in @xmath103 , @xmath104 and @xmath8 we have @xmath105 it is straight forward to show that the perturbation parameters should satisfy the following equations @xmath106 where @xmath107 are the matrix elements of @xmath108 and @xmath109 and @xmath110 are the sine and cosine of the solar neutrino mixing angle . hence one can see that the parameter @xmath104 is proportional to the @xmath8 neutrino mixing angle and is given to first order by @xmath111\theta_{13}e^{-i\delta}c_{\theta}\ ] ] this proves that the matrix element @xmath112 that goes into the leptogenesis formula is directly proportional to the physically observable parameter @xmath8 . this enables us to write @xmath113 . a consequence of this is that if the coefficient of proportionality is chosen to be of order one , then as experimental upper limit goes down , unlike the generic type i seesaw case in section ii , the solar mass difference square starts to dominate for the lma solution to the solar neutrino problem . in this section , we consider the case of two right handed neutrinos which transform into one another under @xmath0 symmetry . the leptogenesis in this model with exact @xmath0 symmetry was discussed in @xcite and was shown that it vanishes . in this model therefore , a vanishing or very tiny @xmath8 would not provide a viable model for leptogenesis . turning this argument around , enough leptogenesis should provide a lower limit on the value of @xmath8 . to set the stage for our discussion , let us first review the argument for the exact @xmath0 symmetry case@xcite . the symmetry under which @xmath114 and @xmath115 whereas the @xmath116 constrains the general structure of @xmath117 and @xmath118 as follows : @xmath119 in order to calculate the lepton asymmetry using eq.(7 ) , we first diagonalize the righthanded neutrino mass matrix and change the @xmath29 to @xmath42 . since @xmath118 is a symmetric complex @xmath74 matrix , it can be diagonalized by a transformation matrix @xmath120 i.e. @xmath121 where @xmath122 are complex numbers . in this basis we have @xmath123 . we can therefore rewrite the formula for @xmath124 as @xmath125 ^ 2_{12 } f(\frac{m_1}{m_2})\end{aligned}\ ] ] now note that @xmath126 has the form @xmath127 which can be diagonalized by the matrix @xmath128 . therefore it follows that @xmath129 . let us now introduce @xmath0 symmetry breaking . if we introduce a small amount of @xmath0 breaking in the right handed neutrino sector as follows : we keep the @xmath29 symmetric but choose the right handed neutrino mass matrix as : @xmath130 after the right handed neutrino mass matrix is diagonalized , the @xmath131 @xmath132 takes the form ( for @xmath133 and in the basis where the light neutrino masses are diagonal ) : @xmath134 here @xmath135 are of order one and @xmath136 . to first order in the small mixing @xmath8 , the complex parameters @xmath137 satisfy the constraint @xmath138 using these order of magnitude values , we now find that @xmath139}{m_2}\end{aligned}\ ] ] where @xmath140 is a function of order one . it is clear that very small values for @xmath8 will lead to unacceptably small @xmath1 . in fig . 1 , we have plotted @xmath141 against @xmath8 for values of the parameters in the model that fit the oscillation data and find a lower bound on @xmath142 for two different values of the cp phases ( figure 1 ) . in this figure , we have chosen , @xmath143 gev . for higher values of @xmath144 the allowed range @xmath8 moves to the lower range . also we note that for values of @xmath145 gev , the baryon asymmetry becomes lower than the observed value . vrs @xmath8 for the case of two right handed neutrinos with approximate @xmath0 symmetry and cp phases @xmath146 and @xmath147 . the values of @xmath8 are predicted to be 0.1 and 0.15 respectively . the horizontal line corresponds to @xmath148@xcite . ] in this section , we present a simple extension of the minimal supersymmetric standard model ( mssm ) by adding to it specific high scale physics that at low energies can exhibit @xmath0 symmetry in the neutrino sector as well as real dirac masses for neutrinos . first we recall that mssm needs to be extended by the addition of a set of right handed neutrinos ( either two or three ) to implement the seesaw mechanism for neutrino masses@xcite . we will accordingly add three right handed neutrinos @xmath149 to mssm . we then assume that at high scale , the theory has @xmath0 @xmath150 symmetry under which @xmath151 are even and odd combinations ; similarly , we have for leptonic doublet superfields @xmath152 and leptonic singlet ones @xmath153 ; two pairs of higgs doublets ( @xmath154 and @xmath155 ) , and a singlet superfields @xmath156 . other superfields of mssm such as @xmath157 as well as quarks are even under the @xmath0 @xmath150 symmetry . now suppose that we write the superpotential involving the @xmath158 fields as follows : @xmath159 then when we give high scale vevs to @xmath160 , then below the high scale there are only the usual mssm higgs pair @xmath161 and @xmath162 that survive whereas the other pair becomes superheavy and decouple from the low energy lagrangian . the effective coupling at the mssm level is then given by : @xmath163 note that the @xmath0 symmetry is present in the dirac neutrino mass matrix whereas it is not in the charged lepton sector as would be required to . we show below that it is possible to have a high scale supersymmetric theory which would lead to real dirac yukawa couplings ( @xmath164 ) if we require the high scale theory to be left - right symmetric . to show how this comes about , consider the gauge group to be @xmath165 with quarks and leptons assigned to left and right handed doublets as usual@xcite i.e. @xmath166 , @xmath167 ; @xmath168 and @xmath169 ; higgs fields @xmath170 ; @xmath171 ; @xmath172 ; @xmath173 and @xmath174 . the new point specific to our model is that we have two sets of the higgs fields with the above quantum numbers , one even and the other odd under the @xmath0 @xmath150 permutation symmetry i.e. @xmath175 , @xmath176 , @xmath177 , @xmath178 and @xmath179 ( plus for fields even under @xmath150 and @xmath180 for fields odd under @xmath150 . ) furthermore , we will impose the parity symmetry under which @xmath181 , @xmath182 , @xmath183 ) , @xmath184 . the higgs sector of the low energy superpotential is determined from this theory after left - right gauge group is broken down to the standard model gauge group by the vev s of @xmath187 . the phenomenon of doublet - doublet spitting leaves only two higgs doublets out of the four in @xmath188 and is determined by a generic superpotential of type @xmath189 where @xmath190 go over @xmath191 and @xmath180 for even and odd and only even terms are allowed by @xmath0 invariance e.g. @xmath192 are nonzero . now suppose that @xmath193 but @xmath194 and @xmath195 . these vevs break the left - right group to the standard model gauge group . it is then easy to see that below the @xmath196 scale , there are only one higgs pair where @xmath197 and @xmath198 . here we have denoted the @xmath199 and @xmath200 . the upshot of all these discussions is that the right handed neutrino yukawa couplings are @xmath0 even and therefore have the form : @xmath201 it is easy to see that redefining the fields appropriately , we can make @xmath29 real . so the only source of complex phase in this model is in the rh neutrino mass matrix , which in this model are generated by higher dimensional couplings of the form @xmath202 as we discuss now . the most general nonrenormalizable interactions that can give rise to right handed neutrino masses are of the form : @xmath203 note that since both @xmath204 acquire vevs , the last term in the above expression will give rise to @xmath0 breaking in the rh neutrino sector while preserving it in the @xmath29 . the associated couplings in the above equations are in general complex . this leads to a realistic three generation model with approximate @xmath0 symmetry as analyzed in the previous sections . in summary , we have studied the implications for leptogenesis in models where neutrino masses arise from the type i seesaw mechanism and where the near maximal atmospheric mixing angle owes its origin to an approximate @xmath0 symmetry . we derive a relation of the form @xmath205 for the case of three right handed neutrinos , which directly connects the neutrino oscillation parameters with the origin of matter . we also show that if @xmath8 is very small or zero , only the lma solution to the solar neutrino puzzle would provide an explanation of the origin of matter within this framework . finally for the case of two right handed neutrinos with approximate @xmath0 symmetry , we predict values for @xmath8 in the range @xmath206 for specific choices of the the high energy phase between @xmath207 and @xmath147 . p. minkowski , phys . lett . * b67 * , 421 ( 1977 ) ; m. gell - mann , p. ramond , and r. slansky , _ supergravity _ ( p. van nieuwenhuizen et al . eds . ) , north holland , amsterdam , 1980 , p. 315 ; t. yanagida , in _ proceedings of the workshop on the unified theory and the baryon number in the universe _ ( o. sawada and a. sugamoto , eds . ) , kek , tsukuba , japan , 1979 , p. 95 ; s. l. glashow , _ the future of elementary particle physics _ , in _ proceedings of the 1979 cargse summer institute on quarks and leptons _ ( m. lvy et al . eds . ) , plenum press , new york , 1980 , pp . 687 ; r. n. mohapatra and g. senjanovi , phys . lett . * 44 * 912 ( 1980 ) . see for instance , a. s. joshipura , e. a. paschos and w. rodejohann , jhep * 08 * , 029 ( 2001 ) ; w. rodejohann and k. r. s. balaji , phys . rev . d * 65 * , 093009 ( 2002 ) [ arxiv : hep - ph/0201052 ] ; s. antusch and s. f. king , hep - ph/0405093 . for reviews of earlier literature see , a. pilaftsis , phys . * d 56 * , 5431 ( 1997 ) ; t. hambye , arxiv : hep - ph/0412053 ; for a general discussion of the connection between leptogenesis and seesaw parameters , see v. barger , d. a. dicus , h - j . he and t. li , hep - ph/0310278 . for the case of quasi - degenerate right handed neutrinjos , see m. flanz , e. a. paschos and u. sarkar , phys . b345 * , 248 ( 1995 ) . c. s. lam , hep - ph/0104116 ; t. kitabayashi and m. yasue , phys.rev . * d67 * 015006 ( 2003 ) ; w. grimus and l. lavoura , hep - ph/0305046 ; 0309050 ; y. koide , phys.rev . * d69 * , 093001 ( 2004 ) ; for examples of such theories , see w. grimus and l. lavoura , hep - ph/0305046 ; 0309050 . r. n. mohapatra , slac summer inst . lecture ; http://www-conf.slac.stanford.edu/ssi/2004 ; hep - ph/0408187 ; jhep , * 0410 * , 027 ( 2004 ) ; w. grimus , a. s.joshipura , s. kaneko , l. lavoura , h. sawanaka , m. tanimoto , hep - ph/0408123 . k. anderson _ et al . _ , arxiv : hep - ex/0402041 ; m. apollonio _ et al . _ , eur . phys . j.c * 27 * , 331 ( 2003 ) arxiv : hep - ex/0301017 ; m. v. diwan _ et al . _ , phys . d * 68 * , 012002 ( 2003 ) arxiv : hep - ph/0303081 ; d. ayrea et al . hep - ex/0210005 ; y. itow et al . ( t2k collaboration ) hep - ex/0106019 ; i. ambats _ et al . _ ( nova collaboration ) , fermilab - proposal-0929 ; m. goodman , hep - ph/0501206 .

we show that in theories where neutrino masses arise from type i seesaw formula with three right handed neutrinos and where large atmospheric mixing angle owes its origin to an approximate leptonic @xmath0 interchange symmetry , the primordial lepton asymmetry of the universe , @xmath1 can be expressed in a simple form in terms of low energy neutrino oscillation parameters as @xmath2 , where @xmath3 and @xmath4 are parameters characterizing high scale physics and are each of order @xmath5 ev@xmath6 . we also find that for the case of two right handed neutrinos , @xmath7 as a result of which , the observed value of baryon to photon ratio implies a lower limit on @xmath8 . for specific choices of the cp phase @xmath9 we find @xmath8 is predicted to be between @xmath10 .

You are an expert at summarizing long articles. Proceed to summarize the following text: non - linear sigma models in 1 + 1 dimensions play an important role in several areas of theoretical and mathematical physics , see e.g. @xcite for a review . they become accessible to analytical methods in particular when the target manifold is a coset manifold ( maximally symmetric ) , such as @xmath4 . this model by construction has a manifest internal @xmath2-symmetry as well as further hidden symmetries that make it integrable . its target space is the riemannian manifold @xmath5 , and its internal symmetry group is compact . string theory is closely related to non - linear sigma models , where the target space comes into play as the spacetime in which the strings propagate . often , it is assumed to be of the form @xmath6 , where @xmath7 is a suitable compact riemannian manifold representing the @xmath8 extra dimensions to have certain well - known special values , depending whether or not one includes fermions . we will not consider such symmetries in this paper , so in this sense the connection to string theory is not a direct one . ] . however , there is no a priori reason not to consider more general target spaces where not just the extra dimensions are curved . for instance , it seems natural to consider target spaces such as desitter spacetime @xmath9 , which , like the sphere , is a coset manifold ( maximally symmetric space ) . this type of non - linear sigma model would be expected to have an internal @xmath1-symmetry ( which is not compact ) together , perhaps , with further hidden symmetries by analogy with the @xmath2 non - linear sigma model . to exploit the hidden symmetries , say , in the @xmath2-model , one may take advantage of the fact that its scattering matrix must be factorizing . in combination with the internal @xmath2 symmetry , natural assumptions about the `` particle spectrum '' ( basically the representation of @xmath2 ) , hints from perturbation theory , and the highly constraining relations imposed by crossing symmetry , yang - baxter relation , analyticity , etc . , one can often guess the form of the 2-body scattering matrix , which then consistently determines the @xmath8-body scattering matrix @xcite . in order to derive from such a scattering matrix quantities associated with the local operators of the theory , one can for example follow the bootstrap - form factor program @xcite . the aim of this program is to determine the matrix elements of local operators between in- and out - states ( form factors ) , which are found using the scattering matrix and various a priori assumptions about the form factors . the program is largely successful but runs into technical difficulties when attempting to compute higher correlation functions in terms of the form factors , or , indeed , when trying to even show that the corresponding series converge . an alternative approach is to construct the operator algebras generated by the local quantum fields by abstract methods ( see @xcite , and @xcite for a review ) . the input is again the scattering matrix , but the procedure is rather different . first , one constructs certain half - local generator fields . these `` left local fields '' @xmath10 play an auxiliary role and are constructed in such a way that with each of them , there is an associated `` right local field '' @xmath11 such that @xmath12=0 $ ] if @xmath13 and @xmath14 are space like related points in 1 + 1 dimensional minkowski space _ and _ @xmath14 is to the right of @xmath13 in a relativistic sense . we will actually work with a similar construction for a `` chiral half '' of a massless theory on a lightray , where @xmath15 = 0 $ ] if @xmath16 with @xmath17 lightray coordinates @xcite . both on two - dimensional minkowski space and in the chiral lightray setting , the left and right local fields generate left and right local operator algebras , and suitable intersections of these algebras then contain the truly local fields @xcite . the latter are thereby characterized rather indirectly , and indeed , the local fields do not have a simple expression in terms of the auxiliary semi - local objects , but rather reproduce the full complexity of the form factor expansion @xcite . in this article , we will show that such constructions also work for non - compact target spaces such as @xmath18 . since the internal symmetry group , @xmath1 , is non - compact , its non - trivial unitary representations must necessarily be infinite - dimensional . this is an obvious major difference , say , to the @xmath2-model , where the basic representation under which the single particle states transform is the fundamental representation , which is finite ( @xmath19- ) dimensional . despite this difference , one may proceed and ask whether the algebraic method can be generalized to non - compact groups such as @xmath1 . for this , one first needs a 2-body scattering matrix ( or rather , an `` @xmath1-invariant yang - baxter operator '' , see sect . [ subsection : invariant - yb - ops ] ) satisfying suitable properties such as yang - baxter - relation , crossing symmetry , analyticity , unitarity , etc . in turns out that the precise algebraic and analytic properties required to make the method work are related to each other in a rather intricate way , and one does not , a priori , see an obvious way to generate simple solutions to the requirements . one result of our paper is to provide such a yang - baxter operator for the spin-@xmath20 principal , complementary and discrete series representations of @xmath1 in sect . [ section : ybops - lorentzgroup ] . this is facilitated by using an invariant geometrical description of the corresponding representations due to bros , epstein , and moschella @xcite ( sect . [ section : uir - lorentzgroup ] ) . our yang - baxter operator @xmath0 can hence be used to define left- and right half - local operator algebras for this model , as we show in a general framework in sect . [ section : crossing ] , and as concrete models in sect . [ subsection : chiral - models ] . if it could be shown that suitable intersections of such algebras are sufficiently large , then this would indeed correspond to ( a chiral half of ) a local 1 + 1 dimensional field theory describing , in some sense , the quantized left ( or right- ) movers of a non - linear sigma model with non - compact target space . the group @xmath1 is not just the isometry group of @xmath21-dimensional desitter spacetime @xmath18 , but also the conformal isometry group of @xmath22-dimensional _ euclidean _ flat space @xmath3 . this dual role becomes geometrically manifest if one attaches a pair of conformal boundaries @xmath23 to @xmath18 . each of these boundaries is isometric to a round sphere @xmath24 , which in turn may be viewed as a 1-point conformal compactification of @xmath3 via the stereographic projection . the induced action of @xmath1 by this chain of identifications provides the action of the conformal group on @xmath3 . this well - known correspondence is at the core of the so - called `` ds / cft - correspondence / conjecture '' @xcite . the idea behind this correspondence is that with the pair of infinities @xmath23 there is associated a pair of euclidean conformal field theories . one way to see this is that unitary representations of @xmath1 do not correspond to unitary representations of @xmath25 via `` analytic continuation '' @xcite . ] acted upon by @xmath1 . on the other hand , with the `` bulk '' @xmath18 , there is associated a corresponding `` string - theory '' with internal symmetry group @xmath1 . the action of the group essentially connects these two theories . inspired by this circle of ideas , one might be tempted to ask whether one can , in our setup , also naturally construct a euclidean conformal field theory on @xmath3 associated with our sigma models with target space @xmath18 . we will demonstrate in the course of this article that this is indeed the case . in our approach , the core datum is a 2-body scattering matrix / yang - baxter operator @xmath0 . in section [ euclqft ] we will outline an abstract procedure how to obtain a corresponding euclidean conformal field theory from such an object . thus , within our framework , there is a sense in which the essentially algebraic quantity @xmath0 can relate a euclidean conformal field theory in @xmath22 dimensions and a kind of `` string theory '' in @xmath21-dimensional desitter target space , and thereby gives a model for the ds / cft correspondence . that correspondence manifests itself more concretely as follows in our construction . if @xmath26 is a lightray coordinate , the left - local chiral fields of the lightray cft are given by [ field1 ] _ r^chir.(u , x ) = _ p , p \{e^iup(xp)^--iz^_r(p , p ) + } , where @xmath27 is a point in desitter space and @xmath28 are `` desitter waves '' of `` momentum '' @xmath29 analogous to plane waves in minkowski spacetime . the creation operators @xmath30 create a `` particle '' of lightray - rapidity @xmath31 and desitter `` momentum '' @xmath29 and obey a generalized zamolodchikov - faddeev algebra , @xmath32 , where @xmath33 is our @xmath0-operator - operator . ] . _ , scaledwidth=95.0% ] the desitter momentum @xmath29 is an element of the projective lightcone @xmath34 . it can be identified with a point @xmath35 in @xmath22-dimensional euclidean space @xmath36 , as depicted in the figure above . this `` celestial sphere '' is identified with @xmath37 . if we restrict @xmath38 to @xmath19 discrete values @xmath39 and set @xmath40 under the correspondence @xmath35 , we can also define a multiplet of @xmath19 _ euclidean _ quantum fields on @xmath3 as [ field2 ] ^eucl._r , j ( ) = z^_r , j ( ) + z_r , j ( ) , j=1 , the idea is that the desired correspondence ( duality ) maps the states created by the operators @xmath41 to the states created by the operators @xmath42 in the `` continuum limit '' @xmath43 . the fields at past null infinity @xmath44 are related by a tcp operator @xmath45 which is introduced in the main text . the main input into all our constructions is a solution of the yang - baxter equation ( ybe ) with additional symmetries . in this section , we consider solutions to the ybe that are compatible with a representation @xmath46 of a group @xmath47 and an associated conjugation @xmath48 . in our subsequent construction of field - theoretic models , we will be interested in the case where @xmath49 , @xmath46 is an irreducible ( spin-0 , principal , complementary or discrete series ) representation of it , and @xmath48 the corresponding tcp operator . in order to compare with @xmath2 sigma models and related constructions , we introduce in section [ subsection : invariant - yb - ops ] yang - baxter operators and functions in the general context of a unitary representation of an arbitrary group @xmath47 , and discuss some examples . the relevant aspects of the representation theory of @xmath50 are recalled in section [ section : uir - lorentzgroup ] in a manner suitable for our purposes , and the connection to the klein - gordon equation on de sitter space is recalled in section [ subsection : klein - gordon ] . these representations are then used in section [ section : ybops - lorentzgroup ] , where examples of invariant yang - baxter operators for the lorentz group are presented . in the following , a _ conjugation _ on a hilbert space means an antiunitary involution , and the letter @xmath51 is reserved for the flip @xmath52 , @xmath53 , on the tensor square of a hilbert space @xmath54 . when it is necessary to emphasize the space , we will write @xmath55 instead of @xmath51 . for identities on various spaces , we write @xmath56 and only use more specific notation like @xmath57 or @xmath58 where necessary . [ definition : invariant - yb - operator ] let @xmath47 be a group , @xmath46 a unitary representation of @xmath47 on a hilbert space @xmath54 , and @xmath48 a conjugation on @xmath54 . 1 . an _ invariant yang - baxter operator _ ( for @xmath59 ) is an operator @xmath60 such that 1 . @xmath0 is unitary . 2 . @xmath61=0 $ ] for all @xmath62 . 3 . @xmath63=0 $ ] . 4 . @xmath64 as an equation in @xmath65 ( i.e. , with @xmath66 ) . + the family of all invariant yang - baxter operators for a given representation @xmath46 and conjugation @xmath48 will be denoted @xmath68 . an _ invariant yang - baxter function _ ( for @xmath46 , @xmath48 ) is a function @xmath69 such that for almost all @xmath70 , * @xmath71 is unitary . * @xmath72=0 $ ] for all @xmath62 . * * @xmath74 as an equation in @xmath65 ( i.e. , with @xmath66 ) . * @xmath75 . + the set of all invariant yang - baxter functions will be denoted @xmath76 . our main interest is in invariant yang - baxter _ operators _ , the yang - baxter functions serve as an auxiliary tool to construct them . independent of @xmath77 , the four operators @xmath78 are always elements of @xmath68 ; these are the trivial unitaries satisfying the constraints ( r1)(r5 ) . the structure of @xmath68 and @xmath76 depends heavily on the representation @xmath46 and group @xmath47 , as we will see later in examples . it is clear from the definition that any ( say , continuous ) function @xmath79 defines an invariant yang - baxter operator @xmath80 , and any operator @xmath81 defines a ( constant ) yang - baxter function @xmath82 . furthermore , any invariant yang - baxter function defines an invariant yang - baxter operator on an enlarged space . this is spelled out in the following construction , which we will use later on in the context of our qft models . [ lemma : ybf->ybo ] let @xmath79 ( for some group @xmath47 , on some hilbert space @xmath54 ) , and consider the enlarged hilbert space @xmath83 , with @xmath47-representation @xmath84 and conjugation @xmath85 . on @xmath86 , define the operator @xmath87 then @xmath88 . the proof of this lemma amounts to inserting the definitions and is therefore skipped . see @xcite for similar results . for later use , we mention that we can also work with a different measure @xmath89 than lebesgue measure @xmath90 in this construction . for example , we can take a finite number @xmath19 of point measures , located at @xmath91 . in that case , @xmath92 , with orthonormal basis @xmath93 , invariant under the conjugation @xmath94 . on vectors @xmath95 , our invariant yang - baxter operators then take the form @xmath96 this construction can also be seen as an example of the partial spectral disintegration formulas considered in @xcite . before presenting examples , we recall how an invariant yang - baxter operator @xmath0 gives rise to an @xmath0-symmetric fock space , following @xcite . in this fock space construction , we consider an invariant yang - baxter operator @xmath0 and call its group representation , conjugation , and hilbert space @xmath97 , @xmath98 , and @xmath99 , as these data enter on the one particle level . as is well known , solutions of the yang - baxter equation ( r4 ) induce representations of the braid group of @xmath8 strands on @xmath100 , by representing the elementary braid @xmath101 , @xmath102 , by @xmath103 . this representation factors through the permutation group because of ( r5 ) , i.e. we have a representation @xmath104 of the symmetric group @xmath105 on @xmath8 letters on @xmath100 . since @xmath0 is unitary ( r1 ) , so are the representations @xmath104 . we denote by @xmath106 the subspace on which @xmath107 acts trivially , i.e. @xmath108 in view of ( r2 ) , the representation @xmath109 of @xmath47 on @xmath100 commutes with the projection @xmath110 , and hence restricts to @xmath111 . we denote this restriction by @xmath112 . furthermore , we define a conjugation @xmath113 on @xmath100 by @xmath114 it is clear that @xmath113 is a conjugation on @xmath100 , and thanks to ( r3 ) , it commutes with @xmath110 and thus restricts to @xmath115 @xcite . we call this restriction @xmath116 . the @xmath0-symmetric fock space over @xmath99 is then defined as @xmath117 we denote depend on @xmath0 . ] its fock vacuum by @xmath118 , the resulting `` @xmath0-second quantized '' representation of @xmath47 by @xmath119 , and the resulting conjugation by @xmath120 . @xmath0-symmetric fock spaces generalize the usual bose / fermi fock spaces , which are given by the special cases @xmath121 . for our purposes , the @xmath0-symmetric spaces ( for non - trivial @xmath0 ) will be convenient representation spaces for our models . we next give some examples of invariant yang - baxter operators and functions . * example 1 : @xmath122 . * we consider the group @xmath123 in its defining representation @xmath46 on @xmath124 , with complex conjugation in the standard basis as conjugation . this is a typical finite - dimensional example , which appears in particular in the context of the @xmath2 sigma models . it is known from classical invariant theory that the @xmath2-invariance constraint ( r2 ) allows only three linearly independent solutions : the identity @xmath56 of @xmath125 , the flip @xmath51 , and a one - dimensional symmetric projection @xmath126 ( * ? ? ? one can then check that ( r1)(r5 ) together only allow for trivial solutions , i.e. @xmath127 . however , non - trivial yang - baxter _ functions _ @xmath128 do exist . a prominent example is @xmath129 which satisfies not only ( r1)(r5 ) , but also the analytic properties ( r6 ) , ( r7 ) that will be introduced in section [ section : crossing ] . this yang - baxter function describes the @xmath2-invariant two - body s - matrix of the @xmath2-sigma model @xcite . * example 2 : the `` @xmath130 '' group @xmath131 inner symmetries . * as an example of a different nature , we consider `` @xmath132 '' group , i.e. the affine group @xmath133 generated by translations @xmath134 and dilations @xmath135 on the real line @xmath136 . the physical interpretation is to view @xmath136 as a lightray , which describes one chiral component of a massless field theory on two - dimensional minkowski space . the group @xmath133 has a unique unitary irreducible representation @xmath137 in which the generator of the translations is positive . we may choose @xmath138 as our representation space , and then have , @xmath139 , @xmath140 where the variable @xmath141 can be thought of as being related to the ( positive ) light like momentum @xmath142 by @xmath143 . the conjugation @xmath144 extends this representation to also include the reflection @xmath145 on the lightray . in this example , the invariant yang - baxter operators @xmath146 can all be computed , and in contrast to the @xmath2 case , many such operators exist . the physically interesting ones are given by multiplication operators of the form @xmath147 as in , where @xmath148 is a scalar function satisfying @xmath149 such `` scattering functions '' include for example the two - body s - matrix of the sinh - gordon model , which is @xcite @xmath150 where @xmath151 is a function of the coupling constant . to generalize to a setting with inner symmetries , we can also , instead of @xmath133 alone , take the direct product @xmath152 of @xmath133 with an arbitrary group @xmath47 , which is thought of as a global gauge group . we then consider a unitary representation @xmath46 of @xmath47 on an additional hilbert space @xmath54 , and form the direct product representation @xmath153 on @xmath154 , i.e. @xmath155 for later application , we stress that @xmath54 can still be infinite - dimensional , as it is the case for the irreducible representations of @xmath49 . to also have a tcp operator in this extended setting , we assume that there exists a conjugation @xmath48 on @xmath54 that commutes with @xmath46 ( i.e. , @xmath46 must be a self - conjugate representation ) , and then consider @xmath156 as tcp operator for @xmath157 . under mild regularity assumptions , one can then show that essentially all invariant yang - baxter operators @xmath158 are again of the form @xmath159 . that is , @xmath160 acts by multiplying with an ( operator - valued ) function @xmath0 , and this function @xmath0 has to exactly satisfy the requirements ( r1)(r5 ) . in particular , @xmath71 commutes with @xmath161 for all @xmath162 , @xmath62 . this example is therefore quite different from the previous @xmath2-example : many invariant yang - baxter operators exist , and they are essentially all given by invariant yang - baxter functions via . both examples can be combined by taking the inner symmetry group as @xmath123 , as one would do for describing the @xmath2-models @xcite . the construction just outlined here can be used to describe the one - particle space of a ( chiral component of ) a massless sigma model with symmetry group @xmath47 . to prepare our construction of such models for @xmath49 , we review some representation theory of this group next . we now turn to the case of central interest for this article , the ( proper , orthochronous ) lorentz group @xmath49 , @xmath164 . in later sections , this group will appear either as the isometry group of @xmath21-dimensional de sitter space @xmath165 or as the conformal group of @xmath3 . in this section , we first give a quick tour dhorizon of some of its representation theory . readers familiar with this subject can skip to the next section . our exposition is in the spirit of @xcite , @xcite ( and references therein ) , and we will use the following notation : capital letters @xmath166 etc . denote points in minkowski space @xmath167 . the dot product of this minkowski spacetime is defined with mostly minuses in this paper , x y = x_0 y_0 - x_1 y_1 - - x_d y_d . points in @xmath3 are denoted by boldface letters , @xmath168 , and their euclidean norm is written as @xmath169 . unitary irreducible representations ( uirs ) of @xmath50 are classified by a continuous or discrete parameter corresponding roughly to the `` mass '' in the minkowski context , and a set of spins corresponding to the @xmath170 casimirs of @xmath171 . in this paper we will only consider the case of zero spin , there is no spin , and our representations exhaust all possibilities , see @xcite . ] and `` principal- '' , `` complementary- '' and `` discrete series '' representations . there are many unitarily equivalent models for these representations in the literature , see e.g. @xcite . the most useful description for our purposes is as follows . first define the future lightcone c_d^+ = \ { p ^d+1 p p = 0 , p_0 > 0 } in @xmath172-dimensional minkowski space . we think of @xmath173 as a ( redundant ) version of momentum space in the minkowski context . on @xmath173 , consider smooth @xmath174-valued `` wave functions '' @xmath175 which are homogeneous , [ hom ] ( p ) = ^- -i(p ) , where at this stage , @xmath176 is arbitrary . as the fraction @xmath177 will appear frequently , we introduce the shorthand @xmath178 . the collection of these wave functions forms a complex vector space which we will call @xmath179 . a linear algebraic representation of @xmath180 is defined by pullback , [ udef ] v _ ( ) : _ _ , ( p ) : = ( ^-1 p ) . in order for this to define a unitary representation , we must equip @xmath179 with an invariant ( under @xmath181 ) positive definite inner product . it turns out that this is possible only for certain values of @xmath182 . these are , see below . ] : 1 . ( principal series ) @xmath183 . ( complementary series ) @xmath184 . ( discrete series ) @xmath185 . two complementary or discrete series representations @xmath186 , @xmath187 are inequivalent for @xmath188 , and two principal series representations @xmath186 and @xmath187 are equivalent if and only if @xmath189 . we now explain what the inner products are in each case . as a preparation , consider first the @xmath21-form @xmath190 and vector field @xmath191 on @xmath173 defined by = , = p_0 + @xmath190 is the natural integration element on the future lightcone , and @xmath191 the generator of dilations . both are invariant under any @xmath180 , i.e. @xmath192 . we then form @xmath193 where @xmath194 is cartan s operator contracting the upper index of the vector @xmath191 into the first index of the @xmath21-form @xmath190 . a key lemma which we use time and again is the following ( * ? ? ? * lemma 4.1 ) : [ lemma : closedform ] suppose @xmath195 is a homogeneous function on the future lightcone @xmath173 of degree @xmath196 . then @xmath197 is a closed @xmath22-form on @xmath173 , @xmath198 . using this lemma , we can now describe the inner products . + * a ) principal series : * here the degree of homogeneity of the wave functions is @xmath199 with @xmath182 real . consequently , the product @xmath200 of two smooth wave functions @xmath201 is homogeneous of degree @xmath196 , so the lemma applies . we choose an `` orbital base '' r0.3 + @xmath202 of @xmath173 ( i.e. a closed manifold intersecting each generatrix of @xmath173 once , see figure on the right , and appendix a for explicit formulas ) , and define a positive definite inner product by [ inn ] ( _ 1 , _ 2(p ) . by lemma [ lemma : closedform ] , this definition is independent of the particular choice of @xmath203 , in the sense that , if @xmath204 is homologous to @xmath203 , then the inner product defined with @xmath204 instead of @xmath203 coincides with . this fact implies at once that the operators @xmath181 are unitary with respect to this inner product . since the type of argument is used time and again , we explain the details : @xmath205 the first equality sign is the definition . in the second equality , it is used that @xmath206 , and in the third equality , a change of variables @xmath207 was made . in the last step we used stokes theorem , noting that the integrand is a closed form , and that @xmath203 and @xmath208 are homologous . the hilbert space of the principal series representation is defined as the completion in the inner product of the space @xmath179 , which we denote by the same symbol . + + pointwise complex conjugation does not leave @xmath179 invariant ( unless @xmath209 ) because the degree of homogeneity is complex , and conjugation changes @xmath182 to @xmath210 . since @xmath211 is real , this implies that @xmath212 is an antiunitary map @xmath213 , intertwining @xmath186 and @xmath214 . to compare @xmath186 and @xmath214 , it is useful to introduce the integral operator [ idef1 ] ( i_)(p ) : = ( 2)^- _ b ( p ) ( p p)^-+ i ( p),_. if @xmath215 , the poles of the gamma function are avoided . the integrand has a singularity at @xmath216 which is integrable if @xmath217 ( see appendix a for explicit formulas arising from particular choices of @xmath203 ) . thus @xmath218 is well - defined as it stands in particular for @xmath219 , corresponding to the case of a complementary series representation , to be discussed below . for the principal series representations , @xmath182 is real , and we may define by replacing @xmath182 with @xmath220 and then taking the limit @xmath221 ( as a distributional boundary value , see for example ( * ? ? ? * chap . 3 ) ) . this adds a delta function term and yields a well - defined operator @xmath218 for @xmath222 . finally , for @xmath209 , one has to take into account the gamma factors in when performing the limit @xmath223 . in this case , one obtains @xmath224 ( this follows from the delta function relation in appendix b ) . after these remarks concerning the definition of @xmath218 , note that for @xmath225 , the value of the integral does not depend on our choice of orbital base @xmath203 , because the integrand clearly has homogeneity @xmath196 in @xmath226 , and is thus a closed form by lemma [ lemma : closedform ] . [ lemma : tpc - principal ] 1 . in the principal series case ( @xmath183 ) , @xmath227 is a unitary intertwining @xmath186 and @xmath214 , i.e. [ intw0 ] v_- ( ) i_= i_v_(),so^(d,1 ) . furthermore , there holds @xmath228 and @xmath229 . 2 . each principal series representation @xmath186 is selfconjugate : @xmath230 is a conjugation on @xmath179 commuting with @xmath186 . @xmath231 by using the invariance @xmath232 and the same type of argument as given in eqs . , @xmath218 is seen to have the intertwining property . the equality @xmath233 follows by a routine calculation . to show @xmath234 ( for @xmath235 ) , it is useful to choose a convenient parameterization of @xmath203 . using the spherical parameterization ( see appendix a ) , the identity @xmath234 follows by application of the composition relation . the special case @xmath224 has been explained above already . @xmath236 it is clear that @xmath237 is an antiunitary operator on @xmath179 . as complex conjugation @xmath238 satisfies @xmath239 , one also sees that @xmath237 is an involution , i.e. @xmath240 . * b ) complementary series : * here the homogeneity of the wave functions is @xmath241 , with @xmath242 , so @xmath182 is _ imaginary_. in this case , we can not apply the same procedure as in the case of the principal series to form a scalar product , because the product @xmath243 of two smooth wave functions @xmath201 is not homogeneous of degree @xmath196 , and consequently , lemma [ lemma : closedform ] does not apply . to get around this problem , we can use the operator @xmath244 , which is well - defined also for @xmath219 . as in the case of the principal series , also here the integral operator @xmath218 has the intertwining property @xmath245 for all @xmath180 . as @xmath182 is imaginary , the intertwining operator @xmath218 ensures that the function @xmath246 on @xmath247 formed from two wave functions @xmath248 is homogeneous of degree @xmath196 , and lemma [ lemma : closedform ] shows that [ idef ] ( _ 1 , _ 2 ) _ : = _ b ( p ) ( i__1)(p ) is again independent of the choice of orbital base @xmath203 . the same argument as that given in eq . then also yields that the inner product just defined is invariant under the representation @xmath186 . for the complementary series , we therefore take as our inner product . we have : for @xmath249 , the inner product is positive definite . since we are free to choose any orbital base in , , we can make a convenient choice . if we choose the spherical model , @xmath202 ( see appendix a for the different canonical models ) then our lemma reduces to lemma 5.5 of @xcite . the proof is however more transparent choosing the flat model , where @xmath250 is parametrized by @xmath251 . using this parametrization , we find ( , ) _ & = c d^d-1 _ 1 d^d-1 _ 2 |_1 - _ 2|^-(d-1)+2i ( _ 2 ) + & = c d^d-1 | |^-2i |()|^2 0 . here @xmath252 are positive numerical constants . in the second line we used the plancherel theorem and a well - known formula for the fourier transform of @xmath253 ( see e.g. ex . vii 7.13 of @xcite ) . taking into account standard properties of the gamma function , we see that the prefactor of the integral is positive if @xmath219 . in the complementary series , the degree of homogeneity is real , and thus complex conjugation is a well - defined operation on @xmath179 . moreover , complex conjugation commutes with @xmath218 for imaginary @xmath182 . thus , if we define @xmath254 , then @xmath237 is an antiunitary involution on @xmath179 in the case of the complementary series . [ lemma : tcp - complementary ] each complementary series representation is selfconjugate : @xmath254 is a conjugation commuting with @xmath186.@xmath255 * c ) discrete series : * here the degree of homogeneity of the wave functions is @xmath199 with @xmath256 . for these values , the gamma - factors in the definition of @xmath218 ( see ) , and hence also in the inner product of the complementary series become singular . thus , one can not , for this reason alone , define an inner product for the discrete series by analytic continuation of . the way out is to pass from @xmath179 to an @xmath50-invariant subspace of wave functions for which the scalar product _ can _ be defined by analytic continuation . since the kernel of @xmath218 is , up to divergent gamma - factors , given by @xmath257 for the discrete series , a natural choice for this subspace is the set of @xmath258 such that [ discr ] ( p ) = ^-(d-1)-n ( p ) , _ b ( p ) ( p p)^n ( p ) = 0 , where the first equality just repeats the homogeneity condition for the case @xmath259 . the second condition is independent of the choice of @xmath203 , and hence indeed @xmath50-invariant . by abuse of notation , we denote the set of @xmath175 satisfying again by @xmath179 . for such @xmath175 , analytic continuation of , to @xmath260 is now possible . since the residue of @xmath48 at @xmath261 is @xmath262 , we find @xmath263 using , one again verifies that the definition of @xmath218 remains independent of @xmath203 , and therefore , by the same argument as already invoked several times , that the inner product is invariant under @xmath181 . since analytic continuation does not usually preserves positivity , it is non - trivial , however , that this inner product is actually positive definite @xcite . for @xmath264 , the inner product is positive definite . since we are free to choose any orbital base in , we can make a convenient choice . we choose the spherical model , @xmath202 ( see appendix a for the different canonical models ) , where @xmath265 , with @xmath266 . we have the series , for @xmath267 ( -1)^n+1 ( 1-x)^n ( 1-x ) = _ m > n x^m . for @xmath268 , this series is absolutely convergent , including the limit as @xmath269 , and all its coefficients are evidently positive . we apply this identity to @xmath270 in the inner product . exchanging integration and summation it follows that ( , ) _ = _ m > n a_m ^2 0 , where @xmath271 is the rank @xmath272 tensor on @xmath273 given by @xmath274 and @xmath275 denotes the norm of such a tensor inherited from the euclidean metric on @xmath273 . the integral form of the triangle inequality gives @xmath276 , so the series is absolutely convergent , meaning that exchanging summation and integration was permissible . the case @xmath277 can be treated e.g. using the flat model and applying a fourier transform we omit the details . as in the complementary series , the degree of homogeneity is real , and thus complex conjugation is a well - defined operation on @xmath179 . moreover , complex conjugation commutes with @xmath218 for imaginary @xmath182 . thus , if we define @xmath254 , then @xmath237 is an antiunitary involution on @xmath179 in the case of the discrete series . this finishes our outline of the representations . in conclusion , we mention that the conjugation @xmath237 , [ thdef ] ( _ ) ( p ) = & + & i(0 , ) or i+ _ 0 , can be interpreted as a tcp operator , at least on the superficial level that it is an antiunitary involution and commutes with the representation , as the reflection @xmath278 on @xmath165 commutes with @xmath50 . these properties are clearly still satisfied if we multiply @xmath237 by a phase factor ( as we shall do in later sections ) . with the help of @xmath237 , one can therefore extend @xmath186 to include the reflection @xmath278 . , we could in fact extend @xmath186 to a ( pseudo - unitary ) representation of the full lorentz group @xmath279 . ] in this short section we recall the relation between the principal and complementary series representations @xmath186 and classical and quantum klein - gordon fields on @xmath21-dimensional desitter spacetime @xmath18 . for our purposes , this space is best defined as the hyperboloid _ d = \ { x ^d+1 x x = -1 } embedded in an ambient @xmath172-dimensional minkowski spacetime @xmath167 . the metric of @xmath165 is simply ( minus ) that induced from the ambient minkowski space . it is manifest from this definition that the group of isometries of @xmath165 is @xmath279 , where group elements act by @xmath281 . on testfunctions @xmath282 , orthochronous lorentz transformations act according to @xmath283 , and we choose an antilinear action of the full spacetime reflection @xmath284 by @xmath285 . the precise relationship between these transformations on @xmath165 and the `` momentum space '' representations @xmath186 from the previous section is , as in flat space , via a special choice of `` plane wave mode functions '' . these mode functions are defined as follows . let @xmath29 be any vector in @xmath173 , choose any time - like vector such as @xmath286 in the ambient @xmath167 , and define @xmath287^{-\alpha - i\nu } \ . \end{aligned}\ ] ] adding a small imaginary part removes the phase ambiguity for @xmath288 when @xmath289 becomes negative . the limit is understood in the sense of a ( distributional ) boundary value . the difference between the `` @xmath290 '' ( `` positive frequency '' ) and `` @xmath291 '' ( `` negative frequency '' ) mode arising when we cross to the other poincar patch is basically a phase . the modes @xmath292 are ( distributional ) solutions to the klein - gordon equation in @xmath27 with mass @xmath293 on the _ entire _ desitter manifold , ( + m^2 ) u^_p = 0 . conversely , if @xmath258 is smooth , then the corresponding `` wave packet '' u_^(x ) = _ b ( p ) ( p)(x p)^--i_is a globally defined , smooth solution to the kg - equation . to make contact with the representations @xmath186 , we define for @xmath282 @xmath294 where @xmath295 is the @xmath279-invariant integration element on @xmath165 ( fourier - helgason transformation ) . with these definitions , we have the following lemma . [ lemma : fourier - helgason ] let @xmath282 , and @xmath296 . then 1 . @xmath297 , 2 . for @xmath298 , there holds @xmath299 . @xmath300 , where @xmath301 is the phase factor @xmath302 @xmath231 the desitter wave is homogeneous of degree @xmath303 in @xmath29 . @xmath236 follows directly from the invariance of @xmath190 . for @xmath304 , we first note @xmath305 . for the complementary series , we have @xmath306 which implies the result in this case . for the principal series , we first recall ( * ? ? ? * lemma 4.1 ) @xmath307 together with , this gives @xmath308 and the claimed result follows . we briefly indicate how these facts can be used to define a covariant klein - gordon quantum field on @xmath165 : denoting by @xmath309 the canonical ccr operators on the fock space over @xmath179 ( this is the special case of the `` @xmath0-twisted '' fock space of section [ subsection : invariant - yb - ops ] given by taking @xmath0 as the tensor flip ) , we define @xmath310 or in the informal notation explained in more detail below in sect . [ section : qft ] @xmath311 . \end{aligned}\ ] ] it then follows immediately that the field @xmath312 is a solution of the klein - gordon equation @xmath313 , which is real , @xmath314 . it transforms covariantly under the second quantization of @xmath186 , namely @xmath315 . furthermore , taking as tcp operator @xmath316 , we have @xmath317 , and hence also the tcp symmetry @xmath318 . we now take the lorentz group @xmath49 in one of the representations @xmath186 from section [ section : uir - lorentzgroup ] , and associated conjugation @xmath237 , and ask for the invariant yang - baxter operators and functions @xmath319 and @xmath320 , analogously to the examples for @xmath123 and @xmath321 considered in section [ subsection : invariant - yb - ops ] . our construction will be clearest in a slightly more general setting : instead of a single representation , we consider two representations @xmath322 , @xmath323 from either the principal , complementary or discrete series , i.e. @xmath324 , and then construct an operator @xmath325 intertwining @xmath326 with @xmath327 . this operator @xmath328 will be an integral operator with distributional kernel @xmath329 , [ rdef ] ( r^_1_2^_1_2)(p_1 , p_2 ) : = _ b b ( p_1 ) ( p_2 ) r^_1_2(p_1 , p_2 ; p_1 , p_2 ) ^_1_2(p_1,p_2 ) . the degrees of homogeneity @xmath330 of the kernel in its four variables @xmath331 will be @xmath332 for @xmath333 in the principal or complementary series , this implies immediately that the integrand of is homogeneous of degree @xmath196 in both @xmath334 and @xmath335 , so that does not depend on the choice of orbital base @xmath203 by lemma [ lemma : closedform ] . furthermore , it follows that @xmath336 , @xmath337 , lies in @xmath338 , i.e. @xmath328 is a map @xmath339 . the same conclusions also hold when one of the parameters @xmath333 ( or both of them ) belong to the discrete series . here one has to check in addition that the constraint is preserved by the integral operator . this follows from the relation . the integral kernel will be taken of the form @xmath340 , where the exponents @xmath341 are complex numbers to be determined . this ansatz presumably does not really imply any serious loss of generality , because a general invariant kernel may be reduced to such expressions via a mellin - transform along the lines of @xcite . since it only contains lorentz invariant inner products , it follows immediately that the corresponding integral operator intertwines @xmath326 with @xmath327 whenever it is well - defined . imposing the above degrees of homogeneity in @xmath342 onto this kernel fixes the powers @xmath343 up to one free parameter , which we call @xmath344 . this then gives the integral kernels [ rdef - kernel ] r_(p_1,p_2 ; p_1,p_2 ) = & c__1 , _ 2 ( ) ( p_1 p_2)^-i- i_1 - i_2 ( p_1 p_1)^-+i+ i_1 - i_2 + & ( p_2 p_2)^-+ i - i_1 + i_2 ( p_1 p_2)^-i+ i_1 + i_2 . as in @xmath218 , there can be singularities whenever the two momenta in an inner product coincide , and the same regularization as discussed earlier is understood also here . the constant @xmath345 is taken to be @xmath346 and @xmath141 is taken to be real . with these definitions , we have [ theorem : r - ds ] let @xmath347 , @xmath162 be such that the poles in are avoided , and @xmath348 the integral operators defined above . then , @xmath70 , 1 . @xmath349 is unitary . @xmath350 for all @xmath351 . @xmath352 @xmath353 with @xmath354 the tensor flip . + @xmath352 @xmath355 4 . on @xmath356 , there holds @xmath357 5 . @xmath358 . in particular , whenever @xmath359 , the function @xmath360 is an invariant yang - baxter function , @xmath361 . there holds the normalization @xmath362 the proof of this theorem is given in appendix b. if we go to one of the canonical models for the orbital base @xmath203 ( described in appendix a ) , we get concrete formulas for @xmath363 and the kernel @xmath33 . in the case of the flat model , our expression for @xmath33 then coincides , up to a phase , with an expression derived previously @xcite ( see also @xcite ) for the case of the principal series representation . these authors also proved the yang - baxter equation ( r4 ) , and a version of the idempotency relation ( r5 ) . their formalism for finding @xmath33 is based on a different model for the representations . as explained before , the exponents @xmath341 are fixed by homogeneity requirements in the variables @xmath364 up to one remaining free parameter , which is @xmath365 . setting this parameter to zero ( as required for the yang - baxter equation ( r4 ) ) leads to a trivial solution . we thus conjecture that @xmath319 contains only trivial operators ( as in the @xmath2 case , example 1 ) . there do however exist many other invariant yang - baxter functions , because there are two operations we may carry out on the above integral operator without violating the properties ( r1)(r5 ) : scaling of @xmath141 and multiplication by suitable @xmath141-dependent scalar factors . [ proposition : tweak - r ] let @xmath366 be the integral operator defined by the kernel , @xmath367 , and @xmath368 a function satisfying @xmath369 then also @xmath370 satisfies ( r1)(r5 ) . the proof consists in a straightforward check of the conditions ( r1)(r5 ) , and is therefore omitted . we conjecture that the operators @xmath370 form essentially all solutions of the constraints ( r1)(r5 ) . the multipliers @xmath371 satisfy exactly the requirements on `` scalar '' yang - baxter functions . in particular , there exist infinitely many functions satisfying the requirements . the freedom of adjusting @xmath0 by rescaling the argument and multiplying with such a scalar function will be exploited in the next section . as explained in section [ subsection : invariant - yb - ops ] , an invariant yang - baxter operator @xmath372 ( def . [ definition : invariant - yb - operator ] ) gives rise to an @xmath0-symmetric fock space on which twisted second quantized versions @xmath59 of the representation @xmath97 and the conjugation @xmath98 act . these `` covariance properties '' are one essential aspect of the yang - baxter operators in our setting . the other essential aspect are locality properties , which are linked to specific analyticity requirements on @xmath0 . these analyticity properties , to be described below , have their origin in scattering theory , where they describe the relation between scattering of ( charged ) particles and their antiparticles @xcite . for the following general discussion , we first consider the conformal sigma models with some arbitrary inner symmetry group @xmath47 ( `` example 2 '' of section [ subsection : invariant - yb - ops ] ) , and later restrict to @xmath49 . our exposition is related to @xcite , where a scalar version of such models was presented , and @xcite , where a massive version with finite - dimensional representation @xmath97 was analyzed . we consider on the one hand the representation @xmath137 of the translation - dilation group @xmath133 of the lightray on @xmath138 , and the conjugation @xmath373 on that space . on the other hand , we consider an arbitrary group @xmath47 , given in a unitary representation @xmath46 with commuting conjugation @xmath48 on a hilbert space @xmath54 . our one - particle space is then @xmath374 , with the representation and the conjugation @xmath375 we pick an invariant yang - baxter function @xmath376 as the essential input into the following construction of quantum fields . these fields will be operators on the @xmath0-symmetric fock space @xmath377 over @xmath99 given by the invariant yang - baxter operator @xmath160 defined by @xmath0 via . the @xmath0-symmetric fock space carries natural creation / annihilation operators : with the help of the projections @xmath378 , we define as in @xcite , @xmath379 , @xmath380 with these definitions , @xmath381 is an annihilation operator ( in particular , @xmath382 ) , and @xmath383 is a creation operator ( in particular @xmath384 ) . it directly follows that , @xmath379 , @xmath62 , @xmath385 where @xmath386 denotes either @xmath387 or @xmath388 . to introduce a field operator on the lightray , we define in is motivated by the fact that we want to study a chiral field , which has an infrared singularity at zero momentum because of the divergence in the measure @xmath389 for @xmath390 . this problem is most easily resolved by passing to the current of this field , which amounts to taking derivatives of test functions . as these derivatives result in the prefactor @xmath391 , subsequent formulas will be easier if we include this factor from the beginning . ] for a test function @xmath392 @xmath393 in analogy to the fourier - helgason transforms , these functions lie in the representation space @xmath138 , and the definition is covariant under @xmath133 in the following sense : @xmath394 analogously , the tcp transformed function @xmath395 yields @xmath396 , where the minus sign is due to the fact that we consider the current . given any vector @xmath397 , we define the field operators ( cf . @xcite ) @xmath398 we may think of the vector @xmath399 as a label for the different `` components '' @xmath400 of the field @xmath401 . note , however , that @xmath54 can be infinite - dimensional , so that @xmath401 can be a field with infinitely many independent components . by proceeding to delta distributions @xmath402 , sharply localized at a point @xmath403 on the lightray , we may also describe this field in terms of the distributions @xmath404 . in the present general setting , one can show that the field operators transform covariantly under @xmath137 and @xmath46 , but not under the tcp operator @xmath45 . furthermore , @xmath405 on an appropriate domain . we do not repeat the calculations from @xcite here ( see , however , section [ section : qft ] for a concrete version of these properties ) , but rather focus on the locality aspects . to begin with , one realizes that @xmath401 is a non - local field unless @xmath406 , in which case it satisfies canonical commutation relation and reduces to a free field . for general @xmath0 , the locality properties of @xmath401 are best analyzed by introducing a second `` tcp conjugate '' field , @xmath407 which in its smeared version is , @xmath392 , @xmath408 this field shares many properties with @xmath401 , and in particular , also transforms covariantly under @xmath137 and @xmath46 . we next want to analyze the commutation relations between @xmath401 and @xmath409 . to this end , one first computes that the `` tcp conjugate creation operator '' acts on @xmath410 according to @xmath411 i.e. in comparison to the left action of @xmath383 , this operator `` creates from the right '' . in particular , @xmath412 , @xmath413&=0\,,\qquad [ z_r(\bxi),\,\theta z_r(\bxi')\theta]=0\,.\end{aligned}\ ] ] to control the mixed commutators @xmath414 $ ] , the following definition is essential . [ definition : crossing - symmetric - yb - operator ] an invariant yang - baxter function @xmath79 ( for some group @xmath47 , on some hilbert space @xmath54 ) is called _ crossing - symmetric _ if it satisfies the following two conditions . * @xmath415 extends to a bounded analytic function on the strip @xmath416 . * there holds the crossing symmetry condition , @xmath162 , @xmath417 in case that @xmath418 is finite - dimensional , these requirements coincide with the corresponding ones in ( * ? ? ? 2.1 ) , where the conjugation was taken as @xmath419 , @xmath420 , with the components @xmath421 referring to a fixed basis and @xmath422 an involutive permutation of @xmath423 . the significance of ( r6)(r7 ) is best explained in terms of tomita - takesaki modular theory ( see , for example , @xcite ) , as we shall do now . note , however , that in the explicit examples to be considered in section [ section : qft ] , we will also give a purely field - theoretic formulation ( theorem [ theorem : commutation - concrete ] ) . for the following argument , we adopt the concept of `` modular localization '' @xcite . the main idea is to anticipate a quantum field theory , defined in terms of a system of local von neumann algebras @xcite , and the connection between the modular data of the algebra corresponding to the half line @xmath424 and the one - parameter group of dilations @xmath425 which leaves @xmath424 invariant , and the reflection @xmath426 , which flips @xmath424 into @xmath427 ( `` bisognano - wichmann property '' , @xcite ) . we define @xmath428 and view this as either an operator on @xmath138 , or on @xmath429 , where it just acts on the left factor . if @xmath392 is supported on the right , @xmath430 , the function @xmath431 has an @xmath432-bounded analytic continuation to the strip @xmath433 , with @xmath434 , @xmath162 . for @xmath435 supported on the left instead , @xmath436 , the same properties hold for @xmath437 ( * ? ? ? * lemma 4.1 ) . in terms of the operator @xmath438 , this means @xmath439 the conditions ( r6)(r7 ) imply that @xmath0 has matching analyticity properties . namely , the matrix elements of @xmath440 are , @xmath441 , @xmath442 and therefore analytically continue to @xmath443 in view of ( r6 ) , with the boundary value @xmath444 these relations imply the following theorem . [ theorem : commutation ] let @xmath445 . then , in the sense of distributions , @xmath446=0\quad\text{for}\quad u < u'\ , , \end{aligned}\ ] ] on the space of vectors of finite fock particle number . the proof follows the same strategy as @xcite , generalized in @xcite . we introduce as a shorthand the vector - valued functions @xmath447 , @xmath448 , with field operators @xmath449 and @xmath450 . here @xmath451 are scalar testfunctions , and to prove the theorem , we have to demonstrate @xmath452=0 $ ] for @xmath430 , @xmath436 . we pick @xmath453 and compute using , , and the definitions @xmath454\,{\boldsymbol{\varphi}}\rangle & = \langle{\boldsymbol{\psi}},\,\left([{z^{\dagger}}_r({{\boldsymbol{g}}}^+),\theta z_r(\bof^-)\theta]+[z_r(\theta{{\boldsymbol{g}}}^-),\theta{z^{\dagger}}_r(\theta\bof^+)\theta]\right)\,{\boldsymbol{\varphi}}\rangle \nonumber \\ & = \langle\langle{{\boldsymbol{g}}}^+,{\boldsymbol{\psi}}\rangle{\omega},\theta\langle \bof^-,\theta{\boldsymbol{\varphi}}\rangle{\omega}\rangle -2\langle p_2^r({\boldsymbol{\psi}}{\otimes}\theta\bof^-),p_2^r({{\boldsymbol{g}}}^+{\otimes}{\boldsymbol{\varphi}})\rangle \nonumber \\ & \quad+ 2\langle p_2^r(\theta{{\boldsymbol{g}}}^-{\otimes}{\boldsymbol{\psi}}),p_2^r({\boldsymbol{\varphi}}{\otimes}\bof^+)\rangle -\langle \theta\langle \theta\bof^+,\theta{\boldsymbol{\psi}}\rangle{\omega},\langle \theta{{\boldsymbol{g}}}^-,{\boldsymbol{\varphi}}\rangle{\omega}\rangle \nonumber \\ & = \langle \theta{{\boldsymbol{g}}}^-{\otimes}{\boldsymbol{\psi}},\,\boldsymbol{r}({\boldsymbol{\varphi}}{\otimes}\bof^+)\rangle - \langle{\boldsymbol{\psi}}{\otimes}\theta\bof^-,\,\boldsymbol{r}({{\boldsymbol{g}}}^+{\otimes}{\boldsymbol{\varphi}})\rangle \ , . \label{eq : mixed - comm } \end{aligned}\ ] ] to show that these terms cancel in case @xmath0 satisfies ( r6 ) and ( r7 ) , we consider the first term in , and insert an identity @xmath455 , @xmath456 , in front of @xmath160 . in view of the invariance ( r2 ) of @xmath160 , we then see that this scalar product equals @xmath457 the vectors in the left and right hand entry of the scalar product are analytic in the strip @xmath443 ( taking into account the antilinearity of the left factor , and the fact that @xmath458 in the right factor ) . as explained above , the same analyticity holds for the operator - valued function @xmath459 . taking into account the boundary values @xmath460 and , we see that coincides with @xmath461 which is identical to the second term in . thus the matrix elements of @xmath462 $ ] between single particle states vanish . using the same arguments as in @xcite , one shows analogously that matrix elements between arbitrary vectors of finite particle number vanish . we may interpret this commutation theorem by regarding @xmath463 as localized in the left halfline @xmath464 , and @xmath465 as localized in the right halfline @xmath466 . this interpretation is consistent with both , covariance and locality . the fields @xmath401 , @xmath409 should however not be regarded as the `` physical '' quantum fields of the model , but rather as auxiliary objects @xcite . to proceed to the physical observables / fields , localized in finite intervals on the lightray , it is helpful to use an operator - algebraic setting . by construction , the field operators satisfy ( on a suitable domain ) @xmath467 if @xmath468 . one can show by the same method as in @xcite that @xmath469 is then essentially selfadjoint . in this case , we can form the unitaries @xmath470 by the functional calculus , and pass to the generated von neumann algebra @xmath471 to conclude the present general section , we point out a few further properties of the algebra @xmath472 , including haag duality with the algebra generated by the reflected field . 1 . the fock vacuum @xmath473 is cyclic and separating for @xmath472 . the modular conjugation of @xmath474 is @xmath475 , and the modular operator is the previously defined @xmath438 . 3 . the commutant of @xmath472 is @xmath476 @xmath231 it follows by standard arguments that @xmath473 is cyclic for @xmath472 and @xmath477 @xcite . as in @xcite , one shows by an analytic vector argument that the unitaries @xmath478 and @xmath479 commute for @xmath430 , @xmath436 . hence @xmath480 , and @xmath473 is also separating for @xmath472 ( and @xmath481 ) . @xmath304 is a straightforward consequence of @xmath236 by tomita - takesaki theory and the definitions . the proof of @xmath236 follows by the same line of arguments as in @xcite . in this section we construct examples of crossing symmetric yang - baxter functions @xmath482 for the @xmath50 representations and conjugations @xmath186 , @xmath237 from section [ section : ybops - lorentzgroup ] . this will be done by exploiting the freedom to adjust our basic @xmath50-invariant yang - baxter function @xmath0 by scaling the parameter @xmath141 and multiplying by a suitable function of @xmath141 ( proposition [ proposition : tweak - r ] ) . we define @xmath483 where @xmath484 as before , and @xmath485 with @xmath486 . @xmath487 is a real parameter chosen below depending on the representations . it is seen that the infinite product converges absolutely in each case considered . it is shown in appendix b that @xmath371 satisfies the requirements of prop . [ proposition : tweak - r ] . [ theorem : crossing ] let @xmath488 label two principal series representations , @xmath162 , and @xmath489 the integral operators defined above with @xmath490 in . then * @xmath491 extends to an analytic bounded function on the strip @xmath492 , and * there holds the crossing symmetry , for the boundary values : if @xmath493 , @xmath494 @xmath495 the same holds true if @xmath496 belong to a complementary or discrete series representation , and we set @xmath497 in . thus for all principal and complementary series representations , @xmath498 is crossing - symmetric in the sense of def . [ definition : crossing - symmetric - yb - operator ] . in appendix b , we also provide an alternative integral representation of the factor @xmath499 . in case we allow for arbitrary combinations of principal , complementary and discrete series representations @xmath333 , the ( rescaled ) gamma factors in can produce poles in the strip @xmath492 , which is the reason for our restriction to two principal series representations , or a single complementary resp . discrete series representation . the function @xmath371 is precisely constructed in such a way that the crossing relation ( r7 ) holds . there remains however a large freedom to modify @xmath371 without violating the conditions ( r1)(r7 ) . in fact , we may multiply @xmath500 by another function @xmath501 , which is analytic and bounded on the strip @xmath492 , and has the symmetry properties @xmath502 there exist infinitely many of such `` scattering functions '' , they are given by all inner functions @xmath503 of the strip @xmath492 with the symmetry properties @xmath504 . using the canonical factorization of inner functions ( see , for example , @xcite ) , one can then write down explicit formulas for @xmath503 . in particular , if @xmath503 contains no singular part , it has the form @xmath505 where @xmath506 , and the zeros @xmath507 have to satisfy @xmath508 and certain symmetry and summability conditions to ensure @xmath504 and convergence of the product @xcite . we now use the crossing symmetric @xmath50-invariant yang - baxter function @xmath509 to build concrete quantum field theoretic models . a first class of models will be constructed within the general setup of section [ section : crossing ] , where now the group is taken as @xmath49 , and we restrict ourselves to a principal or complementary series representation @xmath186 . in a second section , we show that by a variant of this construction , one also gets euclidean conformal field theories in @xmath22 dimensions . our first family of models is a concrete version of the abstract field operators in section [ section : crossing ] . the symmetry group is here the direct product @xmath510 of the translation - dilation group ( acting on lightray coordinates ) , and the lorentz group ( acting on de sitter coordinates ) . group elements will be denoted as @xmath511 in an obvious notation . this group is represented on @xmath512 by the representation @xmath513 as in section [ section : crossing ] , where @xmath186 may belong to the principal or complementary series . that is , the single particle vectors are scalar functions @xmath514 depending on two `` momentum coordinates '' , @xmath515 and @xmath516 , and @xmath517 as tcp operator on this space , we take @xmath518 , where @xmath238 denotes complex conjugation , @xmath237 is defined in and @xmath519 is the phase factor . we then consider the @xmath50-invariant crossing - symmetric yang - baxter function @xmath520 , which defines an @xmath521-invariant yang baxter operator @xmath522 as in section [ subsection : crossing - yb - ops ] . in order not to overburden our notation , we have dropped the tilde from @xmath0 , but still mean _ the rescaled and multiplied version from _ . following the general construction , we then obtain the @xmath0-symmetric fock space @xmath377 . an @xmath8-particle vector @xmath523 is here given by a function of @xmath524 momentum space variables , namely @xmath525 , which is square integrable in each @xmath526 and homogeneous of degree @xmath241 in each @xmath527 ( as well as square integrable on any orbital base @xmath203 of the cone @xmath173 ) . the @xmath0-symmetry is expressed by the equations @xmath528 to be satisfied for each @xmath529 . here , and in the rest of the section , we will usually use the shorthand @xmath530 for an arbitrary orbital base @xmath203 to simplify the notation . the representation @xmath46 of @xmath531 takes explicitly the form , @xmath532 , @xmath533 the explicit form of the tcp symmetry @xmath45 is different for the two series : we have @xmath534 with @xmath535 . for the creation / annihilation operators , we write informally , @xmath379 , z^#_r ( ) = _ b d(p ) z^#_r(,p ) ^#(,p ) , where , informally , we take @xmath536 , so that the integral informally is independent of the choice of @xmath203 , by the argument based on lemma [ lemma : closedform ] . the transformation law then takes the form @xmath537 where the `` @xmath290 '' sign is used for the creation operator @xmath388 , and the `` @xmath291 '' sign for the annihilation operator @xmath387 . the commutation relations are obtained in this informal but efficient notation as [ zccr ] z_r(_1 , p_1 ) z_r(_2 , p_2 ) - r__1-_2 z_r(_2 , p_2 ) z_r(_1 , p_1 ) & = & 0 , + z_r^(_1 , p_1 ) z_r^(_2 , p_2 ) - r__1-_2 z_r^(_2 , p_2 ) z_r^(_1 , p_1 ) & = & 0 , with @xmath349 acting as in . the form of the mixed commutation relation differs according to whether we are in the principal series case ( @xmath183 ) , the complementary ( @xmath249 ) or discrete series case ( @xmath538 ) . in the first case we have z_r(p_1 , _ 1 ) z_r^(p_2 , _ 2 ) - r__2-_1 z_r^(p_2 , _ 2 ) z_r(p_1 , _ 1 ) = ( _ 1-_2)(p_1,p_2 ) 1 where @xmath539 is the dirac delta function on @xmath203 ( relative to the integration measure @xmath363 ) when we integrate this identity against smooth test functions on any orbital base @xmath203 . in the case of the complementary series we have instead z_r(p_1 , _ 1 ) z_r^(p_2 , _ 2 ) - r__2-_1 z_r^(p_2 , _ 2 ) z_r(p_1 , _ 1 ) = c_(_1 - _ 2 ) ( p_1 p_2)^-+i 1 . in the case of the discrete series , we have z_r(p_1 , _ 1 ) z_r^(p_2 , _ 2 ) - r__2-_1 z_r^(p_2 , _ 2 ) z_r(p_1 , _ 1 ) = c_n ( _ 1 - _ 2 ) ( p_1 p_2)^n ( p_1 p_2 ) 1 where @xmath540 . the differences arise from the differences in the definition of the scalar product in each case , see , , respectively . all these exchange relations are generalizations of the zamolodchikov faddeev algebra @xcite . we next describe a concrete version of the quantum field @xmath401 from section [ section : crossing ] , replacing the `` components '' @xmath400 by an additional dependence on a desitter variable . this is done by replacing the vector @xmath399 by a fourier - helgason transform @xmath541 of some testfunction @xmath282 . in view of lemma [ lemma : fourier - helgason ] @xmath304 , this amounts to the field operator , @xmath392 , @xmath282 , @xmath542 where @xmath543 are defined in , and @xmath544 in . in the following , it will be convenient to describe the field operator in terms of its distributional kernels , writing @xmath545 . then we have , informally ( @xmath546 ) , @xmath547 here , the @xmath38-integral is over @xmath136 , and @xmath548 for an arbitrary orbital base @xmath203 , as before . we collect a few properties of this field in the following proposition . [ proposition : phir - props ] the field @xmath401 is an operator - valued distribution on @xmath549 with the following properties . 1 . the field is neutral in the sense that @xmath550 ( on an appropriate domain ) . 2 . the field is @xmath551-covariant , i.e. @xmath552 3 . the field solves the klein - gordon equation of mass @xmath553 on de sitter space , @xmath554 @xmath231 is evident from , , and @xmath236 is a consequence of . note that the prefactor @xmath555 is due to the fact that we consider the current . @xmath304 is satisfied because the de sitter waves @xmath556 are solutions of the klein - gordon equation . the field @xmath401 is seen to fail the usual condition of einstein causality in both , its lightray and its de sitter coordinate , due to the presence of the @xmath0-factors in . so in this sense @xmath401 does not , by itself , straightforwardly define a local quantum field neither on the lightray nor on de sitter space . however , the field satisfies a kind of remnant of the locality condition in the lightray variable . as explained in the abstract setting in section [ section : crossing ] , this is best understood in interplay with its tcp reflected partner field . the definition translates here to @xmath557 it is clear from this definition that also @xmath409 has the properties @xmath231@xmath304 of proposition [ proposition : phir - props ] . explicitly , we have from @xmath558 this is different from @xmath559 because @xmath560 , i.e. the zamolodchikov operators do not transform covariantly under @xmath45 . we have the following concrete version of theorem [ theorem : commutation ] . [ theorem : commutation - concrete ] let @xmath561 be arbitrary , and @xmath182 corresponding to a principal or complementary series representation . then , in the sense of distributions , @xmath562=0\quad\text{for } \;u < u'\ , , \end{aligned}\ ] ] on the space of vectors of finite fock particle number . this theorem follows from theorem [ theorem : commutation ] as a special case . however , we give an independent , explicit argument which illustrates how the properties of the integral operator @xmath0 enter . we focus on the principal series for definiteness . we expand both @xmath563 in terms of the zamolodchikov - faddeev creation / annihilation operators @xmath564 respectively their primed counterparts . the commutator @xmath565 $ ] then gets contributions of the type @xmath566 , [ z_r^\dagger , z_r^{\prime \dagger}]$ ] , as well as @xmath567 , [ z_r , z_r^{\prime \dagger}]$ ] . it is relatively easy to see that the @xmath568 $ ] and @xmath569 $ ] contributions vanish separately ( see ) . this is not the case , however , for the remaining mixed contributions , where a non - trivial cancellation between both terms , called `` @xmath290 '' and `` @xmath291 '' , is required . if we apply these contributions to an @xmath8-particle state @xmath570 , we get a combination of two terms abbreviated as [ 2k ] [ _ r(u , x ) , _r(u,x ) ] _ n = ( ^+ k - ^- k ) _ n . here , each @xmath571 acts as multiplication operator in @xmath572 and as integral kernel ( depending on @xmath573 and @xmath574 ) on @xmath575 . taking into account the @xmath0-symmetry of the wave functions , the explicit form of those kernels is found to be : @xmath576 for `` @xmath290 '' , whereas for `` @xmath291 '' one has @xmath577 to get to the expression for `` @xmath291 '' , we have also used properties r1 ) and r3 , case 2 ) of the kernel @xmath33 . a graphical expression for both kernels in the notation of appendix b is given in the following figure . + here the dots @xmath578 , @xmath579 , indicate integrations over @xmath203 as explained in appendix @xmath203 , and the two remaining integrals over @xmath580 and @xmath581 are written explicitly . the dashed lines connecting the @xmath0-kernels to the desitter waves simply indicate that these parts share the same desitter momentum ( @xmath580 and @xmath581 , respectively ) , and the double lined boxes mean the _ full _ integral kernel of @xmath582 respectively @xmath583 , i.e. including all gamma - factors , the crossing function @xmath584 , and the rescaling @xmath585 . in order to see that the contribution from the `` @xmath290''-kernel cancels precisely that from the `` @xmath291 '' kernel , we now shift the integration contour in @xmath586 upwards to @xmath587 . the shifted contour lies in the domain of analyticity of each @xmath588 , by r6 ) . furthermore , under @xmath589 , we have @xmath590 . by assumption @xmath591 , so the real part of this expression is negative and this provides an exponential damping of the integrand for large values of @xmath592 . the contour shift is thus permissible . for @xmath593 , crossing symmetry r7 ) ( see ) then implies that @xmath594 can also be expressed as here the lines with @xmath595 denote integral operators @xmath596 . since @xmath597 ( lemma [ lemma : tpc - principal ] ) , the inner lines cancel . the two outer operators have the effect of switching the sign of @xmath182 in both the desitter waves . if we now flip the signs on @xmath598 ( taking into account that this changes the @xmath599 prescription on the desitter waves ) , it becomes apparent that the above expression coincides with @xmath600 , so that the two kernels precisely cancel . this concludes the proof . the truly local fields / observables of these model are different from the half - local fields @xmath401 , @xmath409 , and can abstractly be described in an operator - algebraic setting . we therefore proceed from the pair of field operators @xmath401 , @xmath409 to the pair of von neumann algebras @xmath472 , @xmath601 , which by the preceding result are localized in half lines in the lightray coordinate . in order to build from these basic `` half line '' von neumann algebras a net of von neumann algebras @xmath602 , indexed by intervals @xmath603 , one has to translate @xmath604 and form intersections . one defines for the interval @xmath605 @xmath606 then by construction , we have the assignment @xmath602 from open intervals to von neumann algebras forms a @xmath607-covariant local net of von neumann algebras on @xmath377 : 1 . for any open interval @xmath608 , @xmath609 , @xmath610 2 . for two disjoint intervals @xmath611 , we have @xmath612=\{0\}. \end{aligned}\ ] ] the elements of @xmath613 may be understood as the local fields / observables of these models . we do not investigate them here , but just mention two important questions in this context : a ) under which conditions are the algebras @xmath613 `` large '' ( for example in the sense that the fock vacuum @xmath473 is cyclic ) ? and b ) : under which conditions does the net @xmath614 extend from the real line to the circle , transforming covariantly even under the 1-dimensional conformal group @xmath615 , acting by fractional transformation @xmath616 with @xmath617 ? following the same arguments as in the scalar case @xcite ( which build on @xcite ) , one can show that i ) the subspace @xmath618 is independent of the considered interval @xmath619 , ii ) on @xmath620 , the representation @xmath46 extends to psl@xmath621 , and iii ) the net @xmath622 extends to a local conformally covariant net on the circle . a direct characterization of @xmath620 is however difficult in general in particular because the nuclearity criteria @xcite that can be applied to the @xmath2 models @xcite do not apply here because the representation @xmath186 is infinite - dimensional . we leave the analysis of these questions to a future investigation . here we present a variant of our construction which leads to euclidean conformal field theories in @xmath22-dimensions . as before , the construction is based on an @xmath50-invariant yang - baxter function @xmath0 such as . we do not rely on the crossing property in this section . to turn @xmath0 into a yang - baxter operator , we use here the amplification discussed after lemma [ lemma : ybf->ybo ] instead of the coupling to the representation space of the lightray . that is , we pick some @xmath624 and real numbers @xmath625 . for given @xmath182 in the _ complementary series _ representation of @xmath50 ( the conformal group in @xmath22-dimensional euclidean space ) , we define the one particle space as _ 1 = ^n _ , and the invariant yang - baxter operator @xmath626 on @xmath627 , referring to an orthonormal basis @xmath628 of @xmath124 . the @xmath0-symmetric fock space @xmath377 is then defined as before . we now choose the orbital base @xmath629 to be flat ( see appendix a ) , which amounts to parameterizing @xmath516 as [ px ] p = ( ( ||^2 + 1 ) , , ( ||^2 - 1 ) ) in terms of @xmath630 , and results in a one - particle space of the form @xmath631 . vectors in this space are @xmath19-component functions @xmath632 , @xmath633 , and the scalar product is ( see and appendix a ) @xmath634 where @xmath635 and @xmath636 in the complementary series . we therefore have @xmath19 pairs of creation / annihilation operators @xmath637 , and the @xmath19-component quantum fields @xmath638 the @xmath639 ( and @xmath640 ) are operator - valued distributions , but now defined on the _ euclidean space @xmath203 instead of the `` momentum space '' @xmath173_. we we again describe these fields in terms of their distributional kernels @xmath641 and @xmath642 at sharp points @xmath643 , i.e. @xmath644 , etc . as a consequence of the scalar product , the faddeev - zamolodchikov operators satisfy the relations [ zccr1 ] z^_i(_1 ) z_j(_2 ) - r__i-_j z_j(_2 ) z^_i(_1 ) & = & c__ij|_1 - _ 2|^-2 + z^_i(_1 ) z^_j ( _ 2 ) - r__i-_j z^_j(_2 ) z^_i(_1 ) & = & 0 . by construction , the field operators @xmath640 are real , @xmath645 , and transform in the complementary series representation @xmath186 , i.e. v _ ( ) _ r , j ( ) v_()^-1 = j_()^- _ r , j ( ) , where @xmath646 is the usual action of conformal transformations @xmath180 on @xmath647 , and where @xmath648 is the corresponding conformal factor , see appendix a. the scaling dimension of @xmath640 is therefore @xmath636 . by the same arguments as before , the exponentiated fields @xmath649 are then well defined for any @xmath650 and any @xmath651 . we define corresponding `` euclidean '' von neumann algebras _ r(o ) = \ { e^i_r , j(f ) f c^_,0(o ) , j=1 , , n } , for any open , bounded region @xmath652 . by construction , conformal transformations act geometrically on the net @xmath653 in the sense that @xmath654 . if we choose our discretized rapidities @xmath655 to be spaced equidistantly and formally take @xmath43 , then the shifts @xmath656 correspond to symmetries of the net @xmath657 . the simplest case of this construction is given when instead of the integral operators @xmath0 , we take the flip @xmath406 on @xmath658 . in that case , the @xmath33 factor drops out of the commutation relation for the @xmath659 s , and one finds that the vacuum correlation functions of the field @xmath660 are of quasi - free form , i.e. @xmath661 where the sum is over all partitions of the set @xmath662 into ordered pairs , @xmath663 is a real constant , and @xmath664 for all @xmath665 is assumed . these correlation functions correspond to an @xmath19-dimensional multiplet of a generalized bosonic free euclidean field theory . the correlation functions are not reflection positive @xcite ( and so do not define a cft in @xmath22-dimensional minkowski spacetime satisfying the usual axioms ) apart from the limiting case @xmath666 corresponding to the standard free field . locality of the field theory is expressed by the fact that the above correlation functions are symmetric under exchanges @xmath667 . at the level of the von neumann algebras , this amounts to saying that , if @xmath668 and @xmath669 are disjoint , the corresponding von neumann algebras commute [ local ] [ _ f(o ) , _ f(o ) ] = \{0 } . in this sense , the euclidean quantum field theory defined by the assignment @xmath670 is `` local '' . this structure is modified if instead of the flip , we use our integral operators @xmath0 . in that case , the fields @xmath640 are not `` local '' in the sense that the correlation functions are no longer symmetric , and consequently does not hold . to obtain a local euclidean theory , one could consider the algebras _ r(o ) : = _ r(o ) _ r(o) , where @xmath669 denotes the complement of @xmath671 , and where @xmath672 is the commutant of the corresponding von neumann algebra @xmath673 . it follows directly from the definition that the net @xmath674 is local and transforms covariantly under the conformal group @xmath50 in the sense that @xmath675 . local operators @xmath676 in the conformal field theory defined by this new net should be thought of , roughly speaking , as elements in the intersection of @xmath677 for arbitrarily small @xmath668 containing @xmath678 , i.e. in a sense o ( ) _ o _ r(o ) correlation functions @xmath679 of such fields would then again be local in the sense of being symmetric in the @xmath680 . to make such statements precise , one should on the one hand make sure that the size of @xmath677 is sufficiently large , and one should also make precise what is meant by the above intersection , presumably by making a construction along the lines of @xcite . in this work we have constructed yang - baxter @xmath0-operators for certain unitary representations of @xmath50 . the properties of these operators , in particular the yang - baxter equation , unitarity , and crossing symmetry make possible two , essentially canonical , constructions : a ) a 1-dimensional `` light ray '' cft , whose internal degrees of freedom transform under the given unitary representation and b ) a euclidean cft in @xmath22 dimensions in which the group @xmath50 acts by conformal transformations . both a ) and b ) are constructed from one and the same @xmath0-operator ( in a complementary series representation ) . theory b ) depends on a discretization parameter @xmath19 corresponding to a set of @xmath19 discretized `` rapidities '' , @xmath681 . the operator algebras in cases a ) and b ) become formally related when @xmath43 . in fact , the operator algebra a ) is related to certain left- and right local fields on the lightray , and whereas the operator algebra in case b ) to certain fields of the form . they are built from certain generalized creation / annihilation operators @xmath682 in case a ) and @xmath683 in case b ) . these operators satisfy a zamolodchikov - faddeev algebra into which the @xmath0-operators enter . @xmath29 is a desitter `` momentum '' which corresponds to @xmath678 as in fig . 1 resp . the index @xmath684 corresponds to the @xmath684-th discretized rapidity , @xmath526 . the rapidity variable @xmath38 can be thought of roughly speaking as `` dual '' to the lightray variable , @xmath403 ( in the sense of fourier transform ) , whereas @xmath678 is dual to @xmath27 ( desitter points ) ( in the sense of fourier - helgason transform ) . thus , when the spacing between @xmath655 goes to zero , the operator algebras formally coincide , and we think of this isomorphism as a manifestation of a ds / cft - type duality . in line with this interpretation , one is tempted to think of @xmath19 as a `` number of colors '' by analogy with the ads / cft correspondence . however , we note that the algebra in case b ) does not have a corresponding symmetry such as @xmath2 acting on the index @xmath684 . what is restored in the limit as @xmath43 is merely a symmetry corresponding to @xmath685 at finite @xmath19 . we finally note that our construction does _ not _ yield an ordinary local quantum field theory on desitter spacetime . indeed , despite the formal similarities between the expressions for a free desitter quantum field of mass @xmath272 ( see eq . ) and our field ( see eq . ) , we note that these are actually quite different . whereas the former is local in the spacetime sense with respect to the causal relationships in desitter spacetime , the latter is not ( it is only `` half - local '' in the lightray variable @xmath403 , which is absent in eq . ) . it would be interesting to see whether a variant of our method can also produce new local quantum field theories in the ordinary sense in desitter spacetime . we must leave this to a future investigation . * acknowledgements : * the research ( s.h . ) leading to these results has received funding from the european research council under the european union s seventh framework programme ( fp7/2007 - 2013 ) / erc grant agreement no qc & c 259562 . s.h . also likes to thank r. kirschner from leipzig for pointing out to him references @xcite and for explanations regarding rll - relations and related matters . it appears best to perform some of the calculations involving the form @xmath211 and the orbital base @xmath203 of the future lightcone @xmath173 of @xmath167 by using specific choices for @xmath203 . it is known since the times of kepler and newton that there are three canonical choices , which correspond to a flat , hyperbolic , or spherical geometry for @xmath203 . 1 . * ( flat geometry , @xmath687 ) . * we realize @xmath203 as the intersection of @xmath173 with some arbitrary but fixed _ null _ plane in @xmath167 . a parameterization of @xmath203 is in this case given by @xmath688 . the induced geometry is seen to be flat . the point - pair invariant and @xmath22-form @xmath363 are given ( @xmath689 ) in this case by = d^d-1 , p p = |- |^2 . the group of transformations leaving @xmath203 invariant is evidently @xmath690 , the euclidean group . with the choice @xmath250 , we may identify the representation space @xmath179 with a space of square integrable functions on @xmath203 . under this identification , the action of @xmath180 on a wave function @xmath691 is given by ( u _ ( ) ) ( ) = j_()^--i ( ) , where @xmath646 denotes the usual action of a conformal group element on @xmath678 , and where @xmath648 is the conformal factor of this transformation , @xmath692 . 2 . * ( spherical geometry , @xmath693 ) . * we realize @xmath203 as the intersection of @xmath173 with some arbitrary but fixed _ space like _ plane in @xmath167 . a parameterization of @xmath203 is in this case given by @xmath694 . the induced geometry is seen to be a round sphere . the point - pair invariant and @xmath22-form @xmath363 are given ( @xmath689 ) in this case by = d^d-1 p , p p = 1-p p , where we mean the standard integration element of the round sphere . the group of transformations leaving @xmath203 invariant is evidently @xmath695 , the rotational group . 3 . * ( hyperbolic geometry , @xmath696 ) . * we realize @xmath203 as the intersection of @xmath173 with some arbitrary but fixed pair of parallel _ timelike _ planes in @xmath167 . a parameterization of the two disconnected components of @xmath203 is in this case given by @xmath697 . the induced geometry is seen to be hyperbolic for each connected component corresponding to @xmath698 , respectively . the point - pair invariant and @xmath22-form @xmath363 are given ( @xmath689 ) in this case by = , p p = 1 + - , where the integration element is that of hyperbolic space . the group of transformations leaving @xmath203 invariant is evidently @xmath699 . + a ) flat , b ) spherical , and c ) hyperbolic orbital base . the proofs of these theorems make use of the following identities : 1 . ( * symanzik triality relation * ) : + [ eq : triality ] & _ ^d-1 d^d-1 q ( 1-p_1 q)^w_1 ( 1-p_2 q)^w_2 ( 1-p_3 q)^w_3 + & = ( 2)^ ( 1-p_2 p_3)^w_1 ( 1-p_1 p_3)^w_2 ( 1-p_1 p_2)^w_3 , + for complex parameters @xmath700 satisfying @xmath701 , where the dual parameters are defined by @xmath702 . the integral on the left side is defined by analytic continuation in the parameters by the method described e.g. in @xcite . the proof of the identity follows from formula ( 5.104 ) of @xcite ( identical with formula ( b22 ) of @xcite ) , after a suitable analytical continuation in the parameters @xmath703 in the formula and an application of the residue theorem . + the triality relation is best remembered in graphical form as a `` star - triangle relation '' , + ( 0,0 ) ( 90:1 ) node[pos=.7,left]@xmath704 ; ( 0,0 ) ( -150:1 ) node[pos=.7,below]@xmath705 ; ( 0,0 ) ( -30:1 ) node[pos=.8,above]@xmath706 ; ( 0,0 ) circle [ radius=0.06 ] ; ( p1 ) at ( 90:1.5 ) @xmath707 ; ( p2 ) at ( -150:1.5 ) @xmath708 ; ( p3 ) at ( -30:1.5 ) @xmath709 ; ( q ) at ( -90:0.4 ) @xmath126 ; ( =) at ( 0:2 ) @xmath710 ; + ( 90:1 ) ( -30:1 ) node[pos=.7,above]@xmath711 ; ( -30:1 ) ( -150:1 ) node[pos=.5,below]@xmath712 ; ( -150:1 ) ( 90:1 ) node[pos=.3,above]@xmath713 ; ( p1 ) at ( 90:1.5 ) @xmath707 ; ( p2 ) at ( -150:1.5 ) @xmath708 ; ( p3 ) at ( -30:1.5 ) @xmath709 ; + here a line with parameter @xmath714 between two `` momenta '' @xmath715 denotes a `` propagator '' @xmath716 , a dot means integration over that variable wr.t . @xmath717 , and the product over all lines is understood . ( * delta function relation * ) : [ delta ] anal . cont._w 0 \ { ( 1-p_1 p_2)^-+w } = c_n ( 2)^ ^n ( p_1 , p_2 ) , where we mean analytic continuation in the sense of distributions from the domain @xmath718 . for a proof and a mathematically precise explanation of this kind of analytic continuation , see e.g. @xcite . the identity can be demonstrated by applying laplacians to the composition relation below . + the following simple consequence of the triality relation and the delta function relation will be needed for the discrete series representations ( where @xmath719 and @xmath720 ) : + [ eq : triality1 ] & _ ^d-1 d^d-1 q ( 1-p_1 q)^n ( 1-p_2 q)^w ( 1-p_3 q)^-2-n - w + & = c_n ( 2)^ ^n ( p_2,p_3)(1-p_1 p_2)^2+n . + on the right side , we mean by @xmath721 the delta function on @xmath722 relative to the standard integration measure , and by @xmath438 the laplacian on @xmath722 . the actual value of the constant @xmath723 is needed only for @xmath277 , where it is @xmath724 . ( * composition relation * ) : this relation is obtained by taking the limit @xmath725 in the triality relation and using the delta function relation . one obtains the integral identity [ eq : composition - relation ] + & _ ^d-1 d^d-1 q ( 1-p_1 q)^ w_1 ( 1-p_2 q)^w_2 + & = ( 2)^2 ( p_1 , p_2 ) , + where it is assumed that @xmath726 . the composition relation follows in the limit @xmath727 from the triality relation . * ( r3 , part i ) * the flip operator simply exchanges the variables , so that on the level of integral kernels , we have @xmath729 . by inspection of the kernel and the constant @xmath730 , one then sees that the first equation in ( r3 ) holds . * ( r1 ) , ( r3 , part ii ) , ( r5 ) : * for unitarity ( r1 ) and the tcp symmetry ( r3 , part ii ) , we have to distinguish the principal and complementary series , because the scalar products and conjugations are different for the two series . in case @xmath488 both belong to principal series representations , the scalar product is given by , and we therefore have @xmath731 here the second step follows by direct inspection of the kernel , taking into account @xmath488 . we thus have @xmath732 in this case . in case both @xmath733 and @xmath734 belong to the complementary series , the scalar product is more complicated , but the conjugations @xmath735 are simply complex conjugations ( lemma [ lemma : tcp - complementary ] ) , so that the tcp symmetry ( r3 , part ii ) amounts to @xmath736 . this equation holds true because @xmath737 are real for the complementary series . let us consider the tcp symmetry for the principal series , @xmath488 . then the conjugations @xmath738 are given by complex conjugation and the integral operators @xmath739 ( lemma [ lemma : tpc - principal ] ) . inserting the definitions and making use of the graphical notation introduced with the triality relation , one finds that the integral kernel of @xmath740 is given by ( 0,0 ) ( 2,0 ) node[pos=.5,below]@xmath741 ; ( 2,0 ) ( 4,0 ) node[pos=.5,below]@xmath742 ; ( 4,0 ) ( 6,0 ) node[pos=.5,below]@xmath743 ; ( 0,2 ) ( 2,2 ) node[pos=.5,above]@xmath744 ; ( 2,2 ) ( 4,2 ) node[pos=.5,above]@xmath745 ; ( 4,2 ) ( 6,2 ) node[pos=.5,above]@xmath746 ; ( 2,0 ) ( 2,2 ) node[pos=.85,yshift=+1.7ex , sloped , left]@xmath747 ; ( 4,0 ) ( 4,2 ) node[pos=.25,yshift=-1.7ex , sloped , right]@xmath748 ; ( 2,0 ) circle [ radius=0.06 ] ; ( 4,0 ) circle [ radius=0.06 ] ; ( 2,2 ) circle [ radius=0.06 ] ; ( 4,2 ) circle [ radius=0.06 ] ; ( p2 ) at ( -.4,0 ) @xmath708 ; ( q2 ) at ( 6.4,0 ) @xmath749 ; ( p1 ) at ( -.4,2 ) @xmath707 ; ( q1 ) at ( 6.4,2 ) @xmath580 ; ( 0,0 ) ( 2,0 ) node[pos=.5,below]@xmath741 ; ( 2,0 ) ( 4,0 ) node[pos=.5,below]@xmath742 ; ( 4,0 ) ( 6,0 ) node[pos=.5,below]@xmath743 ; ( 0,2 ) ( 4,2 ) node[pos=.5,above]@xmath752 ; ( 2,0 ) ( 4,2 ) node[pos=.5,sloped , above]@xmath753 ; ( 4,2 ) ( 6,2 ) node[pos=.5,above]@xmath746 ; ( 2,0 ) ( 0,2 ) node[pos=.45,sloped , above]@xmath754 ; ( 4,0 ) ( 4,2 ) node[pos=.25,yshift=-1.7ex , sloped , right]@xmath748 ; ( 2,0 ) circle [ radius=0.06 ] ; ( 4,0 ) circle [ radius=0.06 ] ; ( 4,2 ) circle [ radius=0.06 ] ; ( p2 ) at ( -.4,0 ) @xmath708 ; ( q2 ) at ( 6.4,0 ) @xmath749 ; ( p1 ) at ( -.4,2 ) @xmath707 ; ( q1 ) at ( 6.4,2 ) @xmath580 ; ( =) at ( 7,1 ) @xmath710 ; ( = = ) at ( -1,1)@xmath710 ; ( 0,0 ) ( 2,0 ) node[pos=.5,below]@xmath741 ; ( 2,0 ) ( 6,0 ) node[pos=.5,below]@xmath755 ; ( 0,2 ) ( 4,2 ) node[pos=.5,above]@xmath752 ; ( 2,0 ) ( 4,2 ) ; ( 4,2 ) ( 6,2 ) node[pos=.5,above]@xmath746 ; ( 2,0 ) ( 0,2 ) node[pos=.45,sloped , above]@xmath754 ; ( 4,2 ) ( 6,0 ) node[pos=.5,yshift=.2ex , sloped , above]@xmath756 ; ( 2,0 ) circle [ radius=0.06 ] ; ( 4,2 ) circle [ radius=0.06 ] ; ( p2 ) at ( -.4,0 ) @xmath708 ; ( q2 ) at ( 6.4,0 ) @xmath749 ; ( p1 ) at ( -.4,2 ) @xmath707 ; ( q1 ) at ( 6.4,2 ) @xmath580 ; ( 0,0 ) ( 6,0 ) node[pos=.5,below]@xmath745 ; ( 0,0 ) ( 0,2 ) node[pos=.5,sloped , above]@xmath748 ; ( 0,2 ) ( 4,2 ) node[pos=.5,sloped , above]@xmath752 ; ( 0,2 ) ( 6,0 ) node[pos=.5,sloped , below]@xmath757 ; ( 4,2 ) ( 6,2 ) node[pos=.45,sloped , above]@xmath746 ; ( 4,2 ) ( 6,0 ) node[pos=.25,yshift=1.7ex , sloped , right]@xmath756 ; ( 4,2 ) circle [ radius=0.06 ] ; ( p2 ) at ( -.4,0 ) @xmath708 ; ( q2 ) at ( 6.4,0 ) @xmath749 ; ( p1 ) at ( -.4,2 ) @xmath707 ; ( q1 ) at ( 6.4,2 ) @xmath580 ; ( =) at ( 7,1 ) @xmath710 ; ( = = ) at ( -1,1)@xmath710 ; ( 0,0 ) ( 6,0 ) node[pos=.5,below]@xmath745 ; ( 0,0 ) ( 0,2 ) node[pos=.5,sloped , above]@xmath748 ; ( 0,2 ) ( 6,2 ) node[pos=.5,sloped , above]@xmath755 ; ( 0,2 ) ( 6,0 ) ; ( 6,2 ) ( 6,0 ) node[pos=.25,yshift=1.7ex , sloped , right]@xmath747 ; ( p2 ) at ( -.4,0 ) @xmath708 ; ( q2 ) at ( 6.4,0 ) @xmath749 ; ( p1 ) at ( -.4,2 ) @xmath707 ; ( q1 ) at ( 6.4,2 ) @xmath580 ; here the dotted lines result from two propagators with opposite powers that cancel each other . the first dotted line produces a factor of @xmath758 , and the second dotted line produces a factor of @xmath759 . taking into account these factors ( and the @xmath48-factor from the initial diagram ) , one then realizes that the last step represents the integral kernel of @xmath760 . this finishes the proof of ( r3 , part ii ) for two principal series representations . for the mixed case ( one principal series representation and one complementary series representation ) , the proof is similar . returning to ( r1 ) for two complementary series representations , one finds that because of the appearance of the integral operators @xmath218 in the scalar product , the adjoint is given by @xmath761 passing to the graphical notation , this kernel is given by the exact same diagram as in the tcp symmetry proof for the principal series , but with the replacements @xmath762 , @xmath763 , @xmath764 , @xmath765 . converting the last diagram in the earlier calculation into an integral kernel then yields @xmath766 . thus , as in the case of two principal series representations , we have @xmath732 . again , the proof for the mixed case is similar . to finish the proof of ( r1 ) and ( r5 ) , it now remains to show @xmath767 . this follows ( for all combinations of principal / complementary series representations ) by application of the composition relation . * ( r4 ) * the integral identity underlying the yang - baxter relation is the triality relation . one first calculates that the left and right hand sides of coincide on arbitrary @xmath768 if and only if the following two integral kernels ( with the graphical notation introduced for the triality relation ) coincide : note that these integral kernels differ from the ones arising from the yang - baxter equation by factors of @xmath769 and gamma functions of the parameters , but the overall factors are the same for both diagrams . we now use the triality relation to convert these diagrams into a more symmetrical form . beginning with the left diagram , we first convert the interior triangle to a star , and then the three resulting stars into triangles . this shows that the left diagram above coincides with setting all @xmath182-parameters to one and the same value , the properties ( r1)(r5 ) imply the properties ( r1)(r5 ) of an invariant yang - baxter function for the representation @xmath186 and conjugation @xmath237 . ( the properties here are slightly stronger because of the division of ( r3 ) into two separate equalities in ( r3 ) . ) the case when one or more @xmath182-parameters correspond to a discrete series representation can be reduced to the previous cases by perturbing the corresponding @xmath182-parameters slightly from their discrete values along the real axis . then the same arguments as given for the complementary series go through , and the desired identities are obtained in the limit where the relevant @xmath182-parameters go to their discrete values . one has to take care , however , to apply all identities to suitable wave functions , and , for the complementary series variables , use the constraint before taking the limit . one also has to check that this constraint is actually preserved by the @xmath0-operator . this follows from . the integral kernel is entire analytic in @xmath141 for non - coinciding momenta , and for @xmath772 and principal series representations , all singularities are integrable . thus the matrix elements of @xmath349 are analytic functions on the strip @xmath773 , and moreover bounded in @xmath141 on this domain . we now explain the reason for the particular form of the factor @xmath371 . inserting , we see that ( r7 ) is equivalent to @xmath774 since @xmath488 belong to principal series representations , the conjugations are @xmath775 , where @xmath238 denotes pointwise complex conjugation . using this and lemma [ lemma : tpc - principal ] @xmath231 , one checks that amounts on the level of integral kernels to , @xmath162 , @xmath776 , @xmath777 in the graphical notation , the right hand side can be transformed with the triality relation into a factor of @xmath778 has been suppressed in all these diagrams . comparing with the analytically continued matrix elements on the left hand side of , one then finds that holds if @xmath779 in order not to spoil the analyticity of the matrix elements of @xmath328 , we have to choose @xmath371 analytic on the strip @xmath780 . furthermore , @xmath371 must satisfy the requirements of proposition [ proposition : tweak - r ] , i.e. it must be symmetric in @xmath333 , and @xmath781 for @xmath162 . we claim that all requirements are satisfied by @xmath782 where @xmath783 is a real sufficiently large parameter that will be chosen later , and @xmath784 to verify this claim , we need to examine the functions @xmath785 and @xmath786 . one first checks the poles of the gamma functions and sees that @xmath785 is analytic in the strip @xmath780 if @xmath783 is sufficiently large , for instance if we take @xmath787 . its logarithmic derivative can trivially be expressed in terms of the digamma function @xmath788 as @xmath789 clearly , this function @xmath790 is analytic in @xmath780 as well . by taking into account the asymptotic expansion @xmath791 as @xmath792 in @xmath793 , and going through all terms , one also finds that @xmath794 vanishes quadratically in @xmath795 in the strip @xmath780 for @xmath796 . thus @xmath786 is well - defined . furthermore , we have the symmetry properties @xmath797 and @xmath798 , @xmath799 , which imply that @xmath786 is even and real ( for real arguments @xmath142 ) . since @xmath786 also decays fast because it is the fourier transform of a smooth function , we see that @xmath800 is well - defined , odd , and real ( for real @xmath141 ) . it then follows that @xmath371 satisfies the requirements of prop . [ proposition : tweak - r ] . it remains to check that @xmath371 is bounded and analytic in @xmath780 , and that holds . regarding analyticity , the integrand of @xmath800 is entire in @xmath141 , and we may estimate the growth of the sine function by @xmath801 . this growing factor is compensated by the falloff of @xmath802 . clearly @xmath803 , and furthermore @xmath804 decays like @xmath805 as well this latter fact follows from a contour shift in the fourier integral . together with the remaining decay of the digamma functions , this establishes the analyticity of @xmath800 in the strip @xmath780 . by analogous arguments , one also shows that @xmath806 is bounded in the strip . to verify , we compute the fourier transform @xmath807 of @xmath800 . we will use that since @xmath786 is even , we have @xmath808 , and we will also make use of @xmath809 . this gives @xmath810 and after an inverse fourier transformation , we arrive at @xmath811 using the definitions of @xmath371 and @xmath785 , the desired equality then follows . we must also check the analyticity and boundedness of the @xmath48 factors in the definition of @xmath328 and @xmath371 . these follow from the well - known facts that @xmath48 is non - vanishing , has poles at the non - positive integers , and the standard asymptotic formula ( @xmath812 ) |(x+iy)| ~(2)^ |y|^x- e^-|y|/2 . it follows that the @xmath48 factors are bounded by @xmath813 for large @xmath814 . the @xmath38-dependence of coming from the exponentials is analytic @xmath38 once we form the matrix elements of @xmath33 , and these exponentials are also clearly bounded in @xmath38 . if we take matrix elements with smooth wave functions , we get from these factors decay as @xmath815 where @xmath8 is as large as we wish . thus , all pieces in @xmath33 are analytic in the strip @xmath780 and decay faster than any inverse power @xmath815 for @xmath816 if we take matrix elements with smooth wave functions . to derive the infinite product formula for @xmath817 quoted in is rather lengthy , and we only sketch the main steps . first , we expand the digamma functions in the definition of @xmath818 using the well - known series ( z ) = -_e + _ n=0^ ( - ) . substituting this series for each of the terms in the expression for @xmath818 , we find that all contributions from euler s constant @xmath819 and from the sums over @xmath820 cancel each other . we next calculate @xmath821 by performing the integral over @xmath822 separately for each term in the series ( this is admissible , because both the series and the integral are absolutely convergent ) . the resulting integrals all have the form _ -^dt = - e^-(n+)|p| , where the residue theorem was used , and where @xmath823 stands for the various constants that appear . the sum over @xmath8 can then be easily done with the aid of a geometric series , resulting in the expression [ fint ] f__1 _ 2 ( ) = 4 _ 0^dp [ ( _ 12^+ p ) + ( _ 12 ^ -p ) - e^-cp(1+e^p ) ] . in order to perform this integral , we expand out the factors @xmath824 using a geometric series , resulting altogether in a double series indexed by natural numbers @xmath825 . the integral can be pulled inside this double series and can then be performed fairly easily for each term . each such term turns out to be a logarithm , so the double series of these logarithms becomes a logarithm of a doubly infinite product . the end result can be written as @xmath826 the product over @xmath272 can be performed with the aid of the well - known infinite product ( z+1 ) = _ m=1^ ( 1 + ) ^-1 e^z / m , and this results in the formula for @xmath827 quoted in the main text after choosing for @xmath663 the value @xmath787 . in case we consider two coinciding complementary series representations @xmath828 , we have to take into account the different conjugation and different scalar product . one then finds that the same analytic properties , and in particular the same functional equation are required for the factor @xmath829 . the solution is given by the same infinite product formula quoted in the main text , but we now need to choose @xmath830 . this guarantees in particular absolute convergence of the infinite product , as one may see using standard asymptotic expansions of the gamma function . @xmath255 l. d. faddeev . _ quantum completely integrable models in field theory _ , volume 1 of _ mathematical physics reviews _ , 107155 ( 1984 ) . in novikov , s.p . ( ed . ) : mathematical physics reviews , vol . 1 , 107 - 155 g. lechner . _ algebraic constructive quantum field theory : integrable models and deformation techniques_. in : advances in algebraic quantum field theory , r. brunetti et.al . ( eds . ) , 397449 , springer ( 2015 )

we propose a model for the ds / cft correspondence . the model is constructed in terms of a `` yang - baxter operator '' @xmath0 for unitary representations of the desitter group @xmath1 . this @xmath0-operator is shown to satisfy the yang - baxter equation , unitarity , as well as certain analyticity relations , including in particular a crossing symmetry . with the aid of this operator we construct : a ) a chiral ( light - ray ) conformal quantum field theory whose _ internal _ degrees of freedom transform under the given unitary representation of @xmath1 . by analogy with the @xmath2 non - linear sigma model , this chiral cft can be viewed as propagating in a desitter spacetime . b ) a ( non - unitary ) euclidean conformal quantum field theory on @xmath3 , where @xmath1 now acts by conformal transformations in ( euclidean ) _ spacetime_. these two theories can be viewed as dual to each other if we interpret @xmath3 as conformal infinity of desitter spacetime . our constructions use semi - local generator fields defined in terms of @xmath0 and abstract methods from operator algebras .

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Proceed to summarize the following text: the distinguishability of a quantum theory from its classical counterpart is formulated by the heisenberg uncertainty principle @xcite , which bounds our prediction ability for a quantum system . in terms of entropic measures , this uncertainty relation can be recast in a modern information - theoretical form @xcite , which states that @xmath0 , where @xmath1 and @xmath2 denote the shannon entropy for the probability distribution of the measurement outcomes . since the overlap @xmath3 between observables @xmath4 and @xmath5 does not depend on specific states to be measured , the right - hand side ( rhs ) of the inequality provides a fixed lower bound and a more general framework of quantifying uncertainty than the standard deviations @xcite . moreover , once using quantum memory to store information about the measured system , the entropic uncertainty bound could even be violated @xcite . all these entropic uncertainty relations play an important role in many quantum information processing . however , constructed from the probability distribution of a measurement taken , entropic function is still a rather coarse way of measuring the uncertainty of a set of measurements ( see ref . @xcite for a recent review ) . for instance , one can not distinguish the uncertainty inherent in obtaining any combination of outcomes for different measurements . to overcome this defect , a new form of uncertainty relation , i.e. , fine - grained uncertainty relation ( fgur ) , has been proposed recently @xcite . for a set of measurements labeled by @xmath6 , associating with every combination of possible outcomes @xmath7 , there exist a set of inequalities @xmath8 where @xmath9 is the probability of choosing a particular measurement , @xmath10 is the probability that one obtains the outcome @xmath11 after performing measurement @xmath6 on the state @xmath12 . to measure the uncertainty , the maximum in function @xmath13 should be taken over all states allowed on a particular system . it can be proved that one can not obtain outcomes with certainty for all measurements simultaneously when @xmath14 . on the other hand , for the so - called maximally certain state ( mcs ) , the inequality ( [ fgur ] ) can be saturated . since its introduction , many applications have been found for the fgur . for instance , it was shown @xcite that the fgur could be used to discriminate among classical , quantum , and superquantum correlations involving two or three parties . moreover , the uncertainty bound in ( [ fgur ] ) could be optimized once the measured system is assisted by a quantum memory @xcite . dramatically , a profound link between the fgur and the second law of thermodynamics has been found @xcite , which claims that a violation of uncertainty relation implies a violation of thermodynamical law . other studies from various perspectives could be found in @xcite . while most of studies on uncertainty relations are nonrelativistic , a complete account of these relations requires one to understand them in relativistic regime , which may even shed new light on quantum gravity @xcite . in the previous work @xcite , we have shown that , besides the choice on the observers , the entropic uncertainty bound should also depend on the relativistic motion state of the observer who performs the measurement . this new character of quantum - memory - assisted entropic uncertainty relation ( eur ) is a direct result of entanglement generation in a relativistic system . in this paper , we explore the fgur under the decoherence rooting in the relativistic motion of quantum systems , and find the uncertainty bound does depend on the motion state of the system . we first consider observer undergoes an _ uniform _ acceleration relative to an inertial reference . since two frames differ in their description of a given quantum state due to the unruh effect , the concept of measurement becomes observer - dependent , which implies a nontrivial relativistic modification to the fgur . without loss of generality , beyond the single - mode approximation , we show that the unruh effect modifies the uncertainty bound for the measurements in @xmath15 mutually unbiased bases ( mubs ) , which are intimately related to complementarity principle . dramatically , we find that , for a noninertial observer , the measurements in general mubs could be distinguished from each other , while they share same uncertainty bound in inertial frame . we extend these results to an alternative scenario , where , to prevent the unruh decoherence , an observer is restricted in a single rigid cavity which undergoes an _ nonuniform _ acceleration . we show that the uncertainty could be degraded by the nonuniform acceleration of cavity during particular epoch , while the uncertainty bound itself exhibits a periodic evolution with respect to the duration of the acceleration . this phenomenon can be attributed to the entanglement generation between the field modes in single cavity that plays the role of quantum memory . moreover , except the acceleration - duration time with integer periods , the measurements in different mubs are distinguishable by the corresponding uncertainty bounds . we first investigate the fgur for an observer with uniform acceleration @xmath16 , who performs projective measurements on the quantum state constructed from free field modes . for the noninertial observer traveling in , e.g. , right rindler wedge , field modes in left rindler wedge are unaccessible . the information loss associated with the acceleration horizon results in a thermal bath . from quantum information point of view @xcite , this celebrated unruh effect induces a nontrivial evolution of quantum entanglement between field modes which plays a prominent role in most quantum - information protocols . for simplicity , considering fermionic field with few degrees , the most general vacuum state @xmath17 should be annihilated by unruh operators @xmath18 @xcite c_k , u&=&q_rc_k , r^+q_lc_k , l , q_r^2+q_l^2=1 , + c_k , r&=&_k ,- d^_k , , + c_k , l&=&_k ,- d^_k , . where @xmath19 and @xmath20 are real parameters , and @xmath21 . the particle and antiparticle operators @xmath22 and @xmath23 in respective rindler wedge @xmath24 satisfy the usual anti commutation relations . by analytic construction to whole spacetime , the proper unruh modes are symmetric between rindler wedges i and ii . for particular frequency @xmath25 , the unruh vacuum is @xcite & & + ( |1100-|0011 ) [ unruh - f-0 ] and the first excitation is & & + q_l(|1101+|0001 ) [ unruh - f - one ] where we introduce the notations anti - particle vacua are denoted by @xmath26 and @xmath27 . it should be noted that different operator ordering in fermonic systems could lead to nonunique results in quantum information @xcite . for instance , if we rearrange operator ordering in ( [ notation ] ) as @xmath28 , then a new fock basis is defined @xmath29 . this so - called physical ordering @xcite , in which all region i operators appear to the left of all region ii operators , was proposed to guarantee the entanglement behavior of above states would yield physical results . hereafter , we adopt this particular operator ordering . to explore how the relativistic motion of an observer could influence the fgur , we consider a scenario in which the state to be measured should be prepared in an inertial frame . after that , the observer undergoes an uniform acceleration and performs measurements . since the state should be described in the corresponding rindler frame , information would lose via unruh effect . therefore , the uncertainty obtained by the accelerated observer would be motion - dependent . we illustrate above insight by measurements @xmath30 and @xmath31 , behaving as the best measurement basis , where pauli operators @xmath30 and @xmath31 with equal probability @xmath32 are chosen @xcite . remarkably , along with @xmath33 , three set of their eigenvectors form the mubs in hilbert space with dimension @xmath15 , which plays central role to theoretical investigations and practical exploitations of complementarity properties @xcite . for all pure states @xmath34 , @xmath35,\phi\in[0,2\pi)$ ] , the corresponding density matrix @xmath36 should be rewritten according to the transformation ( [ unruh - f-0 ] ) and ( [ unruh - f - one ] ) . since the field modes in rindler wedge ii is unaccessible to observer , after tracing over the modes in wedge ii , the reduced density matrix is _ red&&_|| + & = & |00_00|(^2c^4+^2q_l^2c^2 ) + & & + |11_11|(^2s^4+^2q_r^2s^2 ) + & & + |01_01|(^2s^2c^2 ) + & & + |10_10|(^2s^2c^2+^2q_r^2c^2+^2q_l^2s^2 ) + & & -|00_01|e^iq_lsc^2 + & & + |00_10|e^-iq_rc^3 + & & -|00_11|^2q_rq_lsc + & & + |01_11|e^-iq_rs^2c + & & + |10_11|e^iq_ls^3+(h.c . ) _ with abbreviation @xmath37 . after performing the projective measurements @xmath30 and @xmath31 on particle sector , we have the probabilities for the outcomes @xmath38 p(0^z|_z)_&&(|0^+_0|_red)=c^2(^2+^2q_l^2 ) , + p(0^x|_x)_&&(|+^+_+|_red ) + & = & + q_rc . here , for later convenience , we chose the measurements in basis @xmath39 and @xmath40 which are the eigenstates of pauli matrix @xmath30 and @xmath31 restricted in rindler wedge i. therefore , lhs of fgur ( [ fgur ] ) should be u&&[p(0^z|_z)_+p(0^x|_x ) _ ] + & = & [ q_rc+(q_r^2+q_l^2 + 1)c^2 + 1 ] for a particular unruh mode with fixed acceleration , one can estimate that the maximum of @xmath41 which is @xmath42 should always be obtained with @xmath43 and @xmath44 @xcite . this means that the mcs which saturates the inequality ( [ fgur ] ) is independent of the acceleration of the observer , which indicates that once we choose the bases enabling an optimal uncertainty on average in an inertial frame , the corresponding measurements should maintain their optimality for all noninertial observer . this remarkable phenomenon could be useful in many real quantum process , for instance , the bb84 states @xmath45 and @xmath46 in quantum cryptography @xcite . explicitly , the dependence of the fine - grained uncertainty bound with outcomes @xmath38 on the acceleration parameter and choice of unruh modes could be expressed as _ ( 0^x,0^z)&=&[c^2(1+q_l^2 + q_r^2)+q_rc+1 ] [ bound1 ] above calculation can extend to any other pairs of outcomes @xmath47 , @xmath48 and @xmath49 , which all give the same bound @xmath50 in inertial frame @xcite . dramatically , we find that the nontrivial unruh effect could distinguish these four pairs of measurements and separate them into two categories . for instance , we have p(1^z|_z)_&&(|1_1|_red ) + & = & ^2s^2+^2(q_r^2+q_l^2s^2 ) , + p(1^x|_x)_&&(|-_-|_red ) + & = & -q_rc . which give _ ( 1^x,1^z)&=&[(1+c^2)q_r^2+(1+q_l^2)s^2+q_rc+1 ] + [ bound2 ] for the mcs with @xmath51 and @xmath44 . after straightforward calculations , it can be shown that the uncertainty bound @xmath52 is also dependent on the direction of @xmath53axis , that measurements @xmath47 share the same bound ( [ bound2 ] ) with @xmath49 , while @xmath48 has the same bound ( [ bound1 ] ) with @xmath38 . this is definitely a new feature of fine - grained uncertainty bound triggered by noninertial relativistic motion . to further explore the influence of unruh effect on fugr , we illustrate the uncertainty bounds ( [ bound1 ] ) and ( [ bound2 ] ) for three different choice of unruh modes , as depicted in fig . [ fgur1 ] . is dependent on the acceleration parameter @xmath54 and choice of unruh modes . three set of curves correspond to the choice of unruh modes with @xmath55 ( black solid ) , @xmath56 ( red dashed ) and @xmath57 ( blue dashed - double - dotted ) . ] for the case with @xmath55 , where the minkowskian annihilation operator is taken to be one of the right or left moving unruh modes , the noninertial observer would detect a single - mode state once the field is in a special superposition of minkowski monochromatic modes from an inertial perspective @xcite . under this single - mode approximation ( sma ) , commonly assumed in the old literature on relativistic quantum information @xcite , we recover the standard result @xmath50 for vanishing acceleration . as @xmath54 growing , the value of @xmath52 decreases , indicating an increment on measurement uncertainty . while there is no essential difference in the behavior of measurement uncertainty for outcomes @xmath38 and @xmath48 with various @xmath20 , however , the uncertainty bounds ( [ bound2 ] ) are sensitive with the choice of unruh modes . surprisingly , as @xmath54 growing , for the unruh modes with large @xmath20 , we observe a decrement on measurement uncertainty for outcomes @xmath47 and @xmath49 , which means that unruh effect could even lift the uncertainty bound in ( [ fgur ] ) . finally , as illustrated in fig . [ fgur1 ] , we find that the distinguishability between the measurements in mubs is a common feature for any choice of unruh modes . to explain this , recall that , by definition , a set of orthonormal bases @xmath58 for a hilbert space @xmath59 where @xmath60 is called unbiased iff @xmath61 , @xmath62 holds for all basis vectors @xmath63 and @xmath64 belong to different bases , i.e. , @xmath65 . from an inertial perspective , the mubs are intimately related to complementarity principle @xcite , which indicates that the measurement of a observable reveals no information about the outcome of another one if their corresponding bases are mutually unbiased . however , for a noninertial observer , the bases @xmath58 should be transformed according to proper bogoliubov transformations , which in general breaks the orthonormality . in other word , the mubs in inertial frame would become non - mubs from a noninertial perspective . therefore , for the observer undergoing an uniform acceleration , we conclude that unruh effect could distinguish measurements in mubs . we now discuss an alternative scenario in which observer is localized in a rigid cavity . while the rigid boundaries of the cavity protect the inside observer from the unruh effect , the relativistic motion of the cavity would still affect the entanglement between the free field modes inside @xcite , therefore leading to a motion - dependent uncertainty bound @xcite . for simplicity , we consider a @xmath66-dimensional model , where the cavity with length @xmath67 imposes the dirichlet conditions on the eigenfunctions @xmath68 of the hamiltonian . a typical trajectory of nonuniform - moving cavity contains three segments referred as ( i ) when the cavity maintains its inertial status , then ( ii ) begins to accelerate at @xmath69 , following the killing vector @xmath70 , and finally ( iii ) the acceleration ends at rindler time @xmath71 , and the duration of the acceleration in proper time measured at the center of the cavity is @xmath72 . the dirac field can be expanded in quantized eigenfunctions as @xmath73 in segment i , and similarly be expressed by @xmath74 in segment ii and @xmath75 in segment iii. the nonvanishing anticommutators @xmath76 define the vacuum @xmath77 . any two field modes in distinct regions can be related by bogoliubov transformations like @xmath78 and @xmath79 , where the coefficients can be calculated perturbatively in the limit of small cavity acceleration @xcite . more specifically , by introducing the dimensionless parameter @xmath80 , which is the product of the cavity s length and the acceleration at the center of the cavity , the coefficients can be expanded in a maclaurin series to @xmath81 order , @xmath82 , and similarly @xmath83 . we start from a pure state @xmath84 in segment i. after the uniform acceleration , we can express this state in segment iii by means of the bogoliubov transformations , which contains modes within all frequency . throughout the process , we assume that the observer can only be sensitive to modes in particular frequency @xmath85 . therefore all other modes with frequency @xmath86 should be traced out in the density matrix @xmath87 , which leads to _ red&=&_k|_k_k| + & = & |_k_k|(-f^-_k+f^+_k ) + & & + |_k^++_k|(+f^-_k - f^+_k ) + & & + |_k^+_k|e^-i(g_k+^(2)_kk ) + & & + |_k^+_k|e^i(g_k+^(2)_kk)^ * where the coefficients are @xmath88 and @xmath89 . the probability of measurements @xmath38 are p(0^z|_z)_&&(|_k_k|_red ) + & = & -f^-_k+f^+_k , + p(0^x|_x)_&&(|++|_red ) + & = & \{1+[e^-i(g_k+^(2)_kk)]}. the uncertainty bound should be the maximum of lhs of fgur ( [ fgur ] ) , @xmath90 $ ] . along the analysis before , we know that the acceleration of cavity would not change the mcs with parameters @xmath43 and @xmath44 . therefore , we obtain the uncertainty bound for the cavity system _ ( 0^x,0^z)=[2+(1-f_+)+f_-+(g_k+^(2)_kk ) ] + [ fgur3 ] the coefficients in the bound has been given in @xcite , which are f_+&=&_p=-^|e_1^k - p-1|^2|a^(1)_kp|^2 + & = & [ 4(k+s)^2(q_6(1)-q_6(e_1))+q_4(1)-q_4(e_1)]and @xcite f_- & & f^+_k - f^-_k=(_p0-_p<0)|e_1^k - p-1|^2|a^(1)_kp|^2 + & = & 2(k+s)[q_5(1)-q_5(e_1)]+p(k , s , e_1 ) with @xmath91 characterizing the self - adjoint extension of the hamiltonian . here we use the notation @xmath92 $ ] , li is the polylogarithm and @xmath93 . @xmath94 is a polynomial summing for all terms with odd number @xmath95\ ] ] and ( g_k+^(2)_kk)&=&1-h^2\{(+ ) + & & - } in previous section , we show that an uniformly - accelerating observer can distinguish the measurements in mubs which share the same uncertainty bounds in an inertial frame . here we generalize this to the scenario with rigid cavity . to proceed , we calculate the probabilities of measurements @xmath49 p(1^z|_z)_&&(|_k^++_k|_red ) + & = & + f^-_k - f^+_k , + p(1^x|_x)_&&(|++|_red ) + & = & \{1-[e^-i(g_k+^(2)_kk)]}. which give the uncertainty bound @xmath96 for mcs with @xmath97 _ ( 1^x,1^z)=[2+(1-f_+)-f_-+(g_k+^(2)_kk ) ] + [ fgur4 ] we depict above uncertainty bounds for measurements performed within cavity in fig . [ fgur3 ] . we find that both uncertainty bounds ( [ fgur3 ] ) and ( [ fgur4 ] ) are now periodic in time @xmath98 , which measures the duration of the cavity acceleration , with the period @xmath99 . the evolution of the uncertainty bound can be interpreted as a result of the entanglement generation between the field modes in the single rigid cavity that plays the role of quantum memory @xcite . by properly choosing the parameters to ensure that @xmath100 with @xmath101 , the uncertainty bounds are protected @xcite , recovering the value @xmath102 as in inertial case . while the uncertainty change is very small due to the low acceleration approximation @xmath103 imposed , our results could provide a novel way to detect the relativistic effect by future quantum metrology . depends on the duration time of acceleration of rigid cavity . we choose @xmath104 . for each pair of measurements ( @xmath38 and @xmath48 , @xmath49 and @xmath47 ) , three cures from top to bottom correspond to parameters @xmath105 . the parameter @xmath106 $ ] characterizes the duration time of cavity acceleration . to demonstrate the low acceleration approximation , the uncertainty is estimated under @xmath107 . ] on the other hand , for arbitrary acceleration duration @xmath108 , measurements with outcomes @xmath38 and @xmath49 can be distinguished from each other by the corresponding uncertainty bounds ( [ fgur3 ] ) and ( [ fgur4 ] ) . by a straightforward calculation , it can be shown that measurements @xmath47 share the same bound ( [ fgur4 ] ) with @xmath49 , while @xmath48 has the same bound ( [ fgur3 ] ) with @xmath38 . therefore , same as unruh effect , we conclude that the relativistic motion of a rigid cavity can provoke the distinguishability between the measurements in mubs that share the same bound @xmath102 for an inertial observer . in this paper , we have explorer the nontrivial relativistic modification to the fgur . we have shown that , for the observer undergoes a large acceleration , the associated unruh effect could increase or reduce the fine - grained uncertainty bounds , depending on the choice of unruh modes . nevertheless , the mcs still could be independent of the acceleration , which indicates that once we choose the bases enabling an optimal uncertainty on average in an inertial frame , the corresponding measurements should maintain their optimality for all noninertial observer . moreover , we have shown that the measurements in mubs , sharing same uncertainty bound in inertial frame , could be distinguished from each other when the observer undergoes a nonvanishing acceleration . in an alternative scenario , we have investigated the fgur for the measurements restricted in a single rigid cavity , where the uncertainty bound itself exhibits a periodic evolution with respect to the duration of the acceleration . this phenomenon could be understand by the entanglement generation in a single rigid cavity that plays the role of quantum memory . our results provide a novel way to investigate the relativistic effect from a quantum - information perspective , that could be experimental tested by quantum metrology @xcite . our results could be linked to many interesting issues . for instance , we can generalize above analysis to fundamental mutually unbaised bases in higher dimensional hilbert spaces @xcite , where a @xmath109-qubit can be truncated from free scalar field modes with infinite levels . on the other hand , the fascinating link between fgur and the second thermodynamical law has been explored in @xcite , which proved that a deviation of the fgur implies a violation of the second law of thermodynamics . in this spirit , by investigating the influence of the relativistic motion of observer on a thermodynamical cycle , one could relate the relativistic effect to thermodynamics in an information - theoretic way . in particular , one can investigate the uncertainty relations in general dynamical spacetimes @xcite , e.g. , cosmological background , where the entanglement generation under the evolution of spacetime is expected to play a dramatical role in measurements @xcite . with a proper designed thermodynamical cycle , such investigation might provide us a new perspective on quantum gravity . this work is supported by the australian research council through dp110103434 . j. f. thanks li - hang ren for stimulating discussions . h. f. acknowledges the support of nsfc , 973 program through 2010cb922904 .

one of the most important features of quantum theory is the uncertainty principle . amount various uncertainty relations , the profound fine - grained uncertainty relation ( fgur ) is used to distinguish the uncertainty inherent in obtaining any combination of outcomes for different measurements . in this paper , we explore this uncertainty relation in relativistic regime . for observer undergoes an uniform acceleration who immersed in an unruh thermal bath , we show that the uncertainty bound is dependent on the acceleration parameter and choice of unruh modes . dramatically , we find that the measurements in mutually unbiased bases ( mubs ) , sharing same uncertainty bound in inertial frame , could be distinguished from each other for a noninertial observer . on the other hand , once the unruh decoherence is prevented by utilizing the cavity , the entanglement could be generated from nonuniform motion . we show that , for the observer restricted in a single rigid cavity , the uncertainty exhibits a periodic evolution with respect to the duration of the acceleration and the uncertainty bounds can be degraded by the entanglement generation during particular epoch . with properly chosen cavity parameters , the uncertainty bounds could be protected . otherwise , the measurements in different mubs could be distinguished due to the relativistic motion of cavity . implications of our results for gravitation and thermodynamics are discussed .

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Proceed to summarize the following text: as the quality of photometric data has improved over the years ( largely due to the use of ccds ) , the applicability of a fitting - function which can account for variations in the curvature of a light profile has been demonstrated for elliptical galaxies ( capaccioli 1987 , 1989 ; davies et al . 1988 ; caon , capacciolli & donofrio 1993 ; young & currie 1994 ; graham et al . 1996 ) , and for the bulges of spiral galaxies ( andredakis , peletier & balcells 1995 ; seigar & james 1998 ; moriondo , giovanardi & hunt 1998 ; khosroshahi , wadadekar & kembhavi 2000 ; prieto et al . 2001 ; graham 2001 ; mllenhoff & heidt 2001 ) . these systems are not universally described with either an exponential profile or an @xmath2 law ( de vaucouleurs 1948 , 1959 ) , but rather a continuous range of light profile shapes exist which are well described by the srsic ( 1968 ) @xmath0 model . in ellipticals , the shape parameter @xmath1 from the srsic model is strongly correlated with the other global properties derived independently of the @xmath0 model , such as : total luminosity and effective radius ( caon et al.1993 ; young & currie 1994 , 1995 ; jerjen & bingelli 1997 ; trujillo , graham & caon 2001 ) , central velocity dispersion ( graham , trujillo & caon 2001 ) and also central supermassive black hole mass ( graham et al.2001 ) . additionaly , the spiral hubble type has been shown to correlate with the bulge index @xmath1 such that early type spiral galaxy bulges have larger values of @xmath1 than late type spiral galaxy bulges ( andredakis et al . 1995 ; graham 2001 ) . this correlation arises from the fact that the index @xmath1 is well correlated with the bulge to disc luminosity ratio ( b / d ; see , e.g. simien & de vaucouleurs 1986 ) and this is one of the parameters used to establish morphological type ( sandage 1961 ) . given the abundance of observational work and papers now using the srsic model , it seems timely that a theoretical study is performed on realistic , analytical models following the @xmath0 law . structural and dynamical properties of isotropic , spherical galaxies following @xmath0 models have already been studied in detail in the insightful paper of ciotti ( 1991 ) . however , most elliptical galaxies and bulges of spiral galaxies are known to be non spherical objects . typically , the mass models which have been used for the study of triaxial galaxies have followed analytical expressions which were selected to reproduce the properties of the de vaucouleurs @xmath2 profile ( e.g. jaffe 1983 ; hernquist 1990 ; dehnen 1993 ) , or more recently the modified hubble law ( chakraborty & thakur 2000 ) . for that reason , previous studies based on these kinds of analytical models , although certainly useful , are however unable to probe the full range of properties which are now observed in real galaxies . consequently , it is of importance to know how much room for improvement exits in the study of triaxial objects following the @xmath0 family of profiles . due to the fact that the observed @xmath0 luminosity profiles can not be deprojected to yield analytical expression for the spatial density is an integer . ] , the @xmath0 law has been considered less useful for studies of detailed modelling . an analytical representation ( approximation ) for the mass density profiles which accurately reproduces the observed @xmath0 luminosity profiles would be of great interest for simulations of real galaxies . we have therefore derived just such an analytical expression for the mass density profiles of the srsic family of models . our approximation surpasses the accuracy of both the dehnen models for representing the specific @xmath2 profile and also their extension to the double power law models of zhao ( 1997 ) . in this paper we present a detailed study of how the physical properties of triaxial stellar systems change as a function of the index @xmath1 . an accurate analytical expression for modelling the spatial density is presented in section 2 . in section 3 we explore the axisymmetric and the non axisymmetric components of the potential , forces and torques associated with a srsic light distribution . finally , by using literature available k band observations of a sample of 80 spiral galaxies , the physical basis to the @xmath1@xmath3 ( or @xmath1@xmath4 ) relation is investigated in section 4 . the projected , elliptically symmetric srsic @xmath0 intensity distribution @xmath5 can be written in terms of the projected , elliptical radial coordinate @xmath6 ( see details in trujillo et al . 2001 ) such that : @xmath7 where @xmath8 is the central intensity , and @xmath9 is the effective radius of the projected major axis . curves of constant @xmath6 on the plane of the sky are the isophotes . the quantity @xmath10 is a function of the shape parameter @xmath1 , and is chosen so that the effective radius encloses half of the total luminosity . the exact value is derived from @xmath11@xmath12@xmath13 , where @xmath14 and @xmath15 are the gamma function and the incomplete gamma function respectively ( abramowitz & stegun 1964 , p. 260 ) . the index @xmath1 increases monotonically with the central luminosity concentration of the surface brightness distribution ( trujillo , graham , & caon 2001 ) . the total projected luminosity l associated with this model is given by @xmath16 where @xmath17 is the ellipticity of the isophotes . for a homologous triaxial ellipsoid , the spatial ( deprojected ) luminosity density profile @xmath18 can be obtained by an abel integral equation ( stark 1977 ) : @xmath19(\xi^2-\zeta^2)^{-1/2}d\xi , \ ] ] where @xmath20 is a constant that depends on the three - dimensional spatial orientation of the object ( varela , muoz - tuoz & simonneau 1996 ; simonneau , varela & muoz - tuoz 1998 ) and @xmath21 parametrizes the ellipsoids of constant volume brightness . @xmath20 equals 1 when the proper axis frame of the object has the same orientation as the observer axis frame ( i.e. when the euler angles between the two frames equal zero ) . assume a triaxial object whose mass is stratified over ellipsoids with axis ratios a : b : c ( a@xmath22b@xmath23c ) and the x ( z ) is the long ( short ) axis ( see fig . 1 ) . the symmetry of the problem motivates us to work with ellipsoidal coordinates where : @xmath24 and where @xmath25=b / a and @xmath26=c / a . the mass models considered in this study are the triaxial generalizations of the spherical models discussed in detail by ciotti ( 1991 ) . the mathematical singularities present in eq . 3 were considered and solved by simonneau & prada ( 1999 , eq . substituting eq . 1 into eq . 3 , letting @xmath27 , and multiplying by the mass to light ratio @xmath28m / l , we obtain a similar expression to the one found by these authors : @xmath29 with @xmath30 the dimensionless mass density profiles @xmath31 , where @xmath32 is the total mass , are shown for different values of @xmath1 in fig . it should be noted that the inner density profile decreases more slowly with increasing radius for systems having lower values of @xmath1 . the mass density profiles of the @xmath0 family ( eq . 5 ) can be extremely well approximated by the analytical expression : @xmath33 where @xmath34 , @xmath35 and @xmath36 is the @xmath37th order modified bessel function of the third kind ( abramowitz & stegun 1964 , p. 374 ) . in the appendix a we show the values of the parameters ( @xmath37,p , h@xmath38,h@xmath39,h@xmath40 ) as function of the index @xmath1 . this approximation contains two exact cases : @xmath1=0.5 and @xmath1=1 , and provides relative error less than 0.1% for the rest of the cases ( fig 2b ) in the radial range 10@xmath41 . this approximation surpasses ( by a factor of 10@xmath4210@xmath43 ) the expression presented in lima neto , gerbal & mrquez ( 1999 ) . for three different triaxiality mass distributions : a ) spherical ( @xmath25=@xmath26=1 ) ; b ) moderately triaxial ( @xmath25=0.75 , @xmath26=0.5 ) ; c ) highly triaxial ( @xmath25=0.5 , @xmath26=0.25 ) , we have explored , in detail , the non axisymmetric gravitational field over the z=0 plane ( i.e. the disc plane when studying spiral galaxies ) . we evaluate this quantity by calculating : @xmath44 where @xmath45 and @xmath46 are the m=2 and m=0 component of the gravitational potential , such that the nth order term @xmath47 is evaluated from the gravitational potential on the z=0 plane @xmath48 by using the fourier decomposition ( see , e.g. combes & sanders 1981 ) . gravitational potential and gravitational force expressions are shown on appendix b. the profiles of @xmath49 for different triaxialities and values of @xmath1 are shown in fig . 3 . as it is expected , as the triaxiality increases the non spherical nature of the gravitational field increases . also , we highlight that for a given triaxiality , smaller values of @xmath1 ( i.e less concentrated mass distribution ) give greater non - spherical gravitational fields . the maximum non axisymmetrical behavior of the potential is obtained at radial distances less than 2 @xmath9 . this radial distance is also a function of the index @xmath1 , decreasing a @xmath1 increases and remains quite independent of the triaxiality of the object . for a moderately triaxial object with @xmath1=1 , the non axisymmetrical component of the potential can vary some 6% between r=0 and r=2@xmath9 , and varies some 15% for our highly triaxial model . for an @xmath1=1 model , and starting from our moderately triaxial case ( @xmath25=0.75 , @xmath26=0.50 ) , we increased the value of @xmath26 to 0.75 . the results are shown in fig . 3c and reveal that g(r ) varied only mildly . this figure shows that the non axisymmetric effect ( along the radial distance ) in the z=0 plane is mainly due to how the mass of the bulge is distributed in this plane . the non spherical component of the radial gravitational forces in the z=0 can be estimated by : @xmath50 in fig . 3 the n(r ) profiles ( eq . 9 ) are evaluated for the same cases as was the g(r ) profiles . a remarkable point is that n(r ) reaches its maximum value in the radial range 2 @xmath9@xmath51r@xmath514 @xmath9 . for a spiral bulge structure , this means that the most important non axisymmetric effects take place in a zone which is dominated by the disc . as with the g(r ) parameter , stronger distortions occur as the triaxiality increases and the index @xmath1 decreases . the mechanism which controls this distortion is basically determined by the mass distribution in the z=0 plane ( fig 3c ) . it is noted that the relative ( i.e. % change ) non axisymmetric effects on the radial forces are larger than the relative distortion on the potential . as an example , for a moderately triaxial structure with @xmath1=1 the non axisymmetric component of the radial forces can reach 8% . the torques provoked by the triaxial structures along the angular coordinate are evaluated around the circle of radius @xmath52 where the maximum non axisymmetrical distortion of the radial forces is produced ( i.e. at the peak of the n(r ) profile ) . given the gravitational potential @xmath48 in the z=0 plane , we have at the radius @xmath52 : @xmath53 where @xmath54/r_{max}$ ] represents the amplitude of the tangential force along the angular coordinate at radius @xmath52 , and @xmath55 is the radial force at this radius . due to the symmetry of the ellipsoid , the values of @xmath56 need only be plotted for one quadrant in the z=0 plane ; we use 0@xmath57@xmath5890@xmath57 ( fig . 3 ) . depending on the quadrant , @xmath56 is either negative or positive because the sign of the tangential force changes from quadrant to quadrant . the maximum torque around a circle of radius @xmath52 depends on the triaxiality of the object . as the triaxiality increases the maximum torque tends to be closer to the major axis as one would expect . the position of this peak is quite independent of the value of @xmath1 . the absolute value of the torque for any given triaxiality increases as @xmath1 decreases . for our highly triaxial bulge , @xmath56 ranges from 0.17 ( @xmath1=10 ) to 0.24 ( @xmath1=1 ) , which would be considered a `` bar strength '' class of 2 in the classification scheme of buta & block ( 2001 ) . in the case of our moderately triaxial object , the maximum absolute value of @xmath56 ranges between 0.06 and 0.09 . these values correspond to a `` bar strength '' class of 1 . thus , even a moderately triaxial bulge is capable of provoking non negligible torques on a disk that is to say , a bar is not neccessarily required . a detailed study separating the torque contribution from both bars and bulges would of course be of interest , and it is our intention to add a range of bar potentials to our models in the future . as with the previous parameters , for the range of triaxialities investigated and a given @xmath1 , varying the mass distribution along the z axis ( i.e. varying the triaxiality parameter @xmath26 ) only results in a slight change to @xmath56 ( see fig . 3c ) . for a spherical distribution all above parameters in the previous section we have seen how the non axisymmetrical effects ( in the z=0 plane ) from a triaxial bulge increase as @xmath1 decreases . taken with the correlation between @xmath1 and galaxy type ( andredakis , peletier & balcells 1995 ) shown in fig . 4 , this invokes the natural question : how , if at all , are the structural properties of the bulges ( i.e. @xmath1 ) related ( that is , cause and effect ) with the non axisymmetrical components ( i.e. arms ) observed in the disc ? the results obtained in the previous sections were evaluated without any mention to the relative mass of the bulge and disc . it turns out that the axisymmetrical mass distribution of the disc causes a strong softening of the non axisymmetrical perturbation caused by the non sphericity of the bulge . the degree of `` smoothing '' is an increasing function of the @xmath59 ratio . 5 shows the n(r ) profile for a moderately triaxial bulge with @xmath60 and @xmath4=0.1 and 0.01 , and for a bulge with @xmath61 and @xmath4=1.0 and 0.1 ( fig . n(r ) was evaluated here assuming the disc follows an exponential surface brightness distribution . the ratio between the length scale of the disc and the effective radius of the bulge is assumed to be constant and with a value of @xmath62=5 . in the @xmath63-band is 5 ( graham & prieto 1999 ) . ] the expressions for the potential and the radial force of these structures can be found in binney & tremaine ( 1987 , p. 77 and 78 ) . 5 illustrates that for @xmath4 luminosity ratios typical of real galaxies , the non axisymmetrical effects on the disc largely disappear ( @xmath64 ) . for @xmath4=1 , the values of the n(r ) profile remain basically unchanged to that seen in fig . 3 , but for @xmath4=0.1 these values decrease approximately by a factor 2 , and for @xmath4=0.01 this factor is bigger than 10 . thus , although the non radial effects on the z=0 plane increase as @xmath1 decreases , the smoothing effects of the increasingly dominant disc are stronger . bulges with small values of @xmath1 are unable to produce significant non axisymmetrical effects on a massive disc . to explore the connection between bulges and discs in spiral galaxies ( see , e.g. fuchs 2000 ) , we have used the data from two independent samples of galaxies observed in the k band . the k band provides a good tracer of the mass due to the near absence of dust extinction and the reduced biasing effect of a few per cent ( in mass ) of young stars . we used the data from andredakis et al . ( 1995 ) and the structural analysis of the de jong ( 1996 ) data performed by graham ( 2001 ) . both studies were done fitting a seeing convolved srsic law to the spiral galaxy bulges . in both samples we have removed those objects which contained a clear bar structure , leaving a total of 28 objects from andredakis et al . ( 1995 ) and 52 objects from graham ( 2001 ) . the relations present in fig . 4 between @xmath1 and the @xmath4 luminosity ratio , and @xmath1 versus the morphological type t have a spearman rank order correlation coefficient of @xmath66 and @xmath67 , respectively , for the combined sample . andredakis et al . ( 1995 ) suggest `` although other possibilities can not be excluded , the most straightforward explanation for this trend is that the presence of the disc affects the density distribution of the bulge in such a way as to make the bulge profile steeper in the outer parts . one mechanism to produce such an effect might be that a stronger disc truncates the bulge , forcing its profile to become exponential '' . following this line of thought , via collisionless n body simulations , andredakis ( 1998 ) studied the adiabatic growth of the disc onto an existing @xmath2 spheroid . he found that the disc potential modifies the bulge surface brightness profile , lowering the index @xmath1 . this decrease was larger with more massive and more compact discs . this mechanism , however , saturated at around @xmath1=2 and exponential bulges could not be produced . we believe that this line of reasoning is not the most appropiate explanation for the relation between @xmath1 and @xmath4 . firstly , we find that the index @xmath1 is not only well correlated with the luminous @xmath4 ratio , but is equally well correlated ( @xmath68 ) with bulge luminosity @xmath69 ( fig . additionally , the correlation between @xmath1 and disc luminosity @xmath70 is relatively poor ( @xmath71 ) . secondly , @xmath69 is more strongly correlated ( @xmath72 ) with the @xmath4 ratio than @xmath70 and the @xmath4 ratio ( @xmath73 ) ( fig . hence , it is variations in the bulge which are predominantly responsible for variations in the @xmath4 ratio . these above two correlations seem to indicate that @xmath1 may be related directly with the properties of the bulge rather than with the combined @xmath4 ratio . consequently , as @xmath1 is correlated with the total bulge luminosity , the correlation between @xmath1 and @xmath4 is a result of the more fundamental correlation between @xmath69 and @xmath4 . that is , it is not the relative increase in disc to bulge luminosity which produces bulges with smaller values of @xmath1 , but simply that bulges with larger values of @xmath1 are more luminous ( or vice versa ) and this produces the correlation between @xmath1 and the @xmath4 luminosity ratio . favouring this argument , we note that among the elliptical galaxies ( without the need to invoke any disc ) there exists a strong correlation ( pearson s r=-0.82 ; graham , trujillo & caon , 2001 ) between @xmath1 and the total luminosity of these objects . the index @xmath1 of pressure supported stellar systems are related with the total luminosity of these structures . in agreement with this , aguerri , balcells & peletier ( 2001 ) have found ( using collisionless n body simulations ) that the bulges of late type galaxies can increase their @xmath1 values via dense satellite accretions where the new value of @xmath1 is found to be proportional to the devoured satellite mass . due to the strong correlation between the @xmath4 luminosity ratio and @xmath69 , it might be of interest to ask which one of these quantities is preferred to establish the morphological type t of a galaxy . working from b band images ( which is good for observing the young star population , and consequently the spiral arm structure , which is one of the basic criteria to the hubble galaxy classification ) , simien & de vaucouleurs ( 1986 ) fitted @xmath2 profiles and exponential discs to a sample of 64 spiral galaxies and 34 s0 type galaxies . they presented a good correlation between the bulge to disc luminosity and @xmath3 , but not between @xmath74 and @xmath3 . consequently , their b band observations suggested that the @xmath4 luminosity ratio was preferred to @xmath74 for establishing the morphological type t. in fig . 7 we show the relation between the @xmath4 luminosity ratio and t ( @xmath75 ) and between @xmath69 and t ( @xmath76 ) . thus , from k band observations , and fitting @xmath0 bulge profile models , the change in the luminous mass of the bulge along the hubble sequence appears equally as important as the combined change in the bulge and disc luminosity between the @xmath4 luminosity ratio and t , but only @xmath77 between @xmath74 and t. ] . it would then follow that the luminous mass of the bulge ( i.e. @xmath69 ) is related with the spiral arm structure . the main results of this work are the following : \a ) we have generalised the analysis of the physical properties of spherical stellar systems following the @xmath0 luminosity law to a homologous triaxial distribution . the density distribution , potential , forces and torques are evaluated and compared with the spherical case when applicable ( ciotti 1991 ) . an extremely accurate analytical approximation ( relative error less than 0.1% ) for the mass density profile is provided . \b ) we derive an exact expression showing how the central potential decreases as triaxiality increases . we also show that for a fixed triaxiality , as the index @xmath1 decreases the non axisymmetrical effects in the z=0 plane increase . even for a moderately triaxial object , the non axisymmetrical component of the potential and the radial forces are not negligible for small values of @xmath1 . these components can range from 6 to 8% , respectively , compared to the value of the spherical component . for our highly triaxial model , they can range over some 20% . \c ) the non axisymmetrical effects in the disc plane due to the bulge structure are strongly reduced when an axisymmetrical disc mass is added . for this reason , bulges with smaller values of @xmath1 appear unlikely to produce any significant non axisymmetrical effect on their disc , which is typically 10 to 100 more times more massive than the bulge . in this regard , the @xmath4 mass ratio and the triaxiality of the bulge are more important , that is , can dominate over the effects of small @xmath1 . \d ) the correlation found between @xmath1 and the @xmath4 luminosity ratio found in spiral galaxies is explained here not as a consequence of the interplay between the bulge and the disc , but due to the strong correlation between @xmath1 and @xmath78 , and between @xmath78 and @xmath4 . also , k band data do not support the idea that the @xmath4 luminosity ratio can be preferred over @xmath78 as an indicator to establish galaxy morphological type ( t ) . both parameters present equally good correlations with galaxy type t. 99 abramowitz m. , stegun i. , 1964 , handbook of mathematical functions . dover , new york aguerri , j.a.l . , balcells m. , peletier r. , 2001 , a&a , 367 , 428 andredakis y.c . , peletier r. , balcells m. , 1995 , mnras , 275 , 874 andredakis , y.c . , 1998 , mnras , 295 , 72 binney , j. , tremaine , s. : 1987 , galactic dynamics , princeton university press , princenton buta r. , & block d.l . , 2001 , apj , 550 , 243 block , d.l . , puerari , i. , frogel j.a . , eskridge , p.b . , stockton a. , burkhard f. , 1999 , astroph . and space sci , 269270 , 5 caon n. , capaccioli m. , donofrio 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graham a.w . , prieto m. , 1999 , apj , 524 , l23 graham a.w . , 2001 , aj , 121 , 820 graham a.w . , trujillo , i. , caon , n. , 2001 , aj , 122 , 1707 graham a.w . , erwin , p. , caon , n. , trujillo , i. , 2001 , apj , 563 , l11 hernquist , l. , 1990 , apj , 356 , 359 jaffe , w. , 1983 , mnras , 202 , 995 jerjen , e. , bingelli , b. 1997 , in the nature of elliptical galaxies ; the second stromlo symposium , asp conf . ser . , 116 , 239 khosroshahi h.g . , wadadekar y. , kembhavi a. , 2000 , apj , 533 , 162 lima neto g. b. , gerbal d. & mrquez i. , 1999 , mnras , 309 , 481 mazure a. & capelato h.v . 2002 , a&a , in press mllenhoff c. , heidt j. , 2001 , a&a , 368 , 16 moriondo g. , giovanardi c. , hunt l.k . , 1998 , a&as , 130 , 81 prieto , m. , aguerri , j.a.l . , varela , a.m. & munoz - tunon , c. , 2001 , a&a , 367 , 405 sandage , a. , 1961 . the hubble atlas of galaxies , washington : carnegie institution seigar , m. & james , p. , 1998 , mnras , 299 , 672 srsic j. 1968 . atlas de galaxias australes crdoba : obs . astronmico simien , f. , de vaucouleurs , g. , 1986 , apj , 302 , 564 simonneau e. , varela a. , muoz tuoz c. , 1998 , il nuovo cimento , 113b , 927 simonneau e. , prada f. , 1999 , ( astro ph/9906151 ) stark a. , 1977 , apj , 213 , 368 trujillo , i. , aguerri , j.a.l . , cepa , j. , gutierrez , c.m . , 2001 , mnras , 321 , 269 trujillo , i. , graham , a.w . , caon , n. , 2001 , mnras , 326 , 869 varela a. , muoz tuoz c. , simonneau e. , 1996 , a&a , 306 , 381 young , c.k . , currie , m.j . , 1994 , mnras , 268 , l11 young , c.k . , currie , m.j . , 1995 , mnras , 273 , 1141 zhao h. , 1997 , mnras , 287 , 525 the table a1 shows the values of the parameters that appear in the mass density approximation ( eq . 7 ) . [ cols="<,^,^,^,^,^,^,^",options="header " , ] ( chandrasekhar 1969 , p. 52 , theorem 12 ) , with @xmath81 it follows from eq . ( b1 ) that the potential at an internal point is a result of two contributions : that due to the ellipsoid interior to the point @xmath82 considered , and that due to the homoeoidal shell exterior to @xmath82 : @xmath83,\end{aligned}\ ] ] with f(p , q ) the elliptic integral of the first kind ( abramowitz & stegun 1964 , p. 589 ) , and with the restriction @xmath84 as reported in ciotti ( 1991 ) , the models with low @xmath1 have an inner ( @xmath86 ) potential which is much flatter than models with high @xmath1 . as the triaxiality increases there is no important change to the gradient of the gravitational potential along the semimajor axis ; the main effect is to shift the gravitational potential profile inwards from the spherical case , resulting in a lower potential at intermediate radii . the gravitational forces for a triaxial structure are given by the expression : @xmath87 with @xmath88 a , @xmath89 b , @xmath90 c , and @xmath91 ^ 2}.\ ] ] the restrictions given in eqs . ( 6 ) and ( b4 ) also apply here .

we have investigated the structural and dynamical properties of triaxial stellar systems whose surface brightness profiles follow the @xmath0 luminosity law extending the analysis of ciotti ( 1991 ) who explored the properties of spherical @xmath0 systems . a new analytical expression that accurately reproduces the spatial ( i.e. deprojected ) luminosity density profiles ( error less than 0.1% ) is presented for detailed modelling of the srsic family of luminosity profiles . we evaluate both the symmetric and the non axisymmetric components of the gravitational potential and force and compute the torques as a function of position . _ for a given triaxiality , stellar systems with smaller values of @xmath1 have a greater non axisymmetric gravitational field component_. we also explore the strength of the non axisymmetric forces produced by bulges with differing @xmath1 and triaxiality on systems having a range of bulge to disc ratios . the increasing disc to bulge ratio with increasing galaxy type ( decreasing @xmath1 ) is found to heavily reduce the amplitude of the non axisymmetric terms , and therefore reduce the possibility that triaxial bulges in late type systems may be the mechanism or perturbation for non symmetric structures in the disc . using seeing convolved @xmath0bulge plus exponential disc fits to the k band data from a sample of 80 nearby disc galaxies , we probe the relations between galaxy type , srsic index @xmath1 and the bulge to disc luminosity ratio . these relations are shown to be primarily a consequence of the relation between @xmath1 and the total bulge luminosity . in the k band , the trend of decreasing bulge to disc luminosity ratio along the spiral hubble sequence is predominantly , although not entirely , a consequence of the change in the total bulge luminosity ; the trend between the total disc luminosity and hubble type is much weaker . = = = = = = = = # 1 # 1 # 1 # 1 @mathgroup@group @mathgroup@normal@groupeurmn @mathgroup@bold@groupeurbn @mathgroup@group @mathgroup@normal@groupmsamn @mathgroup@bold@groupmsamn = `` 019 = ' ' 016 = `` 040 = ' ' 336 = " 33e = = = = = = = = # 1 # 1 # 1 # 1 = = = = = = = = [ firstpage ] celestial mechanics : stellar dynamics galaxies : structure of galaxies : photometry galaxies : elliptical and lenticular , cd galaxies : spiral galaxies : kinematics and dynamics

You are an expert at summarizing long articles. Proceed to summarize the following text: given a link @xmath2 and a linear group @xmath3 , a _ tracefree _ ( or _ traceless _ ) @xmath3-representation of @xmath0 means a homomorphism @xmath4 sending each meridian to an element of trace zero . dated back to 1980 , magnus @xcite used tracefree @xmath5-representations to prove the faithfulness of a representation of braid groups in the automorphism groups of the rings generated by the characters functions on free groups . lin @xcite used tracefree @xmath6-representations to define a casson - type invariant of a knot @xmath7 , and showed it to equal half of the signature of @xmath7 . more interestingly , kronheimer and mrowka @xcite observed that for some knots @xmath7 , its khovanov homology is isomorphic to the ordinary homology of the space @xmath8 of conjugacy classes of tracefree representations of @xmath7 . in this context , zentner @xcite determined @xmath8 when @xmath7 belongs to a class of classical pretzel knots . for related works , one may refer to @xcite , @xcite , etc . there are relatively fewer results on tracefree @xmath5-representations . for a knot @xmath9 , nagasato @xcite gave a set of polynomials whose zero loci is exactly the tracefree characters of irreducible @xmath5-representations of @xmath7 . nagasato and yamaguchi @xcite investigated tracefree @xmath5-representations of @xmath10 ( where @xmath7 is a knot in an integral 3-sphere @xmath11 ) , and related to those of @xmath12 , where @xmath13 is the 2-fold cover of @xmath11 branched along @xmath7 . in this paper , for each montesinos link , we completely determine the tracefree @xmath5-representations by given explicit formulas . let @xmath14 . note that each @xmath15 satisfies @xmath16 . by a tangle " we mean an unoriented tangle diagram . given a tangle @xmath17 , let @xmath18 denote the set of directed arcs of @xmath17 , ( each arc gives two directed arcs ) . by a _ ( tracefree ) representation _ of a tangle diagram @xmath17 , we mean a map @xmath19 such that @xmath20 for each @xmath21 and at each crossing illustrated in figure [ fig : crossing ] , @xmath22 . to present such a representation , it is sufficient to give each arc a direction and label an element of @xmath23 beside it . [ h ] at each crossing , title="fig:",scaledwidth=30.0% ] + [ h ] , with the four ends directed outwards , title="fig:",scaledwidth=33.0% ] + let @xmath24 denote the set of tangles @xmath17 with four ends which , when directed outwards , are denoted by @xmath25 , as shown in figure [ fig : general ] . the simplest four ones are given in figure [ fig : basic ] . in @xmath24 there are two binary operations : _ horizontal composition _ @xmath26 and _ vertical composition _ @xmath27 ; see figure [ fig : composition ] . [ h ] $ ] , ( b ) @xmath28 $ ] , ( c ) @xmath29 $ ] , ( d ) @xmath30$],title="fig:",scaledwidth=80.0% ] + [ h ] ; ( b ) @xmath31,title="fig:",scaledwidth=60.0% ] + for @xmath32 , the horizontal composite of @xmath33 copies of @xmath29 $ ] ( resp . [ -1 ] ) is denoted by @xmath34 $ ] if @xmath35 ( resp . @xmath36 ) , and the vertical composite of @xmath33 copies of @xmath29 $ ] ( resp . [ -1 ] ) is denoted by @xmath37 $ ] if @xmath35 ( resp . @xmath36 ) . given integers @xmath38 , the _ rational tangle _ @xmath39,\ldots,[k_{m}]]$ ] is defined as @xmath40\star [ 1/k_{2}]\cdot\ldots\star [ 1/k_{m } ] , & & \text{if\ \ } 2\mid m , \\ & [ k_{1}]\star [ 1/k_{2}]\cdot\ldots\cdot [ k_{m } ] , & & \text{if\ \ } 2\nmid m;\end{aligned}\ ] ] and its _ fraction _ is defined by @xmath41,\ldots,[k_{m}]])=[[k_{1},\ldots , k_{m}]]^{(-1)^{m-1 } } , \label{eq : fraction}\end{aligned}\ ] ] where the _ continued fraction _ @xmath42\in\mathbb{q}$ ] is defined inductively as @xmath43=k_{1 } , \qquad [ [ k_{1},\ldots , k_{m}]]=k_{m}+1/[[k_{1},\ldots , k_{m-1}]].\end{aligned}\ ] ] denote @xmath39,\ldots,[k_{m}]]$ ] as @xmath44 $ ] if the right - hand side of ( [ eq : fraction ] ) equals @xmath45 . a _ montesinos tangle _ is a tangle of the form @xmath46\star\ldots\star [ p_{r}/q_{r}]$ ] . the link obtained by connecting @xmath47 and @xmath48 with @xmath49 and @xmath50 , respectively , is called a _ montesinos link _ and denoted by @xmath51 . given a representation @xmath52 of @xmath53 , denote @xmath54 suppose @xmath42=[p / q]$ ] with @xmath55 . for a representation @xmath52 of @xmath44 $ ] , let @xmath56 denote the elements that @xmath52 assigns to the directed arcs shown in figure [ fig : rational ] . call @xmath57 the _ generating pair _ of @xmath52 , indicating that @xmath52 is determined by @xmath58 and @xmath59 . [ h ] , [ k_{2}],[k_{3}]]$],title="fig:",scaledwidth=45.0% ] + suppose @xmath60 . then @xmath61 , hence each @xmath62 or @xmath63 can be expressed as @xmath64 , where @xmath65 are polynomials in @xmath66 that do not depend on @xmath67 . we take a clever approach to derive formulas for the coefficients @xmath65 . for @xmath68 , put @xmath69 [ lem : standard ] ( i ) if @xmath52 is a representation of @xmath34 $ ] with @xmath70 and @xmath71 , then @xmath72 and @xmath73 . \(ii ) if @xmath52 is a representation of @xmath37 $ ] with @xmath74 and @xmath75 , then @xmath76 and @xmath77 . \(i ) suppose @xmath52 is a representation of @xmath29 $ ] with @xmath70 and @xmath71 , then @xmath78 , and computing directly , @xmath79 applying this repeatedly , we obtain the result . ( ii ) the proof is similar . suppose @xmath80 and @xmath81 . by lemma [ lem : standard ] , @xmath82 , @xmath83 , and in general , @xmath84 with @xmath85 consequently , for @xmath86 , @xmath87 define @xmath88 , @xmath89 , inductively by @xmath90 so that @xmath91 $ ] , @xmath92 $ ] , then @xmath93 put @xmath94 i.e. , @xmath95^{(-1)^{m-1}}. \label{eq : tilde}\end{aligned}\ ] ] one can prove by induction on @xmath96 that @xmath97 we have @xmath98 hence @xmath99 for an integer @xmath9 , denote @xmath100 it can be written as a polynomial in @xmath101 . noticing @xmath102 we obtain @xmath103 and when @xmath104 , @xmath105 as pointed out in the second paragraph of this section , these relations are actually valid for arbitrary @xmath67 . for @xmath106 , denote @xmath107 by @xmath108 . [ rmk : regular ] for @xmath109 , call @xmath110 _ regular _ if there exists @xmath111 such that @xmath112 and @xmath113 with @xmath114 . it is easy to see that @xmath110 is non - regular if and only if @xmath115 and @xmath116 , and under this condition , there exists @xmath111 such that @xmath112 and @xmath117 , where @xmath118 according to ( [ eq : ne])-([eq : se ] ) , the four pairs @xmath119 , @xmath57 , @xmath120 and @xmath121 are simultaneously regular or not . given a representation @xmath52 , say @xmath52 is _ reducible _ if all the elements in @xmath122 have a common eigenvector ; in particular , say @xmath52 is _ abelian _ if @xmath122 is abelian . call @xmath52 _ irreducible _ if it is not reducible . suppose @xmath52 is a representation of @xmath123 . let @xmath124 denote its restriction to @xmath125 $ ] , let @xmath126 denote the generating pair of @xmath124 , and assume @xmath127 . up to conjugacy we may assume @xmath128 . the @xmath129 s have a common value which we denote by @xmath130 , then for each @xmath131 , by ( [ eq : tr - h ] ) , @xmath132 with @xmath133 defined as in ( [ eq : tilde ] ) ; switching @xmath134 with @xmath135 if necessary , we have @xmath136 if @xmath52 is reducible and non - abelian , then @xmath137 is non - regular for at least one of the @xmath131 s , which is , by remark [ rmk : regular ] , equivalent to that the @xmath137 s are all non - regular , hence @xmath138 and @xmath139 with @xmath140 . by ( [ eq : ne ] ) , @xmath141 and @xmath142 are determined by each other ; by ( [ eq : linear ] ) , @xmath143 observing @xmath144 , we have @xmath145 conversely , when @xmath146 satisfy @xmath147 and ( [ eq : reducible ] ) holds , an arbitrary @xmath148 gives rise to @xmath149 and then @xmath150 for all @xmath131 through @xmath151 , ( [ eq : relation ] ) and ( [ eq : ne ] ) . the @xmath150 s combine to define a non - abelian reducible representation of @xmath17 . if @xmath138 and @xmath137 is regular for each @xmath131 , then @xmath152 and @xmath153 is also regular . conversely , given @xmath154 and @xmath155 such that @xmath156 and ( @xmath157 for convention ) @xmath158 for each @xmath131 , there is a unique representation @xmath52 of @xmath17 such that @xmath159 actually , @xmath142 is determined by @xmath160 and @xmath161 as in ( [ eq : sw ] ) : @xmath162 on the other hand , one can show that any given @xmath179 can be written as @xmath180 for some @xmath181 , hence any @xmath182 satisfying ( [ eq : x ] ) can be obtained as follows : take @xmath183 arbitrarily , and then take @xmath184 to fulfill ( [ eq : d - e ] ) . suppose @xmath52 is irreducible and @xmath190 . write @xmath191 , then for each @xmath131 , @xmath192 by ( [ eq : linear ] ) , @xmath193 observing @xmath194 , we are lead to @xmath195 which is , by ( [ eq : p - q ] ) , equivalent to @xmath196 if @xmath197 , then @xmath198 ( which is equivalent to @xmath199 ) , and @xmath200 we may assume @xmath201 where @xmath202 is the numerator of @xmath203 . if @xmath204 , then @xmath205 can be arbitrary , and @xmath206 should satisfy @xmath207 1 . if @xmath52 is abelian , then it is determined by a unique tuple @xmath209 with ( [ eq : s - ell ] ) ; 2 . if @xmath52 is reducible but not abelian , then @xmath204 and up to the two choices @xmath210 , @xmath52 is determined by a unique tuple @xmath209 satisfying ( [ eq : s - ell ] ) and ( [ eq : condition ] ) ; 3 . if @xmath52 is irreducible with @xmath211 , then @xmath52 is determined by @xmath205 and a unique tuple @xmath212 such that ( [ eq : s - ell ] ) and ( [ eq : x ] ) are satisfied and @xmath213 for at least one @xmath131 ; 4 . if @xmath204 and @xmath52 is irreducible with @xmath214 , then @xmath52 is determined by @xmath205 and a unique tuple @xmath215 with ( [ ineq : n ] ) and ( [ eq : final ] ) ; 5 . if @xmath197 and @xmath52 is irreducible with @xmath216 , then @xmath199 and @xmath52 is determined by a unique tuple @xmath215 with ( [ ineq : n ] ) and @xmath217 is not an integral multiple of @xmath203 .

given a link @xmath0 , a representation @xmath1 is _ tracefree _ if the image of each meridian has trace zero . we determine the conjugacy classes of tracefree representations when @xmath0 is a montesinos link .

You are an expert at summarizing long articles. Proceed to summarize the following text: let @xmath1 be the potential energy function of a classical model for a molecular system . a fundamental challenge is to sample the configurational gibbs - boltzmann ( canonical ) distribution with density @xmath2 where @xmath3 where @xmath4 is boltzmann s constant , @xmath5 is temperature , and @xmath6 is a normalization constant so that @xmath7 has unit integral over the entire configuration space . a wide variety of methods are available to calculate averages with respect to @xmath7 ; among these , some of the most popular are based on brownian dynamics or langevin dynamics ( defined in the phase space of positions and momenta ) @xcite . ( in this article we focus exclusively on molecular dynamics techniques ; molecular models can also be sampled using monte - carlo methods , and , more generally , using hybrid algorithms which combine molecular dynamics with a metropolis - hastings test in order to correct averages . for a recent review of such schemes see @xcite . ) recall that brownian dynamics ( overdamped langevin dynamics ) is a system of it - type stochastic differential equations of the form @xmath8 where @xmath9 is the infinitesimal increment of a vector of stochastic wiener processes @xmath10 , and @xmath11 is a positive ( we assume here diagonal ) mass matrix . a simple and popular method for numerical solution of eq . [ bd ] is the euler - maruyama method @xmath12 where @xmath13 is a vector of random variables with standard normal distribution . this produces a sequence of points @xmath14 , which , following a certain relaxation period , are approximately distributed according to the canonical invariant distribution . euler - maruyama has the property that the time averages along discrete trajectories , in the limit of large time ( under appropriate conditions on the potential @xmath15 and assuming no effects from floating point rounding error ) , have error proportional to @xmath16 . one of the observations of this article is that the simple modification @xmath17 provides a second order approximation of stationary averages . we arrive at this scheme by considering the large friction limit in a particular numerical method for langevin dynamics . recall that langevin dynamics is a stochastic - dynamical system involving both positions and momenta @xmath18 of the form @xmath19{\rm d}t + \sigma m^{1/2}{\rm d}w , \label{lang}\ ] ] where @xmath20 is again a vector of @xmath0 independent wiener processes , and @xmath21 is a free parameter , the friction coefficient . the methods that are in fact the primary focus of this paper are splitting integrators that decompose the stochastic vector field of langevin dynamics into simpler vector fields which can be solved exactly . the composition method that results can not be directly related to a stochastic differential equation , so the analogy with backward error analysis for deterministic problems @xcite is incomplete , but nonetheless the invariant measure associated to the numerical method can be derived , using the baker - campbell - hausdorff ( bch ) expansion , as an asymptotic series in two parameters : the stepsize and the reciprocal of the friction coefficient . the superconvergence result alluded to above is then obtained in the high friction limit . the langevin stepsize must be understood to be proportional to the square root of the stepsize that appears in eq . [ bd - mod ] , so in langevin dynamics an effective 4th order approximation is obtained , but only for the marginal configurational invariant distribution , eq . [ configdist ] . our approach also provides a simple method for comparative assessment of the invariant measure of a class of langevin integrators . molecular dynamics is a large family of modelling techniques which is widely used in different application areas and for different purposes @xcite . this article is addressed specifically to the topic of calculating averages with respect to an invariant distribution , and will probably be of highest interest for applications in molecular sampling , in particular the calculation of averages with respect to the configurational density eq . [ configdist ] . the high friction limit renders langevin dynamics unsuitable for dynamical modelling ( except as a method for generating starting configurations for dynamical exploration ) . it is worth noting that invariant measure computations arise frequently in applications other than molecular modelling , and the techniques described here would be of potential use in many of these . in large scale simulations , it should be understood that the statistical error ( dependent on the number of samples used ) is typically the dominant concern . our approach focuses on the truncation error of the invariant distribution , thus the greatest benefit would be seen only when the statistical error is well controlled . nonetheless we observe in our numerical experiments that the least biased scheme from the point of view of the error introduced in configurational sampling is also as efficient as the alternatives and the most robust with respect to variation of the parameters ( stepsize , friction coefficient ) , thus there is effectively no price for the improvement in accuracy . moreover , it would be possible to complement the methods proposed here by procedures such as importance sampling @xcite to further reduce the statistical error . with regard to the approximation of canonical averages , methods have previously been constructed for brownian dynamics with order @xmath22 and for langevin dynamics with order @xmath23 @xcite , but these require multiple evaluations of the force ; for this reason they are not normally viewed as competitive alternatives for molecular sampling @xcite . by contrast all of the methods described in this article use a single force evaluation at each timestep . the approach used here may be compared to other recent works on stochastic numerical methods , and , in particular @xcite . our technique differs from these in ( i ) the direct focus on the stationary configurational distribution , and ( ii ) the use of the bch expansion . other articles ( see e.g. @xcite ) which address the invariant measure in langevin - type stochastic differential equations do not use the backward error analysis ( and do not find the superconvergent scheme ) . the rest of this article is as follows . in section 2 we review necessary background on stochastic differential equations for sampling from the gibbs measure . section 3 presents our expansion of the associated perturbed invariant measure and calculations involving the use of the baker - campbell - hausdorff theorem . section 4 describes the reduction of the methods in the case of overdamped langevin dynamics . section 5 demonstrates the theory obtained using numerical experiments to verify the results . the ornstein - uhlenbeck ( ou ) stochastic differential equation is an it equation of the form @xmath24 where @xmath25 is a random variable defined for each time @xmath26 , @xmath27 and @xmath28 are positive parameters and @xmath29 represents the infinitesimal increment of a wiener process . langevin dynamics ( eq . [ lang ] ) combines the ornstein - uhlenbeck stochastic vector field with conservative dynamics . for stochastic differential equations , the density evolves according to an evolution equation ( the fokker - planck or forward kolmogorov equation ) of the form @xmath30 where @xmath31 is a second order differential operator . in the case of langevin dynamics , the relevant kolmogorov operator is defined by its action on a function @xmath32 of the variables of the system by @xmath33 where @xmath34 is the mass - weighted laplacian in the momenta : @xmath35 . by choosing @xmath36 one easily checks that the gibbs distribution with density @xmath37 is a steady state of the kolmogorov equation , where @xmath38 is a normalising constant and @xmath39 represents the hamiltonian energy function . for later reference , we define averages with respect to the gibbs distribution by @xmath40 assuming @xmath41 is @xmath42 it is possible to demonstrate that the operator @xmath43 is hypoelliptic by using hrmander s criterion @xcite based on iterated commutators , and this implies that the gibbs measure is the unique steady state ( up to normalization ) . even stronger is the result of @xcite which demonstrates the existence of a spectral gap for the operator @xmath44 . many of the challenges related to obtaining formal analytical results for stochastic differential equations relate to singularities of the potential and/or the assumption of an unbounded solution domain . however , with periodic boundary conditions and strong repulsive potentials ( e.g. lennard - jones potentials ) we observe that configurations typically evolve in a bounded set and remain far from singular points ( the radial distribution vanishes in a large interval around the origin ) . indeed it is a simple calculation to demonstrate that for a lennard - jones system with potentials @xmath45 the expected number of samples required to observe @xmath46 at unit temperature would involve a simulation of duration far greater than the age of the universe , due to the steepness of the potential close to the origin . in our simulations of a small lennard - jones cluster in section 5 , we did not observe a separation of two atoms beneath @xmath47 in all of the nearly @xmath48 timesteps performed to gather statistics ; a separation less than @xmath49 could only be seen at very large stepsizes , when instabilities due to other components of model , e.g. harmonic bonds , would have anyway rendered a typical molecular simulation useless . thus it is somewhat of a moot point whether we simply assume that configurations stay well away from singular points ( domain restriction ) or that the potential has been smoothly cut off to remove the singularity ; in no case will we encounter the atomic collision singularity in a simulation of the type envisioned here . for the purpose of deriving practical methods we assume that ( i ) the positions are confined to a periodic simulation box @xmath50 , where is the one - dimensional torus , and ( ii ) @xmath41 is @xmath42 on @xmath51 . these assumptions , which are realistic in most molecular dynamics applications for the reasons mentioned above , allow us to use recent results in hypocoercivity @xcite to establish the regularity properties of @xmath44 . specifically , we have as a consequence of these articles , that * @xmath52 has a unique solution in @xmath53 , i.e. , the gibbs density @xmath54 , * @xmath43 has a compact resolvent @xcite , and the gibbs state is therefore exponentially attracting . note that , if , for some given @xmath55 , there are two solutions of @xmath56 then , using the linearity of the operator these may differ only in a constant scaling of the gibbs density . in a splitting method for a deterministic system @xmath57 , one divides the vector field @xmath58 into exactly solvable parts , i.e. @xmath59 , which are treated sequentially within a timestep . an example of such a splitting method for the hamiltonian system with energy @xmath60 is the `` symplectic euler '' method defined by @xmath61 , @xmath62 . by dividing the vector field as @xmath63 , solving each vector field in turn , we obtain the so - called position verlet method , and by switching the roles of @xmath64 and @xmath65 we obtain the velocity verlet method . splitting methods like these are explicit and this feature is of particular importance in molecular dynamics , where the force calculation is the usual measure of per - timestep computational complexity . in a similar way , langevin dynamics may be treated by splitting @xcite . for example , one may divide the langevin system ( eq . [ lang ] ) into three parts @xmath66= \underbrace{\left [ \!\!\begin{array}{c } m^{-1 } p \\0\end{array}\!\ ! \right ] { \rm d}t}_{\rm a } + \underbrace{\left [ \!\ ! \begin{array}{c } 0 \\-\nabla u\end{array}\!\ ! \right ] { \rm d}t}_{\rm b } + \underbrace{\left [ \!\ ! \begin{array}{c } 0\\ -\gamma p{\rm d}t + \sigma m^{1/2}{\rm d } w \end{array } \!\!\right ] , } _ { \rm o } \label{gla2}\ ] ] and each of the three parts may be solved ` exactly ' . in the case of the ou part ( labelled here simply as o ) we mean by this that we realize the stochastic process by the equivalent formula @xmath67 where @xmath68 is a vector of uncorrelated independent standard normal random processes ( white noise ) . one method based on the splitting in eq . [ gla2 ] is defined by the composition @xmath69 where @xmath70 represents the phase space propagator associated to the ( deterministic or stochastic ) vector field @xmath58 , and we use @xmath71 as the timestep in langevin dynamics ( later we use @xmath16 for the timestep of an associated brownian dynamics ) . the deterministic part is approximated by the position verlet method . this is referred to as a geometric langevin algorithm of order two ( gla-2 ) following @xcite ; an alternative is to use velocity verlet for the hamiltonian part ( @xmath72 ) . a simple generalization of gla methods is obtained by interspersing integrators associated to parts of the hamiltonian vector field with exact ou solves , thus we have @xmath73 and @xmath74 defined in an analogous way . recent work in @xcite similarly uses exact ou solves to give an integrator equivalent to @xmath75 ( in our notation ) , though with a reparameterised timestep . the analysis technique we use can employed to study many such integrators , though for brevity we will limit this article to discussion only on a select few interesting cases . an alternative integrator termed the stochastic position verlet ( spv ) method @xcite , relies on the splitting @xmath76 = \left [ \begin{array}{c } m^{-1 } p \\0\end{array } \right ] { \rm d}t + \left [ \begin{array}{c } 0 \\-\nabla u{\rm d}t -\gamma p{\rm d}t + \sigma m^{1/2 } { \rm d } w \end{array } \right ] .\ ] ] spv is not a ( generalized ) gla - type method , although it , as each of the generalized gla schemes , is _ quasisymplectic _ in the language of @xcite . likewise the commonly used method of brunger , brooks and karplus ( bbk ) @xcite is not of the ( generalized ) gla family . details of all methods examined are given in the appendix . to slightly simplify the presentation that follows , we make the change of variables @xmath77 , @xmath78 , with a corresponding adjustment of the potential ; this is equivalent to assuming @xmath79 . we shall work here with formal series expansions , however we expect that our derivation could be rigorously founded using techniques found in @xcite and @xcite . associated to any given splitting - based method for langevin dynamics , we define the operator @xmath80 that characterises the propagation of density by an expansion of the form : @xmath81 for example , for the method labelled by the string abao , we have @xmath82 the perturbation series may be found by successive applications of the bch expansion @xcite and linearity properties of the kolmogorov operator . however , unlike in the deterministic case , the terms that appear in the series can not be associated to modified vector fields or even sdes @xcite . note that , when iterated @xmath83 times , the method abao produces a sequence of the form @xmath84^n e^{\frac{\delta t}{2 } { \cal l}_{\rm b}}e^{\frac{\delta t}{2 } { \cal l}_{\rm a } } e^ { { \delta t } { \cal l}_{\rm o } } \ ] ] thus , with a minor coordinate transformation , the dynamics sample the same invariant density as baoab . similarly babo and obab are essentially the same method as aboba . for this reason we concentrate in the remainder of this article on aboba and baoab . for these two methods , the symmetry implies that the odd order terms in eq . [ opexp ] vanish identically using the jacobi identity in the bch expansions . after deriving @xmath80 in this way , we seek the invariant distribution which satisfies @xmath85 . for the baoab and aboba methods , we make the ansatz that the invariant measure of the numerical method has the simple form @xmath86).\ ] ] although some technical issues might be encountered , we believe that the existence of such an expansion can be made rigorous using techniques found in @xcite , based on the regularity of the operator @xmath87 . we may rewrite this as @xmath88 this means that the equation @xmath89 becomes @xmath90 equating second order terms in @xmath71 gives @xmath91 the equation @xmath92 has unique solution @xmath93 , up to a constant multiple . hence the homogeneous solution to the above pde is @xmath94 , for some constant @xmath95 ; we therefore require a particular solution @xmath65 of eq . [ eqn::pde ] . according to the fredholm alternative , the equation has a solution provided that , for any solution of @xmath96 we have @xmath97 as the only solutions of @xmath98 are the constants , we require @xmath99 for a symmetric splitting method such as the baoab method , recall that we can use the baker - campbell - hausdorff @xcite formula to find the overall one - step perturbation operator for the scheme . for linear operators @xmath100 , @xmath101 and @xmath6 , we have the relation @xmath102 where @xmath103 \right ] \right . + \left[y , \left [ y , x \right ] \right ] + \left[z , \left [ y , x \right ] \right]+ \left[y , \left [ z , x \right ] \right ] \right . \\ & \left . - \tfrac{1}{2 } \left[y , \left [ y , z \right ] \right ] - \tfrac{1}{2 } \left[x , \left [ x , z \right ] \right ] - \tfrac{1}{2 } \left[x , \left [ x , y \right ] \right ] \right ) + { \cal o}(\delta t^4).\end{aligned}\ ] ] here @xmath104 = xy - yx$ ] is the commutator of @xmath101 and @xmath100 . in the case of the baoab method , we take @xmath105 and @xmath106 to compute the perturbed operator for the method . a similar analysis can be conducted for the other generalised gla - type methods considered in this paper . to compute @xmath80 we simply plug in our choices into the bch formula to obtain : @xmath107 \right ] + \left[{\cal l}^*_{\rm a } , \left [ { \cal l}^*_{\rm a } , { \cal l}^*_{\rm b } \right ] \right ] \right . \\ & & \left . + \left[{\cal l}^*_{\rm o } , \left [ { \cal l}^*_{\rm a } , { \cal l}^*_{\rm b } \right ] \right ] + \left[{\cal l}^*_{\rm a } , \left [ { \cal l}^*_{\rm o } , { \cal l}^*_{\rm b } \right ] \right ] - \tfrac{1}{2 } \left[{\cal l}^*_{\rm a } , \left [ { \cal l}^*_{\rm a } , { \cal l}^*_{\rm o } \right ] \right ] \right . \\ & & \left . - \tfrac{1}{2 } \left[{\cal l}^*_{\rm b } , \left [ { \cal l}^*_{\rm b } , { \cal l}^*_{\rm o } \right ] \right ] - \tfrac{1}{2 } \left[{\cal l}^*_{\rm b } , \left [ { \cal l}^*_{\rm b } , { \cal l}^*_{\rm a } \right ] \right ] \right ) + { o}(\delta t^4),\end{aligned}\ ] ] where recall that @xmath108 and @xmath109 the calculation of the inhomogeneity in ( 8) then amounts to a straightforward computation of the commutator series applied to @xmath110 . the commutators needed are : @xmath111 \right ] \rho_\beta = & 2 \beta p^t u''(x ) \nabla u(x ) \rho_\beta - \rho_\beta \beta \ , p \cdot \nabla_x p^t u''(x ) p , \\ \left[{\cal l}^*_{\rm b } , \left [ { \cal l}^*_{\rm b } , { \cal l}^*_{\rm a } \right ] \right ] \rho_\beta = & -2 \beta p^t u''(x ) \nabla u(x ) \rho_\beta , \\ \left[{\cal l}^*_{\rm o } , \left [ { \cal l}^*_{\rm a } , { \cal l}^*_{\rm b } \right ] \right ] \rho_\beta = & 2 \gamma \left(\delta_x u(x ) - \beta p^t u''(x ) p \right ) \rho_\beta , \\ \left[{\cal l}^*_{\rm a } , \left [ { \cal l}^*_{\rm o } , { \cal l}^*_{\rm b } \right ] \right ] \rho_\beta = & \gamma \beta \left ( |\nabla u(x ) |^2 - p^t u''(x ) p \right ) \rho_\beta , \\ \left[{\cal l}^*_{\rm a } , \left [ { \cal l}^*_{\rm a } , { \cal l}^*_{\rm o } \right ] \right ] \rho_\beta = & 2 \gamma \left ( \beta |\nabla u(x ) |^2 - \delta_x u(x ) \right ) \rho_\beta , \\ \left[{\cal l}^*_{\rm b } , \left [ { \cal l}^*_{\rm b } , { \cal l}^*_{\rm o } \right ] \right ] \rho_\beta = & 0 \\ \left[{\cal l}^*_{\rm o } , \left [ { \cal l}^*_{\rm o } , { \cal l}^*_{\rm det } \right ] \right ] \rho_\beta = & 0,\end{aligned}\ ] ] where we have abbreviated the hessian @xmath112 hence we see directly that @xmath113 giving @xmath114 . \label{fredalt}\end{aligned}\ ] ] observe that eq . [ fa ] is satisfied since the average of the first term is equivalent to a canonical average which vanishes @xmath115 whereas the other terms in eq . [ fredalt ] , being canonical averages of terms which are odd - order in @xmath18 , necessarily also average to zero . an analogous computation can be performed for the aboba method , giving a slightly ( but crucially ) different perturbation operator @xmath116 , where @xmath117\ ] ] here @xmath118 is the hessian matrix of @xmath41 and @xmath119 is the partial laplacian in @xmath120 . equation [ fa ] is again seen to be satisfied . although we are not able to give the general analytical solution to the partial differential equation of eq . [ eqn::pde ] , we can find a solution in an important limiting case : the high friction regime . to do this , we expand the invariant density of the numerical method further , viewing both @xmath71 and @xmath122 as small parameters : @xmath123 ) . \label{densityexpansion}\ ] ] dividing by @xmath27 , we may reduce eq . [ eqn::pde ] to @xmath124(f_{2,0 } + \varepsilon f_{2,1 } + o(\varepsilon^2 ) ) = g_{0 } + \varepsilon g_{1},\ ] ] where @xmath125 and , for baoab , @xmath126 note that this is a singularly perturbed system as @xmath127 is degenerate and it is only the combined operator that has the necessary regularity to define a unique solution . nonetheless , as explained below it is possible to find the leading term @xmath128 by substituting a truncated expansion of fixed degree and solving the resulting equations . equating powers of @xmath129 , we find @xmath130 and @xmath131 truncating at @xmath132 , for example , we find the following solution of these equations : @xmath133 for aboba , @xmath128 the solution would change to @xmath134 we now turn out attention to the configurational marginal distribution obtained by integrating the density expansion eq . [ densityexpansion ] with respect to the momenta . our interest is only in the leading term of this expansion , which defines the sampling behavior for large @xmath27 . ignoring higher order terms in @xmath71 and @xmath129 , we would have the distribution @xmath135 where , for baoab , @xmath136 \right ) $ ] , and @xmath137 . noting that the leading term in the exponent of @xmath138 is quadratic in momenta , we integrate out with respect to @xmath18 to obtain @xmath139 \right ) { \rm d}^n p\\ & = \sqrt{2 \pi k_b t /\det\left ( i + \frac{\delta t^2}{4}u '' \right ) } \exp \left ( -\beta \left [ u - \beta^{-1 } \frac{\delta t^2}{8 } \delta u \right ] \right ) \end{aligned}\ ] ] using the identity @xmath140 , we find @xmath141 \right ) .\ ] ] we then taylor expand the logarithm of @xmath142 and take the trace to obtain a cancellation of the @xmath143 terms , giving @xmath144 the contribution to the configurational distribution error due to @xmath138 is @xmath145 . this means that the overall error in the marginal distribution of @xmath146 ( which includes the neglected factor @xmath147 ) will be @xmath148 . if @xmath129 is small ( or @xmath71 is relatively large ) , the error will be dominated by the quartic term in @xmath71 and we will observe 4th order accuracy in configurational averages . for small @xmath71 the method is always eventually second order . in the case of the aboba method , the remarkable cancellation of the second order errors does not occur and the method always exhibits 2nd order configuration distribution error . we now consider the limit @xmath149 , where the exact solution of the vector ou process reduces to @xmath150 , where @xmath151 is a vector whose components have a standard normal distribution ( gaussian white noise ) . alternatively , we could consider the limit of the particle mass going to @xmath152 , although this requires a reformulation of eq . [ lang ] so that the friction is proportional to the velocity instead of the momentum @xcite . whichever limit is taken , we would expect the ultimate result to be the same . ( here we have reintroduced the masses in order to present the method , since they may be useful scaling parameters in simulation . ) in the configurations it is straightforward to show that the baoab method therefore becomes @xmath153 where the @xmath13 are vectors of i.i.d . standard normal random variables . replacing @xmath154 by @xmath16 we arrive at eq . [ bd - mod ] . since the langevin scheme gives 4th order accurate configurational averages in this limit , we expect the method of eq . [ bd - mod ] to be second order accurate in modified timestep @xmath16 . moreover , since we completely remove the second order term in the langevin dynamics configurational density expansion , we expect to observe this behaviour across all values of @xmath16 . by contrast the aboba scheme gives a much more complicated limit method as @xmath149 which is not in one - step form . in the euler - maruyama method , the random perturbations introduced at each step are independent . in the method of eq . 2 , the random perturbation is a scaling of @xmath155 ; the components @xmath156 of these are independent of each other and decay linearly with timestep : @xmath157 thus , in this new method , we use a colored noise which has characteristics that directly depend on the stepsize , although the noise decorrelates in just a couple of timesteps . this is therefore no longer a markov process , however it can be reformulated as such if one considers the appropriate extended space ( eg . @xmath158 $ ] ) . we implemented the methods aboba , baoab , spv and bbk and compared the accuracy of configurational sampling for different values of @xmath27 and a range of timesteps . a brief analysis shows that the use of the harmonic oscillator leads to special cancellations in the bch series of the splitting schemes , making it a poor test subject . hence , in order to compare the order of accuracy of the different schemes , we first considered an oscillator model in 1d with potential @xmath159 . this was accomplished by introducing @xmath11 intervals ( ` bins ' ) of equal length , and computing the mean error in the observed probability frequency compared to the exact expected frequency ( obtained by integration of the probability density ) . if the observed frequency in bin @xmath160 is @xmath161 , and the exact expected frequency is @xmath162 , then the error calculated is @xmath163 in this one - dimensional example , we used @xmath164 bins to cover the interval from @xmath165 to @xmath166 . the configurational density error is plotted against stepsize in log - log scale . if @xmath167 then we expect this graph to be a line of slope @xmath168 . due to the relative simplicity of this model , we were able to perform highly resolved simulations to calculate accurate error estimates for the configurational distribution . the exact expected value that we compare experimental results against can be computed to arbitrary prescision , and we are able to run as many simulations as needed in order to drive the variance of results to a minimum . the variance in our results , in cases where the stepsize was less than @xmath169 , was consistently below @xmath170 . above a stepsize of @xmath169 some of the methods were found to be unstable . the results of our simulations are summarized in figure [ quartic ] . . the simulation time was fixed for all runs at @xmath171 , and five runs were averaged to further reduce sampling errors . at left , @xmath172 , at right @xmath173 . the graphs are entirely in keeping with the theory presented in the article . [ quartic],title="fig:",width=288 ] . the simulation time was fixed for all runs at @xmath171 , and five runs were averaged to further reduce sampling errors . at left , @xmath172 , at right @xmath173 . the graphs are entirely in keeping with the theory presented in the article . [ quartic],title="fig:",width=288 ] as we can see , when @xmath27 is small , the methods perform somewhat similarly , at least in the qualitative sense , with all showing a 2nd order error in configurational sampling , and the aboba and spv methods essentially identical . as @xmath27 is increased , the substantial difference between baoab and the other methods becomes apparent . in the limit of large @xmath27 the spv method effectively annihilates the force which results in poor sampling . in the graph for @xmath172 , we can see that for larger values of the stepsize , the graph steepens ( indicating that the fourth order term is dominant ) ; as the stepsize is decreased the method exhibits a 2nd order asymptotic decay . with @xmath173 the fourth order behavior is seen for for all indicated data points , although , again , this becomes second order for smaller values of @xmath71 . note that the limit method eq . 2 ( with the substitution @xmath174 ) gives an essentially identical behavior to the @xmath173 case . we also give a comparison of the actual computed configurational distributions , at different stepsizes using each method , for @xmath175 in figure [ 1ddist ] . . the three curves for each of the four methods show the results for three different stepsizes : @xmath176 ( circles ) , @xmath177 ( crosses ) , @xmath178 ( dashed ) , compared to the dark , solid curve representing the exact distribution . the graph shows that the generalized gla methods are superior to spv and bbk in the moderate @xmath27 regime . [ 1ddist],width=576 ] , for morse ( left ) and lennard - jones ( right ) clusters . they also show the choice of bins used in calculating the numerical distributions . [ pic::gofr],title="fig:",width=288 ] , for morse ( left ) and lennard - jones ( right ) clusters . they also show the choice of bins used in calculating the numerical distributions . [ pic::gofr],title="fig:",width=288 ] to examine the performance of the limit method in more detail , we next considered small molecular clusters consisting of seven atoms ( motion restricted to the plane ) , with both morse ( @xmath179 ) and lennard - jones ( @xmath180 ) potentials , given by @xmath181 where we use @xmath182 and @xmath183 . the overall potentials are hence @xmath184 where @xmath185 is the distance between particles @xmath160 and @xmath186 , @xmath187 is the distance between particle @xmath188 and the origin , and a mild harmonic term is included in the case of the lennard - jones system in order to prevent particles being ejected from the cluster . the morse potential gives a smoother dynamics compared to lennard - jones ( the morse forces were on average three times smaller than those for lennard - jones ) and allows a more satisfying determination of the error scaling behavior with stepsize . to quantify the error in configurational sampling , we calculated the radial density @xmath189 by binning the instantaneous interatomic distances at each step into 20 compartments and compared to a calculated reference value ( figure [ pic::gofr ] ) . though not exact , the errors in the reference will be negligible compared with configurational distribution errors at higher stepsizes . for the morse cluster the reference stepsize we used was @xmath190 , whereas for the lennard - jones cluster , we used @xmath191 both with the same integration time that was used in their respective test runs . in both cases the cluster was initialized with 6 particles placed on a regular hexagon with unit side - length and with the remaining particle in the centre , and initial velocities randomly drawn from the canonical distribution . considerable computation is required to achieve the level of accuracy required , due to the dominance of sampling error and the complexity of the system compared to the earlier 1d example . we ran a large number of independent simulations at each stepsize and computed the average radial density plot for both examples , and compared this to the reference result . our results , presented in figure [ morse ] , are entirely consistent with our analysis and show the second order dependence of the configurational sampling error on @xmath16 ( equivalent to fourth order in @xmath71 ) , as compared to the euler - maruyama s first order behavior . these results demonstrate a good agreement with our theoretical results , however even using extensive computation the variances in each experiment were still quite high . if @xmath192 is the density of bin @xmath193 in simulation @xmath194 at stepsize @xmath16 , we calculate the variance as @xmath195 where we used @xmath196 bins for each experiment , @xmath197 for the morse experiment and @xmath198 for the lennard - jones experiment . the variances for the morse experiment were around @xmath170 while the lennard jones experiment had variances around @xmath199 . we expect that completing more simulations ( increasing @xmath0 ) would reduce the variance and give smoother results in figure [ morse ] . nonetheless , in both cases , we observe a significant reduction in the error compared to euler - maruyama . ) . left : morse potential ; right : lennard - jones . for morse we used a temperature of @xmath200 , a fixed time interval of @xmath201 , with stepsizes ranging from @xmath202 to @xmath203 . for lennard - jones the temperature was @xmath204 , @xmath205 and stepsizes ranged from @xmath206 to @xmath207 . in order to drive the variance of the results down , a large number of runs were necessary : for the morse simulation the error is computed using the average histogram computed from 200 independent runs , while the lennard - jones simulation used 1000 independent runs . around two orders of magnitude of improvement are observed in the accurate regime , but , perhaps even more important , the baoab limit method is usable at substantially larger stepsizes than euler - maruyama . [ morse],title="fig:",width=288 ] ) . left : morse potential ; right : lennard - jones . for morse we used a temperature of @xmath200 , a fixed time interval of @xmath201 , with stepsizes ranging from @xmath202 to @xmath203 . for lennard - jones the temperature was @xmath204 , @xmath205 and stepsizes ranged from @xmath206 to @xmath207 . in order to drive the variance of the results down , a large number of runs were necessary : for the morse simulation the error is computed using the average histogram computed from 200 independent runs , while the lennard - jones simulation used 1000 independent runs . around two orders of magnitude of improvement are observed in the accurate regime , but , perhaps even more important , the baoab limit method is usable at substantially larger stepsizes than euler - maruyama . [ morse],title="fig:",width=288 ] it might be a suggested that the improved accuracy seen in the high @xmath27 regime could potentially come at the price of a slower convergence to equilibrium due to a reduced rate of transition between metastable states , hence overall sampling of the configurational distribution might be impaired in favour of local sampling . to address this point , we have plotted in figure [ rateofconv ] the error computed in our lennard - jones simulation as a function of the number of force evaluations ( vertical ) against the friction value @xmath27 used for the simulation ( horizontal ) . gridpoints in the plot are coloured according to the configurational error , computed using eq . [ errorformula ] . the results indicate that the convergence rate for the baoab method is not diminished for large friction coefficient , so it does not appear that we are sacrificing sampling accuracy using this scheme . note that the performance of aboba is also robust in the limit of large @xmath27 , but the achievable accuracy is reduced as it is only second order , consistent with what we have presented in this article . our results confirm the theoretical results of sections 3 and 4 , in particular showing the higher order configurational sampling of the baoab method . the order in its langevin formulation is effectively four in the large @xmath27 limit and large values of @xmath27 do not impair its stability due to the use of exact ornstein - uhlenbeck solves . not only is the order of the method high , the constant multiplying the leading term must be , in the cases looked at here , of modest magnitude , since the errors are relatively small also at the large stepsize stability threshold . since , in the context of molecular dynamics , baoab is a ` cheap ' and easy to implement scheme using only a single force vector per timestep , we stress that there is no price to pay for its improved accuracy . in molecular modelling , there are other errors that play important roles , most importantly errors in the force fields ( or , more fundamentally , the errors due to not modelling quantum mechanics properly ) and sampling errors . obviously these errors may dominate the overall method error and limit the relative benefit to be gained by using one integrator as compared to another , but it is also clear that both of the other types of errors are constantly being reduced through the design of better models and the use of more powerful computers . more important , one can ask the question : how can a practitioner know which part of the error in a given complicated simulation is due to sampling error and which part due to the truncation errors addressed here ? in our experiments with molecular models , even where there was substantial sampling error still present , we nonetheless found the accuracy to be noticeably higher for the baoab method ; it is likely that this improvement in sampling accuracy would be of direct benefit in many real world simulations . finally we point out that the baoab scheme and its limit method ( eq . 2 ) was , in each case studied , stable at a larger stepsize than the alternatives , meaning that longer time intervals are made accessible . this was particularly dramatic in the case of the lennard - jones system . * acknowledgements . * the first author acknowledges the support of a jto fellowship from the institute for computational engineering and sciences at the university of texas . the second author was supported by the centre for numerical algorithms and intelligent software ( funded by epsrc grant ep / g036136/1 and the scottish funding council ) . a. stuart , h. owhadi , and especially g. stoltz made helpful suggestions regarding the presentation of the results . we further acknowledge the comments received from a. abdulle and m. tretyakov which have improved the article and its relation to other works . computations were performed on the university of edinburgh s ecdf compute facility . 99 brnger , a , brooks iii , c , & karplus , m. 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( 2010 ) convergence of numerical time - averaging and stationary measures via poisson equations . _ siam j. num . * 48 * : 552577 . debussche , a. and faou , e. the langevin dynamics methods used for the numerical experiments in this paper are given here . @xmath208 is an @xmath209vector of i.i.d . normal random numbers , with mean @xmath152 and variance @xmath210 . the diagonal mass matrix is denoted @xmath11 , and we assume a timestep @xmath71 is provided . given the parameter @xmath27 , we define useful constants @xmath211 and @xmath212

in this article , we focus on the sampling of the configurational gibbs - boltzmann distribution , that is , the calculation of averages of functions of the position coordinates of a molecular @xmath0-body system modelled at constant temperature . we show how a formal series expansion of the invariant measure of a langevin dynamics numerical method can be obtained in a straightforward way using the baker - campbell - hausdorff lemma . we then compare langevin dynamics integrators in terms of their invariant distributions and demonstrate a superconvergence property ( 4th order accuracy where only 2nd order would be expected ) of one method in the high friction limit ; this method , moreover , can be reduced to a simple modification of the euler - maruyama method for brownian dynamics involving a non - markovian ( coloured noise ) random process . in the brownian dynamics case , 2nd order accuracy of the invariant density is achieved . all methods considered are efficient for molecular applications ( requiring one force evaluation per timestep ) and of a simple form . in fully resolved ( long run ) molecular dynamics simulations , for our favoured method , we observe up to two orders of magnitude improvement in configurational sampling accuracy for given stepsize with no evident reduction in the size of the largest usable timestep compared to common alternative methods . * keywords : * molecular dynamics ; sampling ; langevin dynamics ; brownian dynamics ; stochastic dynamics .

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Proceed to summarize the following text: this paper considers the use of importance sampling for estimating hitting or exit probabilities for stochastic processes . the process model is a @xmath0-dimensional diffusion @xmath1 satisfying the stochastic differential equation ( sde ) @xmath2 where @xmath3 and @xmath4 is a standard @xmath0-dimensional wiener process . of particular interest is the case of gradient flows , @xmath5 , and constant diffusion coefficient , though many aspects of the analysis are more generally applicable . let @xmath6 be an open set and denote by @xmath7 the exit time of @xmath8 from @xmath9 . we are concerned with the estimation of quantities such as the probability that @xmath10 leaves @xmath9 before some time @xmath11 , or that it exits through a particular subset @xmath12 before @xmath13 , and related expected values . the principal novel feature of this work is that the initial point is in the neighborhood of an equilibrium point of the noiseless dynamics . the estimation of such probabilities has several mathematical and computational difficulties . it is related to the estimation of transition probabilities between different metastable states within a given time horizon . as is well known , standard monte carlo sampling techniques lead to exponentially large relative errors as the noise coefficient @xmath14 tends to zero . when rest points are in the domain of interest the situation is even more complicated than usual . this work will focus on this particularly difficult issue . the performance of unbiased estimators for rare event problems is usually measured by the size of the second moment of the estimator based on a single simulation . for a well designed scheme the ratio of this second moment to the quantity of interest will not grow too rapidly as @xmath15 . one measure is the exponential rate of decay of the second moment . if this rate of decay is exactly twice the decay rate for the probability of interest , then the scheme is called asymptotically efficient ( or weakly efficient ) . the notion of _ strong _ efficiency requires that the ratio of the second moment to the square of the probability be bounded above uniformly for all small @xmath3 . while such performance is certainly desirable , it is not common when dealing with models such as ( [ eq : diffusion1 ] ) that involve state dependent dynamics and complicated geometries . as we describe below , in some sense both these measures are inadequate for the situation considered here . a theory based on subsolutions to an associated hamilton - jacobi - bellman ( hjb ) equation has been developed for the design and performance analysis of importance sampling ( see for example @xcite ) . in this approach the change of measure ( which for reasons made evident later on we will call the control ) used in the importance sampling is defined in terms of the gradient of a subsolution , and the performance , as measured by the decay rate of the second moment , is given by the value of the subsolution at the initial location @xmath16 . this theory is the starting point of our analysis of ( [ eq : diffusion1 ] ) , though as mentioned previously the inclusion of rest points will motivate some further developments . for any particular class of process models and events , an essential step in the application of this approach is the construction of appropriate subsolutions . in this paper we will exploit the fact that the freidlin - wentsell quasipotential @xcite can be used to construct various subsolutions for certain _ time independent _ problems related to ( [ eq : diffusion1 ] ) . in addition , for particular but important classes of process models ( e.g. , gradient systems with constant diffusion matrix ) , the quasipotential and hence these subsolutions take explicit and simple forms . as we discuss in detail later on , these subsolutions also give subsolutions for the _ time dependent _ problems , and when @xmath13 is large the value of the subsolution at the starting point ( which now includes time @xmath17 as well as the location ) will be close to the maximal value . it follows that if the final time @xmath13 is large enough , then existing theory implies that the estimator based on this subsolution should have a nearly optimal decay rate for its second moment . while this is a valid statement , there is an important qualitative difference between problems which include a rest point in the domain of interest and those which do not . the distinction turns not on the decay rate , which behaves as expected in both situations , but rather depends on the pre - exponential terms not captured by the decay rate . when the domain does not contain a rest point one has simultaneously good rates of decay and control over the pre - exponential terms . however , when a rest point is present schemes based only on this time independent subsolution keep the desired decay rate but lose the good control over the pre - exponential terms . the qualitative difference is related to the fact that in the former case a subsolution designed on the basis of the @xmath18 problem can be shown to give useful bounds for the problem with @xmath3 , but in the latter case this is no longer true . this qualitative distinction will be made precise later on , when we construct non - asymptotic bounds on the second moment for the two cases . when @xmath3 is small but not too small , the loss of performance due to the large pre - exponential term can be significant , rendering the associated importance sampling scheme little better than ordinary monte carlo . as @xmath15 the exponential decay rate dominates , and importance sampling once again gives much greater performance than ordinary monte carlo . however , the improvement is less than in the case where rest points are not included , and an approach which avoids this loss of performance would certainly be welcome . one can show that in fact there is no time independent subsolution that will resolve this difficulty , and so one must bring in some form of time dependence . it is not at all clear how one might incorporate time dependence throughout the entire domain @xmath19\times\mathcal{d}$ ] . an alternative approach , and the one we follow in this paper , is to combine an explicit solution to an approximating time dependent problem in a neighborhood of the rest point with the time independent subsolution obtained via the quasipotential away from the rest point . as we will show , such an approach will maintain the high decay rate while at the same time properly controlling the pre - exponential term . in the neighborhood of the rest point , one can approximate the dynamics of the diffusion process by a gauss - markov process , i.e. , a process with a constant diffusion matrix and drift that is affine in the state . for these dynamics and appropriate terminal conditions for the localizing problem , the solution to the related pde can be constructed in terms to the famous linear / quadratic regulator problem from optimal control theory . as a consequence an explicit and nearly optimal scheme for a surrogate problem can be identified in the neighborhood of the rest point , which is then merged with the explicit scheme based on the quasipotential in that part of the domain where it is particularly effective . in this paper we analyze the difficulties caused by the presence of rest points in a general setting . we describe and theoretically justify a resolution of these difficulties in the case of dimension one , and present computational data for this case . the construction of the localizing problem is more elaborate in dimension greater than one , and will be presented in a companion paper along with the results of numerical experiments in higher dimensions . the contents of this paper are as follows . in section [ s : ldpandis ] we review the relevant large deviation theory and importance sampling . in section [ s : effectivenessofsubsolutions ] we discuss the effectiveness of time independent subsolutions . in particular , we show that if rest points are not part of the domain of interest then subsolutions lead to both good decay rates and non - asymptotic bounds for the second moment of the corresponding unbiased estimator . however , when rest points are included in the domain of interest , the situation is more complicated and even if the decay rate is good , the prelimit bounds may not be as good as desired . in section [ s : linearproblem ] , we present a change of measure for the problem with a rest point for a quadratic potential function with provably good pre - asymptotic and asymptotic performance and which does not degrade as @xmath13 gets larger . in section [ s : nonlinearproblem ] we extend the discussion to the nonlinear problem with rest points . simulation data , demonstrating the discussions in sections [ s : linearproblem ] and [ s : nonlinearproblem ] are also presented in the corresponding sections . the appendix has some auxiliary lemmas that are used in the main body of the manuscript . in this section we recall well known large deviation results for probabilities of exit times ( subsection [ ss : ldp ] ) , review importance sampling in the context of small noise diffusions ( subsection [ ss : is ] ) , and also recall the notion of subsolutions to certain related hjb equations ( subsection [ ss : roleofsubsolutions ] ) . in most of this paper the following assumptions will be used . the assumptions are stronger than necessary , but simplify the discussion considerably . for example , the non - degeneracy of the diffusion matrix and regularity of the boundary of @xmath9 easily imply that a limit exists for ( [ eq : estimationtarget2 ] ) . they can be weakened , but the existence of the limit then requires conditions that are best addressed in a problem dependent fashion . [ a : mainassumption ] 1 . the drift @xmath20 is bounded and lipschitz continuous . 2 . the coefficient @xmath21 is bounded , lipschitz continuous and uniformly nondegenerate . @xmath9 is an open and bounded subset of @xmath22 , and at all points on its boundary @xmath9 satisfies an interior and exterior cone condition , i.e. , there is @xmath23 such that if @xmath24 , then there exist unit vectors @xmath25 such that @xmath26 and @xmath27 we also assume that if @xmath28 and if @xmath29 solves @xmath30 , then @xmath31 for all @xmath32 . fix @xmath11 and consider an initial point @xmath33 . consider a bounded and class @xmath34 function @xmath35 . let @xmath36 denote expected value given @xmath37 , and define @xmath38 . \label{eq : estimationtargetsmooth1}\ ] ] since @xmath39 scales exponentially it is also useful to define @xmath40 although for now we focus on the case where @xmath41 is bounded and continuous , one is also interested in cases where @xmath41 is discontinuous and takes the value @xmath42 . an example is when for some set @xmath43 , @xmath44 if @xmath45 and @xmath46 if @xmath47 . in this case @xmath48 equals the probability of exiting @xmath9 through @xmath49 before time @xmath13 . for these cases and under mild regularity conditions on @xmath49 , statements analogous to theorem [ t : ldptheorem ] below hold . let @xmath50:\mathbb{r}^{d}\right ) $ ] be the set of absolutely continuous functions on @xmath51 $ ] with values in @xmath22 . we denote the local rate function by@xmath52 \right\rangle \label{eq : localratefunction}\ ] ] where @xmath53 , and the corresponding rate or action functional for @xmath54:\mathbb{r}^{d}\right ) $ ] by @xmath55 for all other @xmath56:\mathbb{r}^{d}\right ) $ ] set @xmath57 . the following large deviations result is well known , e.g. , @xcite . [ t : ldptheorem ] assume condition [ a : mainassumption ] . then for each @xmath58 @xmath59 , \label{eq : ldpth}\ ] ] where @xmath60:\mathbb{r}^{d}):\phi ( t)=x,\phi(s)\in\mathcal{d}\text { for } s\in\lbrack t , t],\phi(t)\in \partial\mathcal{d}\right\ } .\ ] ] we briefly review the use of importance sampling for estimating @xmath48 for a given function @xmath41 . let @xmath61 be any unbiased estimator of @xmath48 that is defined on some probability space with probability measure @xmath62 . thus @xmath63 is a random variable such that @xmath64 where @xmath65 is the expectation operator associated with @xmath62 . in this paper we will consider only unbiased estimators . in monte carlo simulation , one generates a number of independent copies of @xmath63 and the estimate is the sample mean . the specific number of samples required depends on the desired accuracy , which is measured by the variance of the sample mean . however , since the samples are independent it suffices to consider the variance of a single sample . because of unbiasedness , minimizing the variance is equivalent to minimizing the second moment . by jensen s inequality @xmath66 it then follows from theorem [ t : ldptheorem ] that @xmath67 and thus @xmath68 is the best possible rate of decay of the second moment . if @xmath69 then @xmath63 achieves this best decay rate , and is said to be _ asymptotically optimal_. while asymptotic optimality or near asymptotic optimality is desirable , as noted in the introduction one may also desire good behavior of the pre - exponential term . to keep the terminology clear we will avoid the conventional usage of terms such as asymptotic optimality , and refer instead to properties of the decay rate and the pre - exponential term . the unbiased estimators @xmath63 that we consider are all based on measure transformation . consider @xmath70 , a sufficiently integrable and adapted function , such that @xmath71 defines a family of probability measures @xmath72 . then by girsanov s theorem , for each @xmath3 @xmath73 is a brownian motion on @xmath51 $ ] under the probability measure @xmath74 , and @xmath10 satisfies @xmath75 and @xmath76 .\ ] ] for our purposes , @xmath70 is either given as a process that is progressively measurable with respect to a suitable filtration that measures the wiener process ( sometimes called an _ open loop _ control ) , or else it is of _ feedback _ form , in which case there is a suitably measurable function @xmath77\times\mathbb{r}^{d}\rightarrow \mathbb{r}^{d}$ ] such that @xmath78 . of course when implementing importance sampling we consider only the latter form . letting @xmath79 it follows easily that under @xmath72 , @xmath61 is an unbiased estimator for @xmath48 . the performance of this estimator is characterized by its second moment @xmath80 . \label{eq:2ndmoment1}\ ] ] the goal of this paper is to investigate the effect of rest points on @xmath81 and how one can choose controls that guarantee both good decay rates and pre - exponential bounds for @xmath81 . we conclude this section with a review of subsolutions to related hjb equations . such subsolutions are essential for constructing and analyzing good important sampling schemes . let @xmath82 the construction of good importance sampling schemes for a quantity such as ( [ eq : estimationtargetsmooth1 ] ) is closely related to the hjb equation @xmath83@xmath84 and more precisely to its subsolutions . it can be shown that @xmath85 defined in theorem [ t : ldptheorem ] is the unique continuous viscosity solution of ( [ eq : hjbequation1 ] ) and ( [ eq : hjbterm_cond ] ) , see @xcite . [ def : classicalsubsolution ] a function @xmath86\times \mathbb{r}^{d}\mapsto\mathbb{r}$ ] is a classical subsolution to the hjb equation ( [ eq : hjbequation1 ] ) and ( [ eq : hjbterm_cond ] ) if 1 . @xmath87 is continuously differentiable , 2 . @xmath88 for every @xmath58 , 3 . @xmath89 for @xmath90 and @xmath91 for @xmath92 . the connection between subsolutions and the performance of importance sampling schemes has been established in several papers , such as @xcite . these papers either consider classical subsolutions or , more generally , piecewise classical subsolutions . to simplify the discussion , we consider here just classical subsolutions . in the present setting , we have the following theorem regarding asymptotic optimality ( theorem 4.1 in @xcite ) . [ t : uniformlylogefficientregime1 ] let @xmath93 be the unique strong solution to ( [ eq : diffusion1 ] ) . consider a bounded and continuous function @xmath35 and assume condition [ a : mainassumption ] . let @xmath94 be a subsolution according to definition [ def : classicalsubsolution ] and define the control @xmath95 . then @xmath96 since @xmath87 is a subsolution it is automatic that @xmath97 for all @xmath98\times\mathcal{d}$ ] . if @xmath99 then the scheme has the largest possible decay rate . in this section we justify some of the claims made in the introduction regarding the differences in performance between importance sampling schemes when rest points are included in the domain of interest and when they are not . we consider just the problem of estimating the probability of escape from a set before time @xmath13 , and even then consider a particular setup . however , the example will illustrate the difference between the two cases , and also suggest how one might improve the performance of importance sampling when rest points are involved . _ much of the prior application of subsolutions to importance sampling @xcite has involved the estimation of escape probabilities for classes of stochastic networks , in which case the origin is often the unique stable point for the law of large numbers dynamics [ the analogue of ( [ eq : diffusion1 ] ) with _ the event most often studied in this context is that of escape from a set ( i.e. , buffer overflow ) before reaching the origin , after starting near but not at the origin . the analogous event for the diffusion model ( [ eq : diffusion1 ] ) is one of the problems that are the focus of the present work . however , the difficulties that will be described momentarily for the diffusion model do not arise when dealing with the analogous estimation problem for stochastic networks , and indeed in that setting the proximity of the rest point has little impact on either the rate of decay or the pre - exponential term . this is related to the fact that the law of large numbers trajectories for stochastic networks reach the origin in finite time , as opposed to the infinite time it takes for the solution to ( [ eq : diffusion1 ] ) with _ _ @xmath18 _ to reach a stable equilibrium point when not starting at such a point . in turn , this property is responsible for the fact that minimizing trajectories in the definition of the quasipotential are achieved on bounded time intervals for stochastic network models , but take infinite time for processes such as ( [ eq : diffusion1 ] ) . _ for the remainder of this section we concentrate on the special case of @xmath5 and @xmath100 , and on a particular estimation problem . we first argue that if the domain of interest does not include a rest point , then given a time - independent subsolution and associated control not only is a good decay rate obtained , but good bounds on the pre - exponential terms hold as well . we then show why this is not possible when a rest point is included . assume that @xmath101 is the global minimum for @xmath102 , so that @xmath103 , and that @xmath104 for all @xmath105 . without loss we assume that @xmath106 . let @xmath107 and define @xmath108 and @xmath109 . then the problem is to estimate @xmath110 where the initial point @xmath111 is such that @xmath112 . this corresponds to ( [ eq : estimationtargetsmooth1 ] ) , but here @xmath41 is not bounded and smooth , and instead @xmath46 if @xmath113 and @xmath44 if @xmath114 . for this problem one can also identify the rate of decay @xmath115 via ( [ eq : ldpth ] ) . a one - dimensional example is illustrated in figure [ fig : no_rest ] . [ ptb ] no_rest.eps the quasipotential with respect to the equilibrium point @xmath116 is defined by @xmath117:\mathbb{r}^{d}),\phi(0)=0,\phi(t)=x , t\in(0,\infty)\}. \label{eq : quasipotential}\ ] ] it follows from the variational characterization of @xmath118 that @xmath119 is always a weak sense solution to @xmath120 , and therefore by adding an appropriate constant @xmath121 to satisfy any needed boundary and terminal conditions , @xmath122 will always define a weak sense subsolution . in the present case @xmath123 takes the explicit form ( theorem 4.3.1 in @xcite ) @xmath124 and it is easy to check that @xmath125 is a subsolution according to definition [ def : classicalsubsolution ] . indeed , @xmath126 for @xmath113 , while the boundary condition @xmath127 for @xmath114 and terminal condition @xmath127 for @xmath92 hold vacuously . see figure [ fig : sub_no_rest ] . the control ( i.e. , change of measure ) suggested by this subsolution for the importance sampling scheme is @xmath128 . [ ptb ] sub_no_rest.eps the following representation for @xmath129 follows essentially from the arguments of subsection 2.3 of @xcite , which is based on the representation for exponential integrals with respect to brownian motion given in @xcite . the representation is given in terms of the value of a stochastic differential game , where the player corresponding to the importance sampling scheme has already selected their control ( i.e. , @xmath130 ) . the characterization of performance for importance sampling in terms of games was first introduced in @xcite . the only difference between the use here and in @xcite is that there the function @xmath41 is bounded , which is not true here since we consider an escape probability . however , the bounds stated below can be obtained by first replacing @xmath131 by @xmath132 and then letting @xmath133 . let @xmath134 be a filtration satisfying the usual conditions of completion and right continuity and which measures the wiener process . let @xmath135 denote the set of all @xmath134-progressively measurable @xmath0-dimensional processes @xmath136 that satisfy @xmath137 let @xmath3 be fixed and let @xmath138 be the unique strong solution to@xmath139ds\right ] \label{eq : control_diffusion}\ ] ] with initial condition @xmath140 . let @xmath141 denote the first time @xmath138 exits @xmath9 . then @xmath142 . \label{eq : game_rep}\ ] ] it is important to note that ( [ eq : game_rep ] ) provides a non - asymptotic representation for the performance measure @xmath143 . however , to obtain a more concrete statement regarding the performance of the importance sampling scheme , we will want bounds on @xmath143 that are more explicit than the right hand side of ( [ eq : game_rep ] ) . we do this by observing that when viewed as a function of an arbitrary starting point @xmath144 , @xmath145 also satisfies a nonlinear pde of the same general form as ( [ eq : hjbequation1 ] ) [ plus terminal and boundary conditions ] , and thus lower bounds can be obtained by constructing subsolutions for this pde . however , a key difference is that in contrast to ( [ eq : hjbequation1 ] ) , the pde for ( [ eq : game_rep ] ) involves a second derivative term . one can not avoid this issue , in that second derivative information and @xmath14 dependence are needed if one is to obtain non - asymptotic bounds , even when the change of measure is based on a first order equation . we next give the statement of the lower bound as it applies to the special case of this section . a more general statement and the proof will be given in lemma [ l : generalbound ] . however , the proof is an easy consequence of it s formula and the min / max representation @xmath146 .\ ] ] define @xmath147(t , x)=w_{t}(t , x)+\mathbb{h}(x , dw(t , x))+\frac { \varepsilon}{2}d^{2}w(t , x),\ ] ] and let @xmath148 be a subsolution to @xmath149=0 $ ] together with the boundary conditions @xmath150 for @xmath151 , @xmath152 for @xmath153 , and terminal condition @xmath154 for @xmath92 . suppose @xmath130 is the control based on a given smooth function @xmath87 , i.e. , @xmath155 . then @xmath156 \nonumber\\ & \qquad\geq2\bar{w}(0,y)\nonumber\\ & \qquad\qquad+\inf_{v\in\mathcal{a}:\hat{\tau}^{\varepsilon}\leq t\text { w.p.1}}\mathbb{e}\left [ \int_{0}^{\hat{\tau}^{\varepsilon}}2\mathcal{g}^{\varepsilon}[\bar{w}](s,\hat{x}^{\varepsilon}(s))ds-\int_{0}^{\hat{\tau } ^{\varepsilon}}\left\vert d\bar{w}(s,\hat{x}^{\varepsilon}(s))-d\bar { u}(s,\hat{x}^{\varepsilon}(s))\right\vert ^{2}ds\right ] .\nonumber\end{aligned}\ ] ] next we show how ( [ eq : lbq ] ) can be used to obtain bounds that are uniform in @xmath13 . for @xmath157 define@xmath158 where @xmath159 is the subsolution based on the quasipotential for @xmath160 as above , and assume that @xmath159 is twice continuously differentiable . then as with @xmath159 the appropriate boundary and terminal inequalities hold for @xmath161 . we next evaluate the right side of ( [ eq : lbq ] ) when @xmath162 and @xmath163 . a straightforward calculation gives @xmath164 and therefore @xmath165(x)-\frac{1}{2}\left\vert du^{\eta } ( x)-du(x)\right\vert ^{2}=2(\eta-2\eta^{2})\left\vert dv(x)\right\vert ^{2}-\varepsilon(1-\eta)d^{2}v(x).\ ] ] for @xmath3 but smaller than a constant that depends on @xmath166 and @xmath167 , there is @xmath168 with @xmath169 as @xmath170 such that the last display is non - negative . we then obtain from ( [ eq : lbq ] ) the non - asymptotic upper bound _ _ @xmath171 note that this bound is independent of @xmath13 , and also that this argument is not possible when @xmath172 . indeed , since @xmath173 for all @xmath174 , @xmath175(x)\geq0 $ ] is not possible for any choice of @xmath176 when @xmath172 . the quality of the bound obtained by this method depends on the degree to which the subsolution obtained for the pde @xmath149=0 $ ] ( plus boundary and terminal conditions ) accurately approximates the solution to this equation . in this example , we have used quite crude method to produce such a subsolution , which is to simply reduce a given subsolution to the @xmath18 equation by a constant factor of @xmath177 . an examination of the calculations suggest that the bound is not at all tight , which turns out to be true . in fact , in this situation we can construct a better subsolution and hence a tighter bound . for example , so long as the origin is not included @xmath178 can be used to obtain tighter bounds , though this function is not convenient for the time dependent problem . note that the two functions @xmath159 and @xmath179 play very different roles here . one is used to design an importance sampling scheme [ here @xmath159 ] , and one used for its analysis [ here @xmath179 ] . indeed , @xmath179 with @xmath182 is used only for the analysis of the scheme that corresponds to @xmath159 and in particular to derive a bound that is independent of @xmath13 . however , the design of the scheme and thus the simulation algorithm use the control @xmath155 . next we consider the behavior of @xmath143 when @xmath172 . in this case , we claim that @xmath143 grows without bound in @xmath13 for all @xmath3 , and therefore the performance of the control based on the quasipotential degrades as @xmath13 becomes large . to show this is true , we use the game representation to establish a lower bound on @xmath143 . we again examine a particular situation , which is to estimate the probability of escape from @xmath183 before time @xmath13 , after starting at @xmath111 at time @xmath116 . see figure [ fig : with_rest ] . the subsolution is still that of figure [ fig : sub_no_rest ] . [ ptb ] with_rest.eps with the understanding that @xmath184 now represents the time of escape of @xmath138 from @xmath185 , the representation ( [ eq : game_rep ] ) is still valid . suppose that @xmath13 is large and note that , while @xmath186 destabilizes the origin when used to construct the measure used for importance sampling , in the representation ( [ eq : game_rep ] ) it actually _ increases _ the stability of the origin , in the sense that @xmath187 . as a consequence , it is easy to construct a control @xmath188 which shows poor performance as @xmath189 . the construction is suggested in figure [ fig : bad_large_t ] . with @xmath13 large we divide @xmath19 $ ] into an initial part @xmath190 and a final part @xmath191,@xmath192 $ ] , with @xmath193 fixed . during the first part we apply @xmath194 . because the resulting dynamics of @xmath138 are stable about the origin , with very high probability the process settles around @xmath116 for the entire interval @xmath190 . in the game representation there is then a running cost of @xmath195 , which one can check is of order @xmath3 . in the second portion we apply a control which leads to escape prior to time @xmath13 , with a cost that may depend on @xmath193 but is independent of @xmath13 . an example of such a control , at least away from the origin , is as illustrated in figure [ fig : bad_large_t ] . the precise details of the construction in this second part are not important . all that is needed is that such a control exists , which can easily be demonstrated . [ ptb ] bad_large_t.eps when the two parts are combined , we have a control that provides an upper bound of the form @xmath196+c_{2},\ ] ] where @xmath197 and @xmath198 are positive constants . this shows that @xmath199}e^{-\frac{1}{\varepsilon}c_{2 } } , \label{eq : bad_lower_bound}\ ] ] and thus for fixed @xmath3 and large @xmath13 the term we have called the pre - exponential term dominates and the scheme is very far from optimal . we find that in this situation there are two exponential scalings , one in the noise strength and one in the length of the time interval , and the issue of which dominates depends on their relative sizes . these effects are reflected in computational data . in tables [ table1a ] and [ table1b ] we present both estimated values and relative errors for the problem of escape from an interval of the form @xmath200 $ ] , with @xmath201 . the process is a one dimensional gauss - markov model with drift @xmath202 and diffusion coefficient @xmath203 [ see ( [ eq : gm ] ) ] , with @xmath204 . in the tables , values of @xmath13 appear at the top and values of @xmath14 along the left side . each computed value is based on @xmath205 samples . a dash indicates that no samples escaped . to ease the presentation the relative errors are rounded to the nearest integer . owing to the fact that the subsolution based on the quasipotential is far from optimal in any sense when @xmath13 is small , the relative errors are large for small @xmath13 and decrease until approximately @xmath206 . for larger @xmath13 the errors grow rapidly with @xmath13 . note that the estimated relative errors in table [ table1b ] are not necessarily accurate for large @xmath13 , since they are subject to the same errors that can affect the probability estimates , but do indicate the qualitative worsening of estimation accuracy . [ c]|c|c|c|c|c|c|c|c|c|@xmath207 & @xmath208 & @xmath209 & @xmath210 & @xmath211 & @xmath212 & @xmath213 & @xmath214 & @xmath215 + @xmath216 & @xmath217 & @xmath218 & @xmath219 & @xmath220 & @xmath221 & @xmath222 & @xmath223 & @xmath224 + @xmath225 & @xmath226 & @xmath227 & @xmath228 & @xmath229 & @xmath230 & @xmath231 & @xmath232 & @xmath233 + @xmath234 & @xmath235 & @xmath236 & @xmath237 & @xmath238 & @xmath239 & @xmath240 & @xmath241 & @xmath242 + @xmath243 & @xmath244 & @xmath245 & @xmath246 & @xmath247 & @xmath248 & @xmath249 & @xmath250 & @xmath251 + @xmath252 & @xmath253 & @xmath254 & @xmath255 & @xmath256 & @xmath257 & @xmath258 & @xmath259 & @xmath260 + @xmath261 & @xmath253 & @xmath262 & @xmath263 & @xmath264 & @xmath265 & @xmath266 & @xmath267 & @xmath268 + @xmath269 & @xmath253 & @xmath270 & @xmath271 & @xmath272 & @xmath273 & @xmath274 & @xmath275 & @xmath276 + [ c]|c|c|c|c|c|c|c|c|c|@xmath207 & @xmath208 & @xmath209 & @xmath210 & @xmath211 & @xmath212 & @xmath213 & @xmath214 & @xmath215 + @xmath216 & @xmath277 & @xmath278 & @xmath279 & @xmath210 & @xmath210 & @xmath213 & @xmath280 & @xmath281 + @xmath225 & @xmath282 & @xmath213 & @xmath279 & @xmath210 & @xmath210 & @xmath213 & @xmath283 & @xmath284 + @xmath234 & @xmath285 & @xmath286 & @xmath287 & @xmath210 & @xmath210 & @xmath288 & @xmath283 & @xmath289 + @xmath243 & @xmath290 & @xmath291 & @xmath287 & @xmath210 & @xmath210 & @xmath213 & @xmath292 & @xmath293 + @xmath252 & @xmath253 & @xmath294 & @xmath295 & @xmath279 & @xmath210 & @xmath288 & @xmath296 & @xmath297 + @xmath261 & @xmath253 & @xmath298 & @xmath299 & @xmath279 & @xmath210 & @xmath300 & @xmath301 & @xmath302 + @xmath269 & @xmath253 & @xmath303 & @xmath300 & @xmath279 & @xmath210 & @xmath300 & @xmath292 & @xmath304 + tables [ table1c ] and [ table1d ] present the approximated values and relative errors for the problem with the domain @xmath305 $ ] and escape possible therefore only at @xmath306 . the results are of the same qualitative form as before , and carried out only to @xmath307 . [ c]|c|c|c|c|c|c|c|c|@xmath207 & @xmath208 & @xmath209 & @xmath210 & @xmath211 & @xmath212 & @xmath278 & @xmath213 + @xmath216 & @xmath308 & @xmath309 & @xmath310 & @xmath311 & @xmath240 & @xmath312 & @xmath313 + @xmath225 & @xmath314 & @xmath315 & @xmath316 & @xmath317 & @xmath219 & @xmath318 & @xmath319 + @xmath234 & @xmath320 & @xmath321 & @xmath322 & @xmath323 & @xmath324 & @xmath325 & @xmath326 + @xmath243 & @xmath327 & @xmath328 & @xmath256 & @xmath329 & @xmath218 & @xmath330 & @xmath331 + @xmath252 & @xmath253 & @xmath332 & @xmath321 & @xmath333 & @xmath334 & @xmath237 & @xmath218 + @xmath261 & @xmath253 & @xmath335 & @xmath336 & @xmath337 & @xmath321 & @xmath338 & @xmath339 + @xmath269 & @xmath253 & @xmath340 & @xmath341 & @xmath342 & @xmath343 & @xmath344 & @xmath345 + [ c]|c|c|c|c|c|c|c|c|@xmath207 & @xmath208 & @xmath209 & @xmath210 & @xmath211 & @xmath212 & @xmath278 & @xmath213 + @xmath216 & @xmath346 & @xmath213 & @xmath287 & @xmath279 & @xmath279 & @xmath299 & @xmath347 + @xmath225 & @xmath348 & @xmath347 & @xmath287 & @xmath279 & @xmath279 & @xmath295 & @xmath214 + @xmath234 & @xmath349 & @xmath350 & @xmath295 & @xmath279 & @xmath279 & @xmath295 & @xmath214 + @xmath243 & @xmath351 & @xmath352 & @xmath295 & @xmath279 & @xmath279 & @xmath295 & @xmath214 + @xmath252 & @xmath253 & @xmath353 & @xmath354 & @xmath287 & @xmath279 & @xmath295 & @xmath355 + @xmath261 & @xmath253 & @xmath356 & @xmath278 & @xmath287 & @xmath279 & @xmath295 & @xmath357 + @xmath269 & @xmath253 & @xmath358 & @xmath357 & @xmath295 & @xmath279 & @xmath295 & @xmath357 + it is useful to compare the two situations , and identify why uniform control of pre - exponential terms was not possible when @xmath172 . in both cases the control was based on the quasipotential , which is a valid subsolution to the @xmath18 problem . when using ito s formula to bound the second moment of the estimator , we must of course deal with the second derivative term , which is multiplied by @xmath3 . it can happen that this term has a sign that degrades ( increases ) the second moment , and indeed this is always true in a neighborhood of the origin ( this is essentially due to the convexity of @xmath160 near the origin ) . for the case where @xmath359 and for sufficiently small @xmath3 , this could be balanced by using that when @xmath105 @xmath360 and therefore @xmath361 are nonzero . however , this is not possible when the rest point is included in the domain of interest . indeed , the running cost that is accumulated in the construction leading to ( [ eq : bad_lower_bound ] ) corresponds to this term , and as that argument shows it can not be removed . the construction also suggests how the large variance comes about , which is that some trajectories generated under the change of measure defined by @xmath130 remain in a neighborhood of the origin for a long time , in spite of the fact that with such dynamics the origin is an unstable equilibrium point . the likelihood ratios along these trajectories can vary greatly and , even though they are themselves relatively unlikely , they are likely enough to increase the variance of the estimator to the point where it will become worse than standard monte carlo . as such , they are reminiscent of the rogue trajectories which lead to poor performance of non - dynamic forms of importance sampling as discussed in @xcite . it will turn out that to overcome the difficulties introduced by the rest point one must do a much better job of approximating the optimal change of measure than is possible based just on a time and @xmath14-independent subsolution . however , it also turns out that the additional accuracy is needed only near the rest point , where in fact explicit time and @xmath362-dependent solutions can be found . these are then combined with the simple time - independent subsolution based on the quasipotential to produce schemes that are nearly optimal and which protect against both sources of significant variance . an overview of the construction of such schemes is the topic of the next section . in this section we combine a local analysis that produces a time and @xmath14-dependent scheme near the rest point with a scheme based on the quasipotential elsewhere . there are of course few process models and problems for which the related hjb equation can be solved explicitly . however , a class of processes where this is possible are the gauss - markov models , i.e. , sdes with drift that is linear in the state and constant diffusion matrix . for these processes and for terminal conditions of the appropriate form , both the limit ( @xmath18 ) pde and the prelimit ( @xmath3 ) pde have an explicit solution that can be expressed in terms of the value function of a linear - quadratic regulator ( lqr ) control problem . [ ptb ] part_for_sub2.eps our ultimate approach to the construction of importance sampling schemes is suggested by figure [ fig : localization2 ] . the problem of interest is of the form ( [ eq : estimationtargetsmooth1 ] ) or an analogous problem involving escape from @xmath9 prior to @xmath13 . the particular problem will fix the boundary and terminal conditions on @xmath19\times\partial\mathcal{d}$ ] and @xmath363 . in the figure , we are interesting in estimating the exit probability @xmath364 where @xmath116 is the rest point and @xmath365 with @xmath366 . in most of the domain , which is the section outside the curves that terminate at @xmath367 and after time @xmath368 , the control is based on a subsolution @xmath159 constructed in terms of the quasipotential . within the curves the solution to ( [ eq : diffusion1 ] ) is well approximated by a gauss - markov process . hence within this region we will use a control that would be appropriate for a problem if the process were instead the approximating gauss - markov model . the function @xmath369 that defines the pde for this region is therefore the one corresponding to the gauss - markov process . besides the dynamics in this region ( which are determined by the gauss - markov approximation ) , we must choose a terminal condition . this will be given by the minimum of two quadratic functions , one centered at @xmath370 and one centered at @xmath371 . the parameter @xmath370 is a nominal value that for purposes of the present discussion can be taken to be @xmath210 . the parameter @xmath372 plays two roles . one is to determine the size of the region on which the true dynamics are approximated by gauss - markov dynamics . the second role is related to the fact that if the control based on the quadratics centered at @xmath373 is used all the way to @xmath13 then singularities will develop . for this reason , we switch back to the control based on the quasipotential after @xmath368 , and @xmath372 must be chosen so that the subsolution property is preserved across the handoff time @xmath368 . while the discussion above suggests the correct decomposition of the domain , the actual construction is more complex , since the control should transition nicely when moving between the regions , and the construction of a scheme for which rigorous bounds can be proved will require additional mollification and approximations . however , the building blocks are always subsolutions to the indicated pdes . the form of the surrogate problem for the gauss - markov model needs to be explained , as well as various boundary and terminal conditions . to simplify the discussion , we consider a sequence of successively more general problems . in this paper we complete the analysis for the one - dimensional problem . the multi - dimensional problem will be addressed in a companion paper . our first goal is to construct a subsolution for the @xmath18 problem for the processes that will be used in the localization . this can be related to the problem of estimating the probability that the solution to @xmath374 escapes from the interval @xmath200 $ ] before time @xmath13 , a problem that was also used for the computational examples of the last subsection . the parameters @xmath375 and @xmath376 are positive constants . in this subsection we will take @xmath377 , and because of this can postpone the issue regarding singularities in the control at @xmath378 . we thus also take @xmath372 to be zero , and will return to the role of @xmath372 and its selection for a general problem in the next subsection . the corresponding pde for the escape probability is @xmath379 plus the terminal and boundary conditions@xmath380{cc}0 & x=\pm a , t\in\lbrack0,t],\\ \infty & x\in(-a , a),t = t . \end{array } \right.\ ] ] while simple in appearance , this equation does not have an explicit solution . the equation obtained in the limit @xmath381 is more tractable , and the unique viscosity solution can be described as follows . @xmath382 corresponds to the variational problem @xmath383\right\ } .\ ] ] depending on how and when the minimizing trajectory leaves @xmath200 $ ] , the solution takes a particular explicit form . ( in all cases the minimizer can be found by solving the appropriate euler - lagrange equation . ) if for the initial condition @xmath144 the minimizing trajectory leaves before time @xmath13 , then @xmath384 .\ ] ] this is the case when the minimal cost is the negative of the quasipotential , translated by a constant to satisfy the boundary condition at the exit location . such initial conditions satisfy @xmath385 . when @xmath386 @xmath387 the minimizer leaves through @xmath306 at exactly time @xmath13 , and the minimizing value is @xmath388}.\ ] ] one can also interpret @xmath389 as the minimal cost for a linear quadratic regulator with a singular terminal cost applied at time @xmath13 , i.e. , a cost that equals @xmath116 at @xmath306 and @xmath42 otherwise . by symmetry it is clear that when @xmath390 the minimizing trajectory will exit at @xmath391 . define @xmath392 setting @xmath393 , we have @xmath394 note that when @xmath395 @xmath396}=\frac{c}{\bar{\sigma}^{2}}\frac{(a - ae^{2c(t - t)})^{2}}{[1-e^{2c(t - t)}]}=\frac{c}{\bar{\sigma}^{2}}\left [ a^{2}-x^{2}\right ] = f_{1}(ae^{c(t - t)}),\ ] ] and therefore @xmath382 is continuous for all @xmath397 $ ] . in fact more is true , and one can check that @xmath398 and thus @xmath382 has a continuous partial derivative in @xmath16 for @xmath399 . the mapping @xmath400 is convex , and the graph of this mapping lies above that of @xmath401 , which is concave . thus the two graphs intersect only at the point @xmath402/\bar{\sigma}^{2})$ ] , where both functions and their first derivatives in @xmath16 agree . see figure [ fig : eps_zero_soln ] . [ ptb ] eps_zero_soln.eps note also that @xmath403 for @xmath404 $ ] is the solution to the @xmath18 problem with escape from @xmath305 $ ] ( a one - sided version of the problem of escape from @xmath200 $ ] ) . simulation data for the schemes based on these two value functions are presented below . the approximated values are omitted since they are qualitatively similar to those in tables [ table1a ] and [ table1c ] , and only relative errors are presented . table [ table2a ] gives data based on @xmath405 for the problem of two - sided exit , and should be compared to table [ table1b ] . the use of the solution to hjb equation for @xmath18 drastically improves the performance for small @xmath13 , which is due to the fact that the subsolution based on the quasipotential is a very poor approximation to the solution for @xmath3 for such @xmath13 . however , for large @xmath13 the two schemes are comparably bad . [ c]|c|c|c|c|c|c|c|c|c|@xmath207 & @xmath208 & @xmath209 & @xmath210 & @xmath211 & @xmath212 & @xmath213 & @xmath214 & @xmath215 + @xmath216 & @xmath279 & @xmath279 & @xmath210 & @xmath210 & @xmath210 & @xmath213 & @xmath406 & @xmath407 + @xmath225 & @xmath287 & @xmath287 & @xmath210 & @xmath210 & @xmath210 & @xmath213 & @xmath408 & @xmath409 + @xmath234 & @xmath295 & @xmath279 & @xmath210 & @xmath210 & @xmath210 & @xmath288 & @xmath280 & @xmath410 + @xmath243 & @xmath287 & @xmath279 & @xmath279 & @xmath210 & @xmath210 & @xmath288 & @xmath411 & @xmath412 + @xmath252 & @xmath287 & @xmath287 & @xmath279 & @xmath210 & @xmath210 & @xmath288 & @xmath296 & @xmath413 + @xmath261 & @xmath287 & @xmath287 & @xmath279 & @xmath279 & @xmath210 & @xmath288 & @xmath414 & @xmath415 + @xmath269 & @xmath295 & @xmath279 & @xmath279 & @xmath279 & @xmath210 & @xmath300 & @xmath416 & @xmath417 + the data for exit from one side are given in table [ table2b ] . in contrast with the two - sided problem , the performance does not degrade so quickly with large @xmath13 . this suggests that the decline in performance is not so much due to using the approximating @xmath18 pde , but rather the possible lack of regularity of the solution to this equation . [ c]|c|c|c|c|c|c|c|c|@xmath207 & @xmath208 & @xmath209 & @xmath210 & @xmath211 & @xmath212 & @xmath278 & @xmath213 + @xmath216 & @xmath210 & @xmath210 & @xmath210 & @xmath210 & @xmath210 & @xmath279 & @xmath287 + @xmath225 & @xmath210 & @xmath210 & @xmath210 & @xmath210 & @xmath210 & @xmath279 & @xmath295 + @xmath234 & @xmath210 & @xmath210 & @xmath210 & @xmath210 & @xmath210 & @xmath279 & @xmath287 + @xmath243 & @xmath210 & @xmath210 & @xmath210 & @xmath210 & @xmath210 & @xmath279 & @xmath287 + @xmath252 & @xmath210 & @xmath210 & @xmath210 & @xmath210 & @xmath210 & @xmath279 & @xmath287 + @xmath261 & @xmath210 & @xmath210 & @xmath210 & @xmath210 & @xmath210 & @xmath279 & @xmath287 + @xmath269 & @xmath210 & @xmath210 & @xmath210 & @xmath210 & @xmath210 & @xmath279 & @xmath287 + recall that the difficulty with the use of the subsolution based on the quasipotential alone ( here @xmath401 ) was that in a neighborhood of @xmath101 the second derivative was negative , and when @xmath13 is large this leads to poor control of the variance of the associated scheme for both the one - sided and two - sided problems . when using the time - dependent @xmath18 pde as the basis for a scheme for the one - sided problem , the introduction of @xmath418 appears to have largely mitigated the problem due to the second derivative term . note that this function determines the value of @xmath419 when @xmath101 and is convex rather than concave in @xmath16 . in contrast , for the two - sided problem there is a concave singularity at the origin . there the second derivative in @xmath16 is @xmath420 , and the subsolution property for the @xmath3 problem again fails . what is needed is a subsolution that works across the point @xmath101 for the @xmath3 problem . such a subsolution can be constructed using the mollification introduced in the next subsection . simulations based on such a mollified version of the @xmath18 subsolution ( and with mollification parameter @xmath421 ) are presented in table [ table6b ] , and support the claim just made . [ c]|c|c|c|c|c|c|c|c|c|c|@xmath207 & @xmath208 & @xmath209 & @xmath210 & @xmath211 & @xmath212 & @xmath213 & @xmath214 & @xmath215 & @xmath350 + @xmath216 & @xmath210 & @xmath210 & @xmath210 & @xmath210 & @xmath210 & @xmath210 & @xmath287 & @xmath299 & @xmath422 + @xmath225 & @xmath210 & @xmath210 & @xmath210 & @xmath210 & @xmath210 & @xmath210 & @xmath287 & @xmath299 & @xmath423 + @xmath234 & @xmath210 & @xmath210 & @xmath210 & @xmath210 & @xmath210 & @xmath210 & @xmath279 & @xmath295 & @xmath424 + @xmath243 & @xmath279 & @xmath210 & @xmath210 & @xmath210 & @xmath210 & @xmath210 & @xmath279 & @xmath295 & @xmath425 + @xmath252 & @xmath279 & @xmath279 & @xmath279 & @xmath279 & @xmath210 & @xmath210 & @xmath279 & @xmath295 & @xmath426 + @xmath261 & @xmath279 & @xmath279 & @xmath279 & @xmath279 & @xmath210 & @xmath210 & @xmath279 & @xmath287 & @xmath408 + @xmath269 & @xmath279 & @xmath279 & @xmath279 & @xmath279 & @xmath210 & @xmath279 & @xmath279 & @xmath287 & @xmath427 + in spite of the shortcomings we have described in this section , the solution to the @xmath18 problem serves as the starting point for a construction that can be shown to perform well both in theory and in practice . there are three important modifications that are needed . * the first is that , in order to effectively deal with the @xmath3 dynamics and in particular to avoid the degradation still present in table [ table2b ] when @xmath13 is large , the region where the @xmath389 subsolution determines the dynamics must be enlarged . the solution to the @xmath18 problem constructed above leads to a region that vanishes exponentially in @xmath428 [ see ( [ eq : basicfcn1 ] ) ] . * the second modification will be the use a mollification to eliminate singularities such as the one at @xmath101 and help guarantee a global subsolution property for the @xmath3 pde . the particular mollification we use is very convenient , and was first used for importance sampling in @xcite , though with a somewhat different intended use . * the final modification was also alluded to previously , which is to revert to the quasipotential based control in an interval of the form @xmath429 $ ] . all three modifications will be introduced in the next subsection in the context of the gauss - markov process . in this subsection we generalize the construction of the last subsection . as discussed there , the generalizations are needed to address issues that play a role in both the theoretical analysis of the scheme and its practical performance . after introducing these generalizations , we will demonstrate ( numerically and theoretically ) that the suggested change of measure does not degrade in performance as @xmath13 gets larger and is close to optimality not only as @xmath15 , but for fixed @xmath3 as well . we begin by introducing the first generalization , which is to replace the terminal condition @xmath430 , which was used to define @xmath418 , by a terminal condition of the form @xmath431 . the parameter @xmath370 replaces @xmath306 and is a nominal value introduced to disconnect the localization from the boundary . the solution to the lqr that corresponds to @xmath431 , which will be denoted @xmath432 , is automatically smaller than @xmath433 . the motivation for replacing @xmath42 by @xmath434 is because we want the solution to the lqr problem [ i.e. , @xmath432 ] to determine the control near @xmath101 for a set whose width is uniformly ( in @xmath436 ) bounded below away from zero . as discussed in the last section , the second derivative term associated with @xmath437 is of the wrong sign and degrades performance . the neighborhood where @xmath438 does not degenerate as @xmath439 , and its size is decreasing in @xmath440 . the introduction of @xmath440 complicates the construction by also requiring mollification ( in addition to the one that will be needed at @xmath101 ) , since @xmath441 can no longer be smoothly merged with @xmath437 . for @xmath442 the solution to this lqr takes the form @xmath443 recall that @xmath444/\bar{\sigma}^{2}$ ] so that @xmath445 . . it will be important to know which of @xmath447 , @xmath448 , and @xmath437 is smallest , and we note here several properties . let @xmath449 . the first is that there are two real solutions to @xmath450 , and these take the form@xmath451 between these roots @xmath452 , and on the complement of the interval the reverse inequality holds . the limit @xmath453 gives the asymptotic endpoints of the interval where @xmath452 , which are @xmath454 we point out here that a natural scaling for this problem , given that the size of the neighborhood of zero where the quasipotential based subsolution fails for the @xmath3 system scales like @xmath455 , is to ask that this width scale as @xmath456 $ ] . if for example the desired width is @xmath457 , then when @xmath3 is small then we should take @xmath458 . the next adaptation is required so that singularities in the control associated with @xmath459 as @xmath460 do not cause a problem , as well as for purposes of localization . for a parameter @xmath461 we will want @xmath462 for all @xmath463 $ ] , where ( with an abuse of notation ) @xmath464 . this is true if and only if the smaller solution to @xmath465 is less than or equal to zero . this root was found to be @xmath466 and the restriction that this be non positive can be simplified to@xmath467 the inequality @xmath462 will ensure that the subsolution property is preserved if we switch from using @xmath405 for @xmath468 to using @xmath437 for @xmath469 . if @xmath470 this means that we always use @xmath437 , but our interest here is in large @xmath13 . we assume that @xmath471 so that @xmath472 . to guarantee that the smaller root is strictly negative and to conveniently satisfy a bound used later on we assume @xmath473 besides enforcing the subsolution property across the handoff at time @xmath368 , the selection of @xmath372 plays a key role in determining the region used for the localization for the general nonlinear problem . owing to the exponential decay , one can confine the localization to a small region with a modest value of @xmath372 . suppose we consider confining to a region that scales as @xmath457 for all @xmath474 $ ] and arbitrarily large @xmath13 . as discussed previously , for small @xmath436 and large @xmath368 this suggests that @xmath440 be approximately @xmath475 , which means that @xmath476 $ ] . recalling that the loss in performance of the importance sampling scheme when the quasipotential based subsolution is used scales like @xmath477 , this gives a loss over such an interval of the form @xmath429 $ ] as scaling like @xmath478 $ ] . we will return to such considerations in the next section . finally there is the issue of mollification . owing to the replacement of @xmath479 by @xmath459 , there are now several sources of discontinuity in the gradient . the subsolution prior to mollification would in general take the form @xmath480 . however , since we know that @xmath481 is dominated by @xmath401 near @xmath101 , we can also use@xmath482 see figure [ fig : various_functions ] . [ ptb ] domains_for_bounds3.eps we next state a result that will be used to derive performance bounds for schemes based on the mollification . the proof is deferred to the appendix . we consider the general one dimensional process model @xmath483 where @xmath20 and @xmath21 are lipschitz continuous . letting @xmath484 , the relevant @xmath14-dependent pde is@xmath485 ( t , x)=u_{t}(t , x)+du(t , x)b(x)-\frac { 1}{2}\left\vert \sigma(x)du(t , x)\right\vert ^{2}+\frac{\varepsilon}{2}\alpha(x)d^{2}u(t , x)=0 . \label{eq : eps}\ ] ] [ lem : mollification]suppose that the functions @xmath486\times\mathbb{r}$ ] , @xmath487 are twice continuously differentiable in @xmath16 , once continuously differentiable in @xmath436 , and satisfy @xmath488(t , x)\geq\gamma_{i}(x , t,\varepsilon).\ ] ] for @xmath23 define @xmath489 and define the weights@xmath490 then @xmath491 and for @xmath492@xmath493(t , x ) & \geq\frac{1}{2}\left ( 1-\frac{\varepsilon}{\delta}\right ) \left [ \sum_{i=1}^{n}\rho_{i}\left ( t , x;\delta\right ) \left\vert \sigma(x)d\tilde{u}_{i}(t , x)\right\vert ^{2}-\left\vert \sum_{i=1}^{n}\rho_{1}\left ( t , x;\delta\right ) \sigma(x)d\tilde{u}_{i}(t , x)\right\vert ^{2}\right ] \\ & \qquad+\sum_{i=1}^{n}\rho_{i}\left ( t , x;\delta\right ) \gamma _ { i}(x , t,\varepsilon)\\ & \geq\sum_{i=1}^{n}\rho_{i}\left ( t , x;\delta\right ) \gamma_{i}(x , t,\varepsilon).\end{aligned}\ ] ] based on this result we consider the mollification of @xmath494 , which is @xmath495 for reasons that will be made clear in the analysis ( see lemma [ l : region_y_hatx ] ) , we generally use @xmath421 . as discussed previously , we return to the control based on the quasipotential for the last @xmath372 units of time , and so the subsolution takes the form@xmath496{cc}f_{1}(x ) , & t > t - t^{\ast}\\ u^{\delta}(t , x ) , & t\leq t - t^{\ast}\end{array } \right . . \label{eq : controldesignlinear}\ ] ] note that the mollification reduces values , in that @xmath497 , and so the requirement @xmath498 for @xmath463 $ ] holds , since @xmath462 . we next present a rigorous and nonasymptotic bound for the second moment of the importance sampling scheme constructed in the last subsection . to derive a bound that is valid for @xmath3 and uniform in @xmath13 , we use the same representation as in ( [ eq : lbq ] ) . in particular , we choose @xmath499 defined via ( [ eq : controldesignlinear ] ) for the design and @xmath500 for the analysis , where @xmath501{cc}f_{1}(x ) , & t > t - t^{\ast}\\ u^{\delta,\eta}(t , x ) , & t\leq t - t^{\ast}\end{array } \right . , \label{eq : controlanalysislinear}\ ] ] with @xmath502 as discussed in section 3 this choice is driven by the need for a subsolution for the @xmath3 dynamics with an explicit form , and this limitation of the technique leads to conservative bounds on the true performance . to simplify notation , for smooth functions @xmath503 we define @xmath504(t , x)=\mathcal{g}^{\varepsilon}[w](t , x)-\frac { 1}{2}\left\vert \bar{\sigma}\left ( dw(t , x)-du(t , x)\right ) \right\vert ^{2 } , \label{eq : ge_2}\ ] ] where @xmath149 $ ] is defined in ( [ eq : eps ] ) . using that @xmath505=\mathcal{g}^{0}[f_{2}^{m}]=0 $ ] , @xmath506(x)=\frac{\varepsilon}{2}\bar{\sigma}^{2}d^{2}f_{1}(x)=-\varepsilon c\text { and } \mathcal{g}^{\varepsilon}[f_{2}^{m}](t , x)=\frac{\varepsilon}{2}\bar{\sigma}^{2}d^{2}f_{2}^{m}(t , x)=\varepsilon\bar{\sigma}^{2}a^{m}(t ) . \label{eq : values_for_pieces}\ ] ] since the problem is symmetric , it suffices to consider only @xmath507 $ ] . a key role is played by the weights associated with the exponential mollification of the subsolution , which take the forms @xmath508 and @xmath509 to determine which of these dominate at any @xmath144 , the relative sizes of the functions @xmath459 and @xmath401 are required , with smaller functions corresponding to more dominant weights . for this reason the solutions to @xmath510 play an important role , and especially the larger one identified in ( [ eq : roots ] ) . see figure [ fig : various_functions ] . the following bounds on this root will be used to partition the domain in the analysis . let @xmath511 @xmath512 and let @xmath513 denote the larger root in ( [ eq : roots ] ) . then we claim under ( [ eq : bound_on_tstar ] ) that @xmath514\ ] ] and uniformly in @xmath13 . we recall the definition @xmath515 and that @xmath471 , so that @xmath516 . then the smallest possible value of @xmath513 satisfies @xmath517 while the largest satisfies @xmath518 where the first inequality uses @xmath519 . we set @xmath520 . in order to obtain bounds on the performance under the corresponding scheme , we need to bound @xmath521(t , x)$ ] from below in various regions . by ( [ eq : ge_2 ] ) @xmath522(t , x ) & = \mathcal{g}^{\varepsilon}[u^{\delta,\eta}](t , x)-\frac{1}{2}\left\vert \bar{\sigma}\left ( du^{\delta,\eta}(t , x)-du^{\delta}(t , x)\right ) \right\vert ^{2}\nonumber\\ & = \mathcal{g}^{\varepsilon}[u^{\delta,\eta}](t , x)-\frac{1}{2}\eta ^{2}\left\vert \bar{\sigma}du^{\delta}(t , x)\right\vert ^{2}. \label{eq : combinedexpression}\ ] ] we will use the notation @xmath523(x)=-\varepsilon c \label{eq : def_gam0}\ ] ] and @xmath524(t , x)=\varepsilon \bar{\sigma}^{2}a^{m}(t ) . \label{eq : def_gamm}\ ] ] straightforward calculations and some algebra give @xmath525(t , x)\geq(1-\eta)\mathcal{g}^{\varepsilon}[u^{\delta}](t , x)+\frac{1}{2}\left ( \eta-\eta^{2}\right ) \left\vert \bar{\sigma}du^{\delta}(t , x)\right\vert ^{2}.\ ] ] for notational convenience , define @xmath526 ( t , x).\end{aligned}\ ] ] note that by jensen s inequality @xmath527 . we next apply lemma [ lem : mollification ] to @xmath528(t , x)$ ] ( while suppressing the dependence on @xmath529 in the notation for the @xmath530 s ) , and use ( [ eq : combinedexpression ] ) to get@xmath522(t , x ) & \geq ( 1-\eta)\frac{1}{2}\left ( 1-\frac{\varepsilon}{\delta}\right ) \beta _ { 0}(t , x)+(1-\eta)\left [ \rho_{2}^{m,+}(t , x)+\rho_{2}^{m,-}(t , x)\right ] \gamma_{2}^{m}(t)\nonumber\\ & \qquad+(1-\eta)\rho_{1}(t , x)\gamma_{1}+\frac{1}{2}\left ( \eta-2\eta ^{2}\right ) \left\vert \bar{\sigma}du^{\delta}(t , x)\right\vert ^{2 } \label{eq : bound_for_moll}\ ] ] for all @xmath463 $ ] and @xmath474 $ ] . we will partition the domain according @xmath511 @xmath531 and @xmath532 . we consider three cases depending on whether @xmath533 $ ] , @xmath534 $ ] or @xmath535 $ ] if @xmath536 . the case @xmath390 is symmetric . before proceeding with the analysis for each of the cases , we give the definition of exponential negligibility , a concept used frequently in the rest of the paper . [ d : exponentialnegligibility ] a term is called exponential negligible if it is bounded above in absolute value by a quantity of the form @xmath537 , where @xmath538 and @xmath539 . [ l : region_0_y ] assume that @xmath540\times \lbrack0,z]$ ] , @xmath541 and @xmath542 . then , up to an exponentially negligible term @xmath543(t , x)\geq0.\ ] ] in this region @xmath544 , and we claim that the inequality is in fact strict . we have @xmath545 -\frac{c}{k-\bar{\sigma}^{2}e^{2c(t - t)}}\left [ xe^{c(t - t)}-\hat{x}\right ] ^{2}. \label{eq : f1-f2}\ ] ] for each fixed @xmath436 this defines a concave function of @xmath16 . at @xmath101 the value is minimized when @xmath546 . using @xmath547^{4}\leq c / m\bar{\sigma}^{2}$ ] ( since @xmath548 ) and @xmath549 , we obtain the strictly positive lower bound @xmath550\right ] $ ] . since @xmath551 , by concavity there is @xmath552 such that ( [ eq : f1-f2 ] ) is bounded below by @xmath553 for all @xmath540\times\lbrack0,z]$ ] . thus the term in ( [ eq : bound_for_moll ] ) involving the weight @xmath554 is exponentially negligible . since @xmath527 , @xmath555 and @xmath556 , all other terms are non - negative , and the result follows . [ l : region_hatx_a ] assume that @xmath540\times\lbrack h , a]$ ] , @xmath541 and @xmath557 . then letting @xmath558 , we have that for all @xmath559 and with any @xmath560 $ ] , up to an exponentially negligible term@xmath543(t , x)\geq0 . \label{eq : bound2_b}\ ] ] in this region @xmath561 , and it is straightforward that the terms associated with @xmath562 are exponentially negligible . note that @xmath563 is convex , recall that for each @xmath474 $ ] the largest value where the two functions agree is smaller than @xmath564 , and that @xmath565 inserting the largest root for the given @xmath436 gives the value@xmath566 a lower bound on the first term is @xmath567 . using @xmath568^{2}$ ] , the definition of @xmath193 , and @xmath569 to bound the second term from below produces the strictly positive lower bound@xmath570 for all @xmath474 $ ] and @xmath571 . since @xmath572 , this shows that there is @xmath573 such that @xmath574 for all @xmath540\times\lbrack h , a]$ ] . it follows that terms involving @xmath575 are exponentially negligible . since @xmath527 and @xmath576 up to an exponentially negligible term , @xmath522(t , x ) & \geq ( 1-\eta)\rho_{1}(t , x)\gamma^{0}+\frac{1}{2}\left ( \eta-2\eta^{2}\right ) \bar{\sigma}^{2}\left\vert \rho_{1}(t , x)df_{1}(x)\right\vert ^{2}\\ & \geq-(1-\eta)\varepsilon c+2\left ( \eta-2\eta^{2}\right ) \frac{c^{2}}{\bar{\sigma}^{2}}x^{2}\ ] ] up to an exponentially negligible term . choosing @xmath557 gives @xmath543(t , x)\geq-(1-\eta ) \varepsilon c+\eta\frac{c^{2}}{\bar{\sigma}^{2}}x^{2}\ ] ] and for @xmath14 small enough such that @xmath577 $ ] , the last display is non - negative . for this interval to be nonempty imposes the constraint @xmath578 . hence in this region and up to an exponentially negligible term,@xmath543(t , x)\geq0.\ ] ] the final region is the most difficult , since @xmath579 can be either positive or negative . [ l : region_y_hatx ] assume that @xmath540\times\lbrack z , h]$ ] , @xmath557 and set @xmath421 . then up to an exponentially negligible term@xmath543(t , x)\geq\frac{1}{2}\left [ \frac{c^{2}\eta}{2\bar{\sigma}^{2}}\left ( z-\hat{x}e^{c(t - t)}\right ) ^{2}-2\varepsilon c\right ] \wedge0 . \label{eq : bound2_a}\ ] ] while terms corresponding to @xmath562 are exponentially negligible in this region , since @xmath580 changes sign both @xmath581 and @xmath554 may be important . since they are negligible we omit terms corresponding to @xmath562 . by ( [ eq : bound_for_moll ] ) we have up to an exponentially negligible term@xmath522(t , x)\geq & ( 1-\eta)\frac{1}{4}\beta_{0}(t , x)+(1-\eta)\rho_{2}^{m,+}(t , x)\gamma_{2}^{m}(t)+(1-\eta)\rho_{1}(t , x)\gamma_{1}\nonumber\\ & + \frac{1}{2}\left ( \eta-2\eta^{2}\right ) \bar{\sigma}^{2}\left\vert \rho_{2}^{m,+}(t , x)df_{2}^{m}(t , x)+\rho_{1}(t , x)df_{1}(x)\right\vert ^{2 } \label{eq : boundforgreyarea1}\ ] ] as noted previously @xmath527 for all @xmath144 . however , we will exploit the fact that @xmath582 only for points @xmath144 such that @xmath583 . we distinguish two cases depending on whether @xmath584 or @xmath585 . * case i : @xmath584 . * we know that @xmath527 and @xmath586 and can ignore those terms . using @xmath587 , the terms that remain are @xmath588 .\end{aligned}\ ] ] we claim that for @xmath540\times\lbrack z , h]$ ] @xmath589 first , we note that @xmath590 , and thus @xmath591 . therefore @xmath592 second , by ( [ eq : diff_of_deriv ] ) , the definition of @xmath593 and @xmath594 , we also have @xmath595 where the last inequality uses @xmath569 . we conclude that @xmath596 since @xmath597 and @xmath557 , we obtain the bound@xmath543(t , x)\geq-(1-\eta ) \varepsilon c+\frac{1}{16}\eta\bar{\sigma}^{2}|df_{1}(x)-df_{2,+}^{m}(t , x)|^{2}. \label{eq : casei}\ ] ] this gives a bound for case i. * case ii : @xmath585 . * here we will have to use @xmath598 . dropping other terms on the right that are not possibly negative , we obtain from ( [ eq : boundforgreyarea1 ] ) that @xmath543(t , x)\geq(1-\eta)\frac { 1}{4}\beta_{0}(t , x)+(1-\eta)\rho_{1}(t , x)\gamma_{1}.\ ] ] omitting exponentially negligible terms , we note that @xmath599 ( t , x)\\ & = \bar{\sigma}^{2}\rho_{1}\left [ \left\vert df_{1}\right\vert ^{2}-\left\vert df_{2,+}^{m}\right\vert ^{2}-2df_{2,+}^{m}(df_{1}-df_{2,+}^{m})-\rho_{1}(df_{1}-df_{2,+}^{m})^{2}\right ] ( t , x)\\ & = \bar{\sigma}^{2}\rho_{1}(df_{1}-df_{2,+}^{m})\left [ df_{1}+df_{2,+}^{m}-df_{2,+}^{m}-\rho_{1}(df_{1}-df_{2,+}^{m})\right ] ( t , x)\\ & = \bar{\sigma}^{2}\rho_{1}(1-\rho_{1})(df_{1}-df_{2,+}^{m})^{2}(t , x)\\ & \geq\frac{1}{2}\bar{\sigma}^{2}\rho_{1}(df_{1}-df_{2,+}^{m})^{2}(t , x),\end{aligned}\ ] ] where the last inequality uses @xmath585 , and so obtain @xmath543(t , x)\geq(1-\eta)\rho _ { 1}\left [ \frac{1}{8}\bar{\sigma}^{2}|df_{1}(x)-df_{2,+}^{m}(t , x)|^{2}-\varepsilon c\right ] .\ ] ] this gives a bound for case ii . straightforward estimation gives , for all @xmath540\times\lbrack z , h]$ ] , the lower bound@xmath600 using this bound and @xmath557 , we get a lower bound from ( [ eq : casei ] ) in the form @xmath522(t , x ) & \geq\left [ \frac{1}{16}\eta\bar{\sigma}^{2}|df_{1}(x)-df_{2,+}^{m}(t , x)|^{2}-\varepsilon c\right ] \\ & \geq\left [ \frac{c^{2}\eta}{4\bar{\sigma}^{2}}\left ( z-\hat{x}e^{c(t - t)}\right ) ^{2}-\varepsilon c\right ] .\end{aligned}\ ] ] since this is less than the bound for case ii when both terms are negative , the conclusion of the lemma follows . [ t : mainboundlinear ] assume @xmath601 , and that @xmath602 . let @xmath130 be the control based on the function @xmath603 defined in ( [ eq : controldesignlinear ] ) , i.e. , @xmath604 . then up to an exponentially negligible term , we have@xmath605 where @xmath606 \wedge0\right ) \varepsilon.\ ] ] @xmath607 and @xmath608 . although the bound provided by theorem [ t : mainboundlinear ] takes a complicated form , it is important to note that it does not degrade as @xmath189 , and this is also reflected in the simulation data . also , as noted previously there are natural scalings under which @xmath609 and @xmath610 as @xmath381 . using the bound from below given in lemma [ lem : mollification ] and the explicit form of @xmath611 , we obtain @xmath612 hence , we obtain the rate of decay@xmath613\ ] ] uniformly in @xmath13 as @xmath381 . [ proof of theorem [ t : mainboundlinear]]the starting point is the representation ( [ eq : game_rep ] ) [ but rewritten for the more general process model and with time dependent @xmath614 , which is valid for every @xmath3 . we can restrict to @xmath188 such that @xmath615 w.p.1 , obtaining @xmath616 . \label{eq : rep_with_constraint}\ ] ] we can also assume @xmath617 since the bound is straightforward otherwise . we recall that under ( [ eq : bound_on_tstar ] ) the subsolution property is preserved for @xmath618 at @xmath368 , i.e. , that@xmath619 . \label{eq : subsolutionproperty_tstar}\ ] ] next consider any control in the representation such that @xmath620 w.p.1 . we will apply it s formula separately over the intervals @xmath621 and @xmath622 and also use the boundary condition @xmath623 for @xmath624 $ ] . since @xmath625 we obtain @xmath626 1_{\left\ { \hat{\tau}^{\varepsilon}\geq t - t^{\ast } \right\ } } \nonumber\\ & \quad+\mathbb{e}_{0,0}\left [ \bar{u}^{\delta,\eta}((t - t^{\ast})\wedge \hat{\tau}^{\varepsilon},\hat{x}^{\varepsilon}((t - t^{\ast})\wedge\hat{\tau } ^{\varepsilon}))-\bar{u}^{\delta,\eta}(0,0)\right ] . \label{eq : two_partsa}\ ] ] using lemma [ l : generalbound ] and recalling the definition of @xmath627 $ ] in ( [ eq : ge_2 ] ) , the contribution from @xmath621 gives@xmath628 \\ & = \mathbb{e}_{0,0}\int_{0}^{(t - t^{\ast})\wedge\hat{\tau}^{\varepsilon}}\left [ \mathcal{g}^{\varepsilon}[\bar{u}^{\delta,\eta},\bar{u}^{\delta } ] ( s,\hat{x}^{\varepsilon}(s))ds-\frac{1}{4}v(s)^{2}ds+\frac{1}{2}\bar { u}(s,\hat{x}^{\varepsilon}(s))^{2}\right ] ds.\end{aligned}\ ] ] an analogous formula holds for @xmath629 , save that since @xmath630 , the term @xmath631 $ ] simplifies to @xmath632 $ ] . rearranging and using ( [ eq : two_partsa ] ) , @xmath633 \geq2\bar{u}^{\delta,\eta}(0,0)\\ & + \mathbb{e}_{0,0}\int_{0}^{(t - t^{\ast})\wedge\hat{\tau}^{\varepsilon}}2\mathcal{g}^{\varepsilon}[\bar{u}^{\delta,\eta},\bar{u}^{\delta}](s,\hat { x}^{\varepsilon}(s))ds+\mathbb{e}_{0,0}\int_{(t - t^{\ast})\wedge\hat{\tau } ^{\varepsilon}}^{\hat{\tau}^{\varepsilon}}1_{\left\ { \hat{\tau}^{\varepsilon}\geq t - t^{\ast}\right\ } } 2\mathcal{g}^{\varepsilon}[f_{1}](s,\hat{x}^{\varepsilon}(s))ds.\end{aligned}\ ] ] we now replace each term by a lower bound , using lemmas [ l : region_0_y ] , [ l : region_hatx_a ] and [ l : region_y_hatx ] for @xmath634 $ ] . since the bounds are independent of the control process @xmath188 , the representation ( [ eq : rep_with_constraint ] ) implies @xmath635 ds - t^{\ast}2\varepsilon c,\ ] ] where @xmath636 are the times in @xmath637 $ ] where the integrand is negative . we next use the constraint @xmath638 , which guarantees that for @xmath639 sufficiently large the integrand is in fact positive . let @xmath640 .\ ] ] then since the integrand is only negative for @xmath641 , we obtain the lower bound @xmath642 2\varepsilon c$ ] for the integral . adding the remaining @xmath643 then gives the result as stated . in this subsection we present simulation data for the linear problem and make several comments on the application of the algorithm . for comparison purposes , we consider the same two sided problem corresponding to the data from tables [ table1a ] , [ table1b ] , [ table2a ] and [ table6b ] . thus we consider the small noise diffusion process with drift @xmath644 , where @xmath645 , diffusion coefficient @xmath455 , and starting from the stable equilibrium point @xmath101 . the goal is to estimate the probability of exiting the set @xmath646 by a given time @xmath13 . as discussed in subsection [ ss:1dimlinearmodel_algorithm ] , the change of measure for the importance sampling scheme is based on the subsolution ( [ eq : controldesignlinear ] ) . in order to apply it to a given pair @xmath647 , one needs to choose the parameters @xmath648 . before presenting simulation data , we comment on these choices . the analysis in subsection [ s : analysislinearproblem ] assumes @xmath519 and @xmath421 , and we will take @xmath649 . as noted before lemmas [ l : region_0_y]-[l : region_y_hatx ] , it is natural to allow quantities such as @xmath593 and @xmath650 , which characterize the region where the solution to the lqr replaces the subsolution based on the quasipotential , to depend on @xmath14 . one would like the width of this region to scale like @xmath651 , with @xmath652 $ ] , which in turn suggests that @xmath440 scale like @xmath653 . however , the exponential negligibility of certain terms that holds when parameters such as @xmath654 and @xmath440 are independent of @xmath14 need not hold when they depend on @xmath14 . for example , the exponential negligibility of the term @xmath655 appearing in lemma [ l : region_0_y ] should be examined . recall that the exponential rate of decay of terms like @xmath656 is bounded by the smallest value of @xmath579 . a lower bound of the form @xmath550\right ] $ ] was obtained in the proof of lemma [ l : region_0_y ] . inserting the given scaling and approximating for small @xmath14 gives @xmath657 , and upon dividing by @xmath421 gives the exponent @xmath658 . hence exponential negligibility requires @xmath659 , with smaller values of @xmath660 giving a faster rate of decay . note however that the analysis assumes @xmath471 and @xmath638 . with regard to the condition @xmath471 , inserting the given scaling for @xmath440 we get the constraint @xmath661 . this is clearly satisfied for small @xmath3 if @xmath370 is of order @xmath210 . one may also take here @xmath370 to be of order @xmath662 and the constraint will be satisfied for small @xmath14 if @xmath663 . we also remark here that for the nonlinear problem , the condition @xmath471 needs to be strengthened to @xmath664 . with regard to the condition @xmath638 , inserting the given scaling for @xmath440 and recalling the definition @xmath665 , we obtain the constraint @xmath666 . this constraint is satisfied for small enough @xmath14 when @xmath667 , and moreover one can allow @xmath609 as @xmath381 . below we present simulation data for various choices of the parameters as indicated in the corresponding tables . in table [ table1a1 ] , estimated probabilities are reported when @xmath668 and @xmath669 , whereas the related relative error per sample estimates are reported in table [ table1b1 ] ( since the relative errors are consistently smaller we now round to the nearest @xmath670 ) . in tables [ table4b1 ] and [ table5b1 ] relative errors per sample estimates are reported for combinations of @xmath671 that depend on @xmath14 . the related probability estimates are almost identical to those in table [ table1a1 ] . note that the degradation in performance as @xmath13 is gets larger observed previously , is no longer present . this agrees with the theoretical performance bound appearing in theorem [ t : mainboundlinear ] . [ c]|c|c|c|c|c|c|c|c|c|@xmath672 & @xmath211 & @xmath212 & @xmath299 & @xmath278 & @xmath213 & @xmath214 & @xmath215 & @xmath350 + @xmath216 & @xmath221 & @xmath673 & @xmath232 & @xmath674 & @xmath222 & @xmath223 & @xmath675 & @xmath676 + @xmath225 & @xmath229 & @xmath230 & @xmath677 & @xmath242 & @xmath231 & @xmath232 & @xmath678 & @xmath679 + @xmath234 & @xmath238 & @xmath239 & @xmath680 & @xmath681 & @xmath240 & @xmath241 & @xmath319 & @xmath682 + @xmath243 & @xmath247 & @xmath248 & @xmath317 & @xmath683 & @xmath249 & @xmath684 & @xmath685 & @xmath686 + @xmath252 & @xmath315 & @xmath257 & @xmath687 & @xmath688 & @xmath689 & @xmath690 & @xmath691 & @xmath692 + @xmath261 & @xmath264 & @xmath265 & @xmath693 & @xmath256 & @xmath266 & @xmath267 & @xmath268 & @xmath694 + @xmath269 & @xmath272 & @xmath273 & @xmath695 & @xmath696 & @xmath697 & @xmath275 & @xmath698 & @xmath699 + [ c]|c|c|c|c|c|c|c|c|c|@xmath207 & @xmath211 & @xmath212 & @xmath299 & @xmath278 & @xmath213 & @xmath214 & @xmath215 & @xmath350 + @xmath216 & @xmath700 & @xmath701 & @xmath702 & @xmath703 & @xmath704 & @xmath704 & @xmath705 & @xmath703 + @xmath225 & @xmath706 & @xmath707 & @xmath708 & @xmath702 & @xmath709 & @xmath705 & @xmath705 & @xmath709 + @xmath234 & @xmath710 & @xmath711 & @xmath712 & @xmath701 & @xmath713 & @xmath709 & @xmath709 & @xmath709 + @xmath243 & @xmath714 & @xmath715 & @xmath716 & @xmath712 & @xmath717 & @xmath713 & @xmath702 & @xmath703 + @xmath252 & @xmath718 & @xmath718 & @xmath715 & @xmath719 & @xmath707 & @xmath211 & @xmath717 & @xmath708 + @xmath261 & @xmath720 & @xmath721 & @xmath722 & @xmath723 & @xmath714 & @xmath710 & @xmath706 & @xmath724 + @xmath269 & @xmath725 & @xmath726 & @xmath727 & @xmath728 & @xmath729 & @xmath721 & @xmath730 & @xmath731 + [ c]|c|c|c|c|c|c|c|c|@xmath672 & @xmath212 & @xmath299 & @xmath278 & @xmath213 & @xmath214 & @xmath215 & @xmath350 + @xmath216 & @xmath717 & @xmath703 & @xmath705 & @xmath704 & @xmath704 & @xmath705 & @xmath702 + @xmath225 & @xmath211 & @xmath713 & @xmath709 & @xmath705 & @xmath705 & @xmath705 & @xmath703 + @xmath234 & @xmath700 & @xmath708 & @xmath702 & @xmath709 & @xmath705 & @xmath705 & @xmath709 + @xmath243 & @xmath707 & @xmath701 & @xmath708 & @xmath703 & @xmath709 & @xmath705 & @xmath709 + @xmath252 & @xmath716 & @xmath712 & @xmath717 & @xmath713 & @xmath703 & @xmath709 & @xmath709 + @xmath261 & @xmath711 & @xmath724 & @xmath712 & @xmath717 & @xmath713 & @xmath702 & @xmath703 + @xmath269 & @xmath710 & @xmath212 & @xmath706 & @xmath700 & @xmath211 & @xmath717 & @xmath713 + [ c]|c|c|c|c|c|c|c|c|@xmath672 & @xmath212 & @xmath299 & @xmath278 & @xmath213 & @xmath214 & @xmath215 & @xmath350 + @xmath216 & @xmath211 & @xmath703 & @xmath705 & @xmath705 & @xmath703 & @xmath713 & @xmath717 + @xmath225 & @xmath700 & @xmath702 & @xmath709 & @xmath705 & @xmath709 & @xmath702 & @xmath708 + @xmath234 & @xmath707 & @xmath713 & @xmath709 & @xmath709 & @xmath709 & @xmath702 & @xmath702 + @xmath243 & @xmath724 & @xmath713 & @xmath703 & @xmath705 & @xmath703 & @xmath703 & @xmath713 + @xmath252 & @xmath711 & @xmath708 & @xmath703 & @xmath709 & @xmath709 & @xmath703 & @xmath713 + @xmath261 & @xmath710 & @xmath717 & @xmath702 & @xmath709 & @xmath709 & @xmath703 & @xmath713 + @xmath269 & @xmath714 & @xmath211 & @xmath713 & @xmath703 & @xmath709 & @xmath703 & @xmath702 + in this section , we extend the construction of section [ s : linearproblem ] to the general non - linear one dimensional setting . we also generalize the notation and allow the stable equilibrium to be an arbitrary point @xmath732 . consider the process model ( [ eq : diffusion1 ] ) and assume that @xmath733 and that @xmath734 , @xmath735 and @xmath736 for all @xmath737 . thus we can write @xmath644 with unique local minimum at @xmath738 and @xmath739 . it is easy to see that the quasipotential with respect to the equilibrium point @xmath732 takes the form@xmath740 the problem of interest is to estimate the exit probability@xmath741 where @xmath732 is the initial ( and rest ) point such that @xmath742 . furthermore , we assume that @xmath743 for all @xmath744 $ ] and @xmath745 for all @xmath746 . set @xmath747 $ ] . the approach to the nonlinear problem is to merge the linearized dynamics around the equilibrium point with the subsolution based on the quasipotential . this subsolution is @xmath748 observe that the second order approximation to this function around the equilibrium point @xmath732 is@xmath749 where @xmath750 and @xmath751 . let @xmath752 be such that @xmath753 . the appropriate translated version of @xmath447 is @xmath754 and @xmath755 . the subsolution for times less than @xmath368 will be the mollification of @xmath756 . note that since @xmath757 agrees with @xmath401 up to second order it is still the case that @xmath481 will be smallest near @xmath738 . letting @xmath758 and@xmath496{cc}\bar{f}_{1}(x ) , & t > t - t^{\ast}\\ u^{\delta}(t , x ) , & t\leq t - t^{\ast}\end{array } \right . , \label{eq : nonlinearchangeofmeasure}\ ] ] the suggested importance sampling control that is used for the simulation is @xmath759 notice that this construction reduces to the construction of the linear case if the potential is indeed quadratic , since then @xmath760 . in subsection [ ss : simulationnonlinearproblem ] we present simulation data for the nonlinear problem , demonstrating the effectiveness of the suggested change of measure . the analysis and the theoretical bound for the performance of this scheme are completely analogous to the linear problem , modulo the additional error coming from the linearization of the dynamics in the neighborhood of the stable equilibrium point . in subsection [ ss : proofnonlinearproblem ] we rigorously analyze the performance of this algorithm . in this subsection , we present simulation data for the nonlinear problem . we take the drift to be @xmath644 , where the potential function is @xmath761 . this potential function has two stable points at @xmath762 and at @xmath763 , and an unstable equilibrium at @xmath116 . we assume that the starting point is at the right equilibrium point @xmath764 and the exit set is the level set of the potential function @xmath765 , with @xmath766 . thus exit occurs from either of the points @xmath767 or @xmath768 . notice that the local quadratic approximation around the equilibrium point is @xmath769 with @xmath770 . moreover , we have chosen , for simplicity , the diffusion coefficient to be constant @xmath771 . @xmath772 independent trajectories were used for the simulations . we first investigate the performance of a change of measure based on the quasipotential subsolution . thus we change the measure via the control @xmath773 . estimated values and the corresponding estimated relative errors per sample for several values of @xmath647 are in tables [ table1anl ] and [ table1bnl ] , respectively . [ c]|c|c|c|c|c|c|c|c|c|c|@xmath672 & @xmath209 & @xmath210 & @xmath211 & @xmath212 & @xmath295 & @xmath299 & @xmath354 & @xmath300 & @xmath213 + @xmath774 & @xmath775 & @xmath776 & @xmath777 & @xmath778 & @xmath779 & @xmath780 & @xmath781 & @xmath782 & @xmath783 + @xmath784 & @xmath785 & @xmath786 & @xmath787 & @xmath788 & @xmath789 & @xmath790 & @xmath791 & @xmath792 & @xmath793 + @xmath252 & @xmath794 & @xmath795 & @xmath796 & @xmath797 & @xmath798 & @xmath799 & @xmath800 & @xmath801 & @xmath802 + @xmath261 & @xmath803 & @xmath804 & @xmath805 & @xmath806 & @xmath807 & @xmath808 & @xmath809 & @xmath810 & @xmath811 + @xmath269 & @xmath812 & @xmath813 & @xmath814 & @xmath815 & @xmath816 & @xmath817 & @xmath818 & @xmath819 & @xmath820 + @xmath821 & @xmath822 & @xmath823 & @xmath824 & @xmath825 & @xmath826 & @xmath827 & @xmath828 & @xmath829 & @xmath830 + [ c]|c|c|c|c|c|c|c|c|c|c|@xmath672 & @xmath209 & @xmath210 & @xmath211 & @xmath212 & @xmath295 & @xmath299 & @xmath354 & @xmath300 & @xmath213 + @xmath774 & @xmath279 & @xmath279 & @xmath295 & @xmath831 & @xmath832 & @xmath833 & @xmath834 & @xmath835 & @xmath836 + @xmath784 & @xmath279 & @xmath279 & @xmath295 & @xmath215 & @xmath837 & @xmath838 & @xmath839 & @xmath840 & @xmath841 + @xmath252 & @xmath279 & @xmath279 & @xmath295 & @xmath842 & @xmath843 & @xmath844 & @xmath845 & @xmath846 & @xmath847 + @xmath261 & @xmath279 & @xmath279 & @xmath295 & @xmath848 & @xmath423 & @xmath849 & @xmath850 & @xmath851 & @xmath293 + @xmath269 & @xmath279 & @xmath279 & @xmath295 & @xmath286 & @xmath852 & @xmath853 & @xmath854 & @xmath855 & @xmath856 + @xmath821 & @xmath279 & @xmath279 & @xmath287 & @xmath214 & @xmath857 & @xmath858 & @xmath859 & @xmath860 & @xmath861 + as we see from table [ table1bnl ] , even though the quasipotential subsolution performs relatively well for small values of @xmath13 , there is a clear degradation of performance as @xmath13 gets larger . it is also interesting to note that the degradation is uniform across all values of @xmath14 for the same value of @xmath13 . this behavior parallels what was observed for the linear problem . as was mentioned there , the large per sample relative errors for @xmath862 should not be taken as being accurate , but just indicative of poor performance . next we investigate how the suggested change of measure performs . to apply the control , we choose values for the parameters @xmath863 according to the discussion in subsection [ s : simulationresultslinearproblem ] . however , for reasons that will become clearer in the proof of lemma [ l : region_y_hatx_nonlinear ] , we need to strengthen the condition @xmath471 to @xmath664 , say @xmath864 . when we link the other parameters to @xmath14 the @xmath865 as @xmath381 , and because of this we do not explicitly take into account the error from the approximation around the neighborhood of the rest point of the true dynamics by its linearization when selecting the parameters for the implementation of the scheme . estimated values and corresponding estimated relative errors of the exit probabilities of interest for different pairs @xmath647 and different combinations values for @xmath593 are in tables [ table2anl ] through [ table5bnl ] . estimated relative errors for @xmath866 with @xmath867 and @xmath868 are reported in table [ table2anl ] , whereas the related relative errors are reported in table [ table2bnl ] . in tables [ table3bnl ] and [ table4bnl ] , we report only estimated relative errors for the same value of @xmath660 but for @xmath869 and @xmath669 respectively . the related probability estimates are almost identical to those of table [ table2anl ] , so they are not repeated . note that for table [ table3bnl ] , @xmath870 when @xmath871 and for @xmath872 . similarly , for table [ table4bnl ] , @xmath870 when @xmath871 and for @xmath873 when @xmath874 . for such values , the quasipotential subsolution is being used everywhere ( see ( [ eq : nonlinearchangeofmeasure ] ) ) and the numerical results for these values agree with those from table [ table1bnl ] . in order to illustrate the effect when the linear approximation is used over a relatively large region , the data in table [ table5bnl ] are estimated relative errors when @xmath440 is considerably smaller than before , and thus @xmath593 is considerably larger . in particular , we have taken @xmath875 and @xmath876 . comparing tables [ table4bnl ] and [ table5bnl ] , we notice that if the error from the linearization is not confined to a small enough region then the algorithm degrades in @xmath14 though it appears stable in @xmath13 . this is consistent with the theoretical results , which imply a uniformity in @xmath13 but only logarithmic optimality in @xmath14 . that said , one would like to minimize errors associated with linearization as far as possible . as noted in subsection [ s : simulationresultslinearproblem ] , one should choose the scaling parameter @xmath667 . however , with the nonlinear problem minimizing the region over which the approximation is used calls for larger @xmath660 , and so one want it close to but not equal to @xmath877 . for the problems considered here , @xmath878 worked well . [ c]|c|c|c|c|c|c|c|c|c|c|c|@xmath672 & @xmath209 & @xmath210 & @xmath211 & @xmath212 & @xmath295 & @xmath299 & @xmath354 & @xmath300 & @xmath213 & @xmath355 + @xmath774 & @xmath879 & @xmath880 & @xmath881 & @xmath882 & @xmath883 & @xmath884 & @xmath885 & @xmath886 & @xmath887 & @xmath888 + @xmath784 & @xmath889 & @xmath890 & @xmath891 & @xmath892 & @xmath893 & @xmath894 & @xmath895 & @xmath896 & @xmath897 & @xmath898 + @xmath252 & @xmath899 & @xmath900 & @xmath901 & @xmath902 & @xmath903 & @xmath904 & @xmath905 & @xmath906 & @xmath907 & @xmath908 + @xmath909 & @xmath910 & @xmath911 & @xmath912 & @xmath913 & @xmath914 & @xmath915 & @xmath916 & @xmath917 & @xmath918 & @xmath919 + @xmath261 & @xmath803 & @xmath804 & @xmath920 & @xmath921 & @xmath922 & @xmath923 & @xmath924 & @xmath925 & @xmath926 & @xmath927 + @xmath269 & @xmath928 & @xmath813 & @xmath929 & @xmath930 & @xmath931 & @xmath932 & @xmath933 & @xmath934 & @xmath935 & @xmath936 + @xmath937 & @xmath938 & @xmath939 & @xmath940 & @xmath941 & @xmath942 & @xmath943 & @xmath944 & @xmath945 & @xmath946 & @xmath947 + @xmath821 & @xmath948 & @xmath949 & @xmath824 & @xmath950 & @xmath951 & @xmath952 & @xmath953 & @xmath954 & @xmath955 & @xmath956 + [ c]|c|c|c|c|c|c|c|c|c|c|c|@xmath672 & @xmath209 & @xmath210 & @xmath211 & @xmath212 & @xmath295 & @xmath299 & @xmath354 & @xmath300 & @xmath213 & @xmath355 + @xmath774 & @xmath957 & @xmath719 & @xmath707 & @xmath717 & @xmath713 & @xmath958 & @xmath702 & @xmath713 & @xmath717 & @xmath700 + @xmath784 & @xmath722 & @xmath959 & @xmath716 & @xmath211 & @xmath708 & @xmath960 & @xmath713 & @xmath713 & @xmath708 & @xmath211 + @xmath252 & @xmath961 & @xmath962 & @xmath959 & @xmath724 & @xmath211 & @xmath963 & @xmath708 & @xmath708 & @xmath708 & @xmath708 + @xmath909 & @xmath964 & @xmath957 & @xmath965 & @xmath706 & @xmath707 & @xmath966 & @xmath717 & @xmath708 & @xmath708 & @xmath708 + @xmath261 & @xmath967 & @xmath968 & @xmath969 & @xmath719 & @xmath707 & @xmath970 & @xmath211 & @xmath717 & @xmath708 & @xmath708 + @xmath269 & @xmath720 & @xmath971 & @xmath972 & @xmath969 & @xmath212 & @xmath973 & @xmath716 & @xmath700 & @xmath712 & @xmath717 + @xmath937 & @xmath969 & @xmath974 & @xmath971 & @xmath968 & @xmath969 & @xmath965 & @xmath959 & @xmath711 & @xmath716 & @xmath707 + @xmath821 & @xmath965 & @xmath975 & @xmath976 & @xmath977 & @xmath731 & @xmath978 & @xmath979 & @xmath980 & @xmath715 & @xmath710 + [ c]|c|c|c|c|c|c|c|c|c|c|c|@xmath672 & @xmath209 & @xmath210 & @xmath211 & @xmath212 & @xmath295 & @xmath299 & @xmath354 & @xmath300 & @xmath213 & @xmath355 + @xmath774 & @xmath715 & @xmath212 & @xmath716 & @xmath211 & @xmath708 & @xmath958 & @xmath702 & @xmath702 & @xmath702 & @xmath713 + @xmath784 & @xmath959 & @xmath714 & @xmath710 & @xmath707 & @xmath701 & @xmath981 & @xmath713 & @xmath713 & @xmath713 & @xmath713 + @xmath252 & @xmath706 & @xmath957 & @xmath982 & @xmath710 & @xmath724 & @xmath983 & @xmath712 & @xmath701 & @xmath717 & @xmath708 + @xmath909 & @xmath724 & @xmath968 & @xmath979 & @xmath715 & @xmath706 & @xmath984 & @xmath707 & @xmath211 & @xmath701 & @xmath717 + @xmath261 & @xmath724 & @xmath972 & @xmath985 & @xmath980 & @xmath710 & @xmath986 & @xmath724 & @xmath700 & @xmath211 & @xmath701 + @xmath269 & @xmath707 & @xmath967 & @xmath987 & @xmath722 & @xmath969 & @xmath988 & @xmath715 & @xmath719 & @xmath716 & @xmath724 + @xmath937 & @xmath700 & @xmath989 & @xmath990 & @xmath987 & @xmath968 & @xmath957 & @xmath723 & @xmath714 & @xmath715 & @xmath719 + @xmath821 & @xmath716 & @xmath991 & @xmath992 & @xmath993 & @xmath977 & @xmath994 & @xmath972 & @xmath985 & @xmath718 & @xmath982 + [ c]|c|c|c|c|c|c|c|c|c|c|c|@xmath672 & @xmath209 & @xmath210 & @xmath211 & @xmath212 & @xmath295 & @xmath299 & @xmath354 & @xmath300 & @xmath213 & @xmath355 + @xmath774 & @xmath211 & @xmath710 & @xmath959 & @xmath711 & @xmath707 & @xmath970 & @xmath211 & @xmath717 & @xmath708 & @xmath702 + @xmath784 & @xmath211 & @xmath719 & @xmath980 & @xmath715 & @xmath711 & @xmath995 & @xmath724 & @xmath712 & @xmath701 & @xmath717 + @xmath252 & @xmath712 & @xmath711 & @xmath985 & @xmath996 & @xmath969 & @xmath988 & @xmath715 & @xmath710 & @xmath706 & @xmath724 + @xmath909 & @xmath712 & @xmath706 & @xmath968 & @xmath722 & @xmath957 & @xmath997 & @xmath982 & @xmath715 & @xmath212 & @xmath706 + @xmath261 & @xmath707 & @xmath706 & @xmath722 & @xmath998 & @xmath999 & @xmath1000 & @xmath723 & @xmath980 & @xmath715 & @xmath710 + @xmath269 & @xmath707 & @xmath706 & @xmath731 & @xmath989 & @xmath964 & @xmath1001 & @xmath972 & @xmath985 & @xmath718 & @xmath969 + @xmath937 & @xmath724 & @xmath706 & @xmath1002 & @xmath1003 & @xmath1004 & @xmath1005 & @xmath967 & @xmath1002 & @xmath1006 & @xmath985 + @xmath821 & @xmath716 & @xmath706 & @xmath979 & @xmath1007 & @xmath992 & @xmath1008 & @xmath993 & @xmath1009 & @xmath977 & @xmath720 + [ c]|c|c|c|c|c|c|c|c|c|c|c|@xmath672 & @xmath209 & @xmath210 & @xmath211 & @xmath212 & @xmath295 & @xmath299 & @xmath354 & @xmath300 & @xmath213 & @xmath355 + @xmath774 & @xmath211 & @xmath299 & @xmath299 & @xmath295 & @xmath287 & @xmath1010 & @xmath287 & @xmath279 & @xmath279 & @xmath279 + @xmath784 & @xmath211 & @xmath278 & @xmath278 & @xmath354 & @xmath299 & @xmath1011 & @xmath295 & @xmath295 & @xmath287 & @xmath287 + @xmath252 & @xmath712 & @xmath214 & @xmath1012 & @xmath848 & @xmath214 & @xmath1013 & @xmath357 & @xmath1014 & @xmath288 & @xmath300 + @xmath909 & @xmath712 & @xmath831 & @xmath1015 & @xmath1016 & @xmath350 & @xmath1017 & @xmath842 & @xmath848 & @xmath347 & @xmath214 + @xmath261 & @xmath707 & @xmath1018 & @xmath1019 & @xmath280 & @xmath292 & @xmath1020 & @xmath352 & @xmath1018 & @xmath1016 & @xmath1021 + @xmath269 & @xmath707 & @xmath352 & @xmath1022 & @xmath417 & @xmath1023 & @xmath1024 & @xmath1025 & @xmath1026 & @xmath855 & @xmath1027 + @xmath937 & @xmath724 & @xmath299 & @xmath1028 & @xmath1029 & @xmath1030 & @xmath1031 & @xmath1032 & @xmath1033 & @xmath1034 & @xmath1035 + @xmath821 & @xmath716 & @xmath287 & @xmath1036 & @xmath1037 & @xmath1038 & @xmath1020 & @xmath1039 & @xmath1040 & @xmath1041 & @xmath1042 + in this subsection , we present the theoretical analysis of the simulation scheme for general one - dimensional non - linear dynamics and provide rigorous bounds on performance . as for the linear case , the analysis is valid for @xmath3 without degradation as @xmath189 . the analysis and the theoretical bound for the performance of this scheme are completely analogous to the linear problem , modulo the additional error coming from the linearization of the dynamics in the neighborhood of the stable equilibrium point . to distinguish between the linear and the nonlinear problem we need to introduce some notation . for a function @xmath1043\times \mathbb{r})$ ] we define the operator @xmath1044(t , x)=w_{t}(t , x)+\bar{\mathbb{h}}(x , dw(t , x))+\frac{\varepsilon}{2}\sigma^{2}(x)d^{2}w(t , x),\quad \bar{\mathbb{h}}(x , p)=b(x)p-\frac{1}{2}\left\vert \sigma(x)p\right\vert ^{2}.\ ] ] in analogy to ( [ eq : ge_2 ] ) , for smooth functions @xmath503 , we define @xmath1045(t , x)=\bar{\mathcal{g}}^{\varepsilon } [ w](t , x)-\frac{1}{2}\left\vert \sigma(x)\left ( dw(t , x)-du(t , x)\right ) \right\vert ^{2}.\label{eq : ge_2nl}\ ] ] moreover , setting @xmath1046 and @xmath751 we recall that@xmath147(t , x)=w_{t}(t , x)+\mathbb{h}(x , dw(t , x))+\frac { \varepsilon}{2}\bar{\sigma}^{2}d^{2}w(t , x),\quad\mathbb{h}(x , p)=-cxp-\frac { 1}{2}\left\vert \bar{\sigma}p\right\vert ^{2}.\ ] ] the operators with bars correspond to the nonlinear problem , whereas the operators without bars give the corresponding first order approximations . we define an operator measuring the error from the approximation by @xmath1047(t , x)=\bar{\mathcal{g}}^{\varepsilon}[w](t , x)-\mathcal{g}^{\varepsilon}[w](t , x ) \label{eq : errorofoperator}\ ] ] moreover , since @xmath1048 and @xmath1049 are @xmath1050 , we can write for any @xmath737 @xmath1051 where @xmath1052 and @xmath1053 are locally bounded . by assumption we have that @xmath734 and @xmath1054 . as with the linear problem , the subsolution used for the analysis is based on the @xmath1055exponential mollification ( [ eq : second_moll ] ) reduced by the multiplicative factor @xmath177 . we recall that this differs from the subsolution used for the design , which has @xmath1056 . next we proceed with the mathematical analysis of the scheme . the analysis is parallel to what was done for the linear problem , modulo adjustments due to the linearization of the dynamics in the neighborhood of the stable point , and therefore we mainly focus on the differences . in order to simplify the notation we assume without loss of generality ( as it was done in the linear problem ) that the stable equilibrium is @xmath1057 . we write @xmath1058 , and for notational convenience assume that @xmath1059 . as in the linear problem , @xmath1060 and @xmath1061 . the following lemma bounds the error from the approximation . [ l : boundforapproximationofoperatorg ] consider @xmath1062\times\lbrack0,z]$ ] . then , for @xmath1063 @xmath1064(t , x)\right\vert & \leq2a^{m}(t)(z+\hat{x}e^{-c(t - t)})\sup_{x\in\lbrack0,z]}|b(x)+cx|\\ & \quad+\left [ 2\left ( a^{m}(t)(z+\hat{x}e^{-c(t - t)})\right ) ^{2}+\varepsilon a^{m}(t)\right ] \sup_{x\in\lbrack0,z]}|\sigma^{2}(x)-\bar{\sigma}^{2}|\\ & \leq c_{0}\left\ { a^{m}(t)(z+\hat{x}e^{-c(t - t)})z^{2}+\left ( a^{m}(t)(z+\hat{x}e^{-c(t - t)})\right ) ^{2}z+a^{m}(t)\varepsilon z\right\ } , \end{aligned}\ ] ] where @xmath1065}\left [ \frac{\left\vert r_{1}(x)\right\vert } { |x|^{2}}+\frac{\left\vert r_{2}(x)\right\vert } { |x|}\left\vert 2\bar{\sigma } + r_{2}(x)\right\vert \right ] .\ ] ] in addition @xmath1066}|\bar{f}_{1}(x)-f_{1}(x)|\leq c_{1}z^{3},\ ] ] where @xmath197 is a fixed constant . since by assumption @xmath1067 and @xmath1054 , for @xmath1068\times\mathbb{r}$ ] @xmath1069(t , x ) & = \bar{\mathbb{h}}(x , dw(t , x))-\mathbb{h}(x , dw(t , x))+\frac{\varepsilon}{2}(\sigma^{2}(x)-\bar{\sigma}^{2})d^{2}w(t , x)\\ & = r_{1}(x)dw(t , x)-\frac{1}{2}\left ( \left\vert \sigma(x)dw(t , x)\right\vert ^{2}-\left\vert \bar{\sigma}dw(t , x)\right\vert ^{2}\right ) \\ & \quad+\frac{\varepsilon}{2}(\sigma^{2}(x)-\bar{\sigma}^{2})d^{2}w(t , x)\\ & = r_{1}(x)dw(t , x)-\frac{1}{2}r_{2}(x)\left ( 2\bar{\sigma}+r_{2}(x)\right ) \left\vert dw(t , x)\right\vert ^{2}\\ & \quad+\frac{\varepsilon}{2}r_{2}(x)\left ( 2\bar{\sigma}+r_{2}(x)\right ) d^{2}w(t , x).\end{aligned}\ ] ] for all @xmath1070 $ ] , we have @xmath1071 for a constant @xmath121 that depends on @xmath306 . the bound follows by setting @xmath1072 and using @xmath1073 , and the proof of the first part of the lemma is concluded . the proof of the second part goes as follows . observe that@xmath1074 the last display and the bounds in ( [ eq : localizationbounds ] ) imply that @xmath1066}\left\vert \bar{f}_{1}(x)-f_{1}(x)\right\vert \leq c_{1}z^{3}\ ] ] which concludes the proof of the lemma . for notational convenience we identify the quantity appearing in the upper bound for @xmath1075(t , x)\right\vert $ ] as given in lemma [ l : boundforapproximationofoperatorg]:@xmath1076 the following lemma shows that the error term induced by the local approximation of the dynamics in the neighborhood of the stable equilibrium point does not degrade as @xmath13 gets large . [ l : integralofcorrectionterm ] we have that @xmath1077 where , letting @xmath1078 , @xmath1079 \nonumber\\ j_{2}(t^{\ast},t , m ) & = \frac{1}{\bar{\sigma}\sqrt{k}}\left ( 1-\frac{c}{\bar{\sigma}^{2}}\right ) \left [ \log\frac{1+\frac{\bar{\sigma}}{\sqrt{k}}e^{-ct^{\ast}}}{1-\frac{\bar{\sigma}}{\sqrt{k}}e^{-ct^{\ast}}}-\log \frac{1+\frac{\bar{\sigma}}{\sqrt{k}}e^{-ct}}{1-\frac{\bar{\sigma}}{\sqrt{k}}e^{-ct}}\right ] \nonumber\\ & + \frac{c}{2\bar{\sigma}^{4}}\left [ \frac{2\bar{\sigma}^{2}e^{-ct^{\ast}}}{k-\bar{\sigma}^{2}e^{-2ct^{\ast}}}-\frac{2\bar{\sigma}^{2}e^{-ct}}{k-\bar{\sigma}^{2}e^{-2ct}}\right ] \nonumber\\ j_{3}(t^{\ast},t , m ) & = \frac{c}{2\bar{\sigma}^{4}}\left [ \frac{\bar{\sigma } ^{2}}{k-\bar{\sigma}^{2}e^{-2ct^{\ast}}}-\frac{\bar{\sigma}^{2}}{k-\bar{\sigma}^{2}e^{-2ct}}\right ] \nonumber\\ j_{4}(t^{\ast},t , m ) & = \frac{1}{2\bar{\sigma}^{2}}\log\frac{k-\bar{\sigma } ^{2}e^{-2ct}}{k-\bar{\sigma}^{2}e^{-2ct^{\ast}}}.\nonumber\end{aligned}\ ] ] in particular , @xmath1080 . the proof of this lemma follows by straightforward integration of @xmath1081 . moreover , we note that @xmath1082 and that the definition of @xmath1060 implies @xmath1083 [ r : nonlinearremark ] _ the following three lemmas are analogous to lemmas [ l : region_0_y]-[l : region_y_hatx ] from the linear case . the important difference between the nonlinear and the linear case is that the statements involve approximation errors and the statements hold if one confines the linearized dynamics to a small neighborhood of the equilibrium point as dictated by the sizes of @xmath372 and @xmath440 , or equivalently by @xmath372 and @xmath593 . due to the natural scaling of @xmath440 in terms of @xmath14 as indicated in subsection [ s : simulationresultslinearproblem ] , as @xmath14 gets smaller , @xmath593 will get smaller and be confined to a sufficiently small region that the statements of the lemmas are valid . however , the lemmas below are stated for @xmath593 sufficiently small , without referencing to the natural scaling used in the simulation algorithm . _ [ l : region_0_y_nonlinear ] assume that @xmath540\times\lbrack0,z]$ ] , @xmath541 and @xmath542 . then , for sufficiently small @xmath593 we have up to an exponentially negligible term @xmath1084(t , x)\geq-(1-\eta ) c_{0 } r(\varepsilon,\hat{x},m , t).\ ] ] as in the proof of lemma [ l : region_0_y ] we need to show that in this region @xmath1085 is bounded by below away from zero . we have @xmath1086 at @xmath101 , the value is minimized at @xmath546 and , as in lemma [ l : region_0_y ] , we obtain the lower bound @xmath1087>0 $ ] . for @xmath1088 we use the decomposition @xmath1089 + \left [ \bar{f}_{1}(x)-f_{1}(x)\right ] . \label{eq : nonlinearcasei}\ ] ] using the inequality @xmath1090 ^{4}\leq c / m\bar{\sigma}^{2}$ ] , the definition of @xmath593 and equation ( [ eq : f1-f2 ] ) , we obtain that for all @xmath540\times\lbrack0,z]$ ] @xmath1091 -\frac{c}{k-\frac{c}{m}}\left ( ze^{-ct^{\ast}}-\hat { x}\right ) ^{2}\\ & \geq\frac{c}{\bar{\sigma}^{2}}\left [ \frac{\hat{x}^{2}}{m}\frac{c}{\bar{\sigma}^{2}+c / m}-z^{2}\right ] -z^{2}\frac{ce^{-2ct^{\ast}}}{\bar { \sigma}^{2}+c / m}\\ & \geq\frac{c}{\bar{\sigma}^{2}}z^{2}\left [ \frac{3\bar{\sigma}^{2}-c / m}{\bar{\sigma}^{2}+c / m}-\frac{c / m}{\bar{\sigma}^{2}+c / m}\right ] \\ & \geq\frac{c}{\bar{\sigma}^{2}}z^{2}\left [ \frac{2\bar{\sigma}^{2}+3c / m}{\bar{\sigma}^{2}+c / m}-\frac{c / m}{\bar{\sigma}^{2}+c / m}\right ] \\ & = 2\frac{c}{\bar{\sigma}^{2}}z^{2}.\end{aligned}\ ] ] hence , this term is of order @xmath1092 . moreover , by lemma [ l : boundforapproximationofoperatorg ] , we also have that @xmath1066}|\bar{f}_{1}(x)-f_{1}(x)|\leq c_{1}z^{3}.\ ] ] thus , for sufficiently small @xmath593 the first term on the right hand side of ( [ eq : nonlinearcasei ] ) dominates the second term . hence the difference @xmath1085 is bounded from below away from zero if @xmath593 is sufficiently small , and the term involving the weight @xmath554 is exponentially negligible . we next use that @xmath527 , @xmath555 and @xmath556 , lemma [ l : boundforapproximationofoperatorg ] , and also ( [ eq : errorofoperator ] ) and ( [ eq : defofr ] ) . these imply that up to an exponentially negligible term , @xmath1093(t , x ) & \geq(1-\eta)\bar{\mathcal{g}}^{\varepsilon}[f_{2}^{m}](t , x)\\ & = ( 1-\eta)\gamma_{2}^{m}(t)+(1-\eta)r^{\varepsilon}[f_{2}^{m}](t , x)\\ & \geq-(1-\mathbb{\eta})c_{0}r(\varepsilon,\hat{x},m , t),\end{aligned}\ ] ] which concludes the proof of the lemma . [ l : region_hatx_a_nonlinear ] assume that @xmath1094\times\lbrack h , a]$ ] and assume @xmath541 . define @xmath1095\cup[h , a]}\frac { -\varepsilon\sigma^{2}(x)d\left ( b(x)\sigma^{-2}(x)\right ) } { -\varepsilon \sigma^{2}(x)d\left ( b(x)\sigma^{-2}(x)\right ) + b^{2}(x)\sigma^{-2}(x)}\ ] ] let @xmath3 be sufficiently small such that @xmath1096 and consider @xmath1097 . then , for sufficiently small @xmath593 and up to exponentially negligible terms , @xmath1098(t , x)\geq0.\ ] ] as with the linear problem we need to argue that there is a constant @xmath573 such that @xmath1099 for all @xmath1100\times\lbrack h , a]$ ] , which will then imply that the terms involving @xmath1101 are exponentially negligible . we have the decomposition @xmath1102 + \left [ f_{1}(h)-\bar{f}_{1}(h)\right ] + \left [ \bar{f}_{1}(h)-\bar{f}_{1}(x)\right]\ ] ] focusing on @xmath1103 , lemma [ l : region_hatx_a ] guarantees that @xmath1104 > 0 $ ] uniformly in @xmath474 $ ] . moreover , a straightforward computation shows that for @xmath650 sufficiently small , the lower bound of @xmath1105 $ ] is of the order @xmath1106 or equivalently @xmath1092 ( since @xmath1061 ) . by lemma [ l : boundforapproximationofoperatorg ] , the second term @xmath1107 $ ] is of order @xmath1108 . thus , for sufficiently small @xmath650 , or equivalently for sufficiently small @xmath593 , the term @xmath1109 is bounded by below away from zero . this statement , convexity of @xmath1110 and the fact that the third term of the last display is nonnegative for all @xmath535 $ ] , imply that there is a @xmath573 such that @xmath1111 for all @xmath540\times\lbrack h , a]$ ] . hence , terms involving @xmath575 are exponentially negligible . since @xmath527 , up to an exponentially negligible term@xmath1093(t , x ) & \geq(1-\eta)\rho_{1}(t , x)\bar{\mathcal{g}}^{\varepsilon}[\bar{f}_{1}](x)+\frac{1}{2}\left ( \eta-2\eta^{2}\right ) \left\vert \sigma(x)\rho _ { 1}(t , x)d\bar{f}_{1}(x)\right\vert ^{2}\\ & = \varepsilon(1-\eta)\sigma^{2}(x)d\left ( b(x)\sigma^{-2}(x)\right ) + 2\eta(1 - 2\eta)b^{2}(x)\sigma^{-2}(x).\end{aligned}\ ] ] thus for @xmath14 small enough that @xmath1112 we have , up to an exponentially negligible term@xmath1098(t , x)\geq0,\ ] ] concluding the proof of the lemma . [ l : region_y_hatx_nonlinear ] assume that @xmath1094\times\lbrack z , h]$ ] and that @xmath864 . set @xmath1113 and @xmath1114}\sigma^{2}(x)$ ] . then , for sufficiently small @xmath593 we have up to an exponentially negligible term @xmath1093(t , x ) & \geq \frac{\sigma_{\ast}^{2}}{2}\left [ \frac{1}{\bar{\sigma}^{2}}\left ( \frac{c^{2}\eta}{2\bar{\sigma}^{2}}\left ( z-\hat{x}e^{c(t - t)}\right ) ^{2}-2\varepsilon c\right ) + \gamma(t , z , h,\hat{x},\varepsilon,\eta , t)\right ] \wedge0\\ & \qquad - c_{0}(1-\eta)r(\varepsilon,\hat{x},m , t).\end{aligned}\ ] ] where @xmath1115}\left [ \frac{c\eta}{2\bar{\sigma}^{2}}\left ( d\bar{f}_{1}(x)-df_{1}(x)\right ) \left ( z-\hat{x}e^{c(t - t)}\right ) \right . \label{eq : defofgamma}\\ & \qquad\qquad\left . + \frac{1}{8}\eta\left ( d\bar{f}_{1}(x)-df_{1}(x)\right ) ^{2}+2\varepsilon\left ( \frac{c}{\bar{\sigma}^{2}}+d\left ( \frac{b(x)}{\sigma^{2}(x)}\right ) \right ) \right ] \nonumber\end{aligned}\ ] ] and @xmath736 for all @xmath737 . as with the analysis of the linear problem , this region presents difficulties in its analysis , since the term @xmath1085 can be either positive or negative . this means that both @xmath581 and @xmath554 may be important . as with the linear problem , we ignore terms related to @xmath1116 due to exponential negligibility . up to an exponentially negligible term@xmath1093(t , x)\geq & ( 1-\eta)\frac{1}{4}\beta_{0}(t , x)+(1-\eta)\rho_{2}^{m,+}(t , x)\bar{\mathcal{g}}^{\varepsilon}[f_{2,+}^{m}](t , x)+(1-\eta)\rho_{1}(t , x)\bar{\mathcal{g}}^{\varepsilon}[\bar{f}_{1}](x)\\ & + \frac{1}{2}\left ( \eta-2\eta^{2}\right ) \sigma^{2}(x)\left\vert \rho _ { 2}^{m,+}(t , x)df_{2,+}^{m}(t , x)+\rho_{1}(t , x)d\bar{f}_{1}(x)\right\vert ^{2}.\end{aligned}\ ] ] we distinguish two cases depending on whether @xmath584 or @xmath1117 . * case i : @xmath584 . * we know that @xmath527 and due to the positivity of @xmath1118 we have by lemma [ l : boundforapproximationofoperatorg ] @xmath1119(t , x)=\mathcal{g}^{\varepsilon } [ f_{2,+}^{m}](t , x)+r^{\varepsilon}[f_{2,+}^{m}](t , x)\geq - c_{0}r(\varepsilon , \hat{x},m , t).\ ] ] next we need to argue that for @xmath540\times\lbrack z , h]$ ] @xmath1120 lemma [ l : region_y_hatx ] implies that @xmath1121 + \left [ df_{1}(x)-df_{2,+}^{m}(t , x)\right ] \nonumber\\ & \hspace{1cm}\leq\left [ d\bar{f}_{1}(x)-df_{1}(x)\right ] + \left [ \hat { x}\frac{2c}{\bar{\sigma}^{2}}\frac{k}{k-\bar{\sigma}^{2}e^{2c(t - t)}}\left ( \left ( \frac{2c}{m\bar{\sigma}^{2}}\right ) ^{2}-\frac{1}{2}\left ( \frac { c}{m\bar{\sigma}^{2}}\right ) ^{1/2}\right ) \right ] . \label{eq : diff1}\ ] ] it follows easily from the proof of lemma [ l : boundforapproximationofoperatorg ] that @xmath1122}\left\vert d\bar{f}_{1}(x)-df_{1}(x)\right\vert \leq c_{1}h^{2}=100c_{1}z^{2}\ ] ] for some constant @xmath197 which is independent of @xmath593 . for the second term on the right hand side of ( [ eq : diff1 ] ) we have @xmath1123 where we used the definition @xmath1124 and the assumption @xmath1125 . thus , this term is strictly negative and of order @xmath593 for @xmath593 sufficiently small . thus for @xmath593 sufficiently small , the first term on the right hand side of ( [ eq : diff1 ] ) is dominated by the second term which is negative . together , with the non - positivity of @xmath1126 , we get that @xmath1127 for @xmath593 sufficiently small . since @xmath557 , we obtain @xmath1093(t , x ) & \geq(1-\eta)\rho_{1}(t , x)\bar{\mathcal{g}}^{\varepsilon}[\bar{f}_{1}](x)+\frac{1}{16}\eta\sigma^{2}(x)\left\vert d\bar{f}_{1}(x)-df_{2,+}^{m}(t , x)\right\vert ^{2}\\ & \quad-(1-\eta)c_{0}r(\varepsilon,\hat{x},m , t)\\ & \geq\frac{\sigma^{2}(x)}{2}\left [ \frac{1}{8}\eta\left ( d\bar{f}_{1}(x)-df_{2,+}^{m}(t , x)\right ) ^{2}+2\varepsilon d\left ( \frac { b(x)}{\sigma^{2}(x)}\right ) \right ] \\ & \quad-(1-\eta)c_{0}r(\varepsilon,\hat{x},m , t).\end{aligned}\ ] ] * case ii : @xmath585 . * similarly to the linear problem we have @xmath1098(t , x)\geq ( 1-\eta)\rho_{1}(t , x)\sigma^{2}(x)\left [ \frac{1}{8}|d\bar{f}_{1}(x)-df_{2,+}^{m}(t , x)|^{2}+\varepsilon d\left ( \frac{b(x)}{\sigma^{2}(x)}\right ) \right]\ ] ] writing @xmath1128 and using , as in the linear problem , the estimate @xmath1129 we obtain as in the proof of lemma [ l : region_y_hatx ] that @xmath1130(t , x)\\ & \qquad\geq\left\ { \frac{\sigma^{2}(x)}{2}\left [ \frac{1}{8}\eta\left ( d\bar{f}_{1}(x)-df_{1}(x)+\frac{2c}{\bar{\sigma}^{2}}\left ( z-\hat { x}e^{c(t - t)}\right ) \right ) ^{2}+2\varepsilon d\left ( \frac{b(x)}{\sigma^{2}(x)}\right ) \right ] \right\ } \wedge0\\ & \qquad\qquad - c_{0}(1-\eta)r(\varepsilon,\hat{x},m , t)\\ & \qquad\geq\frac{\sigma_{\ast}^{2}}{2}\left [ \frac{1}{\bar{\sigma}^{2}}\left ( \frac{c^{2}\eta}{2\bar{\sigma}^{2}}\left ( z-\hat{x}e^{c(t - t)}\right ) ^{2}-2\varepsilon c\right ) + \gamma(t , z , h,\hat{x},\varepsilon , \eta , t)\right ] \wedge0\\ & \qquad\qquad - c_{0}(1-\eta)r(\varepsilon,\hat{x},m , t).\end{aligned}\ ] ] where @xmath1131 was defined in ( [ eq : defofgamma ] ) . this concludes the proof of the lemma . the performance bound is then summarized in the following theorem . the proof of theorem [ t : mainboundnonlinear ] is the same as the proof of theorem [ t : mainboundlinear ] for the linear case . so , it will not be repeated here . [ t : mainboundnonlinear ] assume @xmath421 , @xmath1132 , @xmath638 and that @xmath864 , where @xmath1133 is as in lemma [ l : region_hatx_a_nonlinear ] . set @xmath1134}\sigma^{2}(x)$ ] . let @xmath130 be the control based on the function @xmath603 defined via ( [ eq : nonlinearchangeofmeasure ] ) , i.e. , @xmath1135 . then , up to an exponentially negligible term , for @xmath1136 such that @xmath1137 and for @xmath593 sufficiently small , we have@xmath1138 1_{\left\ { t\geq t^{\ast}\right\ } } \\ & \hspace{0.1cm}+2i_{2}(\varepsilon , t)1_{\left\ { t < t^{\ast}\right\ } } , \end{aligned}\ ] ] where @xmath1139 ds - t^{\ast}c^{\ast}\varepsilon,\end{aligned}\ ] ] where @xmath636 are the times in @xmath637 $ ] where the integrand is negative , @xmath1140 is as in ( [ eq : defofgamma ] ) , @xmath1141 and @xmath1142}\sigma^{2}(x)\left\vert d(b(x)/\sigma^{2}(x))\right\vert > 0.\ ] ] the bound of theorem [ t : mainboundnonlinear ] takes a complicated form , but as in the linear case , the performance does not degrade as @xmath1143 . this was also reflected by the simulation data in subsection [ ss : simulationnonlinearproblem ] . let us now justify this claim . notice that , by lemma [ l : integralofcorrectionterm ] the term @xmath1144 is finite , uniformly in @xmath13 . next we need to argue , similarly to the linear problem , that when @xmath639 is sufficiently large and @xmath593 is sufficiently small , the integrand of the second term in the definition of @xmath1145 is in fact positive . let us denote the integrand of the second term by @xmath1146 where @xmath1147 . the term @xmath1148 is composed by three terms and we shall argue below they are dominated ( even when they are negative ) , by the second term in the definition @xmath1149 , i.e. , by @xmath1150 when @xmath593 is small enough . this means , as in the case of the linear problem , that when the integral will be finite uniformly in @xmath13 . let us now support the claim just made . it is easy to see that for @xmath1151 $ ] , the first term in the definition of @xmath1152 can be either positive or negative , but it is of order @xmath1153 . the second term in the definition of @xmath1152 is positive and for @xmath1151 $ ] , it is of order @xmath1154 . lastly , the third term in the definition of @xmath1152 , may be positive or negative , but in either cas ! e , it will be of order @xmath1155 for @xmath1151 $ ] . therefore , for @xmath593 sufficiently small , @xmath1152 is dominated by the second term in the definition @xmath1156 , i.e. , by @xmath1157 . hence , the argument that was used for the linear problem in order to show that the integrand of the second term in the definition of @xmath1145 is in fact positive when @xmath639 is large enough , allows us to reach to the same conclusion here as well , given that @xmath593 is chosen sufficiently small . in this appendix we provide proofs of some auxiliary lemmas used in the main body of the manuscript . [ proof of lemma [ lem : mollification]]without loss of generality , we can restrict attention to @xmath1158 . we have @xmath1159@xmath1160 and@xmath1161 \\ & \quad-\rho_{2}(t , x;\delta)\left [ \frac{1}{\delta}d\tilde{u}_{2}(t , x)^{2}-d^{2}\tilde{u}_{2}(t , x)\right ] .\end{aligned}\ ] ] omitting function arguments for notational convenience , for @xmath1162@xmath1163 + \frac{\varepsilon}{2}\alpha d^{2}u^{\delta}\\ & = \rho_{1}\partial_{t}\tilde{u}_{1}+\rho_{2}\partial_{t}\tilde{u}_{2}+\rho_{1}d\tilde{u}_{1}b+\rho_{2}d\tilde{u}_{2}b-\frac{1}{2}\left\vert \sigma\left ( \rho_{1}d\tilde{u}_{1}+\rho_{2}d\tilde{u}_{2}\right ) \right\vert ^{2}\\ & \quad+\frac{\varepsilon}{2}\frac{1}{\delta}\left\vert \sigma\left ( \rho_{1}d\tilde{u}_{1}+\rho_{2}d\tilde{u}_{2}\right ) \right\vert ^{2}-\frac{\varepsilon}{2}\frac{1}{\delta}\left [ \rho_{1}\alpha(d\tilde{u}_{1})^{2}+\rho_{2}\alpha(d\tilde{u}_{2})^{2}\right ] \\ & \quad+\frac{\varepsilon}{2}\left [ \rho_{1}\alpha d^{2}\tilde{u}_{1}+\rho_{2}\alpha d^{2}\tilde{u}_{2}\right ] \\ & = \rho_{1}\left [ \partial_{t}\tilde{u}_{1}+d\tilde{u}_{1}b-\frac{1}{2}\left\vert \sigma d\tilde{u}_{1}\right\vert ^{2}+\frac{\varepsilon}{2}\alpha d^{2}\tilde{u}_{1}\right ] \\ & \quad+\rho_{2}\left [ \partial_{t}\tilde{u}_{2}+d\tilde{u}_{2}b-\frac{1}{2}\left\vert \sigma d\tilde{u}_{2}\right\vert ^{2}+\frac{\varepsilon}{2}\alpha d^{2}\tilde{u}_{2}\right ] \\ & \quad+\frac{1}{2}\left ( 1-\frac{\varepsilon}{\delta}\right ) \left [ \rho_{1}\left\vert \sigma d\tilde{u}_{1}\right\vert ^{2}+\rho_{2}\left\vert \sigma d\tilde{u}_{2}\right\vert ^{2}-\left\vert \sigma\left ( \rho_{1}d\tilde{u}_{1}+\rho_{2}d\tilde{u}_{2}\right ) \right\vert ^{2}\right ] \\ & \geq\frac{1}{2}\left ( 1-\frac{\varepsilon}{\delta}\right ) \left [ \rho_{1}\left\vert \sigma d\tilde{u}_{1}\right\vert ^{2}+\rho_{2}\left\vert \sigma d\tilde{u}_{2}\right\vert ^{2}-\left\vert \rho_{1}\sigma d\tilde{u}_{1}+\rho_{2}\sigma d\tilde{u}_{2}\right\vert ^{2}\right ] + \rho_{1}\gamma _ { 1}+\rho_{2}\gamma_{2}\\ & \geq\rho_{1}\gamma_{1}+\rho_{2}\gamma_{2},\end{aligned}\ ] ] where the last line is due to the convexity of @xmath1164 . [ l : generalbound ] let @xmath1165 and @xmath1166 be two continuously differentiable functions from @xmath19\times\mathbb{r}\mapsto\mathbb{r}$ ] . assume that @xmath20 and @xmath21 are lipschitz continuous . set @xmath1167 , @xmath1168 and let @xmath1169 solve @xmath1170ds\right ] , \quad\hat{x}^{\varepsilon}(0)=y.\ ] ] then for every @xmath3 , @xmath1168 and stopping time @xmath1171 , we have , with probability @xmath210 , @xmath1172 ds\\ & \quad\geq2w(0,y)-2w(\hat{\tau}^{\varepsilon},\hat{x}^{\varepsilon}(\hat{\tau}^{\varepsilon}))+2\sqrt{\varepsilon}\int_{0}^{\hat{\tau } ^{\varepsilon}}dw(s,\hat{x}^{\varepsilon}(s))\sigma(\hat{x}^{\varepsilon } ( s))db(s)\\ & \quad+2\int_{0}^{\hat{\tau}^{\varepsilon}}\mathcal{g}^{\varepsilon } [ w](s,\hat{x}^{\varepsilon}(s))ds-\int_{0}^{\hat{\tau}^{\varepsilon}}\left\vert \sigma(\hat{x}^{\varepsilon}(s))\left ( dw(s,\hat{x}^{\varepsilon } ( s))-du(s,\hat{x}^{\varepsilon}(s))\right ) \right\vert ^{2}ds\end{aligned}\ ] ] assume we use the control @xmath1174 for the design of the scheme and choose @xmath1175 . then @xmath1176 \\ & \quad = dw(t , x)(b(x)+\sigma^{2}(x)du(t , x)-2\sigma^{2}(x)dw(t , x)\\ & \qquad-\frac{1}{2}\left\vert \sigma(x)du(t , x)\right\vert ^{2}+\left\vert \sigma(x)dw(t , x)\right\vert ^{2}\\ & \quad = dw(t , x)b(x)-\frac{1}{2}\left\vert \sigma(x)dw(t , x)\right\vert ^{2}-\frac{1}{2}\left\vert \sigma(x)\left ( dw(t , x)-du(t , x)\right ) \right\vert ^{2}\\ & \quad=\mathbb{h}(x , dw(t , x))-\frac{1}{2}\left\vert \sigma(x)\left ( dw(t , x)-du(t , x)\right ) \right\vert ^{2}.\end{aligned}\ ] ] applying it s formula to @xmath1177 then gives @xmath1178 \right ] ds\nonumber\\ & \quad+\int_{0}^{\hat{\tau}^{\varepsilon}}\frac{\varepsilon}{2}\sigma ^{2}(\hat{x}^{\varepsilon}(s))d^{2}w(s,\hat{x}^{\varepsilon}(s))ds+\int _ { 0}^{\hat{\tau}^{\varepsilon}}\sqrt{\varepsilon}dw(s,\hat{x}^{\varepsilon } ( s))\sigma(\hat{x}^{\varepsilon}(s))db(s)\nonumber\\ & \geq\int_{0}^{\hat{\tau}^{\varepsilon}}\left [ \frac{1}{2}\bar{u}(s,\hat { x}^{\varepsilon}(s))^{2}-\frac{1}{4}v(s)^{2}\right ] ds+\sqrt{\varepsilon } \int_{0}^{\hat{\tau}^{\varepsilon}}dw(\hat{x}^{\varepsilon}(s))\sigma(\hat { x}^{\varepsilon}(s))db(s)\nonumber\\ & \quad+\int_{0}^{\hat{\tau}^{\varepsilon}}\left [ \partial_{s}w(s,\hat { x}^{\varepsilon}(s))+\mathbb{h}(\hat{x}^{\varepsilon}(s),dw(s,\hat { x}^{\varepsilon}(s)))+\frac{\varepsilon}{2}\sigma^{2}(\hat{x}^{\varepsilon } ( s))d^{2}w(s,\hat{x}^{\varepsilon}(s))\right ] ds\nonumber\\ & \quad-\frac{1}{2}\int_{0}^{\hat{\tau}^{\varepsilon}}\left\vert \sigma ( \hat{x}^{\varepsilon}(s))\left ( dw(s,\hat{x}^{\varepsilon}(s))-du(s,\hat { x}^{\varepsilon}(s))\right ) \right\vert ^{2}ds\nonumber\\ & = \int_{0}^{\hat{\tau}^{\varepsilon}}\left [ \frac{1}{2}\bar{u}(\hat { x}^{\varepsilon}(s))^{2}-\frac{1}{4}v(s)^{2}\right ] ds+\sqrt{\varepsilon } \int_{0}^{\hat{\tau}^{\varepsilon}}dw(s,\hat{x}^{\varepsilon}(s))\sigma ( \hat{x}^{\varepsilon}(s))db(s)\nonumber\\ & \quad+\int_{0}^{\hat{\tau}^{\varepsilon}}\mathcal{g}^{\varepsilon } [ w](s,\hat{x}^{\varepsilon}(s))ds-\frac{1}{2}\int_{0}^{\hat{\tau } ^{\varepsilon}}\left\vert \sigma(\hat{x}^{\varepsilon}(s))\left ( dw(s,\hat{x}^{\varepsilon}(s))-du(s,\hat{x}^{\varepsilon}(s))\right ) \right\vert ^{2}ds\nonumber\end{aligned}\ ] ]

we discuss importance sampling schemes for the estimation of finite time exit probabilities of small noise diffusions that involve escape from an equilibrium . a factor that complicates the analysis is that rest points are included in the domain of interest . we build importance sampling schemes with provably good performance both pre - asymptotically , i.e. , for fixed size of the noise , and asymptotically , i.e. , as the size of the noise goes to zero , and that do not degrade as the time horizon gets large . simulation studies demonstrate the theoretical results .

You are an expert at summarizing long articles. Proceed to summarize the following text: a new generation of radio telescopes will map the polarization of cosmic radio sources over a large range of wavelengths , from a few centimetres to several metres . since the plane of polarization of a linearly polarized wave is rotated by an amount that depends on the magnetic field and free - electron distributions and the wavelength ( @xmath4 ) , the resulting data will probe both the synchrotron - emitting sources and any intervening magneto - ionic medium in unprecedented detail . a useful way to characterize the intrinsic properties of magneto - ionic media is the faraday dispersion function , @xmath2 , which contains information on the transverse orientation of the magnetic field ( @xmath5 ) and on the intrinsic polarized emission as a function of faraday depth , @xmath1 . the faraday depth is proportional to the integral along the line of sight @xmath6 of the product of the density of thermal electrons , @xmath7 , and the component of the magnetic field parallel to the line of sight : @xmath8 hence , in principle , @xmath2 can be used to obtain both the perpendicular and the parallel components of the three - dimensional magnetic field . ( our system of coordinates is such that the origin is at the far end of the source and the observer is located at @xmath9 . a magnetic field pointing towards the observer yields a positive faraday depth . ) reconstruction of @xmath2 is usually done by taking advantage of the fourier - transform type relationship between the observed polarized emission and the faraday dispersion function . the _ observed _ complex polarization @xmath10 can be expressed as the integral over all faraday depths of the _ intrinsic _ complex polarization @xmath2 modulated by the faraday rotation @xcite : @xmath11 so that @xmath2 can be expressed in a similar way : @xmath12 @xmath2 is a complex - valued function : @xmath13 where @xmath14 is the fraction of polarized flux that comes from regions of faraday depth between @xmath1 and @xmath15 , @xmath16 is the intrinsic polarization angle ( perpendicular to the transverse component of the magnetic field , @xmath17 ) and may itself depend on @xmath1 . equation ( [ eqf ] ) lies at the heart of methods to recover @xmath2 from multi - frequency observations of the complex polarized intensity ( called rotation measure , rm , synthesis ; ) . the rm synthesis has been used to recover faraday components of compact sources ( e.g. @xcite ) and diffuse structures in the milky way ( e.g. ) , in nearby galaxies ( e.g. ) and in galaxy clusters ( e.g. ) . several techniques have been proposed to deal with the limited @xmath18 coverage provided by real telescopes ( rm - clean ; @xcite ; sparse analysis and compressive sensing ; , @xcite ; multiple signal classification ; @xcite ) and with the missing negative @xmath18 ( e.g. using wavelet transforms ; @xcite , @xcite ) . also used wavelets to analyze the scales of structures in faraday space and emphasized the need to combine data at high and low frequencies . because of the difficulty of the rm synthesis technique to recover multiple faraday components , it has been suggested to use direct @xmath19 and @xmath20 fitting , where @xmath21 and @xmath22 are the @xmath23 and @xmath24 stokes parameters normalised to the total intensity @xmath25 ( @xcite ; @xcite ) . in this paper we show how observations , performed in the various wavelength ranges available at existing and planned radio telescopes , can be used to constrain the variation of @xmath0 ( and therefore the orientation of the magnetic field component perpendicular to the line of sight ) with @xmath1 . we use a fisher matrix analysis to quantify the precision that can be achieved for fitted parameters and investigate the degeneracies that exist between the different constituents of our model . recently , @xcite performed a similar analysis to evaluate the capability of new radio telescopes to constrain the properties of intergalactic magnetic fields through observations of background polarized sources . their work assumed two faraday components , each with a constant @xmath0 , a narrow one ( the compact radio source ) and a broad one ( possibly associated with the milky way ) . here we consider _ a linear variation of @xmath0 with @xmath1 _ and show how the degeneracies between pairs of model parameters can be broken using complementary datasets from different instruments in order to recover @xmath3 , using two simple models of @xmath2 , a constant and a gaussian . in the simple cases we consider , the variation of @xmath26 can be produced by a helical magnetic field . magnetic helicity is a natural consequence of dynamo action and sophisticated statistical methods have been devised to try to infer its presence , although without inclusion of faraday effects ( , ) . anomalous depolarization ( an increase rather than the usual decrease of the degree of polarization with wavelength ) produced by an helical field was discussed by @xcite . helical fields have been invoked to explain the anomalous depolarization properties of the nearby galaxy ngc 6946 ( ) and polarization characteristics of the central part of the starburst galaxy ngc 253 ( ) . helical magnetic fields are also important in galactic and protostellar jets ( e.g. @xcite , @xcite ) . bi - helical fields ( with opposite signs of helicity on small and large scales ) are produced in simulations of galactic dynamos and the signatures of such fields are discussed in a recent paper by @xcite . in this paper , we focus on the detectability of single - helical magnetic fields . we consider observations of the stokes parameters @xmath23 and @xmath24 with the instruments listed in table [ tab1 ] . we used a nominal integration time of 1 h for the low - frequency observations ( giant meterwave radio telescope , gmrt , westerbork synthesis radio telescope , wsrt , low frequency array , lofar ) and 10 min for observations with the more sensitive instruments ( jansky very large array , jvla , australian square kilometre array pathfinder , askap and square kilometre array 1 , ska1 ) . this allows an easy comparison of the sensitivities and makes it possible to display the confidence intervals of the parameters of interest on a common graph ( figs [ fig2 ] and [ fig3 ] ) . we used a channel width of 1 mhz for all instruments except the jvla for which a channel width of 2 mhz is more than sufficient in the wide s - band ( 20004000 mhz ) to resolve the main features of the spectral energy distribution of the polarization . note that all instruments allow the use of narrower channels ; however , there is an obvious trade - off between sensitivity per channel and total integration time . we have varied the channel width over two orders of magnitude between 0.1 and 10 mhz and observed that the resulting precision on the main parameter of interest , @xmath27 ( equation ( [ eqbeta ] ) ) , changes by less than @xmath28 for a same total integration time of the ska1-survey . the quoted sensitivities are indicative since several instruments listed in table [ tab1 ] are still in their design phase . also , some bands , especially the low - frequency ones , will be affected by radio frequency interferences and a fraction of the channels will be missing . with real data at hand it will be straightforward to include the actual frequency coverage and sensitivities in the modeling of a particular data set . we scaled the sensitivities @xmath29 quoted in the literature for a given effective bandwidth @xmath30 and integration time @xmath31 to new values of the channel width @xmath32 and integration time @xmath33 , as given in the table , for a given number of tunings @xmath34 to cover the whole bandwidth : @xmath35 the jvla will be used to carry out sensitive surveys of large parts of the sky . we use figures provided by steven t. myers [ national radio astronomy observatory ( nrao ) ] in the karl jansky vla sky survey prospectus ( see table [ tab1 ] ) for the jvla in its b - configuration . in the s - band ( 24 ghz ) , the effective bandwidth ( free from radio frequency interferences ) is 1500 mhz , and a noise level of 0.1 mjy can be achieved in 7.7 seconds . the size of the synthesized beam is @xmath36 at the centre of the band . in the l - band ( 12 ghz ) , the effective bandwidth is 600 mhz and a noise level of 0.1 mjy can be achieved in 37 seconds of integration . the size of the synthesized beam is @xmath37 . in both cases we assumed a single tuning . we note that @xcite recently submitted a science white paper for a jvla sky survey in the s - band ( in the c - configuration ) , in which they consider several alternatives ranging from a shallow all - sky survey to ultra - deep fields of a few tens of square degrees . they estimate that a shallow all - sky survey of a total of about 3000 hours would lead to the detection of over @xmath38 polarized sources . according to the askap website , a continuum sensitivity of 29 to 37 @xmath39jy beam@xmath40 for beams between 10@xmath41 and 30@xmath41 can be reached in 1 hour for a bandwidth of 300 mhz . four tunings would be required to cover the whole frequency band from 700 to 1800 mhz , so in a total of 1 h a noise level of 1 1.3 mjy per 1 mhz channel would be reached . a major polarization survey with askap ( possum ) is in the design study phase . in its first phase , the ska will observe at low frequencies ( 50 350 mhz , ska1-low ) , mid - frequencies ( 0.35 3.05 ghz , ska1-mid ) and in a survey mode in the 0.65 1.67 ghz range ( ska1-survey ) . in 1 hour of observation and per 0.1 mhz channel , the sensitivity is expected to be 63 @xmath39jy for ska1-mid , 103 @xmath39jy for ska1-low , and 263 @xmath39jy for ska1-survey . we have assumed a single tuning for ska1-low and that for ska1-mid four tunings ( in a 770 mhz bandwidth each ) will be needed to cover the whole band ; for ska1-survey , the maximum bandwidth will be 500 mhz , so two tunings will be needed . the corresponding noise levels per 1 mhz bandwidth and after 10 minutes of observations are given in table [ tab1 ] . for the gmrt 610 mhz band , our sensitivity estimate is based on the figures quoted by @xcite who reached a noise level in @xmath23 and @xmath24 of 36 @xmath39jy per beam of 24@xmath41 in 180 minutes in a 16 mhz band centered at 610 mhz . four tunings would be required to cover the whole band . our estimate of the sensitivity of the gmrt in the 325 mhz band relies on a noise level of 2.7 mjy per beam per 1 mhz channel in 1 hour , based on polarization observations of a pulsar done in 2011 ( farnes , private communication ) and assuming that all 30 antennas would be available . assuming that 2 tunings would be necessary to cover the whole band , this gives a noise level of 3.8 mjy in a total of 1 h. the wsrt also operates in the 320 mhz band ( called the 92 cm band ) . after 1 hour of observation , the theoretical noise level in stokes @xmath25 is 1.2254 mjy beam@xmath40 in a 10 mhz band . this corresponds to about 3.9 mjy beam@xmath40 in a 1 mhz channel . note that confusion noise is expected to be significant in observations of the stokes parameter @xmath25 but it can be neglected @xmath23 and @xmath24 . recently detected polarization with the wsrt towards the andromeda galaxy at 350 mhz . the high - band array ( hba ) of the lofar operates at frequencies between 110 mhz and 250 mhz with a filter between 190 and 210 mhz . lofar has detected polarization in the hba and rotation measures could be inferred ( e.g. in pulsars , , and in polarized sources in the field of m 51 , mulcahy et al . , in prep . ) . however , at the lofar frequencies depolarization is extremely strong and for the fiducial models presented in this paper the measurements at the quoted sensitivities do not provide improved constraints on the parameters related to the magnetic field . the lofar frequency coverage is displayed in fig . [ fig1 ] but the lofar confidence intervals are therefore not shown in the other figures . the fisher analysis is often used in cosmology ( e.g. @xcite , page 94 ) . consider a set of @xmath42 data points and a model with @xmath43 parameters , @xmath44 . the fisher matrix elements @xmath45 are proportional to the second partial derivatives with respect to two given parameters of the likelihood function @xmath46 that the data set derives from the given model . if the measurement errors follow a gaussian probability distribution , then @xmath47 where @xmath48 is defined in eq . ( [ eqchi2 ] ) . denoting @xmath49 and @xmath50 the values of the stokes parameters @xmath23 and @xmath24 for the assumed model , estimated at wavelengths @xmath51 and with noise levels @xmath52 , we have @xmath53 the fisher matrix elements can be written as @xmath54 the covariance matrix is the inverse of the fisher matrix : @xmath55 we consider two simple models for @xmath2 , each with a linearly varying @xmath0 as a function of faraday depth : @xmath56 with some constants @xmath57 and @xmath27 . this is a parametrisation of @xmath16 as a first - order polynomial and , as discussed below , it can also be interpreted as a helical magnetic field . one of the simplest possible models for @xmath2 is the top - hat , @xmath58 where the set of parameters is @xmath59 , @xmath60 is given by eq . ( [ eqpsi0 ] ) and @xmath61 is the top - hat function , with @xmath62 in the range @xmath63 and @xmath64 elsewhere . the complex polarization is @xmath65 e^{2{\rm i } \psi(\lambda^2 ; { \bf{p } } ) } \ , , \label{eqptophat}\ ] ] where @xmath66 and @xmath67 . in a uniform slab , the faraday depth varies linearly with the @xmath6-coordinate for @xmath1 between @xmath68 : @xmath69 where @xmath7 is in @xmath70 , @xmath71 in @xmath39 g , @xmath6 in pc and @xmath1 in rad m@xmath72 . consider a magnetic field with a constant line - of - sight component , but with a rotating component in the plane of the sky : @xmath73 the intrinsic polarization angle clearly varies with the faraday depth : @xmath74 where @xmath75 for an helical field with @xmath76 rad kpc@xmath40 , we have @xmath77 m@xmath78 . note that the sign of @xmath27 depends on the relative orientation of the magnetic field component along the line of sight and the handedness of the helix . a positive @xmath27 means that @xmath71 , which produces the faraday rotation , and the intrinsic rotation of @xmath79 have the same direction . a negative @xmath27 means that the faraday rotation effectively counteracts the intrinsic rotation of the plane - of - sky magnetic field . this effect will be discussed further in sect . 3 . as a simple alternative to the top - hat parametrisation of @xmath2 we also consider a gaussian form , @xmath80 e^{2{\rm i } \psi_0(\phi ; { \bf p})},\ ] ] where @xmath81 and @xmath82 are defined as before . this gives a complex polarization of @xmath83 e^{2{\rm i } \psi(\lambda^2 ; { \bf p } ) } , \label{eqpgaussian}\ ] ] where @xmath84 is given by eq . ( [ psiobs ] ) . the modulus of @xmath10 is a gaussian centered at @xmath85 with variance @xmath86 . in the two previous sections we calculated @xmath10 by integrating eq . ( [ eqpol ] ) analytically . using the properties of the fourier transforms , we now show why any linear variation of @xmath0 with @xmath1 produces a translation of the observed polarized intensity in the @xmath18 space . using the standard expression for fourier transform ( integral over @xmath87 from @xmath88 to @xmath9 of a function @xmath89 times @xmath90 for the direct transform , and times @xmath91 for its inverse ) , eq . ( [ eqpol ] ) can be written as @xmath92 where @xmath93 is the inverse fourier transform . using @xmath94 where @xmath95 is a real - valued function centered at @xmath96 , @xmath97 the factor @xmath98 is independent of @xmath1 and can be taken out of the integral . multiplication becomes a convolution in the fourier ( @xmath18 ) space and convolution becomes a multiplication , so @xmath99 translation in the @xmath1-space gives a rotation in @xmath18-space , and the inverse transform of the term involving @xmath100 becomes a delta - function , giving @xmath101 we then obtain @xmath102 where @xmath103 is the complex polarization corresponding to @xmath95 when @xmath104 . changing the variable from @xmath105 to @xmath18 , we obtain @xmath106 so that @xmath107 this shows that the observed modulus of the polarized intensity of a medium with a given @xmath27 is simply a translation by @xmath27 in @xmath18-space of what would be observed if @xmath27 were equal to zero . @xmath108 is real - valued , by definition ; if it is even in @xmath1 , its inverse fourier transform is also real - valued , and the observed polarization angle will be @xmath109 as found in sect . [ sect231 ] and [ sect232 ] for the top - hat and the gaussian cases . . the askap regions are in blue and the gmrt ones in green . , scaledwidth=48.0% ] we use a fiducial top - hat model of the faraday dispersion function with the following parameters : @xmath110 = 15 rad m@xmath72 , @xmath111 rad m@xmath72 , @xmath112 = 0.1 mjy/(rad m@xmath72 ) , @xmath113 rad , and three values of @xmath27 , 0 and @xmath114 m@xmath78 . the total intrinsic polarized flux density ( integrated over all faraday depths ) is thus @xmath115 mjy . because of depolarization , the signal expected at a given frequency will be weaker ( see figure [ fig1 ] ) but should be detectable within a reasonable amount of observing time by current and future instruments ( see table [ tab1 ] ) . for comparison , we also use a gaussian model of @xmath116 with the same total flux . the dispersion of the gaussian is @xmath117 rad m@xmath72 and the peak flux density per unit of faraday depth @xmath118 mjy/(rad m@xmath72 ) . note that @xmath119 characterises the gaussian profile of the faraday dispersion function for a model with a regular field , not to be confused with the dispersion in rotation measure ( rm ) caused by possible rm fluctuations across an observing beam ( usually denoted @xmath120 ) . in sect . [ sectfaradaydispersion ] we discuss the additional depolarization effect by faraday dispersion produced by a random field . the width of the faraday structure ( that is , the total faraday depth , sometimes denoted @xmath121 , see sect [ sectfaradaydispersion ] ) in our fiducial top - hat model is @xmath122 rad m@xmath72 . in many astrophysical cases this quantity can be larger ( up to @xmath123 rad m@xmath72 in spiral arms , e.g. @xcite ) . a larger total faraday depth translates into a narrower main peak " of the @xmath10 distribution and weaker emission at long wavelengths . [ fig1 ] shows the variation of the polarized intensity with @xmath18 for a top - hat ( left column ) and a gaussian ( right column ) faraday dispersion function . in the former case , the solid line ( @xmath124 ) is the well - known sinc function produced by a uniform slab , which is more usually shown using the linear horizontal axis used in the bottom left panel . note that , for clarity , the graphs in fig . [ fig1 ] do not include any spectral dependence of the intrinsic polarization . figures [ fig2 ] and [ fig3 ] ( the confidence regions of the parameters ) do , on the other hand , include a spectral dependence of the form given by eq . ( [ eqspecdep ] ) . in the gaussian case , the polarized intensity decreases monotonically and no emission is produced in the longer wavebands at which the gmrt and lofar operate ( fig . [ fig1 ] ) . in the bottom right panel we show the variation of the @xmath23 and @xmath24 stokes parameters ( in brown and blue ) for the gaussian @xmath2 , which are the direct observables . they oscillate with a 90@xmath125 phase shift with respect to each other . in the rest of the paper we focus on the top - hat model because it gives stronger emission in longer wavebands for the parameters selected here and it includes the standard case for faraday depolarization calculations of a uniform slab as a special case . it is most interesting to compare the variations of the polarized intensity for a positive and a negative @xmath27 . when @xmath126 , the intrinsic helicity of the magnetic field and the faraday rotation act in the same direction . this results in an increased depolarization at short wavelengths . even for @xmath127 , where faraday depolarization is absent , the emission is significantly depolarized compared to the case of a constant magnetic field orientation ( @xmath124 ) . when @xmath128 , faraday rotation counteracts the intrinsic rotation of the field , which means that the polarized emission peaks at a wavelength different from zero ( dashed lines ) , where @xmath85 ( eqs . ( [ eqptophat ] ) and ( [ eqpgaussian ] ) ) . this effect is similar to the `` anomalous depolarization '' discussed by ( * ? ? ? * section 9 ) . [ fig2 ] shows the 68.3% confidence regions of the five different parameters obtained from the fisher analysis for @xmath124 . the colour code is the same as in fig . [ fig1 ] , with the askap confidence regions in blue and the gmrt ones in green . the plots in the first row show that the central faraday depth @xmath110 is mostly uncorrelated with the other parameters . on the other hand , @xmath129 , which describes the extent of the faraday component in @xmath1-space , is strongly correlated with the parameters related to the intrinsic polarization angle ( @xmath57 and @xmath27 ) and at short wavelengths ( in the askap band ) with the normalization of @xmath2 ( @xmath112 ) . this is because the crucial effect is faraday _ differential _ rotation across the faraday component and not the magnitude of the central faraday depth . an increase in @xmath129 means stronger depolarization due to differential rotation which must be counteracted by a more negative @xmath27 in order to produce a similar fit to the data . vice versa , a lower @xmath129 means weaker depolarization and @xmath27 needs to become positive to increase the depolarization . in other words , @xmath129 and @xmath27 are anti - correlated around @xmath124 . most importantly , the strength of the correlation between pairs of parameters including @xmath57 or @xmath27 varies with the selected waveband . this is what makes it possible to break parameter degeneracies by combining short - wavelength ( like askap ) and long wavelength ( like gmrt ) data sets in order to better constrain the derived parameter values , as we discuss next . [ fig3 ] shows the one - sigma confidence ellipses for @xmath57 , @xmath27 , @xmath112 and @xmath129 for @xmath130 m@xmath78 ( four left panels ) and @xmath131 m@xmath78 ( four right panels ) . as expected , the orientation of the ellipses is similar for the short - wavelength instruments jvla ( s - band in purple and l - band in pink ) , askap ( blue ) and ska1-survey ( black ) . on the other hand , the long - wavelength gmrt data set ( green ) produces a different correlation between parameters ; in some cases the confidence ellipses at short and long wavelengths are almost orthogonal , making it possible to reduce the confidence intervals on the derived parameter values considerably by using both wavelength ranges together ( such as askap and gmrt , shown in yellow ) . this is further illustrated in fig . [ fig4 ] where the precision that can be achieved in @xmath27 for different instruments is shown . a combination of askap and gmrt observations makes it possible to reach an uncertainty @xmath132 m@xmath78 if @xmath128 , for the integration times shown in table [ tab1 ] and the set of model parameters used . if @xmath126 , all signals are weaker because of increased depolarization and the precision on @xmath27 ( and all other model parameters ) is lower . we examined how variations of the intrinsic polarization angle @xmath0 with the faraday depth @xmath1 within a source affect the observable quantities . using simple models for the faraday dispersion @xmath2 and @xmath3 , along with the current and planned properties of the main radio interferometers , we show how degeneracies among the parameters describing the magneto - ionic medium can be minimised by combining observations in different wavebands . in particular we have shown that it may be possible to recover the sign and the magnitude of @xmath27 , a parameter that we have defined and that is related to the relative effect of the helicity of the transverse magnetic field and the faraday rotation due to the parallel component of the magnetic field . since the direction of @xmath133 can be easily inferred from rm measurements , it should be possible to recover the sign ( and , under some assumptions the magnitude ) of the helicity of the magnetic field . however , the additional effect of faraday dispersion by a random component of the magnetic field attenuates the variations of the polarized emission as a function of wavelength and may shift the peak of polarized emission towards shorter wavelengths if @xmath27 is negative . faraday depolarization effects ( both by differential rotation and by dispersion ) will have to be included in the modelling of real data in order to recover information on the helicity of the magnetic field . this approach is complementary to statistical studies of the correlation between the degree of polarization and the rotation measures of cosmic sources which may also provide information on magnetic helicity ( @xcite , @xcite ) . planned surveys of fixed sensitivity will be biased towards radio sources with negative @xmath27 , because of the depolarization produced when @xmath134 . detection of @xmath134 will be more difficult both through @xmath135 and @xmath136 model - fitting and rm synthesis because most of the signal is shifted toward negative @xmath18 . @xcite show that restricting the integral in eq . ( [ eqf ] ) to the positive @xmath18 yields an erroneous reconstruction of @xmath2 when @xmath126 . in this work we used a first - order parametrisation of the variation of intrinsic polarization angle with faraday depth . higher - order representations could be used ( or e.g. chebyshev polynomials ) if the data are of sufficient quality . including a second - order term implies a convolution with an imaginary gaussian and a significantly more complicated expression for @xmath10 . we thank oliver gressel for organizing the stimulating nordita workshop ` galactic magnetism in the era of lofar and ska ' held in stockholm in 2013 september . we also thank axel brandenburg and rodion stepanov for interesting discussions on the topics discussed in this paper and for sharing their related results with us . we are grateful to rainer beck , john h. black , jamie farnes , george heald , anvar shukurov and to the referee for constructive comments . af is grateful to the leverhulme trust for financial support under grant rpg-097 .

wide - band radio polarization observations offer the possibility to recover information about the magnetic fields in synchrotron sources , such as details of their three - dimensional configuration , that has previously been inaccessible . the key physical process involved is the faraday rotation of the polarized emission in the source ( and elsewhere along the wave s propagation path to the observer ) . in order to proceed , reliable methods are required for inverting the signals observed in wavelength space into useful data in faraday space , with robust estimates of their uncertainty . in this paper , we examine how variations of the intrinsic angle of polarized emission @xmath0 with the faraday depth @xmath1 within a source affect the observable quantities . using simple models for the faraday dispersion @xmath2 and @xmath3 , along with the current and planned properties of the main radio interferometers , we demonstrate how degeneracies among the parameters describing the magneto - ionic medium can be minimised by combining observations in different wavebands . we also discuss how depolarization by faraday dispersion due to a random component of the magnetic field attenuates the variations in the spectral energy distribution of the polarization and shifts its peak towards shorter wavelengths . this additional effect reduces the prospect of recovering the characteristics of the magnetic field helicity in magneto - ionic media dominated by the turbulent component of the magnetic field . [ firstpage ] polarization methods : data analysis techniques : polarimetric ism : magnetic fields galaxies : magnetic fields radio continuum : galaxies

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Proceed to summarize the following text: the formation mechanism of jets is a key issue in the study of the magnetospheric structure of compact astrophysical objects . indeed , jets are observed in most compact sources , ranging from active galactic nuclei ( agns ) , quasars , and radio galaxies @xcite to accreting neutron stars , solar - mass black holes ( ss 433 , x - ray novae ) @xcite , and young stellar objects @xcite . moreover , jets have also been recently discovered in young radio pulsars @xcite . at the same time , in most studies devoted to the magnetohydrodynamic ( mhd ) model of such objects @xcite , in which the formation of jets is coupled with the attraction of longitudinal currents flowing in the magnetosphere , the attention was focused on intrinsic collimation in the sense that the effect of the external medium was assumed to be marginal . however , such a situation is possible only for a nonzero total current @xmath2 flowing within the jet @xcite , so the question of its closure in the outer parts of the magnetosphere arises . on the other hand , the longitudinal current is often constrained by the regularity condition at the fast magnetosonic surface , which by no means always leads to the sufficiently large longitudinal currents required for collimation @xcite . in other words , with the exception of the force - free case @xcite , as yet no working model of a jet in which , on the one hand , the total electric current would be zero and , on the other hand , the total magnetic flux @xmath3 in the jet would be finite , has been constructed . however , the force - free approximation , in which , by definition , the particle energy density is disregarded , does not allow the fraction of energy transferred by the outflowing plasma to be determined . at the same time , the question of collimation can not be solved in isolation from the external conditions ( see , e.g. , @xcite ) . in particular , this is clear even from a popular example of the magnetosphere of a compact object with a monopole magnetic field , because for any arbitrarily weak external regular magnetic field , the monopole solution ( for which the magnetic field falls off as @xmath4 ) can not be extended to infinity . moreover , as is well known from an example of moving cosmic bodies , such as jupiter s moons @xcite or artificial earth satellites @xcite , as well as radio pulsars @xcite , the external magnetic field can serve as an effective transfer link , which occasionally determines the general energy losses of the system . for this reason , constructing a consistent magnetospheric model for compact objects immersed in an external magnetic field is , in our view , of undeniable interest , especially since , as was noted above , such a jet model was previously constructed in the force - free approximation @xcite . undoubtedly , the existence of an external regular magnetic field in the vicinity of compact objects is largely open to question . the regular magnetic field in our galaxy , i.e. , the field that is constant on scales comparable to the sizes of our galaxy , is known to be @xmath5 and essentially matches the random magnetic - field component , which varies even on scales of several parsecs @xcite . however , if the collimation is assumed to be actually produced by an external magnetic field , it becomes possible to estimate the jet radius . indeed , assuming the magnetic field in the jet to be similar to the external magnetic field ( [ 1 ] ) , we obtain from the condition for the conservation of magnetic flux @xmath6 where @xmath7 and @xmath8 are the radius and magnetic field of the compact object , respectively . for example , for agns ( @xmath9 , @xmath10 ) , we have @xmath11 which corresponds to the observed jet radii @xcite . one might expect such a picture to be also preserved for an external medium with pressure @xmath12 ; therefore , it seems of interest to consider the internal structure of a one - dimensional jet immersed in an external uniform magnetic field . however , a discussion of the more realistic case of a medium with pressure is beyond the scope of this study . nor do we discuss the collimation itself but only consider the internal structure of observed one - dimensional jets . this issue has become particularly urgent because of the new possibilities offered by space radio interferometry , which enables the internal structure of such jets to be resolved . the effect of an external medium on the internal structure of relativistic jets in the mhd model discussed here was previously studied only by appl and camenzind @xcite . they considered only a special case with a constant angular velocity of the plasma , in which the solution with a zero total electric field flowing inside the jet could not be constructed . as we show below , it is for the case of an angular velocity decreasing toward the jet periphery ( which , incidentally , is typical of all models with a magnetic field passing through the accretion disk ) that the solution with finite magnetic flux @xmath3 and zero total current @xmath13 can be constructed . on the other hand , many authors @xcite obtained a universal solution with a central core for a cylindrical jet : @xmath14 where , in the relativistic case , @xmath15 is the size of the central core , @xmath16 is the angular velocity of the compact object , and @xmath17 is the characteristic lorentz factor of the outflowing plasma . as can be easily seen , such a solution results in a rapid falloff of the poloidal field @xmath18 far from the rotation axis @xmath19 . however , this solution is in conflict with the force - free approximation , in which the poloidal magnetic field remains essentially constant @xcite . indeed , when the energy density of the electromagnetic field exceeds appreciably the plasma energy density ( and it is this case that was considered ) , it would be natural to assume that the internal jet structure must be similar to the force - free one . the examples given above show that a more detailed study with allowance for all possible solutions is required even in the simplest case of a one - dimensional cylindrical jet considered in terms of ideal magnetohydrodynamics . our study aims at a consistent investigation of this issue . several features that we use when studying the structure of relativistic jets typical of agns and radio pulsars should be immediately noted . first of all , the jet radius @xmath20 in all real cases proves to be considerably larger than the light - cylinder radius @xmath21 . this implies that , when the internal structure of jets is investigated , the corresponding equations must be written in complete relativistic form . on the other hand , the gravitational forces can be disregarded in them far from the compact object . finally , for simplicity , we consider below a cold plasma , which is justifiable because the thermal processes in the magnetospheres of radio pulsars play no crucial role . as for the jets from agns , this approximation is applicable here in those magnetospheric regions in which the plasma density is low . in any case , this is true for the field lines passing through the surface of a black hole . the one - dimensional solutions describing collimated jets are obtained in sect . 2 ; analytic and numerical solutions for the basic physical quantities characterizing the structure and physics of jets are given in sect . the problem is solved in straightforward statement ; i.e. , all jets characteristics are determined by a set of parameters in the compact source and , most importantly , by the physical conditions in the external medium . as a result , we have found the conditions under which most of the energy in actual relativistic jets must be transferred by the electromagnetic field , while a region with subsonic flow exists in the central jet regions . we also show that the solution with a central core ( [ 4 ] ) and ( [ 5 ] ) can not be realized in an external magnetic field . finally , some astrophysical implications of the theory developed for one - dimensional jets are discussed in sect . let us consider the structure of a one - dimensional jet where all quantities depend only on radius @xmath22 ; in what follows , the temperature of the matter is assumed to be zero , and @xmath23 . as in the general axisymmetric case , it is convenient to describe the magnetic - field structure in terms of magnetic - flux function @xmath24 , which is related to the longitudinal magnetic field by b_z()=. [ 6 ] accordingly , it is convenient to write the toroidal magnetic field , the electric field , and the 4-velocity vector of the matter as b_()=- , [ 7 ] = - , [ 8 ] = + _ f , [ 9 ] where @xmath25 is the total current within @xmath26 . in the case of a cold plasma , at the cylindrical magnetic surfaces @xmath27 , four `` integrals of motion '' @xcite can be introduced , which should be considered precisely as functions of magnetic flux @xmath28 in the most general statement . these are primarily @xmath29 and @xmath30 in the definitions ( [ 8 ] ) and ( [ 9 ] ) , as well as the @xmath31-component of angular momentum @xmath32 and the energy flux @xmath33 . here , @xmath34 is the relativistic specific enthalpy , which is equal to the mass of particles for a cold plasma . the specific form of the integrals of motion must be determined from boundary conditions in the compact source and from critical conditions at the singular surfaces . as a result , the equilibrium equation for magnetic surfaces far from gravitating bodies ( grad shafranov s equation ) can be written as ( see , e.g. , @xcite ) ( ) + _ f()^2 + ( ) -=0 , [ 10 ] where @xmath35 @xmath36 @xmath37 here , @xmath38 is the square of the mach number with respect to the alfven velocity @xmath39 , and the derivative @xmath40 acts only on the integrals of motion . the remaining jet parameters are given by the well - known algebraic relations ( see , e.g. , @xcite ) : @xmath41 equation ( [ 10 ] ) contains four integrals of motion ; this equation has no singularity at the fast magnetosonic surface , because it depends only on coordinate @xmath22 . as for the alfven surface , @xmath42 , the problem of the boundary conditions generally requires a further study beyond the theory of ideal magnetohydrodynamics . at the same time , for the fairly large currents @xmath43 considered here , a solution continuous across this surface can always be constructed by a small change in the integrals of motion near the alfven surface , @xmath44 . consequently , equation ( [ 10 ] ) requires six boundary conditions . these boundary conditions primarily include the external uniform magnetic field b_z(r_j)=b_ext , [ 14 ] and the regularity condition at the magnetic axis @xmath45 @xmath46 in addition , all four integrals @xmath47 , @xmath48 , @xmath49 , and @xmath50 must be specified . as for the remaining quantities characterizing the flow , such as the jet radius @xmath20 and the outflowing plasma energy , they must be determined as a solution of the problem formulated above . similarly , the solution of the problem must also give an answer to the question of whether the flow in the jet is supersonic . let us now consider the determination of the integrals of motion in more detail . it would be natural to assume that , at the jet boundary where there is no longitudinal motion of the matter , all four integrals of motion become zero _ f(_0 ) = 0 , e(_0 ) = 0 , l(_0 ) = 0 , ( _ 0 ) = 0 . [ 16 ] here , @xmath3 is the finite total magnetic flux concentrated in the jet . this case corresponds to the absence of tangential discontinuities at the jet boundary ; according to ( [ 11 ] ) , the total electric current within the jet is automatically equal to zero . we use the integrals of motion @xmath29 , @xmath51 , and @xmath52 derived by beskin _ et al . _ @xcite for the force - free magnetosphere of a black hole , which satisfy the conditions ( [ 16 ] ) and , consequently , can be directly used to study the jet structure . the only but very important change here is the fact that , for a finite magnetization parameter @xmath53 @xcite , = , [ 17 ] which tends to infinity in the force - free approximation , the particle contribution must be added to the energy integral @xmath54 , because the energy flux of the electromagnetic field near the rotation axis must inevitably vanish . as a result , we have with accuracy up to @xmath55 @xmath56 @xmath57 @xmath58 below , we assume , for simplicity , that @xmath59 we emphasize that @xmath60 in expression ( [ 20 ] ) has the meaning of the injection lorentz factor in the region of the compact object and it is not equal to the lorentz factor of the jet particles . thus , we see from the formula for the energy flux @xmath54 that the contribution by the electromagnetic field becomes dominant only at @xmath61 , where @xmath62 at low values of @xmath28 , most of the energy is transferred by the relativistic particles ; as directly follows from relation ( [ 20 ] ) , their lorentz factor is constant and equal to their initial value @xmath60 . as for the integral @xmath30 , the particle - to - magnetic flux ratio , we chose it in the form ( ) = _ 0(1-/_0 ) , [ 23 ] which satisfies the condition ( [ 16 ] ) . we emphasize that the very possibility of using the integrals of motion obtained by analyzing the inner magnetospheric regions , is not trivial . indeed , the flow outside the fast magnetosonic surface is completely determined by four boundary conditions at the surface of a rotating body . at the same time , a one - dimensional flow can be produced by the interaction with the external medium , which gives rise ( see , e.g. , @xcite ) to perturbations or shock waves propagating from `` acute angles '' and other irregularities . therefore , in regions where the conditions for the validity of ideal magnetohydrodynamics are violated , a significant redistribution of energy @xmath48 and angular momentum @xmath49 is possible ( e.g. a part of them can be lost via radiation ) . nevertheless , we assume here , for simplicity , that the integrals of motion @xmath54 and @xmath63 , functions of flux @xmath28 , remain exactly the same as those in the inner magnetospheric regions . in the one - dimensional case we consider , it is convenient to reduce the second - order equation ( [ 10 ] ) to a set of two first - order equations for @xmath24 and @xmath64 . multiplying equation ( [ 10 ] ) by @xmath65 , we obtain + a()^2 + ( ) -(^2 ^ 2)=0 , [ 24 ] with the derivative @xmath66 acting only on the integrals of motion . finally , we use `` bernoulli s relativistic equation '' @xmath67 , which , given the definitions of the integrals of motion @xmath54 and @xmath63 , can be written as a^2()^2= + - - , [ 25 ] where we introduced the dimensionless variables @xmath68 as a result , substituting the right - hand part of ( [ 25 ] ) into the first term of ( [ 24 ] ) and performing differentiation , we obtain the first first - order differential equation @xmath69\frac{dm^2}{dx}= \frac{m^6}{x^3a}\frac{\omega_{\rm f}^2(0)l^2}{\mu^2\eta^2}- \frac{xm^2}{a}\frac{\omega_{\rm f}^2}{\omega_{\rm f}^2(0 ) } \left(\frac{e^2}{\mu^2\eta^2}-2a\right ) \nonumber \\ + \frac{m^2}{2}\frac{dy}{dx}\left [ \frac{1}{\mu^2\eta^2}\frac{de^2}{dy}+ \frac{x^2}{\omega_{\rm f}^2(0)}\frac{d\omega_{\rm f}^2}{dy}- 2\left(1-\frac{\omega_{\rm f}^2}{\omega_{\rm f}^2(0)}x^2\right ) \frac{1}{\eta}\frac{d\eta}{dy}\right ] . \label{28}\end{aligned}\ ] ] the second first - order differential equation is bernoulli s equation ( [ 25 ] ) , which should now be considered as an equation for the derivative @xmath70 . the set of equations ( [ 25 ] ) and ( [ 28 ] ) allows a general solution to be constructed for a one - dimensional jet immersed in an external magnetic field . we emphasize one important advantage of the set of first - order equations ( [ 25 ] ) and ( [ 28 ] ) over the initial second - order equation ( [ 10 ] ) . the point is that the relativistic equation ( [ 10 ] ) , which is basically the force balance equation , contains the electromagnetic force _ em = _ e*e * + * j * , [ 29 ] in which the electric and magnetic contributions virtually cancel each other far out from the rotation axis @xmath71 . using bernoulli s equation ( [ 25 ] ) , we can derive @xcite ~. [ 30 ] when analyzing ( [ 10 ] ) , we therefore must retain all higher order terms @xmath72 , while the zero - order quantities @xmath73 and @xmath74 in ( [ 28 ] ) are analytically removed using bernoulli s equation , so all terms of this equation are of the same order . finally , it is also important that the exact equation ( [ 28 ] ) has no singularity near the rotation axis . in other words , its solution contains no @xmath75-shaped current @xmath76 flowing along the jet axis ; several authors pointed out to the necessity of it @xcite . let us consider the basic properties of the set of equations ( [ 25 ] ) and ( [ 28 ] ) . as can be easily verified , in the relativistic case under consideration , we may assume @xmath77 with high accuracy . far from the rotation axis , @xmath78 ( @xmath79 ) , equation ( [ 25 ] ) can be rewritten in the limit @xmath80 as @xmath81 or , equivalently , @xmath82 as we see , equation ( [ 31 ] ) does not contain @xmath83 at all and can therefore be integrated independently . this must be the case , because equation ( [ 31 ] ) must coincide with the asymptotics of the force - free equation , which can be derived from ( [ 25 ] ) by going to the limit @xmath84 . assuming now that @xmath85 in ( [ 32 ] ) , we obtain , in particular , for the jet radius @xmath86 consequently , the jet radius is determined by the limit of the @xmath87 ratio as @xmath88 . in particular , for @xmath54 and @xmath29 given by ( [ 18])([20 ] ) , we have @xmath89 so the limit ( [ 33 ] ) does actually exist . as a result , we obtain @xmath90 which essentially coincides with estimate ( [ 2 ] ) . this is no surprise , because we show below that equations ( [ 25 ] ) and ( [ 28 ] ) for the integrals of motion ( [ 18 ] ) , ( [ 20 ] ) , and ( [ 23 ] ) have a constant magnetic field as their solution over a wide range of @xmath22 . let us now consider in more details the behavior of the solution in the inner jet region , where @xmath91 and , hence , the integrals of motion can be approximately written as @xmath92 with @xmath93 and @xmath94 . as a result , @xmath95 , and we can rewrite equations ( [ 25 ] ) and ( [ 28 ] ) as @xmath96 @xmath97 equations ( [ 39 ] ) and ( [ 40 ] ) describing the internal jet structure can be solved analytically . it can be verified by direct substitution that we have the following asymptotics for @xmath98 : m^2(x ) = m_0 ^ 2 = const , [ 41 ] y(x ) = x^2 , [ 42 ] which correspond to a constant magnetic field b_z = b_z(0)= = b(r_l ) = const , [ 43 ] where @xmath99 . here , we assume that @xmath100 , which is the typical for jets from agns and radio pulsars . the solution of ( [ 39 ] ) and ( [ 40 ] ) for @xmath101 , i.e. , at @xmath79 , depends on the relationship between @xmath60 and @xmath102 . for example , at @xmath103 , when , according to ( [ 43 ] ) , the axial magnetic field is fairly weak , the total magnetic flux within @xmath104 @xmath105 can be written as @xmath106 where the flux @xmath107 is given by ( [ 22 ] ) . we see that , if the condition @xmath103 is satisfied , then the total magnetic flux within @xmath104 is lower than @xmath107 ; so , outside this region , the particles also make the main contribution to @xmath54 as before , while the contribution of the electromagnetic field may be neglected . as a result , at @xmath79 , the solution of ( [ 39 ] ) and ( [ 40 ] ) has a quadratic rise of @xmath83 and a power - law falloff of the magnetic field @xcite : m^2(x ) = m_0 ^ 2x^2 , [ 46 ] @xmath108 consequently , the magnetic flux increases very slowly ( logarithmically ) with the distance from the rotation axis : @xmath109 such a behavior of the magnetic field , in turn , shows that the transition flux @xmath110 is reached exponentially far from the rotation axis , in conflict with the estimate ( [ 2 ] ) corresponding to the assumption of jet collimation . we may thus conclude that an external constant magnetic field limits the mach number at the rotation axis above @xmath111 accordingly , as follows from ( [ 43 ] ) , the magnetic field at the rotation axis can not be weaker than @xmath112 if , however , the mach number at the rotation axis does not exceed @xmath60 ( i.e. if @xmath113 ) , then , as for the similar asymptotics @xmath114 , the solution of ( [ 39 ] ) and ( [ 40 ] ) for @xmath115 gives a constant magnetic field ( [ 43 ] ) , which corresponds to the solution y(x ) = x^2 . [ 51 ] at the same time , in this case , we have only a linear increase in the square of the mach number m^2(x ) = m_0 ^ 2x^2 . [ 52 ] then , according to ( [ 27 ] ) and ( [ 51 ] ) , the jet radius can be written as @xmath116 which is equivalent to ( [ 35 ] ) [ and in agreement with ( [ 2 ] ) ] . moreover , as can be easily verified , the constant magnetic field @xmath117 for the invariants ( [ 18])([20 ] ) proves to be an exact solution of ( [ 31 ] ) in the entire jet up to the jet boundary , @xmath118 . here , we may therefore assume @xmath119 . consequently , according to ( [ 43 ] ) , we obtain m_0 ^ 2 = . [ 54 ] using relation ( [ 54 ] ) , we can also express all the remaining jet parameters in terms of the external magnetic field . note that the absence of a declining solution @xmath18 [ see ( [ 4 ] ) ] is associated with the first term in the right - hand part of ( [ 28 ] ) proportional to @xmath120 . this term , which changes appreciably the behavior of the solution , appears to be missed previously . as it was already emphasized above , this is not surprising because the corresponding term in the second - order equation ( [ 10 ] ) is of high order and small . on the other hand , far from the rotation axis @xmath121 , equation ( [ 28 ] ) can be rewritten as + l^2 = 0 , [ 55 ] in which both terms are of the same order . neglecting the term proportional to @xmath120 , we arrive at the solution ( [ 46 ] ) , @xmath122 , for @xmath123 and @xmath124 . the conservation of function h = [ 56 ] was first found by heyvaerts and norman @xcite for conical solutions , when all quantities depend only on spherical coordinate @xmath125 , but has also been repeatedly discussed when analyzing cylindrical flows . however , as we see , @xmath126 is generally not conserved in the cylindrical geometry for relativistic jets . to be more precise , the second term in ( [ 55 ] ) turns out to be significant for all models with a nearly constant density of the longitudinal electric current in the central jet region , where the invariant @xmath63 linearly increases with magnetic flux @xmath28 if @xmath127 . thus , we conclude that the solution with a central core ( [ 4 ] ) can not be realized in the presence of an external medium with a finite regular magnetic field . this conclusion appears to be also valid in the presence of a medium with finite pressure @xmath128 . indeed , since the magnetic flux ( [ 48 ] ) increases very slowly ( logarithmically ) , the solution ( [ 4 ] ) yields an exponentially large jet radius @xmath129 . accordingly , the magnetic energy density must also be low at @xmath130 . however , this configuration can not exist in the presence of an external medium with finite pressure @xmath128 , irrespective of whether it is produced by a magnetic field or by a plasma . we may therefore conclude that the solutions with a central core can be realized only for a special choice of the integral @xmath63 , which increases only slightly with the magnetic flux , and only in the absence of an external medium . for the most natural ( from our point of view ) models with a constant current density in the central jet regions , the solution with a central core can not be realized even in the absence of an external medium . in order to derive now the energy distribution in the jet and the particle lorentz factor , it is convenient to introduce the quantity g(x)=. [ 57 ] since at large distances @xmath79 , according to ( [ 12 ] ) , we have = , [ 58 ] @xmath131 is simply the ratio of the energy flux transferred by particles @xmath132 to the energy flux of the electromagnetic field . as a result , from relation ( [ 52 ] ) for @xmath101 , we obtain x^-11 , [ 59 ] accordingly , from ( [ 57 ] ) and ( [ 58 ] ) for the particle lorentz factor at @xmath101 , we derive @xmath133 finally , expression ( [ 13 ] ) for the @xmath134-component of the 4-velocity vector @xmath135 yields the following toroidal velocity @xmath136 at @xmath78 : @xmath137 we see that the particle energy approaches the universal asymptotic limit ( [ 60 ] ) at @xmath79 . naturally , such a simple asymptotics can also be derived from simpler considerations . indeed , using the `` frozen - in '' equation @xmath138 , we obtain for the drift velocity @xmath139 in our case , however , according to ( [ 6 ] ) and ( [ 7 ] ) , we have @xmath140 @xmath141 as a result , relations ( [ 62])([64 ] ) immediately lead to the exact asymptotics ( [ 60 ] ) . it thus follows that , for example , for electron positron jets from agns , the jet particle energy is typically @xmath142 on the other hand , according to ( [ 57 ] ) , we reach a very important conclusion that , far from the light cylinder , @xmath143 , @xmath144 consequently , according to ( [ 58 ] ) , the particle contribution to the general energy flux balance proves to be minor . for example , at @xmath145 for @xmath146 , we have @xmath147 while in the general case , we obtain @xmath148^{1/2}. \label{68}\ ] ] thus , we reach a fairly nontrivial conclusion that the fraction of energy transferred by particles in a one - dimensional jet must be determined by the parameters of the external medium . let us now discuss the results of exact calculations obtained by numerical integration of equations ( [ 25 ] ) and ( [ 28 ] ) with the integrals of motion ( [ 18])([20 ] ) and ( [ 23 ] ) . in figures [ fig_1](a ) and [ fig_1](b ) , the mach number and the energy flux concentrated in particles @xmath149 are plotted against @xmath150 for @xmath151 , @xmath152 , and @xmath153 . the dashed lines indicate the behavior of these quantities that follows from the analytical asymptotics ( [ 52 ] ) and ( [ 60 ] ) . as we see , at sufficiently small @xmath154 when the integrals of motion ( [ 18])([20 ] ) and ( [ 23 ] ) are similar to ( [ 36])([38 ] ) , the analytical asymptotics match the exact numerical results . on the other hand , as expected , @xmath149 and @xmath83 are zero at @xmath155 , i.e. , at the jet edge . figure [ fig_1](c ) shows the dependence of the poloidal field , @xmath156 , for the inner parts of the jet , @xmath157 . we see from this figure that the magnetic field is nearly constant at @xmath158 , in agreement with the analytic estimates ( [ 43 ] ) and ( [ 51 ] ) . of course , in general , the structure of the poloidal magnetic field is determined by a specific choice of the integrals @xmath54 and @xmath63 . in conclusion , it is of interest to compare the energy of jet particle with the limiting energy acquired by the particles as they outflow from the magnetosphere of a compact object with a monopole magnetic field . according to calculations by beskin et al . @xcite , the particle lorentz factor outside the fast magnetosonic surface @xmath159 in the absence of an external medium can be written as @xmath160 where @xmath161 is given by ( [ 27 ] ) . on the other hand , relations ( [ 51 ] ) and ( [ 60 ] ) for the jet yield @xmath162 as shown in figure [ fig_2](a ) , for @xmath163 , i.e. , for @xmath164 , where @xmath165 the lorentz factor of the jet particles ( [ 71 ] ) is always larger than the lorentz factor acquired by the particles as they outflow from a magnetosphere with a monopole magnetic field , but , of course , is always smaller than the critical lorentz factor @xmath166 which corresponds to the complete transformation of the electromagnetic energy into the particle energy . this implies that , at @xmath164 , the particles must be additionally accelerated during the collimation coupled with the interaction of the outflowing plasma with the external medium . if , alternatively , @xmath167 , then , in the inner jet regions , at @xmath168 the particle energy on a given field line turns out to be even lower than that for a monopole magnetic field , as shown in figure [ fig_2](b ) . the latter result can be easily explained . indeed , for the integrals of motion ( [ 36])([38 ] ) we consider , the factor @xmath169 , whose zero value determines the location of the fast mhd surface ( see @xcite for more detail ) , can be rewritten in the case of a cold plasma as @xmath170 it is easy to show that expression ( [ 76 ] ) for @xmath161 and @xmath83 given by ( [ 51 ] ) and ( [ 52 ] ) is negative at @xmath171 i.e. , at @xmath154 corresponding to ( [ 75 ] ) . consequently , we may reach another important conclusion that , for sufficiently strong external magnetic fields @xmath167 ( [ 73 ] ) when @xmath172 , a region with a subsonic flow inevitably emerges in the inner jet regions @xmath173 , where @xmath174^{3/2}r_{\rm l}. \label{78}\ ] ] at the same time , a region with the subsonic flow can be produced far from the compact object either by a shock wave or by a strong distortion of the magnetic field within the fast magnetosonic surface located in the vicinity of the compact object . in both cases , the magnetic - field perturbation causes the particle energy to decrease . thus , we conclude that the exact equilibrium equations ( [ 25 ] ) and ( [ 28 ] ) do actually allow a construction of a self - consistent model for a jet immersed in an external uniform magnetic field . the advantage of these equations over equation ( [ 10 ] ) results from the fact that all terms in ( [ 28 ] ) are of the same order . in this case , the uniformity of the poloidal magnetic field within the jet ( [ 43 ] ) results from the choice of integrals ( [ 36])([38 ] ) . in general , the poloidal magnetic field depends on the specific form of the integrals . a full analysis of the possible solutions is beyond the scope of this paper . we have shown that the fraction of energy transferred by particles @xmath175 must be largely determined by the parameters of the external medium . in case @xmath176 , where @xmath177^{1/2 } , \label{79}\ ] ] the energy transferred by particles is only a small fraction of the energy flux @xmath178 transferred by the electromagnetic field . consequently , the jet is strongly magnetized ( @xmath179 ) only at sufficiently large @xmath53 . if , however , the magnetization parameter does not exceed @xmath180 , then , in this case , an appreciable part of the energy in the jet is transferred by particles . this , in turn , implies that a considerable part of the energy must be transferred from the electromagnetic field to the plasma particles during jet collimation . it is interesting that @xmath180 turns out to be approximately the same both for agns and for fast radio pulsars : _ cr 10 ^ 5 - 10 ^ 6 . [ 80 ] we have shown that the central part of the jet must be subsonic for sufficiently strong external magnetic fields . thus , the theory gives direct predictions whose validity can be verified by observations . it should also be noted that the results obtained above are applicable both to electron positron and to electron proton jets . however , a direct evidence that the jets in agns are actually electron positron ones has recently appeared @xcite . in our view , an important result is that , if the external regular magnetic field is taken into account , the mhd equations allow a self - consistent model to be constructed for a jet with a zero total longitudinal electric current , @xmath181 . in this case , a uniform magnetic field that matches the external magnetic field can also be a solution for the inner jet regions . as was already emphasized above , the radii of the jets from agns can thus be also explained in a natural way . in addition , since only a small fraction of the electromagnetic - field energy is transformed into the particle energy , the energy transfer from the compact object in the region of energy release can be explained as well . at the same time , extending the mhd solution to the jet region requires very high particle energies ( @xmath182 ) , which have not been recorded yet . however , a consistent discussion of the outflowing - plasma energy requires a proper allowance for the particle interaction with the surrounding medium ( for example , with background radiation ) , which may cause a significant change in particle energy . as for the quantitative predictions about the real physical parameters of jets , they , as we showed above , essentially depend only on the following three quantities : the magnetization parameter @xmath53 ( [ 17 ] ) , the lorentz factor @xmath60 in the generation region , and the external magnetic field @xmath183 . in this case , the main uncertainty for electron positron jets from agns ( @xmath9 , @xmath184 ) is the value of the magnetization parameter . indeed , this quantity depends on the efficiency of pair production in the magnetosphere of a black hole , which , in turn , is determined by the density of hard gamma - ray photons . as a result , if the density of hard gamma - ray photons with energies @xmath185 near the black hole is high enough , then the particles will be produced by direct collisions of photons @xmath186 @xcite . this causes an abrupt increase in the multiplicity parameter @xmath187@xmath188 , where @xmath189 is the characteristic particle density required to shield the longitudinal electric field . using the well - known estimate ( see , e.g. , @xcite ) @xmath190 we obtain @xmath191 on the other hand , for low densities of gamma - ray photons when an electron positron plasma can be produced only in regions with a nonzero longitudinal electric field , which are equivalent to the outer gaps in the magnetospheres of radio pulsars @xcite , the multiplicity of the particle production is fairly small : @xmath192@xmath193 . in this case , we obtain @xmath194 finally , for fast crab- or vela - type radio pulsars ( @xmath195 , polar - cap radius @xmath196 , and @xmath197 ) in which jets are observed , we have @xcite @xmath198 relations ( [ 82 ] ) and ( [ 83 ] ) show that the properties of the jets from agns considerably depend on the magnetization parameter @xmath53 . for example , according to ( [ 79 ] ) , the jet particles for sources with large @xmath199 transfer only a small fraction of the energy compared to the electromagnetic flux , so the flow within the jet differs only slightly from the force - free flow . in addition , in this case the external magnetic field @xmath200 exceeds the critical magnetic field @xmath201 . according to ( [ 43 ] ) and ( [ 73 ] ) , this implies that a subsonic region must exist in the inner regions of such jets . on the other hand , a substantial part of the energy in sources with magnetization parameter @xmath202 during jet collimation must be transferred by plasma particles , and no subsonic region is formed near the rotation axis . as for the fast radio pulsars , the condition @xmath203 is satisfied for them , so a subsonic region in the central parts of pulsar jets is not achieved either . on the other hand , the estimate ( [ 79 ] ) shows that an appreciable part of the jet total energy must be coupled with particles . we emphasize that , since the jet radius ( [ 53 ] ) for agns always exceeds the light - cylinder radius by several orders of magnitude , @xmath204 the toroidal magnetic field @xmath205 within the jet must exceed the poloidal magnetic field @xmath206 in the same proportion , @xmath207 consequently , detection of such a strong toroidal component would be a crucial argument for the picture discussed here . unfortunately , determination of the actual physical conditions in jets currently involves considerable difficulties . nevertheless , not only data on the direct detection of such a structure @xcite but also evidence for the existence of magnetic fields @xmath208 , closely matching the estimate ( [ 86 ] ) @xcite , have recently appeared . we wish to thank a.v . gurevich for interest in this study , a useful discussion , and support . we also wish to thank s.v . bogovalov , l.i . gurvits , and s.a . lamzin for fruitful discussions . this study was supported in part by the intas grant no . 96154 and the russian foundation for basic research ( project number 99 - 02 - 17184 ) . l. malyshkin is also grateful to the international science foundation .

a magnetohydrodynamic model is constructed for a cylindrical jet immersed in an external uniform magnetic field . it is shown that , as in the force - free case , the total electric current within the jet can be zero . the particle energetics and the magnetic field structure are determined in a self - consistent way ; all jet parameters depend on the physical conditions in the external medium . in particular , we show that a region with subsonic flow can exist in the central jet regions . in actual relativistic jets , most of the energy is transferred by the electromagnetic field only when the magnetization parameter is sufficiently large , @xmath0 . we also show that , in general , the well - known solution with a central core , @xmath1 , can not be realized in the presence of an external medium . = -2.0 cm = -3.0 cm = 17.5 cm

You are an expert at summarizing long articles. Proceed to summarize the following text: after the discovery of the mssbauer effect in 1958 , quantitative measurements of relativistic time dilation were carried out in the 1960s based on this effect @xcite-@xcite , and the interest in such measurements lasts to this day @xcite,@xcite . the experiments @xcite-@xcite reported full agreement with the time dilation predicted by einstein s theory of relativity . in the experiments @xcite-@xcite , the mssbauer source was placed at the center of a fast rotating disk and an absorber at the rim of the disk . in the analyses of these experiments , it was assumed that the absorption line of the rotating absorber stays lorentzian with same width as at rest , and is shifted only by the time dilation factor . based on the generalized principle of relativity and the ensuing symmetry , in @xcite it was shown that there are only two possible types of transformations between uniformly accelerated systems . the validity of the _ clock hypothesis _ is crucial to determining which one of the two types of transformations is obtained . the clock hypothesis , as stated in @xcite , maintains that the rate of an accelerated clock is equal to that of a co - moving unaccelerated clock . if the clock hypothesis is not true , then the transformation is of lorentz - type and implies the existence of a universal maximal acceleration @xmath1 and an additional time dilation due to the acceleration of the clock . by _ acceleration _ , we mean the proper acceleration defined @xcite as @xmath2 , where @xmath3 is the proper time . in this case , it was shown in @xcite that a doppler type shift for an accelerated source will be observed . this doppler type shift is similar to the doppler shift due to the velocity of the source . the formulas for this shift are the same as those for the velocity doppler shift , with @xmath4 replaced by @xmath5 . consider an absorber placed on a disk , rotating with angular velocity @xmath6 at the distance @xmath7 from the center . if this absorber is exposed to radiation of frequency @xmath8 through the center of the disk , then , since the velocity of the absorber is perpendicular to the radiation direction , the radiation will undergo a transverse doppler shift @xmath9 due to the time dilation of the absorber . if the conjecture about the existence of a shift due to acceleration is true , there will be an additional shift @xmath10 , which is longitudinal , since the acceleration is in the direction of the radiation . kndig s experiment @xcite measured the transverse doppler shift for a rotating disk by means of the mssbauer effect . in ( only ) this experiment , the absorption line of the rotating absorber was obtained . this experiment , as reanalyzed by kholmetskii _ et al _ @xcite , showed a significant deviation of about @xmath11 of the shift observed from the one predicted by special relativity . this deviation was explained in @xcite by the additional shift due to the acceleration . moreover , the value of the maximal acceleration @xmath1 was estimated to be about @xmath12 . this value of the maximal acceleration implies that the ratio of the new shift ( due to acceleration ) to the transversal one due to the velocity is @xmath13 where @xmath7 is the distance ( @xmath14 in @xcite ) of the absorber from the center of the disk . kndig s experiment was not designed to test the acceleration shift , and the recalculation by kholmetskii _ et al _ is not direct . thus , the corrected kndig s result may serve only as an indication of the existence of a maximal acceleration and an estimate of its value . note that in other time dilation experiments , the value of @xmath7 was about @xmath15 , implying that the ratio of the new shift to the known one is @xmath16 . how was such a deviation not observed in @xcite-@xcite ? the reason for this is that their analyses did not take into account the broadening of the absorption curves during the rotation , as is explained below . we show here that the absorption line of a rotating mssbauer absorber gets broader during the rotation , even in the case when the absorber moves in the perpendicular direction to the radiation . this is due to the fact that the slit for the radiation beam leaving the source and the opening slit of the detector must have a finite width . therefore , the velocity of the absorber is not perpendicular to all individual rays . hence , these rays undergo a longitudinal doppler shift in addition to the expected transversal one . this shift is very significant even for very small slits for the radiation beam and for detector , and leads to broadened non - lorentzian absorption lines . choose the direction of the radiation beam to be in the direction of the @xmath17-axis , and place the origin at center of the source and the detector at @xmath18 . denote by @xmath19 the velocity of the absorber . in order for the transversal doppler shift @xmath20 to be observed , the velocity must be at least @xmath21 a ray leaves the source at the point @xmath22 and enters the detector at a point @xmath23 . the component of the velocity of the absorber in the direction @xmath24 of the ray , for @xmath25 , is @xmath26 , which may be significant even for very small width of the slits ( 2d in figure 1 ) . this implies that most rays will also be exposed to a longtitudal doppler shift and will change completely the shape of the mssbauer absorption line . one should observe a decrease in the absorption amplitude and a non - lorentzian broadening of the absorption spectrum . we choose again the direction of the radiation beam to be in the direction of the @xmath17-axis . we place the origin at the center of the rotating disk ( @xmath27 ) . the radiation leaves the source at the point @xmath28 and enters the detector at a point @xmath23 . the equation of the radiation line is @xmath29 and has the direction @xmath30 , and intersects the @xmath31-axis at the point @xmath32 , where @xmath33 and @xmath34 obviously may have any value between @xmath35 and @xmath36 . denote the distance of the absorber to the center by @xmath7 , and denote the intersection point of our ray with the absorber by @xmath37 . since this point is on our radiation line , it satisfies the equation @xmath38 the velocity of the absorber at @xmath39 is @xmath40 , where @xmath6 is the angular velocity of the disk . thus , from equation ( [ absorbpos ] ) , we get that the component of the velocity of the absorber in the direction of the ray is @xmath41 which , for @xmath25 , is approximately equal to @xmath42 , and even then the velocity @xmath43 is extremely large for @xmath44 and @xmath45 , in comparison to the natural full width at half absorption intensity ( @xmath46 ) of the @xmath47 mssbauer absorption single line spectrum . a typical absorption curve is obtained by placing the source on a transducer and moving it with a velocity @xmath48 in the direction of the @xmath17-axis . this velocity is of order up to several millimeters per second . for such velocities , we may assume that the source velocity in the radiation direction @xmath24 is approximately equal @xmath49 . the absorption curve @xmath50 of an absorber rotating with angular velocity @xmath6 describes the rate of the radiation absorbed as a function of the velocity @xmath49 of the source . for a thin absorber with no chemical shift , a typical absorption curve for an absorber at rest is a lorentzian function @xmath51 , of half width @xmath0 , absorption amplitude @xmath52 , and resonance at @xmath53 , as the one shown in figure 3a for @xmath54 . for a rotating absorber , the total doppler shift that the ray undergoes consists of a longitudinal shift due to the relative velocity of the source and the absorber and a transversal shift due to the time dilation of the absorber . all together are equivalent to a longitudinal shift of a source with velocity @xmath55 since the spread in @xmath43 is increasing with angular velocity , it contributes significantly to the spread of the observed absorption mssbauer line . using ( [ vn ] ) and ( [ tildets ] ) , the absorption curve of the rotating absorber will be @xmath56 where the integral has to be taken over the source emission intensity at @xmath57 and absorption by detector at @xmath58 . this curve can be calculated analytically , if we assume that the distributions along @xmath57 and @xmath58 are uniform on interval @xmath59 $ ] , @xmath25 and @xmath60 . under these assumptions @xmath61 . denoting @xmath62 and defining a function @xmath63 such that @xmath64 , we get @xmath65@xmath66 thus , @xmath67 if @xmath68 and @xmath69 , implying that the absorption curve of the rotating absorber will be @xmath70 , which is a shift of the absorption curve of the non - rotating absorber . this was assumed in the experiments @xcite-@xcite . it is obvious that @xmath71 is symmetric and @xmath72 is symmetric with respect to @xmath73 . the results of the calculated absorption spectra using equation ( [ absorrotatingabs ] ) , using numerical integration , or using the analytic function ( [ aomega ] ) yield the same results , shown as transmission rate spectra line @xmath74 in figure 3 . the simulations show that even for small @xmath75 , if @xmath6 becomes large enough the absorption line broadens drastically and even becomes unobservable . these curves have the same type of widening and shift as the absorption curves obtained experimentally in @xcite . in the experiments @xcite-@xcite,@xcite , the value of @xmath76 was measured only in the case in which the source and the absorber were relatively static , corresponding to @xmath53 . obviously in their experiments @xmath77 , ( they do not specify ) , and their analysis did not consider the changes in shape of the absorption line as a function of @xmath6 . thus we can not rely on their conclusions . if the distributions of source emission intensity is @xmath57 dependent and the detection is @xmath58 dependent , but symmetric in the interval @xmath59 $ ] , one will still observe under the rotation a symmetric broadening . if the distribution is not symmetric and has non - zero average , we will get an additional shift , but such a shift can be compensated for by reversing the direction of the rotation . * conclusions : * in order to measure the transverse doppler shift by conventional msbauer spectroscopy with source mounted on a static transducer and the absorber moving on a fast rotating disc , one should take care of the following : ( a ) : : one must measure the full spectrum of the absorption line . ( b ) : : one must measure this spectrum in both angular directions of the absorber . but most important ; ( c ) : : one must put collimators on source and detector to reduce the slits to minimal dimensions , which will still allow a spectrum measurement in a reasonable time span . only the experiment of w. kndig @xcite obeyed at least ( a ) and ( b ) . in his experiment he used a source of @xmath78 against a @xmath47 enriched iron absorber . thus both were almost identical and in both the iron nuclei were exposed to a strong magnetic field ( 33 t ) causing the emission and absorption spectra to be composed of six line patterns , which lead to observed multiline spectra as shown in fig.4 ( bottom ) , for various rotational frequencies of the absorber . since kndig scanned the spectra in the range of about -1 to + 1 mm / s , he observed only the central line with reduced intensity and line broadening , as the lines simulated in fig.4 ( middle and top ) . thus he actually observed the line broadening due to the finite slits , but interpreted them as due to rotor vibrations affecting the absorber . nevertheless this experiment seems to be still the most accurate transverse doppler shift experiment using the @xmath47 mssbauer effect . his results with the corrected analysis reported in @xcite indicates that the observed shift in this experiment is larger and deviates @xmath79 from that expected by special relativity theory . einstein time dilation formula was verified by several experiments @xcite- . in w. kndig experiment the absorber was exposed to a significant acceleration . in @xcite it was indicated that the deviation of the shift in the experiment may be due to a longitudinal doppler shift caused by the acceleration of the absorber . it led to an estimate of a universal maximal acceleration of order @xmath12 and an indication of non - validity of the einstein clock hypothesis @xcite . for a long time , b. mashhoon argued against the clock hypothesis and developed nonlocal transformations for accelerated observers ( see the review article @xcite ) . note that time dilation mssbauer effect experiments have an advantage in identifying relatively small shift due to acceleration , since in these experiments the shift due to the velocity is of second order while the shift due the acceleration is of first order . thus we recommend mssbauer spectroscopy scientists to perform an accurate experiment measuring the shift of the absorption line for a fast rotating absorber which may reveal a new fundamental law @xcite , with a monumental effect on whole of physics . h. j. hay _ et al _ , phys . lett . * 4 * ( 1960 ) 165 h. j. hay , proc . 2nd conf . on the mssbauer effect , ed a. schoen and d. m. t. compton , wiley , ( 1962 ) 225 t. e. cranshaw , j. p. schiffer and p. a. egelstaff , phys . lett . * 4 * ( 1960 ) 163 t. e. granshaw and h. j. hay , proc . school of physics , enrico fermi " academic press , ( 1963 ) 220 d.c . champeney and p. b. moon , proc . * 77 * ( 1961 ) 350 d.c . champeney , g. r. isaak and a. m. khan , proc . * 85 * ( 1965 ) 583 w. kndig , phys * 129 * ( 1963 ) 2371 a.l . kholmetski , t. yarman and o.v . missevitch , physica scripta * 77 * ( 2008 ) 035302 a. l. kholmetskii , t. yarman , o. v. missevitch and b. i. rogozeva physica scripta * 79 * ( 2009 ) 065007 y. friedman , yu . gofman , physica scripta , * 82 * ( 2010 ) 015004 . a. einstein , ann . * 35 * ( 1911 ) 898 w. rindler , _ relativity : special , general and cosmological _ , oxford university press ( 2004 ) y. friedman , ann . phys . ( berlin ) * 523 * ( 2011 ) 408 j. bailey et al . , nature * 268 * ( 1977 ) 301 r. w. mcgowan , d. m. giltner , s. j. stelnberg and s. au lee , phys . letters * 70 * ( 1993 ) 251 s. reinhardt et al . , nature physics * 3 * ( 2007 ) 861 c. lmmerzah , nature physics * 3*(2007 ) 831 b. mashhoon , annalen phys . * 17*(2008 ) 705 - 727 .

the einstein time dilation formula was tested in several experiments . many trials have been made to measure the transverse second order doppler shift by mssbauer spectroscopy using a rotating absorber , to test the validity of this formula . such experiments are also able to test if the time dilation depends only on the velocity of the absorber , as assumed by einstein s clock hypothesis , or the present centripetal acceleration contributes to the time dilation . we show here that the fact that the experiment requires @xmath0-ray emission and detection slits of finite size , the absorption line is broadened ; by geometric longitudinal first order doppler shifts immensely . moreover , the absorption line is non - lorenzian . we obtain an explicit expression for the absorption line for any angular velocity of the absorber . the analysis of the experimental results , in all previous experiments which did not observe the full absorption line itself , were wrong and the conclusions doubtful . the only proper experiment was done by kndig ( phys . rev . 129 ( 1963 ) 2371 ) , who observed the broadening , but associated it to random vibrations of the absorber . we establish necessary conditions for the successful measurement of a transverse second order doppler shift by mssbauer spectroscopy . we indicate how the results of such an experiment can be used to verify the existence of a doppler shift due to acceleration and to test the validity of einstein s clock hypothesis . _ pacs _ : 76.80.+y , 03.30.+p _ keywords _ : mssbauer effect ; absorption line ; time dilation experiments ; transverse doppler shift ; einstein s clock hypothesis

You are an expert at summarizing long articles. Proceed to summarize the following text: a complementary astrophysical data from type ia supernova , large scale structure ( lss ) and cosmic microwave background ( cmb ) indicate that our universe is currently undergoing a phase of accelerating @xcite . a component which is responsible for this accelerated expansion is usually dubbed `` dark energy '' ( de ) . the simplest candidate for de is the cosmological constant @xcite which is located in the center both from theoretical and observational evidences . however , there are different alternative theories for the dynamical de scenarios which have been proposed to interpret the accelerating universe . one of these models , which has arisen a lot of enthusiasm recently , is hde . it was shown by cohen et al . @xcite that in quantum field theory a short distance cutoff could be related to a long distance cutoff ( ir cutoff l ) due to the limit sets by black hole formation . if the quantum zero - point energy density is due to a short distance cutoff , then the total energy in a region of size @xmath13 should not exceed the mass of a black hole of the same size , namely @xmath14 . the largest @xmath13 is the one saturating this inequality , so that we obtain the hde density as @xcite @xmath15 where @xmath16 is the reduced planck mass . in general the factor @xmath17 in holographic energy density can vary with time very slowly @xcite . by slowly varying we mean that @xmath18 is upper bounded by the hubble expansion rate , @xmath19 , i.e. , @xcite @xmath20 it was also argued that @xmath17 depends on the ir length , @xmath13 @xcite . for the case of @xmath21 , one can take @xmath17 approximately constant in the late time where de dominates ( @xmath22 ) @xcite . since in the present work we consider the late time cosmology where the de dominates , we shall assume the factor @xmath17 to be constant . depending on the ir cutoff , @xmath13 , many hde models have been proposed in the literatures . a comprehensive , but not complete , list of ir cutoffs which have been used includes the particle horizon radius @xmath23 @xcite , the hubble horizon @xmath21 @xcite , the future event horizon radius @xmath24 @xcite , the apparent horizon radius @xmath25 @xcite , the ricci scalar curvature radius @xmath26 @xcite , the so - called granda - oliveros ( go ) cutoff , which is the formal generalization of @xmath27 , namely @xmath28 @xcite , the age of our universe @xmath29 @xcite , the conformal age of our universe @xmath30 @xcite , the radius of the cosmic null hypersurface @xcite , etc . on the other side , in recent years , the theories of large extra dimensions in which the observed universe is realized as a brane embedded in a higher dimensional spacetime , have received a lot of interest . in these theories the cosmological evolution on the brane is described by an effective friedmann equation that incorporates non - trivially with the effects of the bulk onto the brane . one of the well - know picture in the braneworld scenarios was proposed by dvali - gabadadze - porrati ( dgp ) @xcite . in this model our four - dimensional universe is a friedmann - robertson - walker ( frw ) brane embedded in a five - dimensional minkowskian bulk with infinite size . in this model the recovery of the usual gravitational laws on the brane is obtained by adding an einstein - hilbert term to the action of the brane computed with the brane intrinsic curvature . the self - accelerating branch of dgp model can explain the late time cosmic speed - up without recourse to de or other components of energy @xcite . however , the self - accelerating dgp branch has ghost instabilities and it can not realize phantom divide crossing by itself . to realize phantom divide crossing it is necessary to add at least a component of energy on the brane . on the other hand , the normal dgp branch can not explain acceleration but it has the potential to realize a phantom - like phase by dynamical screening on the brane . in the present work we consider the hde model in the framework of dgp braneword with go cutoff , @xmath31 , proposed in @xcite . our work differs from @xcite in that , they studied hde model with go cutoff in standard cosmology , while we investigate this model in the framework of dgp braneword and incorporate the effect of the extra dimension on the evolution of the cosmological parameters on the brane . the main difference between the hde with go cutoff in the framework of dgp braneword , with the one considered in standard cosmology @xcite , is that the equation of state parameter of the hde with go cutoff in standard cosmology is a constant @xcite , namely @xmath32 however , as we shall see in dgp braneworld , due to the bulk effects , @xmath33 becomes a time variable parameter . clearly , a time variable de is more compatible with observations . in particular , the analysis of data from wmap9 or planck-@xmath34 results on cmb anisotropy , bao distance ratios from recent galaxy surveys , magnitude - redshift relations for distant sne ia from snls3 and union2.1 samples indicate that the time varying de gives a better fit than a cosmological constant @xcite . in addition , current data still slightly favor the quintom de scenario with eos across the cosmological constant boundary @xmath35 @xcite . furthermore , we are able to derive explicitly , the cosmological parameters on the brane as functions of redshift parameter and provide a profile of the cosmic evolution on the brane . indeed , this is the advantages of the present model in compared to all other hde models . as far as we know , this is the first model of hde which leads to analytical solution , @xmath36 and @xmath5 . then , we establish the correspondence between our model and scalar field models of de and reconstruct the evolution of scalar fields and potentials . finally , we investigate the stability of this model against perturbation in different cases . we find that for some range of the parameter spaces our model is stable which indicates the viability of this model for explanation of the late time acceleration . we consider a homogeneous and isotropic frw universe on the brane which is described by the line element @xmath37 where @xmath38 represent a flat , closed and open maximally symmetric space on the brane , respectively . the modified friedmann equation in dgp braneworld model is given by @xcite @xmath39 where @xmath40 corresponds to the two branches of solutions @xcite , and @xmath41 stands for crossover length scale between the small and large distances in dgp braneworld defined as @xcite @xmath42 we also assume that there is no energy exchange between the brane and the bulk and so the energy conservation equation holds on the brane , @xmath43 for @xmath44 , the friedmann equation in standard cosmology is recovered @xmath45 recent observations indicate that our universe is spatially flat . for a flat frw universe on the brane , eq . ( [ friedeq01 ] ) reduces to @xmath46 depending on the sign of @xmath47 , there are two different branches for the dgp model . for @xmath48 and in the absence of any kind of energy or matter field on the brane ( @xmath49 ) , there is a de - sitter solution for eq . ( [ friedeq02 ] ) with constant hubble parameter @xmath50 clearly , eq . ( [ de sitter ] ) leads to an accelerating universe with constant equation of state parameter @xmath51 , exactly like the cosmological constant . however , there are some unsatisfactory problems with this solution . first of all , it suffers the well - known cosmological constant problems namely , the fine - tuning and the coincidence problems . besides , it leads to a constant @xmath52 , while many cosmological evidences , especially the analysis of the type ia supernova data indicates that the time varying de gives a better fit than a cosmological constant @xcite . most of these data favor the evolution of the equation of state parameter and in particular it can have a transition from @xmath53 to @xmath54 at recent stage . although some evidence such as the galaxy cluster gas mass fraction data do not support the time - varying @xmath52 @xcite , an overwhelming flood of papers has appeared to understand the @xmath51 crossing in the past decade @xcite . in addition , to arrive at eq . ( [ de sitter ] ) one ignores all parts of energy on the brane including de , dark matter and byronic matter , which is not a reasonable assumption . on the other hand , for @xmath55 and @xmath56 , one can neglect the term @xmath57 in eq . ( [ friedeq02 ] ) to arrive at @xmath58 this is the friedmann equation in spatially flat rs ii braneworld @xcite . clearly , eq . ( [ rs ] ) does not have a self accelerating solution , so it implies the requirement of some kind of de on the brane . in the present paper , we consider the hde model in flat dgp braneworld with go cutoff , which is defined as @xcite @xmath59 with this ir cutoff , the energy density ( [ hde ] ) can be written @xmath60 where @xmath61 and @xmath62 are constants which should be constrained by the recent observational data and we have absorbed the constant @xmath17 in @xmath61 and @xmath62 . hereafter , we work in a unit in which @xmath63 . we also restrict our study to the current cosmological epoch , and hence we are not considering the contributions from matter and radiation by assuming that the dark energy @xmath64 dominates , thus the friedman equation becomes simpler . substituting eq . ( [ rho1 ] ) into friedmann equation ( [ friedeq02 ] ) we can obtain the differential equation for the hubble parameter as + @xmath65 solving this equation , the hubble parameter is obtained as @xmath66 where @xmath67 is constant of integration . since for @xmath68 , the effects of the extra dimension should be disappeared and the result of @xcite , namely @xmath69 must be restored , thus the constant @xmath67 should be chosen as @xmath70 substituting @xmath67 in eq . ( [ hubble1 ] ) , we obtain @xmath71 taking the time derivative of eq . ( [ hubble2 ] ) yields @xmath72 we can also solve eq . ( [ hubble2 ] ) to obtain the scale factor . we find @xmath73 using the fact that @xmath74 , where @xmath3 is the redshift parameter , and combining eqs . ( [ hubble2 ] ) and ( [ scale ] ) we find explicitly the hubble parameter as a function of @xmath3 , @xmath75}{(1+z)^{\frac{1-\alpha}{\beta}}}\nonumber \\ & = & \frac{\epsilon}{r_{\rm c}(1-\alpha)}\left[1+(1+z)^{\frac{\alpha-1}{\beta}}\right].\end{aligned}\ ] ] this equation is valid in two cases . first , for @xmath48 and @xmath10 , and second for @xmath55 and @xmath8 , and has no solution for @xmath76 . inserting eqs . ( [ hubble2 ] ) and ( [ hubbledot ] ) in ( [ rho1 ] ) , we get @xmath77 substituting eqs . ( [ hubble2 ] ) and ( [ hubbledot ] ) in the time derivative of eq . ( [ rho1 ] ) , @xmath78 we arrive at @xmath79.\ ] ] combining eqs . ( [ hubble2 ] ) , ( [ rho ] ) and ( [ rhodot ] ) with ( [ conserveq ] ) , we find the equation of state parameter of hde on the brane as a function of time , @xmath80.\ ] ] in order to see the evolution of @xmath52 during the history of the universe , it is better to express it as a function of the redshift parameter @xmath3 , @xmath81.\ ] ] versus redshift parameter @xmath3 for different model parameters . left panel corresponds to @xmath10 and @xmath11 , while the right panel shows the case @xmath8 and @xmath9 . , title="fig:",width=302 ] versus redshift parameter @xmath3 for different model parameters . left panel corresponds to @xmath10 and @xmath11 , while the right panel shows the case @xmath8 and @xmath9 . , title="fig:",width=302 ] to check the limit of ( [ wdz ] ) in standard cosmology where @xmath82 , we note that , from eq . ( [ scale ] ) and relation @xmath83 we have @xmath84 , as @xmath82 . therefore , ( [ wdz ] ) reduces to @xmath85 which is exactly the result obtained in @xcite . it is worth mentioning that unlike in standard cosmology , where the equation of state parameter of hde with go cutoff becomes a constant @xcite , in dgp braneworld scenario @xmath52 varies with time . thus , it seems that the presence of the extra dimension brings rich physics . for @xmath76 we have @xmath51 , similar to the cosmological constant . let us consider the case with @xmath10 and @xmath8 separately . in the first case where @xmath10 , we find that the equation of state parameter can explain the acceleration of the universe provided @xmath11 . in the second case where @xmath8 , the accelerated expansion can be achieved for @xmath9 . in both cases , our universe has a transition from deceleration to the acceleration phase around @xmath86 , compatible with the recent observations @xcite , and mimics the cosmological constant at the late time . these results can be easily seen in fig . [ w ] which shows the behaviour of @xmath52 versus @xmath3 for both cases . the first and second derivatives of the distance can be combined to obtain the deceleration parameter @xmath2 . it was shown that the zero redshift value of @xmath87 , is independent of space curvature , and can be obtained from the first and second derivatives of the coordinate distance @xcite . it was argued that @xmath87 , which indicates whether the universe is accelerating at the current epoch , can be obtained directly from the supernova and radio galaxy data @xcite . the deceleration parameter is given by @xmath88 using eqs . ( [ hubble2 ] ) and ( [ hubbledot ] ) , the deceleration parameter is obtained as @xmath89}.\ ] ] for @xmath76 we have @xmath90 . although , the equation of state and the deceleration parameters do not depend explicitly on the crossover length scale @xmath41 which is the characterization of the dgp branworld , they depend on @xmath41 via the relation between the scale factor and the redshift parameter as in ( [ scale ] ) . we have plotted the behavior of the deceleration parameter versus @xmath3 in figure [ q ] . form this figures , we see that the transition from deceleration phase to the acceleration phase can be occurred around @xmath86 , which is consistent with the recent observations @xcite . the zero - redshift value of the deceleration parameter is obtained as @xmath91 recent cosmological data from the combined sample of @xmath92 supernovae and @xmath93 radio galaxies , show that @xmath94 , for a window function of width @xmath95 in redshift and a transition from deceleration to the acceleration phase at redshift @xmath96 @xcite . the zero - redshift value ( [ q2 ] ) , for @xmath97 and @xmath98 , leads to @xmath99 , while for @xmath100 and @xmath101 , we get @xmath102 , which is consistent with the observations @xcite . versus redshift parameter @xmath3 for different model parameters . left panel corresponds to @xmath10 and @xmath11 , while the right panel shows the case @xmath8 and @xmath9 . , title="fig:",width=302 ] versus redshift parameter @xmath3 for different model parameters . left panel corresponds to @xmath10 and @xmath11 , while the right panel shows the case @xmath8 and @xmath9 . , title="fig:",width=302 ] the dynamical de proposal is often realized by some scalar field mechanism which suggests that the energy form with negative pressure is provided by a scalar field evolving down a proper potential . scalar fields naturally arise in particle physics including supersymmetric field theories and string / m theory . therefore , scalar field is expected to reveal the dynamical mechanism and the nature of de . however , although fundamental theories such as string / m theory do provide a number of possible candidates for scalar fields , they do not predict its potential , @xmath103 , uniquely . consequently , it is meaningful to reconstruct the potential @xmath103 from some de models possessing some significant features of the quantum gravity theory , such as hde models . famous examples of scalar field de models include quintessence @xcite , k - essence @xcite , tachyon @xcite , phantom @xcite , ghost condensate @xcite , quintom @xcite , and so forth . for a comprehensive review on scalar filed models of dark energy , see @xcite . generically , there are two points of view on the scalar field models of dynamical de . one viewpoint regards the scalar field as a fundamental field of the nature . the nature of de is , according to this viewpoint , completely attributed to some fundamental scalar field which is omnipresent in supersymmetric field theories and in string / m theory . the other viewpoint supports that the scalar field model is an effective description of an underlying theory of de . if we regard the scalar field model as an effective description of such a theory , we should be capable of using the scalar field model to mimic the evolving behavior of the hde and reconstructing the scalar field model according to the evolutionary behavior of hde . in this section we implement a correspondence between hde in dgp braneworld with go cutoff and various scalar field models , by equating the equation of state parameters for these models with the obtained equation of state parameter of eq . ( [ wdz ] ) . let us start with reconstructing the potential and dynamics of quintessence scaler field . the energy density and pressure of the quintessence scalar field are given by @xmath104 thus the potential and the kinetic energy term can be written as @xmath105 where @xmath106 . in order to implement the correspondence between hde and quintessence scaler field , we identify @xmath107 and @xmath108 . using eqs . ( [ wdz ] ) , ( [ vphi ] ) and ( [ dotphi ] ) , we find @xmath109 @xmath110 + \frac{1-\alpha}{3\beta}\left((1+\alpha)(1+z)^{\frac{1-\alpha}{\beta}}+2\right)\big].\ ] ] eq . ( [ phidotq ] ) can also be rewritten @xmath111 using relation @xmath112 one can also obtain @xmath113 integrating , yields @xmath114 for different parameters . in the right panel the behavior of the quintessence scalar field @xmath115 for hde model in dgp braneworld is illustrated for different parameters.,title="fig:",width=302 ] for different parameters . in the right panel the behavior of the quintessence scalar field @xmath115 for hde model in dgp braneworld is illustrated for different parameters.,title="fig:",width=302 ] therefore , we have established holographic quintessence de model and reconstructed the potential and the dynamics of scalar field according to evolutionary behaviour of hde on the brane . theoretically , one may expect to omit the redshift @xmath3 from ( [ vphiq ] ) and ( [ phizq ] ) to derive the potential as a function of the scalar field , namely , @xmath103 . however , due to the complexity of the equations , the analytical form of the potential @xmath116 is hard to be derived . nevertheless , we can plot the reconstructed potential as a function of @xmath117 numerically . the reconstructed quintessence potential @xmath103 and the evolutionary form of the field are plotted in figure [ quint ] , where we have taken the zero redshift value of the scalar field equal to zero , namely , @xmath118 . selected curves are plotted for @xmath10 and the different model parameter @xmath11 . we can also plot the figures for the case @xmath8 and @xmath9 , however , the behaviour is the same as the former case and for the economic reason we have not plotted here the latter case . from these figures we can see the dynamics of the potential as well as the scalar field explicitly . figure [ quint ] shows that the reconstructed quintessence potential is steeper in the early epoch and becomes flat near the present time . in other words , it mimics a cosmological constant at the the present time . the tachyon field is another candidate for de . the equation of state parameter of the rolling tachyon smoothly interpolates between @xmath119 and @xmath120 @xcite . thus , tachyon can be realized as a suitable candidate for the inflation at high energy @xcite as well as a source of de depending on the form of the tachyon potential @xcite . choosing different self - interacting potentials in the tachyon field model lead to different consequences for the resulting de model . due to all the above reasons , the reconstruction of tachyon potential @xmath103 is of great importance . the correspondence between tachyon field and various de scenarios such as hde @xcite and agegraphic de @xcite has been already established . the study has also been generalized to the entropy corrected holographic and agegraphic de models @xcite . the tachyon condensates in a class of string theories and can be described by an effective scalar field with a lagrangian of the form by@xcite @xmath121 where @xmath103 is the tachyon potential . the corresponding energy momentum tensor for the tachyon field can be written in a perfect fluid form @xmath122 where @xmath123 and @xmath124 are , respectively , the energy density and pressure of the tachyon , and the velocity @xmath125 is @xmath126 the energy density and pressure of tachyon field are given by @xmath127 thus the equation of state parameter of tachyon field is given by @xmath128 to establish the correspondence between hde and tachyon field , we equate @xmath52 with @xmath129 . combining eqs . ( [ wdz ] ) , ( [ rhot ] ) and ( [ pt ] ) , we find @xmath130}\frac{(1+\alpha(1+z)^{\frac{1-\alpha}{\beta}})(1+(1+z)^{\frac{1-\alpha}{\beta}})}{(1+z)^{\frac{2(1-\alpha)}{\beta}}},\\ \dot{\phi}^2(z)&=&\frac{\alpha-1}{3\beta } \left[\frac{(1+\alpha)(1+z)^{\frac{1-\alpha}{\beta}}+2}{(1+(1+z)^{\frac{1-\alpha}{\beta}})(1+\alpha ( 1+z)^{\frac{1-\alpha}{\beta}})}\right].\label{phit1}\end{aligned}\ ] ] we can further rewrite ( [ phit1 ] ) as @xmath131^{\frac{1}{2}}.\ ] ] using the fact that @xmath132 we have @xmath133^{\frac{1}{2}}.\ ] ] integrating and setting the constant of integration equal to zero by assuming @xmath118 , we obtain @xmath134^{\frac{1}{2}}dz}.\ ] ] for different parameters . in the right panel the behavior of the tachyon field @xmath115 for hde model in dgp braneworld is illustrated for different parameters.,title="fig:",width=302 ] for different parameters . in the right panel the behavior of the tachyon field @xmath115 for hde model in dgp braneworld is illustrated for different parameters.,title="fig:",width=302 ] in this way we connect the hde in dgp braneworld with a tachyon field , and reconstruct the potential and the dynamics of the tachyon field which describe tachyon cosmology . the reconstructed tachyon field and the evolution of the tachyon potential are plotted in figure [ tachyon ] . again , we see that tachyon potential is steeper in the early epoch and becomes flat near today . thus , the universe enters a de - sitter phase at the late time and the constant potential plays the role of cosmological constant . in this section we would like to investigate the stability of the hde in dgp braneworld against perturbation . it is expected that any viable de model should result a stable de dominated universe . a simple , but not complete , approach to check the stability of a proposed de model is to study the behavior of the square sound speed ( @xmath135 ) @xcite . it was argued that the sign of @xmath136 plays a crucial role in determining the stability of the background evolution . if @xmath137 , it implies that the perturbation of the background energy density is not an oscillatory function and may grow or decay with time , and so we have the classical instability of a given perturbation . on the other hand , the positivity of @xmath138 indicates that the perturbation in the energy density , propagates in the environment and so we expect a stable universe against perturbations . it is important to note that the positivity of @xmath136 is necessary but is not enough to conclude that the model is stable . indeed , the negativity of it shows a sign of instability in the model . the behavior of the squared sound speed for hde @xcite , agegraphic de @xcite and the ghost de model @xcite were investigated . it was found that all these models @xcite are instable against background perturbations and so can not lead to a stable de dominated universe . in the linear perturbation regime , the perturbed energy density of the background can be written as @xmath139 where @xmath140 is unperturbed background energy density . the energy conservation equation ( @xmath141 ) yields @xcite @xmath142 for @xmath12 , eq . ( [ pert2 ] ) becomes a regular wave equation whose solution is given by @xmath143 , which indicates a propagation mode for the density perturbations . for @xmath144 , the perturbation becomes an irregular wave equation and in this case the frequency of the oscillations are pure imaginary and the density perturbation will grow with time as @xmath145 . hence the negative squared speed shows an exponentially growing mode for a density perturbation . thus the growing perturbation with time indicates a possible emergency of instabilities in the background . for hde model in dgp braneworld for different model parameters . left panel corresponds to @xmath10 and @xmath11 , while the right panel shows the case @xmath8 and @xmath9 . , title="fig:",width=302 ] for hde model in dgp braneworld for different model parameters . left panel corresponds to @xmath10 and @xmath11 , while the right panel shows the case @xmath8 and @xmath9 . , title="fig:",width=302 ] the squared speed of sound can be written as @xmath146 where @xmath147 taking the time derivative of ( [ wdz ] ) and substituting the result in ( [ vs1 ] ) , after some calculation , we arrive at @xmath148 - 1.\ ] ] the evolution of @xmath7 versus redshift parameter are plotted in figure [ vs ] . from these figures we see that @xmath7 can be positive provided the parameters of the model are chosen suitably . for example , for @xmath10 , @xmath11 , and also for @xmath8 , @xmath9 the square of sound speed is always positive during the history of the universe , and so in these cases the stable de dominated universe can be achieved . in this paper , we have investigated the holographic model of de in the framework of dgp braneworld . we have chosen the go ir cutoff in the form , @xmath149 , with two free parameters @xmath61 and @xmath62 , which should be constrained by comparing with observations . this cutoff is the generalization of the well - known ricci scalar cutoff in flat universe . we have restricted our study to the current cosmological epoch , and so we have not considered the contributions from matter and radiation by assuming that the dark energy @xmath64 dominates , thus the friedman equation becomes simpler . the main difference between the hde with go cutoff studied in this work in the framework of dgp braneword , with the one considered in standard cosmology @xcite , is that the equation of state parameter of the hde with go cutoff in standard cosmology is a constant @xcite , however , in dgp braneworld , due to the bulk effects , @xmath33 becomes a time variable parameter . the application of the hde with go cutoff , in dgp braneworld allows us to solve the friedmann equation and derive the hubble parameter , @xmath150 , as well as the scale factor , @xmath151 , analytically . we also obtained the equation of state and the deceleration parameters of hde as the functions of the redshift parameter @xmath3 and plotted the evolutionary behaviour of these parameters against @xmath3 . the cosmological quantities depend on the two model parameter @xmath61 and @xmath62 . to show the viability of the model , we studied the zero - redshift values of the cosmological parameters by chosen the suitable values for @xmath61 and @xmath62 . for example , we found that for @xmath97 and @xmath152 , the transition from deceleration phase to the acceleration phase occurred around @xmath153 , which is consistent with recent observations @xcite . the zero - redshift value of the deceleration parameter was obtained @xmath99 for @xmath97 , @xmath98 , and @xmath102 for @xmath100 , @xmath101 which is again compatible with the cosmological data @xcite . besides , for @xmath76 this model mimics a cosmological constant with @xmath51 , and @xmath90 , independent of the redshift parameter @xmath3 . for @xmath8 , @xmath9 and @xmath10 , @xmath11 the equation of state parameter , however , is always larger than @xmath119 , and the universe enters a de - sitter phase at the late time . we have also established a connection between the quintessence / tachyon scalar field and the hde in dgp braneworld . as a result , we reconstructed the corresponding potentials of the scalar field , @xmath103 , and the dynamics of the scalar fields as a function of redshift parameter , @xmath154 , according to the evolutionary behavior of the hde model . finally , we studied the stability of the presented model by studying the evolution of the squared sound speed @xmath7 whose sign determines the sound stability of the model . interestingly enough , we found that @xmath12 provided the parameters of the model are chosen suitably . for example , for @xmath97 , @xmath155 , and @xmath100 , @xmath156 the squared sound speed is always positive during the history of the universe , and so in this case the stable de dominated universe can be achieved . this is in contrast to hde in standard cosmology , which is instable against background perturbations and so can not lead to a stable de dominated universe @xcite . this implies that the presence of the extra dimension in hde model , can bring rich physics and in particular it has an important effect on the stability of the hde model . this issue deserves further investigations . 99 a.g . riess , et al . j. * 116 * , 1009 ( 1998 ) ; + s. perlmutter , et al . , astrophys . j. * 517 * , 565 ( 1999 ) ; + p. debernardis , et al . , nature * 404 * , 955 ( 2000 ) ; + s. perlmutter , et al . , astrophys . j. * 598 * , 102 ( 2003 ) . s. weinberg , rev . * 61 * , 1 ( 1989 ) . a. g. cohen , d. b. kaplan and a. e. nelson , phys . * 82 * , 4971 ( 1999 ) . p. horava and d. minic , phys . lett . * 85 * , 1610 ( 2000 ) ; + s. d. thomas , phys . lett . * 89 * , 081301 ( 2002 ) . s. d. h. hsu , phys . b * 594 * , 13 ( 2004 ) . d. pavon , w. zimdahl , phys . b * 628 * , 206 ( 2005 ) ; + a. sheykhi , phys . d * 84 * , 107302 ( 2011 ) . a. sheykhi , phys . b * 680 * , 113 ( 2009 ) ; + a. sheykhi , phys . b * 682 * , 329 ( 2010 ) 329 ; + a. sheykhi , phys . d * 81 * , 023525 ( 2010 ) ; + ahmad sheykhi , mubasher jamil , phys . b * 694 * , 284 ( 2011 ) ; + a. sheykhi , m. r. setare , int . j. theor . phys . * 49 * , 2777 ( 2010 ) . c. j. gao , nt . j. mod . * 10 * , 95 ( 2012 ) . xia , h. li , x. zhang , phys . d * 88 * , 063501 ( 2013 ) . w. yang , l. xu , phys . d * 89 * , 083517 ( 2014 ) ; + b. novosyadlyj , o. sergijenko , r. durrer , v. pelykh , arxiv:1312.6579 u. alam , v. sahni , a.a . starobinsky , jcap * 0406 * , 008 ( 2004 ) ; + d. huterer , a. cooray , phys . d * 71 * , 023506 ( 2005 ) ; + y.g . gong , int . d * 14 * , 599 ( 2005 ) ; + y.g . gong , class . quantum grav . * 22 * , 2121 ( 2005 ) . b. wang , y. gong and e. abdalla , phys . b * 624 * , 141 ( 2005 ) ; + b. wang , c. y. lin and e. abdalla , phys . b * 637 * , 357 ( 2005 ) ; + m. r. setare , phys . b * 642*,1 ( 2006 ) . p. binetruy , c. deffayet , u. ellwanger , d. langlois , phys . lett . b * 477 * , 285 ( 2000 ) . p. j. e. peebles and b. ratra , astrophys . j. * 325 * , l17 ( 1988 ) ; + b. ratra and p. j. e. peebles , phys . d * 37 * , 3406 ( 1988 ) ; + c.wetterich , nucl . b * 302 * , 668 ( 1988 ) ; + j. a. frieman , c. t. hill , a. stebbins and i. waga , phys . lett . * 75 * , 2077 ( 1995 ) ; + x. zhang , mod . a * 20 * , 2575 ( 2005 ) ; + x. zhang , phys . b * 611 * , 1 ( 2005 ) . a. mazumdar , s. panda , and a. perez - 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we consider the holographic dark energy ( hde ) model in the framework of dgp braneworld with granda - oliveros infrared ( ir ) cutoff , @xmath0 . with this choice for ir cutoff , we are able to derive evolution of the cosmological parameters such as the equation of state and the deceleration parameters , @xmath1 and @xmath2 , as the functions of the redshift parameter @xmath3 . as far as we know , most previous models of hde presented in the literatures , do not gives analytically @xmath4 and @xmath5 . we plot the evolution of these parameters versus @xmath3 and discuss that the results are compatible with the recent observations . with suitably choosing the parameters , this model can exhibit a transition from deceleration to the acceleration around @xmath6 . then , we suggest a correspondence between the quintessence and tachyon scalar fields and hde in the framework of dgp braneworld . this correspondence allows us to reconstruct the evolution of the scalar fields and the scalar potentials . we also investigate stability of the presented model by calculating the squared sound speed , @xmath7 , whose sign determines the stability of the model . our study shows that @xmath7 could be positive provided the parameters of the model are chosen suitably . in particular , for @xmath8 , @xmath9 , and @xmath10 , @xmath11 , we have @xmath12 during the history of the universe , and so the stable dark energy dominated universe can be achieved . this is in contrast to the hde in standard cosmology , which is unstable against background perturbations and so can not lead to a stable dark energy dominated universe .

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Proceed to summarize the following text: in the current studies of ultracold quantum gases , a great deal of interest has been drawn to the study of bose - einstein condensates ( becs ) loaded into optical lattices ( ols ) , i.e. , spatially periodic potentials induced by the interference between counterpropagating laser beams book1,book2,morsch06 . besides playing a crucial role in effectively tuning the interaction strength in the condensate , i.e. , the ratio between the kinetic and interaction energies @xcite , ols offer an extremely useful tool for studies of the transition between the superfluid and mott - insulator states @xcite , and for the investigation of effects in the matter - wave dynamics due to the interplay between nonlinearity and the quasi - discreteness , which is induced by a deep lattice potential @xcite . the mean - field dynamics of the bec loaded into the ol is described by the cubic gross - pitaevskii equation ( gpe ) with the periodic potential book1,book2,morsch06 . the respective bogoliubov s excitation spectrum features a band structure , similar to electronic bloch bands in solid state . if the ol potential is deep enough , the lowest - band dynamics may be approximated by the discrete nonlinear schrdinger ( nls ) equation trombettoni01 . using this correspondence , the bec dynamics was studied in the framework of the nonlinear - lattice theory , see works trombettoni01,abdullaev01,alfimov02 and short review @xcite . the presence of the ol gives raise to energetic and dynamical instabilities , which have been predicted theoretically wu01,smerzi02,konotop02,wu03,menotti03,taylor03,kramer03,kramer05 and studied experimentally @xcite . an important application of the ols is their use for the creation and stabilization of matter - wave solitons . in particular , the periodic potential gives rise to localized gap solitons in the case of _ repulsive _ two - body interactions , as was predicted theoretically @xcite and demonstrated experimentally @xcite ( with the attractive interactions , bright matter - wave solitons were created and observed in condensates of @xmath0li @xcite and @xmath1rb weiman atoms ) . more generally , the use of time- and space - modulated fields acting on atoms is a powerful tool for the control of soliton properties @xcite ; for instance , while the gpe without external potentials admits stable soliton solutions only in the 1d geometry sulem99,ablowitz04 , ol potentials can stabilize solitons in any higher dimension @xcite . unlike 1d solitons , a necessary existence condition for their multidimensional counterparts , stabilized by means of ols , is that the soliton s norm must exceed a certain threshold value . another very useful tool frequently used in experiments with ultracold atomic gases is the control of the strength and sign of two - body interactions by means of an external magnetic field near the feshbach resonance @xcite . further , recent works proposed to exploit the possibility to control the strength of _ three - body _ interactions between atoms , independently from the control of the two - body collisions @xcite . one motivation for such studies is related to the possibility of creating new exotic strongly correlated phases in ultracold gases . indeed , quantum phases , such as topological ones or spin liquids , turn out to be ground - states of the hamiltonian including three- or multi - body - interaction terms , an example being fractional quantum - hall states described by pfaffian wave functions @xcite . in a recent work @xcite , a 1d bose gas with @xmath2-body attractive interactions was studied in the mean - field approximation , with the objective to create highly degenerate ground - states of hamiltonians including many - body terms . for the three - body interactions ( @xmath3 ) , the system is described by a _ gpe , i.e. , the respective term in the energy density is proportional to @xmath4 , where @xmath5 is the single - atom mean - field wave function ( in the general case , a similar term is proportional to @xmath6 ) . soliton solutions can be found for each @xmath2 , but they represent the stable ground - state , with negative energy ( which is defined as per eqs . ( [ e - functional ] ) and ( [ h0 ] ) , see below ) , only for @xmath7 , being unstable excited states with positive energy at @xmath8 . for @xmath3 , soliton solutions are 1d counterparts of the well - known townes solitons @xcite , which play the role of the separatrix between collapsing and decaying localized states . the townes - like solitons with fixed norm ( which is @xmath9 , in the notation adopted below ) exist only at a single critical value of the interaction strength , at which they feature the infinite degeneracy fersino08 : _ all _ the solitonic wave functions , @xmath10 ^{-1/2}$ ] , with arbitrary width @xmath11 ( see eqs . ( [ critical - value ] ) and ( quintic-1 ) below ) , have _ zero energy _ but different values of the chemical potential , @xmath12 . a relevant issue is how this infinite degeneracy is lifted by an external potential , especially by a periodic one corresponding to the ol @xcite . when the two - body interaction is present , the mean - field equation is the gpe with the cubic - quintic ( cq ) nonlinearity @xcite . as said above , it has been shown @xcite that it is possible to tune the strength of the two - body interactions independently from the three - body ones . in addition to that , in the framework of the effective gpe for the bec loaded into a nearly 1d ( cigar - shaped " ) trap with tight transverse confinement , an effective attractive quintic term appears , in the absence of any three - atom interactions , as a manifestation of the residual deviation from the one - dimensionality @xcite . in any case , if the two - body interaction is repulsive while its three - body counterpart is attractive , soliton solutions to the cq gpe can be found in an exact analytical form ( in the free space ) , but they feature an unstable eigenvalue in the bogoliubov - de gennes spectrum of small perturbations around them @xcite , while the instability of the townes - like solitons in the quintic equation is subexponential , being accounted for by a zero eigenvalue . the issue we address in this paper is the possibility to stabilize such solitons by means of the ol potential . previously , the stabilization of originally unstable solitons by means of the ol was considered , in the 2d @xcite and 1d @xcite settings alike , only for localized states of the townes type ( recently , the stabilization of 2d solitons against the _ supercritical collapse _ by the ol was also demonstrated in the cq model in 2d , with both cubic and quintic terms being attractive @xcite ) . it was found that the ol with any value of its strength ( i.e. , with zero threshold ) opens a _ stability window _ around the critical point corresponding to the townes solitons . in this work , we demonstrate that the ol opens a stability window for solitons in the cq model ( with the repulsive cubic and attractive quintic terms ) too , but only if the lattice strength exceeds a _ finite _ threshold value . apart from the context of bec , where the nonlinearity degree is related to the number of atoms simultaneously involved in the contact interaction , nls equations with the power - law and cq nonlinearities are also known as spatial - domain models of the light propagation in self - focusing media kivshar03 ( for a brief overview of optical models based on the cq - nls equation , including references to experimental realizations , see recent works @xcite ) . in the case of the cubic nonlinearity ( the kerr medium ) , effects of imprinted lattices on the transmission of light beams have been investigated both in local @xcite and nonlocal @xcite models . the paper is structured as follows . in section ii , we introduce the cq gpe corresponding to the mean - field description of the 1d bose gas with two - body repulsive and three - body attractive interactions . properties of the ( unstable ) soliton solutions to this equation are also recapitulated in section ii . in section iii , we use the variational approximation ( va ) ( see ref . @xcite for a review ) to discuss effects of the ol on the solitons . we introduce an appropriate ansatz and compute the corresponding energy . the limit of the vanishing two - body interaction is considered too and compared to previous results @xcite . in section iv , the stability region for the soliton solution in the presence of the repulsive two - body interaction and ol is determined and compared with numerical findings . the effect of an additional harmonic - trap potential is studied in section v , showing that the stability region depends on the matching between minima of the periodic potential and the location of the minimum of the harmonic trap . in section vi we present our conclusions . the quantum many - body hamiltonian for the 1d bose gas with @xmath2-body contact attractive interactions is @xmath13 ^{% \mathcal{n}}\left [ \hat{\psi}(x)\right ] ^{\mathcal{n}}\right\ } , \label{ham - contact}\]]where @xmath14 is the bosonic - field operator , @xmath15 is the nonlinearity strength and @xmath16is the single - particle hamiltonian , @xmath17 being the external potential . the case of @xmath7 in the homogeneous limit ( @xmath18 ) corresponds to the integrable lieb - liniger model @xcite . for attractive interactions ( @xmath15 ) , its analytical solution was obtained by means of the bethe ansatz @xcite and for a large number of particles , @xmath19 , the energy of the exact ground - state solution coincides with that obtained in the mean - field approximation calogero75 . in the attractive lieb - liniger model , a finite ground - state energy per particle is provided by fixing product @xmath20 to a constant value @xcite , while for @xmath21 one has to set @xmath22 @xcite . in the heisenberg representation , the equation of motion for field @xmath23 is @xmath24 = \hat{h}_{0}\hat{\psi}-c\left ( \hat{\psi}^{\dag } \right ) ^{\mathcal{n% } -1}\left ( \hat{\psi}\right ) ^{\mathcal{n}-1}\hat{\psi}. \label{dyn}\]]the mean - field approximation reduces eq . ( [ dyn ] ) to the corresponding gpe with the power - law nonlinearity , @xmath25where the macroscopic wave function @xmath26 is normalized to the total number of atoms , @xmath19 , and the nonlinearity degree is related to the order of the multi - body interactions , @xmath2 : @xmath27thus , the usual two - body interaction ( @xmath7 ) corresponds to @xmath28 , and the three - body interaction ( @xmath3 ) to @xmath29 . equation ( [ gnls ] ) conserves the energy , @xmath30 \psi ( x ) , \label{e - functional}\]]which is the classical counterpart of quantum hamiltonian ( [ ham - contact ] ) . in eq . ( [ h0 ] ) , @xmath17 is the external trapping potential , which typically includes a superposition of an harmonic magnetic trap and periodic ol potential , @xmath31 , where the harmonic confining term is @xmath32 . we take the periodic potential as @xmath33 , where @xmath34 is proportional to the power of the laser beams which build the ol , and @xmath35 , with @xmath36 ; here , @xmath37 is the wavelength of the beams , and @xmath38 the angle between them ( the period of the lattice is @xmath39 ) . parameter @xmath40 measures a mismatch between the minimum of the parabolic potential ( at @xmath41 ) and the closest local minimum of the lattice potential : when @xmath42 ( @xmath43 ) a minimum ( maximum ) of @xmath44 coincides with the minimum of @xmath45 . in fact , except for section v , we consider the situation without the parabolic trap ( i.e. , @xmath46 ) , therefore we set @xmath42 in this case . the time - independent power - law gpe corresponding to eq . ( [ gnls ] ) is ( from now on , we use normalized units , with @xmath47 and @xmath48 ) @xmath49 \psi ( x)=\mu \psi ( x ) , \label{gnls - mu}\]]where @xmath50 is the chemical potential , @xmath51 and the norm of the wave function is @xmath9 . in the free - space cubic model ( @xmath28 and @xmath52 ) , eq . ( [ gnls - mu ] ) is the integrable nls equation , whose multi - soliton solutions can be obtained by means of the inverse scattering method @xcite . the commonly known single - soliton nls solution is @xmath53where @xmath54 and @xmath55 is a real amplitude , the respective value of the chemical potential being @xmath56 . for a general value of @xmath57 , the integrability is lost even in the absence of the external potential @xcite ; nevertheless , the respective single - soliton solutions can be found in an explicit form ablowitz81,polyanin04 . for the attractive three - body interactions ( @xmath29 ) , eq . ( [ gnls ] ) is the self - focusing quintic gpe , whose stationary version is @xmath58 \psi ( x)=\mu \psi ( x ) . \label{quintic - mu}\]]for @xmath59 , if one fixes coefficient @xmath60 in front of the interaction term , the townes - like solitons exist for a particular value of the norm of the wave function @xcite . on the other hand , fixing the normalization of the wave function ( recall that the norm is @xmath9 in our units ) amounts , for @xmath61 , to fixing a relation among the chemical potential and the interaction strength @xcite , so that for each @xmath60 it is possible to obtain a single soliton solution ( although , as mentioned above , these solutions provide the ground - state in the infinite system only for @xmath62 , i.e. , for @xmath63 ) . however , for @xmath29 ( i.e. , @xmath3 ) chemical potential @xmath50 remains indefinite , assuming arbitrary negative values , while the soliton solution of the form @xmath64satisfies the unitary normalization condition at a single ( critical ) value of the interaction strength @xcite , @xmath65at @xmath66 , all solutions ( [ quintic-1 ] ) share a common value of the energy , which is simply @xmath67 @xcite , as follows from eqs . ( [ e - functional ] ) and ( [ critical - value ] ) . if the two - body interaction is added to the three - body attraction , the mean - field equation is the gpe with the cq nonlinearity , @xmath68 \psi ( x)=\mu \psi ( x ) . \label{cubic - quintic - mu}\]]as said above , we chiefly focus on the case of the _ repulsive _ two - body interactions , i.e. , @xmath69 . a family of exact soliton solutions to eq . ( [ cubic - quintic - mu ] ) with @xmath59 can be obtained in the exact form @xcite , which , for @xmath69 , is @xmath70where @xmath71 , and the maximum value of the density , at the soliton s center , is @xmath72a simple derivation of eq . ( [ cubic - quintic-1 ] ) is presented in appendix a. obviously , for @xmath73 solution ( [ cubic - quintic-1 ] ) reduces to townes - like soliton ( [ quintic-1 ] ) . imposing the above - mentioned normalization,@xmath74on solution ( [ cubic - quintic-1 ] ) , one arrives at relation@xmath75from where it follows that , for @xmath76 , soliton solutions with @xmath77 satisfying normalization condition ( [ 1 ] ) exist for @xmath78 . however , these solutions are unstable @xcite ( in particular , because they do not satisfy the vakhitov - kolokolov stability criterion kolokolov73 ) . in the following section we discuss how the ol can stabilize such localized solutions . both for @xmath79 and @xmath80 ( @xmath7 and @xmath81 ) , and for the gpe with the mixed cq nonlinearity , the presence of the periodic potential makes it necessary to resort to approximate methods for finding solitons . to this end , we use the va ( variational approximation ) @xcite based on the _ ansatz _ which yields exact soliton solution ( quintic-1 ) of the quintic nls equation in the absence of the external potential : @xmath82here , width @xmath11 is the variational parameter to be determined by the minimization of the energy , while amplitude @xmath55 will be found from normalization condition ( [ 1 ] ) . we expect that ansatz ( [ var ] ) , which does not explicitly include the modulation of the wave function induced by the ol , may give a reasonable estimate of the soliton s energy for sufficiently small values of ol strength @xmath34 in eq . ( [ ol ] ) , cf . the known result for the 2d equation with the cubic nonlinearity ( @xmath79 ) and ol potential @xcite . in the case of the 3d gpe which includes the cubic term and harmonic trap , this approach leads to an estimate for the critical value of the number of atoms above which the condensate collapses , that was found to be in a reasonable agreement with results produced by the numerical solution of the gpe fetter95,ruprecht95 . in 1d , the va based on the gaussian ansatz also provides for quite an accurate approximation to exact soliton solution ( [ cubic-1 ] @xcite . similar analyses carried out in the 1d model including the cubic term and ol @xcite have demonstrated that ( unlike the 2d and 3d cases ) the 1d soliton trapped in the ol potential does not have an existence threshold in terms of its norm ( number of atoms ) . the energy to be minimized in the framework of the va is obtained by inserting ansatz ( [ var ] ) in the gpe energy functional given by eq . ( e - functional ) . the kinetic and quintic - interaction energy terms in the functional both scale as @xmath83 ; then , the energy per particle computed from expression ( [ e - functional ] ) is @xmath84 , \label{e3}\]]@xmath85where @xmath86 is defined in eq . ( [ critical - value ] ) . for @xmath52 ( without the ol ) , the scenario discussed in the previous section for the uniform cq gpe with the attractive three - body and repulsive two - body interactions is recovered , as energy ( [ e3 ] ) reduces in that case to @xmath87for @xmath73 , the energy is positive when @xmath88 ( i.e. , @xmath89 , see eq . ( [ beta ] ) ) and vanishes at @xmath90 ; for @xmath91 ( i.e. , @xmath92 ) one obtains @xmath67 , in agreement with the above - mentioned exact result showing the infinite degeneracy of soliton family ( [ quintic-1 ] ) , while for @xmath93 the energy is negative and diverges ( to @xmath94 ) at @xmath95 , signaling , in terms of the va , the onset of the collapse . with @xmath76 , expression ( [ e3-homog ] ) does not give rise to any minimum of the energy , which agrees with the known fact of the instability of all the solitons in this case @xcite . a detailed study of minima of variational energy ( [ e3 ] ) is presented in appendix b. in the following subsection , we consider the case of the self - focusing quintic gpe in the presence of the ol ( @xmath96 ) , while the discussion of the general case ( @xmath97 ) is given in section iv . here we address the stability of localized variational mode ( [ var ] ) , for different values the ol parameters , strength @xmath34 and wavenumber @xmath98 , keeping @xmath73 . the results of the analysis of minima of the variational energy ( [ e3 ] ) , presented in appendix b , can be summarized as follows ( see also fig . [ fig1 ] ) : for @xmath99 , the infinitely deep minimum of the energy is obtained at @xmath95 , which corresponds to the collapse , as shown in fig . [ fig1](a ) . for @xmath100 , the collapse may be avoided , and three possibilities arise : there exists another special value , @xmath101 , such that for every @xmath60 between @xmath102 and @xmath86 the energy has a minimum at @xmath103 and a maximum at @xmath104 , while for @xmath105 the energy does not have a minimum at any finite value of @xmath11 , see fig . [ fig1](d ) . actually , two different situations should be distinguished for @xmath106 : there exists a specific value ( refer to appendix b ) , @xmath107(with @xmath108 ) such that , for @xmath109 , the energy has a _ global _ minimum at @xmath110 ( which , thus , represents the _ ground - state _ of the boson gas in this situation ) , while , for @xmath111 , the energy minimum at @xmath110 is a _ local _ one . in other words , taking into regard the fact that , as shown by eq . ( [ e3 ] ) , the energy - per - particle approaches value @xmath112 at large @xmath11 , we conclude that , for @xmath113 ( @xmath111 ) , the energy satisfies inequality @xmath114 ( @xmath115 ) , as showed in figs . [ fig1](b , c ) . obtained in the framework of the quintic gpe versus @xmath116 ( in units of @xmath117 ) for @xmath118 ( a ) ; @xmath119 ( b ) ; @xmath120 ( c ) ; @xmath121 ( d ) . in ( a ) the solid ( dotted ) line is the energy for @xmath93 ( @xmath66 ) ; in ( b)-(c ) , points of the energy minimum and maximum , @xmath122 and @xmath123 , are indicated . ] from the above analysis , we infer that for @xmath124 the ground - state is a delocalized one ( although the metastable state , corresponding to the above - mentioned local energy minimum , exists for @xmath111 ) , for @xmath119 the ground - state is represented by a finite - size soliton configuration ( in agreement with ref . @xcite ) and for @xmath93 it is collapsing . equation ( [ c_2_star ] ) shows that the width of the stability region depends on ratio @xmath125 : keeping fixed all other parameters , the decrease of the lattice spacing ( i.e. , the increase of @xmath98 ) leads to a reduction of the stability region . equation ( [ c_2_star ] ) also shows that for @xmath126 the va formally predicts @xmath127 : however , for @xmath128 , the ground - state is delocalized and the variational ansatz ( [ var ] ) can not be used , as it does not take into account the modulation induced by the deep ol potential . in fig . [ deloc-1 ] , we plot the numerically found ground - state of the quintic gpe in a 1d box ( @xmath129 ) . it is seen that , with the increase of @xmath130 , the configuration becomes broader , until a critical value is reached , as discussed in @xcite . in the inset of fig . [ deloc-1 ] we plot the squared width @xmath131 of the numerically found ground - state @xmath132 versus @xmath60 , which makes the delocalization transition evident : for @xmath124 the width @xmath133 is @xmath134 , while around @xmath135 the width suddenly decreases . variational estimate ( [ c_2_star ] ) for the critical value @xmath136 , as predicted by the va ( see eq . ( [ c_2_star ] ) ) , is displayed in fig . [ comparison_quintic ] , together with numerical results . one observes a reasonable agreement between them , especially for small @xmath34 , which is due both to the use of the more adequate ansatz ( [ var ] ) , rather than a gaussian , and also because @xmath137 is found as the value at which the global ( rather than local ) minimum disappears . . solid lines , starting from the narrowest configuration , refer to @xmath138 ( recall that @xmath139 ) , and the dashed line refers to @xmath140 . parameters are @xmath141 , @xmath142 and @xmath143 . inset : squared width @xmath144 of the ground - state as a function of @xmath60 ( the dot - dashed line is a guide to the eye ) . critical value @xmath145 obtained from the numerical analysis is @xmath146 , which should be compared with the corresponding value ( [ c_2_star ] ) predicted by the variational approximation , @xmath147 . ] as a function of @xmath148 , according eq . ( [ c_2_star ] ) ( for the quintic gpe ) , the dashed line corresponding to @xmath149 . discrete symbols represent results obtained from the numerical solution of the quintic gpe . they designate the transition form the localized ground - state to the extended one ( parameters are the same as in fig . ( [ deloc-1 ] ) . according to the variational approximation , the ground - state is delocalized ( @xmath150 ) below the dotted line , and it collapses ( @xmath151 ) for @xmath60 above the dashed line . ] the most interesting situation occurs when the two - body repulsive interaction ( @xmath76 ) competes with the attractive three - body collisions ( @xmath15 ) . as said above , all solitons in the free space ( @xmath52 ) are strongly unstable in this situation @xcite , and the possibility of their stabilization by the ol was not studied before . the analysis of variational energy ( [ e3 ] ) , presented in appendix b , yields the following results for this case . for @xmath93 , the energy does not have a minimum at finite @xmath11 , hence the ol can not stabilize the solitons in this case . if @xmath66 , the energy has a global minimum at a finite value of @xmath11 , when @xmath152for @xmath88 , the energy features a global minimum at finite @xmath11 for @xmath153 , where the modified critical value is @xmath154cf . definition ( [ c_2_star ] ) for @xmath155 . the value @xmath137 depends upon @xmath156 , vanishing for @xmath156 larger than the critical value @xmath157 . this means that , to balance the destabilizing effect of the repulsive two - body interactions , the strength of the periodic potential , @xmath34 , must _ exceed _ its own critical value , @xmath158otherwise , eq . ( [ c_2_star_g ] ) yields @xmath159 , i.e. , the ol can not stabilize the solitons . in fig . [ deloc-2 ] we plot the numerically found ground - state of cq gpe ( [ cubic - quintic - mu ] ) for several values of @xmath34 . it is seen that , at small @xmath34 , the wave function @xmath5 remains delocalized , until a critical value is reached . in the inset of fig . [ deloc-2 ] the squared width of the numerically generated ground - state is plotted versus @xmath34 . in fig . [ comparison_epsilon ] , we compare critical value @xmath160 , as given by eq . ( [ epsilon_critical ] ) , with numerical results : for small @xmath161 , the predicted linear dependence of @xmath160 on @xmath161 is well corroborated by the numerical results , the relative error in the slope being @xmath162 . in principle , the comparison between variational estimate ( [ epsilon_critical ] ) and numerical results might be further improved by choosing a variational wave function which , in the limit of @xmath52 ( uniform space ) would reproduce exact cq soliton ( [ cubic - quintic-1 ] ) . however , the calculations with such an ansatz are extremely cumbersome . . solid lines , starting from the narrowest wave function , refer to @xmath163 , and the dashed line refers to @xmath164 . parameters are @xmath165 , @xmath166 , @xmath142 , @xmath167 . inset : the squared width of the ground - state versus @xmath168 ( the dot - dashed line is a guide to the eye ) . critical value @xmath169 obtained from the numerical data is @xmath170 , which should be compared to the variational prediction given by eq . ( [ epsilon_critical ] ) , which is @xmath171 . ] versus @xmath172 as given by eq . ( [ epsilon_critical ] ) . symbols refer to results obtained from the numerical solution for the ground - state of the cubic - quintic gpe . they represent the delocalization transition . the parameters are the same as in fig . ( [ deloc-2 ] ) . ] in this section we aim to use the variational approximation based on ansatz ( [ var ] ) for examining the combined effect of the parabolic trapping potential acting along with an ol , i.e. , we take eq . ( [ quintic - mu ] ) with external potential@xmath173cf . ( [ ol ] ) , and disregard binary collisions ( @xmath73 ) . value @xmath174 ( @xmath175 ) corresponds to the matching ( largest mismatch ) between the minimum of the harmonic potential and a local minimum of the lattice potential . the respective variational energy is obtained from ( e - functional ) with potential ( [ full - v ] ) : @xmath176 . \label{e3-tot}\ ] ] with @xmath177 , the soliton is stable for @xmath88 , and it collapses otherwise . with @xmath178 , a richer behavior is predicted by the va . the system does stabilize for @xmath88 , while , for @xmath93 , the presence of the mismatched harmonic trap gives rise to a metastability region . since @xmath179 as @xmath95 and @xmath180 as @xmath90 , one can encounter two possibilities : either @xmath181 is positive for all @xmath11 ( and there are no energy minima ) , or equation @xmath182 has two roots , corresponding to a local minimum and a maximum . the equation for the value of @xmath11 at which energy ( [ e3-tot ] ) reaches the local minimum is @xmath183where @xmath184 , and @xmath185@xmath186one can see that , for @xmath66 ( i.e. , @xmath92 ) , eq . ( [ cond - trap ] ) does not have a nonvanishing solution if @xmath98 is smaller than a critical value , @xmath187while it has a nonvanishing solution for @xmath188 . actually , for @xmath93 ( i.e. , @xmath189 ) , eq . ( [ cond - trap ] ) with @xmath190 has two nonvanishing roots , one of which is a local minimum , while such roots do not exist for @xmath191 . for @xmath188 , the right - hand side of eq . ( [ cond - trap ] ) has a maximum value , which fixes the maximum value of @xmath192 , i.e. , the maximum value of @xmath60 , which we refer to as @xmath193 . then , for @xmath194 , the variational energy does not have a local minimum . for @xmath195 there appears a finite metastability region , in terms of wavenumber @xmath98 , as illustrated by fig . [ q ] . in other words , for fixed @xmath60 , metastable states appear at large values of @xmath34 . plane of the model ( including the parabolic trap ) the metastable region from the unstable one . the parameters are @xmath196 , @xmath197 , and @xmath198 . ] in this work we have studied the effect of the ol ( optical lattice ) on the 1d bose gas with attractive three - body and repulsive two - body interactions , described by the gpe ( gross - pitaevskii equation ) with the cq ( cubic - quintic ) nonlinearity . actually , the effective quintic attractive term in the gpe may be induced by the residual deviation of the condensate , tightly trapped in a cigar - shaped confining potential , from the one - dimensionality ( when the three body losses are negligible ) @xcite or by three - body interaction terms between atoms according to recent proposals @xcite . in the absence of an external potential , soliton solutions to this equation with the cq nonlinearity are known in the exact form , but they all are strongly unstable . we have demonstrated that the ol opens a stability window for the solitons , provided that the ol strength , @xmath34 , exceeds a finite minimum value . the size of the stability window depends on @xmath199 , where @xmath98 is the ol s wavenumber . we have also considered effects of the additional harmonic trap , finding that , if the quintic nonlinearity is strong enough ( @xmath99 ) , a metastability region may arise , depending on the mismatch between minima of the periodic potential and harmonic trap . assuming that @xmath200 is real , we look for localized solutions to the cq nls equation , @xmath201with @xmath15 and @xmath69 . interpreting @xmath202 as a formal time variable and @xmath203 as the coordinate of a particle , eq . ( [ cubic - quintic - mu - app ] ) formally corresponds to the newton s equation of motion of this particle , @xmath204where the effective mass is @xmath205 , and the potential is @xmath206with an arbitrary additive constant chosen so as to have @xmath207 . potential ( [ potential - app ] ) for @xmath77 , which corresponds to normalizable solutions , is plotted in fig . [ potential ] . condition @xmath208 yields expression ( [ a - cubic - quintic ] ) for the soliton s amplitude . for @xmath209 . ] further , we make use of the conservation of the corresponding hamiltonian , @xmath210the boundary conditions for localized solutions , @xmath211 , @xmath212 , select @xmath213 in eq . ( [ energy - app ] ) . taking into regard the fact that @xmath214 for @xmath215 , and looking for solutions with @xmath216 at @xmath217 , one obtains from here the soliton solution in an implicit form , @xmath218it further follows from eq . ( [ quadr - app-1 ] ) that@xmath219with @xmath220 . in eq . ( [ quadr - app-2 ] ) , we use notation @xmath221 and @xmath222 . thus , from eq . ( quadr - app-2 ) one obtains @xmath223 ^{2}+4a^{2}a^{4}% \mathcal{e}^{2}}. \label{quadr - app-3}\]]one can easily check that this expression yields @xmath224 for @xmath73 , and that @xmath225 , as it must be . finally , using relation @xmath226 , one obtains eq . ( cubic - quintic-1 ) from eq . ( [ quadr - app-3 ] ) , after a straightforward algebra . in this appendix we aim to study minima of variational energy ( [ e3 ] ) . when @xmath73 , one sees that , for @xmath93 , the energy per particle tends to @xmath94 at @xmath95 , and to @xmath112 at @xmath227 . then , with regard to @xmath228 , no local ( metastable ) minima exist , and variational wave function ( [ var ] ) is not the ground - state for any finite width . for @xmath66 , one obtains the global minimum at @xmath229 , which implies the collapse . for @xmath230 , the situation is different : @xmath231 as @xmath232 ( because @xmath89 ) , and @xmath233 for @xmath234 . then , it is necessary to find the value of @xmath192 at which derivative @xmath181 has two real zeros . introducing the parameter @xmath235with @xmath236 defined as per eq . ( [ beta ] ) , one can write condition @xmath182 as @xmath237where @xmath238 , as defined above . equation ( [ t - cond ] ) can be satisfied if @xmath239 is smaller than a maximum value , @xmath240 , and it then has two roots , @xmath241 and @xmath242 , which correspond , respectively to the minimum at @xmath110 , and maximum at @xmath104 ( see fig . [ fig1 ] ) . for @xmath243 , eq . ( [ t ] ) has no roots , hence the variational energy has no minima at finite values of the soliton s width , @xmath11 . a plot of @xmath241 as a function of @xmath239 is presented in fig . [ fig_theta ] , where the maximum value of @xmath241 is @xmath244 . the energy minimum at @xmath241 is a global one if @xmath245 ; using eq . ( [ e3 ] ) , this condition reads @xmath246as one can see from fig . [ fig_theta ] , condition ( [ condition - app ] ) is satisfied for @xmath247 , where @xmath248 ; then , a global minimum exists only for @xmath249 , while for @xmath250 the minimum is local , corresponding to a metastable state . using the value of @xmath137 and definition ( [ t ] ) , one arrives at eq . ( [ c_2_star ] ) . as a function of parameter @xmath239 ( defined in eq . ( [ t - cond ] ) ) for @xmath73 . the dashed line is the plot of function @xmath251 versus @xmath239 . the maximum value of @xmath252 at @xmath253 is indicated . ] for @xmath76 ( recall it corresponds to the two - body repulsion ) , variational energy ( [ e3 ] ) for @xmath93 does not have a minimum at finite values of @xmath11 . however , for @xmath66 a finite minimum is possible . indeed , with definition of @xmath156 as per eq . ( [ g - cond - text ] ) , condition @xmath182 can be written as @xmath254for @xmath255 , eq . ( [ g - cond ] ) has two roots . by imposing the condition that the value of the energy at @xmath110 be smaller than @xmath112 , one gets @xmath256 . then , similar to the situation considered above , a global minimum exists only for @xmath257 , while for @xmath258 the minimum is local . for @xmath88 , condition @xmath259 reads @xmath260one can see that condition ( [ t - g - cond ] ) is satisfied for @xmath261 , with @xmath262 . then , for @xmath263 , i.e. , for @xmath34 small enough , the variational energy does not have a minimum . imposing the condition that the minimum is global leads to @xmath247 , with @xmath264 . then , for @xmath265 , i.e. for @xmath34 smaller than a critical value , the variational energy can not have a _ global _ minimum at a finite value of @xmath11 , i.e. , localized states can not realize a global minimum . functions @xmath266 and @xmath267 are plotted in fig . [ t - vs - g ] ; in fig . [ theta - max ] , we plot maximum value @xmath268 of @xmath241 for @xmath269 , as a function of @xmath156 . _ acknowledgement : _ we thank l. salasnich for useful comments and discussions . this work was partially supported by esf project instans and by miur projects quantum field theory and statistical mechanics in low dimensions " , and quantum noise in mesoscopic systems " . abdullaev , b. b. baizakov , s. a. darmanyan , v. v. konotop , and m. salerno , phys . a * 64 * , 043606 ( 2001 ) ; i. carusotto , d. embriaco , and g. c. la rocca , _ ibid_. * 65 * , ( 2002 ) 053611 ; b. b. baizakov , v. v. konotop , and m. salerno , j. phys . b * 35 * , ( 2002 ) 5105 ; e. a. ostrovskaya and y. s. kivshar , phys . * 90 * , ( 2003 ) 160407 ; opt . exp . * 12 * , ( 2004 ) 19

we study the effect of an optical lattice ( ol ) on the ground - state properties of one - dimensional ultracold bosons with three - body attraction and two - body repulsion , which are described by a cubic - quintic gross - pitaevskii equation with a periodic potential . without the ol and with a vanishing two - body interaction term , soliton solutions of the townes type are possible only at a critical value of the three - body interaction strength , at which an infinite degeneracy of the ground - state occurs ; a repulsive two - body interaction makes such localized solutions unstable . we show that the ol opens a stability window around the critical point when the strength of the periodic potential is above a critical threshold . we also consider the effect of an external parabolic trap , studying how the stability of the solitons depends on matching between minima of the periodic potential and the minimum of the parabolic trap .

You are an expert at summarizing long articles. Proceed to summarize the following text: single and dilepton probes in heavy ion collisions are of particular interest since such probes , once produced , are largely unaffected by the surrounding qcd medium . they carry valuable information on the particle from which they originate and allow one to assess the properties of the medium formed in the early instants of the collision . the following contributions to the dilepton invariant mass spectrum are discussed here , together with what one might learn from their measurement about the properties of the medium formed in the collision : * low mass dileptons originating from vector meson leptonic decay ( @xmath2 , @xmath3 and @xmath4 ) provide insight on the properties of these mesons in the high temperature expanding fireball produced immediately after the collision , where chiral symmetry may be ( at least partially ) restored @xcite ; * a significant fraction of the virtual and direct photons produced at low @xmath5 ( @xmath6 gev / c ) in heavy ion collisions originates from the thermal black - body radiation of the created fireball @xcite . measuring these photons therefore allows one to quantify the temperature of the fireball ; * open heavy flavors , because of their high mass , allow one to study in - medium energy loss mechanisms in addition to what can be learned from light quarks @xcite ; * heavy quarkonia are of interest because of additional mechanisms that are predicted to occur in the presence of a qgp and that would affect the production of these bound states @xcite . fig . [ low_mass_vector_mesons ] ( left ) shows the correlated dimuons invariant mass distribution at the @xmath2 vacuum mass , measured by the na60 experiment in semi - central in+in collisions @xcite . the @xmath2 mass peak differs significantly from the expected vacuum @xmath2 and can be reasonably well described on the low mass side by the model presented in @xcite . this model includes a detailed description of the baryonic matter created in the collision below the formation temperature of a qgp , @xmath7 . interactions with this baryonic matter are responsible for a broadening of the @xmath2 ( but no modification of its mass ) when approaching chiral symmetry restoration near @xmath8 . [ cols="^,^ " , ] in short : * low mass vector mesons exhibit strong shape modifications with respect to their vacuum properties , that can be well described at sps but not at rhic possibly because some contributions to the dilepton spectrum have not been properly accounted for ; * virtual photons can be used in addition to direct photon measurements to assess the medium temperature averaged over its expansion time and derive its initial temperature ; * a significant suppression of @xmath9 quarks is necessary to describe the observed heavy flavor @xmath10 in a way that is consistent with the b / b+d ratio measured in @xmath11 collisions ; * @xmath12 production in heavy - ion collisions is a puzzle . the situation is more complex than the original picture , due to our poor knowledge of its production mechanism in @xmath11 collisions and to the existence of many cold nuclear matter effects which significantly modify this production even in the absence of a qgp . efforts are being made to better understand the above so that one can quantify the _ hot _ , abnormal effects at both sps and rhic . notably , it appears that the suppression measured at sps in in - in collisions can be entirely described in terms of such cold nuclear matter effects . 00 pisarski r d 1982 , _ phys . lett . _ * 110*b 155 brown g e and rho m 2002 , _ phys . rep . _ * 363 * 85 rapp r and wambach j 2000 , _ adv . * 25 * 1 stankus p 2005 , _ ann . nucl . part . _ * 55 * 517 turbide s _ 2004 , _ phys . _ c*69 * 014903 baier r , schiff d and zakharov b g 2000 , _ annu . nucl . part . _ * 50 * 37 gyulassy m _ nucl - th/0302077 matsui t and satz h 1986,_phys . _ b * 178 * 416 andronic a _ et al . _ 2003 , _ phys . 36 thews r l 2007 , _ nucl . _ a*783 * 301 arnaldi r _ et al . _ ( na60 collaboration ) 2006 , _ phys . lett . _ * 100 * 022302 rapp r 2002 arxiv : nucl - th/0204003 adare a _ et al . _ ( phenix collaboration ) 2007 , arxiv : 0706.3034v1 [ nucl - ex ] turbide s , rapp r and gale 2004 , _ phys . _ c * 69 * 014903 arnold p , moore g d , yaffe l g 2001 , _ jhep _ 0112 , 9 akiba y ( phenix collaboration ) , this proceedings adare a _ et al . _ ( phenix collaboration ) , arxiv : 0804.4168v1 [ nucl - ex ] denterria d and peressounko d 2006 , _ eur . c*46 * 451 aggarwal m m _ et al . _ ( wa98 collaboration ) 2000 , _ phys . lett . _ * 85 * 3595 turbide s , rapp r , gale c 2004 , _ phys . _ c * 69 * 014903 abelev b i _ et al . _ ( star collaboration ) 2007 _ phys . lett . _ * 98 * 192301 adare a _ et al . _ ( phenix collaboration ) 2007 , _ phys . lett _ * 98 * 172301 dion a ( phenix collaboration ) , this proceedings adare a _ et al . _ ( phenix collaboration ) 2009 , _ phys lett . _ b * 670 * 313 engelmore t ( phenix collaboration ) , this proceedings abelev b i _ et al . _ ( star collaboration ) 2005 , _ phys . lett . _ * 94 * 062301 abelev b i _ et al . _ ( star collaboration ) , arxiv:0805.0364 dokshitzer y l and kharzeev d e 2001 , _ phys . lett . _ b * 519 * 199 . armesto n _ _ 2006 , _ phys . _ b * 637 * 262 van hees h , greco v and rapp r 2006 , _ phys . _ c * 73 * 034913 moore g d , teaney d 2005 , _ phys . _ c * 71 * 064904 dunlop j c , this proceedings cacciari m _ _ 2005 , _ phys . lett _ * 95 * 122001 ; private communication da silva c l ( phenix collaboration ) , this proceedings haberzetti h and lansberg j p 2008 , _ phys . rev . lett _ * 100 * 032006 collins j c and soper d e 1977 , _ phys . rev . * 2219 faccioli p , lourenco c , seixas j , woehri h k 2009 , arxiv:0902.4462v1 [ hep - ph ] adare a _ et al . _ ( phenix collaboration ) 2008 , _ phys . _ c * 77 * 024912 eskola k j , paukkunen h , salgado c a 2009 , arxiv:0902.4154v1 [ hep - ph ] scomparin e ( na60 collaboration ) , this proceedings lourenco c , vogt r , woehri h k 2009 , arxiv:0901.3054 [ hep - ph ] arleo f and tram v n 2008 , _ eur . c*55 * 449 liu h ( star collaboration ) , this proceedings linden levy l a , this proceedings

this contribution summarizes the main experimental results presented at the 2009 quark matter conference concerning single and dilepton production in proton and heavy ion collisions at high energy . the dilepton invariant mass spectrum has been measured over a range that extends from the @xmath0 mass to the @xmath1 mass , and for various collision energies at sps , fermilab , hera and rhic . this paper focuses on the various contributions ( photons , low mass vector mesons , open and hidden heavy flavors ) to this spectrum and discuss their implications on our understanding of the matter formed in heavy ion collisions .

You are an expert at summarizing long articles. Proceed to summarize the following text: to reveal the nature of the extremely metal - poor ( emp ) stars in the galactic halo is the key to the understanding of the formation process of the galaxy as well as of the mechanism of star formation in the primordial and very metal - poor gas clouds . because of the very low abundances of iron and other metals , these stars are thought to be survivors from the early days , and hence , are expected to carry the precious information about the early universe when they were born while they reside in our nearby space . for a past decade , a lot of emp stars have been discovered by hk survey @xcite and hamburg / eso ( he s ) survey @xcite , which enables us to use halo emp stars as a probe into the early universe . the number of known emp stars exceeds several hundreds even if we limit the metallicity range below @xmath0}\lesssim -2.5 $ ] @xcite . one of their observed characteristics is very low frequency of stars below the metallicity @xmath0}\simeq -4 $ ] . despite that more than @xmath2 stars have been registered in the metallicity range of @xmath3}\lesssim -3 $ ] by high - dispersion spectroscopy ( e.g. , see saga database ; * ? ? ? * ) , only three stars were found well below this metallicity ; two hyper metal - poor ( hmp ) stars of @xmath0 } < -5 $ ] , he 0107 - 5240 ( @xmath0}= -5.3 $ ] ; * ? ? ? * ) and he 1327 - 2326 ( @xmath0}= -5.4 $ ] ; * ? ? ? * ) , and one ultra metal - poor ( ump ) star of @xmath4 } < -4 $ ] , he 0557 - 4840 ( @xmath0}= -4.8 $ ] ; * ? ? ? has attracted wide interest , in particular , before the discovery of he 0557 - 4840 in - between metallicity of @xmath5 } < -4 $ ] . @xcite points out that such a metallicity cut - off can be interpreted as a result of metal spreading process in the stochastic and inhomogeneous chemical - enrichment model . @xcite then introduce a period of low or delayed star formation due to the negative feedback by the population iii stars , during which metals spread to explain very low iron - abundance of hmp with the carbon yield from rotating stellar models by @xcite . @xcite argues an early infall phase of primordial gas to alleviate the paucity of low - metallicity stars . @xcite adopts a semi - analytic approach for the hierarchical structure formation and presents the model of inhomogeneous galactic chemical evolution in an attempt of reproducing the statistical features of emp stars and the re - ionization of the universe . he addresses the constraints on the imf of population iii stars , arguing high - mass imf of the mean mass at @xmath6 . @xcite also take a similar approach to investigate the chemical evolution of our galaxy with the mass outflow from mini - halos . in these former works , is introduced in rather arbitrary ways , and the proper explanation is yet to be devised about the nature and origin of hmp / ump stars . one of the decisive ingredients in studying the structure formation and chemical evolution of galactic halo is the initial mass function ( imf ) of stars in the early days . most of existent studies have assumed the imf of emp stars more or less similar to that of the metal - rich populations except for hmp and ump stars . from the observations , however , we know that the emp stars have the distinctive feature that than the stars of younger populations @xcite . in addition , it is revealed that the carbon - enhanced extremely metal - poor ( cemp ) stars are divided into two sub - groups , cemp-@xmath7 and cemp - no@xmath7 according to the presence and absence of the enhancement of @xmath7-process elements @xcite . assuming this binary scenario , @xcite argue an imf with the typical mass of @xmath8 for emp stars from the surplus of cemp-@xmath7 stars . previously , @xcite have also asserted an imf peaking in the intermediate - mass range of @xmath9 for population iii stars from the consideration of galactic chemical evolution with the cn enrichment among the emp stars . furthermore , an imf with @xmath10 has been is discussed for the old halo stars from the macho observation in relation to the prospect that the observed micro - lensing may be caused by an alleged population of white dwarfs @xcite . in order to use the carbon - enhancement to constrain the imf , we should properly take into account the evolutionary peculiarity of emp stars . the stars of @xmath0}\lesssim -2.5 $ ] , there are two mechanisms of carbon enhancement , while only one mechanism for the stars of younger populations , pop . i and ii , and also , that a different mode of s - process nucleosynthesis works @xcite . these theoretical understandings , ( * ? ? ? * referred to as paper i in the following ) find that the imf for emp stars has to be high - mass with the typical mass of @xmath11 to explain the observed statistic features of both cemp-@xmath7 and cemp - no@xmath7 stars . that the majority of emp stars , including cemp stars , were born as the low - mass members of binary systems with the primary stars which have shed their envelope by mass loss to be white dwarfs and have exploded as supernovae . the purpose of this paper is twofold , first to demonstrate the robustness of the high - mass imf derived in paper i , and then to discuss the implications to the formation and early evolution of galaxy . in the following , we make a distinction between the total assembly of emp stars that were born in the early galaxy , including massive stars which were already exploded as supernovae , and the low - mass emp stars that are still alive in the nuclear burning stages by calling the former emp population " and the latter emp survivors " . in deriving the constraints on the imf of stars for the emp population , one has to make the assumptions on the binary characteristics , among which the most crucial is the distribution function of mass ratio between the primary and secondary stars in binaries . paper i adopts a flat distribution for simplicity . it seems plausible from the observations of the stellar systems of younger populations @xcite , and yet , it is true that the mass - ratio distribution is yet to be properly established both observationally and theoretically even for the binaries of younger populations . several different mechanisms have been proposed for the binary formation , such as the fragmentation during the collapse and the capture of formed stars , and are thought to give different mass - ratio distributions ( see also e.g. , * ? ? ? * and the references therein ) . the distribution may increase or decrease with the mass - ratio , or the two stars may form in the same imf as suggested for the capture origin . in this paper , we examine the dependence of the resultant imf on the assumed mass - ratio distributions of various functional forms , including the independent coupling of the both stars in the same imf to demonstrate that the high - mass nature of imf of emp population is essentially unaltered . the recent large - scaled surveys of emp stars provide the additional information on the early history of galactic halo . a fairly large number of known metal - poor stars ( 144 and 234 stars of @xmath0 } < -3 $ ] by the hk and he s surveys , respectively ) makes it feasible to discuss the metallicity distribution function @xcite . moreover , the significant coverage of celestial sphere ( 6900 and @xmath12 by the hk and he s surveys , respectively ; * ? ? ? * ; * ? ? ? * ) allows to consider the total number of emp survivors in the galactic halo . we demonstrate that the latter also places an independent constraint on the imf of emp population in combination with the metal yields produced by the emp supernovae if the binary contribution is properly taken into account . we then apply the imf , thud derived , to discuss the chemical evolution in which the stars of emp population take part . the resultant imfs can reproduce the number and slope of observed metallicity distribution functions ( mdf ) for emp stars , and also , to give an explanation to the scarcity and origin of hmp / ump stars with the effects of hierarchical structure formation process included . in this paper , we , and discuss the basic characteristics of hierarchical structure formation by using simple analytic approximations . this paper is organized as follows . in 2 , we discuss the constraints on the imf of emp population from the statistics of cemp stars and from the total number of emp survivors in our galaxies . in 3 , we investigate the metallicity distribution of emp stars in galactic halo with the formation process of the galaxy taken into account . then our conclusions follow with discussion of the origin of observed mdf and also of hmp stars . in appendix , we re - discuss the relationship between the number of emp survivors , estimated from the surveys , and the metal production by the emp supernovae with the binary contribution taken into account , to demonstrate that they entail the same imfs as drawn independently from the statistics of cemp stars . in this section , we revisit the problem of constraining the imf for the stars of emp population from the observations of emp survivors , studied in paper i. the method is based on the analysis of statistics of cemp stars in the framework of binary scenario , and hence , involves the assumptions of emp binary systems . we start with reviewing the method and assumptions used in paper i in deriving the constraints on the imf of emp population stars . we first investigate the dependence of resultant imf on these assumptions , in particular of the mass - ratio distribution of binary members . we then discuss the iron production by emp population stars in relation to the total number of emp survivors , estimated from the hk and he s surveys , to assess the constraints on the imf through the chemical evolution of galactic halo . we give the outline of our method in studying the statistics of cemp stars and chemical evolution of galactic halo with the discussion of the assumptions involved , and a brief summary of the observational facts that our study rely on . [ [ section ] ] consequently , the origins of two sub - groups of cemp stars are identified with these two mechanisms . the cemp-@xmath7 and cemp - no@xmath7 stars stem from the low - mass members of emp binaries with the primaries in the mass ranges of @xmath13 and @xmath14 , respectively . here @xmath15 is the upper limit to initial mass of stars for the formation of white dwarfs . @xmath16 ( * ? ? ? * see also siess 2007 ) , which is also taken to be the lower mass limit to the stars that explode as supernova . this is the fundamental premise of our study . for the formation of cemp stars in the binary systems , the initial separation , @xmath17 , has to be large enough to allow the primary stars to evolve through the agb stage without suffering from the roche lobe overflow , but small enough for the secondary stars to accrete a sufficient mass of the wind to pollute their surface with the envelope matter processed and ejected by the agb companion . the lower bound , @xmath18 , to the initial separation is estimated from the stellar radii of emp stars taken from the evolutionary calculation @xcite , where @xmath19 and @xmath20 are the masses of primary and secondary stars . the agb star is assumed to eject the carbon enhanced matter of @xmath21 with the wind velocity @xmath22 until it becomes a white dwarf , and we define cemp stars as @xmath23 . the upper bound , @xmath24 , is estimated by the amount of accreted matter calculated by applying the bondi - hoyle accretion rate , @xmath25 in the spherically symmetric wind from the companion , and @xmath26 is the relative velocity of the secondary star to the wind . accreted matter is mixed in surface convection of depth @xmath27 and @xmath28 in mass for giants and dwarfs , respectively . e.g. , for the stellar metallicity @xmath29 , the mass of accreted matter has to be larger than @xmath30 and @xmath31 , and hence , the upper bounds are @xmath32 @xmath33 for dwarfs , respectively . if we specify the initial mass function , @xmath34 , and the distributions of binary parameters , therefore , we can evaluate the frequency of cemp-@xmath7 and cemp - no@xmath7 stars , and through the comparison with the observations , we may impose the constraints on the imf and on the binary parameters . the numbers of cemp-@xmath7 and cemp - no@xmath7 stars currently observable in flux - limited samples are given by @xmath35 ) \nonumber\\ \times \int_{0.8 m_\odot}^{3.5 m_\odot } & dm_1 & \xi_b ( m_1 ) \frac{n(q)}{m_1 } \int^{a_m(m_1,m_2)}_{a_{\rm min}(m_1,m_2 ) } f(p ) \frac{dp}{da}da \\ \psi_{{\rm cemp\hbox{-}no}s } & = & f_b \int^{0.8 \ , m_\odot } _ { 0.08 \ , m_\odot}d m_2 n_s ( l[m_2 ] ) \nonumber \\ \times \int_{3.5m_\odot}^{m_{up } } & dm_1 & \xi_b ( m_1 ) \frac{n(q)}{m_1 } \int^{a_m(m_1,m_2)}_{a_{\rm min}(m_1,m_2 ) } f(p ) \frac{dp}{da}da , \end{aligned}\ ] ] where @xmath36 is the binary fraction : @xmath37 is the distribution of the mass - ratio , @xmath38 , and @xmath39 is the distribution of the period of binaries : and @xmath40 is the probability of the stars in the galactic halo with the luminosity @xmath41 in the survey volume of he s survey . similarly the total number of emp survivors is given by @xmath42 ) [ ( 1-f_b ) \xi_s ( m ) + f_b \xi_b ( m ) ] \nonumber \\ & & + f_b \int^{0.8 \ , m_\odot } _ { 0.08 \ , m_\odot}dm_2 n_s ( l[m_2 ] ) \int_{0.8 \ , m_\odot } ^{\infty } dm_1 \xi_b ( m_1 ) \frac{n(q)}{m_1 } , \label{eq : surv}\end{aligned}\ ] ] the rest of the terms give the number of emp survivors formed as binary , @xmath43 . for the form of imf , we may well assume a lognormal function with the medium mass , @xmath44 , and the dispersion , @xmath45 , as parameters @xmath46.\ ] ] in addition , we assume the binary fraction @xmath47 in this paper . our results are little affected by the assumption about @xmath36 since not only the cemp stars but also most of the emp survivors come from the secondary companions of binaries unless @xmath48 , as seen later . as for the binary period , we may adopt the distribution derived for the nearby stars by @xcite , the binary fractions and period distributions of halo stars are observed to be not significantly different from those of nearby disk stars @xcite . additionally , it is shown in paper i that this period distribution is consistent with the observations of cemp stars for the periods of @xmath49 yr confirmed to date ( see fig . 3 in paper i ) . the mass ratio distribution is an essential factor in discussing the evolution of binary systems , and yet , it is not well understood . the mass ratio distribution of metal - poor halo stars is investigated observationally ( e.g. , see * ? ? ? * ; * ? ? ? * ) , and yet , subject to large uncertainties . especially for the binary with intermediate - mass or massive primary stars , it is hard to know the mass ratio distribution from the observations . theoretically , neither the fragmentation of gas cloud nor the accretion process onto proto - binaries are yet well understood even for population i stars ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . in order to test the assumption on the mass - ratio distribution , we investigate the constraints on the imf for different mass - ratio distributions . in paper i , the simplest flat distribution is assumed in paper i among the possible distributions . we may define the coupling mass distribution function , @xmath50 , as the fraction of the binaries with a primary and secondary star in the mass range of @xmath51 $ ] and @xmath52 $ ] @xmath53 and write it in the form ; @xmath54 mass function , @xmath55 , of the primary star is assumed to be the same as imf of single stars : and @xmath37 is the mass ratio distribution , for which we assume both extremities of increase and decrease functional forms in addition to the constant one , adopted in paper i ; @xmath56 & ( \rm{case~b } ) \\ q^{-1 } / \ln ( m_1/0.08 { \ , m_\odot } ) & ( \rm{case~c } ) . \end{array } \right.\ ] ] furthermore , we take up a different type of mass - ratio distribution that the primary and secondary stars independently obey the same imf such as assumed by @xcite . in this case , the coupling mass distribution function is given as a product of the same imf as ; @xmath57 we shall refer to this distribution function as independent " coupling . from the comparison with eq . ( [ eq : masscouple1 ] ) , we may write the mass - ratio function in the form @xmath58 ; it is should be noted , however , that the frequency of binaries with a primary star of mass @xmath19 is not normalized and increases with @xmath19 from zero to 2 , as given by the integral @xmath59 . with these specification and with the assumed mass - ratio distribution function , we may compute the fractions of emp survivors , @xmath60 and of both cemps stars , @xmath61 and @xmath62 , and search the ranges of the imf parameters , medium mass @xmath44 and dispersion @xmath45 , that can reproduce the statistics of cemp stars consistent with observations we can pose another constraint from the total iron yield , @xmath63 , of emp population and the total number , @xmath64 , of the giant emp survivors . for @xmath64 , estimated from the results of existent surveys , the total stellar mass , @xmath65 , of emp population for an assumed imf is given by , @xmath66 where @xmath67 is the fraction of giant emp survivors in all the stellar systems , born as emp population , and @xmath68 is the averaged mass of emp population stars : @xmath69 \delta m_{\rm g } , \label{eq : frac - giants}\ ] ] @xmath70.\ ] ] the first terms of both equations denote the contributions by the stars born as the single stars and as the primary stars in the binaries and the second terms denote the contributions by the stars born as the secondary stars in the binaries . the mass and mass range of emp stars now on the giant branch are taken to be @xmath71 and @xmath72 , based on the stellar evolution calculation of stars with @xmath0}= -3 $ ] , as in paper i. the massive stars of emp population have exploded as supernovae to enrich the interstellar gas with metals . the amount of iron , @xmath73 , ejected by all the supernovae of emp population of the total mass , @xmath65 , is given by @xmath74 where @xmath75 is the fraction of the stars that have exploded as supernovae and given by , @xmath76 : \label{eq : frac - sn}\ ] ] and @xmath77 is the averaged iron yield per supernova , taken to be @xmath78 in the following calculations . with these evaluations and the observed number of emp giants , we can give the total iron yield of stars of emp population as a function of imf parameters . the comparison with the total amount of iron estimated from the chemical evolution of galactic halo may impose constraint on the imf parameters . the first constraint is the number fraction of cemp-@xmath7 stars . the hk and he s observations tell that the cemp stars with @xmath79 account for @xmath80 of emp stars ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? @xcite suggest a slightly lower fraction of @xmath81 with the errors in the abundance analysis taken into account while in this paper , we adopt the observational constraint on the fraction of the cemp-@xmath7 stars at @xmath82 ; @xmath83 the second constraint is the number ratio between cemp - no@xmath7 and cemp-@xmath7 stars . the observed frequency of cemp - no@xmath7 to cemp-@xmath7 stars is @xmath84 or more , ( e.g. , * ? ? ? * ; * ? ? ? @xcite point out that it increases for lower metallicity , reporting the ratio as large as @xmath85 for @xmath0}\le -2.5 $ ] . in addition , emp stars enriched with nitrogen are found in number comparable with , or more than , cemp - no@xmath7 stars ( mixed stars ; * ? ? ? * ) , whose origin can be interpreted in terms of the same mechanism but with more massive primary companions that experience the hot bottom burning ( hbb ) in the envelope of the agb . some other scenarios for cemp stars have been proposed @xcite but we assume all cemp stars are formed in binaries with agb in this paper . we adopt the observational constraint on the relative frequency of cemp - no@xmath7 to cemp-@xmath7 stars at @xmath86 ; @xmath87 we note that the above two constraints are not dependent on the total mass nor on the spatial distribution of the stellar halo because they are concerned with the relative number ratios . the third constraint is the total iron yield from the emp population . the he s survey obtained 234 stars of @xmath0 } < -3 $ ] @xcite as a result of the medium - resolution , follow - up observations of 40% of the candidates , selected by the objective - prism survey of the nominal area @xmath88 @xcite . taking the relative frequency between the giants and dwarfs ( @xmath89 ) and the ratio of the stars of @xmath0}<-3 $ ] and @xmath0}<-2.5 $ ] ( @xmath90 ) from their table 3 , we may estimate the total number of emp giants in the galactic halo in the flux limited sample at : @xmath91 we assume that all giant stars in the survey areas are observed because of the fairly large limiting magnitude of he s survey ( @xmath92 ) , about two magnitude deeper than for the hk survey , and neglect the spatial distribution of emp giant for simplicity since the sufficient information is not yet available ( see 7.3 in paper i for the detail ) . then we have the total number of emp giants @xmath93 in the galaxy . on the other hand , the amount of iron necessary to promote the chemical evolution of the whole gas in galaxy of mass , @xmath94 , up to @xmath0}= -2.5 $ ] is as much as @xmath95 and the supernovae of emp population should have provided this amount of iron unless there were other population(s ) of stars which made iron without producing low - mass stars . @xmath96 for the four mass - ratio distributions , formulated in [ subsec - modelpara ] , we can figure out , as the function of @xmath44 and @xmath45 , the portion of stars that survive to date ( @xmath97 ) , and then , the fractions of stars in these emp survivors that evolved to cemp-@xmath7 and cemp - no@xmath7 stars according to the masses of primary stars and to the orbital separations . figure [ fig : psi ] compares the fractions of cemp-@xmath7 stars in the emp survivors [ @xmath98 and the ratios of cemp - no@xmath7 to cemp-@xmath7 stars [ @xmath99 , predicted with use of these four different mass - ratio distributions , as a function of medium mass @xmath44 of imfs with the dispersion of @xmath100 , taken to be same as the present - day imf of galactic spheroid component @xcite . figures [ fig : s ] and [ fig : nos ] present the contour maps on the @xmath44-@xmath45 diagram for the fractions of cemp-@xmath7 stars and the ratio between the cemp - no@xmath7 and cemp-@xmath7 stars , respectively . cc + left top panels on these figures show the results for the flat mass - ratio distribution of case a , which reproduces the results obtained in paper i. in fig . [ fig : psi ] , the cemp-@xmath7 fraction peaks at @xmath101 , slightly above the upper mass limit of the primary stars for cemp-@xmath7 . note that when the secondary mass is specified , the mass distribution of primary stars peaks at mass smaller than @xmath44 for this mass - ratio function [ @xmath102 , see fig . 12 in paper i ) . two ranges of @xmath44 , @xmath103 and @xmath104 ( light shaded parts ) gives the imfs compatible with the observations , separated by the overproduction of cemp-@xmath7 stars . the relative frequency of cemp - no@xmath7 to cemp-@xmath7 stars is a steep increase function of @xmath44 , and excludes the lower range of @xmath44 compatible with the cemp-@xmath7 fraction . the imfs with @xmath105 ( dark shaded part ) gives compatible ratio with the observations this range of @xmath44 lies in the mass range of primary stars of cemp - no@xmath7 stars or even larger . accordingly , the intersection of the light and dark shaded parts designates the ranges ( @xmath106 ) that can explain the both statistics of cemp stars , and hence , high - mass imfs results for a dispersion @xmath107 . on the @xmath108 diagram of fig.[fig : s ] , the parameter space compatible with the observed cemp-@xmath7 fraction separates into two ranges for the dispersion smaller than @xmath109 , converging to the narrow ranges around @xmath110 and @xmath111 , respectively , as @xmath45 decreases . for larger dispersion , on the other hand , it merges into one part to cover wider range . as for the ratio between the cemp - no@xmath7 and cemp-@xmath7 stars , fig . [ fig : nos ] shows that the medium mass compatible with observed ratio increases with the dispersion to cover wider range , from @xmath112 at @xmath113 through @xmath114 at @xmath115 . accordingly , for the imfs that satisfy the both statistical constraints , the medium mass increases with the dispersion from @xmath116 for @xmath117 and beyond @xmath118 for @xmath119 . for the mass - ratio distribution function increasing with @xmath120 of case b ( right top panel ) , the portion of binaries that have the secondary stars surviving to date decreases with the mass of primary stars in proportion to @xmath121 , more steeply than in proportion to @xmath122 for a flat mass - ratio distribution in case a. since the average mass of the primary stars is smaller for a given emp star , therefore , the fraction of cemp-@xmath7 stars is larger for a given @xmath123 , and the peak shifts to larger @xmath123 , as compared with case a. in fig . [ fig : psi ] , the @xmath123 of imfs compatible with the observed fraction of cemp-@xmath7 stars separates into two ranges , as in case a , but the in - between gap is larger ; the higher mass range shifts upward in mass to greater extent ( @xmath124 ) than the smaller mass range shifts downward ( @xmath125 ) . this also causes a smaller ratio of cemp - no@xmath7 to cemp-@xmath7 stars for a given @xmath44 , and hence , the imfs compatible with the observed ratio shift to a larger mass of @xmath126 , as compared with that for case a. as a result , the imfs consistent with the both statistics of cemp stars turns out to be higher mass by a factor of @xmath127 than for case a ( @xmath128 for @xmath100 ) . in the fig . [ fig : s ] , we see that the range of @xmath44 , compatible with the observed fractions of cemp-@xmath7 star ( shaded area ) , separates into two and the higher range shifts to larger mass for a given @xmath45 . similarly , in the fig . [ fig : nos ] , the observed ratio of the cemp - no@xmath7 to cemp-@xmath7 stars also demands larger @xmath44 , and the range of @xmath44 of imfs compatible with the observation increases rapidly with @xmath45 to exceed @xmath129 for @xmath130 . in order to satisfy the both conditions of cemp star observations , the imfs fall in the range of higher medium mass and in a rather narrow range of dispersion , lying in the parameter space of @xmath131 , larger by a factor of @xmath132 than for case a , and @xmath133 and of @xmath134 for @xmath135 . for a mass ratio function decreasing with @xmath120 of case c , the portion of emp binaries whose low - mass members survive to date depends only weakly on the primary mass ( @xmath136 ) so that the fraction of cemp-@xmath7 stars reduces because of larger contributions from the binaries with more massive primaries . as seen in fig . [ fig : psi ] , the fraction of cemp-@xmath7 stars in the total emp survivors is well below the upper bound of the observations , and hence , the @xmath44 compatible with the observations merges into one narrower range of @xmath137 for @xmath100 . the observed ratio of cemp - no@xmath7 to cemp-@xmath7 stars can be reproduced also by the imfs with a smaller @xmath44 by a factor of @xmath127 than in case a ( @xmath138 ) . accordingly , the @xmath44 for the imfs consistent with the both cemp star statistics are smaller by a factor of @xmath139 than for case a ( the mass range @xmath140 for @xmath141 ) . in the fig . [ fig : s ] , the range of @xmath44 for the imfs , compatible with the observed cemp-@xmath7 fraction varies only little with @xmath45 , and is restricted in the range between @xmath142 , though it separates into two for small @xmath143 . as shown in fig . [ fig : nos ] the dependence of the ratio of cemp - no@xmath7 to cemp-@xmath7 stars on @xmath45 is also weaker than for case a. consequently , the imfs can reproduce the both cemp star statistics with the mass as small as @xmath144 , smaller by a factor of @xmath145 than for case a , but differently from the above two cases , an upper bound is placed at @xmath146 , regardless of the dispersion with a lower bound of @xmath147 . bottom right panels depict the results for the independent " coupling of case d. for this case , the number of emp survivors produced per binary is independent of the mass , @xmath19 , of primary stars , while the binary frequency itself increases with @xmath19 . the former is similarly to case c , and then , the production of emp survivors from the binaries with massive primary poses a severe constraint on the high - mass side of imfs . on the other hand , the latter favors the production of cemp-@xmath7 stars as compared with the low - mass binaries of @xmath148 . the both shift the imfs , compatible with the observed fraction of cemp-@xmath7 stars , to smaller @xmath44 . in addition , the single stars , born in the same number of binaries , contribute to significant fraction of emp survivors , increasing from @xmath149 up to @xmath150 for smaller @xmath44 for @xmath48 since the low - mass binaries are counted as one object . as a result , the maximum fraction of cemp-@xmath7 stars remains below the upper limit of the observed range , which makes the @xmath44 for the imfs that can reproduce the observation lie in a single range within a relatively small upper bound . the observed ratio of cemp - no@xmath7 to cemp-@xmath7 stars demands also lower - mass imfs , as for case c. accordingly , the imfs that can reproduce the both cemp star statistics fall in the narrowest range of @xmath151 with rather small upper mass limit , almost irrespective of the dispersion , on the @xmath152 diagram in fig.[fig : nos ] . the cemp-@xmath7 star faction remains smaller than @xmath153 because of the contribution of the stars born as single . in conclusion , the statistics of cemp stars demand the imfs for the emp population , peaking at the intermediate - mass stars or the massive stars , by far higher mass than those of pop . i and ii stars , irrespectively of the assumed mass - ratio distribution . the presence of cemp - no@xmath7 stars in a significant number of the cemp-@xmath7 stars excludes the imfs of small mass . the derived mass range varies by a factor of @xmath127 , from the highest @xmath131 for the mass - ratio distribution of increase function of the mass ratio ( case b ) to the lowest @xmath154 for the mass - ratio distribution of " independent coupling ( case d ) . this tendency is explained in terms of the difference in the averaged mass of the primary companion of the emp survivors ; if the contributions to the emp survivors decrease rapidly with the mass of primaries , a relatively higher - mass imfs result without an upper mass limit imposed , while if the contribution to the emp survivors are weakly dependent or independent on the primary masses , an upper limit is set with the relatively smaller - mass on the imfs . dditional constraints can be derived the total number of emp survivors and the iron yields from the emp population . figure [ cefig ] shows the contours of the total iron mass , @xmath73 , produced by the massive stars of emp population , the total mass and the amount of iron production of emp population are sensitive both to @xmath44 and @xmath45 , especially for small @xmath45 and large @xmath44 in contrast to with the other cases . for small @xmath45 , therefore , the fraction of low - mass stars varies greatly with @xmath44 , and the both contours of @xmath65 and @xmath155 converge to @xmath156 . as @xmath45 increases , the differences from case a diminish since the imfs tend to extend into the low - mass stars , and in particular , for @xmath157 and @xmath158 , the contours in the both panels resemble each other to run through the similar parameter spaces . from the comparison with the total amount of iron @xmath159 , necessary for the chemical evolution , in this diagram , the parameter space where @xmath160 is excluded by the overproduction of iron or by the underproduction of emp survivors . for the parameter space where @xmath161 , on the other hand , the stars of emp population can leave the number of emp survivors currently observed but are short of iron production , so that the chemical evolution demands other sources of iron production without producing the low - mass stars that survive to date . for a flat mass - ratio distribution , the imfs that can satisfy the condition of iron production coincide the imfs , derived above from the statistics of cemp stars ( shaded area ) in the parameter range of @xmath162 and @xmath163 . for the independent " coupling , the parameter range of imfs that satisfy the condition of iron production also overlap the shaded area of parameter range , derived above from the statistics of cemp stars , but with the mass @xmath164 , slightly smaller than for case a and only for a small dispersion of @xmath165 . for larger @xmath45 , even the highest - mass imfs of @xmath166 result to be slightly short of , or marginally sufficient at the most , iron production . for the two other mass - ratio distributions of @xmath167 ( case b ) and @xmath168 ( case c ) , the iron production , @xmath73 , with a given imf results to be larger or smaller than for case a because of the difference in the number of massive stars exploded as supernova per low - mass survivor ( e.g. , by a factors of 1.38 and 0.47 , respectively , per a star of @xmath169 and the imf of @xmath170 and @xmath171 ) . the iron production then demands smaller - mass ( higher - mass ) imfs for case b ( case c ) as compared with case a , the shift of imfs in an opposite direction , discussed from the statistics of cemp stars . accordingly , for these two extreme cases , the parameter ranges for the imfs derived from the statistics of cemp stars and the iron production are marginally overlapped ( case c ) or are dislocated with a narrow gap ( case b ) , although a definite conclusion waits for future observations , in particular , to improve the estimate of total numbers of emp stars ( see appendix ) . the relative production rate of carbon to iron may also impose additional constraint since the intermediate - mass stars enrich intergalactic matter with carbon through the mass loss on the agb , as discussed by @xcite . in particular , when @xmath45 is small and @xmath44 is in the range of intermediate- and low - masses , the intermediate - mass stars much surpass the massive stars in number and eject more carbon than the latter eject iron . we compute the amount of carbon ejected by agb stars by taking the carbon abundance in the wind ejecta of agb stars at @xmath172 , and the remnant mass at @xmath173 . contours of @xmath174 are plotted in the figure ( dashed lines ) , for which only the carbon from the agb stars are taken into account . the overabundance of carbon excludes the imfs with low dispersion and low medium mass ; it excludes the parameter space in the range of @xmath175 , derived by the cemp star statistics for case d , but has nothing to do with the high - mass imfs derived for case a. we demonstrate that the imfs , derived from the observed properties of cemp stars , have the parameter ranges that can explain the chemical evolution and the production of low - mass stars , consistent with the observations , both for the flat mass - ratio distribution and for the independent " coupling . in appendix we will discuss the converse to demonstrate that the argument based on the total number of emp survivors and the total iron production can potentially provides more stringent constraint on imfs , independent of the argument based on the cemp star statistics . in this paper , therefore , we the different assumptions on the mass - ratio distributions admit the parameter ranges of high - mass imfs that can reproduce the statistics of cemp stars and the chemical evolution , consistent with the existent observations . the predicted mass ranges differ by a factor of 2 or more between @xmath176 . although hardly distinguishable from the observations discussed so far , they surely make the differences in the properties of emp survivors . we discuss the imprints that the mass - ratio distributions have left on the current emp survivors and investigate the possibility of discriminating the mass coupling of binary systems in the emp population , especially for the two distinct distributions of the flat mass - ratio distribution and the `` independent '' coupling . firstly , an obvious difference is the mass distribution function of emp survivors . for a given imf , @xmath177 , the mass distribution , @xmath178 , of emp survivors is given by ; @xmath179 here a low - mass binary , whose components are both less massive than @xmath180 , is counted as one object with the primary star . figure [ fig : survivors ] shows the mass distributions of emp survivors ( @xmath181 for different assumptions of mass - ratio distributions cases a - c . for these mass - ratio functions , the mass distribution of emp survivors is nearly proportional to the mass - ratio distribution @xmath37 because almost all of them come from the secondary stars ; the contribution from the primary components are denoted by thin solid line , and the same contribution comes from the stars born as single . for the independent " coupling , in contrast , the @xmath178 , has the same form as the imf and the number of emp survivors decreases rapidly as the stellar mass decreases . secondly , the fraction of double - lined binary and the contribution of stars born as single among emp survivors may differ according to the mass - ratio distribution . for the flat mass - ratio distribution , this gives a significant fraction of @xmath182 for @xmath183 and @xmath171 , and increases with @xmath45 to @xmath184 for @xmath185 and with decreasing @xmath44 to 18% for @xmath186 , respectively . the number of low - mass binary decreases rapidly for smaller masses while the number of emp survivors , formed as the low - mass members of white dwarf binaries or supernova binaries , remains constant . for the independent " coupling , the fraction of low - mass binaries in the emp survivors reduces to ; @xmath187 , \end{aligned}\ ] ] which gives a much smaller fraction of @xmath188 for @xmath186 and @xmath189 as compared with the flat mass - ratio distribution . the fraction may increase for smaller medium mass , to 5.5% at @xmath190 , and for larger dispersion , to 3.9 % and 9.5% at @xmath191 and 0.7 , respectively , although these may cause underproduction of iron , in particular for smaller @xmath44 , as seen from fig . [ cefig ] ( bottom panel ) . in this case , the proportion of the emp survivors , born as single stars , is fairly large as given by @xmath192.\ ] ] consequently , nearly one third of emp stars were born as single stars , for @xmath193 , which is much larger fraction than in the case of the flat mass - ratio distribution . thirdly , the fraction , @xmath194 , of supernova binaries with the primary stars of mass @xmath195 also differs between the two mass - ratio distributions . for the flat mass - ratio distribution , almost all of the emp survivors belong , or have been belonged , to the binary systems , and the fraction is given by @xmath196 and amounts to @xmath197 . for the independent " coupling , on the other hand , one third of emp survivors are single stars from their birth , and the percentage of supernovae binaries is relatively small , as given by @xmath198,\end{aligned}\ ] ] and turns out to be @xmath153 . the emp survivors from the supernova binaries have experienced a supernova explosion of the erstwhile primary stars at close distances and are thought to suffer from some abundance anomalies , affected by supernova ejecta . accordingly , these stars , in particular from the binaries of sufficiently small separations , may be discriminated by a large enhancement of elements , characteristic to the supernova yields . these differences in the properties of remnant emp survivors may potentially serve as tools to inquire into the nature of emp binaries and to distinguish the mass - ratio distributions . among the emp stars , several double - lined spectroscopic binaries are reported in the literature . if we restricted to the metallicity range of @xmath0}<-3 $ ] , for which the observations with high - resolution spectroscopy may be regarded as unbiased , there are two stars cs22876 - 032 and cs 22873 - 139 with the detailed analyses and one star he 1353 - 2735 ( @xmath0}\simeq -3.2 $ ] , @xmath199 ; * ? ? ? * ) without the binary parameter . so far 39 dwarf stars of @xmath0 } < -3 $ ] are confirmed by the high - resolution spectroscopy ( we define the dwarf as @xmath200 \ge 3.5 $ ] ) , and hence , the fraction of low - mass binaries , composed of two unevolved emp stars , turns out to be @xmath201 . it may be more straightforward to compare our results with the mass distribution function of emp survivors . from the existent observations , however , it is rather hard to determine since the observed dwarfs are mostly concentrated near to the upper end of main sequence . an exception is a carbon dwarf g77 - 61 of @xmath0}= 4.03 $ ] @xcite whose mass is inferred at @xmath202 , but it was found among the proper - motion - parallax stars @xcite , not from the surveys . we have to wait for the larger - scaled surveys in near future to reveal the distribution of emp survivors of low masses . as for the supernovae binaries , they are expected to be related to the large star - to - star variations in the surface elemental abundances , in particular , with those of r - process elements , ranging more than by two orders of magnitude . it is necessary , however , to understand the nature of interactions between the supernova ejecta colliding at very high velocity and the near - by low - mass stars before the meaningful conclusions can be drawn from the observations . we have shown that the high - mass imfs with the binary provide a reasonable explanation of the observed properties of emp stars in the galactic halo , revealed by the recent large - scaled hk and he s surveys . in this section we discuss the consequence of derived imf on the metal enrichment history of galactic halo up to @xmath0}=-2.5 $ ] to study their relevance to the metallicity distribution function ( mdf ) , observed for the emp stars . under the assumption that matter ejected from supernovae spreads homogeneously and is recycled instantaneously , the iron abundance , @xmath203 , of our galaxy of ( baryonic ) mass @xmath204 can simply be related to the cumulative number , @xmath205 , of the stars born before the metallicity reaches @xmath203 as ; @xmath206 where @xmath77 is the averaged iron yield per supernova and @xmath207 is the fraction of emp stars that have exploded as supernovae , defined in eq . ( [ eq : frac - sn ] ) . by differentiating it with respect to @xmath0}= \log ( x_{\rm fe}/ x_{\rm fe , \odot})$ ] , the number distribution of emp survivors is written as a function of metallicity in the form @xmath208 } ) = \frac { d n ( x_{\rm fe } ) } { d{[{\rm fe } / { \rm h } ] } } = \frac{m_h } { \langle y_{\rm fe } \rangle f_{\rm sn } } \ln(10 ) x_{{\rm fe}\odot } 10^{{[{\rm fe } / { \rm h } ] } } .\ ] ] this shows that the number distribution of emp survivors is simply proportional to the iron abundance apart from the variation of @xmath209 through the imf and the latter is small enough to be neglected for @xmath210 ( see fig . [ fig : yield ] in appendix ) . figure [ mdf ] depicts the number distribution of emp survivors and compares it with the observed mdf provided by the he s survey @xcite . we assume stars of mass @xmath211 become type ii supernova and eject @xmath212 of iron . in this figure , the theoretical mdf , @xmath213 , is evaluated under the same flux - limited condition as the observed mdf is derived ; @xmath214 } ) = n ( { [ { \rm fe } / { \rm h } ] } ) f_{\rm g } \times ( 40 \% ) \times ( 8225 \hbox { degree}^2 / 4 \pi \hbox { sr } ) \times 1.93 $ ] . here the fraction of follow - up observation and the sky coverage are taken into account : as for the contribution of to stars , we take the same ratio to the giants as in the observed sample under the assumption that the giant survivors are all reached in the survey area . solid line shows the mdf for the imf with @xmath215 and @xmath171 with the 50% binary fraction , derived above for emp population stars for the flat mass - ratio distribution , and it is similar to the other mass - ratio functions , as discussed in 2.3 . this reasonably reproduces the observed mdf between the metallicity @xmath3}\lesssim -2.5 $ ] , as expected from the discussion in the previous section . in this figure , we also plot the mdf using the low - mass imfs , the salpeter s power - law mass - function as observed among the present - day stellar populations and that derived only from the statistics of cemp-@xmath7 stars by ( * ? ? ? * @xmath216 and @xmath217 ) . they bring about the overproduction of emp survivors by a factor of more than a few hundreds not only from the low - mass members of binaries but also from the primary stars and the single stars ; both the imfs give the similar mdf since our flux - limited samples are dominated by the giants and luminous dwarfs of mass @xmath218 . this means that the emp survivors is by far a small population as compared with the stellar systems of pop . i and ii , and it is only with the high - mass imfs that can make the emp population produce sufficient amount of metals to enrich the early galactic halo without leaving too many low - mass survivors now observable in galactic halo . in addition , we see in this figure that the slope of observed mdf is consistent with the prediction from the simple one - zone approximation at least for @xmath0 } > -4 $ ] . it implies that the imfs have little changed while the galactic halo has evolved through these metallicities . beyond @xmath0}\simeq -2 $ ] , the observed mdf derived from the hk and he s surveys seems to be underestimated since those objects are out of the metallicity range sought after by the survey and subject to imperfect selection . the observed mdf of galactic halo stars has a sudden drop at @xmath219}\lesssim -4 $ ] , and only three stars are found below it}<-4 $ ] ; cd @xmath220 with @xmath0}= -4.19 \pm 0.10 $ ] @xcite and g77 - 61 with @xmath0}= -4.03 \pm 0.1 $ ] @xcite . we will omit these two stars in our discussion since larger abundances of @xmath0}= -4.07 \pm 0.15 $ ] @xcite and @xmath221 @xcite have been reported for the former , and hence , their abundances are closer to the emp stars of @xmath0}\gtrsim -4 $ ] than to the other three hmp / ump stars . ] . we propose the mechanism responsible for this depression of low - metallicity stars from the consideration of the galaxy formation process . in the current cold dark matter ( cmd ) model , galaxies were formed hierarchically . they started from low mass structures and grew in mass through merging and accreting matter , finally to be large - scale structures like our galaxy . in the hierarchical structure formation scenario with @xmath222cdm cosmology , the typical mass of first star forming halos is @xmath223 for the dark matter and @xmath224 for the gas ( e.g. , see * ? ? ? * ; * ? ? ? * ) . in these first collapsed gas clouds , the first stars contain no pristine metals except for lithium . when the first star explodes as supernova , it ejects @xmath225 of iron , which enriches the gas cloud of mass @xmath224 where it was born up to the metallicity of @xmath0}\sim -3.5 $ ] if the ejecta is well mixed in the gas cloud . we call this event the first pollution " . consequently , the 2nd generation stars have the metallicity of @xmath0}\sim -3.5 $ ] . in the course of time , the mini - halos that host the gas clouds merge with each other and accrete the intergalactic gas to form early galactic halo with the baryonic mass of @xmath226 . we may take the metallicity of this early galactic halo to be @xmath0}\simeq -4 $ ] because of the scarcity of stars of metallicity @xmath0}<-4 $ ] . the cumulative number of stars born before the early galactic halo is enriched up to @xmath0}=-4 $ ] is estimated at @xmath227 with taking into account the supernova fraction @xmath207 . if the mini - halos of larger masses stand between the first collapsed halos and the galactic halo , the dilution of iron with unpolluted primordial gas can give birth to the stars of smaller metallicity of @xmath0}\simeq -4 $ ] , and then , the metallicity at the formation of galactic halo can be larger to increase the cumulative number of stars in accordance ( see below ) . we may estimate the fractions of both the first generation stars without metals and the 2nd generation stars of the metallicity @xmath0}\sim -3.5 $ ] , respectively , assuming that stars are born with an equal probability whether in the gas clouds , polluted with metals , or in the primordial gas clouds . accumulated number , @xmath228 , of pop iii stars , born of gas unpolluted by sn ejecta , when the average metallicity reaches @xmath203 , is given by @xmath229 , \label{eq : num - popiii}\ ] ] where @xmath230 is the mass of gas in the first star forming clouds . if we assume the same imf and binary parameters as in the stars of emp population , then , we expect that the number of pop iii stars is @xmath231 and the number of pop iii survivors is @xmath232 \nonumber \\ & & = 1.3 \times 10 ^ 4 , \end{aligned}\ ] ] and similarly we have @xmath233 and @xmath234 of the 2nd generation stars and their survivors , formed before the averaged metallicity of the galaxy reaches @xmath0}=-4 $ ] . the imf of pop . iii stars may differ from emp stars but the existence of the stars with @xmath0}<-5 $ ] suggests that the low - mass stars can be formed before the first pollution . figure [ mdfpop3 ] illustrates an expected mdf with the hierarchical structure formation . after the formation of large galactic halo , the metal enrichment process is thought to follow the argument of the previous subsection . thus , we can explain the cutoff around @xmath0}\sim -4 $ ] naturally . shaded columns indicate the initial distributions of pop . iii stars and of the 2nd generation stars formed in the low - mass clouds . the 2nd stars were mixed and observationally lost their identities among the stars formed in the merged halo . on the other hand , pop . iii stars should form a distinctive class . from the above estimate , we expect @xmath235 pop . iii survivors in the existing flux - limited samples of he s surveys . similarly , the number of second - generation of stars is estimated at @xmath236 in the same flux - limited he s sample , indicative that most of emp stars are formed of mixture of the ejecta from plural supernovae . this has direct relevance to the study of the nucleosynthetic signatures on the emp survivors and the imprints of supernovae of the first and subsequent generations . we have studied the initial mass function ( imf ) and chemical evolution of the galactic halo population on the basis of the characteristics of extremely metal - poor ( emp ) stars , revealed by the recent large - scaled hk and he s surveys ; the observational facts that we make use of are ; ( 1 ) the overabundance of carbon - enhanced emp ( cemp ) stars , ( 2 ) the relative frequencies of cemp stars with and without the enrichment of s - process elements , ( 3 ) the estimate of surface density or total number of emp stars in the galactic halo , and ( 4 ) the metallicity distribution function ( mdf ) . we take into account the contribution of binary stars properly , as expected from the younger populations . in paper i , the high mass imf peaking around @xmath237 is derived for the stars of emp population and it is shown that the binary population plays a major role in producing the low - mass stars that survive to date , but by using the flat mass - ratio distribution between the component stars . in this paper , we examine these properties of the stars of emp population and emp survivors for the different types of mass - ratio distributions and investigate the constraints on the imfs of the stars of emp population and discuss the observational tests of discriminating them . the derived imfs are applied to understand the characteristics of mdf and the nature of emp stars including hmp / ump stars , provided by the surveys . our main conclusions are summarized as follows ; \(1 ) the statistics of cemp stars are explained by the high - mass imfs with the binaries of significant fraction . predicted typical mass is significantly larger than population i or ii stars , ( 2 ) \(3 ) the mass - ratio distribution of binaries in the emp population can be discriminated by the imprints left on the emp survivors such as the mass function , the binary fraction , and the fraction of stars influenced by the supernova explosion of primary stars . \(4 ) the observed mdf of emp survivors is consequent upon the derived imf with the contribution of the binaries . there is no indication of significant change in the imfs between the metallicity of @xmath3}\lesssim -2 $ ] . the depression of stars below @xmath0 } < -4 $ ] is naturally explicable within the current favored framework of the hierarchical structure formation model . then , the pop . iii stars born of primordial gas , and also , the stars in the primordial clouds before they are contaminated by their own supernovae , should form the distinct class other than emp stars , and may have the relevance to hmp and ump stars observed at lower metallicity , as discussed below in this subsection . the feature of our approach is to take into account the stars born in binary systems properly in discussing the low - mass star formation in early universe , based on the finding in paper i. in addition , we make full use of available information from the existent large - scaled surveys and to draw the maximal constraint on the early evolution of our galactic halo . the known emp stars ( @xmath0}\lesssim -2.5 $ ] ) with the detailed stellar parameters amount to @xmath238 in number ( saga database ; * ? ? ? * ) , and allow us to discuss the averaged properties as studied in this paper . discussion in 2.3 through the iron production consistent with the number of emp survivors will be left largely unaffected even quantitatively . in order to improve and sharpen our conclusions , we have to wait for the future larger - scaled surveys such as sdss / segue @xcite and lamost @xcite . also the high dispersion spectroscopy is necessary to understand the characteristics of emp stars . the constraints on the imfs derived in this work may serve as the basis of understanding the formation and early evolution of the galaxy . accordingly , there should be the transition from the high - mass imf to the low - mass one . our result suggests that the transition is postponed until high metallicity even beyond @xmath0}\simeq -2 $ ] is reached . it is likely that the transition may not be simply determined by the metallicity alone , in discussing the primordial stars or hmp / ump stars in the present work , we assume the metallicity at the formation of galactic halo at @xmath0}\simeq -4 $ ] . the detailed chemical evolution with the merger history taken into account is discussed in a subsequent paper ( komiya et al . 2008 , in preparation ) , in the similar ways as done by @xcite and by @xcite , but taking into account the high - mass imf , derived above , and the contribution of binaries . we end by discussing the consequences of the present study on the understanding of the origin of stars found below the cut - off of mdf . in our model , the stars made after the first pollution have the metallicity @xmath0}\simeq -3.5 $ ] and the stars with slightly lower metallicity of @xmath0}\simeq -3.5 - -4 $ ] are made in the merged clouds where metals are diluted with the primordial gas unpolluted by supernova ejecta . after the halos merge , the 2nd generation stars mingle and observationally lose their identities among the stars formed in the merged halo . @xcite argue the effects of surface pollution of pop . iii stars through the accretion of interstellar gas to show that the main - sequence pop . iii stars can be polluted to be @xmath0}\simeq -3 $ ] while the giants to be @xmath0}\simeq -5 $ ] since the pollutant is diluted by the surface convection deepening @xmath239 times in mass on the giant branch . thus , the pop . iii survivors have evolved to giants to be observed as hmp / ump stars . then some of pop . iii stars become carbon - enriched hmp / ump stars with @xmath0}\sim -5 $ ] through binary mass transfer . if the mass of primary star is @xmath240 and @xmath241 , the primary star enhances the surface abundances of carbon and nitrogen though the he - fddm and of carbon and/or nitrogen through tdu and hot bottom burning in the envelope , respectively , which are transferred onto the secondary stars through the wind accretion . it is to be noted that the primary stars of @xmath242 have the accreted pollutants mixed inward into the whole hydrogen - rich envelope at the second dredge - up , and thereafter , evolve like the stars with the pristine metals . at the same time , the accreted matter is diluted in the envelope and the iron abundance is reduced to @xmath0}\sim -5 $ ] in the primary stars . we estimate that @xmath243 of pop . iii stars become carbon - rich hmp / ump stars under the same assumptions on the binary parameters as in paper i. in fig . [ mdfpop3 ] , solid lines denote the expected mdf at the present days with the surface pollution taken into account . the basic form of observed mdf is reproduced , i.e. , the cutoff around @xmath0}\sim -4 $ ] , the scarcity of stars for the metallicity below it and the existence of a few hmp / ump stars . from the above estimates , there should be @xmath235 pop . iii stars in the existent flux - limited samples of he s surveys ; about a half of them may be discovered as giants with the surface metal pollution and one third as carbon stars . the above estimates are made , however , under the assumption that the pop . iii stars are formed in the same imf as emp stars and with the same binary parameters . this may not be warranted and rather we may take that this deficiency may suggest a still higher - mass imf and/or less efficiency of binary formation for pop . iii stars than the emp stars . in the above discussion , we assume the closed box chemistry in the collapsed object before merging . it is shown that the hypernovae , exploded with a large energy of @xmath244 erg , blow off the first collapsed objects of mass @xmath245 @xcite ; if the first stars are sufficiently massive , the metal yields are spread into larger masses , and pollute the ambient gas before they collapse to form mini - haloes , as discussed by @xcite . after that , the first stars in the collapsed clouds are no longer metal - free . nevertheless , those stars which are formed before each collapsed clouds are polluted by their own supernova form a distinct class from those which suffer from the first pollution . further study is necessary to make clear the present appearance of the possible pop iii survivors and to settle the origin of hmp / ump stars , in particular , for tiny amounts of iron - group metals and the overwhelming carbon - enhancement , shared by all these stars known to date . we benefit greatly from discussion with dr . w. aoki . this paper is supported in part by grant - in - aid for scientific research from japan society for the promotion of science ( grant 18104003 and 18072001 ) . one of the important findings of the recent large - scaled surveys is the scarcity of emp stars in the galactic halo . the he s survey gives the total number of emp stars in our galactic halo at @xmath246 ( giants of @xmath247 in eq . ( [ eq : hes - g ] ) plus turn - off stars @xmath248 within the limiting magnitude @xmath249 . similarly , the hk survey gives @xmath250 within the limiting magnitude of @xmath251 ; 114 stars of @xmath0 } < -3 $ ] are found by the medium - resolution , follow - up spectroscopy of 50% of the candidates , selected from the objective prism survey covering the @xmath252 and @xmath253 areas in the northern and southern hemisphere @xcite . because of the significantly large areas covered by these surveys ( @xmath254 of all sky with the follow - up observations ) , we may place reliance on these results , granted that they may not be complete . this also constrains on the imf of stellar population that promoted the chemical evolution , or more specifically , the formation of metals and the low - mass survivors . in the paper , we have discussed the chemical evolution starting with the statistics of cemp stars . in this appendix , we show that the chemical evolution with the total number of emp survivors provides more stringent constraints on the imf of emp population with the aid of the amount of ejecta from supernova models , independently of the statistics of cemp stars . our basic premise is that the same stellar population is responsible both for the production of metals and of low - mass survivors . in discussing the low - mass survivors , it is indispensable to take into account the contribution from the binaries . this is one of the major conclusions in paper i. we assume that the stars are born not only as single stars but also as the members of binaries in an equal number and with the primary stars in the same imf as the single stars . for a given imf , then , the total number , @xmath255 of emp survivors , currently observed in the galactic halo , is related to the cumulative number , @xmath256 , of stars of emp population as ; @xmath257 , \label{eq : frac - surv}\end{aligned}\ ] ] and hence , to the cumulative number of emp supernovae as @xmath258 . these supernovae have to supply the amount of iron , @xmath73 in eq . ( [ eq : emp - ironprod ] ) , in order to enrich the gas in the galaxy of mass @xmath204 with iron to promote the chemical evolution up to the metallicity @xmath0}= -2.5 $ ] . then , we may derive the averaged iron yield , @xmath259 , per supernova of emp population , necessary to explain the chemical evolution of galaxy , by the relation @xmath260 for an assumed imf with the mass - ratio distribution function . we show in figure [ fig : yield ] the averaged yield , @xmath259 , as a function of @xmath44 for @xmath189 : upper panel for the observations of emp stars of different evolutionary stages from the he s survey and of the total emp stars from the hk survey with use of the imfs with the flat mass - ratio function , and lower panel for the different mass - ratio functions with use of the observation of emp giants from the he s survey . in order to compare the stars of different evolutionary stages , we include the effects of the limiting magnitude of the surveys by assuming the de vaucouleurs density distribution , @xmath261 , with the radial distance , @xmath262 , from the galactic center , the same as the stars in the galactic halo and by assigning the luminosity of @xmath263 and @xmath264 to dwarfs and giants , respectively . the amount of iron demanded by the chemical evolution turns out to be a steep decrease function of @xmath44 since in order to leave a fixed number of low - mass survivors , the total number of stars of emp populations , and hence , the supernova fraction increase rapidly with @xmath44 in particular near @xmath265 . the necessary yields computed from the different samples in upper panel show a fairly good agreement with each other . the difference between the giants and dwarfs for the he s samples is indicative of a relatively deficiency of dwarf stars compared with giants by a factor of @xmath266 in number , which may be attributed to rather crude assignment of averaged giant luminosity , and/or to the different efficiency of identifying giants and turn - off stars in the survey plates , and/or to the uncertainties in the spatial density distribution . the results for the hk survey and the he s survey also agree within the difference by a factor of @xmath267 in number despite the difference in the limiting flux by 2 mag , and hence , to the difference in the searched volume by a factor of @xmath268 . the variations with the mass - ratio functions in the lower panel are caused by the difference in the number of supernovae per emp survivor . as compared to the flat mass - ratio function , the mass - ratio function increasing ( decreasing ) with @xmath120 give a larger ( smaller ) number of supernovae to produce one emp survivors ; the difference of which increases for higher - mass imfs . these iron yields necessary to promote the chemical evolution may be compared with the theoretical iron yields predicted from the supernova models . the imf - weighted iron yields , @xmath269 , per supernova is given by using the iron mass , @xmath270 , ejected from a massive star of initial mass @xmath271 as ; @xmath272 } { \int_{m_{up } } dm_1 \xi ( m_1 ) [ 1 + f_b \int_{m_{up}/m_1}^1 n(q ) d q ] } . \label{eq : yieldsn}\ ] ] the imf averaged yield @xmath269 is also shown in this figure , for which the theoretical yields are taken from the metal - deficient supernova models computed by @xcite , and by @xcite and @xcite . it is a slowly increase function of @xmath44 for @xmath210 with the increase in the fraction of more massive stars that ended as supernovae , while beyond it , the gradient grows steeper owing to the contribution of the electron pair - instability supernovae of @xmath273 . the averaged yields , demanded by the chemical evolution , and the theoretical imf - weighted iron yields both meet with each other near @xmath274 and with the iron yield @xmath275 per supernova . as typically seen for the flat mass - ratio distribution , the parameter range coincides with that we have derived for the imfs from the cemp statistics in figs . [ fig : psi]-[fig : nos ] . for higher - mass imfs , the emp stellar population can not produce the sufficient number of low - mass survivors by themselves , while for lower - mass imfs , it results short of iron production . the differences arising from the mass - ratio distributions seem discernible but not large enough to differentiate these mass - ratio distributions in view of the uncertainties of current observations . as compared with the flat distribution , the mass - ratio distribution increasing ( decreasing ) with @xmath120 demands smaller ( larger ) @xmath44 , the opposite tendency derived from the cemp statistics . these distributions prefer smaller ( larger ) number of emp survivors , larger ( smaller ) fraction of cemp-@xmath7 stars and smaller ( larger ) ratio of cemp - no@xmath7 to cemp-@xmath7 stars . in principle , however , we can discriminate the mass - ratio functions in the emp binaries , including those destructed already by the evolution , with use of the survey and observations of emp stars in sufficiently large number and with sufficient accuracy , which waits for future works . in the above discussion , we assume a single log - normal imf with the binary fraction for the stellar population . it is possible to assume the bi - modal imf and to explain the production of iron and the formation of low - mass stars , separately , in terms of the combination of two stellar populations , one with a higher - mass imf responsible for the iron production and the other with a lower - mass imf for the low - mass survivors , respectively . in the case of bi - modal imfs , the constraints , derived here , place an upper mass limit to the imf of lower - mass population and an lower mass limit to the imf of higher - mass population . it is to be noted that the imf with the binary mass function of cases a - c is regarded as a sort of bi - modal imf with the primary plus single stars as the higher - mass population and the secondary stars as the lower - mass population ( see fig . 12 in paper i ) ; 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we exploit the recent observations of extremely metal - poor ( emp ) stars in the galactic halo and investigate the constraints on the initial mass function ( imf ) of the stellar population that left these low - mass survivors of @xmath0}\lesssim -2.5 $ ] and the chemical evolution that took part in . a high - mass nature of imf with the typical mass @xmath1 and the overwhelming contribution of low - mass members of binaries to the emp survivors are derived from the statistics of carbon - enriched emp stars with and without the enhancement of s - process elements ( komiya et al . 2007 , ) . that the same constraints are placed on the imf from the surface density of emp stars estimated from the surveys and the chemical evolution consistent with the metal yields of theoretical supernova models . apply the derived high - mass imf with the binary contribution metallicity distribution function ( mdf ) of emp stars not only for the shape but also for the number of emp stars . in particular , the scarcity of stars below @xmath0}\simeq -4 $ ] is naturally explained in terms of the hierarchical structure formation , and there is no indication of significant changes in the imf for the emp population . the present study indicates that 3 hmp / ump stars of @xmath0 } < -4 $ ] are the primordial stars that were born as the low - mass members of binaries before the host clouds were polluted by their own supernovae .

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Proceed to summarize the following text: the phenomenology of nuclear photoabsorption is governed by two characteristic features . first , all nuclei with mass numbers @xmath5 ranging from 10 to more than 200 obey the same fundamental curve @xmath6 for the total photoabsorption cross section devided by @xmath5 as a function of the photon energy @xmath7 . second , the @xmath1-isobar excitation of the nucleon is responsible for the main properties of this curve in the energy region between 200 and 400 mev . models , which focus on the behaviour of the @xmath1-isobar in a nuclear environment , namely the @xmath1-hole calculations @xcite , have proven to be highly successful in explaining the experimental findings for pion scattering processes @xcite . indeed , on grounds of the @xmath1-hole formalism a wide variety of pion - nucleus reactions can be described within one consistent framework @xcite . in the case of photonuclear reactions , however , serious descrepancies remain , which partially have been accounted for by including non - resonant background terms @xcite . nevertheless , such a procedure , in particular for nuclear photoabsorption , either leads to contradictions with previous @xmath1-hole results or lacks the complete agreement with experimental data @xcite . the question arises , whether theoretical ingredients other than in - medium @xmath1-hole propagations can lead to a similar degree of accuracy . therefore it is natural to address the situation from a different point of view asking to what extend the absorption process can be described , when only a very simple @xmath1-nucleon interaction is used and all additional effects are accounted for in a purely diagrammatic approach . when combined with a simple form of nucleon momentum distribution inside the nucleus , namely a fermi gas model , this leads to analytical expressions , in which different corrections to this lowest - order approximation can be studied . this is the aim of the present article . in our formalism we follow closely wakamatsu and matsumoto @xcite . however , we do not introduce a phenomenological potential to account for the deviation of the nucleon wave functions from plane waves , but study the influence of such corrections in a perturbative way , similar to our previous work @xcite . in addition , our focus is on the energy - dependence of the total photoabsorption , rather than on the differential cross section as a function of the momentum of the outgoing proton . a characteristic feature of wakamatsu s and matsumoto s approach is the technically equal treatment of the ( @xmath8,pn ) and the ( @xmath8,pp ) knock - out process , which allowed them to obtain a natural explanation for the supression of the two - proton knock - out . this characteristic property is also present in our calculation , where it is related to a vanishing trace in spin space . in the last years the ratio ( @xmath8,pp)/(@xmath8,pn ) has been investigated in depth within different formalisms . in an extension of wakamatsu s and matsumoto s work by boato and giannini @xcite , finite - size effects have been calculated and , more recently , a combination of pion exchange and shell - model wave functions was used to investigate this quantity @xcite . currently , two complementary approaches for the description of nuclear knock - out reactions exist . carrasco and oset @xcite used a diagram - oriented many - body expansion in a fermi gas . the evaluation of self - energy diagrams leads to an accuracy for medium effects high enough to study knock - out reactions in great detail . with their primary goal being thus different from ours , their formalism does not yield isolated expressions for the resonant and non - resonant parts of the mechanisms of nuclear photoabsorption a different approach is used by the gent group @xcite , where the main emphasis lies in the construction of realistic shell - model wave functions , rather than on a microscopic description fully based on the evaluation of feynman diagrams . as the nuclear photoabsorption is almost insensitive to structural differences between nuclei , the quality of their approach becomes obvious in the investigation of differential cross sections for nucleon knock - out , rather than of photoabsorption . nuclear photoabsorption provides an interesting tool to study the interplay between one - nucleon and two - nucleon contributions . we obtained analytical expressions for these contributions , as well as for their resonant and non - resonant parts . this set of results can be used as a starting point to include ( and test ) further nuclear or nucleonic effects . in section 2 the basic notations are listed , as well as the interaction terms and the most important model assumptions and approximations , which are present in this calculation . the main results and their most prominent properties , e.g. the effects of relativistic corrections and nuclear structure , are discussed in section 3 , where also the following possible extension of such an approach is considered : as the angular dependence of the two - nucleon process is not very strong ( cf . @xcite ) , the different mechanisms contributing to the photoabsorption curve can also be used to understand qualitative features of experimental data for nucleon knock - out processes as a function of the photon energy . in section 4 some concluding remarks are made , with an emphasis on the applicability of the partial cross sections , whose analytical forms are given in the appendix . the starting point of our investigation is the static hamiltonian for the pion - nucleon interaction , @xmath9 together with a minimal coupling to the photon field . in eq.([s1eq1 ] ) @xmath10 is the pion mass . for all coupling constants we use the same notation and values as given in @xcite , in particular @xmath11=0.08 . in eq.([s1eq1 ] ) underlined symbols denote vectors in ( cartesian ) isospin space , while an arrow indicates a vector in coordinate space . in the static limit the interactions with the @xmath1-isobar excitation of the nucleon are determined by the following hamiltonians ( see e.g. @xcite ) : @xmath12 and @xmath13 with the hermitian conjugate to be added in both cases . here @xmath14 and @xmath15 are the 1/2-to-3/2 transition operators in spin space and isospin space , respectively ; @xmath16 is the proton charge , @xmath17 . for the coupling constants we have @xmath18=2 and @xmath19=0.35 . in all cases , where high momentum transfers occur at the pion - nucleon vertices we regularize the vertex functions by introducing dipole form factors @xmath20 as was also done e.g. in @xcite . the value for the cut - off parameter @xmath21 has been taken to be 800 mev . this value gives the best agreement of our predictions with the experimental data . in addition , a similar value has been used in @xcite . the general expression for the total cross section @xmath22 of the photoabsorption with one nucleon outside the fermi sphere and one pion in the final state ( one - nucleon process ) is of the form @xmath23\ ; . \label{s1eq5}\end{aligned}\ ] ] for the total cross section @xmath24 of the process with two free nucleons in the final state it is given by @xmath25\left [ { 1-n(\vec p_4)\ ; } \right ] \label{s1eq6}\ ; .\end{aligned}\ ] ] the notation for the external momenta is shown in fig.([figa ] ) . the function @xmath26 is the occupation number , @xmath27 is the step function . furthermore , @xmath28 is the nuclear volume , @xmath29 is the fermi momentum , @xmath30 is the mass of the proton , @xmath28 is the nuclear volume , @xmath31 , and @xmath32 is the energy of the outgoing pion . both amplitudes @xmath33 and @xmath34 consist of non - resonant and resonant parts , @xmath35 . diagrammatically this decomposition is shown in fig.([figb ] ) . the second ( crossed ) contribution to the resonant part is small due to the big energy denominator and will be skipped in the following . in the energy @xmath36-function in eq.([s1eq5 ] ) and eq.([s1eq6 ] ) we will usually not consider the smallest term connected with the kinetic energy of the incoming nucleon , as it is smaller than @xmath37mev . indeed , as we expect its contribution to introduce only a small modification of the actual @xmath38-dependence in the integrand in ( [ s1eq5 ] ) , we substitute it by its average value @xmath39 , which results in an overall shift of the absorption cross section . for the one - nucleon process we find first - order relativistic corrections to be essential for a successful treatment of the absorption process . in the case of the resonant contribution , such corrections are accounted for by making the following substitutions in the vertices @xcite : @xmath40 where @xmath41 is the @xmath42-isobar mass and @xmath43 . for the non - resonant part , the corrections give amplitude @xmath33 of the following form : @xmath44-{{2i(\vec \sigma \cdot ( \vec q-\vec k))(\vec \varepsilon \cdot \vec q)}\over { ( \vec q-\vec k)^2+m^2-(\varepsilon _ q-\omega ) ^2 } } \nonumber \\ & -&{{2i(\vec \sigma \cdot \vec q)(\vec \varepsilon \cdot ( \vec k+\vec p-\vec q))}\over { \,\left [ { ( \vec k+\vec p-\vec q)^2 - 2m\omega -(\vec p-\vec q)^2}\right]}}\left . { \matrix{{}\cr { } \cr } } \right\ } , \label{s1eq9}\end{aligned}\ ] ] which corresponds to the production of a @xmath45 . in comparison with @xcite we neglected those terms in ( [ s1eq9 ] ) , which modify the result by less than 2 per cent . it should be noted that no free parameters are present in our approach . a certain model dependence exists , however , in the selection of diagrams . we neglect all those diagrams , which are suppressed by some mechanism . in fig.([figaa ] ) two examples for suppression mechanisms are given . for fig.([figaa]a ) the contribution is small because in the case of infinite nuclear matter due to momentum conservation the four - momentum of the photon should be equal to that of the outgoing pion , which is impossible . for fig.([figaa]b ) let us consider the case , where the upper two nucleons ( incoming and outgoing ) are identified . furthermore , let us select a @xmath1-isobar as an intermediate state . then , one has a vanishing trace in spin space : @xmath46\,=\ , 0\ ] ] at @xmath47 . therefore , the contribution in this case vanishes in the non - relativistic limit considered here . as a result of applying such methods to the various diagrams , we have obtained that only those displayed in fig.([figa ] ) should be taken into acount . explicit evaluation of the diagrams shown in fig.([figa ] ) leads to amplitudes for the one- and two - nucleon contribution to nuclear photoabsorption . squaring these amplitudes and summing over spin and isospin states of the nucleons by using trace methods as previously @xcite one finds the expressions @xmath48 and @xmath49 , which enter into eqs.([s1eq5 ] ) and ( [ s1eq6 ] ) . it has turned out to be convenient to investigate the resonant and non - resonant parts of each of these contributions separately . this can be done by neglecting the interference terms , which are highly suppressed ( cf . fig.([figh ] ) ) . shown here exemplary for the resonant parts , one obtains the following expressions , which serve as a starting point for the integrations with respect to nucleon momenta : @xmath50-q^2-m^2 } \right)\times \nonumber \\ & & n(\vec p-\vec k)\;\left [ { 1-n(\vec p-\vec q ) } \right]\,{{3\left [ { ( \vec q\times \vec k)\cdot \vec \varepsilon } \right]^2+q^2(\vec k\times \vec \varepsilon ) ^2 } \over { \left ( { \omega -\delta - p^2/2 m } \right)^2 + \gamma^2 / 4 } } , \label{s2eq1}\end{aligned}\ ] ] where @xmath51 and @xmath52\left [ { 1-n(\vec p_4)\ ; } \right]\times \nonumber \\ & & \delta \left ( { \omega -{{p_3 ^ 2+p_4 ^ 2 } \over { 2 m } } } \right)\,\delta ( \vec p_1+\vec p_2+\vec k-\vec p_3-\vec p_4)\times \nonumber \\ & & \left\ { { { { 2a^2\left [ { a^2 + 3(\vec \varepsilon \cdot \vec a)^2 } \right ] } \over { ( a^2+m^2)^2}}\,g_\pi ^4(a)+{{2\omega ^2(\vec \varepsilon \cdot \vec a)(\vec \varepsilon \cdot \vec b ) } \over { ( a^2+m^2)(b^2+m^2)}}\,g_\pi ^2(a)g_\pi ^2(b ) } \right\ } \label{s2eq2}\end{aligned}\ ] ] with @xmath53 the analytical expressions for the resulting partial cross sections are given in the appendix . for the total absorption cross section @xmath54 a comparison with experimental data is shown in fig.([figc ] ) . the result of this model calculation compares favorably with the data . the one - nucleon and two - nucleon parts of the cross section are equally important at energies around 250 mev . as can be seen in this plot , significant features of the data , e.g. the position of the peak , are only obtained due to the interplay between the two mechanisms . as mentioned before , each of the contributions to the full curve in fig.([figc ] ) has a resonant and a non - resonant part . in fig.([figd ] ) and ( [ fige ] ) this decomposition is shown for the one - nucleon and the two - nucleon mechanism , respectively . here it is clearly seen that the non - resonant parts give an important contribution at lower energies . in the two - nucleon case the non - resonant part decreases with energy , while in the one - nucleon process it remains almost constant . it is interesting to see , in what way the one - nucleon partial cross sections are affected by the use of only the static limit of the interaction . neglecting the first - order relativistic corrections in the current one has @xmath55}}\,\int\limits_{q_{min}}^{q_{max } } { q\,dq}\,\left ( { 1+s_0(q ) } \right)\times & & \nonumber \\ \left\ { { \left [ { \omega ^2-m^2-{{\omega q^2 } \over m } } \right]+{3 \over 2 } \left ( { q^2-\left [ { q^2\left ( { 1+{\omega \over m } } \right)+m^2 } \right]^2{1 \over { 4\omega ^2 } } } \right ) } \right\ } & & \label{s2eq3}\end{aligned}\ ] ] and @xmath56 ^ 2 } \right\}\left . { \matrix{{}\cr { } \cr } } \right ] , \label{s2eq4}\end{aligned}\ ] ] where the integration limits in both cases are given by @xmath57 and the integrand contains the function @xmath58 in fig.([figf ] ) the corresponding cross sections are compared with those resulting from eq.([s2eq1 ] ) and its non - resonant counterpart . the relativistic corrections lead mainly to a shift of the one - nucleon curve . this effect is essential for obtaining a good agreement with the experimental data . although no relativistic corrections in the current are included , note that in ( [ s2eq3 ] ) and ( [ s2eq4 ] ) terms of that order have been kept in the kinematical contributions to the integrand . as the two - nucleon mechanism gives a comparatively small contribution at higher energies , we neglect relativistic corrections in @xmath59 . this has also been done in @xcite . a more difficult problem is the influence , which nucleon correlations inside the nucleus can have on the nuclear photoabsorption process . we investigate this aspect for the non - relativistic forms ( [ s2eq3 ] ) and ( [ s2eq4 ] ) of the one - nucleon case . the main reason for doing so is the fact that due to the square of the amplitudes we can express part of the integrand via the standard lowest - order central correlation function of a fermi gas ( see e.g. @xcite ) . the object , which is obtained by diagrammatically squaring the one - nucleon contribution of fig.([figa]a ) , can be coupled to an additional nucleon . in the incoherent case of the diagrammatical square also a further pion exchange can be allowed . the modifications of the correlation function , which occur due to such effects , have been investigated analytically in @xcite , where a corrected central correlator @xmath60 has been constructed . the function @xmath61 is given in fig.([figj ] ) . by making the substitution @xmath62 these further correlations can be incorporated effectively . in fig.([figg ] ) the one - nucleon contribution resulting from @xmath61 is compared with the original form , in which @xmath63 has been used . it can be seen that such medium effects modify the result by about 15 per cent . naturally , the use of this substitution technique can only be used to obtain an estimate for such a medium - induced modification . a full examination should involve the inclusion of the additional two - and three - nucleon diagrams in eq.([s1eq5 ] ) . the limit of our approach is certainly reached , when a comparison with differential cross sections is attempted . an extreme case is the comparison with @xmath64he , for which a recent measurement of both , the one - nucleon and the two - nucleon channel exists @xcite . we compare the calculated average cross section with the data for the differential cross section in c.m . frame , as we expect the angular dependence not to be strong in that frame . the corresponding plot is shown in fig.([figi ] ) . although no full agreement is obtained , it is interesting to note that the general features of the two cross sections are well reproduced , such as the peak positions and the relative size of the two processes . by these means it is possible to unambiguously identify the physical mechanisms behind the data points . a similar degree of agreement is obtained for other data @xcite . in the present paper we have developed a diagrammatical description of the nuclear photoabsorption process . the main result of our investigation is that the total photoabsorption cross section can be fully understood in terms of a simple physical picture , where point - like nucleons and @xmath65-isobars interacting via pion exchange are the relevant degrees of freedom . due to the diagram - oriented formalism and the fermi gas model as an approximate description of the nucleons in momentum space , we could obtain analytical expressions for all the relevant contributions to the photoabsorption curve . in this way a flexible and efficient description has been obtained , which can be used as a starting point for the investigation of additional effects . especially in the low - energy part of our calculated curve , the agreement with experiment comes about as a non - trivial interplay between the one - nucleon and two - nucleon contributions . it is worth noting that , as long as a comparatively low cut - off parameter in the vertex form factor is used , there seems to be no need for an explicit diagrammatical inclusion of the @xmath66-meson as an additional mechanism of the nucleon - nucleon interaction . we found relativistic corrections in the case of the one - nucleon process to be crucial for obtaining a good agreement . the aspect of additional nucleon correlations , which can be accounted for as a deviation of the nucleon wave functions from plane waves , deserves some further attention in future investigations . we could estimate the overall effect to be of the order of 15 per cent . we are most grateful to a.i . lvov for useful comments and discussions . one of us ( m.t.h . ) wishes to thank the budker institute , novosibirsk , for the kind hospitality accorded him during his stay , when part of this work was done . here we present the explicit expressions for the four contributions to the photoabsorption curve , which have been used to obtain the figures shown in section 3 . as was mentioned earlier , the interference terms between the resonant and the non - resonant contributions are small ( cf . fig.([figh ] ) ) . therefore , we can write the absorption cross section as a sum of four parts , @xmath67 . first , we deal with the one - nucleon case . in the case of the non - resonant contribution the absorption cross section can be represented in the following form : @xmath68\times \nonumber\\ & \displaystyle g(p , q,\omega)\,\theta(2q\omega - m^2-q^2-\frac{\omega}{m}p^2 ) , \label{app1}\end{aligned}\ ] ] where @xmath69 \label{app4}\ ] ] and @xmath70 $ ] , @xmath71 , @xmath72 . in eq.([app1 ] ) the form factor @xmath73 is the same as in ( [ s1eq4 ] ) . note that the integration with respect to the variable @xmath38 can easily be performed , but the result is too lengthy to be given here explicitely . the resonant part of the one - nucleon process has the following form : @xmath74 ^ 2 + \gamma^2/4}\ ; h(p , y,\omega ) \label{app6}\end{aligned}\ ] ] where @xmath75 and the integrand is given by @xmath76 + \nonumber\\ & \displaystyle [ ( x+1)a_1-{1 \over 8}(x - x^3)a_2]\theta(f-|p - p_f|)\,\theta(p+p_f - f ) \ , , \end{aligned}\ ] ] with @xmath77^{1/2}\ ; , \ ; x=\frac{p^2+f^2-p_f^2}{2pf}\ , , \nonumber \\ & \displaystyle a_1=\omega^2 - 2\omega ay+a^2\ ; , \ ; a_2=\omega^2(1 - 3y^2)+4\omega ay-2a^2\ ; , \nonumber \\ & \displaystyle a=\frac{p\delta}{m}\left(\frac{1+\omega / m}{1+\delta / m}\right)\ , .\end{aligned}\ ] ] the mass difference @xmath1 between the proton and the @xmath78-excitation is @xmath79 mev , the width @xmath80 of the @xmath42-isobar has been taken to be 115 mev . again , the integration with respect to @xmath81 in eq.([app6 ] ) can be performed analytically , but due to its length the result is not presented here . for the partial cross sections of the two - nucleon case , we obtained the following result : @xmath82+\nonumber \\ & \displaystyle \theta(\omega -5\varepsilon ) \,\theta ( 9\varepsilon - \omega ) \,\int\limits_{l_1(q)}^{l_3(q)}\phi(\beta_1,\ , q ) \biggr\}g_\pi ^4(p)\left[g^{(nr)}(p,\omega ) + g^{(r)}(p,\omega ) \right]dp , \nonumber \label{app10}\end{aligned}\ ] ] where @xmath83 , @xmath84 , @xmath85 the elementary function @xmath86 is @xmath87\ , + \nonumber \\ & \displaystyle \sqrt 8pq\,[\frac{1}{3}(2q^2+p^2-p_f^2)\cos ^3x -\frac{2}{5}q^2\cos ^5x\,]\biggr\}\ , \biggl.\biggr|_{x=\beta}^{x=\pi /2 -\beta}\ , . \label{app13}\end{aligned}\ ] ] the functions in the integrand of eq.([app10 ] ) , which characterize the resonant and non - resonant part , are given by @xmath88 and @xmath89 \frac{p^2}{(p^2+m^2)^2 } \label{app15}\ ] ] respectively . in eq.([app10 ] ) the integration limits are @xmath90 and @xmath91 99 m. hirata , j. koch , f. lenz and e. moniz , ann.phys.120 ( 1979 ) 205 e. oset and w. weise , nucl.phys.a329 ( 1979 ) 365 c. garcia - recio , l. salcedo , e. oset , d. strottman and m. lopez - santodomingo , nucl.phys.a526 ( 1991 ) 685 l. salcedo , e. oset , m. vicente - vacas and c. garcia - recio , nucl.phys.a484 ( 1988 ) 557 j. koch , e.j . moniz and n. ohtsuka , ann.phys.154 ( 1984 ) 99 e. oset and w. weise , nucl.phys.a368 ( 1981 ) 375 m. wakamatsu and k. matsumoto , nucl.phys.a392 ( 1983 ) 323 m .- th . htt and a.i . milstein , nucl.phys.a in press l. boato and m. giannini , j.phys.g15 ( 1989 ) 1605 j. ryckebusch , m. vanderhaeghen , l. machenil , m. waroquier , nucl.phys.a568 ( 1994 ) 828 r.c . carrasco and e. oset , nucl.phys.a536 ( 1992 ) 445 m. vanderhaeghen , k. heyde , j. ryckebusch , m. waroquier , nucl.phys.a595 ( 1995 ) 219 j. ryckebusch , l. machenil , m. vanderhaeghen , v. van der sluys and m. waroquier , phys.rev.c49 ( 1994 ) 2704 j. ryckebusch , l. machenil , m. vanderhaeghen , v. van der sluys and m. waroquier , phys.lett.b291 ( 1992 ) 213 s. homma et al . , phys.rev.c36 ( 1987 ) 1623 t. ericson , w. weise , `` pions and nuclei '' , oxford univ . press 1988 d.o . riska , phys.rep.181 ( 1989 ) 207 t. sasakawa , s. ishikawa , y. wu and t. saito , phys.rev.lett.68 ( 1992 ) 3503 a. molinari , phys.rep.64 ( 1980 ) 283 r. wichmann et al . , submitted to phys.lett.b s. homma et al . , phys.rev.lett.53 ( 1984 ) 2538 j. ahrens , nucl.phys.a446 ( 1985 ) 229c [ figa ] notation for three - momenta of the external particles a ) for the one - nucleon process and b ) for the two - nucleon reaction . the wavy lines denote photons and dashed lines denote pions . a circle indicates a bound , an arrow a free nucleon . [ figb ] diagrammatical forms of the resonant part @xmath92 ( first two terms ) and the non - resonant part @xmath93 ( last term ) of the amplitude @xmath94 . these contributions to the @xmath95-interaction enter into the diagrams shown in fig.([figa ] ) . [ figc ] comparison of the calculated curve @xmath96 for nuclear photoabsorption with the experimental data . the dotted curve is the one - nucleon contribution @xmath97 , while the dashed curve represents the two - nucleon mechanism @xmath97 . the data are taken from ref . the empty ( full ) circles correspond to @xmath98pb ( @xmath99c ) data , while the squares represent data on @xmath100o . [ figd ] resonant ( dashed ) and non - resonant ( dotted ) contribution to the one - nucleon reaction ( full line ) . the corresponding analytical expressions @xmath101 , @xmath102 and @xmath97 , respectively , can be found in the appendix . [ figh ] contribution of interference terms . the full ( dashed ) curve corresponds to the one- ( two- ) nucleon case . as can be seen from the overall scale , both for @xmath106 and @xmath107 this contribution is highly suppressed . [ figg ] effect of two- and three - nucleon correlation in the case of the non - relativistic one - nucleon part of the photoabsorption cross section . the dashed curve contains only @xmath63 , while in the full curve @xmath61 ( cf . fig.([figj ] ) ) has been included . [ figi ] approximate description of differential cross sections for helium . the differential cross section with respect to the direction of the outgoing proton is shown as a function of the energy of the incoming photon . data points are taken from @xcite . the full curve and filled circles correspond to the one - nucleon process , while the dashed curve and empty circles represent the two - nucleon case .

the universal curve @xmath0 of nuclear photoabsorption is investigated within a fermi gas model of nuclear matter . an energy range from pion threshold up to 400 mev is considered . the interactions between nucleon , pion , @xmath1-isobar and photon are considered in the non - relativistic approximation with corrections of the order @xmath2 taken into account with respect to proton mass . analytical expressions are obtained , in which the influence of nuclear correlations , two - nucleon contributions and relativistic corrections is studied explicitely . an extension of the model calculation to nucleon knock - out reactions is discussed . contribution of real and virtual pions to nuclear photoabsorption at intermediate energies + m .- th.htt@xmath3 , a.i.milstein@xmath4 and m.schumacher@xmath3 + ( a ) ii . institut der universitt gttingen , gttingen , germany \(b ) budker institute of nuclear physics , 630090 novosibirsk , russia 0em 1.5ex plus 0.5ex minus 0.5ex _ pacs code : _ 25.20.-x _ keywords : _ photoabsorption , mesonic exchange currents , nuclear correlation functions

You are an expert at summarizing long articles. Proceed to summarize the following text: market impact is the expected price change conditioned on initiating a trade of a given size and a given sign . understanding market impact is important for several reasons . one motivation is practical : to know whether a trade will be profitable it is essential to be able to estimate transaction costs , and in order to optimize a trading strategy to minimize such costs , it is necessary to understand the functional form of market impact . another motivation is ecological : impact exerts selection pressure against a fund becoming too large , and therefore is potentially important in determining the size distribution of funds . finally , an important motivation is theoretical : market impact reflects the shape of excess demand , the understanding of which has been a central problem in economics since the time of alfred marshall . in this paper we present a theory for the market impact of large trading orders that are split into pieces and executed incrementally . we call these _ metaorders_. the true size of metaorders is typically not public information , a fact that plays a central role in our theory . the strategic reasons for incremental execution of metaorders were originally analyzed by kyle ( @xcite ) , who developed a model for an inside trader with monopolistic information about future prices . kyle showed that the optimal strategy for such a trader is to break her metaorder into pieces and execute it incrementally at a uniform rate , gradually incorporating her information into the price . in kyle s theory the price increases linearly with time as the trading takes place , and all else being equal , the total impact is a linear function of size . the prediction of linearity is reinforced by huberman and stanzl ( @xcite ) who show that , providing liquidity is constant , to prevent arbitrage permanent impact must be linear . real data contradict these predictions : metaorders do not show linear impact . empirical studies consistently find concave impact , i.e. impact per share decreases with size . it is in principle possible to reconcile the kyle model with concave dependence on size by making the additional hypothesis that larger metaorders contain less information per share than smaller ones , for example because more informed traders issue smaller metaorders . a drawback of this hypothesis is that it is neither parsimonious nor easily testable , and as we will argue here , under the assumptions of our model it violates market efficiency . huberman and stanzl are careful to specify that linearity only applies when liquidity is constant . in fact , liquidity fluctuates by orders of magnitude and has a large effect on price fluctuations . empirical studies find that order flow is extremely persistent , in the sense that the autocorrelation of order signs is positive and decays very slowly . no arbitrage arguments imply either fluctuating asymmetric liquidity as postulated by lillo and farmer ( @xcite ) , or no permanent impact , as discussed by bouchaud et al . ( @xcite ) . the central goal of our model is to understand how order splitting affects market impact . whereas kyle assumed a single , monopolistic informed trader , our informed traders are competitive . they submit their orders to an algorithmic execution service that bundles them together as one large metaorder and executes them incrementally . we show that this leads to a symmetric nash equilibrium satisfying the condition that the final price after a metaorder is executed equals its average transaction price . we call this condition _ fair pricing _ , to emphasize the fact that under this assumption trading a metaorder is a breakeven deal neither party makes a profit as a result of trading . our equilibrium is less general than kyle s in that it assumes uniform execution , but it is more general in that it allows an arbitrary information distribution . this is key because , as we show , there is an equilibrium between information and metaorder size , making it possible to match the metaorder size distribution to empirical data . combining the fair pricing condition with a martingale condition makes it possible to derive the price impact of metaorders as a function of the metaorder size distribution . this allows us to make several strong predictions based on a simple set of hypotheses . for a given metaorder size distribution it predicts the average impact as a function of time both during and after execution . we thus predict the relationship between the functional form of two observable quantities with no a priori relationship , making our theory falsifiable in a strong sense . this is in contrast to theories that make assumptions about the functional form of utility and/or behavioral or institutional assumptions about the informativeness of trades , which typically leave room for interpretation and require auxiliary assumptions to make empirical tests . for example , gabaix et al . ( @xcite ) have also argued that the distribution of trading volume plays a central role in determining impact , and have derived a formula for impact that is concave under some circumstances . however , in contrast to our model , their prediction for market impact depends sensitively on the functional form for risk aversion , where @xmath0 is the standard deviation of profits , the impact will increase with the size @xmath1 of the metaorder as @xmath2 . thus the impact is concave if @xmath3 , linear if @xmath4 ( i.e. if risk is proportional to variance ) , and convex otherwise . for another theory that also predicts concave impact see toth et al . ( @xcite ) . ] . our theory , in contrast , is based entirely on market efficiency and does not depend on the functional form of utility . our work here is related to several papers that study market design . viswanathan and wang ( @xcite ) , glosten ( @xcite ) , and back and baruch ( @xcite ) derive and compare the equilibrium transaction prices of orders submitted to markets with uniform vs. discriminatory pricing . depending on the setup of the model , these prices can be different so that investors will prefer one pricing structure to the other and can potentially be cream - skimmed " by a competing exchange . the fair pricing condition we introduce here forces the average transaction price of a metaorder ( which transacts at discriminatory prices ) to be equal to the price that would be set under uniform pricing . fair pricing , therefore , means investors have no preference between the two pricing structures , and they have no incentive to search out arrangements for better execution . on the surface , this result is similar to the equivalence of uniform and discriminatory pricing in back and baruch ( @xcite ) . however , in their paper , this equivalence results because orders are always allowed to be split , whereas ours is a true equivalence between the pricing of a split vs. unsplit order . in section ii we give a description of the model and discuss its interpretation . in section iii we develop the consequences of the martingale condition and show how this leads to zero overall profits and asymmetric price responses when order flow is persistent . in section iv we show that any nash equilibrium must satisfy the fair pricing condition . in section v we derive in general terms what this implies about market impact . in section vi we introduce specific functional forms for the metaorder size distribution and explicitly compute the impact for these cases . finally , in section vii we discuss the empirical implications of the model and make some concluding remarks . we study a stylized model of an algorithmic trading service combining and executing orders of long - term traders . this can be thought of as a broker - dealer receiving multiple orders on the same security and executing them algorithmically at the same time , or as an institutional trading desk of a large asset manager combining the orders from multiple portfolio managers into one large metaorder . our goal is to model the price impact of a metaorder during a trading period in which it may or may not be present . we set up the model in a stylized manner as a game in which trading takes place across multiple periods , after which final prices are revealed . while this is somewhat artificial in comparison to a real market ( which has no such thing as a final " price ) , the framework is simple enough to allow us to find a solution , and the basic conclusions should apply more broadly . the structure of the model is in many respects similar to the classic framework of kyle ( @xcite ) , but with several important differences . ( a point - by - point comparison is made in the conclusions ) . we begin with an overview . there is a single asset which is traded in each of @xmath1 periods , which can be regarded as a game . there are three kinds of agents , long - term traders , market makers and day traders . the long - term traders have a common information signal that is received before the game starts ; based on this information they formulate their orders and submit them to an algorithmic trading service . the algorithmic service operates mechanically , dividing the bundle of orders ( which we call a _ metaorder _ ) into equal sized lots which are submitted as anonymous market orders in each successive period until the metaorder is fully executed . the day traders , in contrast , receive a new information signal and submit orders based on this signal in every period of the game . in every period each market maker observes the netted order of the long - term and day traders and submits a quote accordingly ; all orders are filled at the best price . the game ends with a final liquidation period at time step @xmath5 in which prices are set exogenously to reflect the accumulated information . the length @xmath1 of the game varies based on the amount of information received , and is unknown to the market makers . we now discuss the setup of the game in more detail , beginning with a description of each of the agents . the _ long - term traders _ receive a common information signal @xmath6 before the game begins , which only they observe . in order to model situations in which there may or may not be a metaorder present , we allow a nonzero probability that @xmath7 . thus with probability @xmath8 the signal @xmath6 is drawn from a distribution @xmath9 , which has nonzero support over a continuous interval @xmath10 , where @xmath11 , and with probability @xmath12 there is no information , i.e. @xmath7 . for simplicity we discuss the case where the draw of @xmath6 happens to be positive , i.e. it causes the final price to increase , but the results apply equally well if @xmath6 is negative . after @xmath6 is revealed the orders of the long - term traders are aggregated together into a _ metaorder _ and executed in a package . the bundling and execution process can be thought of as representing an algorithmic trading firm , broker , or institutional trading desk . there are @xmath13 long - term traders , labeled by an index @xmath14 , where @xmath13 is a large number . after the common information signal is received each long - term trader submits an order of size @xmath15 , where @xmath16 is a large positive integer . each long - term trader decides @xmath17 independently . the individual orders are bundled together into a metaorder of size @xmath18 shares . . for this figure we assume the metaorder is present and show only expected price paths , averaged over the day trader s information . the price is initially @xmath19 ; after the first lot is executed it is @xmath20 . if @xmath21 it is finished and the price reverts to @xmath22 , but if @xmath23 another lot is executed and it rises to @xmath24 . this proceeds similarly until the execution of the metaorder is completed . at any given point the probability that the metaorder has size @xmath25 , i.e. that the order continues , is @xmath26 . if had we followed a typical price path under circumstances when the day trader s noisy information signal is large , rather than the expected price paths shown here , the sequence of prices would be a random walk with a time - varying drift caused by the metaorder s impact . ] the algorithmic trading firm operates purely mechanically , chopping the metaorder into equal pieces and submitting market orders at successive times @xmath27 . these are executed at transaction prices @xmath28 , where @xmath29 , as illustrated in figure [ fig1 ] . we assume the metaorder is executed in lots , which for mathematical convenience are chosen to be of @xmath13 shares . the imposition of a maximum trade size @xmath30 is a technical detail , which induces a bound on @xmath30 , to avoid mathematical problems that occur in the limit @xmath31 . this is explained in the appendix . for the typical situations we have in mind @xmath30 , @xmath13 , and @xmath32 are all large numbers . the choice of lots of size @xmath13 shares is purely a matter of convenience ; the assumption that the lots have a constant size is something we hope can be relaxed in the future . ] ; we assume that @xmath30 is large . if the metaorder is present ( i.e. if @xmath33 ) , trading ends when the metaorder is fully executed , i.e. when @xmath34 . the equilibrium distribution of metaorder lengths is @xmath35 . if @xmath7 we randomly choose a value of @xmath1 from an arbitrary distribution @xmath36 ( which in general differs from @xmath35 ) . the _ day traders _ can be treated as a single representative agent who receives a private information signal @xmath37 at the beginning of each period @xmath38 of the game , and submits a market order ( either to buy or sell ) of size @xmath39 , where @xmath37 is a zero mean iid noise process with an otherwise arbitrary distribution @xmath40 , and @xmath41 is an increasing function is not important here . we assume the day traders do not engage in order splitting , i.e. they trade on the information they receive in given time step only in that time step . this guarantees that in the absence of a metaorder the day trader s order flow provides a sufficient signal for a market maker to infer the new information . ] . there is no restriction on the size of @xmath42 , and in particular we allow the possibility that @xmath37 is large and negative while @xmath6 is positive , so that during the course of execution of the metaorder the combined order flow may change sign , and that at different times the market makers may buy or sell . at each time step the market orders of the long - term traders and the day trader are netted and the single combined order is submitted to the _ market makers _ , who are competitive and profit maximizing . we are not assuming any special institutional privileges , such as those of the specialists in the nyse ; our market makers are simply competitive liquidity providers . at each time step each market maker observes the combined order and submits a quote . the combined order is fully executed by the market maker(s ) offering the best price . the market makers are able to take past order flow and prices into account in setting their quotes . given that @xmath37 is iid , the typical transaction price sequence @xmath43 will look like a random walk , as the market maker responds to the day trader s order flow , with a possible superimposed drift if a metaorder is present . in the final period @xmath44 the combination of the long - term information @xmath6 and the accumulated short term information are revealed . since the long - term traders information signals are independent of those of the day trader , information is additive and the final price is @xmath45 where @xmath19 is the initial price . the average final price is @xmath46 = s_0 + \alpha$ ] , where @xmath47 implies an average over the noise @xmath37 , @xmath48 = \int x \hat{p}(\eta ) d\eta.\ ] ] similarly the average transaction price at time @xmath38 is @xmath49 $ ] . the goal of the paper is to compute the _ average immediate impact _ @xmath50 and the _ average permanent impact _ @xmath51 of the metaorder . the corresponding incremental average impacts are @xmath52 and @xmath53 . as we will show , at the equilibrium the average impacts do not depend on @xmath8 , @xmath54 or @xmath55 . the number of long - term traders @xmath13 is fixed and is common knowledge . the market makers know the initial price @xmath19 , the information distributions @xmath9 and @xmath40 , they can deduce the function @xmath41 relating the day trader s information to their order size , and they observe the net combined order in each period and remember previous order flow . however they do not know the information signals @xmath6 or @xmath37 , and thus they do not know how much of the order flow to ascribe to the long - term trader vs. the day trader . they do not know whether a metaorder is present , and if it is , they do not know its size . perhaps the strongest assumption we have made concerns knowledge of the timing of metaorders . the market makers know the period @xmath38 , and as a result if a metaorder is present , they know when it started . in typical market settings order flow is anonymous and the starting time of a metaorder is uncertain . nonetheless , for long meta - orders and sufficiently high participation rates the starting time can be inferred from the imbalance in order flow , where @xmath56 is the desired number of standard deviations of statistical significance . the accumulated imbalance after @xmath38 steps is @xmath57 , where @xmath58 is the participation rate . equating these gives @xmath59 . hiding an order of size @xmath1 requires @xmath60 . thus larger metaorders need to be executed more slowly to avoid detection . since the time needed to complete execution is inversely proportional @xmath58 , for a large metaorder this can become prohibitive it is impossible to escape detection . ] . assuming a typical participation rate of @xmath61 and two standard deviations to reject the null hypothesis of balanced order flow means that a metaorder can be detected after about @xmath62 timesteps . large metaorders are frequently executed in @xmath63 or even @xmath64 lots share of ibm . ] , implying an error in inferring the starting time of @xmath65 to @xmath66 . thus , it is not unrealistic to imagine that market makers can infer the presence of large metaorders and estimate their starting times . we are not concerned here with the question of whether or not the algorithmic execution service s strategy of splitting the order into equal pieces is an optimal strategy . our goal is instead to assume that this is what they do , and to derive the implications for price impact . our main result is to derive the equilibrium relationship between the metaorder size distribution @xmath35 and the average immediate and permanent impacts @xmath67 and @xmath68 . in addition to taking expectations over the day trader s noise @xmath42 , which we denote by @xmath47 , we must compute expectations about the length of the game . the crux of our argument hinges around the market makers ignorance of @xmath6 ; when @xmath33 this translates into uncertainty about metaorder size . we will use the notation @xmath69 to represent an average over all metaorders of size @xmath70 , and as described above , @xmath47 for averages over @xmath37 . for a generic function @xmath71 the average over metaorder sizes is @xmath72 = \sum_{n = t}^\infty q_n f_n= \frac{\sum_{n = t}^\infty p_n f_n}{\sum_{n = t}^\infty p_{n}}= \frac{\sum_{i=0}^\infty p_{it } f_{it}}{\sum_{i=0}^\infty p_{it } } , e_t[f_n ] = \frac{\sum_{n = t}^m p_n f_n}{\sum_{n = t}^m p_{n}}= \frac{\sum_{i=0}^{m - t } p_{t+i } f_{t+i}}{\sum_{i=0}^{m - t } p_{t+i } } , \label{averages}\ ] ] where in the last term we made the substitution @xmath73 . assuming that a metaorder is present , the likelihood that it will persist depends on the distribution @xmath35 and the number of executions @xmath38 that it has already experienced let @xmath74 be the probability that the metaorder will continue given that it is still active at timestep @xmath38 . this is equivalent to the probability that it is at least of size @xmath38 , i.e. @xmath75 this makes precise how order splitting can make order flow positively autocorrelated . in particular , if @xmath35 has heavy tails than an exponential @xmath26 will increase with time and induce persistence in order flow . in the case where no metaorder is present one can similarly define the probability that the game continues as @xmath76 in this section we introduce a martingale condition and discuss its implications for liquidity and overall profitability . market makers must set prices given only past and present order flow information , without knowing whether the order flow originated from a metaorder or from day traders . their decision function is of the form @xmath77 where @xmath78 $ ] . as we show below , we are able to finesse this difficult problem by imposing a martingale condition and averaging over the day traders noise , which is sufficient for the main goal of this paper of deriving the equilibrium between the impacts and @xmath35 . during the game the market makers may also use information from order flow and prices to update their prior probability @xmath8 for the presence of a metaorder to a more accurate value @xmath79 . we assume that transaction prices are a martingale , so that the current transaction price @xmath28 is equal to the expected future price . define an indicator variable @xmath80 with @xmath81 is the metaorder is present and @xmath82 if it is absent . for the price in the next period it is necessary to average over four possibilities : 1 . with probability @xmath83 the metaorder is present and with probability @xmath26 trading continues . in this case @xmath84 = \tilde{s}'_t + \tilde{r}'_t$ ] , where @xmath85 $ ] . 2 . with probability @xmath83 the metaorder is present and with probability @xmath86 time @xmath38 is the last trading period . in this case @xmath87 = \tilde{s}'_t - { r}'_t$ ] , where @xmath88 $ ] . 3 . with probability @xmath89 the metaorder is not present and with probability @xmath90 trading continues . since the day trader s information @xmath37 is zero mean and iid , the expected average transaction price on the next time step conditioned on the current transaction price must satisfy @xmath91 = \tilde{s}'_t$ ] , i.e. the average transaction price is unchanged . 4 . with probability @xmath89 the metaorder is not present and with probability @xmath92 time @xmath38 is the last trading period . for similar reasons @xmath93 = \tilde{s}'_t$ ] , i.e. the average final price conditioned on the current transaction price is equal to the current transaction price . thus the martingale condition can be written @xmath94 + ( 1 - \mathcal{p}_t)\hat{e } [ { s}'_{t + 1 } | \tilde{s}'_t , m = t ] \right)\\ + ~(1 - \mu'_t ) & \left ( \hat{\mathcal{p}}_t \hat{e } [ \tilde{s}'_{t + 1 } | \tilde{s}'_t , m = f ] + ( 1 - \hat{\mathcal{p}}_t ) \hat{e } [ { s}'_{t + 1 } | \tilde{s}'_t , m = f ] \right)\\\end{aligned}\ ] ] substituting for the average next price conditioned on the current price for cases 1 - 4 gives : @xmath95 the current transaction price @xmath28 and @xmath83 both cancel and this reduces to @xmath96 since @xmath37 is mean zero and iid , @xmath97 = \tilde{r}_t = \tilde{s}_{t+1}- \tilde{s}_t$ ] and similarly for @xmath98 . taking averages over @xmath37 gives @xmath99 equation ( [ shortterm ] ) holds for @xmath100 . if the metaorder has maximal length at the end of the @xmath30th interval by definition @xmath101 , which implies that @xmath102 . let us pause for a moment to digest this result . we started with a martingale condition for realized prices , including fluctuations caused by day traders , and then reduced it to a martingale condition for the average impact due to the presence of a metaorder . the reduced martingale no longer depends on @xmath103 , @xmath36 , or the day trader s information . the fact that we assume a martingale for realized prices implies that arbitrage of the impact is impossible . the ability to average away the day traders is a consequence of our assumption that @xmath6 and @xmath37 are independent , which implies additivity of information . this separates the problem of the metaorder s impact from that of the day trader s impact the metaorder s impact effectively rides on top of the day trader s impact . as we will see , the virtue of this approach is that it allows us to infer quite a lot without needing to solve for the market makers optimal quote setting function @xmath104 . equation ( [ shortterm ] ) can be trivially rewritten in the form @xmath105 where @xmath52 and @xmath53 . thus the martingale condition fixes the ratio of the price responses @xmath106 and @xmath107 , but does not fix their scale . if @xmath74 is large , corresponding to a metaorder that is likely to continue , then @xmath108 small . this means that the price response if the order continues is much less it is than if it stops . to complete the calculation we need another condition to set the scale of the price responses @xmath106 and @xmath107 , which may change as @xmath38 varies . such a condition is introduced in section [ sizeindependencesec ] . even without such a condition , one can already see intuitively that all else equal " , for a heavy - tailed metaorder distribution @xmath35 , the impact will be concave . ( recall that heavy tails in @xmath35 imply that @xmath26 increases with @xmath38 ) . the martingale condition implies that the market makers break even overall , i.e. that their total profits summing over metaorders of all sizes is zero . this is stated more precisely in proposition one . * proposition 1 . * the market makers transact @xmath1 lots at average prices @xmath109 , which later are all valued at a final price @xmath110 . the martingale condition implies zero overall profits , i.e. @xmath111\equiv\sum_{n=1}^\infty p_n \pi_n=0 . \pi = e_1[\pi_n]\equiv\sum_{n=1}^m n \pi_n p_n = 0 , \label{breakevenonaverage}\ ] ] where @xmath112 is the profit per lot transacted . the proof of proposition 1 is given in appendix a. the phrase overall profits " emphasizes that the martingale condition only implies zero profits when averaged over metaorders of all sizes . it allows for the possibility that the market makers may make profits on metaorders in a given size range , as long as they take corresponding losses in other size ranges . surprisingly , proposition 1 is not necessarily true when @xmath30 is infinite . the basic problem is similar to the st . petersburg paradox : as the metaorder size becomes infinite it is possible to have infinitely rare but infinitely large losses . the conditions under which this holds are more complicated , as discussed in appendix a. we now derive the _ fair pricing condition _ , which states that for any @xmath1 @xmath113 under fair pricing the average execution price is equal to the final price . we call this fair pricing for the obvious reason that both parties would naturally regard this as fair " . fair pricing implies that the market makers break even on metaorders of any size , as opposed to the martingale condition , which only implies they break even when averaging over metaorders of all sizes . . * proposition 2 . * _ if the immediate impact @xmath114 has a second derivative bounded below zero , in the limit where the number of informed traders @xmath115 , any nash equilibrium must satisfy the fair pricing condition @xmath116 for @xmath29 . on average market makers profit from orders of length one and take ( equal and opposite ) losses from orders of length @xmath30 . _ this result is driven by competition between informed traders . all informed traders receive the same information signal @xmath6 , and the strategy of informed trader @xmath117 is the choice of the order size @xmath17 . the orders are then bundled together to determine the combined metaorder size @xmath118 . the decision of each informed trader is made without knowing the decisions of others . the derivation has two steps : first we examine the case @xmath119 for @xmath29 , and show that if others hold their strategies constant , providing the impact is concave and @xmath13 is sufficiently large , traders can increase profits by changing strategy . secondly we show that if @xmath116 there is no incentive to change strategy . then we return to examine the cases @xmath21 and @xmath120 , which must be treated separately . the derivation is given in the appendix . in contrast to the martingale condition , which only implies that immediate profits are zero when averaged over size , fair pricing means that they are identically zero for every size . it implies that no one pays any costs or makes any profits simply by trading in any particular size range . the nash equilibrium is symmetric , i.e. all agents make the same decision . ( this must be in any case since there is nothing to distinguish them ) . this means that there is a unique order size @xmath121 for any @xmath6 , and the distribution of information @xmath9 implies the distribution of metaorder size @xmath35 . we can use @xmath35 as a proxy for @xmath9 , which is the key fact allowing us to state our results in terms of the observable quantity @xmath35 rather than @xmath9 , which is much more difficult to observe . although this derivation is based on rationality , the fair pricing condition potentially stands on its own , even if other aspects of rationality and efficiency are violated . in modern markets portfolio managers routinely receive trade cost analysis reports that compare their execution prices relative to the close , which for a trade that takes place over one day is a good proxy for @xmath110 . such reports are typically broken down into size bins , making persistent inconsistencies across sizes clear . execution times for metaorders range from less than a day to several months ( vaglica et al . , @xcite ) , and are much shorter than typical holding times , which for mutual funds are on average a year and are often much more ( schwartzkopf and farmer , @xcite ) . thus the statistical fluctuations for assessing whether fair pricing holds are much smaller than those for assessing informational efficiency . since @xmath122 is fairly well determined , portfolio managers will exert pressure on their brokers to provide them with good execution . as a result we expect the fair pricing condition to be obeyed to a higher degree of accuracy than the informational efficiency condition . although we have derived the nash equilibrium only in the case where the immediate impact is concave , for the remainder of the paper we will simply assume that the martingale condition holds for all @xmath1 and the fair pricing condition holds for @xmath123 . this allows us to derive both the immediate impact @xmath124 and the permanent impact @xmath68 for any given metaorder size distribution @xmath35 . we later argue that for realistic situations the metaorder size distribution gives rise to a concave impact function , consistent with the nash equilibrium . the martingale condition ( eq . [ shortterm ] ) and the fair pricing condition ( eq . [ fairpricing ] ) define a system of linear equations for @xmath125 and @xmath126 at each value of @xmath38 , which we can alternatively express in terms of the price differences @xmath127 and @xmath52 , where @xmath128 . the martingale condition holds for @xmath129 and the fair pricing condition holds for @xmath130 . there are thus @xmath131 homogeneous linear equations with @xmath132 unknowns does not exist , so @xmath133 is not needed . this reduces the number of unknowns by one . ] . because the number of unknowns is one greater than the number of conditions there is necessarily an undetermined constant , which we choose to be @xmath134 . * proposition 3 . * _ the system of martingale conditions ( eq . [ shortterm ] ) and fair pricing conditions ( eq . [ fairpricing ] ) has solution _ @xmath135 the proof is given in appendix a. an important property of the solution is the equivalence of the impact @xmath114 as a function of either time or size . this is in contrast to the prediction of an extended " kyle model under the assumption that traders of different sizes are differently informed , which yields linear impact as a function of time , but allows the slope to vary nonlinearly with size . summing eq . [ solution ] implies that for @xmath136 the immediate impact is @xmath137 for @xmath138 the immediate impact is @xmath139 and for @xmath140 it is @xmath141 . ( the meaning of the undetermined constants @xmath142 and @xmath143 is discussed in a moment ) . the permanent impact @xmath144 is easily obtained . making some simple algebraic manipulations @xmath145 by combining eqs . ( [ final ] ) and ( [ calpdef ] ) we get @xmath146 we have expressed both the permanent and immediate impact purely in terms of @xmath35 and the undetermined constants @xmath143 and @xmath142 . the undetermined constants can in principle be fixed based on the information at the equilibrium . at the equilibrium information signals in the range @xmath147 $ ] will be assigned to metaorders of size one , with an average size @xmath148 , signals in the range @xmath149 $ ] will be assigned to metaorders of size two , with an average size @xmath150 , and so on . the scale of the impact is set by the relations @xmath151 we have used the words in principle " because , unlike metaorder sizes , information is not easily observed . barring the ability to independently measure information , the constants @xmath143 and @xmath142 remain undetermined parameters . @xmath152 plays the important role of setting the scale of the impact . the constant @xmath143 , in contrast , is unimportant it is simply the impact of the first trade , before the metaorder has been detected . the power of the theory developed here is that the impact is predicted in terms of @xmath35 , which is directly measurable ( at least with the proper data ) . the calculation to set the scale shows how the continuous variable @xmath6 maps onto the discrete variable @xmath1 . @xmath153 is a discrete function whose inverse can be written @xmath154 . if there is a continuum limit for large @xmath1 , the two distributions are related by conservation of probability as @xmath155 so for example , if the empirical metaorder size is @xmath35 is asymptotically pareto distributed , as argued in the next section , @xmath156 , @xmath157 , and @xmath158 . based on the empirically observed value @xmath159 , this gives @xmath160 , which means that the cumulative scales as @xmath161 . this is what is typically observed for price returns in american stock markets ( plerou et al . , @xcite ) . we have so far left the metaorder size distribution @xmath35 unspecified . in this section we compute the impact for two examples . the first of these is the pareto distribution , @xmath162 which we argue is well - supported by empirical data means that there exists a constant @xmath163 such that in the limit @xmath164 , @xmath165 . we use it to indicate that this relationship is only valid in the limit of large metaorder size @xmath1 . ] . the second is the stretched exponential distribution , which is not supported by data , but provides a useful point of comparison . there is now considerable accumulated evidence that in the large size limit in most major equity markets the metaorder size @xmath166 is distributed as @xmath167 , with @xmath159 . * _ trade size . _ in many different equity markets for large trades the volume @xmath166 has been observed by several groups to be distributed as a power law ( gopikrishnan et al . ( @xcite ) ; gabaix et al . ( @xcite ) is somewhat controversial , however : eisler and kertesz ( @xcite ) and racz et al . ( @xcite ) have argued that the correct value of @xmath168 . ] . this relationship becomes sharper if only block trades are considered ( lillo et al . , @xcite ) . * _ long - memory in order flow . _ the signs of order flow in many equity markets are observed to have long - memory . we use the term long - memory in its more general sense to mean any process whose autocorrelation function is non - integrable ( beran , @xcite ) . this can include processes with structure breaks , such as that studied by ding , engle and granger ( @xcite ) . ] this means that the transaction sign autocorrelation function @xmath169 decays in time as @xmath170 , where @xmath171 . under a simple theory of order splitting the exponent @xmath172 , which is in good agreement with the data [ ( lillo et al . , @xcite ) ; ( gerig , @xcite ) ; bouchaud , farmer , and lillo ( @xcite ) ] . * _ reconstruction of large metaorders from brokerage data . _ vaglica et al . ( @xcite ) reconstructed metaorders spanish stock exchange using data with brokerage codes and found that @xmath1 is distributed as a power law for large @xmath1 with @xmath173 . there is thus good evidence that metaorders have a power law distribution , though more study is of course needed . in this section we derive the functional form of the impact for pareto metaorder size . we do this by using eq . [ final ] in the limit as @xmath31 , and return in section [ finitesize ] and in appendix b to discuss how this is modified when @xmath30 is finite . while we only care about the asymptotic form for large @xmath1 , for convenience we assume an exact pareto distribution for all @xmath1 , i.e. @xmath174 where the normalization constant @xmath175 is the riemann zeta function . for the pareto distribution the probability @xmath74 that an order of size @xmath38 will continue is @xmath176 where @xmath177 is the generalized riemann zeta function ( also called the hurwitz zeta function ) . the approximations are valid in the large @xmath38 limit . the immediate impact can be easily calculated from eq . ( [ final ] ) . for pareto distributed metaorder sizes , using eqs . ( [ solution2 ] ) and ( [ pareto2eq ] ) , @xmath107 is @xmath178 thus the immediate impact @xmath179 behaves asymptotically for large @xmath38 as @xmath180 the exponent @xmath181 has a dramatic effect on the shape of the impact . for lorenzian distributed metaorder size ( @xmath182 ) the impact is logarithmic , for @xmath183 it increases as a square root , for @xmath184 it is linear , and for @xmath185 it is superlinear . thus as we vary @xmath181 the impact goes from concave to convex , with @xmath184 as the borderline case is special is that for @xmath186 the second moment of the pareto distribution is undefined . under the theory of lillo et al . ( @xcite ) , long - memory requires @xmath186 . ] . ( black circles ) and permanent impact @xmath187 ( red squares ) for @xmath183 . the dashed line is the price profile of a metaorder of size @xmath188 , demonstrating how the price reverts from immediate to permanent impact when metaorder execution is completed . the inset shows a similar plot in double logarithmic scale for a wider range of sizes ( from @xmath189 to @xmath190 ) . the blue dashed line is a comparison to the asymptotic square root scaling . _ bottom panel _ : expected immediate impact @xmath114 as a function of time @xmath38 for tail exponents @xmath191 and @xmath192 , illustrating how the impact goes from concave to convex as @xmath181 increases . , title="fig : " ] ( black circles ) and permanent impact @xmath187 ( red squares ) for @xmath183 . the dashed line is the price profile of a metaorder of size @xmath188 , demonstrating how the price reverts from immediate to permanent impact when metaorder execution is completed . the inset shows a similar plot in double logarithmic scale for a wider range of sizes ( from @xmath189 to @xmath190 ) . the blue dashed line is a comparison to the asymptotic square root scaling . _ bottom panel _ : expected immediate impact @xmath114 as a function of time @xmath38 for tail exponents @xmath191 and @xmath192 , illustrating how the impact goes from concave to convex as @xmath181 increases . , title="fig : " ] figure [ numerical ] illustrates the reversion process for @xmath183 and shows how the shape of the impact varies with @xmath181 . the permanent impact under the pareto assumption is easily computed using eq . ( [ generalpermanentimpact ] ) . a direct calculation shows that @xmath193 eqs . ( [ permanentimpact ] ) and ( [ temporaryimpact ] ) imply that the ratio of the permanent to the immediate impact is @xmath194 for example if @xmath183 the model predicts that on average the permanent impact is equal to two thirds of the maximum immediate impact , i.e. , following the completion of a metaorder the price should revert by one third from its peak value . in the previous section we have assumed that @xmath195 , so that we can treat the problem as if @xmath30 were infinite . in real markets the maximum order size is probably quite large , a significant fraction of the market capitalization of the asset . thus we doubt that the finite support of @xmath1 has much practical importance , except perhaps for extremely large metaorders . from a conceptual point of view , however , having an upper bound on metaorder size creates some interesting effects . as we have already mentioned , proposition 1 fails to hold when @xmath196 , so this must be handled with some care . in appendix c we illustrate how the results change when @xmath197 . what we observe is that when @xmath198 the impact is roughly unchanged from its behavior in the limit @xmath31 , but when @xmath199 the impact becomes highly convex . this is caused by the fact that the market maker knows the metaorder must end when @xmath120 . since by definition @xmath200 , the martingale condition requires that @xmath201 , i.e. there is no reversion when the metaorder is completed . this propagates backward and when @xmath197 it significantly alters the impact , as seen in figure [ figimpact ] . nonetheless , from a practical point of view we do not think this is an important issue , which is why we have relegated the details of the discussion to appendix c. changing the metaorder distribution has a dramatic effect on the impact . while we believe that the pareto distribution is empirically the correct functional form for metaorder size , to get more insight into the role of @xmath35 we compute the impact for an alternative functional form . for this purpose we choose the stretched exponential , which can be tuned from thin tailed to heavy tailed behavior and contains the exponential distribution as a special case . there is no simple expression for the normalization factor needed for a discrete stretched exponential distribution , so we make a continuous approximation , in which the metaorder size distribution is @xmath202 the normalization factor @xmath203 is the incomplete gamma function . the shape parameter @xmath204 specifies whether the distribution decays faster or slower than an exponential . ( @xmath205 implies faster decay and @xmath206 implies slower decay . ) for short data sets , when @xmath207 is small this functional form is easily confused with a power law . it can be shown that the stretched exponential leads to an immediate impact function that for large @xmath38 asymptotically behaves as @xmath208 this is the product of a power law and an exponential ; for large @xmath38 the exponential dominates . the permanent impact is @xmath209 where @xmath210 is the exponential integral function and in the last approximation we have used its asymptotic expansion . the ratio of the permanent to the immediate impact of the last transaction is @xmath211 in contrast to the pareto metaorder size distribution this is not constant . instead the ratio between permanent and immediate impact decreases with size , going to zero in the limit as @xmath212 . the fixed ratio of permanent and immediate impact is a rather special property of the pareto distribution . the three types of agents in our model are similar to kyle s ; his informed trader is replaced by our long - term traders , and his noise traders are replaced by our day traders . in both cases we assume a final liquidation . there are also several key differences . in our model : * our long - term traders do not have a monopoly , but rather have common information and compete in setting the size of their orders . their orders are bundled together and executed as a package by an algorithmic execution service . these two facts are essential to show that the fair pricing condition is a nash equilibrium . * the number of periods @xmath1 for execution is uncertain , and depends on the information of the long - term traders . this is important because the martingale condition is based on the market makers uncertainty about when the metaorder will terminate . * the distribution @xmath9 is arbitrary ( whereas kyle assumed a normal distribution ) . our key result is that information is almost fully reflected in metaorder size , i.e. that @xmath35 and @xmath9 are closely related . this means our results apply to any empirical size distribution @xmath35 . * our day traders respond to ongoing information signals that are permanent in the sense that they affect the final price , in contrast to kyle s noise traders , who are completely uninformed . this means that the information in the final liquidation price is incrementally revealed . * the most important difference is that our model has a different purpose . kyle assumed an information signal and solved for the optimal strategy to exploit it . we assume metaorder execution may or may not be going on in the background , and solve for its impact on prices . the theory presented here makes several predictions with clear empirical implications . in this section we summarize what these are and outline a few of the problems that are likely to be encountered in empirical testing . 1 . the fair pricing condition , eq . [ fairpricing ] , is directly testable , although it requires a somewhat arbitrary choice about when enough time has elapsed since the metaorder has completed for reversion to occur . ( one wants to minimize this time because of the diffusive nature of prices , but one wants to allow enough time to make sure that reversion is complete ) . 2 . the asymmetric price response predicted by eq . [ returnratio ] is testable . however , this only tests the martingale condition , which is the less controversial part of our model . the equivalence of impact as a function of time and size is directly testable . under our theory , for @xmath25 the immediate impact from the first @xmath38 steps is the same , regardless of @xmath1 . this is in contrast to the kyle model which predicts linear impact as a function of time , but can explain concavity in size only by postulating variable informativeness of trades vs. metaorder size . the prediction of immediate and permanent impact based on @xmath35 is directly testable through equations [ final ] and [ generalpermanentimpact ] . if the metaorder distribution is a power law ( pareto distribution ) , then for large @xmath1 the immediate impact scales as @xmath213 and the ratio of the permanent to the immediate impact of the last transaction is @xmath214 . see section [ paretosec ] . prediction ( 2 ) has been tested and confirmed by lillo and farmer ( @xcite ) , farmer et al . ( @xcite ) and gerig ( @xcite ) . preliminary results seem to support , or at least not contradict , prediction ( 5 ) . the only studies of which we are aware that attempted to fit functional form to the impact of metaorders are by torre ( @xcite ) , almgren et al . @xcite , and moro et al . ( @xcite ) ; they find immediate impact roughly consistent with a square root functional form . moro et al . also tested the ratio of permanent to immediate impact and found @xmath215 for the spanish stock market and @xmath216 for the london stock market , with large error bars . to our knowledge the other predictions remain to be empirically tested . though the market makers in our model are uncertain whether or not a metaorder is present , if it is present , they know when its execution begins and ends . the ability to detect metaorders from imbalances in order flow using brokerage codes has been demonstrated [ ( vaglica et al . @xcite ) , ( toth et al @xcite ) ] . a recent study of metaorders based on brokerage code information found average participation rates of @xmath217 for the spanish stock market ( bme ) and @xmath218 for the london stock market , for metaorders whose average size was just under @xmath62 in both markets , making such metaorders difficult to hide . the detection problem introduces uncertainties in starting and stopping times that may affect shape of the price impact . the traditional view in finance is that market impact is just a reflection of information . this point of view often goes a step further and postulates that the functional form of impact is determined by behavioral and institutional factors , such as how informed the agents are who trade with a given volume . this hypothesis is difficult to test because it is inherently complicated and information is difficult to measure independently of impact . within the framework developed here , such anomalies would violate the fair pricing condition . in this paper we embrace the view that impact reflects information , but we show how at equilibrium the trading volume reflects the underlying information and makes it possible to compute the impact . the metaorder size distribution determines the shape of the impact but does not set its scale . metaorder size has the important advantage of being a measurable quantity , and thus predictions based on it are much more testable than those based directly on information . the fair pricing condition that we have derived here may well hold on its own , even without informational efficiency . this could be true for purely behavioral reasons : the fair pricing condition holds because it can be measured reliably , and both parties view it as fair . thus while the main results here are consistent with rationality , they do not necessarily depend on it . we provide an example solution for the pareto distribution for metaorder size because we believe that the evidence supports this hypothesis . this gives the simple result that the impact is a power law of the form @xmath219 , and the ratio of permanent impact to the temporary impact of the last transaction is @xmath214 . however , the bulk of our results do not depend on this assumption . thus the reader who is skeptical about power laws may simply view the results for the pareto distribution as a worked example . the strength of our approach is its empirical predictions . because these involve explicit functional relationships between observable variables they are strongly falsifiable in the popperian sense . a preliminary empirical analysis seems to support the theory , but the statistical analysis so far remains inconclusive . we look forward to more rigorous empirical tests . we would like to acknowledge conversations with jean - 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( [ returnratio ] ) . for @xmath224 the profit per share is @xmath225 by substituting these two last expressions in eq . ( [ appeq1 ] ) we obtain @xmath226=\sum_{n=1}^{m-1}n\tilde r_n \sum_{i = n+1}^{m } p_i-\sum_{n=1}^{m-1}p_n\sum_{i=1}^{n-1}i\tilde r_i - p_m\sum_{i=1}^{m-1 } i\tilde r_i = \nonumber \\ = \sum_{n=1}^{m-1}n\tilde r_n \sum_{j = n+1}^{m } p_j-\sum_{n=1}^{m}p_n\sum_{i=1}^{n-1}i\tilde r_i\end{aligned}\ ] ] by explicitly computing the coefficients of each @xmath227 , it is easy to show they vanish for each @xmath228 , i.e. @xmath229=0 $ ] . * infinite support*. this proposition does not hold when @xmath6 has infinite support . in order to show this let us consider the expected profit for orders of length between @xmath189 and @xmath230 . the following proposition holds : * proposition @xmath231 . * _ the martingale condition for all intervals implies that for any integer @xmath232 , @xmath233 this equation holds both for finite and infinite support ( i.e. @xmath30 can be finite or infinite ) . _ * proof . * from the equation ( [ profpershare ] ) in the previous proposition , we know that martingale condition allows us to write the profit per lot traded as @xmath234 therefore the expected profit for orders shorter or equal to @xmath235 is @xmath236 this is equal to the quantity in eq . ( [ partsum2 ] ) . moreover it is clear that both terms in brackets are non - negative . this means that the market maker typically makes profits on short metaorders . if the support of @xmath35 is infinite then the martingale condition at all intervals implies that @xmath237\equiv\sum_{n=1}^\infty p_n n\pi_n=\lim_{\bar n\to\infty } \left(\sum_{i=\bar n+1}^\infty p_i\right ) \left(\sum_{i=1}^{\bar n } i\tilde r_i\right).\ ] ] in the infinite support case the behavior of the limit in the last term of the above expression depends on the asymptotic behavior of @xmath35 and @xmath238 for large @xmath1 . this is due to the fact that for large @xmath239 the first term in brackets goes to zero while the second term diverges . it is possible to construct examples where @xmath229 $ ] goes to zero , to a finite value , or diverges . this result shows that in the infinite support case the martingale condition does not imply zero overall immediate profits . * proposition 2 . * _ if the second derivative of the immediate impact @xmath114 is bounded strictly below zero , in the limit where the number of informed traders @xmath115 , any nash equilibrium must satisfy the fair pricing condition @xmath116 for @xmath29 . on average market makers profit from orders of length one and take ( equal and opposite ) losses from orders of length @xmath30 . _ the strategy of the proof is to show that counterexamples for which @xmath119 result in contradictions . first consider a candidate equilibrium with @xmath240 for some value of @xmath1 , where @xmath29 . assume a long - term trader @xmath117 buys @xmath241 shares at an average price @xmath242 . after averaging over @xmath37 the shares are subsequently valued at price @xmath243 . the profit is @xmath244 if long - term trader @xmath117 increases her order size by one share while all others hold their order size constant , her profit becomes @xmath245 the change in profit can be written @xmath246 the first term on the right ( @xmath247 ) represents the additional profit if it were possible to trade one extra share at the same average price , and the second term represents the reduction in profit because the average price increases . since there is nothing to distinguish the long - term traders , the equilibrium must be symmetric , i.e. they all make the same decision . thus if there are @xmath13 long - term traders buying @xmath241 shares and a day trader buying a random number of shares @xmath37 , at equilibrium @xmath248 . thus if @xmath13 is large , @xmath1 is also large . in the limit as @xmath13 is large the second term vanishes if @xmath249 this is true providing the second derivative of the function @xmath125 is bounded strictly below zero . thus in this limit @xmath250 and the candidate equilibrium fails because the informed trader has an incentive to deviate . similarly if @xmath251 the informed traders take a loss which can be reduced by trading less . when @xmath252 ( and as before @xmath29 ) no informed trader has an incentive to change her order size . this is clear since in eq . ( [ deltaprofit ] ) with @xmath116 the change in profit is given by the second term alone , which is always negative . a similar calculation shows that this is also true for decreasing order size , i.e. when @xmath116 , @xmath253 causes @xmath254 . the cases @xmath21 and @xmath120 have to be examined separately because in these cases the fair pricing condition is incompatible with the martingale condition and informational efficiency ( i.e. with the conditions on the final price ) . for @xmath21 the market makers profit is @xmath255 and from ( eq . [ shortterm ] ) the martingale condition is @xmath256 thus if @xmath257 , satisfaction of both the martingale condition and the fair pricing condition requires that @xmath258 , or equivalently that @xmath259 = 0 . in other words , if both conditions are satisfied then both the permanent and the temporary impact on the first step are identically zero , which would violate informational efficiency since @xmath260 . in section [ impactsolution ] we show by construction that this holds for all @xmath1 , i.e. it is clear in eq . ( [ final ] ) and ( [ generalpermanentimpact ] ) that the impacts @xmath114 and @xmath68 are identically zero if @xmath261 . to have sensible impact functions we must have @xmath262 , which means that market making is profitable on the first timestep , so that they always make a profit . this is false : the profit from a metaorder of length one is @xmath263 , where @xmath6 is a small number . in contrast , if a market maker participates only in the first trade of a large metaorder , her profit is @xmath264 , where @xmath265 is a large number . thus while @xmath266 , @xmath267 . ] . similarly if @xmath120 the martingale condition implies @xmath268 , i.e. no reversion , and since @xmath125 is an increasing function the market maker takes a loss @xmath269 assuming @xmath116 for @xmath29 , the market makers profit @xmath270 and loss @xmath271 are related by eq . ( [ breakevenonaverage ] ) as @xmath272 for realistic size distributions we expect metaorders of size one to be much more common than those of size @xmath30 , i.e. @xmath273 . the ratio of the total profits is @xmath274 thus the market maker receives frequent but small profits on metaorders of length one and rare but large losses for metaorders of length @xmath30 . long - term traders will rationally abstain from taking a loss on metaorders of length @xmath21 by simply not participating when they receive @xmath6 signals that are too weak ; thus , the trading volume at @xmath21 is due entirely to the day trader . similarly , although @xmath275 , the long - term traders are unable to improve their profits by trading more , since we have bounded the total amount an individual can trade at @xmath32 so they are blocked from further increase . thus the violations of the fair pricing condition when @xmath21 and @xmath120 occur naturally due to the institutional constraints that we have assumed and do not invalidate the equilibrium . * proposition 3 . * _ the system of martingale conditions ( eq . [ shortterm ] ) and fair pricing conditions ( eq . [ fairpricing ] ) has solution @xmath276 _ * proof . * the solution of eq . ( [ solution2a ] ) is a direct consequence of the martingale conditions ( eq . [ shortterm ] ) . the total profit of metaorders of length @xmath222 can be rewritten as ( see proof of proposition 1 ) @xmath277 the fair pricing conditions ( eq . [ fairpricing ] ) state that for @xmath278 it is @xmath279 , i.e. @xmath280 this is a recursive equation which determines @xmath238 once @xmath134 is given ( note that this equation does not hold for @xmath189 because we do not have fair pricing for metaorders of length one ) . the solution of this equation is @xmath281 and we prove it by induction . we assume that the solution holds for @xmath282 and we prove that it is true for @xmath283 . ( [ solreceqa ] ) holds for @xmath284 we can rewrite eq . ( [ receqa ] ) for @xmath283 as @xmath285 now by expanding the sum in brackets it is direct to show that @xmath286 since , by definition , @xmath287 the first two terms in the right hand side cancel and thus one obtains eq . ( [ solreceqa ] ) . this equation is equivalent to eq . ( [ solutiona ] ) . in fact @xmath288 i.e. our thesis , eq . ( [ solutiona ] ) . as already discussed briefly in section [ finitesize ] , if the condition @xmath195 is violated this has an effect on the impact . in this section we consider the exact case of a finite support pareto distribution . we show that when @xmath195 we obtain the same results of the previous section and we discuss what happens when @xmath197 . we assume that the metaorder size distribution is a truncated pareto distribution for all @xmath289 , i.e. @xmath290 where the normalization constant @xmath291 is the harmonic number of order @xmath292 . for the truncated pareto distribution the probability @xmath74 that a metaorder of size @xmath38 will continue is @xmath293 where @xmath177 is the generalized riemann zeta function ( also called the hurwitz zeta function ) . for small @xmath38 the function @xmath74 increases meaning that it is more and more likely that the order continues . in the regime of @xmath294 , @xmath74 is well approximated by the expression of eq . ( [ calpapprox ] ) for an infinite support pareto distribution . however , around @xmath295 , @xmath74 starts to decrease meaning that it becomes more and more likely that the order is going to stop soon , with a corresponding effect on the impact . the immediate impact can be easily calculated once the distribution of metaorder size is known by using eq . ( [ final ] ) . for truncated pareto distributed metaorder sizes , @xmath107 is ( for @xmath296 ) is @xmath297 for large @xmath38 but @xmath298 it is @xmath299 which is the same scaling as the infinite support pareto distribution ( see eq . ( [ temporaryimpact ] ) ) . the same holds true for the permanent impact . we have therefore shown that when @xmath298 the finite support of the metaorder size distribution is irrelevant and we obtain approximately the same results as in section [ paretosec ] . the finite size effects and the role of the finiteness of the support becomes relevant when @xmath300 . figure [ figimpact ] shows the total impact for @xmath301 and different values of @xmath181 . it is clear that the impact is initially described by a power law , but then it becomes strongly convex when the order length becomes comparable with the maximal length .

we develop a theory for the market impact of large trading orders , which we call _ metaorders _ because they are typically split into small pieces and executed incrementally . market impact is empirically observed to be a concave function of metaorder size , i.e. the impact per share of large metaorders is smaller than that of small metaorders . we formulate a stylized model of an algorithmic execution service and derive a fair pricing condition , which says that the average transaction price of the metaorder is equal to the price after trading is completed . we show that at equilibrium the distribution of trading volume adjusts to reflect information , and dictates the shape of the impact function . the resulting theory makes empirically testable predictions for the functional form of both the temporary and permanent components of market impact . based on the commonly observed asymptotic distribution for the volume of large trades , it says that market impact should increase asymptotically roughly as the square root of size , with average permanent impact relaxing to about two thirds of peak impact .

You are an expert at summarizing long articles. Proceed to summarize the following text: let @xmath4 be a finite sequence of positive numbers such that @xmath5 this sequence determines the corner cantor set @xmath6 of generation @xmath7 in @xmath8 , such that the @xmath9-th generation consists of @xmath10 cubes of edge length @xmath11 , each of these cubes contains @xmath12 corner cubes of the @xmath13-th generation , and so on . for brevity , we will call @xmath6 `` a cantor set '' instead of `` a cantor set of generation @xmath7 '' . there is a number of papers on estimates of various capacities , norms of integral transforms and operators , etc . , on such cantor sets . these estimates demonstrate the sharpness of various inequalities where the bounds are attained on cantor sets ; they are also of independent interest . but besides the necessary condition , there are certain additional conditions on @xmath11 in many cases . in the present paper we associate with given numbers @xmath11 satisfying _ only _ the condition , the `` regularized '' sequence @xmath14 such that @xmath15 , @xmath16 , and construct the ( non - corner ) cantor set @xmath6 formed by @xmath17 cubes of edge length @xmath18 . since the corner and non - corner cantor sets have similar structure , it is unimportant for applications which set to use . for a nonnegative finite borel measure @xmath19 in @xmath8 , @xmath20 , and @xmath21 , @xmath22 , define the @xmath23-truncated @xmath24-riesz transform of @xmath19 by @xmath25 where @xmath26 if the limit @xmath27 exists , we shall call it the @xmath24-riesz transform of @xmath19 at @xmath28 . to consider all finite borel measures and all points @xmath29 , one introduces the quantity that always makes sense , namely the so called maximal @xmath24-riesz transform @xmath30 ( note that @xmath31 and @xmath32 are vectors and @xmath33 is a number ) . besides @xmath34 and @xmath35 , we need the @xmath23-truncated @xmath24-riesz operator defined by @xmath36 for every @xmath22 , the operator @xmath37 is bounded on @xmath38 . we set @xmath39 later on we denote by @xmath40 ( without indices ) positive constants which may vary from line to line . let @xmath6 be the corner cantor set generated by a sequence @xmath4 , and consisting of @xmath17 cubes @xmath41 . let @xmath42 the probability measure uniformly distributed on each cube @xmath41 with @xmath43 . mateu and tolsa @xcite proved that if @xmath44 , @xmath45 , and @xmath46 with @xmath47 then @xmath48^{1/2}\le\pmb|\mathfrak{r}_{\mu}^s\pmb|\le c\biggl[\sum_{j=1}^n\t_j^2\biggr]^{1/2},\ ] ] where the constants @xmath49 depend only on @xmath50 and @xmath24 . in fact , mateu and tolsa proved a stronger assertion than the estimate from below : for sufficiently small @xmath23 , @xmath51 this result was refined by tolsa in @xcite , where the condition about monotonicity of densities @xmath52 was dropped . a more general class of cantor sets for @xmath53 ( again under the condition @xmath46 ) was considered in ( * ? ? ? * theorem 3.1 ) . the estimate from above in was also obtained in @xcite by another method . the arguments in @xcite and especially in @xcite are rather complicated . we give two independent proofs of for our `` regularized '' cantor set . the first ( direct ) proof is considerably simpler than in @xcite , @xcite . the second approach gives the desired inequality as a corollary of the following more delicate result . we shall prove that the inequality @xmath54 ( and therefore the analogous estimate for @xmath55 ) holds on a `` big '' portion of @xmath6 . we also consider the related problem in a more general setting and give certain applications . in particular , we establish the two - sided estimate of the riesz capacity associated with @xmath56 . this estimate is a refined version ( for non - corner cantor sets ) of the corresponding results in @xcite . we conclude this section with the construction of `` regularized '' non - corner cantor sets . let @xmath4 be a finite sequence of positive numbers satisfying , and let two parameters @xmath57 and @xmath58 be given . ( later on @xmath59 will depend on @xmath60 and @xmath24 . ) define the set @xmath61 , @xmath62 , of indices inductively in the following way : @xmath63 ; if @xmath64 , @xmath65 , and @xmath66 , then @xmath67 is the least @xmath68 , such that @xmath69 ; if @xmath64 , @xmath65 , and @xmath70 , then @xmath71 . thus , @xmath72 we set @xmath73 clearly ( see ) , @xmath74{c } \s_j\le\ell_j<\a^{-1}\s_j,\ 1\le j < n;\quad \a\s_n\le\ell_n\le\s_n;\\ [ .1 in ] \ell_{j_{p+1}}\le\a2^{-(j_{p+1}-j_p)}\ell_{j_p},\quad p=1,\dots , m-1 . \end{array}\ ] ] hence , @xmath75 . for @xmath76 and @xmath77 let @xmath78 be the cube @xmath79 construct the cantor set @xmath6 recursively as follows . for @xmath80 we set @xmath81 . take @xmath12 closed corner cubes @xmath82 , @xmath83 , of edge length @xmath84 ( i.e. distinct cubes lying inside @xmath85 with edges parallel to the edges of @xmath85 , such that each cube @xmath82 contains a vertex of @xmath85 ) . suppose that the cubes @xmath86 , @xmath87 , of edge length @xmath88 , @xmath89 , are already defined . partition each cube @xmath86 into @xmath90 equal subcubes @xmath91 . ( in the figure above @xmath92 , @xmath93 . ) if @xmath94 , we may consider this partition as @xmath95 sequential partitions of @xmath86 , such that on @xmath9-th step , @xmath96 , we split each cube @xmath97 into @xmath12 cubes @xmath98 of edge length @xmath99 . consider the cubes @xmath100 . remark that by , @xmath101 take @xmath12 closed corner cubes @xmath102 of edge length @xmath103 in each @xmath104 . we get @xmath105 cubes @xmath102 , and set @xmath106 for @xmath107 we obtain the desired set @xmath6 . our first theorem shows that under certain assumptions @xmath3 is comparable with its average value on a set of `` big '' measure , and this property holds not only on cantor sets . it means that the distribution of values of riesz transform is uniform in a certain sense . set @xmath108 , and denote by @xmath109 the class of nonnegative borel measures @xmath110 in @xmath2 such that @xmath111 [ th21 ] suppose that @xmath112 , @xmath113 , @xmath114 exists @xmath110-a.e . , @xmath115 , and @xmath116 then for every @xmath117 we have @xmath118 on the other hand , obviously implies with @xmath119 instead of @xmath120 . the analogous statements hold for @xmath121 . we deduce theorem [ th21 ] in section 3 from a deep result by nazarov , treil and volberg ( theorem 1.1 in @xcite ) . in section 4 we obtain the following estimates for @xmath122 and @xmath123 . as before , we denote by @xmath42 the probability measure uniformly distributed on each cube @xmath41 with @xmath43 . [ th22 ] let an integer @xmath20 and @xmath124 be given . there are constants @xmath57 , @xmath58 , depending only on @xmath60 , @xmath24 , and such that for any positive numbers @xmath125 , satisfying with @xmath126 , and for the corresponding cantor set @xmath6 , @xmath127^{1/2}&\le\|r_{\mu}^s\|_{l^2(\mu)}\le c\biggl[\sum_{j=1}^n\t_j^2\biggr]^{1/2},\quad \t_j=\frac{2^{-dj}}{\ell_j^s},\label{f24}\\ c\biggl[\sum_{j=1}^n\t_j^2\biggr]^{1/2}&\le\pmb|\mathfrak{r}_{\mu}^s\pmb|\le c\biggl[\sum_{j=1}^n\t_j^2\biggr]^{1/2},\label{f25}\end{aligned}\ ] ] where the positive constants @xmath49 depend only on @xmath60 and @xmath24 . ( we use the same notation @xmath52 for values slightly different from the ones in . clearly , the corresponding relations in both cases are equivalent . ) set @xmath128^{-1/2}\mu$ ] . if @xmath129 is small enough then @xmath112 , and @xmath115 by the upper bound in . moreover , the first inequality in implies with @xmath130 . thus , theorem [ th21 ] immediately yields the inequality @xmath131 the existence of cantor - type sets satisfying was established in ( * ? ? ? * section 7 ) using probabilistic arguments . but a concrete set was not presented . the particular case @xmath92 , @xmath132 , @xmath133 , was considered in @xcite . a more general class of plane corner cantor sets was treated in @xcite . clearly , implies the estimates from below in , ( in fact , the estimates from above were obtained in @xcite see section 4 of the present paper for details ) . in section 5 we give a completely different proof of without the use of theorem 1.1 in @xcite and of theorem [ th22 ] . this independent approach allows us to consider a more general class of measures and wider range of @xmath24 . in section 6 we consider the capacity @xmath134 of a compact set @xmath135 defined by the equality @xmath136 where @xmath137 is the class of positive radon measures supported on @xmath138 . for @xmath139 , @xmath140 is the analytic capacity @xmath141 , which is comparable with the analytic capacity @xmath142 by the remarkable result of tolsa @xcite . for @xmath143 , @xmath144 is comparable with the lipschitz harmonic capacity ( see @xcite and ( * ? ? ? * section 10 ) for details and references ) . [ th23 ] let @xmath20 , @xmath124 . for any finite sequence of positive numbers @xmath125 , satisfying with @xmath126 , @xmath145^{-1/2}\le\g_{s,+}(e_n ) \le c\biggl[\sum_{j=1}^n\biggl(\frac{2^{-dj}}{\s_j^s}\biggr)^2\biggr]^{-1/2},\ ] ] where the positive constants @xmath49 and the parameters @xmath146 , @xmath147 of the corresponding cantor set @xmath6 depend only on @xmath60 and @xmath24 . for corner cantor sets and @xmath139 ( that is for analytic capacity @xmath141 ) , the estimates were obtained in @xcite under the additional assumption @xmath148 . a different proof has been given in @xcite . the corresponding inequalities for @xmath142 were proved in @xcite ( before the tolsa s result @xcite about comparability of @xmath142 and @xmath141 ) . the case of the lipschitz harmonic capacity ( i. e. @xmath149 ) was treated in @xcite under the assumptions @xmath45 , and @xmath46 . thus , theorem [ th23 ] is a refined version of these results for regularized " cantor sets . it is noted at the end of @xcite that holds under the same assumptions for the signed riesz capacity @xmath150 as well . in fact , our proof of theorem [ th23 ] is a modification of the arguments in @xcite and @xcite , and these arguments also work for @xmath150 and for regularized " cantor sets under the additional assumption @xmath151 , but without monotonicity of @xmath152 . it is known that @xmath153 for @xmath92 , @xmath132 @xcite , for @xmath154 , @xmath155 ( * ? ? ? * theorem 1.1 ) , and for @xmath156 , @xmath53 @xcite . as far as we know , the validity of this relation in other cases is an open problem . the extension to bilipschitz images of corner cantor sets from @xcite is given in @xcite . in section 7 we give `` the limit case '' of theorem [ th23 ] , when the sequence @xmath157 is infinite . we use the obtained estimates to demonstrate the sharpness of results in @xcite . in particular , we consider the problem of comparison of the capacity @xmath144 and hausdorff content . [ le31 ] if @xmath158 is a non - negative function , @xmath19 is an arbitrary probability measure , and @xmath159 then for @xmath160 we have @xmath161 clearly , @xmath162 assume that @xmath163 then @xmath164 since the left hand side is equal to the same number @xmath165 , we come to a contradiction . by ( * ? ? ? * theorem 1.1 , p. 467468 ) , the uniform boundedness of the cut - off caldern - zygmund operators @xmath166 on @xmath167 implies the boundedness of @xmath147 and of the corresponding maximal singular operator on @xmath168 for every @xmath169 . applying this theorem for @xmath170 , @xmath171 , we get @xmath172 where the constants @xmath173 depend only on @xmath60 and @xmath24 ( the last statement follows from the proof of theorem 1.1 in @xcite ) . lemma [ le31 ] with @xmath174 , @xmath175 , @xmath176 , @xmath177 , @xmath178 , yields , since @xmath179 . the proof of the corresponding statement for @xmath121 is essentially the same . we need some notation . let @xmath180 , @xmath181 be the centers of @xmath86 and @xmath182 correspondingly , and let @xmath183 , @xmath184 be the cubes @xmath86 , @xmath182 , containing @xmath28 . set @xmath185 obviously , @xmath186 [ le41 ] there exists @xmath187 , such that for any @xmath188 and @xmath189 we have @xmath190 with @xmath191 and @xmath192 depending only on @xmath193 . we have @xmath194 by , the cubes @xmath195 in @xmath183 are separated . hence , up to a constant , @xmath196 is majorized by the integral over the measure uniformly distributed on @xmath183 with density @xmath197 , that is @xmath198 suppose that @xmath199 ( i. e. @xmath200 ) . we claim , that for sufficiently big @xmath147 , @xmath201 on the other hand , for @xmath147 big enough we have @xmath202 the lower bound in implies . obviously , @xmath203 whenever @xmath204 and @xmath205 are symmetric with respect to @xmath180 . hence , for `` half '' of the points @xmath199 , the angle between the vectors @xmath206 and @xmath196 is less then or equal to @xmath207 . for these @xmath28 we have @xmath208 we get the lower bound in . the upper bound follows directly from and . [ le42 ] let @xmath209 . then @xmath210 where @xmath192 depends only on @xmath60 and @xmath24 . by symmetry , @xmath211 suppose that @xmath212 . we have @xmath213\,d\mu(x ) . \end{split}\ ] ] by , , @xmath214 , we get @xmath215 as in lemma [ le41 ] , we represent the last integral as the sum of the integrals over @xmath216 and @xmath217 . the second integral is estimated exactly as in . the first integral , as before , is majorized by the integral with uniformly distributed measure . thus , the last bound does not exceed @xmath218 < c'\frac{t\ell_{j_p}}{\ell_{j_{q+1}}^{1-\d}}\cdot\frac{2^{-dj_q}}{\ell_{j_q}^{s+\d}}\\ = c't\bigg(\frac{\ell_{j_p}}{\ell_{j_{q+1}}}\bigg)^{1-\d}\bigg(\frac{\ell_{j_p}}{\ell_{j_{q}}}\bigg)^{\d } \frac{2^{-dj_q}}{\ell_{j_q}^{s } } \le c't\a^{\d(p - q)}2^{-\d(j_p - j_q)}\,\frac{2^{-dj_q}}{\ell_{j_q}^s},\quad c'=c'(d , s)\end{gathered}\ ] ] ( in the last inequality we used the obvious relation @xmath219 and ) . since @xmath220 , we obtain the inequality @xmath221 this inequality together with , , and the obvious relation @xmath222 , imply . set @xmath223 we start from the lower bound in . obviously , @xmath224 ^ 2\,d\mu(x)\\ & = \sum_{p=1}^{m}\int_{e_n}\xi_p(x)^2\,d\mu(x)+\sum_{p\ne q}\int_{e_n}\xi_p(x)\xi_q(x)\,d\mu(x)=:\sigma_1+\sigma_2.\end{aligned}\ ] ] from we have @xmath225 enumerate @xmath226 in decreasing order : @xmath227 . from we derive the estimate @xmath228\\ & < 4ct\a^{\d}\bigg[\sum_{i=1}^{\infty}2^{-\d i}\bigg]\sum_{p=1}^m\t_{p}^2\ , . \end{split}\ ] ] we can choose @xmath146 and @xmath147 in such a way that the constant @xmath229 in is at least twice as big as the constant before the last sum in . we have @xmath230 to get the lower bound in , it remains to note that @xmath231 obviously , the estimate from below obtained in implies the lower bound in . to complete the proof of theorem [ th22 ] , it is enough to get the upper bound in . but the proof of this estimate is literally the same as the proof of the corresponding estimate for corner cantor sets in ( * ? ? ? * corollary 3.5 ) . in this section we develop an independent approach to obtaining the estimate , as well as its generalizations and related results . let a finite sequence @xmath125 , @xmath233 , @xmath234 , and constants @xmath57 , @xmath58 , @xmath235 , be given . for the corresponding cantor set @xmath6 and for a positive measure @xmath19 supported on @xmath6 , set @xmath236 where @xmath237 is the cube @xmath41 containing @xmath28 . define the sets @xmath238 , @xmath239 of cubes by the relations @xmath240^{1/2}\bigg\},\quad \t_j=\frac{2^{-dj}}{\ell_j^s},\label{f51}\\ \tilde\ee&=\bigg\{e_{n , k}:|\tilde r_\nu^s(x_{n , k})|>\frac{c_0}{t^s}\biggl[\sum_{j=1}^n\t_j^2\biggr]^{1/2}\bigg\}\label{f52}.\end{aligned}\ ] ] here as before , @xmath241 is the center of @xmath41 ; in we assume that the values @xmath242 exist . [ th51 ] for every @xmath243 and every integer @xmath20 , there exist constants @xmath244 , @xmath245 , depending only on @xmath193 , with the following properties . fix some @xmath246 , @xmath247 , and a sequence @xmath125 , @xmath233 , @xmath234 . let @xmath19 be a measure supported on the corresponding cantor set @xmath6 , equally ( but not necessarily uniformly ) distributed on each cube @xmath41 , @xmath248 , with @xmath249 . then the number of cubes @xmath41 in @xmath239 is comparable with the number of all cubes in @xmath6 , that is , @xmath250 where @xmath251 is an absolute constant . moreover , if values @xmath242 exist , then @xmath252 the same conclusion holds if we replace @xmath241 in , by any fixed points @xmath253 , such that @xmath254 , @xmath255 . this theorem implies a useful corollary which will be given after the proof . the proofs of , , and of the statement for points @xmath256 , are the same . for definiteness , we consider the case . [ le52 ] for every finite sequence @xmath257 of positive numbers and for given @xmath258 , there is a subsequence @xmath259 , such that @xmath260 where @xmath191 depends only on @xmath261 and @xmath262 . let the maximal index @xmath263 be the least integer for which @xmath264 ( the value of @xmath265 will be determined by the construction in the sequel ) . suppose that indices @xmath266 are already defined , and @xmath267 . then @xmath268 is the least integer satisfying the relation @xmath269 this equality implies that @xmath270 in particular , we get . clearly , @xmath271 . using for @xmath272 with @xmath273 , we get the following estimate : @xmath274 which yields . we use lemma [ le52 ] with @xmath275 , @xmath276 , @xmath277 , in order to extract the future subsequence from @xmath278 ( the numbers @xmath279 are defined by ) . by lemma [ le52 ] , there exists the set @xmath280 of indices such that @xmath281 where @xmath191 is an absolute constant . set @xmath282 . each cube @xmath41 , @xmath248 , can be represented in the form @xmath283 where @xmath284 is the projection of @xmath41 onto the hyperplane @xmath285 , and @xmath286 is the projection of @xmath41 onto the @xmath287-axis . for @xmath288 fixed , we associate the vector @xmath289 with each cube @xmath290 in the following way . the choice of an interval @xmath286 can be viewed as a result of @xmath7 subsequent choices ( steps ) : starting from the interval @xmath291 $ ] , at @xmath9-th step , @xmath292 , we choose the left or the right of two equal subintervals of length @xmath99 in the preceding interval . we set @xmath293 , if we choose the left subinterval at @xmath9-th step , and @xmath294 in the opposite case . thus , we get the one - to - one correspondence between the cubes @xmath290 with fixed @xmath288 , and vectors @xmath295 . [ le53 ] let @xmath246 , @xmath247 , and let the cubes @xmath296 , @xmath297 , be such that @xmath298 suppose that a measure @xmath19 satisfies the conditions of theorem [ th51 ] . then @xmath299 where @xmath300 is the first component of a vector @xmath32 , and the constants @xmath245 , @xmath301 depend only on @xmath193 . in order to simplify notation , we set @xmath302 , @xmath303 ( see the figure in section 1 , where a possible location of @xmath120 , @xmath304 is indicated ) . as before , we denote by @xmath305 , @xmath306 the corresponding cubes containing @xmath28 . obviously , @xmath307 hence , @xmath308 ( for @xmath80 the last sums are absent ) . if @xmath147 is greater than certain @xmath309 then @xmath310 we claim that @xmath311 where @xmath192 is an absolute constant . indeed , by the the left hand side of does not exceed @xmath312 the inequality yields the estimate @xmath313 continuing the estimation , we get : @xmath314 since @xmath315 in the integrals from @xmath316 and @xmath317 , we have @xmath318 thus , @xmath319 as in the proof of lemma [ le41 ] , we majorize the last integrals by the integrals over the measures uniformly distributed on @xmath320 with densities @xmath321 . we get @xmath322 we consider three cases . * case 1 . * @xmath323 . then @xmath324 for @xmath325^{-1}$ ] , where @xmath301 is the constant in . * case 2 . * @xmath53 . by and we have @xmath326 if @xmath327^{-1}$ ] . * @xmath328 . again by , @xmath329 as concerns @xmath317 , we have @xmath330 the estimates for @xmath316 , @xmath317 and yield . the next lemma is a particular case of lemma 7.3 in @xcite . [ le54 ] let @xmath331 , be positive numbers . let @xmath332 , be independent random variables satisfying @xmath333 such that @xmath334 we need the following lemma 10.2 in @xcite . [ le55 ] let positive numbers @xmath331 , be given . for each vector @xmath335 , @xmath336 , with components equal to plus or minus ones we set @xmath337 let @xmath338 be a set of different vectors @xmath339 such that @xmath340 . then there exists a set @xmath341 with the following properties : \1 ) all the vectors in @xmath342 are distinct ; \2 ) @xmath343 ; \3 ) each vector @xmath344 can be obtained from some vector @xmath345 by replacing some negative components of @xmath346 ( depending on @xmath347 ) by positive components while keeping all the positive components of @xmath346 . ( different vectors @xmath347 can be obtained from the same @xmath346 . ) set @xmath348 , @xmath349 , @xmath350 , @xmath351 . fix an integer @xmath288 and the components @xmath352 of vectors @xmath289 . let @xmath353 be the set of cubes @xmath290 with fixed parameters indicated above . clearly , @xmath353 consists of @xmath354 elements depending only on the choice of the components @xmath355 with @xmath356 . we can consider this sampling as a choice of a vector @xmath339 . we introduce the following subsets of @xmath353 ( or equivalently , the subsets of vectors @xmath339 ) : @xmath357 where @xmath358 is the constant in lemma [ le54 ] with @xmath359 , and @xmath301 is the constant in . we shall consider the choice of a vector @xmath339 , @xmath336 , as a random event with the same probability @xmath360 for all vectors . this defines a random variable @xmath361 with values @xmath362 . obviously , we can interpret @xmath361 as the sum of independent random variables @xmath363 , @xmath364 , taking the values @xmath365 and @xmath366 with probability 1/2 . we now use lemma [ le54 ] . in our case , @xmath367 , @xmath359 . since distribution of @xmath361 is symmetric , yields the inequality @xmath368 hence , @xmath369 consider now the set @xmath370 . by @xmath338 we denote also the set of the corresponding vectors @xmath339 . if @xmath371 , then @xmath372 , and we immediately arrive at the estimates below . assume that @xmath373 , and let @xmath342 be the set of vectors corresponding to the set @xmath338 by lemma [ le55 ] . consider an arbitrary vector @xmath344 . it can be obtained from some @xmath345 by the replacement of some negative components of @xmath346 by positive ones . let @xmath374 be the set of indices of the replaced components of @xmath346 . let @xmath375 , @xmath376 , and let @xmath41 , @xmath377 be the cubes associated with @xmath339 and @xmath378 correspondingly . since @xmath379 , it follows that @xmath380 on the other hand , applying lemma [ le53 ] @xmath381 times ( successively for each component in @xmath374 ) we get @xmath382 so that @xmath383 hence @xmath384 . moreover , @xmath385 , since @xmath386 for @xmath387 . bearing in mind that @xmath343 , we obtain @xmath388 thus , for every set @xmath353 ( that is , for each @xmath288 and for each collection of components @xmath355 , @xmath389 ) , we have @xmath390 since there exist @xmath391 values of @xmath288 , and @xmath392 collection of components @xmath355 with @xmath389 , it follows that @xmath393 it remains to note that @xmath394 theorem [ th51 ] is proved . using theorem [ th51 ] we show that for the class of measures @xmath19 satisfying the assumptions of this theorem , the riesz transform @xmath395 is large on a `` big '' portion of @xmath6 . [ co56 ] let an integer @xmath20 and @xmath21 be given . there are constants @xmath57 , @xmath58 , @xmath396 , depending only on @xmath60 , @xmath24 , and such that for any positive numbers @xmath125 , @xmath233 , @xmath126 , and for the corresponding cantor set @xmath6 , @xmath397^{1/2}\bigg\}>\frac{c_1}2\hh^d(e_n)= \frac{c_1}22^{dn}\ell_n^d\,,\quad \t_j=\frac{2^{-dj}}{\ell_j^s}\,.\ ] ] here @xmath398 is @xmath60-dimensional hausdorff measure , and @xmath19 is a measure satisfying the conditions of theorem [ th51 ] , such that @xmath395 exists ( in the sense of principal values ) @xmath398-a . e. on @xmath6 . * remark . * for @xmath399 the condition about existence of @xmath395 holds for any borel measure on @xmath6 . generally speaking , it is not correct for @xmath400 . set @xmath401 , @xmath402 , where @xmath244 and @xmath245 are the constants from theorem [ th51 ] . choose a sufficiently small @xmath146 which will be specified later , and consider the corresponding cantor set @xmath6 . let @xmath403 be the point measure with charges equal to @xmath404 and located at the centers @xmath241 of the cubes @xmath41 . we claim that latexmath:[\[\label{f520 } if @xmath146 is small enough . indeed , let @xmath205 be any two points lying in the same cube @xmath41 ( that is , @xmath406 ) . in the same way as we obtain the inequality @xmath407 let @xmath408 $ ] , and let @xmath409 $ ] be such that @xmath410 . by we get @xmath411 ( we recall that @xmath412 ) . we use with @xmath413 instead of @xmath414 , and with @xmath415 , @xmath416 , @xmath417 . then @xmath418^{1/2 } \le\frac{c_0}{4t^s}\biggl[\sum_{j=1}^n\t_j^2\biggr]^{1/2 } , \end{split}\ ] ] if @xmath419^{-1},\ c''=c''(d , s)$ ] . clearly , @xmath146 depends only on @xmath60 and @xmath24 . the inequality and the same arguments as in yield the estimate @xmath420 let @xmath421 be the set defined by for the measure @xmath403 . fix some cube @xmath422 , and set @xmath423 ( we consider only the points @xmath28 for which @xmath395 exists ) . let @xmath424 be the cube symmetric to @xmath41 with respect to the center @xmath425 of the cube @xmath426 . choose @xmath413 , and let @xmath427 be such that @xmath428 . clearly , @xmath429 . moreover , @xmath430 by the symmetry of the measure @xmath403 with respect to @xmath425 . hence , for at least one couple of vectors @xmath431 , @xmath432 , or @xmath433 , @xmath434 , the angle between these vectors is less or equal to @xmath207 . so , for the sets @xmath435 for @xmath436 we get @xmath437^{1/2}.\end{aligned}\ ] ] analogous estimates hold for @xmath438 . hence , @xmath439^{1/2}\bigg\ } \ge\hh^d(\ee_{n , k})+\hh^d(\ee_{n , k^\ast})\ge \ell_n^d\ ] ] for every @xmath422 . since @xmath440 ( see ) , we get with @xmath441^{-1}$ ] . the inequality is a particular case of corollary [ co56 ] . indeed , note that @xmath442 , where @xmath443 . multiplying both parts of by @xmath444 , we get . as we mentioned in section 2 , our arguments are similar to those in @xcite ( which in turn use the ideas in @xcite ) . on the other hand , there are certain differences as well . for instance , in @xcite the harmonicity of the riesz transform ( outside the support ) was used in an essential way . since @xmath24 is not necessarily equal to @xmath445 , we can not use harmonicity , and we can not work with the measure supported on the boundaries of certain cubes , our measure will be supported also on the interior of these cubes . moreover , and this is more essential , @xcite uses a certain regularity of their cantor sets . we have another type of cantor sets , lacking this regularity ( namely , not all our cubes @xmath446 are separated enough ) . this creates specific difficulties . there is a number of other differences ( for example , we do not use cotlar s inequality ) . also , mateu and tolsa @xcite , while claiming the result for all @xmath124 ( and for their cantor sets ) , give the proof only for @xmath53 . this is why we wish to present a full proof , even though it follows the idea of @xcite . we need the following characterization of @xmath144 obtained in @xcite , chapter 5 : @xmath447 ( see for definition of @xmath109 ) . following @xcite , we introduce the capacity @xmath448 where @xmath42 is the probability measure defined in section 1 . using and , we have @xmath449^{1/2}\approx\tau\biggl[\sum_{p=1}^m\t_p^2\biggr]^{1/2},\quad \t_p=\frac{2^{-dj_p}}{\ell_{j_p}^s}\,.\ ] ] hence , @xmath450^{-1/2}\approx\biggl[\sum_{p=1}^m\t_p^2\biggr]^{-1/2},\ ] ] where the constants of comparison depend only on @xmath60 and @xmath24 . it is easy to see that the measure @xmath451^{-1/2}\mu$ ] with @xmath452 belongs to @xmath109 . now the relation and the upper bound in imply the estimate @xmath453^{-1/2}\approx\g_{s,+}^\mu(e_n),\quad c = c(d , s).\ ] ] thus , it is sufficient to prove that @xmath454 let @xmath19 be a positive radon measure supported on a compact set @xmath138 in @xmath2 , for which @xmath455 . it is shown in @xcite , p. 217 , that the last inequality implies the estimate @xmath456 the arguments in this part of the proof of lemma 4.1 in @xcite are valid not only for @xmath155 , but for @xmath44 as well ( the reference [ p ] , lemma 11 in @xcite should be replaced by [ p ] , lemma 3.1 ) . for @xmath53 , this fact is also noted in @xcite , p. 46 . hence , @xmath457 . by the definition of @xmath144 ( see section 2 ) we have @xmath458 ( the inequality also follows from ( * ? ? ? * lemma 3.2 ) . ) we will prove by induction on @xmath7 . the induction hypothesis is @xmath459 where the constant @xmath460 will be specified later . let @xmath461 suppose that @xmath462 . then by we get @xmath463 . this inequality together with yield the estimate @xmath464 in particular , we get for @xmath465 , if @xmath466 . moreover , we can assume without loss of generality that @xmath467 . hence , there exists @xmath265 , @xmath468 , such that @xmath469 we consider two cases . * for some constant @xmath470 to be determined below , @xmath471 the set @xmath472 is constructed exactly in the same way as @xmath6 , starting with @xmath473 instead of @xmath474 , and with @xmath475 instead of @xmath7 . hence , @xmath476^{-1/2 } = a_0^{-1}c_0c\biggl[\sum_{p = k+1}^m\t_p^2\biggr]^{-1/2}\\ & = a_0^{-1}c_0c[s_m - s_k]^{-1/2}\stackrel{\eqref{f68}}{\le}\sqrt2a_0^{-1}c_0cs_m^{-1/2}\\ & \stackrel{\eqref{f63}}{<}a_0^{-1}c_0c'\g_{s,+}^{\mu}(e_n)<c_0\g_{s,+}^{\mu}(e_n),\end{aligned}\ ] ] if @xmath477 . we get . * case 2 . * for the constant @xmath478 determined above , @xmath479 as in @xcite , we again distinguish two cases , namely @xmath480 and @xmath481 . if @xmath480 , then @xmath482 and we have @xmath483 thus , holds if @xmath460 is sufficiently big . suppose now that @xmath481 . then @xmath484 we consider the measure @xmath485 clearly , @xmath486 . we will show that @xmath487 . assuming this fact for a moment , we get @xmath488 and follows . to prove that @xmath489 is bounded , we will use the local @xmath490 theorem of christ @xcite . according to the main theorem 10 in @xcite , p. 605 , it is enough to prove that @xmath110 satisfies the following conditions : \(i ) @xmath491 ; \(ii ) @xmath492 ; \(iii ) for each ball @xmath493 centered at a point in @xmath494 , there exists a function @xmath495 in @xmath496 , supported on @xmath493 , such that @xmath497 and @xmath498 @xmath110-almost everywhere on @xmath494 , and @xmath499 . first we verify condition ( i ) . let @xmath500 be a ball centered at @xmath501 . if @xmath502 , we have @xmath503 suppose that @xmath504 . let @xmath505 be the least integer for which @xmath506 . then @xmath493 may intersect at most @xmath507 cubes @xmath508 . hence , @xmath509 using the same kind of ideas , it is not difficult to verify condition ( ii ) as well . thus , we only need to check the hypothesis ( iii ) . again , let a ball @xmath510 centered at @xmath511 be given . if @xmath512 , we set @xmath513 . then @xmath514 suppose that @xmath515 . let @xmath516 be the least integer for which there is a cube of generation @xmath516 ( i.e. the cube of @xmath517 ) contained in @xmath518 . clearly , @xmath519 ( since @xmath493 is centered at @xmath511 ) . we will construct the function @xmath495 supported on @xmath493 . by definition of the capacity @xmath144 , there is the positive radon measure @xmath19 supported on @xmath6 such that @xmath455 and @xmath520 . hence , there is the cube @xmath521 for which @xmath522 we need the localization lemma in @xcite . let @xmath523 be any cube ( see for the notation @xmath78 ) . let @xmath524 be an infinitely differentiable function supported on @xmath525 and such that @xmath526 , @xmath527 , @xmath528 , @xmath529 on @xmath530 . by lemma 3.1 in @xcite , p. 207 , @xmath531 to simplify notation , set @xmath532 for @xmath533 . at first we define @xmath495 when @xmath534 . in this case , we set @xmath536 if @xmath537 , we choose any cube @xmath538 , and define @xmath495 by translation of @xmath539 , namely @xmath540 ( we recall that @xmath541 is the center of @xmath542 ) . clearly , @xmath543 ( the last inequality follows from the fact that there are at most @xmath544 cubes of @xmath516-th generation in @xmath493 ) . to prove that @xmath495 is bounded , we apply to a cube @xmath545 ( see section 1 for notations ) . by , the cubes @xmath546 are separated . hence , @xmath547 , and by we have @xmath548 . thus , @xmath549 since @xmath550 consists of @xmath551 cubes @xmath552 , we get the estimate @xmath553 now implies that @xmath554 to complete the proof we only need to check that @xmath555 . the same estimates as in yield the inequality @xmath556 , @xmath557 . integrating by parts , for every @xmath558 we get @xmath559 thus , it is enough to prove for @xmath560 and @xmath561 . set @xmath562 for @xmath563 . then supp@xmath564 . as before , let @xmath532 for @xmath533 . applying to the measure @xmath19 with @xmath565 , and then to the measure @xmath566 instead of @xmath19 and with @xmath567 , we obtain @xmath568 let @xmath569 suppose that @xmath570 . then @xmath571 for which @xmath572 . hence , @xmath573 therefore , @xmath574 this estimate and imply the inequality @xmath575 in the same way we will show that @xmath576 indeed , @xmath577 for @xmath578 , and hence @xmath579 the same arguments as above together with and the property ( i ) of @xmath110 imply . it remains to establish the inequality @xmath580 then will follow from and . for every cube @xmath581 we have @xmath582 hence , the left - hand side of does not exceed @xmath583 since @xmath584 . theorem [ th23 ] is proved . we start with the extension of theorem [ th23 ] to infinite sequences @xmath157 . let @xmath585 , @xmath586 be positive numbers such that @xmath587 and let numbers @xmath57 , @xmath58 be given . we define the set @xmath588 and the `` regularized '' sequence @xmath589 in the same way as in section 1 for @xmath590 . namely , set @xmath63 . if @xmath64 , then @xmath67 is the least @xmath68 for which @xmath69 . possibly , @xmath591 for all @xmath68 . in this case the sequence @xmath592 is finite : @xmath61 , @xmath593 . clearly , @xmath594 furthermore , we set @xmath595 if @xmath592 is finite , then @xmath596 with @xmath597 . the cantor set @xmath138 is defined by the relation @xmath598 where the sets @xmath599 are defined in section 1 . in particular , if @xmath592 is finite then @xmath600 . [ th71 ] let @xmath20 , @xmath124 , and let a sequence @xmath601 satisfies . then @xmath602^{-1/2}\le\g_{s,+}(e ) \le c\biggl[\sum_{j=1}^\infty\biggl(\frac{2^{-dj}}{\s_j^s}\biggr)^2\biggr]^{-1/2},\ ] ] where the positive constants @xmath49 and the parameters @xmath146 , @xmath147 of the cantor set @xmath138 depend only on @xmath60 and @xmath24 . assume that @xmath592 is infinite . since @xmath605^{-1/2},\quad p=1,2,\dots,\ ] ] we get the estimate from above . we also get ( that is @xmath606 ) if the series in diverges . thus , we may assume that this series converges . the definition of @xmath144 and theorem [ th23 ] imply the existence of measures @xmath607 , @xmath608 , such that @xmath609^{-1/2},\quad \|r_{\nu_p}^s\|_{l^\infty(\r^d)}\le1.\ ] ] we may extract a weakly convergent subsequence @xmath610 . denote by @xmath19 the weak limit of this subsequence as @xmath611 . clearly , @xmath612^{-1/2}.\ ] ] for any @xmath613 , we have @xmath614 ( otherwise @xmath615 by continuity of @xmath616 on @xmath617 ) . hence , @xmath618 by , @xmath619 . hence , @xmath620 , and theorem [ th71 ] is proved . in @xcite we obtained estimates for the hausdorff content of the set where the riesz transform @xmath395 is large . we also obtained certain relations between hausdorff content and the capacity @xmath144 . we are going to show that these estimates are attained on the cantor sets defined above . the possibility of considering _ arbitrary _ sequences @xmath157 satisfying enables us to prove the corresponding assertions for _ any _ gauge function @xmath621 ( with the natural assumption that @xmath622 is nonincreasing see the explanation below ) without any additional conditions . the _ hausdorff content @xmath626 _ of a set @xmath627 is defined by @xmath628 where the infimum is taken over all ( at most countable ) coverings of @xmath629 by balls of radii @xmath630 . later on we assume that @xmath622 is nonincreasing . this condition , which may seem to be a regularity condition at the first glance , is actually not a restriction at all . it was proved in @xcite , p. 133 , proposition 5.18 , that for any measure function @xmath621 either @xmath631 for all @xmath627 , or there is another measure function @xmath632 such that @xmath633 is nonincreasing and for which hausdorff contents @xmath634 and @xmath635 coincide up to a constant factor depending only on the dimension @xmath60 . for @xmath636 , set @xmath637 clearly , @xmath638 . let @xmath621 be a measure function , @xmath639 , and let @xmath640 be inverse to @xmath621 . for a measure @xmath19 consisting of @xmath641 point charges , we have obtained the inequality @xmath642^{1/2},\ ] ] where @xmath643 . this implicit estimate can be written in a simpler form using a certain auxiliary function ( see @xcite for details ) . [ pr72 ] for every @xmath644 and @xmath645 , one can find a measure @xmath19 which is a linear combination of @xmath641 dirac point masses , and such that @xmath646 , and @xmath647^{1/2},\quad c = c(d , s)\ge1,\ ] ] for any @xmath648 . in ( * ? * section 7 ) a certain family of random sets and measures was introduced . it was proved that this family contains ( random ) measures @xmath19 with the properties indicated in proposition [ pr72 ] . thus , the existence of a measure with the desired properties was established , but the concrete measure was not presented . below we give a non - probabilistic construction of such a measure . without loss of generality we may assume that @xmath649 . it is sufficient to prove our assertion for @xmath650 . for given @xmath641 , @xmath651 , we introduce the function @xmath652^{1/2},\ \text { where } \ h(a_\s)=n^{-1}h(\s).\ ] ] clearly , @xmath653 as @xmath654 ( we recall that @xmath655 , and @xmath656 ) . since @xmath657 is nonincreasing , @xmath658 in particular , @xmath659 . hence , @xmath660 as @xmath661 thus , there exists @xmath662 such that @xmath663^{1/2}=c_4,\quad \frac1{ph(\s)}\left[\int_{a_{\s}}^{\s}\bigg(\frac{h(t)}{t^s}\bigg)^2\frac{dt}{t}\right]^{1/2}<c_4,\ \s>\s_0,\ ] ] where the constant @xmath664 , depending only on @xmath60 and @xmath24 , will be specified later . define @xmath11 by the equalities @xmath665 by , @xmath666 . let @xmath6 be the cantor set from corollary [ co56 ] , and let @xmath19 be the probability measure consisting of @xmath649 equal point masses @xmath667 located at the centers of the cubes @xmath668 . we will prove that implies . indeed , @xmath669 if @xmath670 is big enough ; here @xmath671 is the constant from , and @xmath672 . define the measure @xmath673 by the equality @xmath674 fix a ball @xmath675 . suppose that @xmath676 . then @xmath677 let @xmath678 , and let @xmath9 be such that @xmath679 ( if @xmath680 , the upper bound is absent ) . then @xmath681 intersects at most @xmath682 cubes @xmath542 of @xmath9-th generation , that is at most @xmath683 cubes @xmath41 . therefore @xmath684 hence , @xmath685 for every set @xmath629 in @xmath2 . we have @xmath686^{1/2}\bigg\}\stackrel{\eqref{f519}}{\ge } ch(\s_0).\end{aligned}\ ] ] thus , for any @xmath648 we have @xmath687 . if @xmath688 , then @xmath689 , and implies with @xmath690 . if @xmath691 , then @xmath692^{1/2}\\ & \ge\frac{c}{c_4p } \left[\int_{h^{-1}(c^{-1}\mathbf m / n)}^{h^{-1}(\mathbf m)}\bigg(\frac{h(t)}{t^s}\bigg)^2\frac{dt}{t}\right]^{1/2},\end{aligned}\ ] ] and we get with @xmath693 and @xmath694 . * remark . * one can see that the relation plays a crucial role in the proof of proposition [ pr72 ] ( as well as in the proof of the assertions below ) . thus , additional assumptions on @xmath157 imply certain conditions on @xmath621 . for instance , the assumption @xmath46 gives the unnatural restriction @xmath695 . for @xmath696 , this means that @xmath697 . moreover , if we assume that @xmath151 , @xmath258 , then @xmath698 , @xmath22 . our assumption that @xmath699 is nonincreasing , does not provide us with this property . thus , we need additional conditions on @xmath621 ( for instance , the stronger assumption that @xmath700 is nonincreasing ) . our next results concern the problem on the comparison of the capacity @xmath144 and hausdorff measure . it was proved in @xcite ( see theorem 10.1 ) that for each compact set @xmath135 , @xmath701^{-1/2},\quad 0<s < d,\ ] ] where @xmath191 depends only on @xmath60 , @xmath24 , and @xmath702 is defined by the equality @xmath703 . this relation is sharp in the following sense . [ pr73 ] for any @xmath704 with @xmath44 , and for any measure function @xmath621 , there is a constant @xmath192 , depending only on @xmath193 , and a compact set @xmath138 , such that @xmath705 , and @xmath706^{-1/2},\ \text { where } \ h(t_2)=m_h(e).\ ] ] fix some @xmath662 , and define the infinite sequence @xmath707 by the equalities . let @xmath138 be the cantor set from theorem [ th71 ] . we claim that @xmath708 the upper bound is obvious . the lower bound for a finite set @xmath592 was proved above . if @xmath592 is infinite , we consider the weak limit @xmath709 of some weakly convergent subsequence of the sequence @xmath710 of the measures defined in . clearly , @xmath711 for any ball @xmath681 ( see ) . hence , @xmath712 . this statement is a supplement to corollary 10.2 in @xcite : for each compact set @xmath135 , @xmath716^{s/\beta},\quad\text{where}\quad 0<s < d,\quad h(t)=t^\beta,\quad\beta > s,\ ] ] and @xmath191 depends only on @xmath60 and @xmath24 . * remark . * it is interesting to compare the condition @xmath718 with the corresponding condition @xmath720 , arising in the analogous problem of the comparison of hausdorff measure and the classical riesz capacity @xmath721 . the latter is generated by potentials with the _ positive _ kernel @xmath722 ( see @xcite , @xcite , sections 1,2 and the references therein , and @xcite , p. 147 for a more general setting ) . clearly , @xmath723 . the exponent 2 in @xmath724 reflects the difference between potentials with the signed kernel @xmath725 , and potentials with the positive kernel @xmath722 .

the aim of this paper is to estimate the @xmath0-norms of vector - valued riesz transforms @xmath1 and the norms of riesz operators on cantor sets in @xmath2 , as well as to study the distribution of values of @xmath1 . namely , we show that this distribution is `` uniform '' in the following sense . the values of @xmath3 which are comparable with its average value are attended on a `` big '' portion of a cantor set . we apply these results to give examples demonstrating the sharpness of our previous estimates for the set of points where riesz transform is large , and for the corresponding riesz capacities . the cantor sets under consideration are different from the usual corner cantor sets . they are constructed by means a certain process of regularization introduced in the paper . .

You are an expert at summarizing long articles. Proceed to summarize the following text: the lhc is expected to directly probe possible new physics beyond the standard model ( sm ) up to a scale of a few tev . while its data should provide answers to several of the major open questions in the present picture of elementary particle physics , it is important to start examining how this sensitivity can be further extended at a next generation of colliders . today we have a number of indications that new physics could be of supersymmetric nature . if this is the case , the lhc will have a variety of signals to discover these new particles and the linear collider ( lc ) will be required to complement the probe of the susy spectrum with detailed measurements . however , beyond supersymmetry there is a wide range of other scenarios invoking new phenomena at , and beyond , the tev scale . they are aimed at explaininig the origin of electro - weak symmetry breaking , if there is no light elementary higgs boson , at stabilising the sm , if susy is not realised in nature , or at embedding the sm in a theory of grand unification . many of such scenarios predict the existence of new particles that would be manifested as rather spectacular resonances in @xmath0 collisions , if the achievable centre - of - mass energy is sufficient . a high energy lc represents an ideal laboratory for studying this new physics @xcite . it also retains an indirect sensitivity , through a precision study of the virtual corrections to electro - weak observables , when their mass exceeds the available centre - of - mass energy . this paper summarises the results of a series of studies aimed at quantifying the potential of a high energy , high luminosity @xmath0 lc in extending to high scales the probe for new physics . while a significant activity has already addressed the issues related to a tev - class collider , we now review the potential of a multi - tev lc , such as clic . the analysis of the lep and slc data has provided a significant experience in the extraction of electro - weak observables , optimising their statistical sensitivity and controlling their systematic uncertainties . at larger centre - of - mass energies , the relevant @xmath1 cross sections are significantly reduced and the experimental conditions at the interaction region need to be taken into account in validating the anticipated accuracies on the cross section @xmath2 , forward - backward asymmetries @xmath3 and left - right asymmetries @xmath4 determination at @xmath5 = 1 tev - 5 tev . since the two - fermion cross section is of the order of only 10 fb , it is imperative to achieve high luminosity by reducing the beam - spot sizes . in this regime the beam - beam effects are important and the primary @xmath0 collision is accompained by several @xmath6 interactions . being mostly confined in the forward regions , this @xmath7 background reduces the polar angle acceptance for quark flavour tagging and dilutes the jet charge separation using jet charge techniques . these experimental conditions require efficient and robust algorithms to ensure sensitivity to flavour - specific @xmath8 production . the statistical accuracy for the determination of @xmath2 , @xmath3 and @xmath4 has been studied , for @xmath9 and @xmath10 , taking the clic parameters at @xmath5 = 3 tev . the simdet parametrised detector simulation has been used and the @xmath6 background , corresponding to 10 overlayed bunch crossings , has been added to @xmath11 , @xmath10 events . @xmath10 final states have been identified using an algorithm based on the sampling of the decay charged multiplicity of the highly boosted @xmath12 hadrons at clic energies @xcite . similarly to lep analyses , the forward - backward asymmetry has been extracted from a fit to the flow of the jet charge @xmath13 defined as @xmath14 , where @xmath15 is the particle charge , @xmath16 its momentum , @xmath17 the jet thrust axis and the sum is extended to all the particles in a given jet . here the presence of additional particles , from the @xmath18 background , causes a broadening of the @xmath13 distribution and thus a dilution of the quark charge separation . the track selection and the value of the power parameter @xmath19 needed to be optimised as a function of the number of overlayed bunch crossings . the results are summarised in terms of the relative statistical accuracies @xmath20 in table [ tab : res ] . another important issue is the accuracy on the luminosity determination , that needs to be controlled to 0.5% , or better . .relative statistical accuracies on electro - weak observables , obtained for 1 ab@xmath21 of clic data at @xmath5 = 3 tev , including the effect of @xmath6 background . [ cols="<,^ " , ] fermion compositeness or exchange of very heavy new particles can be described in all generality by four - fermion contact interactions @xcite . these parametrise the interactions beyond the sm by means of an effective scale , @xmath22 , @xmath23 the strength of the interaction is set by convention as @xmath24 and models can be considered by choosing either @xmath25 or @xmath26 as detailed in table ii . the contact scale @xmath27 can be interpreted as effect of new particles at a mass @xmath28 , @xmath29 . in order to estimate the sensitivity of electro - weak observables to the contact interaction scale @xmath27 , the statistical accuracies discussed in section ii have been assumed for the @xmath30 and @xmath31 final states . the systematics of the assumed 0.5% include the contributions from model prediction uncertainties . results are given in terms of the lower limits on @xmath27 which can be excluded at 95% c.l . , in figure [ fig : ci ] . it has been verified that , for the channels considered in the present analysis , the bounds for the different @xmath22 are consistent . high luminosity @xmath0 collisions at 3 tev can probe @xmath27 at scales of 200 tev , and beyond . for comparison , the corresponding results expected for a lc operating at 1 tev are also shown . beam polarisation represents an important tool in these studies . first , it improves the sensitivity to new interactions , through the introduction of the left - right asymmetries @xmath32 and the polarised forward - backward asymmetries @xmath33 in the electro - weak fits . if both beams can be polarised to @xmath34 and @xmath35 respectively , the relevant parameter is the effective polarisation defined as @xmath36 . in addition to the improved sensitivity , the uncertainty on the effective polarization , can be made smaller than the error on the individual beam polarization measurements . secondly , in the case of a significant deviation from the sm prediction would be observed , @xmath37 and @xmath38 polarization is greatly beneficial to determine the nature of the new interactions . this has been studied in details for a lc at 0.51.0 tev @xcite and those results also apply , qualitatively , to a multi - tev collider . extending the sensitivity to new physics beyond the anticipated reach of the lhc , is a prime aim of future colliders . by accurately measuring electro - weak observables , a lc able to achieve @xmath0 collisions at and beyond 1 tev , with high luminosity , can indirectly probe scales extending from tens to several hundreds tev . giudice , r. rattazzi and j.d . wells , nucl . b544 * ( 1999 ) 3 ; e. a. mirabelli , m. perelstein and m. e. peskin , phys . * 82 * ( 1999 ) 2236 ; t. han , j.d . lykken , r .- j . zhang , phys . * d59 * ( 1999 ) 105006 . see , for example , t.g . rizzo and j.d . wells , phys . rev . * d61 * 016007 ( 2000 ) ; p. nath and m. yamaguchi , phys . rev . * d60 * ( 1999 ) 116006 ; m. masip and a. pomarol , phys . rev . * d60 * ( 1999 ) 096005 ; r. casalbuoni , s. de curtis , d. dominici and r. gatto , phys . b462 * ( 1999 ) 48 ; a. strumia , phys . lett . * b466 * ( 1999 ) 107 .

extending the sensitivity to new physics beyond the anticipated reach of the lhc is a prime aim of future colliders . this paper summarises the potential of an @xmath0 linear collider , at and beyond 1 tev , using a realistic simulation of the detector response and the accelerator induced background . the possible lc energy - luminosity trade - offs offered in probing multi - tev scales for new phenomena with electro - weak observables are also discussed .

You are an expert at summarizing long articles. Proceed to summarize the following text: recently , aschenbach ( @xcite ) reported the discovery of a young supernova remnant ( snr ) designated rx j0852.04622 from high - energy x - ray data from the rosat all - sky survey . this new snr appears near the southeastern boundary of the vela remnant ( e.g. milne @xcite ; aschenbach et al . @xcite ; duncan et al . @xcite ) , appearing in x - rays ( with @xmath2 1.3 kev ) as a nearly circular `` ring '' approximately @xmath3 in angular diameter . around the circumference of this ring are a number of enhancements in the x - ray emission , the most prominent of which appears near the northwestern perimeter . the available x - ray and @xmath1-ray data show the remnant to be comparatively young , with an age of @xmath4 yr ( iyudin et al . @xcite ; aschenbach et al . @xcite ) . following from this x - ray detection , combi ( @xcite ) reported a radio detection of the snr from the 2.42-ghz data of duncan et al . ( @xcite ) . these authors present spatially filtered data from the parkes 2.42-ghz survey , along with results obtained from the 30-m villa elisa telescope at 1.42 ghz ( beamwidth @xmath5 ) . the possibility of providing a more accurate age for this remnant was raised by burgess & zuber ( @xcite ) , who present a re - analysis of nitrate abundance data from an antarctic ice core . these authors find evidence for a nearby galactic sn @xmath6 years ago , in addition to the known historical supernovae ( e.g. clark & stephenson @xcite ) , although it is not possible to link this new sn definitively with rx j0852.04622 . in this paper , we examine three sets of radio continuum data from the parkes telescope , at frequencies of 1.40 , 2.42 and 4.85 ghz . we use these data to further investigate the radio structure of rx j0852.04622 . implications of the radio characteristics of this remnant for statistical studies of snrs are then considered . the radio data presented here come from three principal sources , at frequencies of 4.85 , 2.42 and 1.40 ghz . characteristics of these data are given in table [ table_data ] . first , 4.85-ghz data have been obtained from the parkes - mit - nrao ( pmn ) survey images ( griffith & wright @xcite ) . these images were observed using the 64-m parkes radio telescope , and have an angular resolution of approximately @xmath7 . processing of the pmn observations has removed large - scale information ( @xmath8 ) from the data . nevertheless , the pmn images are a useful source of higher resolution information , and are often able to trace structures of large angular size through associated smaller - scale emission components ( e.g. duncan et al . @xcite ) . second , 2.42-ghz data surrounding rx j0852.04622 have been observed as part of a larger survey presented by duncan et al . ( @xcite ) . these data have a resolution of @xmath9 and include linear polarisation information . some results from these data pertaining to the vela region have been presented by duncan et al . ( @xcite ) . these data were used by combi et al . ( @xcite ) to make the radio detection of rx j0852.04622 . third , 1.40-ghz observations of the region containing the remnant were obtained in 1996 september , as part of a larger survey of the vela region at this frequency . some of these data have already been used by other authors ( e.g.sault et al . @xcite ) . the observing procedure employed for these 1.40-ghz data was analogous to that used for the 2.42-ghz survey ( duncan et al . the telescope was scanned over a regularly - spaced coordinate grid , at a rate of @xmath10 per minute , until the region of interest had been completely covered . this procedure was then repeated , scanning the telescope in the orthogonal direction . stokes-@xmath11 , @xmath12 and @xmath13 data were recorded . the source pks b1934638 was used as the primary gain calibrator for the observations . the flux density of this source was assumed to be 14.90 jy at a frequency of 1.40 ghz . the source 3c138 was also observed , in order to calibrate the absolute polarisation position - angles . the intrinsic polarisation position - angle of 3c138 is @xmath14 ( tabara & inoue @xcite ) . after the calculation and subtraction of appropriate `` baselevels '' from each scan , each pair of orthogonally - scanned maps was combined . ccccc frequency & rms noise & angular & stokes & data origin + ( /ghz ) & ( /mjy ) & resolution & & + 1.40 & 20 & @xmath0 & @xmath15 & this paper + 2.42 & 17 & @xmath9 & @xmath15 & 2.42-ghz survey + 4.85 & 8 & @xmath7 & @xmath11 & pmn survey + the radio emission from rx j0852.04622 is superposed upon a highly structured region of the vela remnant . much of this confusing emission is of similar surface brightness to that seen from the new snr . furthermore , the very bright , thermal region rcw 38 lies almost adjacent to the southeastern boundary of rx j0852.04622 . the peak flux of rcw 38 is approximately 150 jy beam@xmath16 in the 2.42-ghz data . the presence of this confusing radio structure , both thermal and non - thermal , meant that rx j0852.04622 was not recognised as an snr from pre - existing radio observations of the region . prior to the x - ray discovery of rx j0852.04622 the non - thermal emission in this region was thought to emanate from the vela snr . the filtered 2.42-ghz image presented by combi et al . ( @xcite ) clearly shows the snr to have a shell - like radio morphology . this is even apparent in unfiltered maps of the region , such as that presented in fig . [ fig_tpsfull ] . indeed , the emission now known to be associated with rx j0852.04622 can be recognised in the radio images presented by duncan et al . ( @xcite , @xcite ) . combi et al . ( @xcite ) also identify several additional features within their radio image , designated `` a '' through `` d '' , which they suggest may represent extensions to the radio shell . these will be considered in more detail in sect . [ subsection_extensions ] . it should be noted that possibly as a result of their filtering procedure the 2.42-ghz image presented by combi et al . ( @xcite ) does not show either the region rcw 38 , or the bright , non - thermal emission from vela - x to the west . [ fig_tpsfilt ] shows a spatially - filtered image of the region surrounding rx j0852.04622 . this image has been filtered using the `` bgf '' algorithm ( e.g. sofue & reich @xcite ) , implemented within the nod2 software package . a number of filtering resolutions were used , and it was ( qualitatively ) determined that the emission from rx j0852.04622 was optimally enhanced with a filtering resolution of approximately @xmath17 to @xmath18 ( in agreement with combi et al . @xcite ) . a filtering resolution of @xmath18 was used for the radio data presented in fig . [ fig_tpsfilt ] . this figure shows both the emission from rx j0852.04622 and the confusing structure more clearly . comparing fig . [ fig_tpsfilt ] with the unfiltered data presented in fig . [ fig_tpsfull ] , it can be seen that removal of the large - scale structure does not have a major effect on the appearance of the field . the radio image of the new remnant is dominated by two opposing arcs . some fainter radio emission is visible on the remnant s western side , although there is no obvious counterpart to the east . the brightest section of the radio shell lies to the northwest , and appears approximately coincident with the brightest region of the x - ray image . comparing the radio with the x - ray emission ( fig . [ fig_tpsfilt ] , lower panel ) , we see that the distributions of both are generally similar , at least in as much as can be discerned from the cluttered radio field . [ fig_pmn ] shows data from the 4.85-ghz pmn survey from the same region as shown in the previous figures . although this survey is not optimised for extended sources , the northern and southern sections of the limb - brightened shell stand out clearly . the black circle near the centre of fig . [ fig_pmn ] fits the outer boundary of the radio emission from both figs [ fig_tpsfilt ] and [ fig_pmn ] well , and represents what we take to be the outer boundary of the radio emission from rx j0852.046221 . this boundary is @xmath19 in angular diameter , and is centred on the x - ray centre of the snr ( as given by aschenbach @xcite ) . within the uncertainties , this diameter is in agreement with that estimated by combi et al . ( @xcite ) , who quote a value of @xmath20 , based upon the ( lower resolution ) parkes 2.42-ghz data alone . both the radio and x - ray data are consistent with a remnant centred on galactic longitude @xmath21 , latitude @xmath22 . thus , we suggest a galactic designation of g266.201.2 for this snr . as can be seen from fig . [ fig_tpsfilt ] , a good deal of additional radio structure is visible in the vicinity of rx j0852.04622 . over the remnant itself , most of this structure takes the form of two diffuse `` filaments '' , each of which is @xmath23 wide . these filaments are oriented approximately north south over the new snr . beyond the northern boundary of the remnant , the filaments begin to curve towards the west . from larger images of the vela region , such as have been presented by duncan et al . ( @xcite , @xcite ) , these filaments are known to curve around vela - x , forming almost a full quadrant of a circle . the eastern arc ( as seen in fig . [ fig_tpsfilt ] ) is highly polarised ( e.g. duncan et al . @xcite ) , and appears to represent the current boundary of the shock from the vela supernova event . interestingly , the confusing filaments from fig . [ fig_tpsfilt ] are almost completely absent from the pmn data . this is because of the observing and data processing procedures used as part of the pmn survey , coupled with the fact that the confusing filaments lie approximately parallel to the scanning direction of the telescope over the region of sky containing rx j0852.04622 . as mentioned in sect . [ subsection_24ghz ] , combi et al . ( @xcite ) identify a number of additional features within their radio image . these features were designated `` a '' through `` d '' in fig . 1 of their paper , and apparently extend for relatively large angular distances beyond the edge of the rx j0852.04622 shell ( up to almost twice the radius of the remnant ) . combi et al . ( @xcite ) argue that these features may represent extensions to the radio shell of rx j0852.04622 ( c.f . aschenbach et al . it is of interest to consider these in more detail . examining the 2.42-ghz radio image of combi et al . ( @xcite ) , we find that radio features `` a '' and `` c '' appear to be sections of the much more extensive `` arc '' structures discussed in sect . [ subsection_confusion ] . these arcs can be traced in fig . [ fig_tpsfilt ] for several degrees , up to the northern edge of the figure ( i.e. , beyond the boundary of rx j0852.04622 ) . larger radio images of the region show that these features continue for many degrees further , in both total - power and polarised intensity . feature `` b '' appears to be an isolated , slightly extended source with no obvious connection to the new remnant ( even in the unfiltered image presented in fig . [ fig_tpsfull ] ) . finally , feature `` d '' corresponds to the x - ray feature `` d / d@xmath24 '' as identified by aschenbach et al . ( @xcite ) . aschenbach ( @xcite ) notes that this feature is also a source of hard x - rays , but that this emission is associated with a much lower temperature spectrum than that from rx j0852.04622 . we suggest , therefore , that none of the possible `` extensions '' identified by combi et al . ( @xcite ) are associated with rx j0852.04622 . a further argument against an association between these features and the new remnant is that the boundary of the rx j0852.04622 shock is well fitted ( in both the pmn and parkes 2.42-ghz survey data ) by a circle . this is consistent with the radio morphologies of other young shell snrs , such as kepler ( dickel et al . @xcite ) , tycho ( dickel et al . @xcite ) , and the remnant of sn1006 ( reynolds & gilmore @xcite ) , although we caution that rx j0852.04622 is considerably fainter than these and other young remnants . we also note that the higher resolution pmn image ( fig . [ fig_pmn ] ) , although not optimised for extended emission , shows no evidence for any connections between the features noted by combi et al . ( @xcite ) and the shell of the new remnant . the pmn image shows that a point - like ( i.e. unresolved at a resolution of @xmath25 ) source lies approximately @xmath26 east of the apparent centre of the remnant . this source is not coincident with either of the two compact x - ray sources near the centre of the remnant that are discussed by aschenbach et al . ( @xcite ) . the pmn survey source catalogue ( wright et al . @xcite ) lists this source as pmn j08534620 , with a 4.85-ghz flux of @xmath27 mjy . being relatively faint , our ability to detect pmn j08534620 in the 2.42-ghz data is compromised somewhat by confusion , coupled with beam dilution . nevertheless , we can establish the 2.42-ghz flux to be @xmath28 mjy . this leads to a spectral index for this source of @xmath29 ( with @xmath30 ) . the radio spectral indices of pulsar emissions are generally much steeper than the @xmath31 estimated above for the source ( e.g. taylor et al . furthermore , a flux of @xmath27 mjy at a frequency of 4.85 ghz would be exceptionally high for a pulsar . it is much more likely , then , that the pmn j08534620 source is extragalactic in origin , rather than associated with rx j0852.04622 . the presence of the confusing structure noted in sect . [ subsection_confusion ] makes accurate estimates of the integrated remnant flux density difficult . the values for the flux density of the snr given below were estimated by integrating the emission within the boundary of rx j0852.04622 , as defined by the circle seen in fig . [ fig_pmn ] ( lower panel ) . the integrated area extended approximately one beamwidth beyond this circle , in order to include all the flux from the shell . the base level was determined from flux minima near the centre of the remnant , as well as beyond the eastern and southwestern edges of the shell . fluxes contributed by the confusing `` filaments '' seen to the western and eastern sides of the remnant ( as discussed in sect . [ subsection_confusion ] ) were estimated and subtracted from the total , integrated flux . note that the uncertainties in the integrated flux values are dominated by baselevel uncertainty , rather than by uncertainties in the flux estimates of the confusing structure . we estimate the integrated fluxes of rx j0852.04622 at 2.42 and 1.40 ghz to be @xmath32 jy and @xmath33 jy , respectively . these values lead to a very uncertain estimate of the spectral index , with @xmath34 ( @xmath35 ) . to better establish the spectral index of the remnant , the method of `` t t '' plots was used ( e.g. turtle et al . @xcite ) . estimates of the remnant spectral index were made from both filtered and unfiltered images , using the t t plot technique . the northern section of the shell was found to have a consistent , non - thermal index of @xmath36 . the southern section of emission exhibited a much flatter spectrum of @xmath37 . we believe this latter value to be unreliable , due to the proximity of the southern section of the shell to the bright region rcw 38 and its associated emission . at the lower angular resolution of the 1.40-ghz data ( to which the 2.42-ghz data are also smoothed , for the purposes of spectral index calculation ) , some of this thermal emission becomes confused with the southern arc of rx j0852.04622 . we suggest that the value determined for the northern shell section better represents the new remnant s radio emission . extrapolating the measured integrated flux to a frequency of 1 ghz ( using a spectral index of @xmath36 ) , we determine a value of @xmath38 jy , leading to an average surface brightness at this latter frequency of @xmath39 w m@xmath40 hz@xmath16 sr@xmath16 . the above values are summarised in table [ table_values ] . the 2.42-ghz stokes-@xmath12 and @xmath13 data have rms variations of @xmath41 mjy beam@xmath16 at this resolution . the grey - scale image is blanked wherever the polarised intensity falls below 45 mjy beam@xmath16 . orientations of the tangential components of the magnetic fields are also shown . these angles have been calculated from the 1.40- and 2.42-ghz data , assuming that the vector angles vary linearly with @xmath42 . errors in the derived angles are generally @xmath43 . a vector is plotted every @xmath44 , wherever both the 1.40- and 2.42-ghz polarised intensities ( at @xmath0 resolution ) exceed 0.1 jy beam@xmath16 . the grey - scale wedge is labelled in units of jy beam@xmath16.,width=321 ] . filled squares represent positive rm values , while empty squares represent negative values . the size of each square is proportional to the magnitude of the rm , with the maximum size corresponding to @xmath45 rad m@xmath40 . the errors in rm values are @xmath46 rad m@xmath40 . a square is plotted every @xmath47 . data have been blanked wherever the 1.40- or 2.42-ghz polarised intensity ( at @xmath0 resolution ) falls below 0.1 jy beam@xmath16.,width=321 ] we have examined the 1.40- and 2.42-ghz polarimetric data in the field surrounding rx j0852.04622 . these data have been used to estimate the faraday rotation measures ( rms ) across the field , and to calculate the polarisation position - angles at zero wavelength ( thereby estimating the orientations of the tangential components of the magnetic fields within the regions of polarised emission ) . note that these estimates assume the polarisation position - angles to vary linearly with @xmath42 . the polarised intensities from the 2.42-ghz data are shown in fig . [ fig_b ] , at the lower resolution of @xmath0 . superposed upon this grey - scale image are the orientations of the tangential component of the magnetic field . the rms derived across the field are shown in fig . [ fig_rm ] . the large circle shows the boundary of the snr rx j0852.04622 . much of the polarised emission visible within the circle does not appear to be associated with the new snr . for example , the polarisation detected from the eastern half of rx j0852.04622 is clearly associated with the prominent , eastern arc of total - power emission . the only region of rx j0852.04622 to potentially exhibit polarised emission is the northern section of the shell . if associated with the new remnant , the northern arc appears polarised to a level of approximately 20% at 2.42 ghz . other regions of the snr exhibit no polarised emission ( such as the southern section of the shell ) , or are confused with polarised structure from vela . the magnetic field vectors on the north side of the shell appear jumbled , with no clear orientation evident . this is in contrast to the field orientations in other young , shell - type supernova remnants , which are predominantly radial ; e.g. tycho ( wood et al . @xcite ) , kepler ( matsui et al . @xcite ) and sn1006 ( reynolds & gilmore @xcite ) . however , it is questionable whether the detected polarised emission in this northern section of the shell is attributable to rx j0852.04622 . the appearance of much of this rm structure is similar to that seen from the vela emission , beyond the rx j0852.04622 shell s northwestern perimeter . furthermore , there is no discontinuity in either the rm values or the magnetic field orientations near the boundary of the shell . we suggest , therefore , that the polarised emission seen throughout fig . [ fig_b ] originates entirely from the vela remnant . we note that this interpretation is also consistent with the lack of polarisation observed from the southern arc of the rx j0852.04622 shell . the fractional polarisation of the northern arc must then be @xmath48% at 2.42 ghz . as noted above , we detect no polarisation in the vicinity of the bright , southern section of the shell , to a ( @xmath49 ) limit of approximately 5% at 2.42 ghz . we note that these low fractional polarisations are not inconsistent with the polarimetric properties of other young shells , which exhibit fractional polarisations of @xmath50% when the emission is well resolved . ll angular diameter & @xmath19 + integrated 1.40-ghz flux density & @xmath33 jy + integrated 2.42-ghz flux density & @xmath32 jy + @xmath51 ( from t t plots ) & @xmath36 + @xmath52 ( from t t plots ) & @xmath53 ( ? ) + integrated flux density at 1 ghz & @xmath38 jy + surface brightness at 1 ghz & @xmath54 + several observations of rx j0852.04622 suggest that it is a young snr . most notably , both the x - ray temperatures derived by aschenbach ( @xcite ) and the @xmath1-ray work of iyudin et al . ( @xcite ) imply an age of @xmath55 yr . further , the more recent discussion of the @xmath1-ray observations by aschenbach et al . ( @xcite ) suggest the remnant is @xmath56 years old . this is in agreement with the approximately circular radio appearance of the remnant ; a characteristic exhibited by other young , shell - type snrs . even the cautious upper distance limit of approximately 1 kpc provided by aschenbach ( @xcite ) , which leads to a linear diameter of @xmath57 pc , implies the age of this remnant can not be more than a few thousand years . however , there are some radio properties of this remnant are difficult to reconcile with those of other young snrs . * the radio shell is far from complete . the radio image of rx j0852.04622 is composed primarily of two , opposing regions of emission this is in contrast to other young , shell - type snrs , which exhibit essentially complete radio shells . we note a possible resemblance to the structure of sn1006 , however , which also shows opposing arcs of emission . * the radio emission from this new snr is of relatively low surface brightness . the mean radio surface brightness of this new remnant at 1 ghz ( see table [ table_values ] ) is a factor of 5 lower than that of sn1006 . this is significant , because the sn1006 remnant has the lowest surface brightness of all the young , shell - type snrs in current catalogues ( e.g. green @xcite ) see the further discussion in sect . [ subsection_statistics ] . * the radio spectral index of @xmath36 determined herein for rx j0852.04622 is flatter than those of other young shells , which have indices of @xmath58 ( e.g. green @xcite ) . it is possible that some of the unusual radio properties of rx j0852.04622 may result from the snr expanding into a hot , low - density region of the interstellar medium ; aschenbach ( @xcite ) determines an upper limit to this density of approximately 0.06 @xmath59 . if so , this would emphasise the the role played by environmental effects on the detectability of young radio remnants . alternatively , this snr may be just beginning to `` turn on '' at radio wavelengths , although this scenario may be difficult to reconcile with the very incomplete radio shell . further insights into the unusual radio properties of this snr must await more detailed investigations of the characteristics of both the remnant and the environment into which it is expanding . unfortunately , the distance to rx j0852.04622 is highly uncertain . the x - ray data of aschenbach ( @xcite ) provide an upper limit of approximately 1 kpc , based on the lack of absorption , but suggest that the remnant s distance could be as small as 200 pc . this lower limit is based upon a comparison of the new remnant s surface brightness with that of sn1006 . however , sn1006 is atypically faint for known , young , galactic snrs ( this is further discussed in sect . [ subsection_statistics ] ) . the @xmath1-ray data discussed by iyudin et al . ( @xcite ) and aschenbach et al . ( @xcite ) suggest an age of approximately 700 yr , from a comparison of the observed @xmath60ti line flux with that expected from sn models , which corresponds to a distance of approximately 200 pc . however , the interpretation of the @xmath1-ray detection requires the assumption of both the supernova shock velocity and the @xmath60ti yield . iyudin et al . ( @xcite ) note that increases in either of these quantities will lead to an increase in the derived distance of the remnant . we have examined 21-cm observations in the region of rx j0852.04622 , from kerr et al . ( @xcite ) , in an attempt to find any correlating features . however , since the remnant lies in a complex region in vela , no associated features in could be found . the ice core data of burgess & zuber ( @xcite ) may be able to provide an accurate age for rx j0852.04622 , but these data are not able to constrain the distance to the snr without further assumptions . furthermore , as noted above , it is not possible to definitively associate the additional nitrate spike with this snr . if we assume the age of @xmath61 yr determined by burgess & zuber ( @xcite ) to be accurate , then assuming an upper limit to the mean shock velocity of @xmath62 km s@xmath16 places the remnant at a distance of @xmath63 pc , with a linear diameter of @xmath50 pc . a value of @xmath62 km s@xmath16 was nominated as an upper limit to the shock velocity by aschenbach et al . ( @xcite ) , based on their analysis of the x - ray data . should the mean shock velocity exceed this value , the remnant would lie at a distance of @xmath64 pc , with a correspondingly larger linear diameter . in summary , the distance is very poorly constrained by current observational data . unfortunately , since the remnant is faint , is not detected optically , and is in a complex region , many direct techniques used to determine the distance to snrs are not applicable in this case ( e.g. absorption , association with other features or molecular clouds , or optical studies ) . nevertheless , the radio properties of this remnant , even with the present uncertainty in its age and size , have some interesting implications for statistical studies of galactic snrs , which are discussed in the next section . [ fig_sigmadee ] shows a surface - brightness versus diameter plot for galactic snrs for which there are reasonable distances available ( green @xcite , @xcite , @xcite ) . rx j0852.04622 is plotted on this figure with a range of diameters corresponding to distances from 200 pc to 1 kpc . the surface brightness of rx j0852.04622 , of @xmath65 w m@xmath40 hz@xmath16 sr@xmath16 , is faint for a known galactic snr among the faintest 20% of catalogued remnants . this is less than the nominal completeness limit of many radio surveys ( e.g. green @xcite ) . note that although most faint remnants are thought to be old , that the remnant of the sn of ad1006 is also faint , with @xmath66 w m@xmath40 hz@xmath16 sr@xmath16 . we also note that , whilst rx j0852.04622 is one of the fainter remnants to appear on the @xmath67 plot , the only other remnant detected in @xmath60ti @xmath1-ray emission is cas a , which has the highest surface brightness . from fig . [ fig_sigmadee ] , it is clear that the properties of rx j0852.04622 are very unusual if it lies at the smaller distances suggested by the @xmath1-ray data . if the snr is at a distance @xmath68 pc , its diameter is less than 10 pc , but its surface brightness is two or more orders of magnitude less than other known young snrs with similar diameters ( e.g. kepler s sn , tycho s sn , and 3c58 ) . this would have important consequences for statistical studies of galactic snrs ( see green @xcite ) , as the range of @xmath69 or , equivalently , luminosity for a given @xmath70 may be even wider than was previously thought . this in turn would imply that the observational selection effect of faint snrs being difficult to detect is important , not only for old snrs , but also for young snrs . the low radio surface brightness of rx j0852.04622 indicates that a fraction of young snrs may be faint at radio wavelengths . the available sample of young snrs ( i.e. historical events ) is small , however , so it is not possible to meaningfully estimate this proportion . on the other hand , if the remnant is as distant as 1 kpc , then although it is faint for it s diameter of @xmath71 pc , it is not strikingly unusual . we have presented various radio observations of the newly recognised snr rx j0852.04622 , which clarify its size and morphology . several features possibly associated with the remnant by combi et al . ( @xcite ) are examined , and it is concluded that these are probably unrelated to rx j0852.04622 . although the distance and age of rx j0852.04622 are not well determined , its faintness has some interesting implications for the statistical study of snrs , namely that the surface - brightness limits of current radio surveys may miss faint , young remnants , as well as faint , old remnants . clearly an accurate distance determination for this snr is important , although this is difficult , given its faintness and the fact that it lies in a complex region of the sky , confused with emission from the much larger vela snr . the australia telescope is funded by the commonwealth of australia for operation as a national facility managed by csiro . ard is an alexander von humboldt research fellow and thanks the stiftung for their support . the authors gratefully acknowledge p. slane for helpful comments and suggestions on the manuscript . green d.a . , 1998 , _ ` a catalogue of galactic supernova remnants ( 1998 september version ) ' _ , mullard radio astronomy observatory , cambridge , united kingdom ( available on the world - wide - web at `` http://www.mrao.cam.ac.uk/surveys/snrs/ '' ) note added in proof : the authors note a recent paper by schnfelder et al . ( @xcite ) , which concludes that the @xmath1-ray detection of rx j0852.04622 is less statistically significant than was first thought .

we present new radio observations of the recently identified , young galactic supernova remnant ( snr ) rx j0852.04622 ( g266.201.2 ) made at 1.40 ghz with a resolution of @xmath0 . these results , along with other radio observations from the literature , are used to derive the extent , morphology and radio spectrum of the remnant . the possible age and distance to this remnant are discussed , along with the consequences of its properties especially its low radio surface brightness for statistical studies of galactic snrs . the extended features identified by combi et al . ( @xcite ) are considered , and we conclude that these are probably unrelated to the new remnant . if rx j0852.04622 is nearby , as is suggested by the available @xmath1-ray data , then the range of intrinsic radio luminosities for snrs of the same diameter may be much larger than was previously thought .

You are an expert at summarizing long articles. Proceed to summarize the following text: on the 25-th of november in 1915 , einstein presented his field equations , the basic equations of general relativity , to the prussian academy of sciences in berlin . this equation had a tremendous impact on physics , in particular on cosmology . the essence of the theory was expressed by wheeler by the words : _ spacetime tells matter how to move ; matter tells spacetime how to curve_. einsteins theory remained unchanged for about 40 years . then one started to investigate theories fulfilling mach s principle leading to a variable gravitational constant . brans - dicke theory was the first realization of an extended einstein theory with variable gravitational constant ( jordans proposal was not widely known ) . all experiments are , however , in good agreement with einstein s theory and currently there is no demand to change it . general relativity ( gr ) has changed our understanding of space - time . in parallel , the appearance of quantum field theory ( qft ) has modified our view of particles , fields and the measurement process . the usual approach for the unification of qft and gr to a quantum gravity , starts with a proposal to quantize gr and its underlying structure , space - time . there is a unique opinion in the community about the relation between geometry and quantum theory : the geometry as used in gr is classical and should emerge from a quantum gravity in the limit ( planck s constant tends to zero ) . most theories went a step further and try to get a space - time from quantum theory . then , the model of a smooth manifold is not suitable to describe quantum gravity , but there is no sign for a discrete space - time structure or higher dimensions in current experiments @xcite . therefore , we conjecture that the model of spacetime as a smooth 4-manifold can be used also in a quantum gravity regime , but then one has the problem to represent qft by geometric methods ( submanifolds for particles or fields etc . ) as well to quantize gr . in particular , one must give meaning to the quantum state by geometric methods . then one is able to construct the quantum theory without quantization . here we implicitly assumed that the quantum state is real , i.e. the quantum state or the wave function has a real counterpart and is not a collection of future possibilities representing some observables . experiments @xcite supported this view . then the wave function is not merely representing our limited knowledge of a system but it is in direct correspondence to reality ! then one has to go the reverse way : one has to show that the quantum state is produced by the quantization of a classical state . it is , however , not enough to have a geometric approach to quantum gravity ( or the quantum field theory in general ) . what are the quantum fluctuations ? what is the measurement process ? what is decoherence and entanglement ? in principle , all these questions have to be addressed too . here , the exotic smoothness structure of 4-manifolds can help finding a way . a lot of work was done in the last decades to fulfill this goal . it starts with the work of brans and randall @xcite and of brans alone @xcite where the special situation in exotic 4-manifolds ( in particular the exotic @xmath0 ) was explained . one main result of this time was the _ brans conjecture _ : exotic smoothness can serve as an additional source of gravity . i will not present the whole history where i refer to carl s article . here i will list only some key results which will be used in the following * exotic smoothness is an extra source of gravity ( brans conjecture is true ) , see asselmeyer @xcite for compact manifolds and sadkowski @xcite for the exotic @xmath0 . therefore an exotic @xmath0 is always curved and can not be flat ! * the exotic @xmath0 can not be a globally hyperbolic space ( see @xcite for instance ) , i.e. represented by @xmath2 for some 3-manifold . instead it admits complicated foliations @xcite . using non - commutative geometry , we are able to study these foliations ( the leaf space ) and get relations to qft . for instance , the von neumann algebra of a codimension - one foliation of an exotic @xmath0 must contain a factor of type @xmath3 used in local algebraic qft to describe the vacuum @xcite . * the end of @xmath0 ( the part extending to infinity ) is @xmath4 . if @xmath0 is exotic then @xmath4 admits also an exotic smoothness structure . clearly , there is always a topologically embedded 3-sphere but there is no smoothly embedded one . let us assume the well - known hyperbolic metric of the spacetime @xmath4 using the trivial foliation into leafs @xmath5 for all @xmath6 . now we demand that @xmath4 carries an exotic smoothness structure at the same time . then we will get only topologically embedded 3-spheres , the leafs @xmath5 . these topologically embedded 3-spheres are also known as wild 3-spheres . in @xcite , we presented a relation to quantum d - branes . finally we proved in @xcite that the deformation quantization of a tame embedding ( the usual embedding ) is a wild embedding so that the image @xmath7 is an infinite polyhedron or the triangulation needs always infinitely many simplices . ] . furthermore we obtained a geometric interpretation of quantum states : wild embedded submanifolds are quantum states . importantly , this construction depends essentially on the continuum , because wild embedded submanifolds admit always infinite triangulations . * for a special class of compact 4-manifolds we showed in @xcite that exotic smoothness can generate fermions and gauge fields using the so - called knot surgery of fintushel and stern @xcite . in the paper @xcite we presented an approach using the exotic @xmath0 where the matter can be generated ( like in qft ) . * the path integral in quantum gravity is dominated by the exotic smoothness contribution ( see @xcite or by using string theory @xcite ) . the paper is organized as follows . in the following three sections we will explain exotic 4-manifolds and motivate the whole approach by using the path integral for the einstein - hilbert action . here we will also present how to couple the matter and gauge fields to this theory . for a 4-manifold , there are two main invariants the euler and pontrjagin class which determine the main topological invariant of a 4-manifold , the intersection form . in section [ sec : action - induced - by - top ] , we will obtain the einstein - hilbert and holst action by using these two classes . at the first view , this section is a little bit isolated from the previous and subsequent sections but we will use this result later during the study of the scaling . in the main section [ sec : wild - embeddings ] , we will construct the foliation of an exotic @xmath0 of codimension ( equivalent to a lorentz structure ) . following connes , @xcite the leaf space is an operator algebra constructed from the geometrical information of the foliation ( holonomy groupoid ) . this operator algebra is a factor @xmath8 von neumann algebra and we will use the tomita - takesaki modular theory to uncover the structure of the foliation . it is not the first time that this factor was used for quantum gravity and we refer to the paper @xcite for a nice application . states in this operator algebra are represented by equivalence classes of knotted curves ( element of the kauffman bracket skein module ) . the reconstruction of the spatial space from the states gives a wild embedded 3-sphere as geometrical representation of the state . surprisingly , it fits with the properties of the exotic @xmath0 . if one introduces a global foliation of the exotic @xmath0 by a global time then one obtains a foliation into wild embedded 3-spheres . in contrast , if one uses a local but complicate foliation then this wild object can be omitted and one obtains a state given by a finite collection of knotted curves . interestingly , the operator algebra can be understood as observable algebra given by a deformation quantization ( turaev - drinfeld quantization @xcite ) of the classical observable algebra ( poisson algebra of holonomies a la goldman @xcite ) . in section [ sec : action - at - the - quantum - level ] , we will use the splitting of the operator algebra ( [ eq : crossed - product - factor - iii ] ) given by tomita - takesaki modular theory to introduce the dynamics ( see connes and rovelli @xcite with similar ideas ) . finally we will obtain a quantum action ( [ eq : quantum - action ] ) in the quantized calculus of connes @xcite . then the scaling behavior is studied in the next section . for large scales , the action can be interpreted as a non - linear sigma model . the renormalization group ( rg ) flow analysis @xcite gives the einstein equations for large scales . the short - scale analysis is much more involved , yielding for small fluctuations the einstein - hilbert action and a non - minimally coupled scalar field . in particular , we will obtain a @xmath9dimensional fractal structure . in section [ sec : some - properties ] we will present some direct consequences of this approach : the nonlinear graviton @xcite , a relation to lattice gauge field theory with a discussion of discreteness and the appearance of dimensional reduction from 4d to 2d . in section [ sec : where - does - fluctuation ] we will discuss the answer to a fundamental question : where does the quantum fluctuations come from ? the main result of this section can be written as : _ the set of canonical pairs ( as measurable variables in the theory ) forms a fractal subset of the space of all holonomies . then we can only determine the initial condition up to discrete value ( given by the canonical pair ) and the chaotic behavior of the foliation ( i.e. the anosov flow ) makes the limit not predictable . _ this interesting result is followed by a section where we will discuss the collapse of the wavefunction by the gravitational interaction by calculating the minimal decoherence time . furthermore we will discuss entanglement and the measurement process . in section [ sec : some - implications - for.cosmology ] we will list our work in cosmology which uses partly the results of this paper . in the last section [ sec : conclusion - and - open - questions ] , we will discuss some consequences and open questions . some mathematical prerequisites are presented in three appendices . this article is dedicated to my only teacher , carl h. brans for 20 years of collaboration and friendship . he is the founder of this research area . we had and will have many interesting discussions . carl always asked the right question and put the finger on many open points . during the 7 years of writing our book , we had a very fruitful collaboration and i learned so much to complete even this work . carl , i hope for many discussions with you in the future . i m very glad to count on your advice . happy birthday ! why am i going to concentrate on a concept like exotic smoothness ? einstein used the equivalence principle as a key principle in the development of general relativity . every gravitational field can be locally eliminated by acceleration . then , the spacetime is locally modeled as subsets of the flat @xmath0 or the equivalence principle enforces us to use the concept of a manifold for spacetime . together with the smoothness of the dynamics ( usage of differential equations ) , we obtain a smooth 4-manifold as model for the spacetime in agreement with the current experimental situation . a manifold consists of charts and transition functions forming an atlas which covers the manifold completely . _ the smooth atlas is called the smoothness structure _ of the manifold . it was an open problem for a long time whether every topological manifold admits a unique smooth atlas . in 1957 , milnor found the first counterexample : the construction of a 7-sphere with at least 8 different smoothness structures . later it was shown that all manifolds of dimension larger than 4 admit only a finite number of distinct smoothness structures . the real breakthrough for 4-manifolds came in the 80 s where one constructed infinitely many different smoothness structures for many compact 4-manifolds ( countably infinite ) and for many non - compact 4-manifolds ( uncountably infinite ) including the @xmath0 . in all dimensions smaller than four , there is only one smoothness structure ( up to diffeomorphisms ) , the standard structure . the standard @xmath0 is simply characterized by the unique property to split smoothly like @xmath10 . all other distinct smoothness structures are called _ exotic smoothness _ structures . these structures are different , nonequivalent , smooth descriptions of the same topological manifold , a different atlas of charts . in case of the exotic @xmath0 , the difference is tremendous : the standard @xmath0 needs one chart ( and every other description can be reduced to it ) whereas every known exotic @xmath0 admits infinitely many charts ( which can not be reduced to a simpler description ) . so , the spacetime exhibits a much larger complexity by using an exotic smoothness structure , but why is dimension 4 so special ? there is a good description in @xcite and i will give a short account now . at first we have to discuss the question : how do i build an atlas for a smooth manifold ? the answer is given by considering the construction of diffeomorphisms . every diffeomorphism is locally given by the solution of @xmath11 for a real function @xmath12 over the manifold . the fixed points of this equation are the critical points of @xmath12 . in case of isolated critical points , one can reproduce the structure of the manifold ( this is called morse theory ) . every critical point leads to the attachment of a handle , a submanifold like @xmath13 , i.e. the @xmath14handle ( where @xmath15 is the @xmath14disk ) . in many cases , the corresponding structure of the manifold , the handle body , can be very complicated but there are rules ( handle sliding ) to simplify them . in all dimensions except dimension 4 . therefore , two handle bodies can be described by the same 4-manifold topologically but differ in the smooth description . one of the most surprising aspects of exotic smoothness is the existence of exotic @xmath0 s . in all other dimensions @xcite , the euclidean space @xmath16 with @xmath17 admits a unique smoothness structure , up to diffeomorphisms . beginning with the first examples @xcite , taubes @xcite and freedman / demichelis @xcite constructed countably many large and small exotic @xmath0 s , respectively . a small exotic @xmath0 embeds smoothly in the 4-sphere whereas a large exotic @xmath0 can not be embedded in that way . for the following we need some simple definitions : the connected sum @xmath18 and the boundary connected sum @xmath19 of manifolds . let @xmath20 be two @xmath21-manifolds with boundaries @xmath22 . the _ connected sum _ @xmath23 is the procedure of cutting out a disk @xmath24 from the interior @xmath25 and @xmath26 with the boundaries @xmath27 and @xmath28 , respectively , and gluing them together along the common boundary component @xmath29 . the boundary @xmath30 is the disjoint sum of the boundaries @xmath22 . the _ boundary connected sum _ @xmath31 is the procedure of cutting out a disk @xmath32 from the boundary @xmath33 and @xmath34 and gluing them together along @xmath35 of the boundary . then the boundary of this sum @xmath31 is the connected sum @xmath36 of the boundaries @xmath22 . large exotic @xmath0 can be constructed using the failure to arbitrarily split a compact , simply - connected 4-manifold . for every topological 4-manifold one knows how to split this manifold _ topologically _ into simpler pieces using the work of freedman @xcite . donaldson @xcite , however , that some of these 4-manifolds do not exist as smooth 4-manifolds . this contradiction between the continuous and the smooth case produces the first examples of exotic @xmath0 . below we discuss one of these examples . one starts with a compact , simply - connected 4-manifold @xmath37 classified by the intersection form @xcite @xmath38 a quadratic form over the second integer homology group . in the first construction of a large exotic @xmath0 , one starts with the k3 surface as 4-manifold having the intersection form @xmath39 with the the matrix @xmath40 : @xmath41.\ ] ] the work of donaldson @xcite shows that a closed , * smooth * , simply - connected , compact 4-manifold @xmath42 with intersection form @xmath43 does not exist . freedman @xcite showed , however , that there is a topological splitting @xmath44 with the @xmath45times connected sum @xmath46 ( see above ) which fails to be smooth . this splitting means that we glue together the two manifolds @xmath47 and @xmath48 along the common boundary @xmath49 ( @xmath50 is the 4-disk or 4-ball ) . now we define the interior @xmath51 . the splitting ( [ eq : splitting - k3 ] ) gives a way to represent the @xmath52 part of the intersection form ( [ eq : intersection - k3 ] ) by using @xmath37 but that fails smoothly . so , choosing a topological splitting @xmath53\right)\cup\left(\#_{3}(s^{2}\times s^{2})\setminus d^{4}\right)\end{aligned}\ ] ] gives a @xmath54 $ ] inside the @xmath55 . the interior of @xmath54 $ ] defines a manifold @xmath56 glued to a ( topological ) 4-disk @xmath57 along the common boundary , i.e. @xmath58 topologically . @xmath59 is homeomorphic to @xmath0 but the non - existence of the smooth splitting implies that it is an exotic @xmath0 and there is no smooth embedded @xmath60 ( otherwise the topological splitting is smooth ) . this failure for a smooth embedding implies also that such exotic @xmath0 s do not embed in the 4-sphere , i.e. it is a large exotic @xmath0 . the details of the construction can be found in our book @xcite ( section 8.4 ) . gompf @xcite introduced an important tool for finding new exotic @xmath61 from others , the end - sum @xmath62 . let @xmath63 be two topological @xmath61 s . the end - sum @xmath64 is defined as follows : let @xmath65 and @xmath66 be smooth properly embedded rays with tubular neighborhoods @xmath67 and @xmath68 , respectively . for convenience , identify the two semi - infinite intervals with @xmath69 and @xmath70 $ ] leading to diffeomorphisms , @xmath71 and @xmath72\times\mathbb{r}^{3}$ ] . then define @xmath73 as the end sum of @xmath74 and @xmath75 . with a little checking , it is easy to see that this construction leads to @xmath64 as another topological @xmath76 however , if @xmath63 are themselves exotic , then so will @xmath64 and in fact , it will be a `` new '' exotic manifold , since it will not be diffeomorphic to either @xmath74 or @xmath75 . gompf used this technique to construct a class of exotic @xmath61 s none of which can be embedded smoothly in the standard @xmath61 . by an extension of donaldson theory for a special class of open 4-manifolds , so - called end - periodic 4-manifolds , taubes @xcite gives a continuous family of exotic @xmath0 which was extended by gompf to a continuous 2-parameter family @xmath77 . small exotic @xmath0 s are again the result of anomalous smoothness in 4-dimensional topology but of a different kind than for large exotic @xmath0 s . in 4-manifold topology @xcite , a homotopy - equivalence between two compact , closed , simply - connected 4-manifolds implies a homeomorphism between them ( a so - called h cobordism ) , but donaldson @xcite provided the first smooth counterexample , i.e. both manifolds are generally not diffeomorphic to each other . the failure can be localized in some contractible submanifold ( akbulut cork ) so that an open neighborhood of this submanifold is a small exotic @xmath0 . the whole procedure implies that this exotic @xmath0 can be embedded in the 4-sphere @xmath78 . below we discuss the details for one of these examples . in 1975 casson ( lecture 3 in @xcite ) described a smooth 5-dimensional h - cobordism between compact 4-manifolds and showed that they `` differ '' by two proper homotopy @xmath61 s ( see below ) . freedman knew , as an application of his proper h - cobordism theorem , that the proper homotopy @xmath61 s were @xmath61 . after hearing about donaldson s work in march 1983 , freedman realized that there should be exotic @xmath61 s and , to find one , he produced the second part of the construction below involving the smooth embedding of the proper homotopy @xmath61 s in @xmath78 . unfortunately , it was necessary to have a compact counterexample to the smooth h - cobordism conjecture , and donaldson did not provide this until 1985 @xcite . the idea of the construction is simply given by the fact that every such smooth h - cobordism between non - diffeomorphic 4-manifolds can be written as a product cobordism except for a compact contractible sub - h - cobordism @xmath79 , the akbulut cork . an open subset @xmath80 homeomorphic to @xmath81\times{{\mathbb{r}}^{4}}$ ] is the corresponding sub - h - cobordism between two exotic @xmath61 s . these exotic @xmath61 s are called ribbon @xmath61 s . they have the important property of being diffeomorphic to open subsets of the standard @xmath61 . that stands in contrast to the previous defined examples of kirby , gompf and taubes . to be more precise , consider a pair @xmath82 of homeomorphic , smooth , closed , simply - connected 4-manifolds . the transformation from @xmath83 to @xmath84 visualized by a h - cobordism can be described by the following construction . + _ let @xmath59 be a smooth h - cobordism between closed , simply connected 4-manifolds @xmath83 and @xmath84 . then there is an open subset @xmath85 homeomorphic to @xmath81\times{{\mathbb{r}}^{4}}$ ] with a compact subset @xmath86 such that the pair @xmath87 is diffeomorphic to a product @xmath81\times(x_{-}\setminus k , u\cap x_{-}\setminus k)$ ] . the subsets @xmath88 ( homeomorphic to @xmath61 ) are diffeomorphic to open subsets of @xmath61 . if @xmath83 and @xmath84 are not diffeomorphic , then there is no smooth 4-ball in @xmath89 containing the compact set @xmath90 , so both @xmath89 are exotic @xmath61 s . _ + thus , remove a certain contractible , smooth , compact 4-manifold @xmath91 ( called an akbulut cork ) from @xmath83 , and re - glue it by an involution of @xmath92 , i.e. a diffeomorphism @xmath93 with @xmath94 and @xmath95 for all @xmath96 . this argument was modified above so that it works for a contractible _ open _ subset @xmath97 with similar properties , such that @xmath98 will be an exotic @xmath61 if @xmath84 is not diffeomorphic to @xmath83 . furthermore @xmath98 lies in a compact set , i.e. a 4-sphere or @xmath98 is a small exotic @xmath0 . in the next subsection we will see how this results in the construction of handle bodies of exotic @xmath61 . in @xcite freedman and demichelis constructed also a continuous family of small exotic @xmath0 . one of the characterizing properties of an exotic @xmath0 ( all known examples ) is the existence of a compact subset @xmath99 which can not be surrounded by any smoothly embedded 3-sphere ( and homology 3-sphere bounding a contractible , smooth 4-manifold ) . let @xmath100 be the standard @xmath0 ( i.e. @xmath101 smoothly ) and let @xmath1 be a small exotic @xmath0 with compact subset @xmath99 which can not be surrounded by a smoothly embedded 3-sphere . then every completion @xmath102 of an open neighborhood @xmath103 is not bounded by a 3-sphere @xmath104 . however , @xmath1 is a small exotic @xmath0 and there is a smooth embedding @xmath105 in the standard @xmath0 . then the completion of the image @xmath106 has the boundary @xmath107 as subset of @xmath100 . so , we have the strange situation that an open subset of the standard @xmath100 represents a small exotic @xmath1 . in case of the large exotic @xmath0 , the situation is much more complicated . a large exotic @xmath0 does not embed in any smooth 4-manifold which is simpler than the manifold used for the construction of this exotic @xmath0 . above we considered the example of a large exotic @xmath0 constructed from a k3 surface . therefore this large exotic @xmath0 embeds in the k3 surface but not in simpler 4-manifolds like @xmath108 . as of now , we only know of exotic @xmath0 s represented by an infinite number of coordinate patches . this naturally makes it difficult to provide an explicit description of a metric . however , in @xcite , a suggestion to overcome this limitation is provided by the consideration of periodic explicitly described coordinate patches making use of more complex pieces , so - called handles , and even more complex gluing maps . then one also gets infinite structures of handles but with a clear picture : the coordinate patches have a periodic structure . * handles * every 4-manifold can be decomposed using standard pieces such as @xmath109 , the so - called @xmath110-handle attached along @xmath111 to the @xmath112handle @xmath113 . in the following we need two possible cases : the 1-handle @xmath114 and the 2-handle @xmath115 . these handles are attached along their boundary components @xmath116 or @xmath117 to the boundary @xmath60 of the @xmath112handle @xmath50 ( see @xcite for the details ) . the attachment of a 2-handle is defined by a map @xmath118 , the embedding of a circle @xmath119 into the 3-sphere @xmath60 , i.e. a knot . this knot into @xmath60 can be thickened ( to get a knotted solid torus ) . the important fact for our purposes is the freedom to twist this knotted solid torus ( so - called dehn twist ) . the ( integer ) number of these twists ( with respect to the orientation ) is called the framing number or the framing . thus the gluing of the 2-handle on @xmath50 can be represented by a knot or link together with an integer framing . the simplest example is the unknot with framing @xmath120 representing the complex projective space @xmath108 or with reversed orientation @xmath121 , respectively . the 1-handle will be glued by the map of @xmath122 represented by two disjoint solid 2-spheres @xmath123 . akbulut @xcite introduced another description . he observed that a 1-handle is something like a cut - out 2-handle with a fixed framing . we remark that all details can be found in @xcite . now we are ready to discuss the handle body decomposition of an exotic @xmath0 by bizaca and gompf @xcite . * handle decomposition of small exotic @xmath0 * first it is very important to notice that the exotic @xmath0 is the * interior * of the handle body described below ( since the handle body has a non - null boundary and is compact ) . the construction of the handle body can be divided into two parts . the first part is a submanifold consisting of a pair of a 1- and a 2-handle . this pair can be canceled topologically by using a casson handle and we obtain the topological 4-disk @xmath50 with @xmath0 as interior . this submanifold is a smooth 4-manifold with a boundary that can be covered by a finite number of charts . the smoothness structure of the exotic @xmath0 , however , depends mainly on the infinite casson handle . * casson handle * now consider the casson handle and its construction in more detail . briefly , a casson handle @xmath124 is the result of attempts to embed a disk @xmath125 into a 4-manifold . in most cases this attempt fails and casson @xcite looked for a substitute , which is now called a casson handle . freedman @xcite showed that every casson handle @xmath124 is homeomorphic to the open 2-handle @xmath126 but in nearly all cases it is not diffeomorphic to the standard handle @xcite . the casson handle is built by iteration , starting from an immersed disk in some 4-manifold @xmath127 , i.e. an injective smooth map @xmath128 every immersion @xmath129 is an embedding except on a countable set of points , the double points . one can kill one double point by immersing another disk into that point . these disks form the first stage of the casson handle . by iteration one can produce the other stages . finally consider not the immersed disk but rather a tubular neighborhood @xmath115 of the immersed disk including each stage . the union of all neighborhoods of all stages is the casson handle @xmath124 . so , there are two input data involved with the construction of a @xmath124 : the number of double points in each stage and their orientation @xmath130 . thus we can visualize the casson handle @xmath124 by a tree : the root is the immersion @xmath129 with @xmath110 double points , the first stage forms the next level of the tree with @xmath110 vertices connected with the root by edges etc . the edges are evaluated using the orientation @xmath130 . every casson handle can be represented by such an infinite tree . the casson handle @xmath131 having an immersed disk with one ( positively oriented ) self - intersection ( or double point ) is the simplest casson handle represented by the simplest tree @xmath132 having one vertex in each level connected by one edge with evaluation @xmath133 . one of the characterizing properties of an exotic @xmath0 ( all known examples ) is the existence of a compact subset @xmath99 which can not be surrounded by any smoothly embedded 3-sphere ( and homology 3-sphere bounding a contractible , smooth 4-manifold ) . let @xmath100 be the standard @xmath0 ( i.e. @xmath101 smoothly ) and let @xmath1 be a small exotic @xmath0 with compact subset @xmath99 which can not be surrounded by a smoothly embedded 3-sphere . then every completion @xmath102 of an open neighborhood @xmath103 is not bounded by a 3-sphere @xmath104 , but @xmath1 is a small exotic @xmath0 and there is a smooth embedding @xmath105 in the standard @xmath0 . then the completion of the image @xmath106 has the boundary @xmath107 as subset of @xmath100 . so , we have the strange situation that an open subset of the standard @xmath100 represents a small exotic @xmath1 . now we will describe @xmath1 . historically it was constructed by using a counterexample of the smooth h - cobordism theorem @xcite . then the compact subset @xmath134 is given by a non - canceling 1-/2-handle pair . the attachment of a casson handle @xmath124 cancels this pair topologically . then one obtains the 4-disk @xmath50 with interior @xmath100 , but this cancellation of the 1/2-handle pair can not be done smoothly and one obtains a small exotic @xmath1 which is schematically given by @xmath135 . remember @xmath1 is a small exotic @xmath0 , i.e. @xmath1 is embedded into the standard @xmath100 by definition . the completion @xmath136 of @xmath137 has a boundary given by the 3-manifold @xmath138 . there is also the possibility to construct @xmath138 directly as the limit @xmath139 of a sequence @xmath140 of 3-manifolds . to construct this sequence of 3-manifolds @xcite , one can use the kirby calculus , i.e. one represents the compact subset @xmath134 by 1- and 2-handles pictured by a link say @xmath141 where the 1-handles are represented by a dot ( so that surgery along this link gives @xmath134 ) @xcite . then one attaches a casson handle to this link @xcite . as an example see figure [ fig : link - picture - for - k ] . the casson handle is given by a sequence of whitehead links ( where the unknotted component has a dot ) which are linked according to the tree ( see the right figure of figure [ fig : building - block - simplest - ch ] for the building block and the left figure for the simplest casson handle given by the unbranched tree ) . for the construction of a 3-manifold which surrounds the compact @xmath134 , one considers @xmath142stages of the casson handle and transforms the diagram to a real link ( the dotted components are changed to usual components with framing @xmath143 ) . by handle manipulations one obtains a knot so that the @xmath21-th ( untwisted ) whitehead double of this knot represents the desired 3-manifold ( by using surgery ) . then our example in figure [ fig : link - picture - for - k ] will result in the @xmath21-th untwisted whitehead double of the pretzel knot @xmath144 , figure [ fig : pretzel - knot ] ( see @xcite for the handle manipulations ) . then this sequence of 3-manifolds @xmath145 characterizes the exotic smoothness structure of @xmath1 . the limit of this sequence @xmath139 gives a wild embedded 3-manifold @xmath138 whose physical relevance will be explained later . here , we will motivate the appearance of exotic smoothness by discussing the path integral for the einstein - hilbert action . for simplicity , we consider general relativity without matter ( using the notation of topological qft ) . space - time is a smooth oriented 4-manifold @xmath127 which is non - compact and without boundary . from the formal point of view ( no divergences of the metric ) one is able to define a boundary @xmath146 at infinity . the classical theory is the study of the existence and uniqueness of ( smooth ) metric tensors @xmath147 on @xmath127 satisfying the einstein equations subject to suitable boundary conditions . in the first order hilbertpalatini formulation , one specifies an @xmath148-connection @xmath149 together with a cotetrad field @xmath150 rather than a metric tensor . fixing @xmath151 at the boundary , one can derive first order field equations in the interior ( now called _ bulk _ ) which are equivalent to the einstein equations provided that the cotetrad is non - degenerate . the theory is invariant under space - time diffeomorphisms @xmath152 . in the particular case of the space - time @xmath153 ( topologically ) , we have to consider smooth 4-manifolds @xmath154 as parts of @xmath127 whose boundary @xmath155 is the disjoint union of two smooth 3-manifolds @xmath156 and @xmath157 to which we associate hilbert spaces @xmath158 of 3-geometries , @xmath159 . these contain suitable wave functionals of connections @xmath160 . we denote the connection eigenstates by @xmath161 . the path integral , @xmath162\right)\label{eq : path - integral}\ ] ] is the sum over all connections @xmath149 matching @xmath163 , and over all @xmath150 . it yields the matrix elements of a linear map @xmath164 between states of 3-geometry . our basic gravitational variables will be cotetrad @xmath165 and connection @xmath166 on space - time @xmath127 with the index @xmath167 to present it as 1-forms and the indices @xmath168 for an internal vector space @xmath79 ( used for the representation of the symmetry group ) . cotetrads @xmath150 are square - roots of metrics and the transition from metrics to tetrads is motivated by the fact that tetrads are essential if one is to introduce spinorial matter . @xmath165 is an isomorphism between the tangent space @xmath169 at any point @xmath170 and a fixed internal vector space @xmath79 equipped with a metric @xmath171 so that @xmath172 . here we used the action @xmath173=\intop_{m_{i , f}}\epsilon_{ijkl}(e^{i}\wedge e^{j}\wedge\left(da+a\wedge a\right)^{kl})+\intop_{\partial m_{i , f}}\epsilon_{ijkl}(e^{i}\wedge e^{j}\wedge a^{kl})\label{eq : action - with - boundary}\ ] ] in the notation of @xcite . the boundary term @xmath174 is equal to twice the trace over the extrinsic curvature ( or the mean curvature ) . for fixed boundary data , ( [ eq : path - integral ] ) is a diffeomorphism invariant in the bulk . if @xmath175 are diffeomorphic , we can identify @xmath176 and @xmath177 i.e. we close the manifold @xmath154 by identifying the two boundaries to get the closed 4-manifold @xmath178 . provided that the trace over @xmath179 can be defined , the partition function , @xmath180\right)\label{eq : path - integral-1}\ ] ] where the integral is now unrestricted , is a dimensionless number which depends only on the diffeomorphism class of the smooth manifold @xmath178 . in case of the manifold @xmath154 , the path integral ( as transition amplitude ) @xmath181 is the diffeomorphism class of the smooth manifold relative to the boundary . the diffeomorphism class of the boundary , however , is unique and the value of the path integral depends on the topology of the boundary as well on the diffeomorphism class of the interior of @xmath154 . therefore we will shortly write @xmath182 and consider the sum of manifolds like @xmath183 with the amplitudes @xmath184 where we sum ( or integrate ) over the connections and frames on @xmath185 ( see @xcite ) . then the boundary term @xmath186=\intop_{\sigma_{f}}\epsilon_{ijkl}(e^{i}\wedge e^{j}\wedge a^{kl})=\intop_{\sigma_{f}}h\sqrt{h}d^{3}x\label{eq : boundary - term - eh}\ ] ] is needed where @xmath187 is the mean curvature of @xmath157 corresponding to the metric @xmath188 at @xmath157 ( as restriction of the 4-metric ) . in the path integral ( [ eq : path - integral ] ) , one integrates over the frames and connections . the possibility of singular frames was discussed at some places ( see @xcite ) . the cotetrad field @xmath189 changes w.r.t . the smooth map @xmath190 by @xmath191 . the transformation matrix @xmath192 has maximal rank @xmath193 for every regular value of the smooth map , but at the critical points @xmath194 of @xmath12 , some derivatives vanish and one has a smaller rank at the point @xmath194 , called a singular point . then there is no inverse frame ( or tetrad field ) at this point . usually singular frames are of this nature and one can decompose every singular frame into a product of a regular frame and a ( singular ) transformation induced by a smooth map . how can one interpret these singularities ? at this point one needs some differential topology . a homeomorphism can be arbitrarily and accurately approximated by smooth mappings ( see @xcite , theorem 2.6 ) , i.e. in a neighborhood of a homeomorphism one always finds a smooth map . secondly , there is a special class of smooth maps , the stable maps . here , two smooth maps are stable equivalent if both maps agree after a diffeomorphism of the corresponding manifolds @xcite . here we are interested into smooth mappings from 4-manifolds into 4-manifolds . by a deep result of mather @xcite , stable mappings for this dimension are dense in all smooth mappings of 4-manifolds . in @xcite , we analyzed this situation : the approximation of a homeomorphism by a stable map . if this smooth map has no singularities then we can perturb them to a diffeomorphism . for a singular map , however , we showed that it induces a change of the smoothness structure . then , a singular frame corresponds to a regular frame in a different smoothness structure . the path integral changed the domain of integration : @xmath195 we remark that this change is unique for dimension four . no other dimension has this plethora of smoothness structures which can be used to express the singular frames . the inclusion of exotic smoothness changed the description of trivial spaces like @xmath0 completely . instead of a single chart , we have now an infinite sequence of charts or an infinite sequence of 4-dimensional submanifolds . we will describe it more completely later . each submanifold is bounded by a 3-manifold ( different from a 3-sphere ) and we obtain a sequence of 3-manifolds @xmath196 characterizing the smoothness structure . the sequence of 3-manifolds divides the path integral into a product @xmath197 and we have to think about the boundary term ( [ eq : boundary - term - eh ] ) . in @xcite we analyzed this term : the boundary @xmath198 seen as embedding into the spacetime @xmath127 can be described locally as spinor @xmath199 and one obtains for the boundary term @xmath200 the dirac action with the dirac operator @xmath201 and @xmath202(see @xcite for the construction of @xmath199 ) . in particular we obtained the eigenvalue equation @xmath203 , i.e. the mean curvature is the eigenvalue of the dirac operator which has compact spectrum ( from the compactness of @xmath198 ) or we obtained discrete levels of geometry . this result enforced us to identify the 3-manifolds ( or the parts ) with the matter content . furthermore the path integral of the boundary can be carried out by an integration along @xmath199 ( see @xcite).with some effort @xcite , one can extend this boundary term to a tubular neighborhood @xmath204 $ ] of the boundary @xmath198 . however , the relation ( [ eq : boundary - term - dirac ] ) is only true for simple ( i.e. irreducible ) 3-manifolds , i.e. for complements of a knot admitting hyperbolic structure . for more complex 3-manifolds , we have the following simple scheme : the knot complements are connected by torus bundles ( locally written as @xmath205 $ ] ) . therefore we also have to describe these bundles by using the boundary term . in @xcite we described this situation by using the geometrical properties of these bundles and we will give a short account of these ideas in subsection [ sub : the - graviton - propagator ] . simply expressed , in this bundle one has a flow of constant curvature along the tube . the constant curvature connections are given by varying the chern - simons functional . now following floer @xcite , the 4-dimensional version of this flow equation is the instanton equation ( or the self - dual equation ) leading to the correct yang - mills functional ( chern - simons gives the pontrjagin class and the instanton equation makes it to the yang - mills functional ) . more importantly , the three possible types of torus bundles fit very good into the current scheme of three gauge field interactions ( see @xcite ( section 8) ) . now we have the following picture : fermions as hyperbolic knot complements and gauge bosons as torus bundles . both components together are forming an irreducible 3-manifold which is connecting to the remaining space by a @xmath206boundary ( see the prime decomposition in appendix b ) . this connection via @xmath207 $ ] ( @xmath206bundle ) is the only connection between matter and space . here , there is only one interpretation : this @xmath206bundle must be interpreted as gravity . in this section we will support this conjecture and construct the corresponding action . at first we will fix the model , i.e. let @xmath208 and @xmath209 be the 3-manifolds for matter and space , respectively . the connected sum @xmath18 of both components represents the whole spatial component @xmath210\right)\cup_{s^{2}}\sigma_{s}=\sigma_{m}\#s^{3}\#\sigma_{s}\ ] ] of the spacetime . the decomposition above showed the geometry of the@xmath206bundle ( in the sense of thurston , see appendix b ) to be the spherical geometry with isometry group @xmath211 . the idea of the following construction can be simply expressed : the 2-sphere @xmath212 explores locally the curvature of the space where the curvature is given by the inverse volume @xmath213 of the 2-sphere @xmath212 . the 2-sphere can be written as a homogenous space @xmath214 also known as hopf bundle . as mentioned above , the geometry of the bundle @xmath207 $ ] ( interpreted as an equator region of @xmath60 ) is the spherical geometry with isometry group @xmath211 . so , as a local model we have an embedding of a 3-manifold ( as the spatial component for a fixed time ) into the spacetime with local lorentz symmetry ( represented by @xmath215 ) . from the mathematical point of view , it is a reductive cartan geometry@xcite over the homogenous space @xmath216 , the 3-dimensional hyperbolic space . for the moment , let us extend this symmetry to the spacetime @xmath127 itself . a cartan connection @xmath149 decomposes as a @xmath217valued connection @xmath218 ( @xmath219 denotes the lie algebra of @xmath211 ) and a coframe field @xmath150 ( with values in @xmath220 ) as @xmath221 by using the scale @xmath222(in agreement with the physical units ) and with curvature @xmath223 then for the spacetime ( as 4-manifold ) , we interpret the cartan connection @xmath149 as the connection of the frame bundle ( with respect to the lorentz structure ) . now we have to think about what characterizes the @xmath206bundle in a 4-manifold , i.e. a surface bundle over a surface ( at least locally ) . it is known that a surface bundle over a surface is topologically described by the euler class as well as the pontrjagin class ( via the hirzebruch signature theorem ) . therefore we choose the sum of the euler and pontrjagin class for the frame bundle as action @xmath224 where the pontrjagin class is weighted by a parameter @xmath225 . using the rules above , we obtain @xmath226 the einstein - hilbert action with cosmological constant and the holst action with immirizo parameter as well the euler and pontrjagin class for the reduced bundles . in this model , the curvature is changed locally by adding a @xmath206bundle . then the scale @xmath227 has to agree with the volume of the @xmath212 . in the action we have the coupling constant @xmath228 which has to agree with @xmath229 ( @xmath230 planck length ) to get in contact with einsteins theory , i.e. we must set @xmath231 the agreement with the einstein - hilbert action showed that this approach can describe gravity but it does not describe the global geometry . later we can show , however , that it must be the de sitter space @xmath232 globally . in this section we will support our main hypothesis that an exotic @xmath0 has automatically a quantum geometry , but as noted in the introduction , we must implicitly assume that the quantum - geometrical state is realized in the exotic @xmath0 . interestingly , it follows from the physically motivated existence of a lorentz metric which is induced by a codimension - one foliation . therefore we will construct the foliation and the corresponding leaf space as the space of observables ( using ideas of connes ) . this leaf space is a non - commutative @xmath233algebra with observable algebra a factor @xmath3 von neumann algebra . a state in this algebra can be interpreted as a wild embedding which is also motivated by the exotic smoothness structure . the classical state is the tame embedding . then , the deformation quantization of this tame embedding is the wild embedding ( see @xcite ) . in principle , the wild embedding determines the @xmath233algebra completely . this algebra is generated by holonomies along connections of constant curvature . it is known from mathematics that this algebra ( forming a so - called character variety @xcite ) determines the geometrical structure of the 3-manifold ( along the way of thurston @xcite ) . the main structure in this approach is the fundamental group , i.e. the group of closed , non - contractible curves in a manifold . the quantization of this group ( as an expression of the classical geometry ) gives the so - called skein algebra of knots in this manifold . we will relate this skein algebra to the leaf space above . on the way to show this relation , we will obtain the generator of the translation from one 3-manifold into another 3-manifold , i.e. the time together with the hamiltonian . in section [ sec : pathint ] , we described the sequence of 3-manifolds @xmath145 characterizing the exotic smoothness structure of @xmath1 . then @xmath112framed surgery along this pretzel knot produces @xmath234 whereas the @xmath21-th untwisted whitehead double will give @xmath198 . for large @xmath21 , the structure of the casson handle is contained in the topology of @xmath198 and in the limit @xmath139 we obtain @xmath138 ( which is now a wild embedding @xmath235 in the standard @xmath100 given by the embedding of the small exotic @xmath1 , see above ) . what do we know about the structure of @xmath198 or @xmath138 in general ? the compact subset @xmath134 is a 4-manifold constructed by a pair of one 1-handle and one 2-handle which topologically cancel . the boundary of @xmath134 is a compact 3-manifold having the first betti number @xmath236 this information is also contained in @xmath138 . by the work of freedman @xcite , every casson handle is topologically @xmath126 ( relative to the attaching region ) and therefore @xmath138 must be the boundary of @xmath50 ( the casson handle trivializes @xmath134 to be @xmath50 ) , i.e. _ @xmath138 is a wild embedded 3-sphere @xmath60_. then we obtain two different descriptions of @xmath1 : 1 . as a sequence of 3-manifolds @xmath198 ( all having the first betti number @xmath237 ) as boundaries of the neighborhood of @xmath134 with increasing size and 2 . as a global hyperbolic space of @xmath238 written as @xmath239 where @xmath240 is a wild embedded 3-sphere ( which looks differently for different @xmath6 ) the first description gives a non - trivial but smooth foliation but there is no global spatial space . in contrast to this highly non - trivial foliation , the second description gives a global foliated spacetime containing a global spatial component , the wild embedded 3-sphere . in the first description we have a complex , relational description with no global time - like slices . here , there only is a local coordinate system ( with its own eigenzeit ) . this relational view has the big advantage that the simplest parts are also simple submanifolds ( only finite surfaces with boundary ) . in contrast , the second description introduces a global foliation into equal time slices . then the complexity is contained into the spatial component which is now a wild embedding ( i.e. a space with an infinite number of polygons ) . this second approach will be described in the next subsection . so , lets concentrate on the first approach . every 3-manifold @xmath198 admits a codimension - one ( @xmath241invariant ) foliation ( see @xcite for the details ) . by the description of the exotic @xmath1 using the sequence of 3-manifolds @xmath242 we also get a foliation of the exotic @xmath1 . the foliation on @xmath198 is defined by a @xmath241invariant one - form @xmath218 which is integrable @xmath243 and defines another one - form @xmath244 by @xmath245 . then the integral @xmath246 is known as godbillon - vey number @xmath247 with the class @xmath248 . from the physics point of view , it is the abelian chern - simons functional . the godbillon - vey class characterizes the codimension - one foliation for the 3-manifold @xmath198 ( see the appendix b for more details ) . the foliation is very complicated . in @xcite the local structure was analyzed . let @xmath249 be the curvature and torsion of a normal curve , respectively . furthermore , let @xmath250 be the frame formed by this vector field dual to the one - forms @xmath251 and let @xmath252 be the second fundamental form of leaf . then the godbillon - vey class is locally given by @xmath253 where @xmath254 for @xmath255 invariant foliations i.e. @xmath256=z$ ] , @xmath257=t$ ] and @xmath258=n$ ] . recall that a foliation @xmath259 of a manifold @xmath127 is an integrable subbundle @xmath260 of the tangent bundle @xmath261 . the leaves @xmath262 of the foliation @xmath259 are the maximal connected submanifolds @xmath263 with @xmath264 . we denote with @xmath265 the set of leaves or the leaf space . now one can associate to the leaf space @xmath265 a @xmath266-algebra @xmath267 by using the smooth holonomy groupoid @xmath268 of the foliation ( see connes @xcite ) . according to connes @xcite , one assigns to each leaf @xmath269 the canonical hilbert space of square - integrable half - densities @xmath270 . this assignment , i.e. a measurable map , is called a random operator forming a von neumann @xmath271 . a deep theorem of hurder and katok @xcite for foliations with non - zero godbillon - vey invariant states that this foliation has to contain a factor @xmath8 von neumann algebra . as shown in @xcite , the von neumann algebra for the foliation of @xmath198 and for the exotic @xmath1 is a factor @xmath272algebra . for the construction of this algebra , one needs the concept of a holonomy groupoid . foliations are determined by the holonomies of closed curves in a leaf and the transport of this closed curve together with the holonomy from the given leaf to another leaf . now one may ask why one considers only closed curves . let @xmath273 the space of all paths in a manifold then this space admits a fibration over the space of closed paths @xmath274 ( also called loop space ) with fiber the constant paths ( therefore homeomorphic to @xmath127 ) , see @xcite . then , a curve is determined up to deformation ( i.e. homotopy ) by a closed path . consider now a closed curve @xmath225 in a leaf @xmath222 and let act a diffeomorphism on @xmath222 . then the curve @xmath225 is modified as well to @xmath275 but @xmath225 and @xmath275 are related by a ( smooth ) homotopy . therefore to guarantee diffeomorphism invariance in this approach , one has to consider all closed curves up to homotopy . this structure can be made into a group ( using concatenation of paths as group operation ) called the fundamental group @xmath276 of the leaf . above we spoke about holonomy but a holonomy needs a connection of some bundle which we did not introduce until now . but connes @xcite described a way to circumvent this difficulty : given a leaf @xmath222 of @xmath259 and two points @xmath277 of this leaf , any simple path @xmath225 from @xmath278 to @xmath279 on the leaf @xmath222 uniquely determines a germ @xmath280 of a diffeomorphism from a transverse neighborhood of @xmath278 to a transverse neighborhood of @xmath279 . the germ of diffeomorphism @xmath280 only depends upon the homotopy class of @xmath225 in the fundamental group of the leaf @xmath222 , and is called the holonomy of the path @xmath225 . all fundamental groups of all leafs form the fundamental groupoid . the holonomy groupoid of a leaf @xmath222 is the quotient of its fundamental groupoid by the equivalence relation which identifies two paths @xmath225 and @xmath275 from @xmath278 to @xmath279 ( both in @xmath222 ) iff @xmath281 . then the von neumann algebra of the foliation is the convolution algebra of the holonomy groupoid which will be constructed later for the wild embedding . remember a von neumann algebra is an involutive subalgebra @xmath127 of the algebra of operators on a hilbert space @xmath187 that has the property of being the commutant of its commutant : @xmath282 . this property is equivalent to saying that @xmath127 is an involutive algebra of operators that is closed under weak limits . a von neumann algebra @xmath127 is said to be hyperfinite if it is generated by an increasing sequence of finite - dimensional subalgebras . furthermore we call @xmath127 a factor if its center is equal to @xmath283 . it is a deep result of murray and von neumann that every factor @xmath127 can be decomposed into 3 types of factors @xmath284 . the factor @xmath285 case divides into the two classes @xmath286 and @xmath287 with the hyperfinite factors @xmath288 the complex square matrices and @xmath289 the algebra of all operators on an infinite - dimensional hilbert space @xmath187 . the hyperfinite @xmath290 factors are given by @xmath291 , the clifford algebra of an infinite - dimensional euclidean space @xmath292 , and @xmath293 . the case @xmath8 remained mysterious for a long time . now we know that there are three cases parametrized by a real number @xmath294 $ ] : @xmath295 the krieger factor induced by an ergodic flow @xmath59 , @xmath296 the powers factor for @xmath297 and @xmath298 the araki - woods factor for all @xmath299 with @xmath300 . we remark that all factor @xmath8 cases are induced by infinite tensor products of the other factors . one example of such an infinite tensor space is the fock space in quantum field theory . the modular theory of von neumann algebras ( see also @xcite ) has been discovered by m. tomita @xcite in 1967 and put on solid grounds by m. takesaki @xcite around 1970 . it is a very deep theory that , to every von neumann algebra @xmath301 acting on a hilbert space @xmath179 , and to every vector @xmath302 that is cyclic , i.e. @xmath303 , and separating , i.e. for @xmath304 , @xmath305 , associates : * a one - parameter unitary group @xmath306 * and a conjugate - linear isometry @xmath307 such that : @xmath308 where the commutant @xmath309 of @xmath310 is defined by @xmath311_{-}\!=0,\forall\ , a\!\in\!\mathcal{b}(\mathcal{h})\}$ ] . more generally , given a von neumann algebra @xmath310 and a faithful normal state is faithful if @xmath312 ; it is normal if for every increasing bounded net of positive elements @xmath313 , we have @xmath314 . ] ( more generally for a faithful normal semi - finite weight ) @xmath218 on the algebra @xmath310 , the modular theory allows to create a one - parameter group of @xmath315-automorphisms of the algebra @xmath310 , @xmath316 such that : * in the gelfand namark segal representation @xmath317 induced by the weight @xmath218 , on the hilbert space @xmath318 , the modular automorphism group @xmath319 is implemented by a unitary one - parameter group @xmath320 i.e. we have @xmath321 , for all @xmath322 and for all @xmath6 ; * there is a conjugate - linear isometry @xmath323 , whose adjoint action implements a modular anti - isomorphism @xmath324 , between @xmath325 and its commutant @xmath326 , i.e. for all @xmath322 , we have @xmath327 . the operators @xmath328 and @xmath329 are called respectively the modular conjugation operator and the modular operator induced by the state ( weight ) @xmath218 . we will call `` modular generator '' the self - adjoint generator of the unitary one - parameter group @xmath330 as defined by stone s theorem i.e. the operator @xmath331 the modular automorphism group @xmath319 associated to @xmath218 is the unique one - parameter automorphism group that satisfies the kubo martin schwinger ( kms - condition ) with respect to the state ( or more generally a normal semi - finite faithful weight ) @xmath218 , at inverse temperature @xmath332 , i.e. @xmath333 and for all @xmath334 . using tomita - takesaki - theory , one has a continuous decomposition ( as crossed product ) of any factor @xmath8 algebra @xmath127 into a factor @xmath335 algebra @xmath336 together with a one - parameter group is the group of positive real numbers with multiplication as group operation also known as pontrjagin dual . ] @xmath337 of automorphisms @xmath338 of @xmath336 , i.e. one obtains @xmath339 that means , there is a foliation induced from the foliation producing this @xmath335 factor . connes @xcite ( in section i.4 page 57ff ) constructed the foliation @xmath340 canonically associated to the foliation @xmath341 of factor @xmath3 above having the factor @xmath335 as von neumann algebra . in our case it is the horocycle flow : let @xmath342 the polygon on the hyperbolic space @xmath343 determining the foliation above . @xmath342 is equipped with the hyperbolic metric @xmath344 together with the collection @xmath345 of unit tangent vectors to @xmath342 . a horocycle in @xmath342 is a circle contained in @xmath342 which touches @xmath346 at one point , but from the classification of factors , we know that @xmath335 is also splitted into @xmath347 so that every factor @xmath8 is determined by the factor @xmath348 . the factor @xmath287 are the compact operators in the hilbert space . with an important observation we will close this intermezzo . the factor @xmath335 admits an action of the group @xmath349 by automorphisms so that the crossed product ( [ eq : crossed - product - factor - iii ] ) is the factor @xmath3 . the corresponding invariant , the flow of weights @xmath350 , was determined by connes @xcite to be the godbillon - vey invariant . therefore _ the modular generator above is given by the godbillon - vey invariant , i.e. this invariant is the hamiltonian of the theory . _ then the @xmath266-algebra @xmath351 of the foliation @xmath259 is the @xmath266-algebra @xmath352 of the smooth holonomy groupoid @xmath268 . for completeness we will present the explicit construction ( see @xcite sec . the basic elements of @xmath351 ) are smooth half - densities with compact supports on @xmath268 , @xmath353 , where @xmath354 for @xmath355 is the one - dimensional complex vector space @xmath356 , where @xmath357 , and @xmath358 is the one - dimensional complex vector space of maps from the exterior power @xmath359 , @xmath360 , to @xmath283 such that @xmath361 for @xmath362 , the convolution product @xmath363 is given by the equality @xmath364 then we define via @xmath365 a @xmath315-operation making @xmath366 into a @xmath315-algebra . for each leaf @xmath262 of @xmath259 one has a natural representation of @xmath366 on the @xmath367 space of the holonomy covering @xmath368 of @xmath262 . fixing a base point @xmath369 , one identifies @xmath368 with @xmath370and defines the representation @xmath371 the completion of @xmath366 with respect to the norm @xmath372 makes it into a @xmath266-algebra @xmath351 . among all elements of the @xmath266-algebra , there are distinguished elements , idempotent operators or projectors having a geometric interpretation in the foliation . for later use , we will construct them explicitly ( we follow @xcite sec . ii.8.@xmath373 closely ) . let @xmath374 be a compact submanifold which is everywhere transverse to the foliation ( thus @xmath375 ) . a small tubular neighborhood @xmath376 of @xmath336 in @xmath127 defines an induced foliation @xmath340 of @xmath376 over @xmath336 with fibers @xmath377 . the corresponding @xmath266-algebra @xmath378 is isomorphic to @xmath379 with @xmath380 the @xmath266-algebra of compact operators . in particular it contains an idempotent @xmath381 , @xmath382 , where @xmath12 is a minimal projection in @xmath380 . the inclusion @xmath383 induces an idempotent in @xmath351 which is given by a closed curve in @xmath127 transversal to the foliation . in case of the foliation above ( of the 3-manifolds @xmath198 ) , one has the foliation of the polygon @xmath342 in @xmath343 and a circle @xmath119 attached to every leaf of this foliation . therefore we have the leafs @xmath384 $ ] and the @xmath119is the closed curve transversal to the foliation . then every leaf defines ( using the isomorphism @xmath385)=\pi_{1}(s^{1})=\mathbb{z}$ ] ) an idempotent represented by the fiber @xmath119 forming a base for the gns representation of the @xmath233algebra . now we are able to construct a state in this algebra . a state is a linear functional @xmath386 so that @xmath387 and @xmath388 . elements of @xmath351 are half - densities with a support along some closed curve ( as part of the holonomy groupoid ) . in a first step , one can use the gns - representation of the @xmath389algebra @xmath351 by a map @xmath390 in to the bounded operators of a hilbert space . by the theorem of frchet - riesz , every linear functional can be represented by the scalar product of the hilbert space for some vector . to determine the linear functionals , we have to investigate the geometry of the foliation . the foliation was constructed to be @xmath241invariant , i.e. fixing the upper half space @xmath343 . then we considered the unit tangent vectors of the tangent bundle over @xmath343 defining the @xmath391geometry . but more is true . every part of the 3-manifold @xmath198 is a knot / link complement with hyperbolic structure with isometry group @xmath392 where the other geometric structures like @xmath393 and @xmath255 embed . here we remark the known fact that every @xmath392-geometry lifts uniquely to @xmath394 ( the double cover ) . therefore , to model the holonomy , we have to choose a flat @xmath395connection and write it as the well - known integral of the connection 1-form along a closed curve . the linear functional is the trace of this integral ( seen as matrix using a representation of @xmath394 ) known as wilson loop . one can use the well - known identity @xmath396 for @xmath394 which goes over to the wilson loops . let @xmath397 $ ] be the wilson loop of a connection @xmath149 along the closed curve @xmath225 . then the relation of the wilson loops @xmath398\cdot w_{\eta}[a]=w_{\gamma\circ\eta}[a]+w_{\gamma\circ\eta^{-1}}[a]\ ] ] for two intersecting curves @xmath225 and @xmath244 is known as the mandelstam identity for intersecting loops , see fig . [ fig : mandelstam - identity - as - skein ] for a visualization . and subsection [ sub : drinfeld - turaev - quantization]).[fig : mandelstam - identity - as - skein ] ] this relation is also known from another area : knot theory . there , it is the kauffman bracket skein relation used to define the kauffman knot polynomial . therefore we obtain a state in the @xmath389algebra by a closed curve in the leaf which extends to a knot ( an embedded , closed curve ) in a submanifold of the 3-manifold defined up to the skein relation . finally : @xmath399}\ ] ] we will later explain this correspondence as a deformation quantization . we will close this subsection by some remarks . every representation @xmath400 defines ( up to conjugacy ) a flat connection . at the same time it defines also a hyperbolic structure on @xmath198 ( for @xmath401 ) . by the argumentation above , the quantized version of this geometry ( as defined by the @xmath389algebra of the foliation ) is given by the skein space ( see subsection [ sub : drinfeld - turaev - quantization ] for the definition of the skein space ) . our previous work implied that the transition from the standard @xmath100 to a small exotic @xmath1 has much to do with quantum gravity ( qg ) . therefore one would expect that a submanifold in the standard @xmath100 with an appropriated geometry represents a classical state . before we construct this state , there is a lot to say about the wild embedded 3-sphere as a quantum state . to describe this wild 3-sphere , we will construct the sequence of @xmath198 by using the example of @xcite which was already partly explained in subsection [ sub : small - exotic - r4-sequnce-3mf ] . at first we remark that the interior of the handle body in figure [ fig : handle - picture - of - exotic - r4 ] is the @xmath1 . the casson handle for this @xmath1 is given by the simplest tree @xmath402 , one positive self - intersection for each level . the compact 4-manifold inside of @xmath1 can be seen in figure [ fig : link - picture - for - k ] as a handle body . the 3-manifold @xmath198 surrounding this compact submanifold @xmath134 is given by surgery ( @xmath112framed ) along the link in figure [ fig : handle - picture - of - exotic - r4 ] with a casson handle of @xmath142levels . in @xcite , this case is explicitly discussed . @xmath198 is given by @xmath112framed surgery along the @xmath21-th untwisted whitehead double of the pretzel @xmath403 knot ( see figure [ fig : pretzel - knot ] ) . obviously , there is a sequence of inclusions @xmath404 with the 3-manifold @xmath405 as limit . let @xmath406 be the corresponding ( wild ) knot , i.e. the @xmath407-th untwisted whitehead double of the pretzel knot @xmath144 ( or the knot @xmath408 in rolfson notation ) . the surgery description of @xmath405induces the decomposition @xmath409 where @xmath410 is the knot complement of @xmath406 . in @xcite , the splitting of knot complements was described . let @xmath411 be the pretzel knot @xmath144 and let @xmath412 be the whitehead link ( with two components ) . then the complement @xmath413 has one torus boundary whereas the complement @xmath414 has two torus boundaries . now according to @xcite , one obtains the splitting @xmath415 and we will describe each part separately ( see figure [ fig : splitting - of - knot - complement ] ) . at first the knot @xmath411 is a hyperbolic knot , i.e. the interior of the 3-manifold @xmath413 admits a hyperbolic metric . by the work of gabai @xcite , @xmath413admits a codimension - one foliation . the whitehead link is a hyperbolic link but we need more : the whitehead link is a fibered link of genus @xmath416 . that is , there is a fibration of the link complement @xmath417 over the circle so that @xmath418 is a surface of genus @xmath416 ( seifert surface ) for all @xmath419 . now we will also describe the changes for a general tree . at first we will modify the whitehead link : we duplicate the linked circle , i.e. there are as many circles as branching in the tree to get the link @xmath420 with @xmath421 components . then the complement of @xmath420 has also @xmath421 torus boundaries and it also fibers over @xmath119 . with the help of @xmath420 we can build every tree @xmath422 . now the 3-manifold @xmath423 is given by @xmath112framed surgery along the @xmath407-th untwisted ramified ( usage of @xmath420 ) whitehead double of a knot @xmath110 , denoted by the link @xmath424 . the tree @xmath422 has one root , then @xmath423 is given by @xmath425 and the complement @xmath426 splits like the tree into complements of @xmath420 and one copy of @xmath427 ( see figure [ fig : splitting - of - knot - complement ] ) . using a deep result of freedman @xcite , we obtain : + _ @xmath423 is a wild embedded 3-sphere @xmath240 . _ our result about the existence of a codimension - one foliation for @xmath423 can be simply expressed : foliations are characterized by the holonomy properties of the leafs . this principle is also the corner stone for the usage of non - commutative geometry as description of the leaf space . in the previous subsection , we already characterized the state as an element of the kauffman skein module . here we are interested in a reconstruction of the underlying space but now assuming a global foliation so that we will obtain the whole spatial space . starting point is the state constructed in the subsection [ sub : construction - of - state ] . here , we got a relation between the state @xmath218 as linear functional over the algebra and the kauffman skein module . using this relation , we consider a leaf @xmath428 $ ] and the 3-dimensional extension as solid torus @xmath117 . the kauffman skein module @xmath429 is polynomial algebra with one generator ( the loop around @xmath119 ) . now we consider one 3-manifold @xmath198 with the corresponding foliation . using the splitting above , the kauffman skein module @xmath430 is determined by the skein module for the parts , i.e. by the knot complements . therefore we have to consider the skein module for hyperbolic 3-manifolds . hyperbolic 3-manifolds contain special surfaces , called essential or incompressible surfaces , see appendix c. it is known @xcite that the skein module of 3-manifolds containing essential surfaces is not finitely generated . therefore , the state itself is not finitely generated . if we use the leaf @xmath117 as a local model for one generator then we will obtain an infinitely complicated 3-manifold made from pieces @xmath117 so that the corresponding generators are not related to each other . an example of this structure is the whitehead manifold having a non - finitely generated kauffman skein module @xcite . in general we will obtain a wild embedded 3-manifold by using this simple pieces . by the argumentation in the previous subsection we know that this wild embedded 3-manifold is the wild embedded 3-sphere @xmath423 . finally we obtain : @xmath431 the state @xmath218 is realized by some wild embedded 3-sphere . following @xcite we will construct a @xmath233algebra from the wild embedded 3-sphere . let @xmath432 be a wild embedding of codimension - one so that @xmath433 . now we consider the complement @xmath434 which is non - trivial , i.e. @xmath435 . now we define the @xmath233algebra @xmath436 ) associated to the complement @xmath437 with group @xmath438 . if @xmath439 is non - trivial then this group is not finitely generated . from an abstract point of view , we have a decomposition of @xmath440 by an infinite union @xmath441 of level sets @xmath442 . then every element @xmath443 lies ( up to homotopy ) in a finite union of levels . the basic elements of the @xmath233algebra @xmath436 ) are smooth half - densities with compact supports on @xmath440 , @xmath444 , where @xmath354 for @xmath443 is the one - dimensional complex vector space of maps from the exterior power @xmath445 ( @xmath446 ) , of the union of levels @xmath262 representing @xmath225 , to @xmath283 such that @xmath447 for @xmath448 , the convolution product @xmath363 is given by the equality @xmath364 with the group operation @xmath449 in @xmath439 . then we define via @xmath365 a @xmath315operation making @xmath450 into a @xmath315algebra . each level set @xmath442 consists of simple pieces ( in case of alexanders horned sphere , we will explain it below ) denoted by @xmath451 . for these pieces , one has a natural representation of @xmath450 on the @xmath367 space over @xmath451 . then one defines the representation @xmath452 the completion of @xmath450 with respect to the norm @xmath453 makes it into a @xmath266-algebra @xmath454 ) . finally we are able to define the @xmath233algebra associated to the wild embedding . using a result in @xcite , one can show that the corresponding von neumann algebra is the factor @xmath3 . among all elements of the @xmath266-algebra , there are distinguished elements , idempotent operators or projectors having a geometric interpretation . for later use , we will construct them explicitly ( we follow @xcite sec . @xmath455 closely ) . let @xmath456 be the wild submanifold . a small tubular neighborhood @xmath376 of @xmath423 in @xmath0 defines the corresponding @xmath266-algebra @xmath457 is isomorphic to @xmath458 with @xmath380 the @xmath266-algebra of compact operators . in particular it contains an idempotent @xmath381 , @xmath459 , where @xmath12 is a minimal projection in @xmath380 . it induces an idempotent in @xmath460 . by definition , this idempotent is given by a closed curve in the complement @xmath434 . these projection operators form the basis in this algebra in this section we will describe a way from a ( classical ) poisson algebra to a quantum algebra by using deformation quantization . therefore we will obtain a positive answer to the question : does the @xmath233algebra of the foliation ( as well of a wild ( specific ) embedding ) comes from a ( deformation ) quantization ? of course , this question can not be answered in all generality , but for our example we will show that the enveloping von neumann algebra of foliation and of this wild embedding is the result of a deformation quantization using the classical poisson algebra ( of closed curves ) of the tame embedding . this result shows two things : the wild embedding can be seen as a quantum state and the classical state is a tame embedding . in this section we will describe the formal structure of a classical theory coming from the algebra of observables using the concept of a poisson algebra . in quantum theory , an observable is represented by an hermitean operator having the spectral decomposition via projectors or idempotent operators . the coefficient of the projector is the eigenvalue of the observable or one possible result of a measurement . at least one of these projectors represents ( via the gns representation ) a quasi - classical state . thus , to construct the substitute of a classical observable algebra with poisson algebra structure , we have to concentrate on the idempotents in the @xmath266-algebra . now we will see that the set of closed curves on a surface has the structure of a poisson algebra . let us start with the definition of a poisson algebra . let @xmath342 be a commutative algebra with unit over @xmath461 or @xmath283 . poisson bracket _ on @xmath342 is a bilinearform @xmath462 fulfilling the following 3 conditions : 1 . anti - symmetry @xmath463 2 . jacobi identity @xmath464 3 . derivation @xmath465 . then a _ poisson algebra _ is the algebra @xmath466 . now we consider a surface @xmath467 together with a closed curve @xmath225 . additionally we have a lie group @xmath268 given by the isometry group . the closed curve is one element of the fundamental group @xmath468 . from the theory of surfaces we know that @xmath468 is a free abelian group . denote by @xmath469 the free @xmath470-module ( @xmath470 a ring with unit ) with the basis @xmath468 , i.e. @xmath469 is a freely generated @xmath470-module . recall goldman s definition of the lie bracket in @xmath469 ( see @xcite ) . for a loop @xmath471 we denote its class in @xmath468 by @xmath472 . let @xmath473 be two loops on @xmath467 lying in general position . denote the ( finite ) set @xmath474 by @xmath475 . for @xmath476 denote by @xmath477 the intersection index of @xmath478 and @xmath479 in @xmath480 . denote by @xmath481 the product of the loops @xmath473 based in @xmath480 . up to homotopy the loop @xmath482 is obtained from @xmath483 by the orientation preserving smoothing of the crossing in the point @xmath480 . set @xmath484=\sum_{q\in\alpha\#\beta}\epsilon(q;\alpha,\beta)(\alpha_{q}\beta_{q})\quad.\label{eq : lie - bracket - loops}\ ] ] according to goldman @xcite ( theorem 5.2 ) , the bilinear pairing @xmath485:z\times z\to z$ ] given by ( [ eq : lie - bracket - loops ] ) on the generators is well defined and makes @xmath469 a lie algebra . the algebra @xmath486 of symmetric tensors is then a poisson algebra ( see turaev @xcite ) . the whole approach seems natural for the construction of the lie algebra @xmath469 but the introduction of the poisson structure is an artificial act . from the physical point of view , the poisson structure is not the essential part of classical mechanics . more important is the algebra of observables , i.e. functions over the configuration space forming the poisson algebra . for the foliation discussed above , we already identified the observable algebra ( the holonomy along closed curves ) as well the corresponding group to be @xmath394 . therefore for the following , we will set @xmath487 . now we introduce a principal @xmath268 bundle on @xmath467 , representing a geometry on the surface . this bundle is induced from a @xmath268 bundle over @xmath488 $ ] having always a flat connection . alternatively one can consider a homomorphism @xmath489 represented as holonomy functional @xmath490 with the path ordering operator @xmath491 and @xmath218 as flat connection ( i.e. inducing a flat curvature @xmath492 ) . this functional is unique up to conjugation induced by a gauge transformation of the connection . thus we have to consider the conjugation classes of maps @xmath493 forming the space @xmath494 of gauge - invariant flat connections of principal @xmath268 bundles over @xmath467 . now ( see @xcite ) we can start with the construction of the poisson structure on @xmath494 , based on the cartan form as the unique bilinearform of a lie algebra . as discussed above we will use the lie group @xmath487 but the whole procedure works for every other group too . now we consider the standard basis @xmath495 of the lie algebra @xmath496 with @xmath497=h,\,[h , x]=2x,\,[h , y]=-2y$ ] . furthermore there is the bilinearform @xmath498 written in the standard basis as @xmath499 now we consider the holomorphic function @xmath500 and define the gradient @xmath501 along @xmath12 at the point @xmath149 as @xmath502 with @xmath503 and @xmath504 the calculation of the gradient @xmath505 for the trace @xmath506 along a matrix @xmath507 is given by @xmath508 given a representation @xmath509 of the fundamental group and an invariant function @xmath510 extendable to @xmath511 . then we consider two conjugacy classes @xmath512 represented by two transversal intersecting loops @xmath513 and define the function @xmath514 by @xmath515 . let @xmath516 be the intersection point of the loops @xmath513 and @xmath517 a path between the point @xmath278 and the fixed base point in @xmath468 . then we define @xmath518 and @xmath519 . finally we get the poisson bracket @xmath520 where @xmath521 is the sign of the intersection point @xmath278 . thus , _ the space @xmath511 has a natural poisson structure ( induced by the bilinear form ( [ eq : lie - bracket - loops ] ) on the group ) and the poisson algebra _ @xmath522 _ of complex functions over them is the algebra of observables . _ now we introduce the ring @xmath523 $ ] of formal polynomials in @xmath188 with values in @xmath283 . this ring has a topological structure , i.e. for a given power series @xmath524 $ ] the set @xmath525 $ ] forms a neighborhood . now we define a _ quantization _ of a poisson algebra @xmath342 is a @xmath523 $ ] algebra @xmath526 together with the @xmath283-algebra isomorphism @xmath527 so that \1 . the module @xmath526 is isomorphic to @xmath528 $ ] for a @xmath283 vector space @xmath79 \2 . let @xmath529 and @xmath530 be @xmath531 , @xmath532 then @xmath533 one speaks of a deformation of the poisson algebra by using a deformation parameter @xmath188 to get a relation between the poisson bracket and the commutator . therefore we have the problem to find the deformation of the poisson algebra @xmath534 . the solution to this problem can be found via two steps : 1 . at first find another description of the poisson algebra by a structure with one parameter at a special value and 2 . secondly vary this parameter to get the deformation . fortunately both problems were already solved ( see @xcite ) . the solution of the first problem is expressed in the theorem : _ the skein module @xmath535)$ ] ( i.e. @xmath536 ) has the structure of an algebra isomorphic to the poisson algebra @xmath534 . _ _ ( see also @xcite ) _ then we have also the solution of the second problem : _ the skein algebra @xmath537)$ ] is the quantization of the poisson algebra @xmath534 with the deformation parameter @xmath538.(see also @xcite ) _ to understand these solutions we have to introduce the skein module @xmath539 of a 3-manifold @xmath127 ( see @xcite ) . for that purpose we consider the set of links @xmath540 in @xmath127 up to isotopy and construct the vector space @xmath541 with basis @xmath540 . then one can define @xmath542 $ ] as ring of formal polynomials having coefficients in @xmath541 . now we consider the link diagram of a link , i.e. the projection of the link to the @xmath543 having the crossings in mind . choosing a disk in @xmath543 so that one crossing is inside this disk . if the three links differ by the three crossings @xmath544 ( see figure [ fig : skein - crossings ] ) inside of the disk then these links are skein - related . .[fig : skein - crossings ] ] then in @xmath542 $ ] one writes the skein relation . ] @xmath545 . furthermore let @xmath546 be the disjoint union of the link with a circle and one writes the framing relation @xmath547 . let @xmath548 be the smallest submodule of @xmath542 $ ] containing both relations . then we define the kauffman bracket skein module by @xmath549/s(m)$ ] . we list the following general results about this module : * the module @xmath550 for @xmath536 is a commutative algebra . * let @xmath467 be a surface , then @xmath537)$ ] carries the structure of an algebra . the algebraic structure of @xmath537)$ ] can be simply seen by using the diffeomorphism between the sum @xmath488\cup_{s}s\times[0,1]$ ] along @xmath467 and @xmath488 $ ] . then the product @xmath551 of two elements @xmath552)$ ] is a link in @xmath488\cup_{s}s\times[0,1]$ ] corresponding to a link in @xmath488 $ ] via the diffeomorphism . the algebra @xmath537)$ ] is in general non - commutative for @xmath553 . for the following we will omit the interval @xmath81 $ ] and denote the skein algebra by @xmath554 . in subsection [ sub : construction - of - state ] , we described the state as an element of the kauffman skein module @xmath555 of the leaf @xmath222 . now we obtained also that the observable algebra is the kauffman skein module again . how does this whole story fit into the description of the observable algebra for the foliation as factor @xmath3 ? in @xcite , it was shown that the kauffman bracket skein module of a cylinder over the torus embeds as a subalgebra of the noncommutative torus . however , the noncommutative torus can be seen as the leaf space of the kronecker foliation of the torus leading to the factor @xmath335 . then by using ( [ eq : crossed - product - factor - iii ] ) , we obtain the factor @xmath3 back . we will use this relation in the next section to get the quantum action . above , we used the foliation to get quantum states which agreed with the deformation quantization of a classical state . central point in our argumentation is the construction of the @xmath233algebra with the corresponding von neumann algebra as observable algebra . this von neumann algebra is a factor @xmath3 . by using the tomita - takesaki modular theory , there is a relation to the factor @xmath335 by using an action of the group @xmath349 by automorphisms of a lebesgue measure space leading to the decomposition of the factor @xmath3 . this action is related to an invariant , the flow of weights mod(m ) . the main property of the factor @xmath3 is the constant flow of weights mod(m ) . connes @xcite described the flow of weights as a bundle of densities over the leaf space , i.e. the @xmath349 homogeneous space of nonzero maps . in case of foliation considered above , this density is constant and we can naturally identify this density with the volume of the submanifold defining the foliation . by definition , this volume is given by the godbillon - vey invariant ( see eqn . ( [ eq : gv - number - thurston - foliation-1 ] ) in appendix b , the circle in the fiber has unit size ) . this invariant can be seen as an element of @xmath556 with the holonomy groupoid @xmath268 of the foliation . as shown by connes @xcite , the godbillon - vey class @xmath557 can be expressed as a cyclic cohomology class ( the so - called flow of weights ) @xmath558 of the @xmath233algebra for the foliation . then we define an expression @xmath559 uniquely associated to the foliation ( @xmath560 is the dixmier trace ) . the expression @xmath467 generates the action on the factor by @xmath561 so that @xmath467 is the action or the hamiltonian multiplied by the time ( see ( [ eq : modular - generator ] ) ) . it is an operator which defines the dynamics by acting on the states . for explicit calculations we have to evaluate this operator . one way is the usage of the relation between the foliation and the wild embedding . this wild embedding is determined by the fundamental group @xmath439 of its complement . in @xcite , we discussed the properties of this group @xmath439 . it is a perfect group , i.e. every element is generated by a commutator . then a representation of this group into some other group like @xmath562 ( the limit of @xmath563 for @xmath139 ) reduces to the representation of the maximal perfect subgroup . for that purpose we consider the representation of the group @xmath439 into the group @xmath564 of elementary matrices , which is the perfect subgroup of @xmath562 . then we obtain matrix - valued functions @xmath565 as the image of the generators of @xmath439 w.r.t . the representation @xmath566 labeled by the dimension @xmath567 of the embedding space @xmath568 . via the representation @xmath569 , we obtain a cyclic cocycle in @xmath570 generated by a suitable fredholm operator @xmath341 . here we use the standard choice @xmath571 with the dirac operator @xmath201 acting on the functions in @xmath572 . then the cocycle in @xmath570 can be expressed by @xmath573[f , x^{\nu}]\ ] ] using a metric @xmath574 in @xmath1 via the pull - back using the representation @xmath569 . finally we obtain the action @xmath575[f , x_{\mu}])=tr_{\omega}([d , x^{\mu}][d , x_{\mu}]|d|^{-2})\label{eq : quantum - action}\ ] ] which can be evaluated by using the heat - kernel of the dirac operator @xmath201 . the appearance of the heat kernel is a sign for a relation to quantum field theory where the heat kernel is a very convenient tool for studying one - loop divergences , anomalies and various asymptotics of the effective action . away from this operator expression for the godbillon - vey invariant , there are geometrical evaluations which are not defined on the leaf space but rather on the whole manifold . as mentioned above , this invariant admits values in the real numbers and we can evaluate them according to the type of the number : for integer values one obtains the euler class and for rational numbers the pontrjagin class ( for the corresponding bundles ) . therefore using the ideas of section [ sec : action - induced - by - top ] , we obtain the einstein - hilbert and the holst action but also a correction given by irrational values of the godbillon - vey number . a good test for the theory is the dependence of the action ( [ eq : quantum - action ] ) on the scale . the theory has a strong geometrical flavor and therefore the scaling behavior can be understood by a geometrical construction using the exotic @xmath1 . as explained above , the central point in the construction is the casson handle . from the scaling point of view , the casson handle contains disks of any size ( with respect to the embedding @xmath576 ) . the long scales are given by the first levels of the casson handle whereas the small scales are represented by the higher levels of the casson handle . let us consider the small exotic @xmath1 . from the physics point of view , the large scale is given by the first levels of the casson handle . in the construction of the foliation of @xmath1 , the first levels describe a polygon in the hyperbolic space @xmath343 with a finite and small number of vertices . the godbillon - vey number of this foliation is given by the volume of this polygon . in principle , it is also true for the inclusion of the higher levels ( and also for the whole casson handle ) but every higher level gives only a very small contribution to the godbillon - vey number . therefore , the first levels of the casson handle can be simply characterized by the godbillon - vey number , i.e. by the size of the polygon in the scale @xmath577 . then the godbillon - vey number is given by @xmath578 . in @xcite we analyzed this situation and found the relation @xmath579 to the godbillon - vey number . here we integrate over the disk ( equal to the polygon ) which is used to define the foliation . this model is the non - linear sigma model ( for the embedding of the disk into @xmath198 with metric @xmath147 ) depending on the scale @xmath580 . the scaling behavior of this model was studied in @xcite and one obtains the rg flow equation @xmath581 reducing to the ricci flow equations for large scales ( @xmath582 ) . the fixed point of this flow are geometries of constant curvature ( used to prove the thurston geometrization conjecture ) . therefore in the classical limit of large scales , we obtain a geometry of the 3-manifold of constant curvature whereas for small scales one has to take into account higher curvature corrections . on the spacetime , one has also flow equations from one 3-manifold of constant curvature to another 3-manifold of constant curvature . this flow equation is equivalent to the ( anti-)self - dual curvature ( or instantons ) by using the gradient flow of the chern - simons functional @xcite . this approach has much in common with the non - linear graviton of penrose @xcite . we will explain these ideas in subsection [ sub : the - graviton - propagator ] . for the short scale , we need the full power of the casson handle . as a first step we can evaluate the action ( [ eq : quantum - action ] ) so that the dirac operator @xmath201 acts on usual square - integrable functions , so that @xmath583=dx^{\mu}$ ] is finite . then the action ( [ eq : quantum - action ] ) reduces to @xmath584 where @xmath585 is the index for the coordinates on @xmath1 and @xmath586 represents the index of the disk ( inside of the casson handle ) . now we will choose a small fluctuation @xmath587 of a fixed embedding of the disk in the casson handle given by @xmath588 with @xmath589 . then we obtain @xmath590 and we use a standard argument to neglect the terms linear in @xmath591 : fluctuations have no preferred direction and therefore only the square contributes . then we have @xmath592 for the action . by using a result of @xcite one obtains for the dixmier trace @xmath593 with the first coefficient @xmath594 of the heat kernel expansion @xcite @xmath595 and the action simplifies to @xmath596 for the main contributions where @xmath74 is the scalar curvature of the embedded disk @xmath125 . again , but now for small fluctuations , we obtain the flow equation ( [ eq : rg - flow - equation ] ) but we have to consider the small case @xmath597 . then we have to take arbitrary curvature contributions into account . this short calculation showed that the short - scale behavior is given by a two - dimensional action . in the next section we will understand this behavior geometrically . for small fluctuations we obtained a disk but what happens for larger fluctuations ? then we have to take even the higher levels of the casson handle into account . these higher levels form a complicated surface with a fractal structure ( a generalization of the cantor set ) . then the action ( [ eq : action - small - fluctuations ] ) has to be replaced by an integral over this fractal space . for the evaluation of the quantum action ( [ eq : quantum - action ] ) one can use the ideas of noncommutative geometry as used for fractals and quasi - fuchsian groups , see @xcite ( section iv.3 ) . in this section we will present some properties of the theory . for an impression , it is enough to present the main ideas . the details will be published separately . by using the large scale behavior in subsection [ sub : long - scale - behaviour ] , we have to consider ricci - flat spaces and an easy calculation gives the well - known propagator in the linearized version , however , we are not interested in the linearized version . grt is a highly non - linear theory and therefore one has to take this non - linearity into account . the ricci - flatness of the spacetime goes over to the 3-manifold as the spatial component where it implies a 3-manifold of constant curvature ( as fixed point of the ricci flow ) . then as shown by witten @xcite , the 3-dimensional einstein - hilbert action @xmath598 is related to the chern - simons action @xmath599 with respect to the ( levi - civita ) connection @xmath149 and the length @xmath262 . by using the stokes theorem we obtain @xmath600)=\intop_{m_{t}}tr(f\wedge f ) \ , , \ ] ] i.e. the action for the 4-manifold @xmath601 $ ] ( as local spacetime ) with the curvature @xmath602 , i.e. the action is the ( topological ) pontrjagin class of the 4-manifold . from the formal point of view , the curvature 2-form @xmath602 is generated by a @xmath215 connection @xmath149 in the frame bundle , which can be lifted uniquely to a @xmath394- ( spin- ) connection . according to the ambrose - singer theorem , the components of the curvature tensor are determined by the values of holonomy which is in general a subgroup of @xmath394 . thus we start with a suitable curvature 2-form @xmath602 with values in the lie algebra @xmath603 of the lie group @xmath268 as subgroup of the @xmath394 . the variation of the chern - simons action gets flat connections @xmath604 as solutions . the flow of solutions @xmath605 in @xmath601 $ ] ( parametrized by the variable @xmath606 , the time ) between the flat connection @xmath607 in @xmath608 to the flat connection @xmath609 in @xmath610 will be given by the gradient flow equation ( see for instance @xcite ) @xmath611 where the coordinate @xmath606 is normal to @xmath336 . therefore we are able to introduce a connection @xmath612 in @xmath601 $ ] so that the covariant derivative in @xmath606-direction agrees with @xmath613 . then we have for the curvature @xmath614 , where the fourth component is given by @xmath615 . thus we will get the instanton equation with ( anti- ) self - dual curvature @xmath616 it follows @xmath617\times n)=\intop_{n\times[0,1]}tr(\tilde{f}\wedge\tilde{f})=\pm\intop_{n\times[0,1]}tr(\ \tilde{f}\wedge*\tilde{f})\,,\ ] ] ( i.e. the macdowellmansouri action ) . we remark the main point in this argumentation : we obtain a self - dual curvature as gradient flow between two 3-manifolds of constant curvature . of course , ( anti-)self - dual curvatures are also solutions of einsteins equation ( but the reverse is not true ) . following penrose @xcite , we call these solutions the nonlinear graviton . above we constructed the observable algebra from the foliation leading to the kauffman bracket skein module . in the subsection we will discuss the relation to lattice gauge field theory . main source for this discussion is the work of bullock , frohman and kania - bartoszyska @xcite . in this paper the authors realize that gauge fields come from the restricted dual of the hopf algebra on which the theory is based . this leads to a coordinate free formulation . then they comultiply connections in a way that implies the usual exchange relations for fields while preserving their evaluability . their new foundations allow them to compute wilson loops and many other operators using a simple extension of tangle functors . then they analyzed the structure of the algebra of observables . in their viewpoint , the observables correspond to quantum groups seen as rings of invariants of n - tuples of matrices under conjugation . the connection with lattice gauge field theory is that each n - tuple of matrices corresponds to a connection on a lattice with one vertex and n - edges , with the gauge fields based on a classical group . the construction given in this paper leads to an algebra of characters of a surface group with respect to any ribbon hopf algebra . the algebras are interesting from many points of view : they generalize objects studied in invariant theory ; they should provide tools for investigating the structure of the mapping class groups of surfaces ; and they should give a way of understanding quantum invariants of @xmath618-manifolds . the algebra of observables based on the enveloped lie algebra @xmath619 is proved to be the ring of @xmath268-characters of the fundamental group of the associated surface . then , given the ring of @xmath268-characters of a surface group , they showed that the observables based on the corresponding drinfeld - jimbo algebra form a quantization with respect to the usual poisson structure . furthermore they proved for the classical groups that the algebra of observables is generated by wilson loops . finally , invoking a quantized cayley - hamilton identity , they obtain a new proof , that the @xmath620-characters of a surface are exactly the kauffman bracket skein module of a cylinder over that surface . the power of lattice gauge field theory is that it places the representation theory of the underlying manifold and the quantum invariants in the same setting . ultimately the asymptotic analysis of the quantum invariants of a @xmath618-manifold in terms of the representations of its fundamental group should flow out of this setting . the identification of the representation theory of a quantum group with that of a compact lie group leads to rigorous integral formulas for quantum invariants of @xmath618-manifolds . this should in turn lead to a simple explication of the relationship between quantum invariants and more classical invariants of @xmath618-manifolds . this relation to lattice gauge field theory seems to imply an underlying discrete structure of the space and/or spacetime , but the approach in the paper uncovers the reason @xcite : the kauffman bracket skein module is discrete structure containing only a finite amount of information . therefore , any description has to be discrete as well including the approach via gauge fields . this idea can be extended to the 4-manifold . as explained above , every smooth 4-manifold can be effectively described by handles and one only needs a finite number to describe every compact 4-manifold . then the handles can be simply triangulated by using simplices to end up with a piecewise - linear ( or pl ) structure . the surprising result of cerf for manifolds of dimension smaller than 7 was simple : pl - structure ( or triangulations ) and smoothness structure are the same . this implies that every pl - structure can be smoothed to a smoothness structure and vice versa . therefore _ the discrete approach ( via triangulations ) and the smooth approach to defining a manifold are the same _ ! so , our spacetime admits a kind of duality : it contains discrete information in its handle structure but it is a continuous space at the same time . both approaches are interchangeable . therefore the underlying structure of the spacetime is discrete but the spacetime itself is a smooth 4-manifold . or , the information contained in a smooth 4-manifold is finite . in @xcite we describe an exotic black hole by constructing a smooth metric for the interior . here we will present the main argument shortly . in @xcite the existence of an exotic black hole ( as exotic kruskal space ) using an exotic @xmath0 was suggested . the idea was simply to consider the complement @xmath621 where @xmath622 was only understood topologically . in case of the exotic small @xmath0 given by a casson handle , we can reproduce our construction of an exotic @xmath623 by using a casson handle . therefore we will here concentrate on the representation of the exotic @xmath623 by using the casson handle @xmath124 to get @xmath624 in @xcite the analytical properties of the casson handle were discussed . the main idea is the usage of the theory of end - periodic manifolds , i.e. an infinite periodic structure generated by @xmath59 glued along a compact set @xmath134 to get @xmath625 the end - periodic manifold . the definition of an end - periodic manifold is very formal ( see @xcite ) and we omit it here . all casson handles generated by a balanced tree have the structure of end - periodic manifolds as shown in @xcite . by using the theory of taubes @xcite one can construct a metric on @xmath626 by using the metric on @xmath59 . then a metric @xmath147 in @xmath627 transforms to a periodic function @xmath628 on the infinite periodic manifold @xmath629 where @xmath630 is the building block @xmath59 at the @xmath631-th place . to reflect the number of the building block , we have to extend @xmath628 to @xmath632by using a metric @xmath633holomorphic in @xmath634 with @xmath635 where @xmath631 identifies the two boundaries of @xmath59 . from the formal point of view we have the _ generalized fourier - laplace transform ( or fourier - laplace transform for short ) _ @xmath636 where the coefficient @xmath637 represents the building block @xmath638 in @xmath639 . without loss of generality we can choose the coordinates @xmath278 in @xmath127 so that the @xmath143-th component @xmath640 is related to the integer @xmath641 $ ] via its integer part @xmath642 $ ] . using the inverse transformation we can construct a smooth metric @xmath147 in @xmath639 at the @xmath21-th building block via @xmath643 for @xmath644 , @xmath645 , @xmath641 $ ] with the projection @xmath646 ( mathematically : @xmath639 is the universal cover of @xmath647 like @xmath461 is the universal cover of @xmath119 ) . in the case of the kruskal space we have the metric @xmath648 in the usual units with a singularity at @xmath649 used for the whole space @xmath623 . the coordinates @xmath650 together with the relation @xmath651 represent @xmath543 and the angles @xmath652 the 2-sphere @xmath212 parametrized by the radius @xmath577 . clearly this metric can be also used for each building block @xmath59 having the topological structure @xmath653 with two attaching regions topologically given by @xmath654 forming the boundary @xmath655 ( see the description of a casson handle above ) . remember that the casson handle is topologically the subset @xmath656 . now we consider the decomposition @xmath657 and the part @xmath658 will be later the @xmath543 part of the casson handle . the size of the @xmath125 is parametrized by @xmath577 as above . then we obtain the metric ( [ eq : kruskal - metric ] ) for the building block @xmath59 . our model of the black hole based on the implicit dependence of the two coordinates @xmath650 on the parameter @xmath577 , the radius of the 2-sphere . therefore we choose for the coordinate @xmath659 the relation @xmath660 and obtain a metric @xmath628 on @xmath632 . so , we make the assumptions : 1 . the coordinate @xmath661 is related to the radius by @xmath660 . only the @xmath650 part of the metric is periodic and we do not change the other component @xmath662 of the metric . the integer part @xmath663 $ ] of the coordinate @xmath664 gives the number of the building block @xmath638 in the casson handle ( seen as end - periodic manifold ) . the metric on @xmath665 is given by a fourier transformation ( [ eq : fl - trafo ] ) of the @xmath650 part of the metric in the building block @xmath59 . some more comments are in order . the number @xmath663 $ ] is related to the coordinate @xmath664 as substitute of time . the metric @xmath147 in @xmath639 is smooth with respect to @xmath664 and we obtain the number of the building block by @xmath663 $ ] . to express this property we have to identify @xmath650 with the coordinates of @xmath658 . then we obtain the metric on @xmath665 by the generalized fourier - laplace transformation of the metric on @xmath635 using the metric of the building block @xmath59 and the coordinate @xmath661 similar to ( [ eq : fl - trafo ] ) @xmath666 especially the singular part of the metric ( i.e. the @xmath650 part ) on the building block @xmath59 @xmath667 transforms to the heaviside jump function @xmath668 using the relation ( [ eq : kruskal - relation ] ) , having no singularity . the metric vanishes , however , for large values of @xmath664 in the interior of the black hole . this sketch of some arguments gives a hint that the transformation of the smoothness to exotic smoothness could possibly smooth out some singularity in the black hole case . this metric vanishes along two directions or _ one obtains a dimensional reduction from 4d to 2d . _ but , is there a geometrical reason for this reduction ? a hyperbolic 3-manifold @xmath127 admits a hyperbolic structure by fixing a homomorphism @xmath400 ( up to conjugation ) . from the physics point of view , this homomorphism is given by the holonomy along a closed curve ( as element in @xmath669 ) for a flat connection . a sequence of these holonomies does not converge but it is possible to compactify the space of flat @xmath394 connections . this limit can be understood geometrically : the hyperbolic 3-manifold is triangulated by tetrahedrons . however , because of the hyperbolic geometry , the edge between two vertices is not the usual line but rather a geodesics in the hyperbolic geometry . the curvature of this geodesics depends on the hyperbolic structure . in the limit , all geodesics of the tetrahedron meet and one obtains a tree instead of tetrahedrons . therefore in the limit of large curvature , one obtains a reduction from 3d ( = tetrahedrons ) to 1d ( = tree ) . [ fig : degen - hyp - tree ] visualizes the transition from 2d(triangle ) to 1d(tree ) . ] in a purely geometrical theory , one has to answer this question . it can not be shifted to assume the appearance of quantum fluctuations . instead we have to understand the root of these quantum fluctuations . starting point of our approach is the foliation of the exotic @xmath1 by using the anosov flow . main point in the argumentation above is the appearance of the hyperbolic geometry in 3- and 4-dimensional submanifolds . the foliation can , however , be interpreted differently : a foliation defines a dynamics at a manifold leading to a splitting into leafs ( the integral curves of the dynamics ) . therefore , a tiny variation in the initial conditions will lead to a strong variation of the corresponding integral curve . this chaotic behavior is a natural consequence of the exotic smoothness structure ( leading to the non - trivial @xmath241foliation ) . for completeness we will describe this dynamics , called the anosov flow . for that purpose we consider the standard basis @xmath670 of the lie algebra @xmath671 with @xmath672=x\quad\quad[j , y]=-y\quad\quad[x , y]=2j\ ] ] leading to the exponential maps @xmath673 defining right - invariant flows on the unit tangent bundle @xmath674 of the hyperbolic space . the connection to the anosov flow comes from the realization that @xmath675 is the geodesic flow on @xmath676 . with lie vector fields being ( by definition ) left invariant under the action of a group element , one has that these fields are left invariant under the specific elements @xmath675 of the geodesic flow . this flow goes over to a surface @xmath677 defined by a subgroup @xmath678 with @xmath679 . now the geodesic flow @xmath675 acts on the exponential maps @xmath680 so that the geodesic flow itself is invariant , @xmath681 , but the other two shrink and expand : @xmath682 and @xmath683 . then the bundle @xmath684 splits into three subbundles @xmath685 where one bundle @xmath686 expands , one bundle @xmath687 contracts and one bundle @xmath688 is invariant w.r.t . geodesic flow . this property is crucial for the following discussion . because of the expanding behavior of one subbundle , the anosov flow is the generator of a chaotic dynamics . therefore , two geodesics diverge exponentially in this foliation , but this behavior goes over to the holonomies characterizing the geometry . the transport of a holonomy along two diverging geodesics can lead to totally different holonomies . currently this dynamics is deterministic , i.e. if we choose exactly the same initial condition then we will end at the state ( seen as limit point ) . this situation changes if we are unable to choose the initial condition exactly ( by choosing real numbers ) but instead we can only choose a rational number where this rational number is the characterizing property of the state . then all initial conditions ( represented by all real numbers ) in this class represent the same state but have totally different limit points of the corresponding dynamics . now we will describe this dynamics . starting point is the observable algebra @xmath689 , i.e. the space of holonomies @xmath690 ( i.e. homomorphisms ) up to conjugation , see subsection [ sub : the - observable - algebra ] . the deformation quantization ( see subsection [ sub : drinfeld - turaev - quantization ] ) is the kauffman bracket skein module . here we made use of the identity @xmath691 between two elements @xmath692 ( w.r.t . a representation ) . using the group commutator @xmath693=aba^{-1}b^{-1}$ ] one also obtains @xmath694)=(tr(a))^{2}+(tr(b))^{2}+(tr(ab))^{2}-tr(a)tr(b)tr(ab)\ ] ] according to deformation procedure , pairs of elements @xmath692 coming from closed curves via the holonomy and fulfilling @xmath695)=\pm2\ ] ] can be a canonical pair w.r.t . the symplectic structure . the sign is purely convention and we choose @xmath696)=-2 $ ] . then the canonical pair has to fulfill the equation @xmath697 which can be written in a more familiar form @xmath698 by using @xmath699 . because of the discreteness of @xmath468 , we have to look for rational solutions of this equation ( diophantine equation ) . the solutions of the equation are markoff triples forming a binary tree ( see fig . [ fig : binary - tree - of - markoff ] ) . binary tree of markoff numbers as solution of equation ( [ eq : markoff - equation ] ) . ] the set of these elements @xmath700 corresponding to discrete groups is known to be fractal in nature @xcite . it is the large class of quasi - fuchsian groups having a fractal curve ( julia set ) as limit set . then we have the desired behavior : + _ the set of canonical pairs ( as measurable variables in the theory ) forms a fractal subset of the space of all holonomies . then we can only determine the initial condition up to discrete value ( given by the canonical pair ) and the chaotic behavior of the foliation ( i.e. the anosov flow ) makes the limit not predictable . _ + at the end of this section , one remark about the role of the canonical pair . it is always possible to construct a classical continuous random field that has the same probability density as the quantum vacuum state . furthermore it is known that a random field can be generated by a chaotic dynamics . there is , however , a large difference between the classical random field and a quantum field : there are pairs of not equally accurate measurable observables ( mostly the canonical pairs ) for quantum fields impossible for the classical random fields . with our approach , we showed the same behavior for the canonical pairs . our geometrical approach should also lead to a description of the measurement process ( including the collapse of the wave function ) . in section [ sec : wild - embeddings ] , we constructed the geometrical expression for a quantum state given by a wild embedding ( the wild @xmath60 ) . the reduction of the quantum state ( as linear combination ) to an eigenstate ( or the collapse of the wave function ) is equivalent to a reduction of the wild embedding to a tame embedding . therefore we need a mechanism to reduce the wild embedding to a tame one . the construction of the wild @xmath60 is strongly related to the casson handle . the exoticness of the smooth structure of @xmath1 and the wildness of the @xmath60 depend both on the self - intersection of some disk . if we are able to remove these self - intersections then we will obtain the desired reduction . according to the discussion in subsection [ sub : small - exotic ] , one needs a casson handle for the cancellation . how many levels of the casson handle are needed to cancel the self - intersection ? this question was answered by freedman @xcite : one needs three levels ( a three - level casson tower ) ! at the same time , however , one produces more self - intersections in the higher levels . therefore one needs a little bit more : a casson tower where a complete casson handle can be embedded . then this casson handle is able to cancel the self - intersection and we will obtain a tame embedding or a classical state . as shown by freedman @xcite and gompf / singh @xcite , one needs a 5-stage casson tower so that a casson handle with the same attaching circle can be embedded into this 5-stage tower . we obtain a process which is the collapse of the wave function . what is the cause of this collapse ? as explained above , we can not choose a single disk to remove the self - intersections . instead we have to choose a casson tower where each stage is a boundary - connected sum of @xmath701 , i.e. its boundary is the sum @xmath702 where the number of components is equal to the number of self - intersections . so , every piece @xmath703 of the boundary is given by the identification of the two boundary components for @xmath207 $ ] . in section [ sec : action - induced - by - top ] , we identified this 3-manifold with the graviton or _ the collapse of the wave function is caused by a gravitational interaction_. the corresponding process is known as decoherence . in the following we will calculate the minimal decoherence time for the gravitational interaction . the 5-stage casson tower can be also understood as a cobordism between the 3-manifold @xmath704 ( the @xmath119 defines the attaching circle ) and a 3-manifold having the same homology . in case of the simplest casson tower , it is given by five complements of the whitehead link @xmath705 closed by two solid tori , i.e. @xmath706 and this manifold can be very complicated for more complex towers . now we will add some geometry to calculate the decoherence time . as shown by witten @xcite , the action @xmath707 for every 3-manifold ( in particular for @xmath708 and @xmath709 denoted by @xmath710 ) is related to the chern - simons action @xmath711 . the scaling factor @xmath262 is related to the volume by @xmath712{vol(\sigma_{0,1})}$ ] and we obtain formally @xmath713 by using @xmath714 with the ( unit ) volume @xmath715 . if @xmath710 is a hyperbolic 3-manifold then the ( unit ) volume is a topological invariant which can not be normalized to 1 . together with @xmath716 one can compare the kernels of the integrals of ( [ eq : witten - relation ] ) and ( [ eq : cs - integral - relation ] ) to get for a fixed time @xmath717 this gives the scaling factor @xmath718 where we set @xmath719 in the following . the hyperbolic geometry of the cobordism is best expressed by the metric @xmath720 also called the friedmann - robertson - walker metric ( frw metric ) with the scaling function @xmath721 for the ( spatial ) 3-manifold . mostow rigidity enforces us to choose @xmath722 in the length scale @xmath262 of the hyperbolic structure . in the following we will switch to quadratic expressions because we will determine the expectation value of the area . a second reason for the consideration of quadratic expressions is again the hyperbolic structure of @xmath343 . we needed this structure for the construction of the foliation which is given by a polygon in @xmath343 . this polygon defines a compact surface of genus @xmath723 . then the foliation of the polygon induces a foliation of the small exotic @xmath1 . the area of the polygon is mainly the godbillon - vey invariant of the foliation . it is known that foliations of surfaces are given by quadratic differentials of the form defined below . here , there are deep connections to trees and @xmath394 flat connections , i.e. a tree defines a quadratic differential and vice versa @xcite . using the previous equation , we obtain @xmath724 with respect to the scale @xmath725 . by using the tree of the casson handle , we obtain a countable infinite sum of contributions for ( [ eq : scale - quadratic - expansion ] ) . before we start we will clarify the geometry of the casson handle . a casson handle admits a hyperbolic geometry . therefore the tree corresponding to the casson handle must be interpreted as a metric tree with hyperbolic structure in @xmath343 and metric @xmath726 the embedding of the casson handle in the cobordism is given by the rules 1 . the direction of the increasing levels @xmath727 is identified with @xmath728 and @xmath729 is the number of edges for a fixed level with scaling parameter @xmath725 . the contribution of every level in the tree is determined by the previous level best expressed in the scaling parameter @xmath725 . 3 . an immersed disk at level @xmath21 needs at least one disk to resolve the self - intersection point . this disk forms the level @xmath421 but this disk is connected to the previous disk . so we obtain for @xmath730 at level @xmath421 @xmath731 up to a constant . by using the metric @xmath726 with the embedding ( @xmath732 , @xmath733 ) we obtain for the change @xmath734 along the @xmath735direction ( i.e. for a fixed @xmath279 ) @xmath736 . this change determines the scaling from the level @xmath21 to @xmath421 , i.e. @xmath737 and after the whole summation ( as substitute for an integral in case of discrete values ) we obtain for the relative scaling @xmath738 with @xmath739 . the chern - simons invariant for @xmath708 vanishes and we are left with @xmath740 and the complements @xmath705 are hyperbolic 3-manifolds with @xmath741 by using the software snapea . finally for the scaling we obtain @xmath742 and for the time we have to choose @xmath743 using the well - known relation @xmath744 between length and time , i.e. we see one coordinate along the casson handle as time axis . the time @xmath745 has to be identified with the planck time @xmath746 ( see section [ sec : action - induced - by - top ] ) so that @xmath747 is the minimal decoherence time for the gravitational interaction . now we also discuss the entanglement which has to be also geometrically expressed . a quantum state is an element of the skein algebra @xmath554 for @xmath488 $ ] . for two disjoint surfaces @xmath748 one has @xmath749 now let us choose a knot @xmath750 in @xmath751 $ ] as element of @xmath752 as well a knot @xmath753 in @xmath754 . then @xmath755 is an element of @xmath756 . furthermore we can assume that the knots @xmath753 and @xmath750 can be also an element of @xmath752 and @xmath754 , respectively . then the element @xmath757 exists but now as an element of @xmath554 with @xmath758 . using the skein relations in @xmath554 , see fig . [ fig : skein - crossings ] , we obtain a linking between the corresponding knots , i.e. @xmath750 and @xmath753 forming a link . fig . [ fig : entangled - circle ] visualizes the transition from disjoint circles ( = disjoint states ) to linked circles ( = entangled states ) . ] then entanglement is reduced to a linking ! next we have to think about the measurement which reduces the entangled state to one product state . here we will only present some rough ideas for the description of the measurement process , but at first we have to define a measurement device . in this proposal , it is a union of casson handles which can be used to unlink two linked components . at the level of skein algebras , the casson handle is also given by elements of a skein algebra ( given by closed , knotted curves at the levels ) . the particular structure of the casson handle is not determined ( see also section [ sec : where - does - fluctuation ] ) . now a given quantum state is linked to this casson handle . the limit point of the casson handle ( i.e. the leafs of the tree ) give the result of the unlinking . all limit points of the casson handle have a fractal structure ( a cantor set ) expressing our inability to know the outcome of the measurement . the tree structure of the casson handle has also another effect : the limit points are exponentially separated from each other and can be seen as classical states . with these speculations , we will close this section . in the last section we will collect some implications for a cosmological model . let us assume the topology @xmath4 for the spacetime but with an exotic smoothness structure @xmath759 . one can construct this spacetime from the exotic @xmath1 by @xmath760 . from previous work , we know : * cosmological anomalies like dark matter and dark energy are ( conjecturally ) rooted in exotic smoothness @xcite . * the initial state of the cosmos must be a wild 3-sphere representing a quantum state @xcite . * then there is an inflationary phase @xcite driven by a decoherence which can be described by the starobinsky model . + in this model , we have a topological transition from a 3-manifold @xmath708 to another 3-manifold @xmath709 . both 3-manifolds are homology 3-spheres . therefore let us describe this change ( a so - called homology cobordism ) between two homology 3-spheres @xmath708 and @xmath709 . the situation can be described by a diagram @xmath761 which commutes . the two functions @xmath199 and @xmath762 are the morse function of @xmath708 and @xmath709 , respectively , with @xmath763 . the morse function over @xmath710 is a function @xmath764 having only isolated , non - degenerated , critical points ( i.e. with vanishing first derivatives at these points ) . a homology 3-sphere has two critical points ( located at the two poles ) . the morse function looks like @xmath765 at these critical points . the transition @xmath766 represented by the ( homology ) cobordism @xmath767 maps the morse function @xmath768 on @xmath708 to the morse function @xmath769 on @xmath709 . the function @xmath770 represents also the critical point of the cobordism @xmath767 . as we learned above , this cobordism has a hyperbolic geometry and we have to interpret the function @xmath771 not as an euclidean form but change it to the hyperbolic geometry so that @xmath772 i.e. we have a preferred direction represented by a single scalar field @xmath773 . therefore , the transition @xmath774 is represented by a single scalar field @xmath773 and we identify this field as the moduli . finally we interpret this morse function in the interior of the cobordism @xmath767 as the potential ( shifted away from the point @xmath143 ) of the scalar field @xmath775 with lagrangian @xmath776 with two free constants @xmath777 and @xmath778 . for the value @xmath779 and @xmath780 we obtain the starobinski model @xcite ( by a conformal transformation using @xmath775 and a redefinition of the scalar field @xcite ) @xmath781 with the mass scale @xmath782 much smaller than the planck mass . from our discussion above , the appearance of this model is not totally surprising . it favors a surface to be incompressible ( which is compatible with the properties of hyperbolic manifolds ) . * this inflationary phase is followed by another exponentially increasing phase leading to a hyperbolic 4-manifold with constant curvature which is rigid by mostow rigidity @xcite . here , we obtained the global geometry of the spacetime : it is a de sitter space @xmath232 with a cosmological constant which is the curvature of the spacetime . * this constant curvature can be identified with the cosmological constant in good agreement with the planck satellite results @xcite . the cosmological constant is constant by mostow rigidity ( but now for the 4-manifold ) . * the topology of the spatial component ( seen as 3-manifold ) is strongly restricted @xcite by the smoothness of the spacetime . * the inclusion of matter can be done naturally as direct consequence of exotic smoothness @xcite . * the interior of black holes can be described by exotic smoothness where the singularity is smoothed out @xcite . smooth quantum gravity , the usage of exotic smoothness structures on 4-manifolds , are the attempt to obtain a consistent theory of quantum gravity without any further assumptions . for us , the change of the smoothness structure is the next step in extending general relativity , where non - euclidean geometry was used to describe gravity and all accelerations . then , two different smoothness structures represent two different physical systems . in particular i think that the standard smoothness structure represents the case of a spacetime without matter and non - gravitational fields . in this paper we are going a more radical way to construct a quantum theory without quantization but by using purely geometrical ideas from mathematical topics like differential and geometric topology . the flow of ideas can be simply described by the following points : * an exotic @xmath0 is given by an infinite handlebody ( so one needs infinitely many charts ) and one finds also the description by an infinite sequence of 3-manifolds together with 4-dimensional cobordisms connecting them . * every 3-manifold admits a codimension - one foliation which goes over to the 4-dimensional cobordisms . the leaf space of this foliation is an operator algebra with a strong connection to algebraic quantum field theory . * the states ( as linear functionals in the algebra ) depend on knotted curves and are elements of the kauffman bracket skein algebra . the reconstruction of the spatial space gives a wild embedded 3-sphere which is therefore related to the state , or the quantum state can be identified with the wild embedding . the classical state is a tame ( i.e. usual ) embedding where the deformation quantization of a tame embedding is a wild embedding . * the structure of the operator algebra can be analyzed by the tomita - takesaki modular theory . then it is possible to construct the quantum action by using the quantized calculus of connes . * for large scales , one gets the einstein - hilbert action . whereas for small scales , one obtains a dimensional reduced action . * the foliation is given by a hyperbolic dynamics having a chaotic behavior . for our states , one gets an unpredictable behavior so that the dynamics can generate the quantum fluctuations . this list shows the current state but there are many open points , where we list only the most important here : * what is the hamiltonian of the theory ? in principle we constructed this operator but have a problem connecting to loop quantum gravity . * what are the states seen as knots ? the states are knots but the skein and mandelstam identities give a class of knots : the states are conjecturally the concordance class of knots . * is the state a solution of the hamiltonian ? here we conjecture that the concordance class of the knot lies already in the kernel of the hamiltonian ( therefore it is a solution of the hamiltonian constraint ) a lot is done but there are also many open problems . + * happy birthday carl ! * i have to thank carl for 20 years of friendship and collaboration as well numerous discussions . special thanks to jerzy krl for our work and many discussions about fundamental problems in math and physics . now i understand the importance of model theory . let us now consider the basic construction of the casson handle @xmath124 . let @xmath127 be a smooth , compact , simply - connected 4-manifold and @xmath783 a ( codimension-2 ) mapping . by using diffeomorphisms of @xmath125 and @xmath127 , one can deform the mapping @xmath12 to get an immersion ( i.e. injective differential ) generically with only double points ( i.e. @xmath784 ) as singularities @xcite . but to incorporate the generic location of the disk , one is rather interesting in the mapping of a 2-handle @xmath115 induced by @xmath785 from @xmath12 . then every double point ( or self - intersection ) of @xmath786 leads to self - plumbings of the 2-handle @xmath115 . a self - plumbing is an identification of @xmath787 with @xmath788 where @xmath789 are disjoint sub - disks of the first factor disk . in complex coordinates the plumbing may be written as @xmath790 or @xmath791 creating either a positive or negative ( respectively ) double point on the disk @xmath792 . consider the pair @xmath793 and produce finitely many self - plumbings away from the attaching region @xmath794 to get a kinky handle @xmath795 where @xmath796 denotes the attaching region of the kinky handle . a kinky handle @xmath795 is a one - stage tower @xmath797 and an @xmath798-stage tower @xmath799 is an @xmath21-stage tower union of kinky handles @xmath800 where two towers are attached along @xmath801 . let @xmath802 be @xmath803 and the casson handle @xmath804 is the union of towers ( with direct limit topology induced from the inclusions @xmath805 ) . a casson handle is specified up to ( orientation preserving ) diffeomorphism ( of pairs ) by a labeled finitely - branching tree with base - point * , having all edge paths infinitely extendable away from * . each edge should be given a label @xmath133 or @xmath806 and each vertex corresponds to a kinky handle ; the self - plumbing number of that kinky handle equals the number of branches leaving the vertex . the sign on each branch corresponds to the sign of the associated self plumbing . the whole process generates a tree with infinite many levels . in principle , every tree with a finite number of branches per level realizes a corresponding casson handle . the simplest non - trivial casson handle is represented by the tree @xmath807 : each level has one branching point with positive sign @xmath133 . the reverse construction of a casson handle @xmath808 by using a labeled tree @xmath422 can be found in the appendix a. let @xmath809 and @xmath810 be two trees with @xmath811 ( it is the subtree ) then @xmath812.given a labeled based tree @xmath813 , let us describe a subset @xmath814 of @xmath115 . now we will construct a @xmath815 which is diffeomorphic to the casson handle associated to @xmath813 . in @xmath115 embed a ramified whitehead link with one whitehead link component for every edge labeled by @xmath133 leaving * and one mirror image whitehead link component for every edge labeled by @xmath806(minus ) leaving * . corresponding to each first level node of @xmath813 we have already found a ( normally framed ) solid torus embedded in @xmath816 . in each of these solid tori embed a ramified whitehead link , ramified according to the number of @xmath133 and @xmath806 labeled branches leaving that node . we can do that process for every level of @xmath813 . let the disjoint union of the ( closed ) solid tori in the @xmath21-th family ( one solid torus for each branch at level @xmath21 in @xmath813 ) be denoted by @xmath817 . @xmath813 tells us how to construct an infinite chain of inclusions : @xmath818 and we define the whitehead decomposition @xmath819 of @xmath813 . @xmath820 is the whitehead continuum @xcite for the simplest unbranched tree . we define @xmath814 to be @xmath821 alternatively one can also write @xmath822 where @xmath823 is the cone of a space @xmath824/(x,0)\sim(x',0)\qquad\forall x , x'\in a\ ] ] over the point @xmath825 . as freedman ( see @xcite theorem 2.2 ) showed @xmath814 is diffeomorphic to the casson handle @xmath826 given by the tree @xmath813 . in @xcite thurston constructed a foliation of the 3-sphere @xmath60 which depends on a polygon @xmath342 in the hyperbolic plane @xmath343 so that two foliations are non - cobordant if the corresponding polygons have different areas . for later usage , we will present the main ideas of this construction only ( see also the book @xcite chapter viii for the details ) . starting point is the hyperbolic plane @xmath343 with a convex polygon @xmath827 having @xmath110 sides @xmath828 . assuming the upper half plane model of @xmath343 then the sides are circular arcs . the construction of the foliation depends mainly on the isometry group @xmath255 of @xmath343 realized as rational transformations ( and this group can be lifted to @xmath829 ) . the followings steps are needed in the construction : 1 . the polygon @xmath134 is doubled along one side , say @xmath830 , to get a polygon @xmath831 . the sides are identified by ( isometric ) transformations @xmath832 ( as elements of @xmath829 ) . 2 . take @xmath833-neighborhoods @xmath834 with @xmath835 sufficient small and set @xmath836 having the topology of @xmath837 and we set @xmath838 . 3 . now consider the unit tangent bundle @xmath839 , i.e. a @xmath840bundle over @xmath343 ( or the tangent bundle where every vector has norm one ) . the restricted bundle over @xmath841 is trivial so that @xmath842 . let @xmath843 be circular arcs ( geodesics ) in @xmath343 ( invariant w.r.t . @xmath829 ) starting at a common point which define parallel tangent vectors w.r.t . the metrics of the upper half plane model . the foliation of @xmath841 is given by geodesics transverse to the boundary and we obtain a foliation of @xmath844 ( as unit tangent bundle ) . this foliation is given by a @xmath829-invariant smooth 1-form @xmath218 ( so that @xmath845defines the leaves ) which is integrable @xmath243 . ( @xmath846invariant foliation @xmath847 ) 4 . with the relation @xmath848 , we obtain @xmath849 or the gluing of @xmath850 solid tori to @xmath844 gives a solid tori . every glued solid torus will be foliated by a reeb foliation . finally using @xmath851 ( the heegard decomposition of the 3-sphere ) again with a solid torus with reeb foliation , we obtain a foliation on the 3-sphere . the construction of this foliation @xmath852 ( thurston foliation ) will be also work for any 3-manifold . thurston @xcite obtains for the godbillon - vey number @xmath853 and @xmath854 so that _ any real number can be realized by a suitable foliation of this type_. furthermore , two cobordant foliations have the same godbillon - vey number ( but the reverse is in general wrong ) . let @xmath855\in h^{3}(s^{3},\mathbb{r})$ ] be the dual of the fundamental class @xmath856 $ ] defined by the volume form , then the godbillon - vey class can be represented by @xmath857\label{eq : godbillon - vey - class - thurston - foliation}\ ] ] the godbillon - vey class is an element of the derham cohomology @xmath858 . now we will discuss the general case of a compact 3-manifold carrying a foliation of the same type like the 3-sphere above . the main idea of the construction is very simple and uses a general representation of all compact 3-manifolds by dehn surgery . here we will use an alternative representation of surgery by using the dehn - lickorish theorem ( @xcite corollary 12.4 at page 84 ) . let @xmath859 be a compact 3-manifold without boundary . there is now a natural number @xmath860 so that any orientable 3-manifold can be obtained by cutting out @xmath110 solid tori from the 3-sphere @xmath60 and then pasting them back in , but along different diffeomorphisms of their boundaries . moreover , it can be assumed that all these solid tori in @xmath60 are unknotted . then any 3-manifold @xmath859 can be written as @xmath861 where @xmath862 is the gluing map from each boundary component of @xmath863 to the boundary of @xmath864 . this gluing map is a diffeomorphism of tori @xmath865 ( where @xmath866 ) . the dehn - lickorish theorem describes all diffeomorphisms of a surface : every diffeomorphism of a surface is the composition of dehn twists and coordinate transformations ( or small diffeomorphisms ) . the decomposition @xmath867 of the 3-sphere can be used to get a decomposition of @xmath859 by @xmath868 which will guide us to the construction of a foliation on @xmath859 : * construct a foliation @xmath869 on @xmath844 using a polygon @xmath342 ( see above ) and * glue in @xmath110 reeb foliations of the solid tori using the diffeomorphisms @xmath870 . finally we get a foliation @xmath871 on @xmath859 . according to the rules above , we are able to calculate the godbillon - vey number @xmath872 therefore for any foliation of @xmath60 , we can construct a foliation on any compact 3-manifold @xmath859 with the same godbillon - vey number . both foliations @xmath852 and @xmath871 agree for the common submanifold @xmath844 or there is a foliated cobordism between @xmath873 and @xmath874 . of course , @xmath60 and @xmath859 differ by the gluing of the solid tori but every solid torus carries a reeb foliation which does not contribute to the godbillon - vey number . a connected 3-manifold @xmath336 is prime if it can not be obtained as a connected sum of two manifolds @xmath875 neither of which is the 3-sphere @xmath60 ( or , equivalently , neither of which is the homeomorphic to @xmath336 ) . examples are the 3-torus @xmath876 and @xmath703 but also the poincare sphere . according to @xcite , any compact , oriented 3-manifold is the connected sum of a unique ( up to homeomorphism ) collection of prime 3-manifolds ( prime decomposition ) . a subset of prime manifolds are the irreducible 3-manifolds . a connected 3-manifold is irreducible if every differentiable submanifold @xmath467 homeomorphic to a sphere @xmath212 bounds a subset @xmath201 ( i.e. @xmath877 ) which is homeomorphic to the closed ball @xmath123 . the only prime but reducible 3-manifold is @xmath703 . for the geometric properties ( to meet thurstons geometrization theorem ) we need a finer decomposition induced by incompressible tori . a properly embedded connected surface @xmath878 is called 2-sided then correspond to the components of the complement of @xmath467 in a tubular neighborhood @xmath488\subset n$ ] . ] if its normal bundle is trivial , and 1-sided if its normal bundle is nontrivial . a 2-sided connected surface @xmath467 other than @xmath212 or @xmath125 is called incompressible if for each disk @xmath879 with @xmath880 there is a disk @xmath881 with @xmath882 . the boundary of a 3-manifold is an incompressible surface . most importantly , the 3-sphere @xmath60 , @xmath883 and the 3-manifolds @xmath884 with @xmath885 a finite subgroup do not contain incompressible surfaces . the class of 3-manifolds @xmath884 ( the spherical 3-manifolds ) include cases like the poincare sphere ( @xmath886 the binary icosaeder group ) or lens spaces ( @xmath887 the cyclic group ) . let @xmath888 be irreducible 3-manifolds containing incompressible surfaces then we can @xmath336 split into pieces ( along embedded @xmath212 ) @xmath889 where @xmath890 denotes the @xmath21-fold connected sum and @xmath885 is a finite subgroup . the decomposition of @xmath336 is unique up to the order of the factors . the irreducible 3-manifolds @xmath891 are able to contain incompressible tori and one can split @xmath888 along the tori into simpler pieces @xmath892 @xcite ( called the jsj decomposition ) . the two classes @xmath268 and @xmath187 are the graph manifold @xmath268 and the hyperbolic 3-manifold @xmath187 ( see figure [ fig : torus - decomposition ] ) . the hyperbolic 3-manifold @xmath187 has a torus boundary @xmath895 , i.e. @xmath187 admits a hyperbolic structure in the interior only . in this paper we need the splitting of the link / knot complement . as shown in @xcite , the whitehead double of a knot leads to jsj decomposition of the complement into the knot complement and the complement of the whitehead link ( along one torus boundary of the whitehead link complement ) . one property of hyperbolic 3-manifolds is central : mostow rigidity . as shown by mostow @xcite , every hyperbolic @xmath142manifold @xmath896 with finite volume has this property : _ every diffeomorphism ( especially every conformal transformation ) of a hyperbolic @xmath142manifold with finite volume is induced by an isometry . _ therefore one can not scale a hyperbolic 3-manifold and the volume is a topological invariant . together with the prime and jsj decomposition @xmath897 we can discuss the geometric properties central to thurstons geometrization theorem : _ every oriented closed prime 3-manifold can be cut along tori ( jsj decomposition ) , so that the interior of each of the resulting manifolds has a geometric structure with finite volume . _ now , we have to clarify the term geometric structure s . a model geometry is a simply connected smooth manifold @xmath37 together with a transitive action of a lie group @xmath268 on @xmath37 with compact stabilizers . a geometric structure on a manifold @xmath336 is a diffeomorphism from @xmath336 to @xmath898 for some model geometry @xmath37 , where @xmath899 is a discrete subgroup of @xmath268 acting freely on @xmath37 . t is a surprising fact that there are also a finite number of three - dimensional model geometries , i.e. 8 geometries with the following models : spherical @xmath900 , euclidean @xmath901 , hyperbolic @xmath902 , mixed spherical - euclidean @xmath903 , mixed hyperbolic - euclidean @xmath904 and 3 exceptional cases called @xmath905 ( twisted version of @xmath906 ) , nil ( geometry of the heisenberg group as twisted version of @xmath907 ) , sol ( split extension of @xmath543 by @xmath461 , i.e. the lie algebra of the group of isometries of 2-dimensional minkowski space ) . we refer to @xcite for the details . h. abchir . invariants at infinity for the whitehead manifold . in c. gordan , v.f.r . jones , l. kauffman , s. lambropoulou , and j.h . przytycki , editors , _ knots in hellas 98 _ , pages 117 , singapore , 1998 . world scientific . t. asselmeyer - maluga and j. krl . small exotic smooth @xmath909 and string theory . in _ international congress of mathematicians icm 2010 short communications abstracts book _ , ed . r. bathia , page 400 , hindustan book agency , 2010 . t. asselmeyer - maluga and r. mader . exotic @xmath909 and quantum field theory . in c. burdik et . al . , editor , _ 7th international conference on quantum theory and symmetries ( qts7 ) _ , page 012011 , bristol , uk , 2012 . iop publishing . arxiv:1112.4885 , doi:10.1088/1742 - 6596/343/1/012011 .

over the last two decades , many unexpected relations between exotic smoothness , e.g. exotic @xmath0 , and quantum field theory were found . some of these relations are rooted in a relation to superstring theory and quantum gravity . therefore one would expect that exotic smoothness is directly related to the quantization of general relativity . in this article we will support this conjecture and develop a new approach to quantum gravity called _ smooth quantum gravity _ by using smooth 4-manifolds with an exotic smoothness structure . in particular we discuss the appearance of a wildly embedded 3-manifold which we identify with a quantum state . furthermore , we analyze this quantum state by using foliation theory and relate it to an element in an operator algebra . then we describe a set of geometric , non - commutative operators , the skein algebra , which can be used to determine the geometry of a 3-manifold . this operator algebra can be understood as a deformation quantization of the classical poisson algebra of observables given by holonomies . the structure of this operator algebra induces an action by using the quantized calculus of connes . the scaling behavior of this action is analyzed to obtain the classical theory of general relativity ( grt ) for large scales . this approach has some obvious properties : there are non - linear gravitons , a connection to lattice gauge field theory and a dimensional reduction from 4d to 2d . some cosmological consequences like the appearance of an inflationary phase are also discussed . at the end we will get the simple picture that the change from the standard @xmath0 to the exotic @xmath1 is a quantization of geometry . _ on the occasion of the 80-th birthday of carl h. brans _

You are an expert at summarizing long articles. Proceed to summarize the following text: nonleptonic @xmath7-meson decays are of crucial importance to deepen our insights into the flavor structure of the standard model ( sm ) , the origin of cp violation , and the dynamics of hadronic decays , as well as to search for any signals of new physics beyond the sm . however , due to the non - perturbative strong interactions involved in these decays , the task is hampered by the computation of matrix elements between the initial and the final hadron states . in order to deal with these complicated matrix elements reliably , several novel methods based on the naive factorization approach ( fa ) @xcite , such as the qcd factorization approach ( qcdf ) @xcite , the perturbation qcd method ( pqcd ) @xcite , and the soft - collinear effective theory ( scet ) @xcite , have been developed in the past few years . these methods have been used widely to analyze the hadronic @xmath7-meson decays , while they have very different understandings for the mechanism of those decays , especially for the case of heavy - light final states , such as the @xmath0 decays . presently , all these methods can give good predictions for the color allowed @xmath8 mode , but for the color suppressed @xmath9 mode , the qcdf and the scet methods could not work well , and the pqcd approach seems leading to a reasonable result in comparison with the experimental data . in this situation , it is interesting to study various approaches and find out a reliable approach . as the mesons are regarded as quark and anti - quark bound states , the nonleptonic two body meson decays concern three quark - antiquark pairs . it is then natural to investigate the nonleptonic two body meson decays within the qcd framework by considering all feynman diagrams which lead to three effective currents of two quarks . in our considerations , beyond these sophisticated pqcd , qcdf and scet , we shall try to find out another simple reliable qcd approach to understand the nonleptonic two body decays . in this note , we are focusing on evaluating the @xmath0 decays . the paper is organized as follows . in sect . ii , we first analyze the relevant feynman diagrams and then outline the necessary ingredients for evaluating the branching ratios and @xmath3 asymmetries of @xmath10 decays . in sect . iii , we list amplitudes of @xmath0 decays . the approaches for dealing with the physical - region singularities of gluon and quark propagators are given in sect . finally , we discuss the branching ratios and the @xmath3 asymmetries for those decay modes and give conclusions in sects . v and vi , respectively . the detail calculations of amplitudes for these decay modes are given in the appendix . we start from the four - quark effective operators in the effective weak hamiltonian , and then calculate all the feynman diagrams which lead to effective six - quark interactions . the effective hamiltonian for @xmath12 decays can be expressed as @xmath13+{\rm h.c.},\ ] ] where @xmath14 and @xmath15 are the wilson coefficients which have been evaluated at next - to - leading order @xcite , @xmath16 and @xmath17 are the tree operators arising from the @xmath18-boson exchanges with @xmath19 where @xmath20 and @xmath21 are the @xmath22 color indices . based on the effective hamiltonian in eq . ( [ heff ] ) , we can then calculate the decay amplitudes for @xmath23 , @xmath24 , and @xmath25 decays , which are the color - allowed , the color - suppressed , and the color - allowed plus color - suppressed modes , respectively . all the six - quark feynman diagrams that contribute to @xmath26 and @xmath27 decays are shown in figs . [ tree]-[annihilation ] via one gluon exchange . as for the process @xmath28 , it does nt involve the annihilation diagrams and the related feynman diagrams are the sum of figs . [ tree ] and [ tree2 ] . based on the isospin symmetry argument , the decay amplitude of this mode can be written as @xmath29 . the explicit expressions for the amplitudes of these decay modes are given in detail in next section . the decay amplitudes of @xmath11 decay modes are quite different . for the color - allowed @xmath8 mode , it is expected that the decay amplitude is dominated by the factorizable contribution @xmath30 ( from the diagrams ( a ) and ( b ) in fig . [ tree ] ) , while the nonfactorizable contribution @xmath31 ( from the diagrams ( c ) and ( d ) in fig . [ tree ] ) has only a marginal impact . this is due to the fact that the former is proportional to the large coefficient @xmath32 , while the latter is proportional to the quite small coefficient @xmath33 . in addition , there is an addition color - suppressed factor @xmath34 in the nonfactorizable contribution @xmath31 . in contrast with the @xmath8 mode , the nonfactorizable contribution @xmath31 ( from ( c ) and ( d ) diagrams in fig . [ tree2 ] ) in the @xmath9 mode is proportional to the large coefficient @xmath32 , and even if with an additional color - suppressed factor @xmath34 , its contribution is still larger than the factorizable one @xmath30 ( from ( a ) and ( b ) diagrams in fig . [ tree2 ] ) which is proportional to the quite small coefficient @xmath33 . thus , it is predicted that the decay amplitude of this mode is dominated by the nonfactorizable contribution @xmath31 . as for the @xmath35 mode , since its amplitude can be written as the sum of the ones of the above two modes , it is not easy to see which one should dominate the total amplitude . the branching ratio for @xmath0 decays can be expressed as follows in terms of the total decay amplitudes @xmath36 where @xmath37 is the lifetime of the @xmath7 meson , and @xmath38 is the magnitude of the momentum of the final - state particles @xmath39 and @xmath40 in the @xmath7-meson rest frame and given by @xmath41\ , \left[m_b^2-(m_{d}-m_{\pi})^2\,\right]}\,.\end{aligned}\ ] ] as is well - known , the direct @xmath3 violation in meson decays is non - zero only if there are two contributing amplitudes with non - zero relative weak and strong phases . the weak - phase difference usually arises from the interference between two different topological diagrams . for three @xmath0 decays , it is seen from the feynman diagrams in figs . [ tree]-[annihilation ] that there are no weak - phase differences , and hence no direct @xmath3 violation in all these three modes , we shall then consider the mixing - induced @xmath3 violation . as the final states @xmath42 can be produced both in the decays of @xmath43 meson via the cabibbo - favored ( @xmath44 ) and in the decays of @xmath45 meson via the doubly cabibbo - suppressed ( @xmath46 ) tree amplitudes . the relative weak - phase difference between these two amplitudes is @xmath47 and , when combining with the @xmath48 mixing phase , the total weak - phase difference is @xmath49 to all orders in the small ckm parameter @xmath50 . thus , the @xmath51 decays can in principle be used to measure the weak phase @xmath52 , since the weak phase @xmath53 has been measured with high precision . the time - dependent @xmath3 asymmetry of such decay modes is defined as : @xmath54 where @xmath55 is the mass difference of the two eigenstates of @xmath56 mesons , and @xmath57 and @xmath58 are given as @xmath59 with @xmath60 where the rephase - invariant quantities @xmath61 , @xmath62 and @xmath63 @xcite characterize the indirect , direct and mixing - induced cp violations respectively . as @xmath64 for neutral @xmath7 system , we have @xmath65 which characterizes direct cp violation . defining @xmath66 and @xmath67 as the amplitudes of @xmath68 and @xmath69 decay modes , respectively , we can further express these two cp asymmetries as @xmath70 where @xmath71 , and @xmath72 represents the relative strong - phase difference between the two amplitudes @xmath66 and @xmath67 . similarly , we can define another two @xmath3-violating parameters @xmath73 and @xmath74 for the @xmath75 decays @xmath76 with the parameter @xmath77 defined as @xmath78 are the charge conjugations of the amplitudes @xmath67 and @xmath66 . since the magnitude of the cabibbo - suppressed decay amplitude @xmath79 is much smaller than that of the cabibbo - favored decay amplitude @xmath80 , the ratio @xmath81 should be quite small and is found to be about @xmath82 in our framework . thus , to a very good approximation , @xmath83 , and the coefficients of the sine terms are given by @xmath84 to compare with the current experimental data , one usually define the following two quantities , which are given by the combination of two @xmath3-violating parameters @xmath57 and @xmath74 , @xmath85 which can provide constraints on the weak phase @xmath6 and the strong phase @xmath86 . using the methods given in the appendix , we can get the @xmath0 decay amplitudes , which are composed of three parts : the factorizable contribution @xmath30 , the nonfactorizable contribution @xmath31 , and the annihilation contribution @xmath87 . the amplitude of @xmath8 mode is found to be @xmath88 with @xmath89 where @xmath90 , @xmath91 and @xmath92 . @xmath93 are the wave functions of mesons . for the @xmath7-meson wave function , we shall take the form given in @xcite @xmath94,\ ] ] with @xmath95 , and @xmath96 being a normalization constant . the @xmath39 meson distribution amplitude is given by @xmath97,\ ] ] with the shape parameter @xmath98 . for the @xmath40 meson light cone wave functions , we use the asymptotic form as given in refs . @xcite : @xmath99 with @xmath100 . @xmath101 where @xmath102 and @xmath103 . @xmath104\nonumber\\ & + & ( c_1+\frac{c_2}{n_c})\phi_b(x)\biggl[\biggl(\big((\bar x - y)m_b+m_b \big)m^2_b\phi(z)+ \mu_\pi m_d \big((2x-\bar y - z)m_b \nonumber\\&-&4m_b \big)\phi_\pi(z ) + \mu_\pi m_dm_b(y-\bar z)\frac{\phi_\sigma'(z)}{6}\biggl ) \frac{m_b}{d_{ba } k_a^2}\nonumber\\ & + & \biggl((x-\bar z)m^2_b\phi(z)-\mu_\pi m_d\big((y-\bar z)\frac{\phi_\sigma'(z)}{6}+ ( 2x - y-\bar z)\phi_\pi(z)\big)\biggl)\frac{m^2_b}{d_{da } k_a^2}\biggl]\biggl\},\label{anni1}\end{aligned}\ ] ] where @xmath105 , @xmath106 , @xmath107 , @xmath108 , @xmath109 and @xmath110 . the annihilation contribution is found to be much smaller than the ones from the factorizable and the nonfactorizable diagrams . numerically , it is negligible . for the color - suppressed @xmath111 decay , its amplitude can be written as @xmath112 with @xmath113 , \label{ckms1}\end{aligned}\ ] ] here @xmath114 , @xmath115 and @xmath116 . @xmath117\,,\label{ckms2}\end{aligned}\ ] ] where @xmath118 and @xmath103 . for the annihilation amplitude @xmath87 , it is the same as the one in eq . ( [ anni1 ] ) since the two modes @xmath119 and @xmath42 have the same annihilation topological diagrams . for the doubly cabibbo - suppressed decay mode @xmath120 , its decay amplitude can be written as @xmath121 here , @xmath122 and @xmath87 can be obtained from the ones of decay mode @xmath123 by simply exchanging the wilson coefficients @xmath14 and @xmath15 . for the @xmath124 decay , its amplitude can be yielded by using the isospin relation @xmath125 . to perform a numerical calculation of the decay amplitudes of @xmath126 decays , the light - cone projectors of mesons are found to be very useful , and the details of these quantities are presented in the appendix . where one encounters the endpoint divergences stemming from the convolution integrals of the meson distribution amplitudes with the hard kernels , which is caused by the collinear approximation . to regulate such an infrared divergence , we may introduce an intrinsic mass scale realized in the symmetry - preserving loop regularization@xcite . at the tree level , it is equivalent to adopt an effective dynamical gluon mass in the propagator . practically , such a gluon mass scale has been used to regulate the infrared divergences in the soft endpoint region @xcite @xmath127^{-\frac{12}{11}},\ ] ] the use of this effective gluon propagator is supported by the lattice @xcite and the field theoretical studies @xcite , which have shown that the gluon propagator is not divergent as fast as @xmath128 . taking the hadronic scale @xmath129 , the dynamical gluon mass scale can be determined from one of the well measured decay mode . numerically , we will see that taking @xmath130 mev , the dynamical gluon mass scale is around @xmath131 . another physical - region singularity arises from the on mass - shell quark propagators . it can be easily checked that each feynman diagram contributing to a given matrix element is entirely real unless some denominators vanish with a physical - region singularity , so that the @xmath132 prescription for treating the poles becomes relevant . in other words , a feynman diagram will yield an imaginary part for the decay amplitudes only when the virtual particles in the diagram become on mass - shell , thus the diagram may be considered as a genuine physical process . the cutkosky rules @xcite give a compact expression for the discontinuity across the cut arising from a physical - region singularity . when applying the cutkosky rules to deal with a physical - region singularity of quark propagators , the following formula holds @xmath133-i\pi\delta[(k_1-k_2-k_3)^2],\label{quarkd}\\ \frac{1}{(p_b - k_2-k_3)^2-m_b^2+i\epsilon}&=&p\biggl[\frac{1 } { ( p_b - k_2-k_3)^2-m_b^2}\biggl]-i\pi\delta[(p_b - k_2-k_3)^2-m_b^2 ] , \label{quarkb}\end{aligned}\ ] ] where @xmath134 denotes the principle - value prescription . the role of the @xmath86 function is to put the particles corresponding to the intermediate state on their positive energy mass - shell , so that in the physical region , the individual feynman diagram satisfies the unitarity condition . equations ( [ quarkd ] ) and ( [ quarkb ] ) will be applied to the quark propagators @xmath135 and @xmath136 in equation ( [ anni1 ] ) , respectively . it is then seen that the possible large imaginary parts arise from the virtual quarks across their mass shells as physical - region singularities . it is seen that for theoretical predictions it depends on many input parameters , such as the wilson coefficient functions , the ckm matrix elements , the hadronic parameters , and so on . to carry out a numerical calculation , we take the following input parameters @xcite @xmath137 the wolfenstein parameters of the ckm matrix elements are taken as @xcite : @xmath138 , with @xmath139 . the coefficient of the twist-3 distribution amplitude of the pseudoscalar @xmath40 meson is chosen as @xmath140 @xcite . with the above values for the input parameters , we are able to calculate the contributions of different amplitudes for each decay mode . our final results at @xmath141 scale are presented in table [ amplitude ] . 0.8pt 0.15 in .[amplitude ] numerical results at @xmath141 scale of the amplitudes for different diagrams in @xmath11 decays . amplitudes @xmath122 , and @xmath87 represent the factorizable ( ( a)and ( b ) diagrams in figs . [ tree ] or [ tree2 ] ) , the non - factorizable ( ( c ) and ( d ) diagrams in figs . [ tree ] or [ tree2 ] ) , and the annihilation ( diagrams in fig . [ annihilation ] ) contributions , respectively . [ cols="<,^,^,^,^",options="header " , ] in summary , we have calculated the decay amplitudes , strong phases , branching ratios , and @xmath3 asymmetries for the @xmath0 decays , including both the color - allowed and the color - suppressed modes . it has been shown that these decay modes are theoretically clean as there are no penguin contributions . as a consequence , direct @xmath3 violations are absent . the contributions from the factorizable diagrams dominate all the decay amplitudes except for the @xmath142 process . all our predictions for branching ratios are consistent with the existing measurements . for the @xmath143 mode , our predictions will be faced with the future experiments as no data are available at present . due to small interference effects between the cabibbo - suppressed and the cabibbo - favored amplitudes , the non - zero @xmath3-violating parameters @xmath57 and @xmath74 have been predicted in the @xmath144 decay modes . it has been shown that the @xmath3-violating parameters have a strong dependence on the weak phase @xmath6 , but they are not sensitive to the dynamical gluon mass scale . with the angle @xmath6 varying within the range @xmath145 , almost all of the values for the cp - violating parameters @xmath146 and @xmath147 are within the range of the current experimental data . thus no constraints on the weak phase @xmath6 could be obtained through those parameters based on the current experiment data , and more precise measurements are needed in future experiments . in this paper , we have further shown that the divergence treatments used in our previous work @xcite are reliable . namely , the endpoint divergence caused by the soft collinear approximation in gluon propagator could be simply avoided by adopting the cornwall prescription for the gluon propagator with a dynamical mass scale . note that when the intrinsic mass is appropriately introduced , it may not spoil the gauge symmetry as shown recently in the symmetry - preserving loop regularization @xcite . meanwhile , for the physical - region singularity of the on - mass - shell quark propagators , it can well be treated by applying for the cutkosky rules . the combination of these two treatments for the endpoint divergences of gluon propagator and the physical - region singularity of the quark propagators enables us to obtain reasonable results , which are consistent with the existing experimental data and also in agreement with the ones @xcite obtained by using the pqcd approach . however , this is different from the treatment of the latter , where @xmath148 and sudakov factors have been used to avoid the endpoint divergence . it is noted that the resulting predictions for the branching ratios are in general scale dependence on the dynamical gluon mass which plays the role of the ir cut - off . this dependence should in principle be compensated from the possible scale in the wave functions which characterizes the nonperturbative effects . in our approach , the dynamical gluon mass may be regarded as a universal scale to be fixed from one of the decay modes . for instance , in our present considerations , if the decay mode @xmath149 is taken to extract the dynamical gluon mass scale , we have @xmath150 mev , and the resulting predictions for other decay modes can serve as a consistent check . within the current experimental errors and theoretical uncertainties for some relevant parameters , it is seen that our treatment is reliable . in order to further check the validity of the gluon - mass regulator method adopted to deal with the endpoint divergence , it is useful to extend this method to more decay modes . anyway , the treatments presented in this paper may enhance its predictive power for analyzing non - leptonic @xmath7-meson decays . this work was supported in part by the national science foundation of china ( nsfc ) under the grant 10475105 , 10491306 , 10675039 and the project of knowledge innovation program ( pkip ) of chinese academy of sciences . to evaluate the hadronic matrix elements of @xmath0 decays , the meson light - cone distribution amplitudes play an important role . in the heavy quark limit , the light - cone projectors for @xmath7 , @xmath39 and @xmath40 mesons in momentum space can be expressed , respectively , as @xcite @xmath151_{\alpha\beta } , \nonumber\\ { \cal m}_{\alpha\beta}^d & = & \frac{i f_d\,}{4}\ , [ ( { \not\ ! p_2 \,}+ m_d ) \,\gamma_5 \,\phi_d(y)]_{\alpha\beta } , \nonumber\\ { \cal m}_{\delta\alpha}^\pi & = & \frac{i f_p}{4}\,\biggl\{{\not\ ! p_3 \,}\gamma_5 \,\phi(u)-\mu_p\gamma_5 \biggl(\phi_p(u)-i\sigma_{\mu \nu}n_-^\mu v^\nu \frac{\phi^{\prime } _ \sigma ( u)}{6 } + i \sigma _ { \mu \nu } p_3^{\mu } \frac{\phi_\sigma(u)}{6 } \frac{\partial}{\partial k_{\bot\nu}}\biggl)\biggl\}_{\delta\alpha},\label{projector } \end{aligned}\ ] ] from the feynman diagrams shown in figs . [ tree]-[annihilation ] , we can get the amplitudes for each decay mode using the relevant feynman rules and the light - cone projectors listed in eqs . ( [ projector ] ) . for the tree diagrams of @xmath8 mode shown in fig . [ tree ] , the amplitudes of each diagrams can be written as @xmath152\frac{-ig_{\alpha\beta}\delta_{ab}}{k^2},\nonumber\\ & = & -i f_\pi g^2_s\frac{c_f}{n_c}\frac{1}{d_b k^2}\mathrm{tr}\big[{\cal m}^b\gamma^\alpha{\cal m}^d{\not\ ! p_3 \ , } ( 1-\gamma_5)({\not\ ! k_b \,}+m_b)\gamma_\alpha\big]\nonumber\\ a^{\ref{tree}b}&=&if_\pi p_3^\mu\mathrm{tr}\big[{\cal m}^b(-ig_s\gamma^\alpha t^a_{ij}){\cal m}^d(-ig_s\gamma^\beta t^b_{kl})\frac{i}{{\not\ ! k_c \,}-m_c}\gamma_\mu ( 1-\gamma_5)\big]\frac{-ig_{\alpha\beta}\delta_{ab}}{k^2},\nonumber\\ & = & -if_\pi g^2_s\frac{c_f}{n_c}\frac{1}{d_c k^2}\mathrm{tr}\big[{\cal m}^b\gamma^\alpha{\cal m}^d\gamma_\alpha({\not\ ! k_c \,}+m_c){\not\ ! p_3 \ , } ( 1-\gamma_5)\big]\nonumber\\ a^{\ref{tree}c}&=&\mathrm{tr}\big[{\cal m}^\pi(-ig_s\gamma^\alpha t^a_{ij})\frac{i}{{\not\ ! k_d \,}}\gamma^\mu ( 1-\gamma_5)\big]\mathrm{tr}\big[{\cal m}^b(-ig_s\gamma^\beta t^b_{kl}){\cal m}^d\gamma_\mu ( 1-\gamma_5)\big]\frac{-ig_{\alpha\beta}\delta_{ab}}{k^2},\nonumber\\ & = & -g^2_s\frac{c_f}{n_c}\frac{1}{d_d k^2}\mathrm{tr}\big[{\cal m}^\pi\gamma^\alpha{\not\ ! k_d \,}\gamma^\mu ( 1-\gamma_5)\big]\mathrm{tr}\big[{\cal m}^b\gamma_\alpha{\cal m}^d\gamma_\mu ( 1-\gamma_5)\big]\nonumber\\ a^{\ref{tree}d}&=&\mathrm{tr}\big[{\cal m}^\pi\gamma^\mu ( 1-\gamma_5)\frac{i}{{\not\ ! k_u \,}}(-ig_s\gamma^\alpha t^a_{ij})\big]\mathrm{tr}\big[{\cal m}^b(-ig_s\gamma^\beta t^b_{kl}){\cal m}^d\gamma_\mu ( 1-\gamma_5)\big]\frac{-ig_{\alpha\beta}\delta_{ab}}{k^2}\nonumber\\ & = & -g^2_s\frac{c_f}{n_c}\frac{1}{d_u k^2}\mathrm{tr}\big[{\cal m}^\pi\gamma^\mu ( 1-\gamma_5){\not\ ! k_u \,}\gamma^\alpha \big]\mathrm{tr}\big[{\cal m}^b\gamma_\alpha{\cal m}^d\gamma_\mu ( 1-\gamma_5)\big],\label{amplitude1}\end{aligned}\ ] ] where @xmath153 stands for the @xmath21th(@xmath154 ) diagrams in fig.[tree ] , @xmath155 and @xmath156 the momentum of @xmath157 quark propagator and gluon propagator , respectively . furthermore , @xmath158 and @xmath159 represent for the @xmath157 quark propagator and gluon propagator , respectively . in fig . [ tree](a ) , the @xmath40 meson can be written as a decay constant since it originates from the vacuum . inversing the fermi lines and writing down the @xmath7 meson projector @xmath160 , gluon vertex @xmath161 , @xmath39 meson projector @xmath162 , the four quark vertex @xmath163 , b quark propagator @xmath164 and another gluon vertex @xmath165 in a trace one by one , and finally the gluon propagator @xmath166 , we can get the amplitude @xmath167 . @xmath168 can be calculated in a similar way . in fig . [ tree](c ) , the @xmath40 meson can no longer be written as a decay constant any more since it exchanges a gluon with the spectator quark . writing down the @xmath40 meson projector @xmath169 , gluon vertex @xmath170 , @xmath171 quark propagator @xmath172 and the four quark vertex @xmath173 in turn in one trace , and writing down the @xmath7 meson projector @xmath160 , gluon vertex @xmath165 , @xmath39 meson projector @xmath174 and the four quark vertex @xmath163 in the other trace one by one , and finally the gluon propagator @xmath166 , we can get the amplitude @xmath175 . similarly , we can get the amplitude @xmath176 . summing up the former and the latter two quantities in eq . ( [ amplitude1 ] ) , we can get the factorizable part @xmath30 ( eq . ( [ favor2 ] ) ) and the nonfactorizable @xmath31 ( eq . ( [ favor3 ] ) ) , respectively . as for the annihilation diagrams for @xmath177 in fig . [ annihilation ] , the amplitudes can be written as @xmath178\frac{-ig_{\alpha\beta}\delta_{ab } } { k^2_a},\nonumber\\ & = & -i f_b g^2_s\frac{c_f}{n_c}\frac{1}{d_{ca } k^2_a}\mathrm{tr}\big[{\cal m}^d\gamma^\alpha ( { \not\ ! k_{ca } \,}+m_c){\not\ ! p_1 \,}(1-\gamma_5){\cal m}^\pi\gamma_\alpha\big]\nonumber\\ a^{\ref{annihilation}b}&=&if_b p_1^\mu \mathrm{tr}\big[{\cal m}^d\gamma_\mu ( 1-\gamma_5)\frac{i}{{\not\ ! k_{ua } \,}}(-ig_s\gamma^\alpha t^a_{ij}){\cal m}^\pi(-ig_s\gamma^\beta t^b_{kl})\big]\frac{-ig_{\alpha\beta}\delta_{ab}}{k^2_a},\nonumber\\ & = & -i f_b g^2_s\frac{c_f}{n_c}\frac{1}{d_{ua } k^2_a}\mathrm{tr}\big[{\cal m}^d{\not\ ! p_1 \ , } ( 1-\gamma_5){\not\ ! k_{ua } \,}\gamma^\alpha{\cal m}^\pi\gamma_\alpha\big],\nonumber\\ a^{\ref{annihilation}c}&=&\mathrm{tr}\big[{\cal m}^b(-ig_s\gamma^\alpha t^a_{ij})\frac{i}{{\not\ ! k_{da } \,}}\gamma^\mu ( 1-\gamma_5)\big]\mathrm{tr}\big[{\cal m}^d\gamma_\mu ( 1-\gamma_5){\cal m}^\pi(-ig_s\gamma^\beta t^b_{kl})\big]\frac{-ig_{\alpha\beta}\delta_{ab}}{k^2_a}\nonumber\\ & = & - g^2_s\frac{c_f}{n_c}\frac{1}{d_{da } k^2_a}\mathrm{tr}\big[{\cal m}^b\gamma^\alpha { \not\ ! k_{da } \,}\gamma^\mu ( 1-\gamma_5)\big]\mathrm{tr}\big[{\cal m}^d\gamma_\mu ( 1-\gamma_5){\cal m}^\pi\gamma_\alpha \big],\nonumber\\ a^{\ref{annihilation}d}&=&\mathrm{tr}\big[{\cal m}^b\gamma^\mu ( 1-\gamma_5)\frac{i}{{\not\ ! k_{ba}-m_b \,}}(-ig_s\gamma^\alpha t^a_{ij})\big]\mathrm{tr}\big[{\cal m}^d\gamma_\mu ( 1-\gamma_5){\cal m}^\pi(-ig_s\gamma^\beta t^b_{kl})\big]\frac{-ig_{\alpha\beta}\delta_{ab } } { k^2_a},\nonumber\\ & = & - g^2_s\frac{c_f}{n_c}\frac{1}{d_{ba } k^2_a}\mathrm{tr}\big[{\cal m}^b\gamma^\mu ( 1-\gamma_5)({\not\ ! k_{ba } \,}+m_b)\gamma^\alpha \big]\mathrm{tr}\big[{\cal m}^d\gamma_\mu ( 1-\gamma_5){\cal m}^\pi\gamma_\alpha \big],\label{amplitude1a}\end{aligned}\ ] ] where @xmath179 and @xmath180 stand for the momentum of @xmath157 quark propagator and gluon propagator , and @xmath181 and @xmath159 represent for the @xmath157 quark propagator and gluon propagator in these annihilation diagrams , respectively . summing up the four quantities in eq . ( [ amplitude1a ] ) , we can get the annihilation contribution @xmath87 ( eq . ( [ anni1 ] ) ) of this decay mode . similarly , as for the tree diagrams of @xmath111 decay mode in fig [ tree2 ] , its amplitudes can be written as follows @xmath182\frac{-ig_{\alpha\beta}\delta_{ab}}{k^2},\nonumber\\ & = & -if_d g^2_s\frac{c_f}{n_c}\frac{1}{d_{b } k^2}\mathrm{tr}\big[{\cal m}^b\gamma^\alpha{\cal m}^\pi{\not\ ! p_2 \ , } ( 1-\gamma_5)({\not\ ! k_b \,}+m_b)\gamma_\alpha\big],\nonumber\\ a^{\ref{tree2}b}&=&if_d p_2^\mu \mathrm{tr}\big[{\cal m}^b(-ig_s\gamma^\alpha t^a_{ij}){\cal m}^\pi(-ig_s\gamma^\beta t^b_{kl})\frac{i}{{\not\ ! k_d \,}}\gamma_\mu ( 1-\gamma_5)\big]\frac{-ig_{\alpha\beta}\delta_{ab}}{k^2},\nonumber\\ & = & -if_d g^2_s\frac{c_f}{n_c}\frac{1}{d_{d } k^2}\mathrm{tr}\big[{\cal m}^b\gamma^\alpha{\cal m}^\pi\gamma_\alpha { \not\ ! k_d \,}{\not\ ! p_2 \,}(1-\gamma_5)\big],\nonumber\\ a^{\ref{tree2}c}&=&\mathrm{tr}\big[{\cal m}^d(-ig_s\gamma^\alpha t^a_{ij})\frac{i}{({\not\ ! k_c \,}-m_c)}\gamma^\mu ( 1-\gamma_5)\big]\mathrm{tr}\big[{\cal m}^b(-ig_s\gamma^\beta t^b_{kl}){\cal m}^\pi\gamma_\mu ( 1-\gamma_5)\big ] \frac{-ig_{\alpha\beta}\delta_{ab}}{k^2},\nonumber\\ & = & -ig^2_s\frac{c_f}{n_c}\frac{1}{d_{c } k^2}\mathrm{tr}\big[{\cal m}^d\gamma^\alpha({\not\ ! k_c \,}+m_c)\gamma^\mu ( 1-\gamma_5)\big]\mathrm{tr}\big[{\cal m}^b\gamma_\alpha{\cal m}^\pi\gamma_\mu ( 1-\gamma_5)\big],\nonumber\\ a^{\ref{tree2}d}&=&\mathrm{tr}\big[{\cal m}^d\gamma^\mu ( 1-\gamma_5)\frac{i}{{\not\ ! k_u \,}}(-ig_s\gamma^\alpha t^a_{ij})\big]\mathrm{tr}\big[{\cal m}^b(-ig_s\gamma^\beta t^b_{kl}){\cal m}^\pi\gamma_\mu ( 1-\gamma_5)\big]\frac{-ig_{\alpha\beta}\delta_{ab}}{k^2}\nonumber\\ & = & -ig^2_s\frac{c_f}{n_c}\frac{1}{d_{u } k^2}\mathrm{tr}\big[{\cal m}^d\gamma^\mu ( 1-\gamma_5){\not\ ! k_u \,}\gamma^\alpha \big]\mathrm{tr}\big[{\cal m}^b\gamma_\alpha{\cal m}^\pi\gamma_\mu ( 1-\gamma_5)\big].\label{amplitude2}\end{aligned}\ ] ] we can get the factorizable contribution @xmath30 ( eq . ( [ ckms1 ] ) ) and the nonfactorizable part @xmath31 ( eq . ( [ ckms2 ] ) ) by summing up the formerand the latter two quantities in eq . ( [ amplitude2 ] ) . as for the annihilation diagrams for @xmath9 decay , its amplitude is the same as the one in eq . ( [ amplitude1a ] ) since the two modes @xmath119 and @xmath42 have the same annihilation topological diagrams , which are shown in fig . 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the branching ratios and cp violations of the @xmath0 decays , including both the color - allowed and the color - suppressed modes , are investigated in detail within qcd framework by considering all diagrams which lead to three effective currents of two quarks . an intrinsic mass scale as a dynamical gluon mass is introduced to treat the infrared divergence caused by the soft collinear approximation in the endpoint regions , and the cutkosky rule is adopted to deal with a physical - region singularity of the on mass - shell quark propagators . when the dynamical gluon mass @xmath1 is regarded as a universal scale , it is extracted to be around @xmath2 mev from one of the well - measured @xmath0 decay modes . the resulting predictions for all branching ratios are in agreement with the current experimental measurements . as these decays have no penguin contributions , there are no direct @xmath3 asymmetries . due to interference between the cabibbo - suppressed and the cabibbo - favored amplitudes , mixing - induced @xmath3 violations are predicted in the @xmath4 decays to be consistent with the experimental data at 1-@xmath5 level . more precise measurements will be helpful to extract weak angle @xmath6 .

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Proceed to summarize the following text: everyday experience tells us that thin elastic sheets _ crumple _ , when confined into a small volume , _ e.g. _ a sheet of paper `` confined '' by ones hands . crumpling also plays an important role in the mechanical behavior of packaging material and in the dissipation of the energy of collisions by the `` crumple zones '' of automobiles . crumpling is therefore a problem of much intrinsic interest , but understanding this behavior is complicated by the complex morphology of a typical crumpled sheet . despite the complicated appearance of a crumpled sheet , the crumpling phenomenon is in itself very robust . it is easily observed in thin sheets made from a variety of materials , suggesting that it can be studied using simplified or idealized models that capture the essential features of thin elastic sheets . this approach leads one to consider a crumpled sheet as a minimum energy configuration for a simple elastic energy functional for thin sheets , _ viz . _ the fpl von karman ( fvk ) energy . using this approach of elastic energy minimization , the crumpling response is now understood as a result of the elastic energy of the sheet concentrating on a small subset of the entire sheet @xcite . the energy in a crumpled sheet is concentrated on a network of thin line - like creases ( ridges ) that meet in point - like vertices . recent work has resulted in quantitative understanding of both the vertices @xcite and the ridges @xcite . scaling laws governing the behavior of crumpled sheets have been obtained in the physics literature @xcite . minimum energy configurations for the fvk energy have also been studied in the context of the _ blistering problem _ , _ viz . _ the buckling of membranes as a result of isotropic compression along the boundary . the blistering problem is relevant to the delamination of thin films that are chemically deposited at high temperatures , as well as the mechanical behavior of micro - fabricated thin - film diaphragms @xcite . there is a considerable body of mathematical work focused on the blistering problem @xcite . upper and lower bounds have been obtained for approximations to the elastic energy @xcite , for the fvk energy @xcite and for full three dimensional nonlinear elasticity @xcite . the fvk energy and full three dimensional nonlinear elasticity give the same scaling for the upper and the lower bounds . this yields a rigorous scaling law for the energy of a blister . as we discuss below , the scaling laws for the blistering problem are different from those for the crumpling problem , even for the _ same energy functional_. this indicates that the energy minimizing configurations for the fvk energy show an interesting dependence on the boundary conditions . the minimization of the fvk elastic energy is an example of a non - convex variational problem that is regularized by a singular perturbation @xcite . it is well known that this can lead to a variety of multiple - scale behaviors including energy concentration and/or small scale oscillations in the minimizers @xcite . multiple scale behaviors , both _ microstructure _ and _ singularities _ , are ubiquitous in condensed matter systems . the crumpling phenomenon appears to be a particularly simple and tractable instance of multiple scale behavior . for this reason , there has been much recent interest in physics literature about the nature of the crumpling phenomenon @xcite . asymptotic analysis @xcite and scaling arguments @xcite show that the ridge energy scales as @xmath3 where @xmath1 is the thickness of the sheet , and @xmath4 is the length of the ridge . ben - amar @xcite and mahadevan @xcite show that the energy of a @xmath5-cone scales as @xmath6 where @xmath7 is the radius of the core associated with the vertex . it is clear that the ridge energy is asymptotically larger than the vertex energy as @xmath8 . however , which of these two energies is important for a given sheet depends on the relation between the nondimensional thickness @xmath9 of the sheet , and the `` crossover thickness '' @xmath10 which is determined by setting @xmath11 . @xmath12 in turn depends on the values of the multiplicative constants for these scaling laws . these multiplicative constants can not be determined by scaling arguments . mahadevan @xcite estimate the constant for the vertex energy numerically using an ansatz for the shape of a sheet near the vertex , and they find that @xmath13 where @xmath14 is the complement of the tip angle of the cone @xcite . based on lobkovsky s work @xcite , boudaoud @xcite estimate the value of the constant in the ridge energy as @xmath15 . this implies , for the experiments in refs . @xcite and @xcite , the vertex energy dominates the ridge energy . it would be useful to prove these scaling laws , and determine the multiplicative constants rigorously , that is , in an ansatz - free manner . in this paper , we propose a model problem that yields structures analogous to a single ridge in a crumpled sheet . we prove a rigorous lower bound ( with a numerical value for the multiplicative constant ) for the elastic energy in our model problem . this is a step toward rigorously proving the scaling law of lobkovsky for the ridge energy @xcite . we also discuss how the techniques in this paper can be extended to prove similar results for a `` real '' crumpled sheet , as opposed to our model problem . this paper is organized as follows in sec . [ sec : setup ] , we describe the problem of interest , set up the relevant energy functional and determine the appropriate boundary conditions . we also rescale the various quantities to a form that is suitable for further analysis , and recast the problem in terms of the rescaled quantities . in sec . [ sec : l_bnd ] , we prove our main result , _ viz . _ a lower bound for the elastic energy for our boundary conditions . we present a concluding discussion in sec . [ sec : discussion ] . we are interested in a _ minimal ridge _ , _ i.e. _ , the single crease that is formed when a long rectangular elastic strip is bent through an angle by clamping the lateral boundaries to a bent frame . this situation is depicted in figure [ fig : ridge ] . from the symmetry of the problem , it is clear that we only need to consider one half of the strip . this is represented schematically in figure [ fig : coords ] . we will use ( the material ) coordinates @xmath16 on the reference half strip @xmath17 , @xmath18 . also , we associate a @xmath19 coordinate systems in space , so that as @xmath20 , the half - strip is asymptotically in the @xmath21 plane , as depicted in figure [ fig : coords ] . the grid in the figure is generated by the lines @xmath22 and @xmath23 that are straight in the reference ( material ) coordinates . @xmath24 represents the out of plane deformation , and the in - plane coordinate are chosen so that the @xmath25 and the @xmath26 axes are asymptotically parallel to the @xmath27 and @xmath28 axes respectively as @xmath29 . since the sheet is bent through an angle @xmath30 , as @xmath31 , the sheet will lie in the plane @xmath32 . the symmetry of the two halves implies that the the line @xmath33 maps into the plane @xmath34 , which bisects the angle between the planes @xmath21 ( the asymptote as @xmath20 ) and @xmath32 ( the asymptote as @xmath35 ) . the symmetry between the two halves also implies that the tangent to the lines @xmath22 at @xmath33 should be perpendicular to the plane @xmath36 . consequently @xmath37 and @xmath38 at @xmath33 . a mathematically justified way to obtain the elastic energy of the deformed sheet is to treat the sheet as a three dimensional ( albeit thin ) object and use a full nonlinear three dimensional elastic energy functional for the energy density . this approach however does not take advantage of the `` thinness '' of the sheet . in particular , we would like to treat the thin sheet as a two dimensional object . this will greatly reduce the complexity of the problem . the derivation of reduced dimensional descriptions of thin sheets has a long history . there is a classical theory for thin elastic sheets built on the work of euler , cauchy , kirchoff , fpl and von karman @xcite . many modern authors have considered the problem of deriving a reduced dimensional theory @xcite from three dimensional elasticity in a mathematically rigorous fashion through @xmath39 convergence @xcite . convergence is the appropriate notion for convergence in variational problems . in the context of thin sheets , roughly speaking , finding the @xmath39 limit amounts to identifying an appropriate two dimensional energy functional whose minimizers give the limiting behavior of the minimizers of the full three dimensional energy in the limit the thickness @xmath40 . this problem hasnt been solved in its entirety . there are two situations for which reduced dimensional theories have been derived rigorously as @xmath39 limits of full three dimensional elasticity . membrane theories @xcite are applicable in situations where the stretching is essentially uniform through the thickness of the sheet and bending theories @xcite are applicable in situations where the the sheet is essentially unstretched , and all the elastic energy is due to strains that are first order in the thickness of the sheet . neither of these theories are applicable for the minimal ridge . for the minimal ridge , both the stretching ( membrane ) energy and the bending energy are important . in fact , a ridge is a result of the competition between these two energies . so we turn to the classical fpl von karman _ ansatz _ for an appropriate definition of the elastic energy @xcite . although the fvk energy is not rigorously derived from three dimensional elasticity , it can be thought of as the sum of the membrane and the bending energies that have been derived rigorously in different limiting situations . we will further assume that , for the minimal ridge , the deflections @xmath41 and @xmath42 are small compared to the natural length scale @xmath4 . the strains are of the order of the square of the maximum deflection divided by @xmath4 and they are small . after some rescaling , the energy of the deformed sheet is given by the linearized fvk energy @xmath43 dx dy \nonumber \\ & & + \sigma^2 \int ( w_{xx}^2 + 2 w_{xy}^2 + w_{yy}^2 ) \ , dx dy . \label{eq : unscaled } \end{aligned}\ ] ] where @xmath27 and @xmath28 are reference coordinates on the sheet , @xmath25 and @xmath26 are in - plane coordinates , @xmath24 is the out of plane displacement and @xmath1 is the thickness of the sheet . the integrand for the first integral is given by the squares of the linearized strains , @xmath44 the blistering of thin films is also described by the elastic energy in ( [ eq : unscaled ] ) . a similar energy also describes multiple scale buckling in _ free _ elastic sheets ( _ i.e. _ sheets that are not forced through the boundary conditions ) that are not intrinsically flat @xcite . the difference between the blistering problem and a minimal ridge is in the boundary conditions , which we describe below . since the strains are assumed to be small , @xmath45 . if the bending half - angle @xmath46 , so that @xmath47 , we get @xmath48 . since the deformation goes to zero far away from the bend , we have @xmath49 also , the lateral boundaries at @xmath50 are clamped to the frame . therefore , @xmath51 note that the appropriate boundary condition for the minimal ridge at @xmath33 is a free boundary condition , subject to the symmetry requirement @xmath52 . we are going to replace this free boundary condition with a dirichlet boundary condition @xmath53 where we will leave the functions @xmath54 and @xmath55 unspecified , except for a size condition . defining @xmath56}(|v_0(x)|,|w_0(x)|),\ ] ] we impose the size condition by requiring that @xmath7 be `` small '' . the relevant small parameter in the problem is the non - dimensional thickness of the sheet , @xmath9 . following lobkovsky @xcite , we introduce the rescaled coordinates and displacements by @xmath57 and @xmath58 since @xmath59 all have dimensions of a length , it is clear that the rescaled quantities @xmath60 are all dimensionless . with these rescalings , the dimensionless energy @xmath61 is given by @xmath62 \nonumber \\ & & + \left [ w_{yy}^2 + 2 \epsilon^{2/3 } w_{xy}^2 + \epsilon^{4/3 } w_{xx}^2 \right]dx dy . \label{eq : scaled } \end{aligned}\ ] ] our quest for rigorous scaling results for the energy @xmath63 reduces to the following show that the rescaled energy @xmath64 , of a minimizer @xmath65 , is bounded above and below by positive constants _ uniform _ in the dimensionless thickness parameter @xmath66 , as @xmath67 . setting @xmath68 the rescaled quantities satisfy the boundary conditions @xmath69 and @xmath70 we have the lateral boundary conditions @xmath71 also , the deformation at @xmath72 satisfies @xmath73 and @xmath74 where @xmath75 . in this section , we prove a lower bound for the linearized elastic energy in eq . ( [ eq : unscaled ] ) , provided that the length scale @xmath7 associated with the boundary conditions satisfies a size condition . @xmath76 is as defined in eq . ( [ eq : unscaled ] ) . there is a constant @xmath77 such that , for all @xmath78 and @xmath79 satisfying the boundary conditions @xmath80 and the size condition @xmath81}(|v_0(x)|,|w_0(x)| ) = a \leq b \sigma^{1/3 } l^{2/3 } \alpha^{2/3},\ ] ] we have the lower bound @xmath82 we will prove the theorem by proving the scaled version of the statement , _ viz . _ , the rescaled boundary conditions along with the rescaled size condition @xmath83 imply that @xmath84 the hypothesis for the theorem includes a size condition on the displacement at @xmath33 . this is somewhat unsatisfying , as it is not _ a priori _ obvious that a ridge in a `` real '' crumpled sheet will satisfy this condition . in our search for a lower bound , we can assume @xmath85 w.l.o.g . from eq . ( [ eq : scaled ] ) , it follows that for @xmath86 , @xmath87 . by the standard trace theorems , the boundary conditions @xmath88 and @xmath89 are therefore assumed pointwise for almost every @xmath90 . since @xmath91 at @xmath92 , it follows that @xmath93 . using this together with jensen s inequality we get @xmath94dx dy \nonumber \\ & \geq & \int_0^{\infty } \left [ \frac{1}{2 } \left ( \int_{-1}^1 w_x^2 dx \right)^2 + \int_{-1}^1 w_{yy}^2 dx \right ] dy \label{eq : onlyw}\end{aligned}\ ] ] thus the functional @xmath95 dy\ ] ] bounds @xmath96 from below . hence it suffices to obtain a lower bound for @xmath97 with the given hypothesis to prove the theorem . let @xmath98 and @xmath99 denote the quantities @xmath100 and @xmath101 although @xmath98 and @xmath99 are only lower bounds for the `` true '' bending and the stretching energies @xmath102 and @xmath103 , we will nonetheless call @xmath98 and @xmath99 the bending and the stretching energy for convenience . for every @xmath90 , we define @xmath104^{-1},\ ] ] and for every @xmath105 , we define @xmath106 since @xmath107 , @xmath108 ( a.e . ) and @xmath109 ( a.e . ) . for any function @xmath110 depending on @xmath90 and @xmath105 , let @xmath111 denote @xmath112 , so that , @xmath113 @xmath114 is a `` local '' ( in @xmath90 ) measure of the bending energy , and @xmath115^{-1}$ ] can be thought of as the bending energy density in @xmath90 that is obtained by integrating out the @xmath105 dependence . we can also think of @xmath114 as the length scale associated with the bending energy as a function of @xmath90 , _ viz . _ , we expect that the bending energy density in @xmath105 decays rapidly for @xmath116 ( see fig . [ fig : ridge ] ) . likewise , @xmath117 ^ 2 $ ] is a local ( in @xmath105 ) measure of the stretching energy . in terms of @xmath114 and @xmath118 , the bending an the stretching energies are given by @xmath119 ^ 2 dy.\ ] ] we now begin our proof of the theorem . the idea behind the proof is to show that the stretching energy @xmath99 can be bounded from below by a negative power of the bending energy @xmath98 , so that the total energy @xmath120 tends to @xmath121 as @xmath122 and @xmath123 . this ensures the existence of a positive lower bound for @xmath97 ( and consequently also for @xmath64 ) . for every @xmath105 , we have the inequality @xmath124 if @xmath108 , @xmath125 is a @xmath126 function by the sobolev embedding theorem and the boundary conditions imply that @xmath88 and @xmath127 . consequently , for such an @xmath90 , @xmath128 we will estimate @xmath129 from this equation using the elementary inequalities @xmath130 and @xmath131 for all @xmath132 . by our hypothesis on the boundary conditions , @xmath133 by jensen s inequality @xmath134 combining these estimates , we get @xmath135,\ ] ] for all positive @xmath136 and @xmath137 . in particular , setting @xmath138 and @xmath139 yields @xmath140 integrating this inequality in @xmath90 we obtain @xmath141 proving the lemma . our proof is based on demonstrating that a small bending energy @xmath98 will lead to a large stretching energy . this idea is quantified by the following lemma where we use eq . ( [ eq : integral_bound ] ) to estimate the stretching energy @xmath99 from below in terms of the bending energy @xmath98 . let @xmath142 there is a constant @xmath143 such that , if @xmath144 , the stretching energy @xmath99 satisfies @xmath145 [ lem : essential ] we have , @xmath146 by the poincare inequality . ( [ eq : integral_bound ] ) now yields @xmath147.\ ] ] we will now extract the appropriate scaling dependence of @xmath118 and @xmath99 on @xmath148 and @xmath98 . setting @xmath149 , we deduce that a characteristic scale @xmath150 for @xmath105 is given by @xmath151 rescaling @xmath105 in terms of @xmath150 , we obtain @xmath152,\ ] ] where @xmath153 is as defined above , _ i.e. _ @xmath154 making the change of variables @xmath155 , and using @xmath156 , we see that @xmath157 ^ 2 dy \geq \frac{\pi^4 \alpha^{14 } k(\mu)}{512 e_b^5},\ ] ] where @xmath158 ^ 2 dz.\ ] ] @xmath159 is clearly a continuous function of @xmath153 and @xmath160 . since @xmath161 , there is an @xmath162 such that for all @xmath144 , @xmath163 . the lemma follows . we can now prove the theorem . let @xmath164 . the hypothesis imply @xmath165 if @xmath166 , there is nothing to prove . therefore , we can assume that @xmath167 . then , it follows that @xmath168 consequently @xmath169 . minimizing @xmath120 , we see that @xmath170 the theorem follows by `` undoing '' our rescaling to express the results in terms of @xmath171 and @xmath63 . in the appendix , we show that @xmath172 therefore , we can choose @xmath173 and @xmath174 . we see that @xmath175 is not exceedingly small . rather it is @xmath176 . also , it is not the best constant for this theorem , and we can get a better value by optimizing our choices for the constants ( @xmath136 and @xmath137 which we set to be @xmath177 and @xmath15 respectively ) . we have proved a rigorous lower bound for the elastic energy of a ridge that is consistent with the results obtained by lobkovsky @xcite using scaling arguments , and by lobkovsky @xcite using matched asymptotics . in order to prove these scaling laws rigorously , we also need analogous upper bounds that are consistent with the same scaling . this approach has been used successfully for a variety of other variational problems @xcite . the upper bound is usually obtained by constructing a test solution that yields the bound . one is often guided by scaling arguments in the construction of the appropriate test solution . this is in contrast to the lower bounds , where one needs to obtain functional analysis type inequalities that captures the competition between distinct energies in the problem ( _ e.g. _ lemma [ lem : essential ] ) . it is this competition that determines the scaling behavior of the problem . we have not constructed the upper bounds , because we believe that they will follow from the scaling ansatz in lobkovsky s work @xcite . we also believe that the upper bound will scale in the same manner as the lower bound , thereby giving us a rigorous scaling law for the energy of a single ridge . as we remark before proving the theorem , our result is not directly applicable to the minimal ridge since we have the extra hypothesis @xmath178 we need to replace this restriction with a free boundary condition at @xmath33 subject to the symmetry requirement @xmath34 , and the boundary condition @xmath38 . despite this caveat , we claim that the analysis in this paper captures the essential features of the minimal ridge problem , namely the scaling in lemma [ lem : essential ] , and consequently the scaling law for the energy of the ridge . we will show that this is indeed the case by investigating the problem with the `` true '' boundary conditions elsewhere . a harder problem is to show that the scaling laws also hold for a real crumpled sheet , where the forcing is not through clamping the boundaries to a frame , but rather through the confinement in a small volume . in this case , there are interesting global geometric and topological considerations , some of which are explored in refs . @xcite and @xcite . as lobkovsky and witten @xcite argue , the boundary condition that the deformation goes to zero far away from the ridge implies that the ridges do not interact with each other significantly . the ridges can be considered the _ elementary excitations _ of a crumpled sheet . in our quest for upper and lower bounds , this translates to the statement that the test solutions for the upper bound can be constructed by piecing together local solutions near each ridge . thus the upper bound is obtained relatively easily . the hard part is to show that the confinement actually leads to the formation of ridges , and that the competition between the bending and the stretching energy for this situation has the same form as in lemma [ lem : essential ] . in this context , we expect that global topological considerations , as well as the non self - intersection of the sheet will play a key role in the analysis , as they do in the analysis of elastic rods ( one dimensional objects ) @xcite . as we note above , the blistering problem is described by the same elastic energy ( eq . ( [ eq : unscaled ] ) ) , but with different boundary conditions . our results show an interesting contrast with results for the blistering problem . ben belgacem _ et al . _ have shown that @xcite , for an isotropically compressed thin film , the energy of the minimizer satisfies @xmath179 where @xmath4 is a typical length scale of the domain , and @xmath180 is the compression factor . a construction for the upper bound strongly suggests that the minimizers develop an infinitely branched network with oscillations on increasingly finer scales as @xmath40 . in contrast , our results indicate that the energy of a minimal ridge satisfies @xmath181 and the energy concentrates in a region of width @xmath182 . this shows that the nature of the solution of the variational problem for the elastic energy in ( [ eq : unscaled ] ) depends very strongly on the boundary conditions . in particular the very nature of the energy minimizers is different for the two problems for the blistering problems , as @xmath8 the minimizers develop a branched network of folds refining towards the boundary . the minimizers therefore display the problem of small scale _ oscillations _ @xcite . the minimal ridge problem on the other hand displays the _ concentration _ phenomenon @xcite as @xmath8 , with the energy concentrating on a region of width @xmath183 . it would be interesting to explore this issue further , and in particular , to understand the mechanisms by which the boundary conditions determine the nature of the minimizer . in this appendix , we prove the inequality @xmath158 ^ 2 dz \geq \frac{1}{105 } - \frac{\mu}{6}.\ ] ] let @xmath184 . then , for @xmath185 , we have @xmath186 ^ 2 = f^2 - 2 \mu f + \mu^2 \geq f^2 - 2 \mu f.\ ] ] for @xmath187 , we have @xmath186 ^ 2 = 0 \geq f^2 - 2 \mu f.\ ] ] using these inequalities in the definition of @xmath159 we obtain @xmath188 ^ 2 - 2 \mu z^2 ( 1 - z ) dz \geq \frac{1}{105 } - \frac{\mu}{6}.\ ] ] i wish to thank brian didonna , bob kohn , l. mahadevan , stefan mller and tom witten for helpful and enlightening conversations . this work was started during a visit to the max - planck institute for mathematics in the sciences in leipzig , and i would like to thank the mpi for their hospitality . this work was supported in part by the national science foundation through its mrsec program under award number dmr-9808595 , and by a nsf career award dms-0135078 . this work is also supported in part by a research fellowship from the alfred p. sloan jr . foundation . 10 h. ben belgacem , s. conti , a. desimone , and s. mller , `` energy scaling of compressed elastic films - three - dimensional elasticity and reduced theories , '' _ arch . _ , vol . 164 , pp . 137 , 2002 . h. le dret and a. raoult , `` le modle de membrane non linaire comme limite variationnelle de llasticit non linaire tridimensionnelle , '' _ c. r. acad . paris sr . _ , vol . 317 , no . 2 , pp . 221226 , 1993 . b. a. didonna , t. a. witten , s. c. venkataramani , and e. m. kramer , `` singularities , structures , and scaling in deformed @xmath190-dimensional elastic manifolds , '' _ phys . e ( 3 ) _ , vol . 65 , no . 1 , part 2 , pp . 016603 , 25 , 2002 . o. gonzalez , j. h. maddocks , f. schuricht , and h. von der mosel , `` global curvature and self - contact of nonlinearly elastic curves and rods , '' _ calc . var . partial differential equations _ , vol . 14 , no . 1 , pp . 2968 , 2002 . l. tartar , `` compensated compactness and applications to partial differential equations , '' in _ nonlinear analysis and mechanics : heriot - watt symposium , vol . iv _ , pp . 136212 , boston , mass . : pitman , 1979 .

we study the linearized fpl von karman theory of a long , thin rectangular elastic membrane that is bent through an angle @xmath0 . we prove rigorous bounds for the minimum energy of this configuration in terms of the plate thickness @xmath1 and the bending angle . we show that the minimum energy scales as @xmath2 . this scaling is in sharp contrast with previously obtained results for the linearized theory of thin sheets with isotropic compression boundary conditions , where the energy scales as @xmath1 .

You are an expert at summarizing long articles. Proceed to summarize the following text: according to recent estimates , 8090% of the matter in the universe is non - baryonic . with the presence of extended dark matter halos in galaxies of all morphological types now well - established , attention is focusing on comparing the detailed mass distribution with theoretical predictions of galactic dark halo structure . the shape of the dark matter distribution is determined by the initial density perturbation spectrum , the coupled dynamical evolution of baryonic and non - baryonic constituents , and the nature of the dark matter itself . therefore , its measurement represents a powerful diagnostic of fundamental astrophysical processes and parameters . the standard cold dark matter ( cdm ) model is highly successful in explaining the distribution of mass in the universe on scales ranging from galaxies on up , but is undergoing a critical re - examination due in large part to its confrontation with measurements of _ late - type _ galaxy mass distributions indicating that dark matter is less concentrated than expected . in this work @xcite , we use a recent _ chandra _ x - ray observatory observation of ngc 4636 to constrain its dark matter distribution , with the goal of testing whether this `` concentration crisis '' applies to this ( _ early - type _ ) elliptical galaxy . hot , extended distributions of x - ray emitting hot gas in hydrostatic equilibrium are crucial tracers of the gravitational potential in individual elliptical galaxies @xcite ; gravitational lensing is useful only in a statistical sense for a large sample of galaxies . the physical properties of the hot gas can be accurately measured out to large radii where dark matter dominates and optical techniques are infeasible . the effective resolution of the mass estimate essentially corresponds to that of the gas temperature profile ( see below ) . the unprecedented _ chandra _ angular resolution enables one to do x - ray imaging spectroscopy of the hot ism on scales comparable to optical studies of the stars for the first time , thus yielding the relative distributions of luminous and dark matter . given a mass distribution , and the hot gas density distribution obtained from the measured x - ray surface brightness distribution , we solve the equation of hydrostatic equilibrium to derive the corresponding gas temperature profile . the total mass model consists of three components an @xmath4 m@xmath2 central supermassive black hole @xcite , stars distributed as measured from _ hubble space telescope _ and ground - based photometry , and a dark matter density distribution parameterized by an asymptotic slope at the origin ( @xmath5 ) , a scale radius characterizing the transition to an @xmath6 decline ( as determined by the results of numerical simulations ) at large radius , and a normalization . for a given @xmath5 , the dark matter scale and normalization , and the stellar mass - to - light ratio , are varied to obtain a match with the composite temperature profile measured by _ chandra _ and the _ xmm - newton _ observatory ( figure 1 ) . dark matter density cusp ; solid curve ) , and best - fit model with dark matter density core ( broken curve ) . ] models with constant mass - to - light ratios and those where the dark matter density inner slope , @xmath5 , is as steep as the singular isothermal sphere value of @xmath7 are clearly ruled out . we derive accurate constraints on the total mass distribution from 0.735 kpc that are robust to the assumed value of @xmath5 . the total mass increases as @xmath0 ( more quickly than the stars ) to a good approximation over this range in radii , attaining a total of @xmath8 m@xmath2 ( corresponding to @xmath3 in solar units ) at the outermost point we consider ( figure 2 ) . there are degeneracies with respect to the relative distributions of dark and luminous matter . we find acceptable models with dark matter cusps ( e.g. , @xmath9 ) and with cores ( @xmath10 ) . the former imply lower stellar mass to light ratios ( the allowed range : @xmath11 , consistent with studies of the stellar population ) and , in some cases , the predominance of dark matter all the way into the nucleus of the galaxy . the dark matter fraction within the half - light radius ( @xmath12 kpc ) can be as high as 0.8 and `` central '' ( actually , the average over the inner 700 pc ) dark matter density as high as 4.0 m@xmath2 pc@xmath13 ; the cored models define lower limits to these quantities of 0.5 and 0.16 m@xmath2 pc@xmath13 , respectively . even the lowest allowed central density is an order of magnitude greater than the previously estimated values for other types of galaxies ( figure 3 ) that precipitated the consideration of alternatives to cdm . comparison of non - parametric measures of the dark matter concentration with expectations based on standard cdm numerical simulations indicate that dark matter in ngc 4636 is more compact than average for its mass scale , but within the expected scatter . this is consistent with the findings of statistical studies of the gravitational lensing properties of elliptical galaxies , though it appears that ngc 4636 is unusually dark matter dominated . many of the proposed scenarios for reducing the dark matter concentration in late - type galaxies are at odds with our results . it is instructive to consider the two classes of models illustrated in the above figures from this perspective those with flat dark matter cores , and those with cuspy dark matter cores ( that provide better fits to the x - ray data ) interpreted as contracting from initially flat cores due to baryonic infall . for the flat - core models , the high central dark matter mass density and large cores are contrary to the expected scaling relations for self - interacting dark matter ( sidm ) and other models where dark matter structure is driven by dark matter particle interaction . for the cuspy - core models the central phase - space density , conserved during adiabatic contraction , is too high . alternatively , the velocity - dependent cross section can be fine - tuned so that interactions are ineffective at the higher mass ( and velocity ) scale of ellipticals ; although here too , one may run into phase space difficulties if ellipticals form from mergers of lower mass systems with interaction - induced dark matter cores . scenarios where the mechanism for reducing the dark matter concentration becomes less effective with increasing mass scale are not ruled out , since ellipticals such as ngc 4636 represent the most massive galaxies with the deepest potential wells . these include models that invoke warm dark matter , or the expansion of dark matter due to coupling with a powerful protogalactic starburst - driven baryonic outflow . if this transitional mass is indeed on the giant elliptical galaxy scale , cuspy dark matter distributions in galaxy clusters are implied . our result that the central dark matter density in ngc 4636 is @xmath14 the typical value for less massive galaxies contradicts the naive expectations of bottom - up hierarchical clustering where the most massive systems form latest and reflect the relatively low average density in the universe at that epoch . a cold dark matter initial perturbation spectrum that is tilted or otherwise lacking in small - scale power may help explain ly-@xmath15 forest data @xcite , but likely exacerbates this contradiction . recent work on the relation between the merging histories of galaxies and their morphology @xcite may be the key to resolving the dark matter concentration problem . a significant dispersion in dark matter concentration is predicted by numerical simulations at any given mass range , but particularly on galaxy scales . this results from the stochasticity of the merging process , and introduces the following bias . galaxies of low central dark matter density represent relatively recently formed systems , and/or particularly fragile galaxies that experienced relatively tranquil assembly histories these may correspond to disk galaxies where low dark matter central densities are indeed measured . conversely , giant elliptical galaxies such as , or perhaps particularly , ngc 4636 may form at relatively high redshifts and undergo exceptionally prominent merger histories . possible observational inconsistencies of the ( non - interacting ) cdm paradigm resulted in a renewed scrutiny and a proliferation of alternative models . careful analysis of the observational situation in light of more detailed , and more deeply understood , theoretical models , as well as the shortcomings of the proposed alternatives seem to indicate that cdm is withstanding these recent challenges to remain the most viable model of structure in the universe ( see j. primack s contribution to this volume ) . our results on the high concentration of dark matter in ngc 4636 evidently strengthens the case for cdm , while introducing other puzzles such as why ngc 4636 seems to be exceptionally dark matter dominated . although one must exercise caution in generalizing from our results , the most stringent constraints on alternative models for dark halo structure , that presumably universally apply , emerge from studying such a , possibly extreme , system . results , in progress , of similar investigations for other galaxies ( e.g. , ngc 1399 , ngc 4472 ) are anticipated with great interest . 9 loewenstein , m. and mushotzky , r. f. , astrophysical journal , submitted . loewenstein , m. and white , r. e. , iii , astrophysical journal , 518 ( 1999 ) , 50 . merritt , d. and ferrarese , l. , monthly notices of the royal astronomical society , 320 ( 2001 ) , l30 . alam , s. m. k. , bullock , j. s. , and weinberg , d. h. , astrophysical journal , submitted . wechsler , r. h. , bullock , j. s. , primack , j. r. , kravtsov , a. , and dekel , a. , astrophysical journal , 569 ( 2002 ) , 52 .

we determine the total enclosed mass profile from 0.7 to 35 kpc in the elliptical galaxy ngc 4636 based on the hot interstellar medium temperature profile measured using the _ chandra _ x - ray observatory , and other x - ray and optical data . the total mass increases as @xmath0 to a good approximation over this range in radii , attaining a total of @xmath1 m@xmath2 ( corresponding to @xmath3 ) at 35 kpc . we find that at least half , and as much as 80% , of the mass within the optical half - light radius is non - luminous , implying an exceptionally low baryon fraction in ngc 4636 . the large inferred dark matter concentration and central dark matter density , consistent with the upper end of the range expected for standard cold dark matter halos , imply that mechanisms proposed to explain low dark matter densities in less massive galaxies are not effective in elliptical galaxies .

You are an expert at summarizing long articles. Proceed to summarize the following text: the capacity of the interference channel remains one of the most challenging open problems in the domain of network information theory . the capacity region is not known in general , except for a specific range of channel parameters . for the two - user scalar gaussian interference channel , where the interference alignment is not required , the approximate capacity region to within one bit is known @xcite . for the channels where interference alignment is required such as the @xmath0-user gaussian interference channel @xcite and the gaussian x - channel @xcite , a tight characterization of the capacity region is not known , even for symmetric channel cases . a tractable approach to the capacity of interference channels is to consider partial connectivity of interference links and analyze the impact of topology on the capacity . topological interference management @xcite approach gives important insights on the degrees - of - freedom ( dof ) of partially connected interference channels and their connection to index coding problems @xcite . it is shown that the symmetric dof of a partially connected interference channel can be found by solving the corresponding index coding problem . in this paper , we consider a class of three - user partially connected interference channels and characterize approximate capacity regions at finite snr . we focus on the impact of interference topology , interference alignment , and interplay between interference and noise . we choose a few representative topologies where we can achieve clear interference alignment gain . for these topologies , z - channel type outer bounds are tight to within a constant gap from the corresponding inner bound . for each topology , we present an achievable scheme based on rate - splitting , lattice alignment , and successive decoding . lattice coding based on nested lattices is shown to achieve the capacity of the single user gaussian channel in @xcite . the idea of lattice - based interference alignment by decoding the sum of lattice codewords appeared in the conference version of @xcite . this lattice alignment technique is used to derive capacity bounds for three - user interference channel in @xcite . the idea of decoding the sum of lattice codewords is also used in @xcite to derive the approximate capacity of the two - way relay channel . an extended approach , compute - and - forward @xcite enables to first decode some linear combinations of lattice codewords and then solve the lattice equation to recover the desired messages . this approach is also used in @xcite to characterize approximate sum - rate capacity of the fully connected @xmath0-user interference channel . the idea of sending multiple copies of the same sub - message at different signal levels , so - called zigzag decoding , appeared in @xcite where receivers collect side information and use them for interference cancellation . the @xmath0-user cyclic gaussian interference channel is considered in @xcite where an approximate capacity for the weak interference regime ( @xmath1 for all @xmath2 ) and the exact capacity for the strong interference regime ( @xmath3 for all @xmath2 ) are derived . our type 4 and 5 channels are @xmath4 cases in _ mixed _ interference regimes , which were not considered in @xcite . we consider five channel types defined in table [ tab : txrxsignals ] and described in fig . [ fig : channeltype ] ( a)(e ) . each channel type is a partially connected three - user gaussian interference channel . each transmitter is subject to power constraint @xmath5\leq p_k = p$ ] . let us denote the noise variance by @xmath6 $ ] . without loss of generality , we assume that @xmath7 . the side information graph representation of an interference channel satisfies the following . * a node represents a transmitter - receiver pair , or equivalently , the message . * there is a directed edge from node @xmath8 to node @xmath9 if transmitter @xmath8 does not interfere at receiver @xmath9 . the side information graphs for five channel types are described in fig . [ fig : channeltype ] ( f)(j ) . we state the main results in the following two theorems , of which the proofs will be given in the main body of the paper . for the five channel types , if @xmath10 is achievable , it must satisfy @xmath11 for every subset @xmath12 of the nodes @xmath13 that does not include a directed cycle in the side information graph over the subset . + for any rate triple @xmath10 on the boundary of the outer bound region , the point @xmath14 is achievable . .five channel types [ cols="^,<",options="header " , ] in this section , we present an outer bound on the capacity region of type 5 channel defined by @xmath15 = \left [ { \begin{array}{*{20}c } 1 & 1 & 0\\ 0 & 1 & 1\\ 1 & 0 & 1\\ \end{array } } \right ] \left [ { \begin{array}{*{20}c } x_1\\ x_2\\ x_3\\ \end{array } } \right ] + \left [ { \begin{array}{*{20}c } z_1\\ z_2\\ z_3\\ \end{array } } \right].\ ] ] this is a cyclic gaussian interference channel @xcite . we first show that channel type 5 is in the mixed interference regime . by normalizing the noise variances , we get the equivalent channel given by @xmath16 = \left [ { \begin{array}{*{20}c } \frac{1}{\sqrt{n_1 } } & \frac{1}{\sqrt{n_1 } } & 0\\ 0 & \frac{1}{\sqrt{n_2 } } & \frac{1}{\sqrt{n_2}}\\ \frac{1}{\sqrt{n_3 } } & 0 & \frac{1}{\sqrt{n_3}}\\ \end{array } } \right ] \left [ { \begin{array}{*{20}c } x_1\\ x_2\\ x_3\\ \end{array } } \right ] + \left [ { \begin{array}{*{20}c } z_1'\\ z_2'\\ z_3'\\ \end{array } } \right].\ ] ] we can see that @xmath17 we state the outer bound in the following theorem . the capacity region of type 5 channel is contained in the following outer bound region : @xmath18 @xmath19 where we used the fact that @xmath20 . @xmath21 where we used the fact that @xmath22 . @xmath23 for ease of gap calculation , we also derive relaxed outer bounds . first , we can see that for @xmath24 , @xmath25 five outer bound theorems in this section , together with this inequality , give the sum - rate bound expression in theorem 1 . next , we can assume that @xmath26 for @xmath27 . otherwise , showing one - bit gap capacity is trivial as the capacity region is included in the unit hypercube , i.e. , @xmath28 . for @xmath26 , @xmath29 the resulting relaxed outer bounds @xmath30 are summarized in table [ tab : outerbounds ] . given @xmath31 ^ 2 $ ] , the rate region @xmath32 is defined by @xmath33 where @xmath34 . and , @xmath35 is achievable where @xmath36 is convex hull operator . lattice @xmath37 is a discrete subgroup of @xmath38 , @xmath39 where @xmath40 is a real generator matrix . quantization with respect to @xmath37 is @xmath41 . modulo operation with respect to @xmath37 is @xmath42{\textrm { mod } \lambda}=\mathbf{x}-q_{\lambda}(\mathbf{x})$ ] . for convenience , we use both notations @xmath43 and @xmath44{\textrm { mod } \lambda}$ ] interchangeably . fundamental voronoi region of @xmath37 is @xmath45 . volume of the voronoi region of @xmath37 is @xmath46 . normalized second moment of @xmath37 is @xmath47 where @xmath48 . lattices @xmath49 , @xmath50 and @xmath37 are said to be nested if @xmath51 . for nested lattices @xmath52 , @xmath53 . we briefly review the lattice decoding procedure in @xcite . we use nested lattices @xmath54 with @xmath55 , @xmath56 , and @xmath57 . the transmitter sends @xmath58{\textrm { mod } \lambda}$ ] over the point - to - point gaussian channel @xmath59 where the codeword @xmath60 , the dither signal @xmath61 , the transmit power @xmath62 and the noise @xmath63 . the code rate is given by @xmath64 . after linear scaling , dither removal , and mod-@xmath37 operation , we get @xmath65{\textrm { mod } \lambda}= \left[\mathbf{t}+\mathbf{z}_e\right]{\textrm { mod } \lambda}\end{aligned}\ ] ] where the effective noise is @xmath66 and its variance @xmath67=(\beta-1)^2 s+\beta^2 n$ ] . with the mmse scaling factor @xmath68 plugged in , we get @xmath69 . the capacity of the mod-@xmath37 channel @xcite between @xmath70 and @xmath71 is @xmath72 where @xmath73 and @xmath74 are mutual information and differential entropy , respectively . for reliable decoding of @xmath70 , we have the code rate constraint @xmath75 . with the choice of lattice parameters , @xmath76 , @xmath77 and @xmath78 , @xmath79 thus , the constraint @xmath75 can be satisfied . by _ lattice decoding _ @xcite , we can recover @xmath70 , i.e. , @xmath80 with probability @xmath81 where @xmath82\ ] ] is the probability of decoding error . if we choose @xmath37 to be poltyrev - good @xcite , then @xmath83 as @xmath84 . we present an achievable scheme for the proof of _ theorem 8_. the achievable scheme is based on rate - splitting , lattice coding , and interference alignment . message @xmath85 is split into two parts : @xmath86 and @xmath87 , so @xmath88 . transmitter 1 sends @xmath89 where @xmath90 and @xmath91 are coded signals of @xmath92 and @xmath93 , respectively . transmitters 2 and 3 send @xmath94 and @xmath95 , coded signals of @xmath96 and @xmath97 . in particular , @xmath90 and @xmath95 are lattice - coded signals . we use the lattice construction of @xcite with the lattice partition chain @xmath98 , so @xmath99 are nested lattices . @xmath100 is the coding lattice for both @xmath90 and @xmath95 . @xmath49 and @xmath101 are shaping lattices for @xmath90 and @xmath95 , respectively . the lattice signals are formed by @xmath102{\textrm { mod } \lambda}_1\\ & & \ \mathbf{x}_3=[\mathbf{t}_3+\mathbf{d}_3]{\textrm { mod } \lambda}_3\end{aligned}\ ] ] where @xmath103 and @xmath104 are lattice codewords . the dither signals @xmath105 and @xmath106 are uniformly distributed over @xmath107 and @xmath108 , respectively . to satisfy power constraints , we choose @xmath109=n\sigma^2(\lambda_1)=(1-\alpha_1 ) np$ ] , @xmath110={\alpha}_1 np$ ] , @xmath111={\alpha}_2 np$ ] , @xmath112=n\sigma^2(\lambda_3)=np$ ] . with the choice of transmit signals , the received signals are given by @xmath113+\mathbf{x}_2+\mathbf{z}_2'\\ & & \mathbf{y}_3=\mathbf{x}_3+\mathbf{z}_3'.\end{aligned}\ ] ] where @xmath114 $ ] is the sum of interference , and @xmath115 and @xmath116 are the effective gaussian noise . the signal scale diagram at each receiver is shown in fig . [ fig : signalscale ] ( a ) . at the receivers , successive decoding is performed in the following order : @xmath117 at receiver 1 , @xmath118 at receiver 2 , and receiver 3 only decodes @xmath95 . note that the aligned lattice codewords @xmath119 , and @xmath120{\textrm { mod } \lambda}_1 \in\lambda_c\cap\mathcal{v}(\lambda_1)$ ] . we state the relationship between @xmath121 and @xmath122 in the following lemmas . the following holds . @xmath123{\textrm { mod } \lambda}_1=\mathbf{t}_f\ ] ] where @xmath124 . @xmath125{\textrm { mod } \lambda}_1\\ & & = [ m_{\lambda_1}(\mathbf{t}_{11}+\mathbf{d}_{11})+m_{\lambda_3}(\mathbf{t}_3+\mathbf{d}_3)-\mathbf{d}_f]{\textrm { mod } \lambda}_1\\ & & = [ m_{\lambda_1}(\mathbf{t}_{11}+\mathbf{d}_{11})+m_{\lambda_1}(\mathbf{t}_3+\mathbf{d}_3)-\mathbf{d}_f]{\textrm { mod } \lambda}_1\\ & & = [ \mathbf{t}_{11}+\mathbf{d}_{11}+\mathbf{t}_3+\mathbf{d}_3-\mathbf{d}_f]{\textrm { mod } \lambda}_1\\ & & = [ \mathbf{t}_{11}+\mathbf{t}_3]{\textrm { mod } \lambda}_1\\ & & = \mathbf{t}_f\end{aligned}\ ] ] the second and third equalities are due to distributive law and the identity in the following lemma . for any nested lattices @xmath126 and + any @xmath127 , it holds that @xmath128{\textrm { mod } \lambda}_1=[\mathbf{x}]{\textrm { mod } \lambda}_1.\ ] ] @xmath129{\textrm { mod } \lambda}_1\\ & & = [ \mathbf{x}-\lambda_3]{\textrm { mod } \lambda}_1\\ & & = [ m_{\lambda_1}(\mathbf{x})-m_{\lambda_1}(\lambda_3)]{\textrm { mod } \lambda}_1\\ & & = [ m_{\lambda_1}(\mathbf{x})-\lambda_3 + q_{\lambda_1}(\lambda_3)]{\textrm { mod } \lambda}_1\\ & & = [ m_{\lambda_1}(\mathbf{x})]{\textrm { mod } \lambda}_1\\ & & = [ \mathbf{x}]{\textrm { mod } \lambda}_1\end{aligned}\ ] ] where @xmath130 , thus @xmath131 . the following holds . @xmath132{\textrm { mod } \lambda}_1=[\mathbf{x}_f]{\textrm { mod } \lambda}_1.\ ] ] @xmath133{\textrm { mod } \lambda}_1\\ & & = [ m_{\lambda_1}(\mathbf{t}_{11}+\mathbf{t}_3)+\mathbf{d}_f]{\textrm { mod } \lambda}_1\\ & & = [ \mathbf{t}_{11}+\mathbf{t}_3+\mathbf{d}_f]{\textrm { mod } \lambda}_1\\ & & = [ m_{\lambda_1}(\mathbf{t}_{11}+\mathbf{d}_{11})+m_{\lambda_1}(\mathbf{t}_3+\mathbf{d}_3)]{\textrm { mod } \lambda}_1\\ & & = [ m_{\lambda_1}(\mathbf{t}_{11}+\mathbf{d}_{11})+m_{\lambda_3}(\mathbf{t}_3+\mathbf{d}_3)]{\textrm { mod } \lambda}_1\\ & & = [ \mathbf{x}_{11}+\mathbf{x}_3]{\textrm { mod } \lambda}_1\\ & & = [ \mathbf{x}_f]{\textrm { mod } \lambda}_1\end{aligned}\ ] ] receiver 2 does not need to recover the codewords @xmath134 and @xmath135 but the real sum @xmath121 to remove the interference from @xmath136 . since @xmath137 , we first recover the modulo part and then the quantized part to cancel out @xmath121 . this idea appeared in @xcite as an achievable scheme for the many - to - one interference channel . the mod-@xmath49 channel between @xmath122 and @xmath138 is given by @xmath139{\textrm { mod } \lambda}_1\\ & & \ \ \ \ = [ \mathbf{x}_f-\mathbf{d}_f + \mathbf{z}_{e2}]{\textrm { mod } \lambda}_1\\ & & \ \ \ \ = [ \mathbf{t}_f + \mathbf{z}_{e2}]{\textrm { mod } \lambda}_1\end{aligned}\ ] ] where the effective noise @xmath140 . note that @xmath141=(\bar{\alpha}_0 + 1)np$ ] , and the effective noise variance @xmath142=(\beta_2 - 1)^2(\bar{\alpha}_0 + 1)p+\beta_2 ^ 2 n_{e2}$ ] where @xmath143 . with the mmse scaling factor @xmath144 plugged in , we get @xmath145 . the capacity of the mod-@xmath49 channel between @xmath122 and @xmath138 is @xmath146 for reliable decoding of @xmath122 at receiver 2 , we have the code rate constraint @xmath147 . this also implies that @xmath148 . by lattice decoding , we can recover the modulo sum of interference codewords @xmath122 from @xmath138 . then , we can recover the real sum @xmath121 in the following way . * recover @xmath149 by calculating @xmath150{\textrm { mod } \lambda}_1 $ ] ( lemma 3 ) . * subtract it from the received signal , @xmath151 where @xmath152 . * quantize it to recover @xmath153 , @xmath154 with probability @xmath81 where @xmath155\ ] ] is the probability of decoding error . if we choose @xmath49 to be simultaneously rogers - good and poltyrev - good @xcite with @xmath156 , then @xmath83 as @xmath84 . * recover @xmath121 by adding two vectors , @xmath157 we now proceed to decoding @xmath94 from @xmath158 . since @xmath94 is a codeword from an i.i.d . random code for point - to - point channel , we can achieve rate up to @xmath159 at receiver 1 , we first decode @xmath90 while treating other signals @xmath160 as noise . the effective noise in the mod-@xmath49 channel is @xmath161 with variance @xmath162=(\beta_1 - 1)^2 \bar{\alpha}_0 p+\beta_1 ^ 2 n_{e1}$ ] where @xmath163 . for reliable decoding , the rate @xmath164 must satisfy @xmath165 where the mmse scaling parameter @xmath166 . similarly , we have the other rate constraints at receiver 1 : @xmath167 at receiver 3 , the signal @xmath95 is decoded with the effective noise @xmath168 . for reliable decoding , @xmath169 must satisfy @xmath170 in summary , * @xmath90 decoded at receivers 1 and 2 @xmath171 where @xmath172 . * @xmath91 decoded at receiver 1 @xmath173 * @xmath94 decoded at receivers 1 and 2 @xmath174 * @xmath95 decoded at receivers 2 and 3 @xmath175 where @xmath176 . note that @xmath177 , @xmath178 , and @xmath179 . putting together , we can see that the following rate region is achievable . @xmath180 where @xmath181 thus , _ theorem 8 _ is proved . we choose the parameter @xmath182 , which is suboptimal but good enough to achieve a constant gap . this choice of parameter , inspired by @xcite , ensures making efficient use of signal scale difference between @xmath183 and @xmath184 at receiver 1 , while keeping the interference of @xmath91 at the noise level @xmath184 at receiver 2 . by substitution , we get @xmath185 since @xmath186 $ ] , it follows that @xmath187 , and @xmath188 . starting from @xmath189 from table [ tab : outerbounds ] , we can express the two - dimensional outer bound region at @xmath190 as @xmath191 depending on the bottleneck of @xmath192 expressions , there are three cases : * @xmath193 * @xmath194 * @xmath195 . at @xmath196 , the outer bound region is @xmath197 depending on the bottleneck of @xmath192 expressions , we consider the following three cases : * @xmath198 * @xmath199 * @xmath200 . _ case i _ ) @xmath198 : the outer bound region at @xmath196 is @xmath201 for comparison , let us take a look at the achievable rate region . the first term of @xmath202 is lower bounded by @xmath203 we get the lower bounds : @xmath204 for fixed @xmath205 and @xmath206 , the two - dimensional achievable rate region is given by @xmath207 _ case ii _ ) @xmath199 : the outer bound region at @xmath196 is @xmath208 now , let us take a look at the achievable rate region . we have the lower bounds : @xmath209 for fixed @xmath205 and @xmath206 , the two - dimensional achievable rate region is given by @xmath210 _ case iii _ ) @xmath200 : the outer bound region at @xmath196 is @xmath211 for this range of @xmath205 , the rate @xmath190 is small , i.e. , @xmath212 , and @xmath213 and @xmath169 are close to single user capacities @xmath214 and @xmath215 , respectively . let us take a look at the achievable rate region . the first term of @xmath202 is lower bounded by @xmath216 we get the lower bounds : @xmath217 for fixed @xmath205 and @xmath206 , the following two - dimensional rate region is achievable . @xmath218 in all three cases above , by comparing the inner and outer bound regions , we can see that @xmath219 , @xmath220 and @xmath221 . therefore , we can conclude that the gap is to within one bit per message . given @xmath222 $ ] , the region @xmath32 is defined by @xmath223 and @xmath224 is achievable . for this channel type , rate splitting is not necessary . transmit signal @xmath225 is a coded signal of @xmath226 . in particular , @xmath94 and @xmath95 are lattice - coded signals using the same pair of coding and shaping lattices . as a result , the sum @xmath227 is a dithered lattice codeword . the power allocation satisfies @xmath228=\alpha_1 np$ ] , @xmath111=np$ ] , and @xmath112=np$ ] . the received signals are @xmath229+\mathbf{x}_1+\mathbf{z}_1\\ & & \mathbf{y}_2=\mathbf{x}_2+\mathbf{x}_1+\mathbf{z}_2\\ & & \mathbf{y}_3=\mathbf{x}_3+\mathbf{x}_1+\mathbf{z}_3.\end{aligned}\ ] ] the signal scale diagram at each receiver is shown in fig . [ fig : signalscale ] ( b ) . decoding is performed in the following way . * at receiver 1 , @xmath230 $ ] is first decoded while treating @xmath231 as noise . next , @xmath232 is decoded from @xmath233=\mathbf{x}_1+\mathbf{z}_1 $ ] . for reliable decoding , the code rates should satisfy @xmath234 * at receiver 2 , @xmath94 is decoded while treating @xmath235 as noise . similarly at receiver 3 , @xmath95 is decoded while treating @xmath236 as noise . for reliable decoding , the code rates should satisfy @xmath237 putting together , we get @xmath238 where @xmath239 starting from @xmath189 from table [ tab : outerbounds ] , we can express the two - dimensional outer bound region at @xmath213 as @xmath240 depending on the bottleneck of @xmath192 expressions , there are three cases : * @xmath241 * @xmath242 * @xmath243 . at @xmath244 , the region can be expressed as @xmath245 depending on the bottleneck of @xmath192 expressions , we consider the following three cases . _ case i _ ) @xmath246 : the two - dimensional outer bound region at @xmath244 is @xmath247 for fixed @xmath248 and @xmath249 , the following two - dimensional region is achievable . _ case ii _ ) @xmath251 : the two - dimensional outer bound region at @xmath244 is @xmath252 for fixed @xmath248 and @xmath249 , the following two - dimensional region is achievable . @xmath253 _ case iii _ ) @xmath254 : the two - dimensional outer bound region at @xmath244 is @xmath255 for fixed @xmath248 and @xmath249 , the following two - dimensional region is achievable . @xmath256 in all three cases above , by comparing the inner and outer bounds , we can see that @xmath257 , @xmath258 , and @xmath259 . we can conclude that the inner and outer bounds are to within one bit . given @xmath260 $ ] , the region @xmath32 is defined by @xmath261 and @xmath262 is achievable . for this channel type , neither rate splitting nor aligned interference decoding is necessary . transmit signal @xmath225 is a coded signal of @xmath226 . the power allocation satisfies @xmath228=\alpha np$ ] , @xmath111=\alpha np$ ] , and @xmath112=np$ ] . the received signals are @xmath263 the signal scale diagram at each receiver is shown in fig . [ fig : signalscale ] ( c ) . decoding is performed in the following way . * at receiver 1 , @xmath95 is first decoded while treating @xmath231 as noise . next , @xmath232 is decoded from @xmath264 . for reliable decoding , the code rates should satisfy @xmath265 * at receiver 2 , @xmath95 is first decoded while treating @xmath266 as noise . next , @xmath94 is decoded from @xmath267 . for reliable decoding , the code rates should satisfy @xmath268 * at receiver 3 , @xmath95 is decoded while treating @xmath269 as noise . for reliable decoding , the code rates should satisfy @xmath270 putting together , we get @xmath271 where @xmath272 starting from @xmath189 from table [ tab : outerbounds ] , we can express the two - dimensional outer bound region at @xmath169 as @xmath273 depending on the bottleneck of @xmath192 expressions , there are two cases : @xmath274 and @xmath275 . we assume that @xmath275 , equivalently @xmath276 . we also assume that @xmath277 , equivalently @xmath278 . the other cases are trivial . the two - dimensional outer bound region at @xmath279 is @xmath280 for @xmath276 , the two - dimensional outer bound region is @xmath281 for @xmath278 , the two - dimensional achievable rate region at @xmath282 is @xmath283 by comparing the inner and outer bounds , we can see that @xmath284 , @xmath285 , and @xmath286 . we can conclude that the inner and outer bounds are to within one bit . the relaxed outer bound region @xmath30 given by @xmath287 the cross - sectional region at a given @xmath213 is described by @xmath288 depending on the bottleneck of @xmath192 expressions , there are three cases : * @xmath241 * @xmath242 * @xmath243 . in this section , we focus on the third case . the other cases can be proved similarly . if the sum of the righthand sides of @xmath190 and @xmath169 bounds is smaller than the righthand side of @xmath289 bound , i.e. , @xmath290 then the @xmath289 bound is not active at the @xmath213 . this condition can be expressed as a threshold on @xmath213 given by @xmath291 for this relatively large @xmath213 , the cross - sectional region is a rectangle as described in fig . [ fig : outerboundcrosssection4 ] ( a ) . in contrast , for a relatively small @xmath213 , when the threshold condition does not hold , the cross - sectional region is a mac - like region as described in fig . [ fig : outerboundcrosssection4 ] ( b ) . in the rest of the section , we present achievable schemes for each case . given @xmath292 ^ 3 $ ] , the region @xmath32 is defined by @xmath293 where @xmath294 and @xmath295 , and @xmath262 is achievable . we present an achievable scheme for the case of @xmath296 . message @xmath85 is split into three parts : @xmath87 , @xmath86 and @xmath297 , so @xmath298 . we generate the signals in the following way : @xmath90 and @xmath299 are differently coded signals of @xmath92 , and @xmath91 and @xmath300 are coded signal of @xmath93 and @xmath301 , respectively . the transmit signal is the sum @xmath302 the power allocation satisfies @xmath110=\alpha_0 np$ ] , @xmath109=\alpha_2 np$ ] , @xmath303=\alpha_1 np$ ] , and @xmath304=(1-\alpha_0-\alpha_1-\alpha_2 ) np$ ] . the transmit signals @xmath94 and @xmath95 are coded signals of the messages @xmath96 and @xmath97 , satisfying @xmath111=\alpha_2 np$ ] and @xmath112=np$ ] . the signals @xmath299 and @xmath95 are lattice - coded signals using the same coding lattice but different shaping lattices . as a result , the sum @xmath305 is a dithered lattice codeword . the received signals are @xmath306+\mathbf{x}_{12}+\mathbf{x}_{11}+\mathbf{x}_{10}+\mathbf{z}_1\\ & & \mathbf{y}_2=\mathbf{x}_{11}'+\mathbf{x}_{12}+\mathbf{x}_{11}+\mathbf{x}_{2}+\mathbf{x}_{10}+\mathbf{z}_2\\ & & \mathbf{y}_3=\mathbf{x}_3+\mathbf{x}_2+\mathbf{z}_3.\end{aligned}\ ] ] the signal scale diagram at each receiver is shown in fig . [ fig : signalscale4 ] ( a ) . decoding is performed in the following way . * at receiver 1 , @xmath307 $ ] is first decoded while treating other signals as noise and removed from @xmath308 . next , @xmath300 , @xmath90 , and @xmath91 are decoded successively . for reliable decoding , the code rates should satisfy @xmath309 where @xmath310 and @xmath311 . note that @xmath177 , @xmath178 , and @xmath179 . * at receiver 2 , @xmath299 is first decoded while treating other signals as noise . having successfully recovered @xmath92 , receiver 2 can generate @xmath90 and @xmath299 , and cancel them from @xmath136 . next , @xmath300 is decoded from @xmath312 . finally , @xmath94 is decoded from @xmath313 . for reliable decoding , the code rates should satisfy @xmath314 * at receiver 3 , @xmath95 is decoded while treating @xmath168 as noise . reliable decoding is possible if @xmath315 putting together , we can see that given @xmath316 $ ] , the following rate region is achievable . @xmath317 where @xmath318 we choose @xmath319 , @xmath248 and @xmath205 such that @xmath320 , that @xmath321 , that @xmath322 , and that @xmath323 . it follows that @xmath324 , that @xmath325 , and that @xmath326 . we get the lower bounds for each term of @xmath202 expression above . @xmath327 and @xmath328 since @xmath329 , @xmath330 putting together , @xmath331 given @xmath248 , we choose @xmath205 that satisfies @xmath332 . as a result , we can write @xmath333 , and also @xmath334 since @xmath335 , @xmath336 the following rate region is achievable . @xmath337 for fixed @xmath248 and @xmath338 , the two - dimensional rate region , given by @xmath339 is achievable . in comparison , the two - dimensional outer bound region at @xmath340 , given by @xmath341 as discussed above , the sum - rate bound on @xmath289 is loose for @xmath213 larger than the threshold , so the rate region is a rectangle . by comparing the inner and outer bound rate regions , we can see that @xmath342 and @xmath343 . therefore , we can conclude that the gap is to within one bit per message . given @xmath292 ^ 3 $ ] , the region @xmath32 is defined by @xmath344 where @xmath345 and @xmath346 , and @xmath262 is achievable . for the case of @xmath347 , we present the following achievable scheme . at transmitter 1 , we split @xmath348 into @xmath93 and @xmath92 , so @xmath349 . the transmit signal is the sum @xmath350 the power allocation satisfies @xmath110=\alpha_0 np$ ] , @xmath109=(\alpha_1-\alpha_0 ) np$ ] , and @xmath304=(1-\alpha_1)np$ ] at receiver 1 , @xmath111=\alpha_2 np$ ] at receiver 2 , and @xmath112=np$ ] at receiver 3 . the signals @xmath299 and @xmath95 are lattice codewords using the same coding lattice but different shaping lattices . as a result , the sum @xmath305 is a lattice codeword . the received signals are @xmath351+\mathbf{x}_{11}+\mathbf{x}_{10}+\mathbf{z}_1\\ & & \mathbf{y}_2 = \mathbf{x}_{11}'+\mathbf{x}_{11}+\mathbf{x}_2+\mathbf{x}_{10}+\mathbf{z}_2\\ & & \mathbf{y}_3 = \mathbf{x}_3+\mathbf{x}_2+\mathbf{z}_3.\end{aligned}\ ] ] the signal scale diagram at each receiver is shown in fig . [ fig : signalscale4 ] ( b ) . decoding is performed in the following way . * at receiver 1 , @xmath307 $ ] is first decoded while treating other signals as noise and removed from @xmath308 . next , @xmath90 and then @xmath91 is decoded successively . for reliable decoding , the code rates should satisfy @xmath352 where @xmath353 and @xmath354 . note that @xmath177 , @xmath178 , and @xmath179 . * at receiver 2 , @xmath299 is first decoded while treating other signals as noise . having successfully recovered @xmath92 , receiver 1 can generate @xmath90 and @xmath299 , and cancel them from @xmath136 . next , @xmath94 is decoded from @xmath313 . at receiver 2 , @xmath91 is not decoded . for reliable decoding , the code rates should satisfy @xmath355 * at receiver 3 , @xmath95 is decoded while treating @xmath168 as noise . reliable decoding is possible if @xmath315 putting together , we can see that given @xmath356 $ ] , the following rate region is achievable . @xmath357 where @xmath358 we choose @xmath319 , @xmath248 , and @xmath205 such that @xmath359 , that @xmath360 , that @xmath322 , and that @xmath361 . it follows that @xmath325 and that @xmath362 . @xmath363 and @xmath364 putting together , @xmath365 let us define @xmath366 by the equality @xmath367 . if we choose @xmath368 , then @xmath369 , and @xmath370 we can see that the following rate region is achievable . @xmath371 for fixed @xmath372 $ ] and @xmath373 , the two - dimensional rate region @xmath32 , given by @xmath374 is achievable . the union @xmath375}\mathcal{r}_\alpha$ ] is a mac - like region , given by @xmath376 this region is described in fig . [ fig : maclike ] ( a ) . in comparison , the two - dimensional outer bound region at @xmath377 , given by @xmath378 since @xmath379 , @xmath380 and @xmath381 , we can conclude that the gap is to within one bit per message . let us consider the relaxed outer bound region @xmath30 given by @xmath382 the cross - sectional region at a given @xmath190 is described by @xmath383 depending on the bottleneck of @xmath192 expressions , there are three cases : * @xmath193 * @xmath194 * @xmath195 . in this section , we focus on the third case . the other cases can be proved similarly . if the sum of the righthand sides of @xmath213 and @xmath169 bounds is smaller than the righthand side of @xmath384 bound , i.e. , @xmath385 then the @xmath384 bound is not active at the @xmath190 . by rearranging , the threshold condition is given by @xmath386 note that @xmath387 is roughly half of @xmath388 . for this relatively large @xmath190 , the cross - sectional region is a rectangle as described in fig . [ fig : outerboundcrosssection5 ] ( a ) . in contrast , for a relatively small @xmath213 , when the threshold condition does not hold , the cross - sectional region is a mac - like region as described in fig . [ fig : outerboundcrosssection5 ] ( b ) . in the following subsections , we present achievable schemes for each case . given @xmath389 ^ 3 $ ] , the region @xmath32 is defined by @xmath390 where @xmath391 and @xmath392 , and @xmath262 is achievable . we present an achievable scheme for the case of @xmath393 . message @xmath96 for receiver 2 is split into two parts : @xmath394 and @xmath395 , so @xmath396 . we generate the signals in the following way : @xmath397 and @xmath398 are differently coded signals of @xmath399 , and @xmath400 is a coded signal of @xmath401 . the transmit signal is the sum @xmath402 the power allocation satisfies @xmath228=\alpha_1 np$ ] , at receiver 1 , @xmath403=\alpha_2 ' np$ ] , @xmath404=\alpha_2 np$ ] , and @xmath405=(1-\alpha_2-\alpha_2')p$ ] at receiver 2 , and @xmath112=np$ ] at receiver 3 . the signals @xmath398 and @xmath95 are lattice codewords using the same coding lattice but different shaping lattices . as a result , the sum @xmath406 is a lattice codeword . the received signals are @xmath407+\mathbf{x}_{22}+\mathbf{x}_{21}+\mathbf{z}_2\\ & & \mathbf{y}_3=\mathbf{x}_3+\mathbf{x}_1+\mathbf{z}_3.\end{aligned}\ ] ] the signal scale diagram at each receiver is shown in fig . [ fig : signalscale5 ] ( a ) . decoding is performed in the following way . * at receiver 1 , @xmath398 is first decoded while treating other signals as noise . having successfully recovered @xmath399 , receiver 1 can generate @xmath397 and @xmath398 , and cancel them from @xmath308 . next , @xmath400 is decoded from @xmath408 . finally , @xmath232 is decoded from @xmath231 . for reliable decoding , the code rates should satisfy @xmath409 * at receiver 2 , @xmath410 $ ] first decoded while treating other signals as noise and removed from @xmath136 . next , @xmath400 and @xmath397 are decoded successively . for reliable decoding , the code rates should satisfy @xmath411 where @xmath412 and @xmath413 . note that @xmath414 , @xmath415 , and @xmath179 . * at receiver 3 , @xmath95 is decoded while treating @xmath236 as noise . reliable decoding is possible if @xmath416 putting together , we can see that given @xmath417 $ ] , the following rate region is achievable . @xmath418 where @xmath419 we choose @xmath248 and @xmath205 such that @xmath420 , that @xmath198 , that @xmath421 , and that @xmath422 . it follows that @xmath423 . we get the lower bounds for each term of @xmath424 expression above . @xmath425 the first entry of @xmath192 in @xmath426 is lower bounded as follows . @xmath427 the second entry of @xmath428 is lower bounded as follows . @xmath429 putting together , we get the lower bound @xmath430 given @xmath205 , we choose @xmath248 that satisfies @xmath431 . as a result , we can write @xmath432 . we also have @xmath433 putting together , we can see that the following rate region is achievable . @xmath434 for fixed @xmath205 and @xmath435 , the two - dimensional rate region , given by @xmath436 is achievable . in comparison , the two - dimensional outer bound region at @xmath437 is given by @xmath438 as discussed above , the sum - rate bound on @xmath384 is loose for @xmath190 larger than the threshold , so the rate region is a rectangle . by comparing the inner and outer bound rate regions , we can see that @xmath439 and @xmath440 . therefore , we can conclude that the gap is to within one bit per message . given @xmath441 ^ 2 $ ] , the region @xmath32 is defined by @xmath442 where @xmath443 and @xmath444 , and @xmath262 is achievable . for the case of @xmath445 , we present the following scheme . at transmitter 2 , rate splitting is not necessary . the transmit signal is the sum @xmath446 where @xmath397 and @xmath398 are differently coded versions of the same message @xmath96 . the power allocation : @xmath228=\alpha_1 np$ ] at receiver 1 , @xmath403=\alpha_2 np$ ] , and @xmath405=(1-\alpha_2)np$ ] at receiver 2 , and @xmath112=np$ ] at receiver 3 . the signals @xmath398 and @xmath95 are lattice codewords using the same coding lattice but different shaping lattices . as a result , the sum @xmath406 is a lattice codeword . the received signals are @xmath447+\mathbf{x}_{21}+\mathbf{z}_2\\ & & \mathbf{y}_3=\mathbf{x}_3+\mathbf{x}_1+\mathbf{z}_3.\end{aligned}\ ] ] the signal scale diagram at each receiver is shown in fig . [ fig : signalscale5 ] ( b ) . decoding is performed in the following way . * at receiver 1 , @xmath398 is first decoded while treating other signals as noise . having successfully recovered @xmath399 , receiver 1 can generate @xmath397 and @xmath398 , and cancel them from @xmath308 . next , @xmath232 is decoded from @xmath231 . for reliable decoding , the code rates should satisfy @xmath448 * at receiver 2 , @xmath410 $ ] first decoded while treating other signals as noise and removed from @xmath136 . next , @xmath397 is decoded from @xmath449 . for reliable decoding , the code rates should satisfy @xmath450 where @xmath451 and @xmath452 . note that @xmath414 , @xmath415 , and @xmath179 . * at receiver 3 , @xmath95 is decoded while treating @xmath236 as noise . reliable decoding is possible if @xmath416 putting together , we get @xmath453 where @xmath454 we choose @xmath248 and @xmath205 such that @xmath420 , that @xmath198 , that @xmath422 , and that @xmath455 . it follows that @xmath423 . we get the lower bound @xmath456 and @xmath457 let us define @xmath458 by the equality @xmath459 . if we choose @xmath460 , then @xmath461 . we can see that the following rate region is achievable . @xmath462 for fixed @xmath463 $ ] and @xmath435 , the two - dimensional rate region @xmath32 , given by @xmath464 is achievable . the union @xmath465}\mathcal{r}_\alpha$ ] is a mac - like region , given by @xmath466 in comparison , the two - dimensional outer bound region at @xmath437 is given by @xmath467 since @xmath468 , @xmath469 and @xmath470 , we can conclude that the gap is to within one bit per message . we presented approximate capacity region of five important cases of partially connected interference channels . the outer bounds based on @xmath471-channel type argument are derived . achievable schemes are developed and shown to approximately achieve the capacity to within a constant bit . for future work , the channels with fully general coefficients may be considered . in this paper , we presented different schemes for each channel type although they share some principle . a universal scheme is to be developed for unified capacity characterization of all possible topologies . the connection between interference channel and index coding problems is much to explore . in particular , the results on the capacity region for index coding in @xcite seem to have an interesting connection to our work . at transmitter 1 , message @xmath348 is split into three parts @xmath472 , and the transmit signal is @xmath473 . the signals satisfy @xmath303=n(p - n_2-n_3)$ ] , @xmath109=nn_3 $ ] , and @xmath110=nn_2 $ ] . at transmitter 2 , message @xmath474 is split into three parts @xmath475 , and the transmit signal is @xmath476 . the signals satisfy @xmath403=n(p - n_3)$ ] and @xmath477=nn_3 $ ] . rate - splitting is not performed at transmitter 3 , and @xmath112=np$ ] . the top layer codewords @xmath478 are from a joint random codebook for @xmath479 . the mid - layer codewords @xmath480 are from a joint random codebook for @xmath481 . the bottom layer codeword @xmath91 is from a single - user random codebook for @xmath93 . the received signals are @xmath482 decoding is performed from the top layer to the bottom layer . at receiver 1 , simultaneous decoding of @xmath483 is performed while treating other signals as noise . and then , @xmath90 and @xmath91 are decoded successively . at receiver 2 , simultaneous decoding of @xmath484 is performed while treating other signals as noise . and then , simultaneous decoding of @xmath480 is performed . at receiver 3 , simultaneous decoding of @xmath485 is performed while treating other signals as noise . for reliable decoding , code rates should satisfy @xmath486 at receiver 1 , @xmath487 at receiver 2 , @xmath488 at receiver 3 . putting together , @xmath489 at the top layer , @xmath490 at the mid - layer , @xmath491 at the bottom layer . note that the rate variables are not coupled between layers . we get the achievable rate region @xmath492 this region includes the following region . @xmath493 therefore , we can conclude the capacity region to within one bit . transmit signal construction is the same as the one for channel type 4 . the received signals are @xmath494 decoding is performed from the top layer to the bottom layer . at receiver 1 , simultaneous decoding of @xmath484 is performed while treating other signals as noise . and then , simultaneous decoding of @xmath90 and @xmath495 is performed . lastly , @xmath91 is decoded . at receiver 2 , simultaneous decoding of @xmath496 is performed while treating other signals as noise . and then , @xmath495 is decoded . at receiver 3 , simultaneous decoding of @xmath497 is performed while treating other signals as noise . and then , @xmath90 and @xmath91 are decoded successively . for reliable decoding , code rates should satisfy @xmath498 at receiver 1 , @xmath499 at receiver 2 , @xmath500 at receiver 3 . putting together , @xmath501 at the top layer , @xmath502 at the mid - layer , @xmath503 at the bottom layer . note that the rate variables are not coupled between layers . we get the achievable rate region @xmath492 this region includes the following region . @xmath504 therefore , we can conclude the capacity region to within one bit . s. sridharan , a. jafarian , s. vishwanath , s. jafar , and s. shamai , `` a layered lattice coding scheme for a class of three user gaussian interference channel , '' _ 46th annual allerton conference on communication , control , and computing , _ pp . 531538 , 2008 . g. bresler , a. parekh , and d. n. c. tse , `` the approximate capacity of the many - to - one and one - to - many gaussian interference channels , '' _ ieee trans . theory , _ vol . 9 , pp . 45664592 , sep . 2010 . a. s. motahari , s. o .- gharan , m .- a . maddah - ali , and a. k. khandani , `` real interference alignment : exploiting the potential of single antenna systems , '' _ ieee trans . theory , _ vol . 47994810 , aug . 2014 . m. p. wilson , k. narayanan , h. pfister , and a. sprintson , `` joint physical layer coding and network coding for bidirectional relaying , '' _ ieee trans . theory , _ vol . 56415654 , nov .

we derive inner and outer bounds on the capacity region for a class of three - user partially connected interference channels . we focus on the impact of topology , interference alignment , and interplay between interference and noise . the representative channels we consider are the ones that have clear interference alignment gain . for these channels , z - channel type outer bounds are tight to within a constant gap from capacity . we present near - optimal achievable schemes based on rate - splitting and lattice alignment . interference channel , interference alignment , nested lattice code , side information graph , topological interference management .

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Proceed to summarize the following text: establishing an observational mapping between the properties of core - collapse supernovae ( ccsne ) explosions ( and related luminous outbursts ) and local populations of massive stars is key for constraining stellar evolution theory ( e.g. , langer 2012 ) . arguably , the best link so far comes from the direct detections of red - supergiants with main - sequence masses @xmath6 m@xmath7 as progenitors of type iip supernovae ( e.g. , smartt 2009 , and references therein ) , the most common kind of ccsne . however , direct detections of sn progenitors with higher masses ( @xmath8 m@xmath7 ) has been elusive ( e.g. , kochanek et al . 2008 ; smartt 2009 ) . the detection of a very luminous star ( @xmath9 mag ) identified in pre - explosion images at the location of the type iin sn 2005gl provided the first direct evidence for a very massive h - rich star ( @xmath10 m@xmath7 ) that exploded as a luminous ccsn ( gal - yam et al . 2007 ; gal - yam & leonard 2009 ) . these observations lend support for the connection made initially by smith et al . ( 2007 ) between one of the most luminous type iin sn 2006gy and very massive ( @xmath11 m@xmath7 ) luminous blue variable ( lbv ) stars like @xmath12 carinae . there is now mounting evidence of this connection ( e.g. , sn 2006jc , pastorello et al . 2007 ; sn 2010jl , smith et al . 2011a ; sn 1961v , kochanek et al . 2011 , smith et al . 2011b ) , which challenges massive stellar evolution theory in at least two important ways : 1 ) the most massive stars are not expected to be h - rich at the time when they explode as ccsne , at least with normal mass - loss prescriptions ; and 2 ) these stars have to loose up to a few solar masses of h - rich material in eruptions just before ( months to decades ) the core - collapse explosion in order to make them visible as type iin sne . on this last point of timing strong eruptive mass - loss with the core - collapse explosion , there is some very interesting theoretical work , but its still very early days ( e.g. , woosley et al . 2007 ; arnett & meakin 2011 ; quataert & shiode 2012 ; chevalier 2012 ) . luckily , in the last month nature has been kind and has provided the best case to date of a very massive star exploding as a bright core - collapse supernova : sn 2009ip . this transient was discovered on 2009 aug . 29 by the chase survey in the outskirts of the galaxy ngc 7259 ( maza et al . 2009 ) and was initially given a sn name by the iau . however , detailed photometric and spectroscopic studies by smith et al . ( 2010 ) and foley et al . ( 2011 ) showed that it never quite reached supernova status . it was actually a supernova impostor ( e.g. , humphreys & davidson 1994 , kochanek et al . 2012 ; with peak absolute mag . @xmath13 , similar to @xmath12 carinae s great eruption , but significantly shorter timescale ) , in this case a massive lbv with estimated mass @xmath14 m@xmath7 ( constrained from pre - discovery imaging ) that had an eruptive mass - loss event . the catalina real - time transient survey ( crts ) discovered another outburst of comparable amplitude during 2010 ( drake et al . 2010 ) and a third outburst on 2012 july 24 ( drake et al . 2012 ) , the brightest so far . the initial optical spectra obtained by foley et al . ( 2012 ) on aug . 26 showed narrow emission lines of h and he i , consistent with the spectra obtained in the 2009 outburst . however , an unprecedented change of the spectrum was reported by smith & mauerhan ( 2012a ) . 15 and 16 they had detected very broad emission lines with p - cygni profiles , consistent with features observed in real type ii sne ; the massive lbv star had probably exploded as a supernova in real time ! another dramatic change in the spectral properties was reported by smith & mauerhan ( 2012b ) , by sep . 28 the broad p - cygni features had mostly disappeared , leaving behind narrow features characteristic of type iin sne . all the details of their discovery and their follow - up observations can be found in mauerhan et al . ( 2012b ; hereafter m12b ) . pastorello et al . ( 2012 ; hereafter p12 ) has also presented detailed follow - up observations of sn 2009ip and propose that the latest outburst might not be the final core - collapse explosion . in this paper we present high - cadence optical observations of sn 2009ip that clearly resolve the brightening initially reported by brimacombe ( 2012 ) . in section [ sec1 ] we discuss the observations , data reduction , and extraction of optical light curves using difference imaging photometry . in section [ sec2 ] we present an analysis of the different phases of the light curve . in section [ sec3 ] we present a discussion of the observed properties . in section [ sec4 ] we present the conclusions of this work . we adopt a distance of 20.4 mpc ( @xmath15 mag ) and galactic extinction of @xmath16 mag towards sn 2009ip ( m12b ) throughout the paper . all the dates presented in this paper are ut . imaging of the field of sn 2009ip was obtained by one of us ( j. b. ) between 2012 sep . 23.6 and oct . 9.6 from the coral towers observatory ( cairns , australia ) . the data were collected from two different telescopes using different broad - band filters : 33-cm rcos ( with an @xmath17 filter ) and 41-cm rcos ( with an ir luminance filter , which has sharp blue cutoff at @xmath18 nm ) . the ccd cameras used in both telescopes are identical 3k@xmath192k sbig stl6k , which give a total field of view of @xmath20 ( 1/pix platescale binned @xmath21 ) . all the images were obtained using an exposure time of 900 sec . a section of an ir image of sn 2009ip obtained on sep . 24 is shown in figure 1 . we used the software maxim dl ( version 4.62 ) to complete the initial data reduction , which consists of bias subtraction , dark subtraction , and flat - fielding . after visually inspecting all images and rejecting frames affected by bad tracking , we kept 85 ( @xmath17 ) and 118 ( ir ) frames for further photometric analysis . we used the image subtraction package isis2 ( alard 2000 ) to extract the fluxes of sn 2009ip in individual frames following the steps described in hartman et al . ( 2004 ) . as reference images , we used stacks of @xmath22 images with good seeing obtained on sep . we carried out psf photometry with the daophot ii package ( stetson 1992 ) to subtract the sn flux and used the resulting sn - subtracted images as the final reference frames . the sn fluxes obtained with isis2 in the @xmath17 and ir images were calibrated to standard @xmath17 and @xmath23 magnitudes , respectively , using data from the aavso photometric all - sky survey ( apass ) . we obtained from apass calibrated sloan r and i magnitudes of 5 isolated stars in the field of sn 2009ip . we converted these magnitudes in the sdss photometric system to standard @xmath24 ( johnson / kron - cousins ) photometry using the transformation equations obtained by r. lupton . in order to estimate the zeropoints , we performed aperture photometry of the 5 local standard stars in the reference images using a 9radius aperture with the iraf task phot . the resulting average zeropoints in the @xmath24 filters have a standard deviation of 0.02 mag . the final photometry of sn 2009ip calibrated using these zeropoints is presented in table 1 and the light curves are shown in figure 1 . the errors in each magnitude estimate include poisson errors from isis2 as well as the estimated standard error in the photometric zeropoints . our photometry is in excellent agreement with the results presented in p12 , with mean differences during the period of observations of @xmath25 mag in @xmath17 and @xmath26 mag in @xmath23 . the high cadence and number of observations allow us to clearly resolve the brightening of sn 2009ip , initially reported in brimacombe ( 2012 ) and prieto et al . the optical light curves presented in figure 2 show well - defined phases : 1 ) approximately constant magnitude ( sep . 23 ) ; 2 ) very rapid brightening in a period of hours ( sep . @xmath27 ) ; 3 ) turn - over to significantly slower brightening ( sep . @xmath28 ) ; and 4 ) slow brightening , reaching peak magnitude ( sep . 30 @xmath29 oct . 9 ) . we describe these phases in more detail below . between sep . 23.56 and 23.66 , the magnitude of sn 2009ip is consistent with being constant at @xmath30 mag ( @xmath2 ) . however , between sep . 23.66 and 24.45 sn 2009ip brightened by @xmath31 mag and continued a very rapid brightening throughout the 6 hrs of continuous imaging obtained during the night of sep . 24 . in this period , it brightened between @xmath32 ( sep . 24.45 ) and @xmath33 mag ( sep . 24.70 ) . a linear fit to the @xmath23-band rise on sep . 24 gives a slope of @xmath34 mag day@xmath35 . between sep . 24.70 and 25.38 the sn brightened by @xmath36 mag , reaching @xmath37 mag ( @xmath38 ) . the light curve turned over to a slower brightening on sep . a linear fit to the data obtained during the night of sep . 25 gives a slope of @xmath39 mag day@xmath35 . the turn - over can be seen clearly in both the @xmath17 and @xmath23-band photometric data obtained between sep . 25 and sep . 28 ( see figure 2 ) , reaching @xmath40 mag ( @xmath41 ) by sep . 28.47 . the latest part of the light curve between sep . 30 and oct . 7 shows a slow and steady brightening at a rate of @xmath42 mag day@xmath35 ( @xmath43 mag ) . however , in our most recent images obtained on oct . 9.6 the sn faded by @xmath44 mag with respect to the images obtained on oct . the magnitude of sn 2009ip on oct . 9.6 was @xmath45 mag ( @xmath4 ) . the @xmath46 color is consistent with being constant ( @xmath47 ) during the period between sep . 25 and oct . our photometric observations , obtained during a 16-day period between sep . 23.6 and oct . 9.6 show that the type iin sn 2009ip brightened by 3.7 mag in the optical to an absolute magnitude @xmath48 ( see also margutti et al . 2012b ; m12b ; p12 ) and it has reached peak magnitude around oct . 7 ( see figure 2 ) . its peak luminosity is in the observed range of type ii sne ; in fact it is more luminous than most type iip sne ( e.g. , li et al . the comparison to other type iin sne is less straightforward since the rates are lower and the samples are incomplete . however , there are several well - studied type iin sne with peak absolute magnitudes around @xmath49 ( e.g. , kiewe et al . 2012 ; stritzinger et al . 2012 ; roming et al . 2012 ; mauerhan et al . 2012a ) . we can estimate the total radiated energy during the period of the observations presented here . we need an estimate of the bolometric correction to convert the @xmath23-band luminosities to bolometric luminosities and we do this in two ways . first , we use the observed constant @xmath46 color , corrected for galactic extinction , and fit a black - body to estimate the luminosity ratio @xmath50 and @xmath51 k. second , we use the longer wavelength coverage ( from near - uv to @xmath52-band ) of the swift observations presented in margutti et al . ( 2012b ) and fit a black - body to the extinction - corrected fluxes , obtaining @xmath53 and @xmath54 k. we integrate the @xmath23-band luminosities estimated from the extinction - corrected magnitudes . assuming a constant bolometric correction during the 16-day observing period , we obtain a total ( optical ) radiated energy of @xmath55 erg and luminosities of @xmath56 erg s@xmath35 as of oct . 9 , depending on the correction used . this is already comparable to the estimated radiated energy of the type iin sn 2011ht during its @xmath57 day plateau phase ( roming et al . 2012 ; mauerhan et al . 2012a ) . the evolution of sn 2009ip since the discovery of its latest recorded eruption on 2012 jul . 24 ( drake et al . 2012 ) is quite remarkable ( figure 3 ; see also m12b and p12 ) , as it is unlike any other type iin sne studied to date . the absolute magnitude was approximately constant at @xmath58 ( @xmath52 to @xmath23-bands ) for a period of @xmath59 days after jul . 24 , and even faded to @xmath60 ( at least between sep . 18 and sep . 22.5 , martin et al . 2012 ; margutti et al . 2012a ) before the rapid brightening that ( given our observations ) most likely started between sep . 23.7 and 24.5 . if we assume the maximum bolometric correction estimated above as an upper limit and a plateau phase at @xmath61 for @xmath62 days , we constrain the total radiated energy during this period to be @xmath63 erg . however , we should note that the brightening could have started earlier than jul . 24 , in which case this could be a lower limit in the total radiated energy during this phase . the spectroscopic evolution of sn 2009ip since its latest recorded eruption have also been unique . foley et al . ( 2012 ) reported an optical spectrum obtained on aug . 26 , which showed a blue continuum with narrow ( @xmath64 km s@xmath35 ) emission lines of h and he i , similar to the spectra obtained during the 2009 eruption ( smith et al . 2009 ; foley et al . 2012 ) . vinko et al . ( 2012 ) reported that an independent spectrum obtained on aug . 26 also showed broad p - cygni features with very fast velocities of @xmath65 km s@xmath35 , consistent with sn shock speeds . these broad features seem to have persisted , and even strengthened , until ( at least ) sep . 23 ( m12b ) . however , the high velocity features and p - cygni profiles had mostly gone away by sep . 26 - 27 , leaving behind strong , narrow ( @xmath66 km s@xmath35 ) emission features characteristic of many type iin sne spectra ( m12b ; burgasser et al . 2012 ; vinko et al . 2012 ; gall et al . 2012 ; p12 ) . as shown in figure 3 , the transition in spectroscopic properties is coincident with the very rapid brightening observed in the optical light curves . this , most likely , marks the start of a strong interaction between the fast sn shock and a dense csm environment likely formed from material ejected in the previous lbv outbursts , as proposed by m12b . the timing of the rapid optical brightening is also coincident with the detection of x - ray emission from sn 2009ip ( campana & margutti 2012 ) , which is consistent with this picture . another possible signature of a strong ejecta - csm interaction is the decline of @xmath67 mag observed right before the spectral changes were seen and the light curve started to rise rapidly . moriya & maeda ( 2012 ) propose that this is an unavoidable consequence of a fast sn shock breaking out of a dense csm . figure 4 shows a zoom - in view of the evolution of the optical luminosity of sn 2009ip after sep . 23.5 , which is approximately when it started to transition to a type iin - dominated spectrum . for comparison , we also show the optical luminosity evolution of three type iin sne from the literature that have relatively well - sampled light curves before and after peak brightness . this sample includes the very luminous sn 2003ma ( rest et al . 2011 ) and sn 2006gy ( smith et al . 2007 ) , and the relatively low - luminosity sn 2011ht ( roming et al . 2012 ; mauerhan et al . 2012a ; but see humphreys et al . 2012 ) . the very early rapid optical brightening of the type iin sn 2009ip is consistent with @xmath68 during a short 2-day phase , although this naturally depends on the @xmath69 used . this is the evolution expected for an homologously expanding ( optically thick ) black - body photosphere at constant expansion velocity and effective temperature . the luminous type iin sn 2003ma and sn 2006gy show relatively similar evolution in their optical luminosity at early times before peak . in fact , smith & mccray ( 2007 ) modeled the early rise of sn 2006gy with c model . sn 2011ht rises at a slower rate in the optical , although it was observed to rise much faster in the uv - range ( roming et al . 2012 ) . given the simplified assumptions that go into the @xmath70 scaling , we plan to explore more realistic models ( e.g. , moriya et al . 2012 ) to fit the early time light curve of sn 2009ip in a future publication . at later times , close to peak brightness , the optical luminosity and light curve shape of sn 2009ip becomes more similar to sn 2011ht than sn 2003ma or sn 2006gy . mauerhan et al . ( 2012a ) proposed that sn 2011ht belongs to a growing sub - class of type iin sne ( type iin - p ; including also sn 2009kn , kankare et al . 2012 , and sn 1994w , sollerman et al . 1998 ) with long - lasting post - peak plateaus of @xmath57 days ( @xmath71 mag ) , and faint late - time decay slopes consistent with low @xmath72ni yield production ( @xmath73 m@xmath7 ) . perhaps we are seeing something similar in the case of sn 2009ip ( but see p12 ) , although there are important differences in spectroscopic properties compared to the small group of type iin - p sne ( e.g. , sn 2009ip shows significantly hotter continuum and the narrow lines do not show p - cygni profiles ) . an interesting related issue that comes out from the recent observations of sn 2009ip before the rapid brightening and fast transition to narrow - line dominated spectrum , is whether surveys might be missing a relatively low - luminosity phase ( at least compared to the main peak ) in which the broad sn shock velocities are revealed . given the typical depth of transient surveys , this is certainly possible ( e.g. , see fig . 17 in kiewe et al . in fact , smith et al . ( 2011b ) proposed such an origin for the @xmath57 day plateau at @xmath74 mag observed before the main peak in the light curve of sn 1961v . we have presented high - cadence optical photometric monitoring of the remarkable , young type iin supernova sn 2009ip using data obtained between 2012 sep . 23 and oct . 9 with @xmath75-@xmath76 meter aperture telescopes from the coral towers observatory ( cairns , australia ) . using difference imaging analysis , we obtain precise @xmath17 and @xmath23-band light curves and are able to resolve well - defined brightening phases . in particular , we see a very rapid brightening early on ( 0.5 mag in 6 hr ) that quickly turns over after a couple of days . the sn brightened between absolute magnitude @xmath77 and @xmath78 in 16 days . the changes that we observe in the light curves are correlated with the reported spectroscopic changes and the x - ray detection , and , most likely , mark the start of a strong interaction between the fast sn ejecta and the dense csm formed from material that was ejected in the pre - supernova lbv eruptions observed since 2009 . our study highlights the importance of organized , high - cadence observations by amateur astronomers with small - aperture telescopes and the impact they can have in transients and sne research , in this case with direct important connections to massive stellar evolution . we thank s. de mink , o. graur , j. mauerhan , r. quimby , and a. rest for stimulating discussions about sn 2009ip , and l. watson for detailed comments on an earlier version of this manuscript . j. l. p. acknowledges support from a carnegie - princeton fellowship . lcccc sep 24.536400 & 195.040909 & 16.549 & 0.047 & @xmath17 + sep 24.552130 & 195.056641 & 16.494 & 0.045 & @xmath17 + sep 24.567801 & 195.072327 & 16.382 & 0.040 & @xmath17 + sep 24.583472 & 195.087997 & 16.403 & 0.043 & @xmath17 + sep 24.599144 & 195.103668 & 16.356 & 0.044 & @xmath17 + sep 24.614826 & 195.119370 & 16.256 & 0.039 & @xmath17 + sep 24.633634 & 195.138153 & 16.211 & 0.040 & @xmath17 + sep 24.644965 & 195.149475 & 16.259 & 0.043 & @xmath17 + sep 24.656285 & 195.160797 & 16.256 & 0.040 & @xmath17 + sep 24.667593 & 195.172134 & 16.250 & 0.038 & @xmath17 + & & & & @xmath17 +

recent observations by mauerhan et al . have shown the unprecedented transition of the previously identified luminous blue variable ( lbv ) and supernova impostor sn 2009ip to a real type iin supernova ( sn ) explosion . we present high - cadence optical imaging of sn 2009ip obtained between 2012 ut sep . 23.6 and oct . 9.6 , using @xmath0 meter aperture telescopes from the coral towers observatory in cairns , australia . the light curves show well - defined phases , including very rapid brightening early on ( 0.5 mag in 6 hr observed during the night of sep . 24 ) , a transition to a much slower rise between sep . 25 and sep . 28 , and a plateau / peak around oct . 7 . these changes are coincident with the reported spectroscopic changes that , most likely , mark the start of a strong interaction between the fast sn ejecta and a dense circumstellar medium formed during the lbv eruptions observed in recent years . in the 16-day observing period sn 2009ip brightened by 3.7 mag from @xmath1 mag on sep . 23.6 ( @xmath2 ) to @xmath3 mag ( @xmath4 ) on oct . 9.6 , radiating @xmath5 erg in the optical wavelength range . currently , sn 2009ip is more luminous than most type iip sne and comparable to other type iin sne .

You are an expert at summarizing long articles. Proceed to summarize the following text: recently , verlinde @xcite presented a remarkable new idea that gravity can be explained as an entropic force caused by the information changes when a material body moves away from the holographic screen . this idea implies that gravity is not fundamental . with the holographic principle and the equipartition law of energy , verlinde showed that newton s law of gravitation can arise naturally and unavoidably in a theory in which space is emergent through a holographic scenario , and a relativistic generalization leads to the einstein equations . in fact , the similar idea can be traced back to sakharov s work @xcite . on the other side , using the equipartition law of energy for the horizon degrees of freedom together with the thermodynamic relation @xmath3 , padmanabhan also obtained the newton s law of gravity @xcite . subsequently , with the idea of entropic force , some applications have been carried out . the friemdann equations and the modified friedmann equations for friedmann - robertson - walker universe in einstein gravity @xcite , @xmath4 gravity @xcite , deformed horava - lifshitz gravity @xcite , and braneworld scenario @xcite were derived with the help of holographic principle and the equipartition rule of energy . the newtonian gravity in loop quantum gravity was derived in ref . @xcite . in ref . @xcite the coulomb force was regarded as an entropic force . in ref . @xcite , it was shown that the holographic dark energy can be derived from the entropic force formula . it was pointed out in @xcite that verlinde s entropic force is actually the consequence of a specific microscopic model of spacetime . the similar ideas were also applied to the construction of holographic actions from black hole entropy @xcite . while ref . @xcite showed that gravity has a quantum informational origin . on the other hand , a modified entropic force in the debye s model was gave in @xcite . in ref . @xcite , some of the problems of verlinde s proposal on the thermodynamical origin of the principle of relativity was presented , and the thermodynamic origin of the principle of relativity was explained by hidden symmetries of thermodynamics . other applications can be seen in @xcite . according to verlinde s idea , the holographic screens locate at equipotential surfaces , where the potential is defined by a timelike killing vector . then the local temperature on a screen can be defined by the acceleration of a particle that is located very close to the screen . the energy on the screen is calculated by the holographic principle and the equipartition rule of energy @xmath5 with @xmath6 the bit density of information on the screen . in this paper , we apply these formulas to investigate the temperature and energy on holographic screens for 4-dimensional static spherically symmetric and the kerr black hole , and look for experiment methods for testing the entropic force . the paper is organized as follows . in sec . [ sec2 ] , we review verlinde s idea about the temperature and the energy from an entropic force in general relativity . then , with the idea of entropic force together with the equipartition law of energy , we calculate the temperature and the energy for 4-dimensional static spherically symmetric black holes and the kerr black hole in sec . [ sec3 ] and sec . [ sec4 ] , respectively . finally , we give a brief discussion and present that the entropic force could be tested by experiments . we first review the idea of verlinde about the temperature and the energy from an entropic force in general relativity . consider a static background with a global time like killing vector @xmath7 . one can relate the choice of this killing vector field with the temperature and the energy . in order to define the temperature , we first need to introduce the potential @xmath8 via the timelike killing vector @xmath7 : @xmath9 where @xmath10 satisfies the killing equation @xmath11 the redshift factor is denoted by @xmath12 and it relates the local time coordinate to that at a reference point with @xmath13 , which we will take to be at infinity . the potential is used to define a foliation of space , and the holographic screens are put at surfaces of constant redshift . so the entire screen has the same time coordinate . the four velocity @xmath14 and the acceleration @xmath15 of a particle that is located very close to the screen can be expressed in terms of the killing vector @xmath7 as @xmath16 note that because the acceleration is the gradient of the potential , it is perpendicular to screen @xmath17 . so we can turn it into a scalar quantity by contracting it with a unit outward pointing vector @xmath18 normal to the screen @xmath17 and to @xmath7 . with the normal vector @xmath19 , the local temperature @xmath20 on the screen is defined by @xmath21 where a redshift factor @xmath12 is inserted because the temperature @xmath20 is measured with respect to the reference point at infinity . we will call the temperature defined in ( [ t ] ) as unruh - verlinde temperature . assuming that the change of entropy at the screen is @xmath22 for a displacement by one compton wavelength normal to the screen , one has @xmath23 now the entropic force , which is required to keep a particle with mass @xmath24 at fixed position near the screen , is turned out to be @xmath25 where @xmath26 is the relativistic analogue of newton s acceleration , and the additional factor @xmath12 is due to the redshift , which is a consequence of the entropy gradient . we now consider a holographic screen on a closed surface of constant redshift @xmath8 . the number of bit @xmath27 of the screen is assumed to proportional to the area of the screen and is given by @xcite @xmath28 then , by assuming that each bit on the holographic screen contributes an energy @xmath29 to the system , and by using the equipartition law of energy , we get @xmath30 with @xmath6 the bit density on the screen . inserting the expressions of @xmath20 and @xmath27 into ( [ energy_a ] ) results in @xmath31 in the following , we will calculate the unruh - verlinde temperature and the energy on holographic screens for 4-dimensional black holes with static spherically symmetric metric and stationary axisymmetric metric , respectively . we first consider the case of 4-dimensional general static spherically symmetric black holes . in the general case , the metric can be taken as the form @xmath32 where @xmath33 is the metric on a unit 2-sphere . we assume @xmath34 for ensuring that the metric is asymptotically flat . the event horizon radius @xmath35 is usually determined by the largest solution of @xmath36 . by using the killing equation @xmath37 and the static spherically symmetric properties of the metric ( [ general1 ] ) @xmath38 as well as the condition @xmath39 at infinity , the timelike killing vector of the general black hole is solved as @xmath40 which is zero at the event horizon . according to ( [ phi ] ) , ( [ amu ] ) and ( [ t ] ) , the potential , the acceleration and the unruh - verlinde temperature on the holographic screen put at the spherical surface with radius @xmath41 are calculated as in the first version of our manuscript . ] @xmath42 the energy on the screen is @xmath43 next , we focus on the schwarzschild black hole and the rn black hole . for the schwarzschild black hole , the metric has the form of ( [ general1 ] ) with @xmath44 the event horizon of the balk hole is located at @xmath45 , and the hawking temperature @xmath46 on the event horizon is given by @xmath47 the acceleration and the unruh - verlinde temperature on the screen are calculated as @xmath48 note that the unruh - verlinde temperature on the event horizon @xmath49 is just the hawking temperature @xmath46 : @xmath50 then , with the expressions ( [ e_spherically ] ) and ( [ f(r)_schwarz ] ) , we obtain the energy on the screen : @xmath51 it is interesting to note that this energy is independent of the radius of the screen . in fact this is a consequence of the gauss law . as an important example of spherically symmetric black holes , the rn black hole has the metric form of ( [ general1 ] ) with @xmath52 there are two horizons , the event horizon with radius @xmath53 and the cauchy horizon with radius @xmath54 , where @xmath55 the extremal rn black hole corresponds to the case @xmath56 or , equivalently , @xmath57 . the hawking temperatures @xmath58 on the horizons are @xmath59 the potential , the acceleration and the unruh - verlinde temperature on the screen read @xmath60 on the horizons @xmath61 , we have @xmath62 which shows that the unruh - verlinde temperatures on both the horizons equal to the hawking temperatures @xmath58 . the solution of @xmath63 is @xmath64 . so for @xmath65 , i.e. , @xmath66 , the maximum of @xmath20 in the range @xmath67 is at @xmath68 . for @xmath69 , the maximum of @xmath20 in the range @xmath67 is at @xmath70 and is given by @xmath71 the energy on the screen is @xmath72 which is the well - known komar energy of the rn black hole . now this energy is dependent on the radius of the screen , which is also a consequence of the gauss laws of gravity and electrostatics . for the screen at infinity , the energy is reduced to that of the schwarzschild black hole : @xmath73 . it is interesting to note that the energies on the horizons @xmath74 are the same : @xmath75 for the extremal rn black hole with @xmath76 , the corresponding unruh - verlinde temperature and energy are @xmath77 they vanish at the horizon @xmath78 . this result is in agreement with the statements that the extremal black hole has a unique internal state @xcite and its temperature is zero due to the vanishing of surface gravity on horizon . in this section , we extend this work to the kerr black hole . the kerr solution describes both the stationary axisymmetric asymptotically flat gravitational field outside a massive rotating body and a rotating black hole with mass and angular momentum . the kerr black hole can also be viewed as the final state of a collapsing star , uniquely determined by its mass and rate of rotation . moreover , its thermodynamical behavior is very different from the schwarzschild black hole and the rn black hole , because of its much more complicated causal structure . hence its study is of great interest in understanding physical properties of astrophysical objects , as well as in checking any conjecture about thermodynamical properties of black holes . in terms of boyer lindquist coordinates , the euclidean kerr metric reads @xmath79 where @xmath80 is the kerr horizon function @xmath81 and @xmath82 here @xmath83 is the angular momentum for unit mass as measured from the infinity ; it vanishes in the schwarzschild limit . the nonextremal kerr black hole has the event horizon @xmath84 and the cauchy horizon @xmath54 at @xmath85 the extreme case corresponds to @xmath86 or @xmath87 . the hawking temperatures on the horizons are @xmath88 the solution for the timelike killing vector @xmath10 is read as @xcite @xmath89 where @xmath90 . the corresponding potential is @xmath91 which shows that equipotential surfaces are dependent with two parameters @xmath41 and @xmath92 , hence the holographic screens are not spherically symmetric but axisymmetric . however , when @xmath93 or @xmath94 , the holographic screens are approximate spherically symmetric , which can be seen from the contour plots of the potential @xmath8 in the @xmath95-@xmath96 plane showed in figs . [ fig : contourphia ] and [ fig : contourphib ] . it can be seen that , when @xmath97 , the effect of the angular momentum are remarkable near the horizons . when far away from the event horizon , the equipotential line can be approximated as a spherical surface . -4 mm -4 mm the non - zero components of the acceleration are @xmath98 } , \nonumber \\ a^\theta & = & \frac{mra^2(a^2+r^2)\sin(2\theta ) } { \varrho^4[2mr(a^2+r^2)+\delta\varrho^2]}. \label{a_kerr}\end{aligned}\ ] ] the unruh - verlinde temperature is given by @xmath99 ^ 2 } { \varrho ^4 \left[2 m r \left(a^2+r^2\right ) + \delta \varrho ^2\right]^3 } \right . \nonumber \\ & & \quad\quad~ + \left . \frac{\delta \left[a^2 m r ( a^2+r^2 ) \texttt{sin}(2\theta ) \right]^2 } { \varrho ^4 \left[2 m r \left(a^2+r^2\right ) + \delta \varrho ^2\right]^3 } \right\ } ^{\frac{1}{2 } } . \label{t_kerr}\end{aligned}\ ] ] on the horizons @xmath61 ( @xmath100 ) , the unruh - verlinde temperatures become @xmath101 so , the unruh - verlinde temperatures on both horizons are equal to the hawking temperatures . for vanishing @xmath83 , the above result is reduced to @xmath102 , which is just the hawking temperature ( [ t_schwarzschild ] ) on the horizon of the schwarzschild black hole . contour plots of the temperature @xmath20 in the @xmath95-@xmath96 plane are shown in figs . [ fig : contourta ] and [ fig : contourtb ] . -4 mm -4 mm now , by using the equipartition law of energy and the holographic principle @xmath103 , we get the energies on the horizons @xmath104 which is the reduced mass @xmath1 of the kerr black hole . when far away from the event horizon , or equivalently @xmath93 , the screen is approximated as a sphere and the energy can be expressed as @xmath105 for @xmath106 , the result is @xmath73 , which is independent of the angular of the kerr black hole . for @xmath107 , the expression ( [ e_kerr_approx ] ) reduce to @xmath73 , which is just the energy expression ( [ e_schwarzschild ] ) for the schwarzschild black hole . in this paper , with the holographic principle and the equipartition law of energy , we investigate the unruh - verlinde temperature and energy on holographic screens for several 4-dimensional black holes with static spherically symmetric metric and stationary axisymmetric metric . on the event horizon of a static spherically symmetric black hole , the hawking temperature @xmath46 is related to the surface gravity @xmath108 by the relation @xmath109 , while the unruh - verlinde temperatures @xmath20 is related to the acceleration @xmath110 by @xmath111 , which shows that the unruh - verlinde temperatures @xmath20 for a static spherically symmetric black hole is identical to the hawking temperature . hence , outside the horizons , the unruh - verlinde temperature @xmath20 can be considered as a generalized hawking temperature on the holographic screen . for the kerr black case , we also consider @xmath20 as the generalized hawking temperature . on the screen located at the infinity , the surface gravity or the acceleration of a particle ( which is free there ) vanishes , hence the temperature is zero . the entropic force ( [ fmu ] ) for a static spherically symmetric black hole can be calculated as @xmath112 . the magnitude of the force is @xmath113 . for the schwarzschild black hole , the entropic force @xmath114 is just the newton force @xmath115 here we recover the newton s gravitational constant @xmath116 . while , for the rn case , the entropic force is turned out to be @xmath117 which shows that there is a corrected term caused by the charge of the black hole . this force is related to the pure gravitational effect . the gravity or the geometry near the event horizon of the black hole is affected by the energy of the electric field . when far away form the horizon , where the electric field becomes weak , the entropic force will recover to the newton one . note that , since @xmath118 , we have @xmath119 and @xmath120 . similarly , the angular momentum of the kerr black hole will also affect the structure of the spacetime near the event horizon . the entropic force of the kerr black hole is read as @xmath121 and @xmath122 at @xmath68 and @xmath93 , respectively . it also reduces at large @xmath41 to the newton force because the effect of the field " caused by the angular momentum of the black hole becomes weaker with the increase of @xmath41 . the energy got from the holographic principle is indeed the komar energy on the screen . with the same explanation above , we also can understand that the energy @xmath123 on the screen will trend to the adm mass @xmath0 when @xmath106 for anyone of the black holes considered in the paper . this is because the field " produced by the charge or the angular momentum is strong on the screen near the event horizon and weak when far away from the black hole . on the screen at infinity , the effect of them vanishes and the energy is naturally equal to the black hole mass @xmath0 according to the gravitational gauss law . we see clearly from ( [ e_rn_h ] ) and ( [ e_kerr_h ] ) that when applying the entropic force idea to the black hole horizon , it is the reduced mass @xmath1 that takes the place of the black hole mass @xmath0 . for the schwarzschild black hole , there is no other parameter beside @xmath0 , so the reduced mass @xmath1 is just the black hole mass @xmath0 . for the rn black hole , the reduced mass is read from ( [ e_rn ] ) as @xmath124 , which is @xmath125 on the event horizon . for the kerr case , the reduced mass is @xmath126 on the event horizon and is @xmath127 at @xmath93 . therefore , with the black hole mass @xmath0 replaced by the reduced mass @xmath1 , we can rewrite the entropic force of a black hole as @xmath128 which has the same form as the newton s law of gravity . this formula shows that the force between two neutral particles is exactly the newton s force of gravity , while the force between a neutral particle and a charged particle would departure from the newton s law of gravity . hence , the idea of entropic force could be tested by experiments according to eq . ( [ f_entropic ] ) . explicitly , the force between a neutral particle and a charged particle is @xmath129 according entropic force idea . therefore , by comparing the high accurate experiment results of the force between a neutral particle and a charged particle with the theoretical predictions of the entropic force , one could check whether the entropic idea is right or wrong . this work was supported by the program for new century excellent talents in university , the national natural science foundation of china ( no . 10705013 ) , the key project of chinese ministry of education ( no . 109153 ) , and the fundamental research funds for the central universities ( no . lzujbky-2009 - 54 , no . lzujbky-2009 - 122 and no . lzujbky-2009 - 163 ) .

we investigate the temperature and energy on holographic screens for 4-dimensional black holes with the entropic force idea proposed by verlinde . we find that the `` unruh - verlinde temperature '' is equal to the hawking temperature on the horizon and can be considered as a generalized hawking temperature on the holographic screen outside the horizons . the energy on the holographic screen is not the black hole mass @xmath0 but the reduced mass @xmath1 , which is related to the black hole parameters . with the replacement of the black hole mass @xmath0 by the reduced mass @xmath1 , the entropic force can be written as @xmath2 , which could be tested by experiments .

You are an expert at summarizing long articles. Proceed to summarize the following text: systems of differential equations in both finite- and infinite - dimensional settings close to an ahb have been subject to intense research due to their dynamical complexity and importance in applications . the latter range from models in fluid dynamics @xcite to those in the life sciences , in particular , in computational neuroscience @xcite . when the proximity to the ahb coincides with certain global properties of the vector field , it may result in a very complex dynamics @xcite . the formation of the smale horseshoes in systems with a homoclinic orbit to a saddle - focus equilibrium provides one of the most representative examples of this type @xcite . canard explosion in relaxation systems affords another example @xcite . recent studies of relaxation systems , motivated mainly by applications in the life sciences , have revealed that the proximity to an ahb has a significant impact on the system dynamics . it manifests itself as a family of multimodal periodic solution that are composed of large - amplitude relaxation oscillations ( generated by the global structure of the vector field ) and small - amplitude nearly harmonic oscillations ( generated by the vector field near the equilibrium close to the ahb ) @xcite ( see figure [ f.1 ] ) . these families of solutions possess rich bifurcation structure . a remarkable example of an infinite - dimensional system close to the ahb has been recently studied by frankel and roytburd @xcite . they derived and systematically studied a model of solid fuel combustion in the form of a free boundary problem for a @xmath1 heat equation with nonlinear conditions imposed at the free boundary modeling the interface between solid fuel mixture and a solid product . the investigations of this model revealed a wealth of spatial - temporal patterns ranging from a uniform front propagation to periodic and aperiodic front oscillations . the transitions between different dynamical regimes involve a variety of nontrivial bifurcation phenomena including period - doubling cascades , period - adding sequences , and windows of chaotic dynamics . to elucidate the mechanisms responsible for different dynamical regimes and transitions between them , frankel and roytburd employed pseudo - spectral techniques to derive a finite - dimensional approximation for the interface dynamics in the free boundary problem @xcite . as shown in @xcite , a system of three ordinary differential equations captured the essential features of the bifurcation structure of the infinite - dimensional problem . the numerical bifurcation analysis of the finite - dimensional approximation revealed a rich family of multimodal periodic solutions similar to those reported in the context of relaxation systems near the ahb @xcite . the bifurcation diagrams presented in @xcite and in @xcite share a striking similarity , despite the absence of any apparent common structures in the underlying models ( except to the proximity to the ahb ) . in particular , in both models , topologically distinct multimodal periodic solutions are located on isolas , closed curves in the parameter space . the methods of analysis of the mixed - mode solutions in @xcite used in an essential way the relaxation structure present in these problems . these approaches can not be applied directly to analyzing the model in @xcite , because it is not a priori clear what creates the separation of the time scales in this model , in spite of the evident fast - slow character of the numerical solutions . this is partly due to the spectral method , which was used to derive the system of equations in @xcite : while it has captured well the finite - dimensional attractor of the interface dynamics , it has disguised the structure of the physical model . one of the goals of the present paper is to identify the structure responsible for the generation of the multimodal oscillations in a finite - dimensional model for the interface dynamics and to relate it to those studied in the context of relaxation oscillations . the family of flows in @xcite includes in a natural way two types of the ahbs . depending on the parameter values , the equilibrium of the system of ordinary differential equations in @xcite undergoes either a sub- or a supercritical ahb . a similar situation is encountered in certain neuronal models ( see , e.g. , @xcite ) . in either case , the global multimodal periodic solutions are created after the ahb . however , in the case of a supercritical bifurcation , they are preceded by a series of period - doubling bifurcations of small amplitude limit cycles , arising from the ahb . on the other hand , in the subcritical case , the ahb gives rise to multimodal solutions , whose lengths and time intervals between successive large amplitude oscillations can be very long . in the present paper , we perform a detailed asymptotic analysis of the trajectories in a class of systems motivated by the problem in @xcite . our analysis includes both cases of the sub- and supercritical ahbs . we also investigate the dynamical regimes arising near the border between the regions of sub- and supercritical ahb . this region in the parameter space contains a number of nontrivial oscillatory patterns including multimodal trajectories with substantial time intervals between successive spikes , irregular , and possibly chaotic oscillations , as well as a family of periodic orbits undergoing a cascade of period - doubling bifurcations . our analysis shows that these dynamical patterns and the order in which they appear under the variation of the control parameters are independent on the details of the model , but are characteristic to the transition from sub- to supercritical ahb . the outline of the paper is as follows . after introducing the model and rewriting it in the normal coordinates , we present a set of the numerical experiments to be explained in the remainder of the paper . then we state our results for each of the following cases : supercritical ahb , subcritical ahb , and the transition layer between the regions of sub- and supercritical ahb . in section 3 , we analyze the local behavior of trajectories near a weakly unstable saddle - focus . the local expansions used in this section are similar to those used in @xcite for analyzing the ahb using the method of averaging . however , rather than establishing existence of periodic solutions , the goal of the present section is to approximate the trajectories near the saddle - focus . for this , after rescaling the variables and recasting the system into cylindrical coordinates , we reduce the system dynamics to a @xmath2 slow manifold . by integrating the leading order approximation of the reduced system , we obtain necessary information about the local behavior of trajectories . the results of this section are summarized in theorem 3.1 . in section 4 , we study oscillatory patterns generated by the class of systems under investigation . it is divided into three subsections devoted to the oscillations triggered by the supercritical ahb ( 4.1 ) , subcritical ahb ( 4.2 ) , and those found in the transition region between sub- and supercritical ahb ( 4.3 ) . in the supercritical case , we show that the oscillations just after the ahb are already bimodal . generically , one needs to use two harmonics to describe the limit cycle born at the supercritical ahb in @xmath0 . we give a geometric interpretation of this effect , which we call the frequency doubling , due to the fact that the second frequency is twice as large as that predicted by the hopf bifurcation theorem @xcite . we also compute the curvature and the torsion of the periodic orbit as a curve in @xmath0 . the latter is useful for the geometric explanation of the frequency doubling . in subsection 4.2 , we study the subcritical case . we show that under two general assumptions on the global vector field , the presence of the return mechanism ( * g1 * ) and the strong contraction property ( * g2 * ) , the subcritical ahb results in sustained multimodal oscillations . even though the oscillations may not be periodic ( our assumptions do not warrant periodicity ) , the time intervals between consecutive spikes of the resultant motion comply to the uniform bounds given in theorem 4.1 this subsection also contains the definitions of the multimodal oscillations and the precise formulation of the assumptions on the global vector field . the proof of theorem 4.1 is relegated to section 5 . subsection 4.3 describes the transition between sub- and supercritical ahb . this transition contains a distinct bifurcation scenario , when a small positive real part of the complex conjugate pair of eigenvalues is positive and fixed and the first lyapunov coefficient changes sign . the numerical simulations show that this results in the formation of the chaotic attractor followed by the reverse period - doubling cascade . to explain this bifurcation sequence , we derive a @xmath1 first return map . the asymptotic analysis in this subsection is complemented by numerical extension of the map to the region inaccessible by local asymptotic expansions . the first return map obtained by the combination of the analytic and numerical techniques reveals the principal traits of the transition from sub- to supercritical ahb . finally , the discussion of the results of the present paper and their relation to the previous work is given in section 6 . in the present section , we formulate the model and present a set of numerical results , which motivated our study . the following system of three odes was derived in @xcite , as a finite - dimensional approximation for the interface dynamics in a free boundary problem modeling solid fuel combustion : @xmath3 here , @xmath4 approximates the velocity of the interface between the solid fuel and the burnt material . functions @xmath5 and @xmath6 are the first two coefficients in the series expansion for the spatial temperature profile with respect to the basis of chebyshev - laguerre polynomials . equations ( [ 6.1])-([6.3 ] ) are obtained by projecting the original infinite - dimensional problem onto a finite - dimensional function space and using the method of collocations for determining the unknown coefficients . an important ingredient of the model , nonlinear kinetic function @xmath7 is given by k(v)=(1-v)^p-(1-v)^-1p+1 . * a * + * c * + * e * + * g * it reflects the dependence of the velocity of propagation on the temperature in the front . this relation is very complex and is not completely understood at present . to account for a range of possible kinetic mechanisms , the model includes two control parameters : @xmath8 and @xmath9 . both the interface velocity and the temperature profile are calculated in the uniformly moving frame of reference . therefore , system of equations ( [ 6.1])-([6.3 ] ) describes the deviations of the interface dynamics from that of a front traveling with constant speed . in particular , periodic and aperiodic oscillations generated by ( [ 6.1])-([6.3 ] ) correspond to complex spatio - temporal patterns in the infinite - dimensional model . the numerical results presented in @xcite show a remarkable similarity between the complex patterns generated by the infinite- and finite - dimensional models and between the scenarios for transitions between different regimes in both models . for more information about the derivation of the free - boundary problem for solid fuel combustion and its finite - dimensional approximation , we refer the reader to @xcite and bibliography therein . a simple inspection of ( [ 6.1])-([6.3 ] ) shows that it has an equilibrium at the origin @xmath10 for all values of @xmath8 and @xmath11 . we linearize ( [ 6.1])-([6.3 ] ) about the equilibrium at the origin @xmath12 near @xmath13 , jacobian matrix @xmath14 has a negative eigenvalue @xmath15 and a pair of complex conjugate eigenvalues @xmath16 and @xmath17 . the latter crosses the imaginary axis transversally at @xmath18 : @xmath19 therefore , the equilibrium of ( [ 6.1])-([6.3 ] ) undergoes an ahb . in the neighborhood of @xmath20 , @xmath21 defines a smooth invertible function . we shall use @xmath22 as a new control parameter . after a linear coordinate transformation , system of equations ( [ 6.1])-([6.3 ] ) has the following form : x = ( ccc -1 & 0 & 0 + 0 & & -b ( ) + 0 & b ( ) & ) x + h(x , ) , x=(x_1 , x_2 , x_3)^t , h=(h_1 , h_2 , h_3)^t . here , by @xmath23 we denote the imaginary part of @xmath24 as a function of the new control parameter . nonlinear function @xmath25 is a smooth function such that @xmath26 and @xmath27 for values of @xmath22 close to @xmath28 . more precisely , @xmath29 stands for a family of functions parametrized by @xmath9 ( see ( [ 6.4 ] ) ) . to keep the notation simple , we omit the dependence of @xmath29 on the second control parameter @xmath9 . * a * it turns out that in the range of parameters of interest , the ahb of the equilibrium for @xmath30 can be either subcritical , or supercritical , or degenerate depending on the value of @xmath9 . this gives rise to several qualitatively distinct oscillatory regimes generated by ( [ 6.5 ] ) for small values of @xmath31 . the corresponding bifurcation scenarios for ( [ 6.1])-([6.3 ] ) for fixed values of @xmath9 and varying @xmath8 were studied numerically in @xcite . below , we reproduce some of these numerics for the system in new coordinates and supplement them with a set of new numerical experiments relevant to the analysis of this paper . after that we state our results for each of the following cases : supercritical ahb , subcritical ahb , and the transition layer between the regions of sub- and supercritical bifurcations . * a * we start with discussing the supercitical case . it is well known that a supercritical ahb produces a stable periodic orbit in a small neighborhood of the bifurcating equilibrium . the period of the nascent orbit is approximately equal to @xmath32 where @xmath33 is the imaginary part of the pair of complex conjugate eigenvalues of the matrix of the linearized system at the bifurcation . the numerical time series of @xmath34 and @xmath35 show that , while @xmath35 oscillate with the frequency prescribed by the andronov - hopf bifurcation theorem @xcite , the former oscillates with twice that frequency ( see figure [ f.1]a , b ) . the asymptotic analysis of section 3 explains this counter - intuitive effect and shows that this is , in fact , a generic property of the periodic orbits born from a supercritical bifurcation in @xmath36 , @xmath37 . note that the difference in frequencies of oscillations in @xmath34 and @xmath35 can not be understood from the topological normal form from the ahb @xcite , because the latter does not contain the information about the geometry of the periodic orbit . in section 4.1 , we compute two geometric invariants of the periodic orbits as a curve in @xmath0 : the curvature and the torsion . the latter shows that generically the orbit born from a supercritical ahb is not planar . this together with certain symmetry properties of the orbit explains the frequency doubling of the oscillations in @xmath34 . as was noted in @xcite , for increasing values of @xmath38 , the small periodic orbit born from the supercritical ahb undergoes a cascade of period - doubling bifurcations , the first of which is shown in figure [ f.1]c , d . already at the first period - doubling bifurcation , the periodic orbit lies outside the region of validity of the local power series expansions , developed in the present paper . therefore , our analysis does not explain the period - doubling bifurcations for increasing values of @xmath39 . in section 4.3 we complement our analytical results with the numerical construction of the @xmath1 first return map . the latter explains the mechanism for period - doubling cascade and the window of complex dynamics reported in @xcite . * a * * a * we next turn to the subcritical case . the dynamics resulting from the subcritical ahb depends on the properties of the vector field outside of a small neighborhood of the equilibrium . we conducted a series of numerical experiments to study the global properties of the vector field ( see caption of figure [ f.2 ] for details ) . based on these numerical observations , we identify two essential features of the vector field : * ( g1 ) * the return mechanism and * ( g2 ) * the strong contraction property . specifically , ( g1 ) : : for a suitably chosen cylindrical crossection , @xmath40 , placed sufficiently close to the origin , and another crossection , @xmath41 , transverse to @xmath42 ( see figure [ f.2]a ) , the flow - defined map @xmath43 is well - defined for @xmath44 and depends smoothly on the parameters of the system . ( g2 ) : : there is a region @xmath45 , adjacent to @xmath41 and containing a subset of @xmath42 ( see figure [ f.2]a ) , in which the projection of the vector field in direction transverse to @xmath42 is sufficiently stronger than that in the tangential direction . in section 4.2 , conditions * ( g1 ) * and * ( g2 ) * are made precise . properties * ( g1 ) * and * ( g2 ) * guarantee that for small values of @xmath39 , the trajectories leaving a small neighborhood of the saddle - focus , after some relatively short time enter @xmath45 . in @xmath45 , the trajectories approach @xmath42 closely and then follow it to a sufficiently small neighborhood of the unstable equilibrium . for small @xmath39 , the trajectories starting from a sufficiently small neighborhood of the saddle - focus , remain in some larger ( but still small ) neighborhood of the origin for a long time . eventually , they hit @xmath40 and the dynamics described above repeats . therefore , under conditions * ( g1 ) * and * ( g2 ) * , the subcritical ahb bifurcation results in a sustained motion consisting of long intervals of time that the trajectory spends near a weakly unstable saddle - focus and relatively brief excursions outside of a small neighborhood of the origin ( see figure [ f.2]a ) . in terms of the time series , the dynamical variables undergo series of small amplitude oscillations alternating with large spikes ( figure [ f.1]e - h ) . for increasing values of @xmath38 , the number of small amplitude oscillations decreases and so do the time intervals between consecutive spikes , so - called interspike intervals ( isis ) . the multimodal solutions arising near the subcritical ahb are not necessarily periodic . nonetheless , the isis may be characterized in terms of the control parameters present in the system , regardless of whether the underlying trajectories are periodic or not . specifically , in section 4.2 , we derive an asymptotic relation for the duration of the isis : ~12 ( 1 + ( p)o(^- ) ) , 4 , where the first lyapunov coefficient @xmath46 reflects the distance of the system from the transition from sub- to supercritical ahb , with @xmath46 being positive when the ahb is subcritical and equal to zero at the transition point . finally , @xmath47 is a small parameter and positive @xmath48 . the role of @xmath49 will become clear later . we illustrate ( [ 6.6 ] ) with numerically computed plots of the isis under the variation of control parameters in figure [ f.3 ] . the isis provide a convenient and important characteristics of the oscillatory patterns involving pronounced spikes . for example , it is widely used in both theoretical and experimental neuroscience for description of patterns of electric activity in neural cells . an important aspect of the isis , is that the dependence of the isis on the control parameters in the system ( which often can be directly established experimentally ) reveals the bifurcation structure of the system . therefore , the analytical characterization of the isis in terms of the bifurcation parameters , such as given in ( [ 6.6 ] ) , is important in applications . for the problem at hand , the isis depend on the interplay of the two parameters @xmath22 and @xmath46 , which reflect the proximity of the system to a codimension 2 bifurcation . the dependence of the isis on @xmath22 was studied in @xcite for a model problem near a hopf - homoclinic bifurcation . our analysis extends the formula for the isis obtained in @xcite to a wide class of systems and emphasizes the role of @xmath46 , which is important for the present problem . we also note that the structure of the global vector field suggests that ( [ 6.5 ] ) is also close to a homoclinic bifurcation , which can also influence the isis . however , under the variation of the control parameters @xmath22 and @xmath9 the system remains bounded away from the homoclinic bifurcation , so that the latter effectively does not affect the isis . in fact , ( [ 6.6 ] ) can be easily extended to include the distance from the homoclinic bifurcation as well . as follows from ( [ 6.6 ] ) , the isis increase for decreasing values of @xmath50 ( see figure [ f.3]b ) . this observation prompted our interest in investigating the transition in the system dynamics as @xmath51 crosses @xmath28 . numerical simulations show that as @xmath51 approaches @xmath28 from the positive side , the oscillations become very irregular and very likely to be chaotic ( figure [ f.4 ] ) . when @xmath52 is fixed and @xmath51 crosses 0 , the irregular dynamics is followed by the reverse period - doubling cascade and terminates with the creation of a regular small amplitude periodic orbit , which can be then tracked down to a nondegenerate supercritical ahb ( figure [ f.5]a - h ) . this scenario presents a certain interest as it suggests the formation of the chaotic attractor in a well - defined bifurcation setting . namely , the transition from the region of subcritical to supercritical ahb for small @xmath52 leads to the formation of a chaotic attractor and a reverse period - doubling cascade . as in the case of the complex dynamics appearing for increasing values of @xmath22 for supercritical ahb , the irregular oscillations in the present case can not be understood using the local analysis alone . therefore , we complemented our analysis with the study of numerically constructed first return map . the latter gives a clear geometric picture of the origins of the chaotic dynamics and the period - doubling cascade of periodic orbits in this parameter regime . in the present section , we study trajectories of ( [ 6.5 ] ) in a small neighborhood of the equilibrium at the origin . assume that @xmath53 is a smooth function in a small neighborhood of @xmath54 , so that all power series expansions below are justified . we are interested in the dynamics of ( [ 6.5 ] ) for values of @xmath22 close to @xmath28 . the role of @xmath22 in our analysis is twofold . on the one hand , @xmath22 is a control parameter in this problem , on the other hand , the smallness of @xmath22 is used in the asymptotic analysis below . to separate these two roles , we use the following rescaling = ^2 , where @xmath47 is a fixed sufficiently small constant and @xmath55 varies in a certain interval of size @xmath56 . we introduce cylindrical coordinates in @xmath0 ( x_1,x_2,x_3)(x_1 , , ) ^+s^1,x_2=,x_3= , and define d=\ { ( x_1 , , ): | x_1 | m^2 ^ 2 , ( 0,2 m ] } , d_0=\ { ( x_1 , , ): | x_1 to characterize the trajectories of ( [ 6.5 ] ) in @xmath57 we will use an exponentially stable slow manifold , @xmath58 , whose leading order approximation is given by s_0= \ { ( x_1 , , ) : x_1=u()^2 , 02 m } , where u()= a + a(2- ) . and @xmath59 and @xmath60 are computable constants ( see appendix a ) . below we show that the trajectories of ( [ 6.5 ] ) with initial data from @xmath61 approach an @xmath62 neighborhood of @xmath63 in time @xmath64 and stay in this neighborhood as long as they remain in @xmath57 . the reduction of the system dynamics to @xmath58 yields a complete description of the trajectories of ( [ 6.5 ] ) in @xmath57 . the qualitative character of the solution behavior in @xmath57 depends on the sign of _ the first lyapunov coefficient _ = 12_0 ^ 2 ( u ( ) |q_2()+ |q_3 ( ) ) d , where @xmath65periodic trigonometric polynomial @xmath66 are given in appendix a. finally , by @xmath33 we denote the absolute value of the imaginary parts of the complex conjugate pair of eigenvalues at the ahb ( see ( [ 6.5 ] ) ) . the results of the analysis of this section are summarized in * theorem 3.1 * suppose @xmath67 and @xmath68 . let @xmath47 denote a small parameter . then for @xmath48 and for sufficiently large ( independent of @xmath49 ) @xmath69 , the trajectories with initial conditions in @xmath61 enter an @xmath62 neighborhood of @xmath63 in time @xmath70 and remain there as long as they stay in @xmath57 . in the neighborhood of @xmath63 , the trajectories can be uniformly approximated on any interval of time @xmath71 $ ] of length @xmath56 : x_1(t)=^2 u()+o(^3 ) , x_2(t)=+o(^2),x_3(t)=+o(^2 ) , where @xmath72 and @xmath51 are defined in ( [ 5.4 ] ) and ( [ 5.5 ] ) , respectively , and @xmath73 * remark 3.1 * a ) : : from theorem 3.1 , one can easily deduce the existence of a periodic orbit @xmath74 , when @xmath75 . @xmath74 is stable ( unstable ) if @xmath39 ( @xmath76 ) . the leading order approximation for @xmath74 follows from ( [ 5.7 ] ) : x_1=|^2 u()+o(^3 ) , x_2=|+o(^2 ) , x_3=|+o(^2 ) , [ 0 , 2 ) , where @xmath77 . equations ( [ 5.4 ] ) and ( [ 5.8 ] ) imply that the frequency of oscillations in @xmath34 is twice that of oscillations in @xmath35 , provided @xmath78 in ( [ 5.4 ] ) ( see figure [ f.1]a , b ) . in section 4.1 , we give a geometric interpretation of this frequency doubling effect . b ) : : if @xmath39 and @xmath50 , theorem 3.1 describes the trajectories with initial conditions near a weakly unstable equilibrium . it shows that along these trajectories , @xmath35 undergo approximately harmonic oscillations , whose amplitude grows at the rate @xmath79 . in addition , they satisfy the following scaling relation : |x_1(t ) ~a^2(t ) , where @xmath80 stands for the average value of @xmath81 over one cycle of oscillations . c ) : : the domain of validity of the asymptotic expansions in ( [ 5.7 ] ) extends much farther than @xmath57 . as will follow from the proof of theorem 3.1 , @xmath82 in the definition of @xmath57 may be taken up to @xmath83 . this implies that , with the error term @xmath84 , ( [ 5.7 ] ) remains valid for the ranges of @xmath34 and @xmath35 up to @xmath85 and @xmath86 , respectively . in the remainder of this section , we prove theorem 3.1 . by expanding @xmath53 into finite taylor sum with the reminder term , we have @xmath87 where @xmath88 to study system of equations ( [ 5.1 ] ) in a small neighborhood of the origin , we rewrite it in cylindrical coordinates ( [ cl ] ) and rescale variables x_1=^2 = r. in new coordinates , we have @xmath89 where @xmath90 and @xmath91 are homogeneous trigonometric polynomials . using @xmath92 as a new variable , from ( [ 5.11 ] ) , we obtain @xmath93 where @xmath94 and @xmath95 are trigonometric polynomials of period @xmath96 . functions @xmath97 and @xmath98 are bounded and continuous in @xmath99 $ ] . here @xmath100 stands for the domain of the rescaled variable @xmath101 when the original variable @xmath102 : d^= \ { ( , r ): | | 4m^2 , 0<r2 m } . similarly , we define d^_0=\ { ( , r ) : | | m^2 , 0<rm}. to determine the slow manifold for ( [ 5.1 ] ) , we introduce the following notation . by @xmath103 and @xmath104 we denote the mean value and the oscillating part of the trigonometric polynomial @xmath95 in ( [ 5.11a ] ) , respectively : p_1()-p_1^()1 + 4 ^ 2 . we first prove an auxiliary lemma , which characterizes the trajectories of ( [ 5.11a ] ) and ( [ 5.12 ] ) in @xmath105 . in the following two lemmas , we will keep track of how the remainder terms in the asymptotic expansions depend on the size of @xmath105 . this will be used later to determine the domain of validity for the asymptotic approximation of the slow manifold . below @xmath106 means that there exist positive constants @xmath107 and @xmath108 ( independent of @xmath49 ) such that @xmath109 for @xmath110 $ ] . in the remainder of this section , all estimates are valid for sufficiently large @xmath69 independent of @xmath49 ( as stated in the theorem 3.1 ) , but also allow the possibility that @xmath111 grows as @xmath112 . * lemma 3.1 * suppose @xmath113 remains in @xmath105 for @xmath114 $ ] . then ( ) = _ 0()r^2 ( ) + e^-(-_0)\ { ( _ 0)-_0(_0 ) r^2(_0 ) } + o(m^3 ) , , where @xmath115 is defined in ( [ 5.16 ] ) . * proof : * using ( [ 5.13 ] ) , we integrate ( [ 5.11a ] ) over @xmath116 $ ] e^()=e^_0(_0)+ |p_1 _ _ 0^e^s r^2(s)ds + _ _ 0^e^s r^2(s ) p_1(s ) ds+ _ _ 0^e^s ( ( s ) , r(s),s , ) ds . using integration by parts , we have @xmath117 from ( [ 5.12 ] ) and ( [ dprime ] ) , we find that for @xmath118 @xmath119 where @xmath120 is independent of @xmath49 . therefore , _ _ 0^e^s r^2(s)ds = e^r^2()-e^_0 r^2(_0 ) + e^-_0o(m^3 ) . similarly , using integration by parts in the second integral on the right hand side of ( [ 5.17 ] ) , we obtain @xmath121 applying integration by parts in the last integral on the right hand side of ( [ a.2 ] ) , we have @xmath122 by direct verification , one finds that @xmath123 the integrals involving @xmath124 on the right hand sides of ( [ a.2 ] ) and ( [ a.3 ] ) are bounded by @xmath125 since , by ( [ 5.12 ] ) , @xmath126 . using these observations , from ( [ a.2 ] ) and ( [ a.3 ] ) we obtain _ _ 0^e^s r^2(s ) p_1(s)ds = e^r^2()p_1()-e^_0 r^2(_0)p_1(_0)+ e^-_0o(m^3 ) , where @xmath127 is given by ( [ 5.16 ] ) . the combination of ( [ 5.17 ] ) , ( [ 5.18 ] ) , and ( [ a.4 ] ) yields ( [ 5.15 ] ) . + @xmath128 the following lemma shows that the trajectories of ( [ 5.11a ] ) and ( [ 5.12 ] ) , which stay in @xmath105 for sufficiently long time enter a small neighborhood of @xmath58 no later than in time @xmath129 and remain in this neighborhood as long as they stay in @xmath105 ( see ( [ dprime ] ) ) . * lemma 3.2 * let @xmath130 denote a trajectory of ( [ 5.11a ] ) and ( [ 5.12 ] ) , which stays in @xmath105 for @xmath131 $ ] , @xmath132 , @xmath133 . then ( ) = _ 0 ( ) r^2 ( ) + o(m^3 ) , for @xmath134 and as long as @xmath135 . * proof : * denote @xmath133 . by plugging in @xmath136 into ( [ 5.15 ] ) , we have |(_1 ) - _ 0(_1)r^2(_1)|c_2 , for some @xmath137 independent of @xmath47 . using lemma 3.1 again with @xmath138 , we obtain @xmath139 the expression in the curly brackets is @xmath62 by ( [ 5.21 ] ) . + @xmath128 * remark 3.2 * lemmas 3.1 and 3.2 show that the trajectories of ( [ 5.11a ] ) and ( [ 5.12 ] ) converge to an exponentially stable manifold @xmath58 , whose leading order approximation is given in the definition of @xmath63 . the method , which we used in lemmas 3.1 and 3.2 to obtain the leading order approximation of the slow manifold , can be extended to calculate the higher order terms in the expansion for @xmath58 . having shown that the trajectories approach an @xmath62 neighborhood of @xmath63 in time @xmath70 , next we reduce the dynamics of ( [ 5.11a ] ) and ( [ 5.12 ] ) to the slow manifold . for this , we define i()=(1r()+q_1())^2 , where @xmath140 and @xmath141 is a trigonometric polynomial on the right hand side of ( [ 5.12 ] ) . for the sake of definiteness , we choose @xmath142 such that @xmath143 the following lemma provides the desired reduction . * lemma 3.3 * for @xmath134 and as long as @xmath144 , @xmath145 satisfies the following system of equations @xmath146 where @xmath147 is a trigonometric polynomial with period @xmath148 and zero mean @xmath149 . the expression for @xmath51 is given in ( [ 5.5 ] ) . * proof : * the change of variables r()=1j()-q_1(),q_1^()=q_1 ( ) , in equation ( [ 5.12 ] ) yields j = -^2j((+q_2 ( ) ) j^2 + q_3())+o(m^2 ^ 3 ) . after another change of variables , @xmath150 , we obtain = -2 ^ 2 ( ( + q_2 ( ) ) i + q_3 ( ) ) + o(m^3 ^ 3 ) , by lemma 3.2 , for @xmath134 ( ) = _ 0()r^2 ( ) + o(m^3 ) . by plugging in ( [ 5.29 ] ) into ( [ 5.28 ] ) , we obtain = -2 ^ 2 ( i + _ 0()q_2()+q_3())+ o(m^3 ^ 3 ) . equation ( [ 5.24 ] ) follows by rewriting ( [ 5.30 ] ) = -2 ^ 2 ( i + + q())+ o(m^3 ^ 3 ) , where = 1_0^(_0()q_2()+q_3())d _ 0^ q()d=0 , and @xmath115 is given in ( [ 5.16 ] ) . equation ( [ 5.25 ] ) follows from the last equation in ( [ 5.11 ] ) and the definition of @xmath151 . + @xmath128 the statements in theorem 3.1 can now be deduced from equations ( [ 5.24 ] ) and ( [ 5.25 ] ) . we only need to show that a trajectory with initial condition in @xmath152 remains in @xmath105 for times longer than @xmath70 . this follows from the fact that @xmath153 in @xmath105 . consequently , it takes time @xmath154 for @xmath155 to undergo @xmath56 change necessary for leaving @xmath105 from @xmath156 . we omit any further details . in conclusion , we note that the domain of validity of the asymptotic analysis of this section extends much further beyond @xmath105 . indeed , from ( [ 5.20 ] ) we observe that the remainder term tends to @xmath28 with @xmath112 provided @xmath157 . this means that the expansions for @xmath158 and @xmath159 can be controlled in regions of size up to @xmath160 and @xmath161 respectively . for the original variables @xmath162 @xmath34 , these estimates translate into @xmath163 and @xmath164 respectively . we end this section by deriving a useful estimate for the time of flight of trajectories passing close to the saddle - focus . for this , consider an initial value problem for system of equations ( [ 5.11a ] ) and ( [ 5.12 ] ) . suppose that the initial condition implies that @xmath165 . we would like to know how long it takes for @xmath155 to reach a given value @xmath166 , @xmath167 . we assume that @xmath168 and @xmath169 are sufficiently separated , e.g. , @xmath170 . to estimate @xmath171 , note that = _ 1+_2 , where @xmath172 is time necessary to reach an @xmath62 neighborhood of @xmath63 from @xmath168 . denote i_1:= i(_1)=i_0+o(^2|| ) , _ 1=_0+_1 . the second term on the right hand side of ( [ 5.51 ] ) can be estimated by integrating ( [ 5.24 ] ) over @xmath173:$ ] |i= i_1 e^-2_2+(1- e^-2_2 ) -2 ^ 2_0^_2 e^-2(_2-s ) q(_0+s)ds + o(^3 ) . the integral on the right hand side of ( [ 5.53 ] ) is bounded uniformly for @xmath174 . this observation combined with ( [ 5.52 ] ) implies |i= i_0 e^-2_2-(1- e^-2_2 ) + o(^2 ) . note that the contribution of @xmath175 term in ( [ 5.52 ] ) to ( [ 5.54 ] ) is negligible provided @xmath176 is sufficiently large . from ( [ 5.54 ] ) , we obtain the desired estimate = 12 ^ 2(i_0 + |i + + o(^2 ) ) . in the present section , we use the local analysis of section 3 to study certain oscillatory patterns arising in the model of solid fuel combustion ( [ 6.1])-([6.3 ] ) . by plugging in the values of the parameters of ( [ 6.1])-([6.3 ] ) into the expression for @xmath51 ( [ 5.5 ] ) , we find that @xmath46 is a quadratic function with two zeros at @xmath177 and @xmath178 . these values are in a good agreement with those obtained by numerical bifurcation analysis in @xcite . the quadratic character of @xmath46 is explained by the fact that @xmath51 is determined by the second order terms in the taylor expansions of @xmath179 . we concentrate on the parameter region around @xmath180 , where @xmath46 changes its sign . for small positive @xmath22 , there are two dynamical regimes : for values of @xmath9 lying to the left and to the right of some @xmath62 neighborhood of @xmath181 corresponding to the supercritical and subcritical ahbs . the transition region between these two parameter regimes adds to the repertoire of qualitatively distinct dynamical behaviors . below , we study these three cases in more detail . it follows from theorem 3.1 that for @xmath182 and sufficiently small @xmath39 , ( [ 6.5 ] ) has a stable limit cycle @xmath74 , whose leading order approximation is given by x()=(|^2(a+a 2 ) , | , | ) , |=,[0 , 2 ) ( see figure [ f.6]a ) . moreover , the analysis of section 3 shows that all trajectories starting from a sufficiently small neighborhood of the origin and not belonging to @xmath42 , converge to @xmath74 . the leading order approximation of @xmath74 in ( [ 4.1 ] ) reveals a remarkable property of the oscillations generated by the limit cycle born from the supercritical ahb : the frequency of oscillations in @xmath34 is twice as large as that of the oscillations in @xmath35 . to explain the frequency doubling effect , we recall that = + o ( ) . therefore , ( [ 4.1 ] ) implies that , unless @xmath183 , the frequency of oscillations in @xmath34 is @xmath184 , while that of oscillations in @xmath35 is @xmath185 . the latter coincides with the frequency of the bifurcating periodic solution . below , we complement the analytical explanation of the frequency doubling with the geometric interpretation . the oscillations in @xmath35 can be well understood using the topological normal form of the ahb @xcite . indeed , since at the bifurcation the center manifold at the origin is tangent to the @xmath186 plane , a standard treatment of the ahb using the center manifold reduction @xcite shows that the projection of the bifurcating limit cycle onto @xmath186 plane to leading order is a circle ( figure [ f.6]b ) and the projection of the vector field is given by the equation of the angular variable ( [ 4.2 ] ) . therefore , the oscillations in @xmath35 are approximately harmonic with the period equal approximately to @xmath187 . the topological normal from , however , does not describe the oscillations in @xmath34 . for this , one needs to take into account the geometry of the bifurcating periodic orbit as a curve in @xmath0 . the geometry of @xmath74 is fully determined by the two geometric invariants : the curvature , @xmath188 , and the torsion , @xmath189 . for the purposes of the present discussion , we need only the latter , but we compute both invariants for completeness . after some algebra , the parametric equation for the periodic orbit ( [ 4.1 ] ) yields @xmath190\right|\over \left|\dot x\right|^3}= \sqrt{1 + 2a^2\left(3\cos 4\theta + 5\right)}+\mbox{h.o.t},\\ \lbl{4.3 } \kappa(\theta ) & = & { \left(\dot x , \ddot x , \dddot x\right)\over\left|\left[\dot x , \ddot x\right]\right|^2}= { -\gamma\over\alpha } { 6a\sin 2\theta\over 1 + 2a^2\left(3\cos 4\theta + 5\right ) } + \mbox{h.o.t},\quad \theta\in[0 , 2\pi).\end{aligned}\ ] ] the geometry of the leading order approximation of the periodic orbit is determined by the three parameters @xmath191 @xmath192 and @xmath193 . the former two parameters are the same as the parameters in the topological normal form of the nondegenerate ahb @xcite ; while the latter captures the geometry of the slow manifold ( or unstable manifold ) near the origin . from the geometrical viewpoint the bifurcation is degenerate if either @xmath51 or @xmath193 is equal to zero . the latter condition holds if and only if the slow manifold is either a plane or a circular paraboloid near the origin . in this case , equation ( [ 4.3 ] ) implies that ( to leading order ) the bifurcating orbit lies in a plane . generically , @xmath74 is not planar ( see figure [ f.6]a ) . the fact that the periodic orbit is generically not planar combined with the symmetry of the orbit about the @xmath194axis implies that its projection onto any plane containing @xmath194axis has to have a self - intersection ( see figure [ f.6]c ) . from figure [ f.6]c , it is clear that as the phase point goes around @xmath74 once , @xmath34 has to trace its range at least twice . therefore , the frequency of oscillations in @xmath34 has to be at least twice as high as that of the oscillations in @xmath35 . the above discussion implies that the frequency doubling of oscillations in @xmath34 is a generic geometric property of the ahb . * a * if the ahb is subcritical ( @xmath50 ) the loss of stability of the equilibrium at the origin of ( [ 6.5 ] ) results in the creation of multimodal trajectories which spend a considerable amount of time near a weakly unstable equilibrium . to describe the resultant dynamics we give the following definition . * definition 4.1 * we say that a trajectory of ( [ 6.5 ] ) undergoes multimodal oscillations for @xmath195 if there exist positive constants @xmath196 and @xmath197 independent of @xmath49 and an unbounded sequence of times @xmath198 such that a. : : @xmath199 , b. : : @xmath200\;\;\forall t^\prime \in \left(t_{2i-1 } , t_{2i+1}\right).$ ] the time intervals @xmath201 are called isis . the proximity of ( [ 6.5 ] ) to the ahb alone is clearly not sufficient to account for the appearance of the multimodal oscillations in ( [ 6.5 ] ) . below , we formulate two additional assumptions on the vector field outside of the small neighborhood of the equilibrium , which are relevant to ( [ 6.1])-([6.3 ] ) . under these conditions , we show that the subcritical ahb results in sustained multimodal oscillations . in addition , we determine the asymptotics of the isis for positive @xmath48 . for ( [ 6.5 ] ) to generate multimodal oscillations , it is necessary that the trajectories leaving an @xmath79 neighborhood of the origin reenter it after some interval of time . therefore , the vector field in @xmath56 neighborhood of the origin must provide a return mechanism . our second assumption on the global vector field of ( [ 6.5 ] ) is that the trajectories approaching the origin along a @xmath1 stable manifold , @xmath42 , are subject to a strong contraction toward @xmath42 , i.e. , in a small neighborhood of an @xmath56 segment of @xmath42 , the projection of the vector field onto a plane transversal to @xmath42 is sufficiently stronger than that along @xmath42 ( figure [ f.2]a ) . this guarantees that the trajectories entering such region of strong contraction approach @xmath42 very closely and follow it to an @xmath79 neighborhood of the unstable equilibrium , from where they are propelled away along the unstable manifold , @xmath202 . due to the proximity of ( [ 6.5 ] ) to the ahb , the motion away from the origin is very slow . this results in the pronounced intervals of the small amplitude oscillations . below , we summarize these observations into two formal assumptions on the global vector field * ( g1 ) * and * ( g2)*. for this , we first need to introduce some auxiliary notation . for analytical convenience , we assume that in an @xmath56 neighborhood of the origin , @xmath42 can be and has been straightened by a smooth change of coordinates . more specifically , the nonlinear terms in ( [ 6.5 ] ) satisfy h_2,3 ( x_1,0,0,)=0,w(x_1,)=-x_1+h_1(x_1,0,0,)>0 , x_1[d_1,0 ) , for some @xmath203 . note that @xmath204 is assumed to be sufficiently far away from the origin ( see figure [ f.8 ] ) . to describe the mechanism of return , we introduce two crossections : ^+=[-c_1 ^ 2 , c_1 ^ 2]d_^-=\{d_1}d__1 , where @xmath205 denotes a disk of radius @xmath206 centered at the origin : d_=\ { y^2:|y|}. positive constants @xmath206 and @xmath207 are sufficiently small ( see figure [ f.2]a ) and @xmath208 is chosen so that @xmath40 intersects the slow manifold @xmath58 transversally . in addition , we require that @xmath209 to guarantee that @xmath40 belongs to the region of validity of the local analysis of section 3 ( see remark 3.1c ) . let @xmath210 and consider a trajectory of ( [ 6.5 ] ) starting from @xmath211 . we assume that for sufficiently small @xmath47 and @xmath48 , every such trajectory intersects @xmath41 from the left . denote the point of the first intersection by @xmath212 . we assume that * ( g1 ) * the first return map @xmath213 depends smoothly on @xmath49 and @xmath214 . to measure the rate of contraction toward @xmath42 , we consider a @xmath215 matrix a(x_1,)= ( cc f_2x_2 & f_2x_3 + f_3x_2 & f_3x_3 ) _ ( x_1,0,0 , ) . let @xmath216 denote the eigenvalues of the symmetric matrix @xmath217 denote @xmath218 and @xmath219 . we assume that for sufficiently small @xmath47 and @xmath48 , * ( g2 ) * @xmath220;&\\ \lbl{4.7 } \exists\ ; d_2 \in ( d_1,0):\ ; \bar\lambda(x_1,\alpha)>0 , & x_1\in[d_1,d_2]&\quad \ & \quad \min_{x_1\in[d_1 , d_2 ] } { \bar\lambda(x_1,\alpha)\over w(x_1,\alpha ) } = o\left(\left|\ln\epsilon\right|\right),\end{aligned}\ ] ] * ( g3 ) * @xmath221 under these conditions , we have * theorem 4.1 * let @xmath50 , conditions in ( [ 4.3a ] ) , * ( g1 ) * and * ( g2 ) * hold . then for sufficiently small @xmath47 and @xmath222 a trajectory of ( [ 6.5 ] ) with initial condition from an @xmath79 neighborhood of the origin and not belonging to @xmath42 undergoes multimodal oscillations . the isis are uniformly bounded from below _ i^-=12(1+^4c^- ) , i=2,3 , . if , in addition , * ( g3 ) * holds then the isis satisfy two - sided bounds ^-_i ^+=12(1+^4+c^+ ) , i=2,3 , , for some @xmath223 . positive constants @xmath224 do not depend on @xmath225 @xmath191 and @xmath51 . * remark 4.1 * ( a ) : : the principal assumptions on the global vector field are formulated in * ( g1 ) * and ( [ 4.7 ] ) . the condition in ( [ 4.6 ] ) makes the derivation of certain estimates in the proof of the theorem easier and is used for analytical convenience . likewise , * ( g3 ) * is not essential for the proposed mechanism . however , without this condition obtaining the two - sided estimates for the isis requires an additional argument in the proof . condition * ( g3 ) * means that the two eigenvalues of @xmath193 have the same order of magnitude . this condition is not restrictive . ( b ) : : for fixed @xmath50 , the estimates in ( [ 4.8a ] ) and ( [ 4.8b ] ) can be rewrittten as _ i , with constants @xmath226 independent from @xmath22 . similarly , for fixed @xmath22 and varying @xmath50 inequalities in ( [ 4.8a ] ) and ( [ 4.8b ] ) imply |c^- -12_i |c^+ -12 , where constants @xmath227 do not depend on @xmath50 . the numerical experiments presented in section 2 show that the transition from subcritical to supercritical ahb contains a distinct bifurcation scenario involving the formation of chaotic attractor via a period - doubling cascade . the analytical explanation of these phenomena is outside the scope of the present paper . below we comment on the difficulties arising in the analytical treatment of this problem . in the present section , we use a combination of the analytic and numerical techniques to elucidate the origins of the complex dynamics arising in the parameter regime near the border between regions of sub- and supercritical ahb . the principal features of this bifurcation scenario are summarized in figures [ f.4 ] and [ f.5 ] . in these numerical experiments , we kept @xmath22 fixed at a small positive value and varied @xmath51 . by taking progressively smaller values of @xmath50 , one first observes that the multimodal patterns exhibit an increase in the isis ( figure [ f.4]a , b ; see also figure [ f.3]b ) . we consider these oscillatory patterns regular ( even if they are not periodic ) , because the timings of the spikes remain within narrow bounds in accord with ( [ 4.8a ] ) . for smaller values of @xmath50 , the oscillatory patterns become irregular and are characterized by long intervals of oscillations of small and intermediate amplitudes between successive spikes ( figure [ f.4 ] c , d ) . as @xmath51 becomes negative , the trajectories lose spikes and consist of irregular oscillations ( figure [ f.5 ] a , b ) . further decrease of @xmath51 leads the system through the reverse cascade of period - doubling bifurcations ( figure [ f.5 ] c - h ) . the period - doubling cascade terminates with the creation of the limit cycle , which can be followed to a nondegenerate supercritical ahb by letting @xmath228 ( figure [ f.5 ] g , h ) . to account for the bifurcation scenario described above , we construct the first return map . since near the origin the trajectories spend most of the time in the vicinity of the @xmath2 slow manifold , the first return map is effectively one - dimensional . the distinct unimodal structure of the @xmath1 first return map affords a lucid geometric interpretation for the bifurcation scenario in the @xmath229 systems of differential equations near the transition from sub- to supercritical ahb . specifically , we show that the mechanism for generating complex dynamics in the continuous system in this parameter regime is the same as in the classical scenario of the period - doubling transition to chaos in the one - parameter families of unimodal maps @xcite . * a * + * c * we next turn to the derivation of the first return map . we start with extending the asymptotic analysis of section 3 to cover the case of @xmath230 . this requires computing additional terms on the right hand side of the reduced equation ( [ 5.24 ] ) , because already for @xmath231 , the term involving @xmath51 on the right hand side of ( [ 5.24 ] ) is comparable with the @xmath232 remainder term . we then use the more accurate approximation for the equation for @xmath155 to compute a @xmath1 mapping : g : i()i(+ 2 ^ -1 ) , which describes how @xmath155 changes after one cycle of oscillations . as noted above , the reduced equation ( [ 5.24 ] ) is not suitable for analyzing the case of small @xmath233 and one needs to include more terms in the expansion on the right hand side of ( [ 5.24 ] ) . in analogy with the topological normal form for the degenerate ahb @xcite , we expect that terms up to @xmath234 ) are needed in ( [ 5.24 ] ) to resolve the dynamics for small @xmath233 . this can be achieved by a straightforward albeit tedious calculation . below we describe the formal procedure of obtaining the required expansions . the justification of these expansions can be done in complete analogy with the analysis of section 3 . first , we compute terms on the right hand sides of ( [ 5.11a ] ) and ( [ 5.12 ] ) up to @xmath232 and @xmath235 respectively . then we look for a solution of ( [ 5.11a ] ) in the following form : ( ) = _ 0(r,)+_1(r,)+^2_2(r,)+o(^3 ) . by taking an initial condition from @xmath61 and using anzats ( [ 9.1 ] ) in ( [ 5.11a ] ) , we recover @xmath236 ( see ( [ 5.16 ] ) ) and find the next two terms in the expansion of @xmath237 : @xmath238 and @xmath239 . thus , we obtain the approximation of the slow manifold with the accuracy @xmath232 . next , we plug in ( [ 9.1 ] ) into ( [ 5.12 ] ) and collect terms multiplying equal powers of @xmath49 to obtain r = r_1(r,)+ +^5 r_5(r,)+o(^6 ) , where @xmath240 are @xmath187 periodic functions of the second argument . finally , by rewriting ( [ 9.2 ] ) in terms of @xmath155 ( see ( 3.25 ) ) , integrating it over one period of oscillations , @xmath241 , and disregarding @xmath234 terms , we obtain a map for the change in @xmath155 after one cycle of oscillations i_n+1=g(i_n ) , g(i)i-2 ^ 2(i + + ^2ci ) , = 2 ^ -1 . except for the last term in the definition of @xmath242 , the map in ( [ 9.4 ] ) follows from the reduced system ( [ 5.24 ] ) and ( [ 5.25 ] ) . the last term is only needed if the value of @xmath233 does not exceed @xmath62 . our calculations show that the second lyapunov coefficient , @xmath243 , is negative for the values of parameters used in ( [ 6.1])-([6.3 ] ) . therefore , for sufficiently small @xmath47 , ( [ 9.4 ] ) defines a unimodal map . away from an @xmath79 neighborhood of @xmath28 , the graph of @xmath244 is almost linear with a weakly attracting slope @xmath245 for @xmath246 ( figure [ f.7 ] ) . for @xmath246 , @xmath242 has a unique fixed point : |i=\ { cc - + c^2+o(^3 ) , & 0 . . for negative values of @xmath247 , @xmath248 is a stable fixed point of @xmath242 , which corresponds to a stable limit cycle born from a supercritical ahb ( see figure [ f.5 ] g , h ) . it follows from ( [ 9.4 ] ) that for increasing values of @xmath51 , the graph of the map moves down . consequently , the fixed point moves to the left and eventually looses stability via a period - doubling bifurcation ( figure [ f.5 ] e , f ) . further increase in @xmath51 , yields more period - doubling bifurcations ( figure [ f.5 ] c , d ) . given the unimodal character of @xmath242 , one expects that this sequence of period - doubling bifurcations eventually leads to the formation of a chaotic attractor ( figure [ f.5 ] a , b ) . the unimodal character of the map combined with the manner of its dependence on @xmath51 suggest a clear geometric mechanism for the formation of the chaotic attractor and the subsequent period - doubling cascade arising near the transition from sub- to supercritical ahb ( figure [ f.5]a - h ) . we now outline the limitations of the asymptotic analysis of this section and difficulties arising in the justification of ( [ 9.4 ] ) . according to ( [ 9.5 ] ) already at the moment of the first period - doubling bifurcation , fixed point @xmath169 belongs to an @xmath79 neighborhood of @xmath28 . in this neighborhood , @xmath249 ( see ( [ 5.22 ] ) ) and , therefore , @xmath169 lies outside of the region of validity of the asymptotic analysis . in this region , ( [ 9.4 ] ) may only be considered as a formal asymptotic expression . a rigorous justification of the from of the map in the boundary layer meets substantial analytical difficulties : it requires introducing an additional set of intermediate asymptotic expansions and matching them with those obtained in section 3 . we do not address this problem in the present work . below , we resort to using numerical techniques to verify the principal features of the first return map suggested by the asymptotic analysis : the unimodality of @xmath242 and its dependence on @xmath51 . the numerics confirmed our predictions about the form of the first return map and it also revealed certain additional features . for numerical construction of the first - return map , we fix angle @xmath250 and define a crossection in cylindrical coordinates : @xmath251 let @xmath252 be a trajectory of ( [ 6.5 ] ) and @xmath253 denote a sequence of times , at which @xmath254 . then , we define : ( t_k ) ( t_k+1)(t ) = . by a suitable choice of @xmath255 , we can achieve that a trajectory of ( [ 6.5 ] ) , intersects @xmath256 once during one cycle of oscillations . therefore , for a trajectory lying in a @xmath2 slow manifold , ( [ 9.6 ] ) defines a @xmath1 first - return map . note that maps ( [ 9.4 ] ) and ( [ 9.6 ] ) are related via rescaling of the coefficients , since to leading order @xmath257 . therefore , the graphs of @xmath258 and @xmath259 are similar in the domain , where ( [ 9.4 ] ) is valid . we computed the first return maps for fixed @xmath260 and for several values of @xmath51 . the representative plots are shown in figure [ f.8 ] a - c . the numerically computed maps confirmed our expectations about the graph of @xmath261 in the boundary layer near @xmath28 : in this region , the map is decreasing with a strongly expanding slope . away from @xmath28 , the graph of @xmath261 contains an almost linear branch , whose slope is very close to @xmath262 as predicted by ( [ 9.4 ] ) ( see lower branches in figure [ f.8]b , c ) . these numerics also confirm that under the variation of @xmath51 , the graph of the first return map is translated in vertical direction ( figure [ f.8 ] a - c ) . therefore , the family of the first return maps possesses two principal ingredients ( the unimodality and the additive dependence on @xmath51 ) , which are necessary for the qualitative explanation of the bifurcation scenario given in figure [ f.7 ] . in addition , our numerical experiments reveal a new feature of the first return map : for small @xmath263 , the graph of the map has another almost linear branch in the outer region away from the origin ( see figure [ f.8]b ) . this branch of the graph of the map also has an attracting slope . the presence of this branch in the first return map indicates the existence of another ( branch of the ) slow manifold different from that described in section 3 . a possible explanation for the appearance of the upper branch in the first return map is due to the unstable manifold of the periodic orbit . for small @xmath263 , the first - return map ( [ 9.6 ] ) is multivalued in the outer region . however , since both the upper and the lower branches have positive slopes less than @xmath264 , the qualitative dynamics for the map shown in figure [ f.8]b does not depend on the exact mechanism for the selection between the branches in ( [ 9.6 ] ) . although explaining the multivaluedness of the first return maps shown in figures [ f.8]b , c presents an interesting problem , it is not critical for the qualitative explanation of the bifurcation scenario arising during the transition from the subcritical to supercritical ahb . the latter was the main goal of the present subsection . * remark 4.2 * in addition , to the region in the parameter space containing the border between the regions of sub- and supercritical ahb , there is another parameter regime resulting in the complex dynamics . in @xcite , it was shown numerically that the limit cycle born from the supercritical ahb in ( [ 6.5 ] ) undergoes a period - doubling cascade leading to the formation of the chaotic attractor for increasing values of @xmath22 . the first period - doubling bifurcation in this cascade is shown in figure [ f.1]c , d . this bifurcation scenario is consistent with the form of the first - return map constructed in this subsection . indeed , it is easy to see from ( [ 9.5 ] ) that for increasing values of @xmath55 the fixed point , @xmath169 , moves to the left . therefore , the explanation given above for the period - doubling cascade resulting from the variation of @xmath51 near @xmath28 also applies to the case of increasing @xmath22 and fixed @xmath265 ( see figure [ f.8]d ) . in the present section , we show that under the assumptions of theorem 4.1 the trajectories of ( [ 2.5 ] ) with initial conditions in @xmath41 enter @xmath61 . the local analysis in section 3 describes the behavior of trajectories from the moment they reach @xmath61 and until they leave @xmath57 . in particular , it shows that the dynamics near the origin has two phases : the fast approach to the slow manifold and the slow drift away from the origin along the slow manifold . upon leaving @xmath57 , the trajectories are reinjected back to @xmath41 by the return mechanism postulated in * ( g1)*. this scenario implies that the system undergoes multimodal oscillations as stated in theorem 4.1 . we start with presenting several auxiliary estimates , which will be needed for the proof . by ( [ 4.3a ] ) , one can choose @xmath120 such that w(x_1 , ) c_1 |x_1|,x_1 and for sufficiently small @xmath44 . next , we note that @xmath266 . this equation and ( [ 4.6 ] ) , by the implicit function theorem , imply that there exists @xmath267 such that @xmath268 . \end{array } \right.\ ] ] using ( [ 4.6 ] ) and ( [ 7.1 ] ) , we have @xmath269,\ ] ] where @xmath270 is defined in ( [ 4.3a ] ) . let @xmath271 be such that ( * g1 * ) and ( * g2 * ) hold for @xmath272 $ ] . in the remainder of this section , unless stated otherwise , it is assumed that @xmath273 $ ] . to simplify notation , we will often omit the dependence of various functions on @xmath22 . using ( [ 4.3a ] ) , we rewrite ( [ 6.5 ] ) in the following form : @xmath274 where @xmath275 and @xmath276 are defined in ( [ 4.3a ] ) and ( [ 4.5 ] ) respectively ; @xmath277 , and w(0)=0 , ( x_1,y)=o(|y|^2 ) , _ 1,2(x_1,y)=_1,2(x_1)+o(|y|),x . let d_3=-m^2 ^ 2 denote the @xmath194coordinate on the left lateral boundary of @xmath61 ( see ( [ 5.2 ] ) ) and @xmath278\times d_{\delta_1}$ ] ( see figure [ f.9 ] ) . from ( [ 6.5 ] ) and ( [ 4.3a ] ) , one can see that for @xmath279 sufficiently small ( independent of @xmath49 ) , the right hand side of ( [ 7.2 ] ) @xmath280 therefore , in @xmath281 , ( [ 7.2 ] ) and ( [ 7.3 ] ) may be rewritten as follows = a(x_1 ) y + ( x_1,y ) , where @xmath282 by ( [ 7.1 ] ) and ( [ 7.4 ] ) , we have |(x_1,y)|,(x_1,y)__1 , for some @xmath137 independent of @xmath49 . to follow the trajectories from @xmath41 to @xmath61 , we introduce two regions : @xmath283\times d_{\delta_1}\quad\mbox{and}\quad\pi_{\delta_2}^2=[d_2,d_3]\times d_{\delta_2}\;\;\mbox{(see figure \ref{f.9})}.\ ] ] recall that @xmath284 denotes the disk of radius @xmath285 ( see ( [ 4.4a ] ) ) . positive constant @xmath207 is the same as in ( [ 4.4 ] ) and @xmath286 will be specified later ( figure [ f.9 ] ) . recall that @xmath287 denotes the value of the @xmath194coordinate on the the left lateral boundary of @xmath61 ( see ( [ 5.2 ] ) ) . by taking @xmath69 large enough , we can arrange @xmath288 for @xmath289 $ ] . by * ( g1 ) * , the vector field in @xmath290 is sufficiently strong so that the trajectories entering @xmath291 through @xmath41 get into a narrow domain @xmath292 and remain there until they reach @xmath61 . the following lemma allows to control the trajectories in @xmath278\times d_{\delta_1}$ ] . * lemma 5.1 * let @xmath293\subseteq[d_1,d_3],\ ; 0<\bar\delta\le { \delta_1},$ ] and -=_x_1_|y|=1(a(x_1)y , y ) < 0 . then |y(_1)|||y(x_1)|2|e^-(x_1-_1 ) , x_1 , provided 1 . * proof : * we may assume that @xmath294 $ ] , since , otherwise , by the uniqueness of solution of the initial value problem for ( [ 7.5 ] ) , @xmath295 @xmath296 $ ] , and ( [ 7.8 ] ) holds . from ( [ 7.5 ] ) , ( [ 7.6 ] ) , and ( [ 7.7 ] ) , we have -|y(x_1)| + @xmath297 is defined in ( [ 7.7 ] ) , and @xmath298 denotes the euclidean norm of @xmath299 . let @xmath300 denote the solution of the initial value problem |y^=-|y + |(x_1,|y ) , |y(_1)=| . it is sufficient to show that ( [ 7.8 ] ) holds for @xmath300 . we represent @xmath300 as the limit of the sequence of successive approximations |y^(n+1)(x_1)=|e^-(x_1-_1)+__1^x_1 e^-(x_1-s ) @xmath301 we use induction to show ||y^(n)(x_1)| 2 |e^-(x_1-_1),x , n=0,1,2, . inequality ( [ 7.13 ] ) holds for @xmath302 . we show that @xmath303 . using the definition of @xmath304 in ( [ 7.10 ] ) and @xmath305 , we have @xmath306 using ( [ 7.14 ] ) and ( [ 7.9 ] ) , from @xmath305 we obtain @xmath307 . by induction , ( [ 7.13 ] ) holds . by taking @xmath308 in ( [ 7.13 ] ) , we have ( [ 7.15 ] ) and the standard theory for differential inequalities @xcite . + @xmath128 in the following lemma , we determine the size of @xmath309 . * lemma 5.2 * there exists positive @xmath310 , such that the trajectories of ( [ 7.2 ] ) and ( [ 7.3 ] ) entering @xmath309 from @xmath311 remain in @xmath309 until they reach @xmath61 ( see figure [ f.9 ] . ) * proof : * denote @xmath312}\sup_{\left|y\right|=1}\left(\tilde a(x_1)y , y\right).\ ] ] recall @xmath313 therefore , by ( [ 4.6 ] ) , @xmath314 is positive . in addition , @xmath315 } \sup_{\left|y\right|=1}\left(\tilde a(x_1)y , y\right)= \min_{x_1\in[d_2,d_3]}{-\sup_{\left|y\right|=1}\left ( a(x_1)y , y\right)\over w(x_1)}\\ & = & { \min_{x_1\in[d_2,d_3 ] } \bar\lambda(x_1)\over\max_{x_1\in[d_2,d_3 ] } w(x_1)}= { \bar\lambda(d_3)\over\max_{x_1\in[d_2,d_3 ] } w(x_1)}= o(\epsilon^2 ) . \end{aligned}\ ] ] and @xmath316 . next we apply lemma 5.1 with @xmath317 @xmath318 , and @xmath319 . note that for small @xmath320 the inequality ( [ 7.9 ] ) can be rewritten as 1 . since @xmath321 ( see ( [ 7.4a ] ) ) and @xmath322 , one can choose @xmath310 so that ( [ 7.16 ] ) holds . + @xmath128 having found the size of @xmath309 , we now determine the rate of contraction in @xmath290 sufficient to funnel the trajectories entering @xmath290 through @xmath41 to @xmath309 . * lemma 5.3 * the trajectories of ( [ 7.2 ] ) and ( [ 7.3 ] ) entering @xmath290 through @xmath41 remain in @xmath290 until they reach @xmath309 . * proof : * denote @xmath323}\sup_{\left|y\right|=1}\left(\tilde a(x_1)y , y\right).\ ] ] from * ( g2 ) * , one finds that @xmath324 is positive . lemma 5.3 now follows from lemma 5.1 with @xmath325 @xmath326 , and @xmath327 . indeed , for @xmath328 , we have _ 1 . inequality ( [ 7.17 ] ) is sufficient for ( [ 7.9 ] ) to hold . by lemma 5.1 , we have |y(d_2)|2_1 e^-_1(d_2-d_1 ) . with @xmath328 , by ( [ 7.19 ] ) , we can achieve @xmath329 . + @xmath128 lemmas 5.2 and 5.3 imply that the trajectories entering @xmath290 through @xmath41 stay in @xmath330 until they reach @xmath61 . moreover , the inequality in * ( g1 ) * guarantees that such trajectories are bounded away from @xmath42 . thus , we can use the analysis of section 3 to describe the evolution of trajectories from the moment they reach @xmath61 until they leave @xmath57 . by remark 3.1c , this description extends to any region where @xmath331 i.e. , the trajectories can be controlled until they hit @xmath40 . this is followed by the return to @xmath291 according to * ( g1 ) * , and the next cycle of the multimodal oscillations begins . the analysis of this section applies to any trajectory starting from @xmath61 and not belonging to @xmath42 . it remains to estimate the isis . for this , we compute the time needed for the trajectory starting in an @xmath79 neighborhood to return back to this neighborhood after making one global excursion . since the time of flight of the trajectory outside a small neighborhood of the origin depends regularly on the control parameters , the duration of the very long isis is determined by the time spent in the neighborhood of the origin . to estimate the latter , we note that at the moment a trajectory enters @xmath61 , we have |y(d_3)|c_3 ^2 . this follows from lemma 5.3 , since @xmath332 and @xmath333 to obtain the lower bound on @xmath334 , recall that by * ( g1 ) * , we have |y(d_1)|>0 . as follows from * ( g3 ) * , the maximal rate of contraction in @xmath281 does not exceed @xmath70 in absolute value . this combined with ( [ 7.20b ] ) implies that |y(d_3)|c_4 ^ 2 + 2 for some @xmath335 and @xmath336 independent of @xmath49 . we omit the proof of ( [ 7.20c ] ) , because it is completely analogous to that of lemma 5.1 . let @xmath337 denote the moment of time when a trajectory of ( [ 6.5 ] ) enters @xmath61 from @xmath309 . after switching back to the original parametrization of @xmath338 by time , @xmath339 , we rewrite ( [ 7.20a ] ) and ( [ 7.20c ] ) : c_3 ^ 2 + 2|y(t^-)|c_4 ^ 2 . for @xmath340 the trajectory approaches and remains close to the slow manifold as long as @xmath341 ( see remark 3.1c ) . let @xmath342 and denote t^+=\{t > t^-:|y(t)|=^j}. from ( [ 7.20 ] ) and ( [ 7.21 ] ) , we have the following bounds for @xmath343 and @xmath344 : c_6 ^ -2-i_0c_7 ^ -2 i_1=o(^2(1-j ) ) , j(23 , 1 ) . for these ranges of values of @xmath168 and @xmath345 , from ( [ 5.55 ] ) we have _ in = t^+-t^-=12 ^ 2((1+i_0 ) ( 1+o(1 ) ) ) . the combination of ( [ 7.22 ] ) and ( [ 7.23 ] ) yields two - sided bounds for @xmath346 : ^-_in_in_in^+ , where @xmath347 where positive constants @xmath348 can be chosen independent of @xmath49 . inequalities ( [ 7.24 ] ) provide bounds for the time that a multimodal trajectory spends near the unstable equilibrium . on the other hand , the time of flight outside a small neighborhood of the origin , @xmath349 , depends regularly on @xmath49 , by * ( g2)*. therefore , by adjusting constants @xmath350 in ( [ 7.24 ] ) if necessary , one can obtain uniform bounds on the isis @xmath351 for sufficiently small @xmath47 , positive @xmath51 and @xmath55 from bounded intervals , as stated in theorem 4.1 . in the present paper , we investigated a mechanism for generation of multimodal oscillations in a class of systems of differential equations close to an ahb . our analysis covers both cases of sub- and supercritical ahb . for the supercritical case , we identified a novel geometric feature of the bifurcating limit cycle , the frequency doubling effect . it turns out that generically in the normal system of coordinates the oscillations in one of the variables are twice faster than in the remaining two variables . therefore , the leading order approximation of the limit cycle bifurcating from the supercritical ahb requires two harmonics . the asymptotic analysis of the present paper explains the frequency - doubling . in addition , we provide a complementary geometric interpretation to this counterintuitive effect . in particular , we showed that it is a consequence of the geometry of the limit cycle . the latter is not captured by the topological normal form of the ahb . the analysis of the multimodal oscillations arising from the subcritical ahb requires additional assumptions on the global behavior of trajectories . we identified two principal properties of the global vector field : the mechanism of return and the strong contraction property . in the presence of this global structure , the subcritical ahb produces sustained multimodal oscillations combining the small amplitude oscillations near the unstable equilibrium with large amplitude spikes . the resultant motion is recurrent in a weak sense : it may not be periodic but nevertheless the timings of the spikes possess certain regularity . we have shown that the isis have well - defined asymptotics near the ahb and comply to the two - sided bounds , which depend on the principal bifurcation parameters . our estimates show that near the ahb , the isis can be extremely long and can change greatly under relatively small variation of the bifurcation parameters . the ability of the system to exhibit such extreme variability in the isi duration is important in many applications , in particular , in the context of neuronal dynamics . previous studies investigated different possible mechanisms for generating multimodal patterns with very long isis @xcite . for the finite dimensional approximation of the model of solid - fuel combustion ( [ 6.1])-([6.3 ] ) , the main motivating example for our work , the proximity to the homoclinic bifurcation was suggested in @xcite as a possible mechanism for prolonged isis . our conclusions confirm the importance of the proximity to the homoclinic bifurcation for explaining the oscillatory patterns in ( [ 6.1])-([6.3 ] ) . the proximity to the homoclinic bifurcation is implicitly reflected in our assumptions on the global vector field . however , we emphasize the critical role of the ahb : the duration of the isis can be effectively controlled by the parameters associated with the ahb without changing the distance of the system to the homoclinic bifurcation . in all our numerical experiments , the system remained bounded away from the homoclinic bifurcation , nevertheless it exhibited patterns with very long isis whose duration was amenable to control . our analysis extends the estimate for the isis obtained in @xcite to a wide class of problems . it also emphasizes the proximity of the system to the border between sub- and supercritical ahb , as another factor in creating oscillatory patterns with long isis . we show that as this border is approached from the subcritical side the isi grow logarithmically . this observation is important for explaining the oscillatory patterns generated by ( [ 6.1])-([6.3 ] ) since in this model the ahb can change its type under the variation of the second control parameter @xmath9 . this situation is not specific to the model of solid fuel combustion . recent studies suggest that there is a class of neuronal models close to the ahb whose type may change with the values of parameters @xcite . therefore , it is important to understand the principal features of the transition from sub- to supercritical ahb . we found that when the border between the regions of sub- and supercritical bifurcation is approached from the subcritical side ( while the distance from the ahb remains fixed ) , the oscillations become chaotic . the regime of irregular oscillations is then followed by the reverse period doubling cascade . to understand the nature of this bifurcation scenario we used a combination of analytic and numerical techniques . using the insights gained from the asymptotic analysis , we constructed a @xmath1 first - return map . the map provides a clear geometric interpretation for the bifurcation scenario near the transition from sub- to supercritical ahb . our study suggests that the formation of the chaotic attractor via a period - doubling cascade is a universal feature of this transition . for example , the bifurcation scenarios reported for the hodgkin - huxley model in @xcite are very similar to those studied in the present paper and are likely to share the same mechanism . mixed - mode oscillations similar to those studied in the present paper , have been studied in for a class of the slow - fast systems in @xmath0 near the ahb . although , the work toward developing a complete mathematical theory for such oscillations is still in progress , the general mechanism for their generation and the bifurcation structure of the problem have been greatly elucidated recently @xcite . the present paper shows the relation between the mechanisms for the mixed - mode oscillations in the model in @xcite and for those in the slow - fast systems . the latter possess a well - defined structure of the global vector field due to the presence of the disparate timescales in the governing equations @xcite . the analyses of the mixed - mode oscillations in @xcite use in an essential way the relaxation structure of the problem . the model in @xcite is an example of the mixed - mode generating system , which does not possess an explicit relaxation structure . in fact , it is hard to expect such structure in a system obtained via projecting an infinite - dimensional system onto a finite - dimensional subspace . in formulating the assumptions on the global vector - field ( * g1 * ) and ( * g2 * ) , we were looking for the minimal requirements on the system near an ahb that guarantee the existence of the mixed - mode solutions . due to the lack of the information about the global vector field of ( [ 6.1])-([6.3 ] ) , it appears impossible to verify these conditions analytically . however , the numerical simulations clearly show that system of equations ( [ 6.1])-([6.3 ] ) possesses the qualitative structure required by ( * g1 * ) and ( * g2 * ) ( figure [ f.2]b ) . on the other hand , we expect that conditions ( * g1 * ) and ( * g2 * ) should be possible to verify analytically for a wide class of slow - fast systems . therefore , we believe that our results will be useful for understanding mixed - mode oscillations in such systems . in particular , it would be interesting to apply this approach to the modified hodgkin - huxley system @xcite . the numerical results reported in @xcite strongly suggest that the mechanism proposed in the present paper is responsible for the generation of the very slow rhythms and chaotic dynamics in the hodgkin - huxley model . * acknowledgments . * we thank victor roytburd for introducing us to this problem and to michael frankel and victor roytburd for helpful conversations . this work was partially supported through national science foundation award no 0417624 . [ sec : c ] in this appendix , we list explicit expressions of various constants and trigonometric polynomials , which appear in the definitions in of the slow manifold , @xmath58 , and the first lyapunov coefficient , @xmath51 . all expressions are given in terms of the coefficients of the power expansions on the right hand side of ( [ 5.1 ] ) . by @xmath352 we denote @xmath353 @xmath354 . the following constants are used to define the leading order approximation of the slow manifold in ( [ 5.3 ] ) and ( [ 5.4 ] ) : @xmath355 the following trigonometric polynomials enter the right hand sides of ( [ 5.11a ] ) and ( [ 5.12 ] ) : @xmath356 functions @xmath357 are used in the calculation of the first lyapunov coefficient @xmath51 ( see ( [ 5.5 ] ) ) . j. drover , j. rubin , j. su , and b. ermentrout , analysis of a canard mechanism by which excitatory synaptic coupling can synchronize neurons at low firing frequencies , _ siam j. appl . _ , @xmath360 , 6992 , 2004 . e.i . volkov and d.v . volkov , multirhythmicity generated by slow variable diffusion in a ring of relaxation oscillators and noise - induced abnormal interspike variability , _ phys . e _ , @xmath360 , 2002 .

one- and two - parameter families of flows in @xmath0 near an andronov - hopf bifurcation ( ahb ) are investigated in this work . we identify conditions on the global vector field , which yield a rich family of multimodal orbits passing close to a weakly unstable saddle - focus and perform a detailed asymptotic analysis of the trajectories in the vicinity of the saddle - focus . our analysis covers both cases of sub- and supercritical ahb . for the supercritical case , we find that the periodic orbits born from the ahb are bimodal when viewed in the frame of coordinates generated by the linearization about the bifurcating equilibrium . if the ahb is subcritical , it is accompanied by the appearance of multimodal orbits , which consist of long series of nearly harmonic oscillations separated by large amplitude spikes . we analyze the dependence of the interspike intervals ( which can be extremely long ) on the control parameters . in particular , we show that the interspike intervals grow logarithmically as the boundary between regions of sub- and supercritical ahb is approached in the parameter space . we also identify a window of complex and possibly chaotic oscillations near the boundary between the regions of sub- and supercritical ahb and explain the mechanism generating these oscillations . this work is motivated by the numerical results for a finite - dimensional approximation of a free boundary problem modeling solid fuel combustion .

You are an expert at summarizing long articles. Proceed to summarize the following text: this note is a write - up of talks given at the workshop `` vertex operator algebras in mathematics and physics '' at the fields institute in toronto in october 2000 and at some other occasions before . its purpose is to explain how results in low dimensional quantum field theory obtained from different perspectives and with different motivations can be put into a uniform picture by generalizing the notions of discriminant forms and genera from integral quadratic forms to vertex operator algebras , indicating a rich underlying arithmetical theory . the relation between two - dimensional conformal field theory and three - dimensional topological quantum field theory has been quite known from the beginning . for example , witten relates in his well - known paper @xcite the wess - zumino - witten models with chern - simons theories and the jones polynomial of knots . the basic idea is to use the monodromy properties of correlation functions on surfaces to obtain invariants for knots and three - dimensional manifolds . being not overly precise , one can say that a quantum field theory is a functor from some bordism category of oriented manifolds to some category of linear spaces satisfying some basic gluing axioms . @xmath0 @xmath1 @xmath2 @xmath3 @xmath4 for two - dimensional conformal field theories , one takes two - dimensional surfaces with a conformal structure on it and one is forced to take infinite dimensional linear spaces . using instead riemannian surfaces ( which have a naturally defined conformal structure ) , one reaches the more restricted but essential notion of chiral algebras . for three - dimensional topological quantum field theories ( 3d - tqfts ) , one takes three - dimensional topological manifolds ( together with extra structures like framed links ) and only finite dimensional linear spaces . it took some time before precise and efficient mathematical formulations for both structures , originating partly from physics , emerged . chiral algebras have been axiomatized by the algebraic structure of _ vertex algebras _ ; see @xcite for some approaches . 3d - tqfts can be constructed from _ modular braided tensor categories _ ; see @xcite . in this paper , we will restrict ourselves mainly to vertex operator algebras ( voas ) ; these are vertex algebras which have a virasoro element and we assume that they are non - negatively graded ; we consider only simple and rational voas and we make some further strong assumptions . there are many different kinds of examples of voas known , but it is quite unclear yet how a general structure theory for them may look like . an important class of voas can be constructed from positive definite even integral lattices . for integral lattices , there is a deeply developed structure theory available leading to invariants like rational equivalence , the genus , or the spinor genus . the first purpose of this paper is to explain how the notion of the _ genus _ can be generalized in a natural way from lattices to voas by using modular braided tensor categories so that the definition is compatible with the notion of the genus for lattices . secondly , we show that new voas can be constructed from given ones only by using the associated modular braided tensor category , generalizing thereby the well - known discriminant technique of quadratic form theory ; cf . @xcite . it should be mentioned that there are papers , which do not distinguish between voas and the associated modular braided tensor category . usually , the authors make assumption [ hauptvermutung ] given below and work in the setting of 3d - tqfts . the example of the lattice voas associated to the two positive definite even unimodular lattices @xmath5 and @xmath6 in dimension @xmath7 @xcite shows that there are different voas which can lead to the same 3d - tqft and which have the same central charge . the paper is organized as follows . in the second section , some basic definitions and results from the theory of integral quadratic forms are recalled . in section three , we define the genus of a vertex operator algebra and try to formulate theorems or at least conjectures analogous to the ones known for quadratic forms . in the final section , we reformulate the extension problem for vertex operator algebras in terms of the associated modular braided tensor category as already outlined in my paper @xcite . this extension problem can be considered as a kind of problem in coding theory over 3d - tqfts . the solution for voas defining an abelian intertwining algebra is given . also , complete results for self - dual framed voas or , equivalently , self - dual codes over the 3d - tqft associated to the ising model , are presented . there are many open problems about the structure of vertex algebras compared to what is known for quadratic forms . but the theory of quadratic forms is more than two hundred years old . it seems to me that from the fifteen years old theory of vertex algebras we can see only the tip of the iceberg of such a structure theory . i like to thank the organizers of the conference for inviting me to the fields institute and for the possibility to present these ideas . i am grateful to markus rosellen for his comments on an early version of the paper and for discussions about abelian intertwining algebras as well as to geoffrey mason for reminding me of some references . i also like to thank jim lepowsky and the referee for their many valuable suggestions . finally , my thanks goes to supun koranasophonpun for proofreading the manuscript . we collect some results from the arithmetic of integral quadratic forms . the theory of genera for binary forms was developed by gau in @xcite . instead of the more classical notion of quadratic forms in coordinates , we use the language of lattices . we denote in this section by @xmath8 the additive group of the integers modulo @xmath9 and by @xmath10 the ring of @xmath11-adic integers . recall that a map @xmath12 from an abelian group @xmath13 into an additively written group @xmath14 is called a quadratic form if the expression @xmath15 is additive in @xmath16 and in @xmath17 , i.e. , bilinear . a _ finite quadratic space _ @xmath18 is a finite abelian group @xmath19 together with a quadratic form @xmath20 , such that the induced bilinear form @xmath21 , @xmath22 is nondegenerate . note that different quadratic forms can induce the same bilinear form . a finite quadratic space can be decomposed uniquely as a direct sum : @xmath23 where the @xmath24 are the @xmath11-sylow subgroups of @xmath19 and @xmath25 is the quadratic form @xmath12 restricted to @xmath24 . furthermore , each @xmath26 is the direct sum of quadratic spaces as follows : for primes @xmath27 , it is the sum of spaces @xmath28 , with @xmath29 , @xmath30 and quadratic form @xmath31 , @xmath32 , where @xmath33 is an integer with jacobi symbol @xmath34 . the only relations in the abelian semigroup generated by the @xmath28 for fixed @xmath11 arise from @xmath35 , with @xmath36 , @xmath37 . for @xmath38 , one takes @xmath39 , there are also two other kind of generators and further relations ; cf . @xcite , prop . 1.8.1 and 1.8.2 . * definition 2.b * an even lattice is a free @xmath40-module of finite rank together with a nondegenerate symmetric bilinear map @xmath41 , such that @xmath42 for @xmath43 . the map @xmath44 can be linearly extended to @xmath45 for any ring @xmath46 . every lattice @xmath47 defines a quadratic space @xmath18 by letting @xmath48 , where @xmath49 is the dual lattice , and @xmath50 is the quadratic form @xmath51 , called the discriminant form . the _ signature @xmath52 _ of @xmath47 is the pair @xmath53 , where @xmath54 respectively @xmath55 are the maximal dimension of a positive respectively negative definite subspace of @xmath56 . by sylvester s law of inertia , @xmath57 , the rank of @xmath47 . the _ genus of @xmath47 _ is the collection of the local lattices @xmath58 , @xmath11 a prime number , including @xmath59 for @xmath60 . it follows easily that the genus of @xmath47 determines @xmath61 and @xmath52 . also , the converse is true : the discriminant form @xmath61 and the signature @xmath52 determine the genus of @xmath47 . not every pair consisting of a finite quadratic space @xmath18 and a pair @xmath53 of nonnegative integers can be realized as the genus of a lattice . first , there is a condition modulo @xmath62 : for a lattice @xmath47 with discriminant form @xmath61 and signature @xmath53 , one has @xmath63 the next result shows that for ranks large enough , this is the only condition that the quadratic space and the signature have to satisfy . an even lattice @xmath47 with discriminant form @xmath18 and signature @xmath53 exists if condition ( [ milgram ] ) is satisfied and @xmath64 . note that @xmath65 is clearly a necessary condition . for @xmath66 , one also has a precise but more complicated condition for the existence of a lattice ; see @xcite , th . 1.10.1 . the set @xmath67 is finite . the proof uses the result that there are only finitely many lattices @xmath47 of fixed discriminant @xmath68 ; see for example @xcite , ch . 9 , th . 1.1 . much more precise information about the number of lattices in a genus is known . we restrict ourselves to positive definite lattices , i.e. , lattices @xmath47 of rank @xmath69 where @xmath70 and the @xmath11-component @xmath71 . this result was obtained independently by h. j. s. smith @xcite and h. minkowski @xcite . the number of embeddings of lattices @xmath72 of rank @xmath73 into @xmath47 for all @xmath72 can be encoded in the siegel theta series @xmath74 of genus @xmath75 , a function on the siegel upper half plane @xmath76 of genus @xmath75 : @xmath77 it is a modular form of weight @xmath78 for a congruence subgroup @xmath79 of @xmath80 acting on @xmath81 . @xmath82 where @xmath83 is an explicitly given _ eisenstein series _ for @xmath79 , depending only on the genus of @xmath47 . this theorem was first proven by c. l. siegel @xcite and reformulated in an adelic picture by a. weil , cf . @xcite , see also @xcite for a general setting . there is a slightly finer invariant for lattices , namely m. eichler s _ spinor genus _ , cf . the set of lattices in a genus can be decomposed into @xmath84 spinor genera , also very often @xmath85 . the sum @xmath86 over all lattices @xmath87 in the same spinor genus as a positive definite lattice @xmath47 is for all spinor genera in the genus of @xmath47 the same if @xmath88 . for indefinite lattices of rank at least @xmath89 , there is only one lattice in each spinor genus . this is one way to see that there is only one even unimodular ( i.e. , of discriminant @xmath90 ) lattice @xmath91 of signature @xmath92 if @xmath93 . furthermore , one can deduce that two lattices @xmath47 and @xmath87 are in the same genus if and only if @xmath94 and @xmath95 are isomorphic . there is also the concept of _ rational equivalence_. two lattices @xmath47 and @xmath87 are called rational equivalent if the rational quadratic spaces @xmath96 and @xmath97 are isomorphic . an equivalent more geometric formulation is to say that both @xmath47 and @xmath87 have an isomorphic sublattice @xmath72 of finite index . from the hasse principle , it follows that @xmath47 and @xmath87 are rational equivalent if and only if all the local quadratic spaces @xmath98 and @xmath99 , for @xmath11 a prime including @xmath60 , are isomorphic . in particular , lattices in the same genus are rational equivalent . finally , we remark that the results explained in this section hold similarly for non - even integral lattices . in this section , we explain how the concepts from the arithmetic of quadratic forms as explained in the last section can , at least partially , be generalized to vertex operator algebras . a good starting point for the description of three - dimensional topological quantum field theories are modular braided tensor categories @xcite . with the help of the kirby calculus @xcite , one can use them to define invariants of @xmath89-manifolds , cf . we will give the basic definitions , for details see @xcite . a _ monoidal _ category is a category @xmath100 together with a functorial associative tensor product @xmath101 and a neutral object @xmath102 for @xmath103 . we assume that the tensor category is _ strict _ , so for all objects @xmath104 , @xmath105 , @xmath106 of @xmath107 the products @xmath108 and @xmath109 are identical and not just isomorphic objects . a _ ribbon category _ is a monoidal category together with : 1 . functorial isomorphisms @xmath110 ( the _ braiding _ ) such that @xmath111 2 . functorial isomorphisms @xmath112 ( the _ twist _ ) such that @xmath113 3 . a triple @xmath114 , which associates to any @xmath105 a _ dual object _ @xmath115 and morphisms @xmath116 ( the _ evaluation _ ) and @xmath117 ( the _ coevaluation _ ) , such that @xmath118 @xmath119 these axioms can be visualized by labeled bands in @xmath120 . in a ribbon category , one can define for any endomorphism @xmath121 the _ trace _ of @xmath122 as an element of @xmath123 : @xmath124 the _ quantum dimension _ @xmath125 of @xmath105 is the trace of the identity morphism of @xmath105 . a _ modular braided tensor category _ ( over @xmath126 ) , or _ modular category _ for short , is a ribbon category which has also the structure of an abelian category over @xmath126 such that : 1 . the tensor product is @xmath126-linear and @xmath127 ; 2 . every object is a direct sum of a finite set of simple objects ; 3 . the isomorphism classes of simple objects form a finite set @xmath128 and an object @xmath105 is simple precisely if @xmath129 ; 4 . the matrix @xmath130 is invertible . we call the number @xmath131 the _ discriminant_. in @xcite , a square root @xmath132 is called a rank . the @xmath133-matrix is defined as the matrix @xmath134 . we also need @xmath135 . here , the numbers @xmath136 are defined by @xmath137 . the @xmath138-matrix is the diagonal matrix @xmath139 with the @xmath140 , @xmath141 , on the diagonal . the @xmath133- and @xmath138-matrix satisfy the relations @xmath142 , @xmath143 and @xmath144 . these are the same relations as for the generators @xmath145 and @xmath146 of @xmath147 , the mapping class group of a genus @xmath90 surface . therefore the @xmath133- and @xmath138-matrix define a complex representation @xmath148 of @xmath147 of dimension @xmath149 . if we omit the factor @xmath150 in the definition of the @xmath138-matrix , we obtain a more natural projective representation of @xmath147 , cf . @xcite , ii.3.9 . in the context of modular categories , one can prove that the @xmath151 and @xmath152 are roots of unity ; cf . @xcite . finally , there is the notation of a _ unitary modular braided tensor category _ where one has natural maps @xmath153 identical to complex conjugation for @xmath154 , @xmath141 , such that @xmath155 for all @xmath156 . two modular categories are called equivalent , if there is a functorial isomorphism between them , carrying over all structures . one can also consider nonstrict monoidal categories and define modular categories by using them . every monoidal category is equivalent to a strict one by maclane s coherence theorem and this is also true for modular braided tensor categories , cf . @xcite , xi.1.4 . the categories arising from conformal field theory are in general not strict . we formulate two versions of the _ verlinde formula _ which are theorems in the context of modular categories . first , let @xmath157 be the structure constants of the _ fusion algebra _ defined on the free vector space @xmath158 $ ] by the product @xmath159 then one has @xmath160 where @xmath161 is the label defined by @xmath162 and @xmath163 is the label for @xmath164 . secondly , let @xmath165 be the dimension of the vector space associated by the @xmath166-tqft to a genus @xmath75 surface decorated with labels @xmath167 , @xmath168 , @xmath169 . then one has @xmath170 we mention three important classes of examples of modular categories : \1 ) the modular category @xmath171 associated to a finite quadratic space @xmath18 , see @xcite , i.1.7.2 and ii.1.7.2 . we have @xmath172 , @xmath173 , where the maps @xmath174 , @xmath175 ( @xmath176 denotes the @xmath177-torsion subgroup of @xmath19 ) used in @xcite are up to equivalence identical to the map @xmath12 , see @xcite , sec . 7.5 . since the fusion algebra of @xmath171 is just the group ring @xmath178 $ ] of the abelian group @xmath19 , we call such a modular category abelian . any modular category with such an abelian group ring @xmath178 $ ] as fusion algebra arises in such a way , see @xcite , ch . 7 . \2 ) the chern - simons theory associated to a simple lie group @xmath14 and positive integral level @xmath179 . this modular category can be constructed using quantum groups , cf . @xcite . \3 ) discrete chern - simons theories constructed from a finite group @xmath14 and a class @xcite . find a good description of the set of equivalence classes of modular braided tensor categories over @xmath180 . for two such categories @xmath107 and @xmath181 , it is easy to define a product category @xmath182 , such that its simple objects are the products of the simple objects of the factors . one can decompose a category into indecomposable pieces @xmath183 under the product : @xmath184 . an additional problem compared to the situation for quadratic spaces is that it is not possible to decompose @xmath107 into a product of local pieces @xmath185 for different primes @xmath11 such that , for example , the numbers @xmath151 for @xmath185 are @xmath186-th root of unity . a conjecture for modular braided tensor categories ( also usually formulated for rational voas ) says that the above introduced representation @xmath187 has a congruence subgroup as kernel or , more precisely , @xmath188 where @xmath9 is the smallest natural number such that @xmath189 . away from the prime @xmath38 , this has been proven by coste and gannon @xcite . bantay @xcite has recently reduced the problem to a property of permutation orbifolds . assuming this conjecture , eholzer @xcite has classified the indecomposable modular categories ( or at least the fusion algebras together with the @xmath138-matrix and @xmath152 ) with @xmath190 using the representation theory of @xmath191 . a lot of work has been done on the classification of fusion algebras and also compatible tensor categories , cf . for example @xcite . for the purpose of this paper , the following definition of a restricted class of vertex algebras seems to be most adequate . a _ vertex operator algebra of central charge @xmath192 _ is a graded complex vector space with @xmath193 together with a linear map @xmath194 $ ] ( _ the state field correspondence _ ) such that for @xmath195 the coefficients @xmath196 of @xmath197 are of degree @xmath198 and @xmath199 for fixed @xmath200 and @xmath201 large enough , and there are two distinguished elements @xmath202 ( the _ vacuum _ ) and @xmath203 ( the _ virasoro element _ ) subject to the following axioms : * @xmath204 and for @xmath205 , one has @xmath206 for @xmath207 , @xmath208 . * for @xmath209 , @xmath205 , there is a @xmath210 with @xmath211 in @xmath212 $ ] . * the coefficients of @xmath213 define a representation of the _ virasoro algebra _ of central charge @xmath192 : @xmath214=l_{m+n}+\frac{m^3-m}{12}\delta_{m+n,0}\cdot c\cdot { \rm id}_v$ ] and for @xmath205 , one has @xmath215 . these axioms can be distilled from the general axioms of axiomatic quantum field theory applied to the case of riemannian surfaces as the space time . to obtain a more geometric formulation as indicated in the introduction from a voa , further conditions have to be satisfied . for the terms we use below and for details , we refer to the literature : there is the notion of modules and intertwining spaces @xmath216 for a voa , see @xcite . a voa is usually called _ rational _ , if there are only finitely many nonisomorphic irreducible modules and any module can be decomposed into a finite direct sum of irreducible ones . ( the first condition can be deduced from the second , cf . @xcite . ) the _ conformal weight _ @xmath217 of a module is its smallest @xmath218-eigenvalue . let @xmath219 be a complete set of nonisomorphic irreducible modules . the @xmath138-matrix is defined as the diagonal matrix @xmath220 with the numbers @xmath221 , @xmath222 , on the diagonal . zhu has shown @xcite that under some conditions on the rational voa , there is a representation of @xmath223 on some space of genus @xmath90 correlation functions . the generator @xmath224 of @xmath223 defines the matrix @xmath225 . using the tensor product theory for modules of voas , huang and lepowsky associated a ( nonstrict ) braided tensor category to a suitable class of voas , see @xcite and the references in @xcite . finally , there is a definition for the dimension @xmath226 of the space of vacua on the genus @xmath75 surface with labels @xmath167 , @xmath168 , @xmath169 by zhu @xcite . i assume further that the voa is _ unitary _ , this means the voa can be defined over the real numbers and the natural invariant symmetric form on it is positive definite , although this will exclude many interesting examples . also , the voa should be _ simple _ , i.e. , @xmath105 is irreducible as a module over itself . we like to use the above mentioned results and some other properties believed to be true for many interesting voas . we therefore make the following assumption on the voas considered : [ hauptvermutung ] the voa @xmath105 satisfies the conditions needed in the construction of the braided tensor category on the category of modules as in @xcite and also satisfies the conditions used in @xcite . furthermore , the braided tensor category of @xmath105-modules has the structure of a modular braided tensor category @xmath227 such that @xmath228 one problem is certainly to find simple sufficient conditions on the voa so that the assumption becomes a theorem . sometimes other conditions in the definition of rationality are added . assuming that assumption [ hauptvermutung ] is true would directly imply that the central charge and conformal weights are rational numbers . property ( iv ) follows from ( iii ) if one can prove the sewing relations @xmath229 and @xmath230 for important examples of voas , at least parts of the assumption are known to be true ; for details we refer to the references : \1 ) voas defining an abelian intertwining algebra ; cf . @xcite . \2 ) wess - zumino - witten models , i.e. , the voas defined on the highest weight representations of level @xmath179 for an affine kac - moody lie algebra @xmath231 ; cf . @xcite . \3 ) holomorphic orbifolds ; cf . @xcite . \4 ) minimal models ; cf . @xcite . one could try to generalize the above mentioned results to rational vertex algebras . in their definition one allows @xmath40-graded vector spaces @xmath105 and the pieces @xmath232 are not assumed to be finite - dimensional . vertex algebras should be considered as the analog of indefinite even lattices . for a vertex algebra @xmath105 of central charge @xmath192 , one can define the number @xmath233 as the largest central charge of a subvoa @xmath106 of @xmath105 , @xmath234 as @xmath235 and call the pair @xmath236 the signature of @xmath105 . for the rest of this section , we consider only voas for which assumption [ hauptvermutung ] is satisfied . the _ genus _ of a rational voa @xmath105 of central charge @xmath192 is defined as the pair consisting of the associated modular braided tensor category @xmath227 and the central charge @xmath192 . it is not known much about the set of central charges which can occur . from the discussion of modular categories , we know that @xmath192 has to be rational . the theory of unitary highest weight representations of the virasoro algebra shows that @xmath192 belongs to the minimal series @xmath237 , @xmath238 , @xmath239 , @xmath168 , for @xmath240 . there is the following analogue of milgrams theorem for modular braided tensor categories . [ thvoamilgram ] for a rational voa @xmath105 with associated modular category @xmath227 and central charge @xmath192 , one has @xmath241 * proof : * the result is an immediate consequence of assumption [ hauptvermutung ] and the existence of the @xmath223-representations . we have @xmath242 and @xmath243 because of the @xmath147-representations . using properties ( i ) and ( ii ) in assumption [ hauptvermutung ] , we get @xmath244 . let @xmath107 be a modular braided tensor category . does there exist rational voas @xmath105 of central charge @xmath192 with @xmath245 if condition ( [ voamilgram ] ) is satisfied and @xmath192 is large enough ? what are the possible values of @xmath192 for abelian @xmath246 ? for voas there are more central charges @xmath192 possible than one can realize by lattice voas . the bound @xmath247 does not hold in general : the fixpoint subvoa @xmath248 of the lattice voa @xmath249 for the involution @xmath250 lifted from the reflection at the origin of @xmath251 ( cf . @xcite ) has central charge @xmath62 , but @xmath252 , where @xmath12 is isomorphic to @xmath253 copies of the hyperbolic plane . the set @xmath254 is finite . _ example : _ let @xmath107 be the trivial modular category , i.e. , there is only one simple object . the corresponding voas are called self - dual or holomorphic . by theorem [ thvoamilgram ] , the central charge @xmath192 has to be divisible by @xmath62 . an interesting case is @xmath255 . schellekens has found a list of @xmath256 candidates for the corresponding genus @xcite . for the @xmath257 lattice voas constructed from the genus of the positive definite even unimodular lattices in dimension @xmath257 , one has an existence and uniqueness theorem . @xmath258 other voas are completely constructed as @xmath259-orbifolds , cf . one of them is the moonshine module @xmath260 , cf . the fact that @xmath260 belongs to the genus was proven in @xcite . the uniqueness conjecture @xcite that @xmath260 is the only voa @xmath105 in the genus with @xmath261 is still open . for the voa with @xmath262 as the affine kac - moody subvoa , we can show uniqueness , cf . @xcite and next section . conformal field theory suggests that for @xmath263 it may be possible to define a ( possibly vector valued ) genus @xmath75 partition function @xmath264 on the genus @xmath75 teichmller space @xmath265 with automorphic properties for the mapping class group of a genus @xmath75 surface . for @xmath266 this follows from zhu s work @xcite . are there good mass formulas for the genus of a voa of the form @xmath267 one problem is to find a good definition for @xmath268 if @xmath269 is infinite ; one could try to take the tamagawa measure if @xmath269 is a reductive lie group defined over @xmath270 by using voas defined over @xmath271 . for a kind of relative mass formula of this type involving a nonabelian extension problem , see theorem [ relmass ] in the next section . the definition of the genus of a voa was modeled along the notation of the genus of an even lattice @xmath47 such that the following diagram is commutative @xmath272 the upper horizontal arrow is the lattice construction of voas @xcite , the left down arrow gives the genus of a lattice via the discriminant form , the right down arrow comes from the abelian intertwining algebra structure on the modules of @xmath273 @xcite and the lower horizontal arrow is the map associating the abelian modular category to a finite quadratic space as explained as an example above . regarding genera of vertex algebras , a structure theory seems to be more complicated than for indefinite lattices : the vertex algebras @xmath274 and @xmath275 ( here , @xmath276 denotes the leech lattice ) are not isomorphic since both define in a uniform way different generalized kac - moody lie algebras , namely the monster @xcite and the fake monster lie algebra @xcite . ( they are different because all cartan subalgebras of a generalized kac - moody lie algebra are conjugate by inner automorphisms as shown by u. ray @xcite . ) but both should be self - dual vertex algebras of signature @xmath277 . another such vertex algebra is studied in @xcite . a natural generalization of the notion of _ rational equivalence _ from lattices to voas would be as follows : we take the smallest equivalence relation such that two rational voas @xmath105 and @xmath106 of central charge @xmath192 are equivalent if both have isomorphic rational subvoas with the same virasoro element . this means that there exist voas @xmath278 , @xmath279 , @xmath168 , @xmath280 and voas @xmath281 , @xmath168 , @xmath282 such that @xmath283 is a common subvoa of @xmath284 and @xmath285 for @xmath286 , @xmath168 , @xmath287 . just to say there exists a common isomorphic subvoa would nt give an equivalence relation : take for @xmath105 and @xmath106 the fixpoint voas @xmath288 and @xmath289 where @xmath104 is the @xmath290 lattice voa and @xmath14 and @xmath291 are finite subgroups of @xmath292 which ca nt be conjugated into the same larger finite subgroup of @xmath292 ( like the binary icosahedral and the binary tetrahedral group ) . also , one can generalize the discussion of this section to vertex operator super algebras ( svoas ) . the central charge @xmath192 of a self - dual svoa has to be half - integral @xcite , th . for an interesting example with @xmath293 and the classification for small @xmath192 see also @xcite . in this section , we explain how extension problems for suitable voas can be formulated in terms of the associated modular braided tensor category and sometimes solved by this methods . * the even overlattices @xmath72 of an even lattice @xmath47 are in one - to - one correspondence to the isotropic subspaces @xmath294 of the associated quadratic space @xmath61 . * the quadratic space for @xmath72 is given by the pair @xmath295 . * two such extensions give isomorphic lattices @xmath72 exactly if the corresponding subspaces can be transformed into each other by an isometry of @xmath61 induced from an automorphism of @xmath47 . our aim is to present similar results for extensions of voas . let @xmath105 be a rational voa . we call a voa @xmath296 an _ extension _ of @xmath105 if it contains a subvoa isomorphic to @xmath105 and has the same vacuum and virasoro element as @xmath105 . we have @xmath297 for a decomposition of @xmath106 into irreducible @xmath105-modules @xmath298 , where @xmath299 is a complete set of irreducible modules of @xmath105 and the nonnegative integers @xmath300 are the multiplicities of @xmath298 in @xmath106 . also , we assume that we have fixed an embedding @xmath301 . we call two extensions @xmath302 and @xmath303 isomorphic if there is an voa - isomorphism @xmath304 and an automorphism @xmath250 of @xmath105 such that @xmath305 . for @xmath306 and @xmath307 , we can define in this way the _ automorphism group _ @xmath308 of the extension @xmath302 . part ( a ) of theorem [ latticeext ] implies that the overlattices of a lattice can be determined from the structure of its discriminant form . for voas we have as an analogous result : [ voaext ] the voa - extensions @xmath106 of a rational voa @xmath105 satisfying assumption [ hauptvermutung ] can be determined completely in terms of the associated modular braided tensor category @xmath227 . * proof : * fix a @xmath105-module isomorphism @xmath309 extending @xmath310 . the vertex operator @xmath311 is an element of @xmath312 for @xmath296 to be a voa - extension of @xmath105 , the modules @xmath298 with @xmath313 have to be of integral conformal weight and @xmath311 has to satisfy the commutativity axiom . both conditions are completely controlled by the twist and the braiding of the associated modular braided tensor category . @xmath314 similarly , one can see that the structure of the intertwining algebra of an extension is completely determined by @xmath227 and @xmath311 . ( see th . 6.2 of @xcite ) using notation from @xcite : ( a ) the extensions @xmath106 of @xmath105 are in one - to - one correspondence to rigid @xmath227-algebras @xmath19 on which the twist @xmath315 is trivial . ( b ) the category of representations of @xmath106 can be identified with @xmath316 . ] we expect that any rational voa @xmath105 has only finitely many nonisomorphic extensions @xmath296 . in the case that @xmath227 is abelian , i.e. , @xmath317 for a finite quadratic space @xmath18 , one could expect in generalization of theorem [ latticeext ] ( b ) that the voa - extensions are determined by the isotropic subspaces of @xmath19 . this is indeed the case . recall that a simple module @xmath298 is called a _ simple current _ if for each simple module @xmath318 there is a another simple module @xmath319 such that @xmath320 holds in the fusion algebra . [ simplecurrent ] let @xmath105 be a rational voa which has an abelian intertwining operator algebra structure on the direct sum of the simple currents . let @xmath294 be a subgroup of the abelian group @xmath321 of labels of the simple currents for which the modules @xmath322 , @xmath323 , have integral conformal weight . then there exists a unique simple voa - extension @xmath296 of the form @xmath324 , @xmath325 , and one has @xmath326 . * proof : * _ existence : _ choose a non - zero vector @xmath327 for and define @xmath328 . recall @xcite that the first cohomology groups @xmath329 of the second eilenberg - maclane space @xmath330 ( which is up to homotopy the unique cw - complex with homotopy groups @xmath331 , and @xmath332 for @xmath333 ) can explicitly described by the following cochain complex : @xmath334 the coboundary maps @xmath335 which we need are defined by @xmath336 and @xmath337 , where @xmath338 denotes the usual group cohomology coboundary operator . note that we can work with normalized cochains , i.e. , cochains with @xmath339 , @xmath340 , respectively @xmath341 and @xmath342 for @xmath192 , @xmath343 . the commutativity and associativity properties for abelian intertwining algebras @xcite associate to @xmath311 a cocycle @xmath344 . the operator @xmath311 defines a voa structure on @xmath106 if @xmath345 . the cohomology group @xmath346 can be identified via @xmath347 with the space of quadratic forms @xmath348 on @xmath294 , see @xcite , th . 3 . in our situation , we have @xmath349 since the modules @xmath322 , @xmath323 , have integral conformal weight and @xmath350 equals the exponential function applied to @xmath351 times the conformal weight of @xmath322 . therefore , there exists a @xmath352 with @xmath353 . replacing @xmath354 by @xmath355 , we see that @xmath296 is a voa . _ uniqueness : _ assume @xmath296 is an extension . similar as in the proof of prop . 2.5 ( 4 ) of @xcite , one gets @xmath356 for all @xmath192 , @xmath343 . but @xmath357 since @xmath358 must have multiplicity one . one has @xmath359 since by proposition 11.9 of @xcite one has @xmath360 if @xmath209 and @xmath361 are not @xmath362 . we can replace @xmath354 by @xmath363 where @xmath364 is a @xmath89-cocycle , i.e. , @xmath365 . a @xmath105-module isomorphism of the space @xmath366 defines for any irreducible @xmath322 , @xmath323 , a scalar @xmath367 , i.e. , a two cochain @xmath368 , and @xmath122 is changed to @xmath369 . but from the universal coefficient theorem , @xmath370 and @xmath371 since the @xmath40-module @xmath372 is divisible . so , voas @xmath296 for different choices of cocycles @xmath122 are isomorphic . _ remarks : _ in terms of the associated modular braided tensor category , the simple currents define a subcategory @xmath373 which is abelian , i.e. , @xmath374 for a finite quadratic space @xmath18 ( we allow here that @xmath12 is degenerate , i.e. , @xmath171 has not to be modular ) : the space of quadratic forms on @xmath19 can be identified as before with the cohomology group @xmath375 and one has @xmath376 . furthermore , @xmath377 since the modules @xmath322 , @xmath323 , have integral conformal weight , i.e , @xmath294 is an isotropic subspace of @xmath18 . the theorem completely solves the extension problem for voas @xmath105 for which on the direct sum of a complete set of nonisomorphic irreducible modules there is the structure of an abelian intertwining algebra . in this case , the modular braided tensor category @xmath378 of an extension @xmath106 are described as in theorem [ latticeext ] for lattices . the structure of the modular braided tensor category @xmath378 for general simple current extensions was investigated in @xcite , where a conjecture for the set of irreducible @xmath106-modules and the corresponding @xmath133-matrix was made . this conjecture was proven in @xcite . there is a natural action of the dual group @xmath379 on @xmath106 . let @xmath106 be a voa and @xmath380 be a finite subgroup of the automorphism group . let @xmath381 be the fixpoint voa . then the extensions @xmath104 of @xmath105 which are contained in @xmath106 are in one - to - one correspondence to the subgroups of @xmath14 , i.e. , there is a subgroup @xmath382 such that @xmath383 . from the perspective of modular braided tensor categories , this situation was analyzed independently by bruguires @xcite and mger @xcite . both give a purely topological characterization of the above situation using a characterization result for neutral tannakian categories obtained by doplicher and roberts @xcite and deligne @xcite , respectively . both authors describe how to construct the modular braided tensor category @xmath378 from @xmath227 and the representation subcategory for the group @xmath14 . mger also proves the galois correspondence , see @xcite , sec . 4.2 . as a final example , we completely solve the extension problem for self - dual framed vertex operator algebras ( fvoas ) . fvoas were introduced in @xcite and are voas which have a tensor product @xmath384 of @xmath385 central charge @xmath386 rational virasoro voas as subvoa with the same virasoro element , i.e. , they are the extensions of @xmath384 . recall that @xmath387 has three irreducible modules @xmath387 , @xmath388 and @xmath389 of conformal weight @xmath362 , @xmath390 and @xmath391 and @xmath388 is a simple current . from the decomposition @xmath392 one gets the two codes @xmath393 and it was shown in @xcite that this codes are linear , @xmath394 ( @xmath395 denotes the code orthogonal to @xmath294 ) , @xmath107 is even , and all weights of vectors in @xmath181 are divisible by @xmath62 . theorem [ simplecurrent ] says that there is a unique fvoa of the form @xmath396 for every linear even code @xmath107 , a result proven in @xcite ( prop . 2.16 ) and also obtained in @xcite . as mentioned above , the associated modular category for @xmath397 is described in @xcite . the set of irreducible modules is described in prop . 5.2 there , a result first proven in @xcite . in special cases , the fusion algebra and the @xmath133-matrix and the intertwining algebra were determined in @xcite . [ relmass ] for @xmath398 , one has @xmath401 where the sum on the left hand side of the equation runs over all isomorphism classes ( in the sense of extensions ) @xmath105 of self - dual fvoas of central charge @xmath402 and @xmath403 is the number of codes @xmath181 of length @xmath385 with @xmath404 , @xmath405 and * sketch of proof : * since @xmath397 is a simple current extension , its modular category @xmath406 can be expressed completely in terms of the known modular category @xmath407 and the code @xmath107 . for @xmath105 self - dual , all quantum dimensions of @xmath406 are @xmath90 , so it is an abelian modular category @xmath408 with an explicitly given abelian group @xmath19 and quadratic form @xmath12 . one has a short exact sequence @xmath409 and the self - dual extensions of @xmath397 are in one - to - one correspondence with sections @xmath410 for which @xmath411 is isotropic . counting them gives the result . let us check the theorem for the first case @xmath412 . in @xcite , the five fvoas of central charge @xmath62 together with their automorphism group have been determined . for every @xmath413 , @xmath177 , @xmath89 , @xmath239 , @xmath253 , there is an up to equivalence unique code @xmath414 of dimension @xmath179 and a unique fvoa @xmath415 with code @xmath416 and automorphism group @xmath417 . by @xcite , main theorem 2 , one has @xmath418 and for the sum @xmath419 we obtain : @xmath420 in agreement with theorem [ relmass ] . the methods of the proof can be used to give a construction of the moonshine module @xmath260 as the self - dual fvoa with @xmath421 the lexicographic code of length @xmath422 and minimal weight @xmath239 ; see @xcite , sect . 5 for the precise structure of the virasoro module decomposition . the only input from voa - theory which one needs is the construction of the virasoro voa of central charge @xmath386 and the structure of its intertwining algebra . all previous constructions use at some place the lattice vertex operator algebra construction ; cf . @xcite . igor b. frenkel , yi - zhi huang , and james lepowsky , _ on axiomatic approaches to vertex operator algebras and modules _ , memoirs of the ams , band 104 , nr . 494 , american mathmeatical society , providence , 1993 . robert griess and gerald hhn , _ virasoro frames and their stabilizers for the @xmath424 lattice type vertex operator algebra _ , to appear in journal fr die reine und angewandte mathematik , math.qa/0101054 ( 2001 ) . yi - zhi huang , _ a nonmeromorphic extension of the moonshine module vertex operator algebra _ , moonshine , the monster , and related topics ( south hadley , ma , 1994 ) , amer . soc . , providence , ri , 1996 , hep - th/9406190 , pp . 123148 . saunders maclane , _ cohomology theory of abelian groups _ , proceedings of the international congress of mathematicians , cambridge , mass . , 1950 , vol . 2 ( providence , r. i. ) , amer . soc . , 1952 , pp . 814 . akihiro tsuchiya , kenji ueno , and yasuhiko yamada , _ conformal field theory on universal family of stable curves with gauge symmetries _ , integrable systems in quantum field theory and statistical mechanics , adv . pure math . , 19 , academic press , boston , ma , 1989 , pp . 459566 . yongchang zhu , _ vertex operator algebras , elliptic functions , and modular forms _ , thesis , yale university , 1990 , appeared as : _ modular invariance of characters of vertex operator algebras , _ soc * 9 * ( 1996 ) .

the notion of the _ genus _ of a quadratic form is generalized to vertex operator algebras . we define it as the modular braided tensor category associated to a suitable vertex operator algebra together with the central charge . statements similar as known for quadratic forms are formulated . we further explain how extension problems for vertex operator algebras can be described in terms of the associated modular braided tensor category .

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Proceed to summarize the following text: with the wide application , the charged particles acceleration based on the laser - plasma interaction have been rapidly concerned by many simulations and experiments @xcite . in recent years , the proton acceleration generated by an ultraintense laser irradiating a solid target has been investigated extensively due to the potential applications in ions fast ignition ( fi ) of inertial confinement fusion ( icf ) @xcite , compact proton sources for cancer therapy @xcite , laboratory astrophysics @xcite and so on . for instance , the fi by proton beams to transport a few of meters to the dense core usually requires a high beam quality @xcite . however , the coupling efficiency from laser to protons is too low to achieve the enough ignition energy . therefore , how to gain a high quality beam with high energy , low energy spread and small size is a challenge topic currently . in the past years , several acceleration schemes have been proposed and studied for obtaining higher quality proton beams by many researchers . according to the parameters variety of the laser and targets , the acceleration approaches are usually classified into target normal sheath acceleration ( tnsa ) @xcite , breakout afterburner acceleration ( boa ) @xcite , shock wave acceleration @xcite , and radiation pressure acceleration ( rpa ) @xcite , etc .. the tnsa , which has been studied rapidly and implemented in the experiments , is that protons can be accelerated to high energy through the electrostatic sheath field created by the hot electrons expanding into vacuum at the rear side of the target . nevertheless , the practical application is limited because of the obtained protons possessing a large divergence , large energy spread and low number density . moreover , in order to obtain a dense and monoenergetic beam , the rpa is proposed to efficiently accelerate the protons . this scheme is mainly associated to the intense space - charge field , which is created by the radiation pressure of the laser . yet , multi dimensional simulations show that the rpa is not enough stable due to the excited transverse instability , such as weibel instability and rayleigh - taylor - like instability @xcite . in order to suppress the instability , many solutions are presented and studied in detail like of with a properly tailored laser @xcite or / and with a shaped foil target @xcite , etc .. recently zou _ et al . _ @xcite proposed first a cone target used to focus laser and guide fast electrons in fi @xcite . while the cone target can suppress the transverse expansion by a co - moving focusing electric field and achieve a better beam quality , however , the proton beams will diffuse following the laser field when they fly out from the right exit of the cone . this deficiency makes the protons not be accelerated availably to a farther distance . in this work , a gold cone with a capillary attached to the behind is proposed and involved problems are studied by using the two - dimensional particle - in - cell ( pic ) simulations . the results show that this cone - capillary target can availably collimate and guide the protons , then achieve a better beam quality with a density @xmath0 , monoenergetic peak @xmath1 , spatial emittance @xmath6 with divergence angle @xmath3 and diameter @xmath4 in a farther distance . the enhanced reasons are mainly attributed to the focus by the transverse electric field @xmath7 generated in the cone as well as the capillary . besides , in the process of exploring the capillary parameters , one finds that the capillary lengths act as a important role in the protons acceleration . this paper is organized as follows . section ii outlines the target configurations , simulation parameters and results . besides , the enhanced reasons of protons acceleration are also discussed and analyzed in detail . the capillary parameters are examined in section iii , and the optimal capillary length is given . finally a brief summary is presented in sec . for the purpose to demonstrate the enhancement of proton acceleration , we have designed two kinds of targets by using the gold cone and gold cone - capillary . we have investigated the created protons quality by them via the two - dimensional pic simulations . two different cases of targets are illustrated in fig . [ fig:1 ] whose the case ii by fig . [ fig:1](b ) represents the gold cone with a capillary attached behind the cone and as a comparison the case i by fig . [ fig:1](a ) stands for the gold cone only . .,width=566 ] the simulation is performed with the pic code epoch @xcite . the simulation box is @xmath8 and the grid size is @xmath9 with @xmath10 grid cells . the preionized cone or / and cone - capillary consist of electrons and au ions whose density is established to be @xmath11 . the cone is located from @xmath12 to @xmath13 with the cone - angle of @xmath14 from horizon . the thickness of its wall is @xmath15 and the diameter of the left opening is @xmath16 . a proton - rich foil , which is full ionized with density @xmath17 and thick @xmath18 , is situated at a distance @xmath19 away from the left boundary of the cone . a capillary with length @xmath20 is attached behind the cone . here @xmath21 is the laser wavelength and @xmath22 ( @xmath23 laser frequency , @xmath24 electron mass and @xmath25 electron charge ) is the critical electron plasma density . besides , each cell consists of @xmath26 weighted particles per species in our simulations . a circular polarized ( cp ) laser pulse is normally incident from the left boundary of the box . the laser has a peak intensity of @xmath27 , and rises to the maximum value in @xmath28 then remains constant for @xmath29 , where @xmath28 is the laser period . the intensity profile of the laser is gaussian with spot size of @xmath30 [ full width at half maximum ( fwhm ) ] . besides , we use the absorption boundary in @xmath31 direction and @xmath32 direction for both fields and particles , respectively . and the case ii [ ( b ) and ( d ) ] at @xmath33 [ ( a ) and ( b ) ] and @xmath34 [ ( c ) and ( d ) ] . here the proton density is normalized by @xmath35.,width=566 ] [ ( a ) , ( c ) ] and @xmath36 [ ( b ) , ( d ) ] . blue and red lines represent the case i and the case ii , respectively.,width=566 ] now let us to see the results of proton acceleration in the two different kinds of targets . figure [ fig:2 ] shows the distributions of the proton - rich foil density in both cases . at @xmath33 , the transverse expansion of the proton - rich foil is suppressed greatly in both cases , which is good for the transverse collimation in the acceleration process , as shown in fig . [ fig:2](a ) and [ fig:2](b ) . these results are in accordance with the ref . meanwhile in comparison with the case i and ii the proton density is more compacter for the case i. but as time goes on the situation is reversed since proton density becomes more and more compact for the case ii . this point can be seen at @xmath36 in fig . [ fig:2](c ) and ( d ) that a dense proton source can be gained with a density @xmath0 and bunch transverse diameter @xmath37 in case ii , but in contrast the proton density decreases significantly and bunch also diffuses seriously in the transverse direction in the case i. to demonstrate the enhanced acceleration effect by using the cone - capillary , the energy spectra and angular distribution of the protons are shown in fig . [ fig:3 ] . for the energy spectra they are plotted in fig . [ fig:3](a ) and [ fig:3](b ) at different time . it shows that there are a monoenergetic proton bunches in both cases at @xmath33 . similarly as in the density the better quality seems to be got in case i at this relative earlier time . for example , see fig . [ fig:3](a ) , the bunch quality of a peak energy @xmath38 with energy spread @xmath39 in the case i is indeed better than that of a peak energy @xmath40 with energy spread @xmath41 in the case ii . however with time increasing , the situation is reversed again for the energy spectra . in fact the monoenergetic peak structure disappears evenly in the case i at @xmath36 while it remains a good quality by a peak energy @xmath1 with energy spread @xmath42 in the case ii , see fig . [ fig:3](a ) , even if the protons have moved out from the capillary . on the other hand the protons number at the monoenergetic peak in the case ii is almost three times higher than that in the case i. for the angular distribution they are plotted in fig . [ fig:3](c ) and [ fig:3](d ) . the momentum angle of protons is defined by @xmath43 , where @xmath44 and @xmath45 are the momentum of the protons in the transverse @xmath32-axis and longitudinal @xmath31-axis , respectively . in the case ii , it is surprising to see that the fwhm of the proton divergence distribution maintains @xmath5 so that the spatial emittance is about @xmath46 for the bunch of @xmath47 diameter at time of @xmath48 , which is greatly beneficial for the practical applications . in the case i , however , the proton divergence angle is large , which decreases slightly from about @xmath49 at @xmath50 to @xmath51 at @xmath34 . evenly the corresponding spatial emittance increases from @xmath52 at @xmath50 to @xmath53 at @xmath34 . these results obviously indicate that the case ii is benefit to accelerate the protons to a farther distance and get a improved high quality protons bunch . thus it is necessary to clarify what is behind this improvement . we will make some analysis and discussion on the electric and magnetic fields for the both cases in the next subsection , which will show their important roles , in particular the role by the transverse electric field , on the enhancement of proton quality . for the case i [ ( a ) and ( c ) ] and case ii [ ( b ) and ( d ) ] at @xmath50 [ ( a ) and ( b ) ] and @xmath34 [ ( c ) and ( d ) ] . here the electric field is normalized by @xmath54.,width=566 ] at @xmath50 ( a ) and @xmath34 ( b ) in the position @xmath55 , respectively . blue and red lines represent the case i and the case ii , respectively . here the electric field is normalized by @xmath54.,width=566 ] in the above subsection we have exhibited the enhanced acceleration effect of protons by using the cone - capillary . now let us to see how this enhancement is affected mainly by the electric and magnetic fields in different target structures which plays a key role to the improved collimation of produced protons bunch . the longitudinal electric fields @xmath56 at @xmath33 and @xmath34 are plotted respectively in fig . [ fig:4 ] . at @xmath33 [ fig . [ fig:4](a ) and [ fig:4](b ) ] , in the region of about @xmath57 , one can note that the @xmath56 for the case i is obviously stronger than that for the case ii . thus , the protons can be accelerated to higher energy , corresponding to a higher monoenergetic peak structure in the case i. but as time goes on [ fig . [ fig:4](c ) and [ fig:4](d ) ] , the @xmath56 diffuses along transverse and becomes quite weak in the region of about @xmath58 . however what is interesting for the case ii , a stronger @xmath56 is built in a farther distance due to the focus of the capillary at @xmath36 , which can provide driving force for the proton acceleration in long time . in order to show this feature clearer , the slices of the @xmath56 in position of @xmath55 are plotted in fig . [ fig:5 ] for both cases . one can see that , for the case i , there exists a stronger @xmath56 to accelerate the protons forward at early time , while decreases significantly at later time , see the blue lines in fig . [ fig:5](a ) and [ fig:5](b ) . for the case ii , however , the @xmath56 becomes stronger and stronger with time increasing , see the red lines in fig . [ fig:5](a ) and [ fig:5](b ) , which is prevailing to the protons acceleration in longer time because of the focus by the capillary . and the case ii [ ( b ) and ( d ) ] at @xmath33 [ ( a ) and ( b ) ] and @xmath34 [ ( c ) and ( d ) ] . here the electron density is normalized by @xmath35.,width=566 ] for the case i [ ( a ) and ( c ) ] and case ii [ ( b ) and ( d ) ] at @xmath50 [ ( a ) and ( b ) ] and @xmath34 [ ( c ) and ( d ) ] . here the electric field is normalized by @xmath54.,width=566 ] for the case i [ ( a ) and ( c ) ] and case ii [ ( b ) and ( d ) ] at @xmath50 [ ( a ) and ( b ) ] and @xmath34 [ ( c ) and ( d ) ] . here the magnetic field is normalized by @xmath54.,width=566 ] when the laser propagates into the cone or / and cone - capillary , surface electrons are first pulled out into the vacuum by laser electric fields , then they are accelerated forward by the lorentz force of @xmath59 , as shown in fig . [ fig:6 ] . the electrons expanded into the vacuum continue to propagate along the cone wall and induce a strong transverse electric field @xmath7 , shown in fig . [ fig:7 ] , and meanwhile an intense quasistatic magnetic field @xmath60 , shown in fig . [ fig:8 ] . the induced quasistatic magnetic field plays a role of driving the electrons out from the cone surface , while the @xmath7 acts to draw electrons back into the inside of the cone . moreover , it is worth to note that the moving @xmath7 can also play the part of focusing the protons in the transverse , which has been proposed by zou _ this results in a balance between the electric field and the magnetic field @xcite that leads to the collimation and guidance of the electrons along the cone or / and cone - capillary . at @xmath33 , the protons can collimated by the transverse electric field @xmath7 for both cases , as shown in fig . [ fig:7](a ) and [ fig:7](b ) . however , as time goes on , the @xmath7 disappears in the region of @xmath61 for the case i. while it is still quite strong due to the existence of the capillary for the case ii , which ensures to focus sustainedly the protons in a farther distance . from the preceding discussion , we have researched how the protons acceleration is enhanced by the cone - capillary . one can conclude that the @xmath7 generated by the capillary acts as the crucial role in guiding and collimating the protons . accordingly , the length of capillary is the key parameter in dominating the beam quality . this reminds us to make more simulations to see whether there exists an optimal capillary length for protons quality when the other parameters are fixed . ( black curve ) and the number density locating the peak ( red curve ) as the function of the length @xmath62 of the capillary , respectively.,width=566 ] , ( b ) @xmath63 , ( c ) @xmath64 and ( d ) @xmath36 , respectively.,width=566 ] the proton divergence angle @xmath65 and number density of the peak energy are plotted as the function of the capillary length @xmath62 , as shown in fig . [ fig:9 ] . obviously the length @xmath66 represents the case i. it shows that the @xmath65 decreases with @xmath62 , see the black curve in fig . [ fig:9 ] , and the number density around of the peak energy increases with @xmath62 , see the red curve in fig . [ fig:9 ] . certainly these results are mainly attributed to the collimation of the @xmath7 generated by the capillary . on the other hand , it should be emphasized that the number density increases appreciably when @xmath67 . we believe the reason is that the bulk protons have not escaped out from the right opening of the capillary . moreover , fig . [ fig:10 ] exhibits the proton energy spectra with different capillary lengths at different moments . as time goes on , it shows that the shorter the length of the capillary , the earlier the monoenergetic peak of the proton beam disappears . thus this can be summed up that the shorter capillary , for example @xmath68 , @xmath69 and @xmath70 , is not conducive to accelerating protons to a farther distance . we can also note that for the longer capillary such as @xmath20 and @xmath71 , the monoenergetic peak are approximately coincident at early time , however , at later time , see fig . [ fig:10](d ) , comparing the cases of @xmath20 with the length @xmath67 , the monoenergetic peak in both of energy value and number density are better for the former case . this is not surprising because that the longer capillary would make the longitudinal electric field @xmath56 decreasing , which would result in the decrease of the total conversion efficiency from the laser to protons . overall , the extensive simulation results indicate an optimal proper capillary length existence , e.g. @xmath20 shown in fig . [ fig:10 ] , by which one can achieve a high quality proton source even in longer time . in summary , we have investigated the enhanced proton acceleration in the cone - capillary by using the 2d3v pic simulations performance . compared with the cone without the capillary , our results show that the protons can be accelerated and guided to a farther distance by using the cone - capillary . this enhancement has been analyzed and discussed in detail . first , when the protons enter into the cone , they can be collimated in transverse direction by the @xmath7 generated in the cone for both cases . afterwards , they can be continually accelerated and guided by the @xmath7 generated in the capillary , which results in a higher quality proton source in longer time for the case ii . yet it decreases significantly and diffuses along transverse direction for the case i. as a result , a dense proton source can be achieved with a density @xmath0 , monoenergetic peak @xmath1 , spatial emittance @xmath2 with divergence angle @xmath3 and diameter @xmath72 by the cone - capillary . lastly , the capillary lengths are also optimized and analyzed . these results may have many important implications such as the ions fast ignition , the medical applications as well as the laboratory astrophysics and so on . this work was supported by the national natural science foundation of china ( nsfc ) under grant nos . 11475026 , 11305010 , 11365020 and the nsaf of china under grant no . the computation was carried out at the hscc of the beijing normal university . the authors are particularly grateful to cfsa at university of warwick for allowing us to use the epoch . m. roth , t. e. cowan , m. h. key , s. p. hatchett , c. brown , w. fountain , j. johnson , d. m. pennington , r. a. snavely , s. c. wilks , k. yasuike , h. ruhl , f. pegoraro , s. v. bulanov , e. m. campbell , m. d. perry , and h. powell , phys . 86 * , 436 ( 2001 ) . r. a. snavely , m. h. key , s. p. hatchett , t. e. cowan , m. roth , t. w. phillips , m. a. stoyer , e. a. henry , t. c. sangster , m. s. singh , s. c. wilks , a. mackinnon , a. offenberger , d. m. pennington , k. yasuike , a. b. langdon , b. f. lasinski , j. johnson , m. d. perry , and e. m. campbell , phys . lett . * 85 * , 2945 ( 2000 ) . j. fuchs , p. antici , e. dhumieres , e. lefebvre , m. borghesi , e. brambrink , c. a. cecchetti , m. kaluza , v. malka , m. manclossi , s. meyroneinc , p. mora , j. schreiber , t. toncian , h. ppin , and p. audebert , nat . * 2 * , 48 ( 2006 ) . m. s. wei , s. p. d. mangles , z. najmudin , b. walton , a. gopal , m. tatarakis , a. e. dangor , e. l. clark , r. g. evans , s. fritzler , r. j. clarke , c. hernandez - gomez , d. neely , w. mori , m. tzoufras , and k. krushelnick , phys . 93 * , 155003 ( 2004 ) . x. r. hong , w. j. zhou , b. s. xie , y. yang , l. wang , j. m. tian , r. a. tang and w. s. duan , _ realization of the radiation pressure acceleration in the near critical density regime _ , phys . plasmas ( under review 2016 ) . c. a. j. palmer , j. schreiber , s. r. nagel , n. p. dover , c. bellei , f. n. beg , s. bott , r. j. clarke , a. e. dangor , s. m. hassan , p. hilz , d. jung , s. kneip , s. p. d. mangles , k. l. lancaster , a. rehman , a. p. l. robinson , c. spindloe , j. szerypo , m. tatarakis , m. yeung , m. zepf , and z. najmudin phys . lett . * 108 * , 225002 ( 2012 ) . r. kodama , p. a. norreys , k. mima , a. e. dangor , r. g. evans , h. fujita , y. kitagawa , k. krushelnick , t. miyakoshi , n. miyanaga , t. norimatsu , s. j. rose , t. shozaki , k. shigemori , a. sunahara , m. tampo , k. a. tanaka , y. toyama , t. yamanaka , and m. zepf , nature ( london ) * 412 * , 798 ( 2001 ) . t. d. arber , k. bennett , c. s. brady , a. lawrence - douglas , m. g. ramsay , n. j. sircombe , p. gillies , r. g. evans , h. schmitz , a. r. bell , and c. p. ridgers , plasma phys . fusion * 57 * , 113001 ( 2015 ) . r. kodama , y. sentoku , z. l. chen , g. r. kumar , s. p. hatchett , y. toyama , t. e. cowan , r. r. freeman , j. fuchs , y. izawa , m. h. key , y. kitagawa , k. kondo , t. matsuoka , h. nakamura , m. nakatsutsumi , p. a. norreys , t. norimatsu , r. a. snavely , r. b. stephens , m. tampo , k. a. tanaka , and t. yabuuchi , nature ( london ) * 432 * , 1005 ( 2004 ) .

a scheme with gold cone - capillary is proposed to improve the protons acceleration and involved problems are investigated by using the two - dimensional particle - in - cell simulations . it is demonstrated that the cone - capillary can efficiently guide and collimate the protons to a longer distance and lead to a better beam quality with a dense density @xmath0 , monoenergetic peak energy @xmath1 , spatial emittance @xmath2 with divergence angle @xmath3 and diameter @xmath4 . the enhancement is mainly attributed to the focusing effect by the transverse electric field generated by the cone as well as the capillary , which can prevent greatly the protons from expanding in the transverse direction . comparable to without the capillary , the protons energy spectra have a stable monoenergetic peak and divergence angle near to @xmath5 in longer time . besides , the efficiency of acceleration depending on the capillary length is explored , and the optimal capillary length is also achieved . such a target may be benefit to many applications such as ions fast ignition in inertial fusion , proton therapy in medicine and so on .

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Proceed to summarize the following text: the study of multilinear pseudodifferential operators goes back to the pioneering works of r. coifman and y. meyer , @xcite , @xcite , @xcite and @xcite . since then , there has been a large amount of work on various generalisations of their results , as well as studies of bilinear operators with symbols satisfying different conditions to those in the standard bilinear coifman - meyer classes . the literature in this area of research is vast and any brief summary of it here would not do the authors justice . therefore we confine ourselves to mention only those works with a direct connection to the present paper . r. coifman and y. meyer , in @xcite and @xcite , proved the boundedness from @xmath2 to @xmath3 of multilinear pseudodifferential operators with symbols in the class @xmath4 ( see definition [ bilinear multiplier defn ] below ) for @xmath5 and @xmath6 with @xmath7 . in the seminal paper @xcite , l. grafakos and r. torres systematically developed the theory of multilinear caldern - zygmund operators . they proved a multilinear @xmath8-theorem which they applied to generalise the result above to @xmath9 . as a further application , they demonstrated the boundedness in lebesgue spaces of multilinear pseudodifferential operators which , together with each of the adjoint operators , belonged to @xmath10 ( see definition [ def4 ] ) . however , in @xcite , a. bnyi and r. torres showed that there exist symbols in @xmath11 that do not give rise to bilinear operators which are bounded from @xmath12 to @xmath3 for @xmath13 such that @xmath14 . in particular , there is no analogue of the caldern - vaillancourt theorem in the bilinear setting . moreover , the class of operators @xmath15 is not closed under transposition . in contrast , @xcite demonstrates that @xmath16 is closed under transposition . recently , in @xcite , a. bnyi , d. maldonado , v. naibo and r. torres proved that @xmath17 is closed under transposition for @xmath18 and @xmath19 . in particular , given an operator in @xmath20 , its adjoints are also in @xmath20 . since @xmath21 , it follows that symbols in @xmath22 give rise to bounded operators , by applying the result of @xcite quoted above . in summary , we see that @xmath23 are bounded on appropriate lebesgue spaces when @xmath24 ( that is , the caldern - zygmund case ) , but in general they fail to be bounded when @xmath25 . the purpose of this paper is to address the following question , which is of interest for @xmath26 in - between these values , ` given @xmath27 , what @xmath28 is sufficient to ensure that symbols in @xmath29 give rise to bounded operators ? ' this question is in the spirit of questions asked in @xcite . we will study this question for two different symbol classes . first , in section [ main1 ] , we will consider a larger symbol class which does not require any differentiability in the spatial variable at all . that is , we study the multilinear symbol class @xmath30 ( see definition [ def2 ] ) which , in particular , contains @xmath29 for any @xmath31 . our main result in this context is theorem [ multi_one ] , which generalises a result obtained by the present authors in @xcite regarding the linear case . the study of such symbol classes originates in @xcite , where c. kenig and the third author studied linear operators . in the context of multilinear operators , results regarding mildly regular bilinear operators have been proved previously . in particular , d. maldonado and v. naibo established in @xcite boundedness properties of bilinear pseudodifferential operators on products of weighted lebesgue , hardy , and amalgam spaces . the regularity they require in the spatial variables is only that of dini - type . section [ mixed norms ] deals with linear operators on mixed - norm lebesgue spaces , and is a corollary to the proof of theorem [ multi_one ] . the second topic we will study is the bilinear symbol class @xmath32 . in section [ lpsmrho and applications ] we adapt methods used to study symbols in @xmath30 to weaken the requirement on @xmath33 necessary to prove boundedness on lebesgue spaces of operators in @xmath17 for @xmath34 . this is formulated as theorem [ smooth pseudo bilinear ] . in section [ outwith ] , although we can not show boundedness for general operators arising from symbols in @xmath35 , we can prove boundedness on a suitable subclass . this is stated as theorem [ modulationtype ] , which is a result of the same flavour as that proved by f. bernicot and s. shrivastava in @xcite regarding a subclass of @xmath36 , albeit proved by more straight - forward methods . a related result regarding @xmath37 was also proved in @xcite . we begin the main body of the paper with section [ prelim ] where we set out some definitions , fix some notation and recall some well - known results that we will use later . we study the following type of _ multilinear pseudodifferential operator_. given a function @xmath38 we define the @xmath39-linear operator @xmath40 to act on @xmath39 functions @xmath41 belonging to the schwartz class @xmath42 as @xmath43 here @xmath44 are all variables in @xmath45 , @xmath46 and @xmath47 denotes the fourier transform @xmath48 of @xmath49 . we refer to the function @xmath50 , which has @xmath51 variables , as the _ symbol _ of the operator @xmath40 . we set @xmath52 with @xmath53 , and define @xmath54 where @xmath55 denotes the standard euclidean norm of @xmath56 also , here and in the sequel we shall use @xmath57 to denote the set of nonnegative integers . + we will use a standard littlewood - paley partition of unity @xmath58 in @xmath59 by letting @xmath60 be a smooth radial function which is equal to one on the unit ball centred at the origin and supported on its concentric double . setting @xmath61 and @xmath62 for @xmath63 we have @xmath64 and @xmath65 for @xmath66 . one also has , for all multi - indices @xmath67 and @xmath68 , @xmath69 , and @xmath70 [ def2 ] given @xmath71 , @xmath72 and @xmath27 the symbol @xmath73 is said to belong to @xmath30 when for each multi - index @xmath67 there exists a constant @xmath74 such that @xmath75 in the case @xmath76 we also use the notation @xmath77 for the class of symbols of the linear pseudo - pseudodifferential operators , see @xcite . [ def4 ] given a class of symbols @xmath78 , operators which arise from elements in @xmath78 are denoted by @xmath79 . that is , we say @xmath80 when there exists a symbol @xmath81 such that @xmath82 , as defined in . consequently , for @xmath83 we say @xmath84 . for a non - negative function @xmath85 , which we refer to as a _ weight _ , we define @xmath86 to be the closure of @xmath87 in the norm @xmath88 when @xmath89 we write simply @xmath90 to mean @xmath91 and @xmath92 is the class of functions which belong to @xmath93 for each @xmath85 which is the characteristic function of a compact set . we wish to study the boundedness from @xmath94 to @xmath95 of the operator @xmath40 , initially defined for schwartz functions @xmath96 via , for particular exponents @xmath97 and weights @xmath98 . although the integral in may not be absolutely convergent for @xmath41 which do not decay sufficiently rapidly , if we can prove bounds on the operator norm which depend only on @xmath99 and @xmath50 , then it is a straight - forward exercise to show that @xmath40 has a unique extension to @xmath94 which agrees with for @xmath100 . this is the sense in which we will refer to the boundedness of @xmath40 . given @xmath101 , the @xmath102 maximal function @xmath103 is defined by @xmath104 where the supremum is taken over balls @xmath105 in @xmath106 containing @xmath107 . clearly then , the hardy - littlewood maximal function is given by @xmath108 an immediate consequence of hlder s inequality is that @xmath109 for @xmath110 . we shall use the notation @xmath111 for the average of the function @xmath112 over @xmath105 . one can then define the class of muckenhoupt @xmath113 weights as follows . [ weights ] let @xmath114 be a positive function . one says that @xmath115 if there exists a constant @xmath116 such that @xmath117 one says that @xmath118 for @xmath119 if @xmath120 the @xmath113 constants of a weight @xmath118 are defined by @xmath121_{a_1}:= \sup_{b\ , \textrm{balls in}\,\ , \r^{n}}\,w_{b}\vert w^{-1}\vert_{l^{\infty}(b)},\ ] ] and @xmath121_{a_p}:= \sup_{b\ , \textrm{balls in}\,\ , \r^{n}}\,w_{b}(w^{-\frac{1}{p-1}})_{b}^{p-1}.\ ] ] the following results are well - known and can be found in , for example , @xcite . [ maxweight ] for @xmath122 , the hardy - littlewood maximal operator is bounded on @xmath123 if and only if @xmath124 . consequently , for @xmath125 , @xmath126 is bounded on @xmath123 if and only if @xmath127 [ convolve ] suppose that @xmath128 is integrable non - increasing and radial . then , for @xmath129 , we have @xmath130 for all @xmath131 . we will need the following multilinear version of the hausdorff - young theorem due to a. benedek and r. panzone @xcite . [ hausdorff - young ] suppose that @xmath132 and @xmath133 then @xmath134 as is common practice , we will denote constants which can be determined by known parameters in a given situation , but whose value is not crucial to the problem at hand , by @xmath135 . such parameters in this paper would be , for example , @xmath33 , @xmath26 , @xmath136 , @xmath137 , @xmath138_{a_p}$ ] , and the constants @xmath74 in definition [ def2 ] . the value of @xmath135 may differ from line to line , but in each instance could be estimated if necessary . we also write @xmath139 as shorthand for @xmath140 . the following lemma will be useful in obtaining pointwise estimates for the kernel of operators in @xmath142 for @xmath143 with @xmath144 and @xmath145 with @xmath146 we define @xmath147 [ zlemma ] let @xmath148 with @xmath149 and @xmath150 $ ] . given any @xmath151 such that the set @xmath152 \,|\ , |z_j|{\geqslant}1\ } \neq \emptyset$ ] , one has @xmath153 for all @xmath154 and @xmath155 setting @xmath156 , and using the definition of @xmath157 , inequality and the leibniz rule we see that @xmath158 where in we have also used the assumption @xmath27 . we claim that @xmath159 for all @xmath160 . integrating by parts and using with @xmath161 yields @xmath162 now summing both sides of the above estimate over all @xmath163 with @xmath164 and using the straightforward inequality @xmath165 , we obtain . for the integrals containing @xmath166 , integration by parts and yield @xmath167 therefore , if @xmath168 then @xmath169 from this and the definition of the set @xmath170 , by taking @xmath168 and @xmath171 , it follows that @xmath172 for @xmath173 and @xmath174 such that @xmath175 . the estimate for @xmath176 follows by combining the estimates and . this proves the lemma . as an immediate corollary we have the following kernel estimates [ zlemma2 ] let @xmath177 and suppose @xmath107 and @xmath178 are such that @xmath179 \,|\ , |x - y_j|{\geqslant}1\ } \neq \emptyset$ ] . then one has @xmath180 provided either @xmath181 and @xmath72 , or @xmath182 and @xmath183 . when @xmath184 , this follows from lemma [ zlemma ] by setting @xmath185 . an examination of the proof of lemma [ zlemma ] reveals that it can be easily modified for the case @xmath25 provided @xmath183 . the following theorem is the main result of this section . [ multi_one ] fix @xmath186 $ ] for @xmath187 and let @xmath188 with @xmath189 and @xmath190 . then there exists a constant @xmath135 , depending only on @xmath137 , @xmath191 , @xmath33 , @xmath26 and a finite number of the constants @xmath74 in definition @xmath192 , such that latexmath:[\[\label{pointws } for all @xmath131 . consequently , for @xmath194 and @xmath195 such that @xmath196 , @xmath40 is a bounded operator from @xmath94 to @xmath197 whenever @xmath198 and @xmath199 . the boundedness follows immediately from the pointwise estimate by theorem [ maxweight ] . to prove we use the littlewood - paley partition of unity as in , we decompose the symbol as @xmath200 with @xmath201 , @xmath202 . first we consider the operator @xmath203 . we have @xmath204 with @xmath205 now estimate yields @xmath206 for each @xmath207 and hence for @xmath208 . therefore theorem [ convolve ] implies @xmath209 . now let us analyse @xmath210 for @xmath66 . we note , just as before , that @xmath211 can be written as @xmath212 with @xmath213 one observes that @xmath214 where the weight functions @xmath215 will be chosen momentarily . therefore , hlder s inequality yields @xmath216 where @xmath217 . now for an @xmath218 , we define @xmath215 by @xmath219 we now wish to estimate @xmath220 by splitting each @xmath221-integral as integration over @xmath222 and @xmath223 . considering an arbitrary case of the @xmath224 possibilities , we can estimate this portion of the integral by @xmath225 where the sum is taken over multi - indices @xmath163 ( each with @xmath137 components ) such that @xmath226 if @xmath222 and @xmath227 if @xmath223 . without loss of generality we may assume @xmath228 , so by theorem [ hausdorff - young ] and the estimate , this in turn is majorised by @xmath229 furthermore , once again using theorem [ convolve ] , we have @xmath230 with a constant that only depends on the dimension @xmath137 . combining these facts with yields @xmath231 summing in @xmath232 , and using and , we obtain @xmath233 we observe that the series above converges if @xmath234 . this proves and , with it , the theorem . we remark in passing that the case @xmath235 theorem [ multi_one ] follows from its linear predecessor , that is theorem 3.3 in @xcite , with @xmath236 replacing @xmath45 . in this section we show that a modification of the proof of theorem [ multi_one ] yields a mixed norm boundedness result for a class of linear pseudodifferential operators . let @xmath237 with @xmath238 for @xmath239 to a symbol @xmath240 we associate a linear operator @xmath241 a - priori defined on functions @xmath112 in @xmath242 given by @xmath243 where @xmath244 is the fourier transform of @xmath112 in @xmath236 : @xmath245 we define the space @xmath246 to be the mixed norm space which is the closure of @xmath247 in the norm @xmath248 we will also need the following notation . for a function @xmath249 we define @xmath250 to be the @xmath102 maximal function acting only in the @xmath251-variables . that is , @xmath252 for @xmath187 , where the supremum is taken over balls @xmath105 in @xmath45 containing @xmath251 . [ multi_two ] fix @xmath186 $ ] for @xmath187 and let @xmath253 with @xmath189 and @xmath190 . then there exists a constant @xmath135 , depending only on @xmath137 , @xmath191 , @xmath33 , @xmath26 and a finite number of the constants @xmath74 in definition @xmath192 , such that @xmath254 . consequently , @xmath40 is a bounded operator from @xmath255 to @xmath256 whenever @xmath257 @xmath258@xmath187@xmath259 and @xmath260 . we repeat the prove of theorem [ multi_one ] , but with the linear operator above and observe that is replaced by @xmath261 we control the first factor on the right - hand side of as before . to control the second factor in we can use theorem [ convolve ] to show that it is majorised by @xmath262 by combining these estimates we obtain . using the boundedness of the maximal function and minkowski s inequality repeatedly , we obtain the boundedness of @xmath40 on the mixed norm space defined above . we now consider linear pseudodifferential operators acting on functions on @xmath45 . [ def5 ] let @xmath85 be a weight @xmath258that is , a non - negative function@xmath259 , and @xmath264 , @xmath72 and @xmath27 be parameters . a symbol @xmath265 belongs to the class @xmath266 if for each multi - index @xmath267 there exists a constant @xmath74 such that @xmath268 when the weight @xmath269 then we use the notation @xmath270 for @xmath271 . [ lrsym ] suppose @xmath272 , @xmath273 $ ] and @xmath274 with conjugate @xmath275 @xmath258for which @xmath276@xmath259 satisfy the relation @xmath277 . suppose further that @xmath27 and @xmath278 . let @xmath279 and @xmath85 and @xmath280 be weights with @xmath281 . then there exists a constant @xmath135 , depending only on @xmath137 , @xmath33 , @xmath26 , @xmath136 , @xmath282 , @xmath138_{a_{q / p'}}$ ] and a finite number of @xmath74 from @xmath283 @xmath284 , such that 1 . [ weightedlp ] if @xmath285 and @xmath286 , then @xmath287 where @xmath288 ; and 2 . [ unweightedlp ] if @xmath289 or @xmath290 , then @xmath291 where @xmath292 . first , we define @xmath293 it is then easy to check that @xmath294 provided @xmath295 is sufficiently large . we consider the littlewood - paley pieces @xmath296 of the operator @xmath297 , where @xmath298 . using hlder s inequality , the hausdorff - young inequality and theorem [ convolve ] , we compute @xmath299 consequently , under the hypotheses of @xmath300 , hlder s inequality with exponents @xmath301 and @xmath302 , the weighted boundedness of the hardy - littlewood maximal function and minkowski s inequality show us that @xmath303 we can then sum in @xmath232 to find that @xmath304 which completes the proof of @xmath300 . the proof of @xmath305 is similar . when @xmath289 , @xmath306 , so we can not use the boundedness of the hardy - littlewood maximal function in . so before applying theorem [ convolve ] in , we instead first take the @xmath307 norm of the inequality , apply hlder s inequality as before , and then apply young s inequality to the factor involving @xmath308 . when @xmath290 , @xmath309 , so once again we can not use the weighted boundedness of the hardy - littlewood maximal function , but we can use its boundedness on @xmath310 . as an application of theorem [ lrsym ] we now establish our main result regarding the boundedness of smooth bilinear pseudodifferential operators that fall outwith the scope of the bilinear caldern - zygmund theory . we shall prove a further result concerning a subclass of these operators in section [ outwith ] . [ bilinear multiplier defn ] given @xmath311 , @xmath312 and @xmath313 $ ] , a symbol @xmath73 belongs to the class @xmath314 if , for each pair of multi - indices @xmath67 , and @xmath315 , there exists a constant @xmath316 such that @xmath317 in particular , a symbol @xmath318 belongs to the class @xmath319 if , for each triple of multi - indices @xmath320 @xmath321 and @xmath322 , there exists a constant @xmath323 such that @xmath324 define the adjoint operators @xmath325 and @xmath326 of a bilinear operator @xmath327 via the identities @xmath328 , where @xmath329 denotes the dual pairing on @xmath102 ( @xmath330 ) . we will use theorem 1 in @xcite repeatedly , so we record it here . [ 1in2 ] assume that @xmath18 , @xmath19 and @xmath331 . then @xmath332 for some @xmath333 and @xmath334 . [ smooth pseudo bilinear ] let @xmath331 , with @xmath18 , @xmath19 and @xmath335 for @xmath336 $ ] @xmath258see figure @xmath337@xmath259 . then @xmath338 for @xmath136,@xmath282 and @xmath339 such that @xmath277 . observe that the condition @xmath340 ensures that we can not have both @xmath136 and @xmath282 less than @xmath341 . theorem [ 1in2 ] tells us that the adjoint operators @xmath342 and @xmath342 are operators in the same class as @xmath40 . considering these adjoint operators if necessary , we can reduce the proof of the theorem to the special case @xmath343 . we consider a littlewood - paley partition of unity @xmath58 defined as in with @xmath344 we then set @xmath345 , so @xmath346 . we can write @xmath347 where @xmath348 is a linear operator for each fixed @xmath349 . in order to take advantage of the smoothness of the symbol while viewing the operator as written in , we must deal with each piece of the symbol @xmath350 depending on how the size of @xmath351 relates to the size of @xmath352 . more precisely , for @xmath353 we have that @xmath354 on the @xmath355-support of @xmath350 . fixing an @xmath356 sufficiently small , we can conclude that @xmath357 for @xmath358 such that @xmath359 . thus , by @xcite , if @xmath360 then we have that @xmath361 this shows that @xmath362 therefore , using the fact that @xmath363 an application of theorem [ lrsym ] with the assumption @xmath364 yields @xmath365 for the case @xmath366 , we can repeat the same argument , but reverse the roles of @xmath351 and @xmath352 , and @xmath136 and @xmath282 , to obtain once again . we then sum in @xmath367 and @xmath232 to obtain boundedness from @xmath368 to @xmath3 provided @xmath369 this result can be improved by using duality and once again applying theorem [ 1in2 ] . we are concerned with triples of reciprocals of exponents @xmath370 such that @xmath371 and @xmath336 $ ] . the set of such triples is a closed triangle with vertices @xmath372 , @xmath373 and @xmath374 in the plane @xmath375 , which itself lies in @xmath376 ( see figure [ figure ] ) . considering the edge of the triangle with end - points @xmath372 and @xmath373 ( that is , where @xmath377 ) we have proved that @xmath40 maps @xmath378 to @xmath379 ( of course , here we require @xmath380 ) for @xmath381 this agrees with the statement of the theorem for these exponents . equally the theorem in the case @xmath382 ( corresponding to the point @xmath374 ) is also included in the condition . considering the adjoint @xmath326 , using duality and applying theorem [ 1in2 ] allows us to conclude that @xmath40 maps @xmath383 to @xmath384 for the same range on @xmath33 . once again , this agrees with the statement of the theorem for these exponents , but this time corresponds to triples @xmath370 on the line with end - points @xmath372 and @xmath374 . similarly , considering the adjoint @xmath325 , using duality and applying theorem [ 1in2 ] allows us to conclude that @xmath40 maps @xmath385 to @xmath386 , again , for the same range of @xmath33 . yet again , this agrees with the statement of the theorem for these exponents , but now corresponds to triples @xmath370 on the line with end - points @xmath373 and @xmath374 . thus , we have proved the theorem on the edges of the triangle . finally , the bilinear version of the riesz - thorin interpolation theorem ( see @xcite ) allows us to complete the proof on the interior of the triangle . in this section we will establish the boundedness of a subclass of bilinear pseudodifferential operators with symbols in the class @xmath388 with @xmath389 $ ] . with this goal in mind , the following lemma will prove to be useful . [ notquite ] given a smooth function @xmath390 , define an operator @xmath170 by @xmath391 where @xmath392 denotes the fourier transform in @xmath393 @xmath258that is , with @xmath394@xmath259 . the operator @xmath170 is bounded from @xmath395 to @xmath396 if and only if @xmath397 moreover , @xmath398 the boundedness of @xmath170 is equivalent to the boundedness of @xmath399 , where @xmath400 is the adjoint operator of @xmath170 : @xmath401 we can readily see that @xmath402 is given by @xmath403 so @xmath404 thus , @xmath405 is a multiplier and the condition of the lemma is exactly the condition required for its boundedness on @xmath396 . [ modulationtype ] if @xmath406 and @xmath407 are such that , for each multi - index @xmath408 , @xmath409 where @xmath410 is the unit ball centred at @xmath411 , then @xmath40 is bounded operator from @xmath412 to @xmath413 for @xmath414 $ ] and @xmath415 $ ] such that @xmath371 . this corresponds to @xmath370 contained in the closed triangle of @xmath416 @xmath337 with vertices @xmath417 , @xmath418 and @xmath419 . first , let us observe that it suffices to prove the boundedness from @xmath420 to @xmath379 . indeed , assuming this and using theorem [ 1in2 ] , together with duality arguments and the multilinear riesz - thorin theorem ( see @xcite ) , yield the theorem . let us now suppose that the symbol @xmath50 has compact @xmath107-support , say contained in the unit ball @xmath421 . we follow @xcite and write @xmath422 and so @xmath423 where @xmath424 and @xmath425 is a multiplier operator for each @xmath426 . therefore , since @xmath427 we have that @xmath428 and so @xmath429 this means that , by our hypotheses , @xmath430 for any @xmath431 . consequently , by lemma [ notquite ] , @xmath432 once again using the support properties of @xmath50 , we conclude that @xmath433 which proves the theorem under the extra hypothesis that @xmath50 has compact @xmath107-support . + to remove the hypothesis that @xmath50 has compact @xmath107-support , we follow the argument of @xcite . observe that it suffices to show that , for each @xmath434 , @xmath435 for all @xmath436 indeed , choosing @xmath437 , integrating in @xmath411 , using the cauchy - schwarz inequality and interchanging the order of integration produces the estimate @xmath438 to prove we introduce the cut - off function @xmath439 , which is identically one on @xmath440 and zero outside its concentric double @xmath441 . define @xmath442 and @xmath443 and @xmath444 . let @xmath445 be a second nonnegative cut - off function which is identically one on @xmath446 and supported in @xmath440 . using the cut - off function @xmath447 , we can write @xmath448 where @xmath449 now using the cut - off function @xmath450 our previous boundedness result concerning bilinear operators with compact spatial support yields @xmath451 and this is , in turn , controlled by @xmath452 for any @xmath68 , because of the support properties of @xmath453 and @xmath454 . to estimate the contribution of the remaining term in , we use the kernel estimate of corollary [ zlemma2 ] . we need to estimate @xmath455 but since @xmath456 is supported outside @xmath457 , corollary [ zlemma2 ] yields that for all @xmath458 , is majorised by @xmath459 and using the cauchy - schwarz inequality and the fact that @xmath460 , we have @xmath461 provided @xmath462 this completes the proof of and with it , the theorem .

we consider two types of multilinear pseudodifferential operators . first , we prove the boundedness of multilinear pseudodifferential operators with symbols which are only measurable in the spatial variables in weighted lebesgue spaces . these results generalise earlier work of the present authors concerning linear pseudo - pseudodifferential operators . secondly , we investigate the boundedness of bilinear pseudodifferential operators with symbols in the hrmander @xmath0 classes . these results are new in the case @xmath1 , that is , outwith the scope of multilinear caldern - zygmund theory .

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Proceed to summarize the following text: ultracold fermi gases have been subject of many experimental and theoretical studies during recent years ( see e.g. @xcite ) . they provide a unique system to study key concepts of condensed matter theory . this is because in these systems many parameters such as the particle density , the fermi energy , the confinement potential , or the interaction strength between the fermions , which in a solid state system are typically fixed quantities , can be externally controlled in a wide range @xcite . in particular , magnetic - field feshbach resonances provide the means for controlling the interaction strength between fermions by varying an external magnetic field . the tunability of the s - wave scattering length , which is the dominant interaction channel , makes ultracold fermi gases ideal for exploring different regimes of interacting many - body systems in a single system . this includes the limiting regimes of weakly attracting fermions , which condense into cooper pairs forming a bardeen - cooper - schrieffer ( bcs ) phase below a certain temperature @xmath0 , and repulsive dimers formed by two fermions , which can undergo a bose - einstein condensation ( bec ) . these two limiting regimes are separated by the strongly interacting bcs - bec crossover regime where the scattering length diverges and the system exhibits unitary properties @xcite . in addition to the variable interaction strength , ultracold atomic gases offer a unique opportunity to explore the influence of a confinement on the pairing correlations , because dimensionality and confinement can be precisely controlled by tuning external parameters @xcite . varying the confinement , which is often well approximated by a harmonic confinement potential , allows one to access new degrees of freedom . restricting the fermi gases to quasi - low dimensionality may , e.g. , offer the possibility for an experimental evidence of unconventional phases , like the fulde - ferrell - larkin - ovchinnikov ( fflo ) state @xcite . moreover it may help to study and get experimental insight into shape resonances , theoretically predicted for quasi - low dimensional conventional superconductors @xcite . in ref . @xcite the first quantitative measurements of the transition from 2d to quasi-2d and 3d in a weakly interacting fermi gas has been reported . at low atom numbers , the shell structure associated with the filling of individual transverse oscillator states has been observed . on the theoretical side the ground state properties of a @xmath1li gas confined in a cigar - shaped laser trap have been investigated predicting size - dependent resonances of the superfluid gap @xcite , similar to the case of superconducting nanowires @xcite , yielding an atypical bcs - bec crossover . an important effort is now devoted to the exploration of the out - of - equilibrium behavior of trapped ultracold atomic fermi gases and , in particular , to the determination of their dynamical properties . the dynamics has been studied in the normal as well as in the condensed phase , observing second sound @xcite and soliton trains @xcite , and showing a low - frequency oscillation of the cloud after a change of the system confinement or optical excitation @xcite . furthermore , state - of - the - art technology allows one to change the coupling constant on such short time scales that it is possible to explore the regime where the many - body system is governed by a unitary evolution with nonequilibrium initial conditions . in ultracold atomic fermi gases the dynamics may be initiated by readjusting the pairing interaction through switching an external magnetic field in the region of a feshbach resonance ( i.e. , a quantum quench ) or by a rapid change of the confinement potential of the trap @xcite . due to the small energies in the trapping potential the dynamics in the fermi gases take place on a millisecond timescale . therefore , in contrast to metallic superconductors , where sub - picosecond excitations are required to achieve non - adiabatic dynamics @xcite , in atomic gases the non - adiabatic regime can be reached already by excitations in the sub - millisecond range . spontaneous symmetry breaking gives rise to collective modes of the order parameter which are classified into the higgs amplitude mode , and the goldstone mode , the latter corresponding to a phase oscillation of the gap @xcite . the higgs mode has been subject of intensive theoretical and experimental @xcite research efforts in the past . on the theoretical side the non - adiabatic temporal response of the order parameter to ( quasi-)instantaneous perturbations has been studied . different regimes of an oscillatory temporal behavior of the pairing potential were theoretically predicted in homogeneous fermionic condensates @xcite . it was shown that the amplitude of the order parameter oscillates without damping when the coupling constant is increased above a certain critical value @xcite . on the other hand , the gap vanishes when the coupling constant is decreased below another critical value . in between these two limiting regimes the amplitude exhibits damped dephased oscillations and the system goes to a stationary steady state with a finite gap @xcite . in extended systems the approach to a stationary state occurs in an oscillatory way with an inverse square root decay in time of the amplitude of the oscillations @xcite . a similar evolution was predicted for conventional bulk superconductors @xcite where the non - adiabatic regime is reached by excitation with short , intense terahertz pulses . an experimental realization was reported in @xcite . in contrast , in finite length superconducting nanowires a breakdown of the damped oscillation and a subsequently rather irregular dynamics has been predicted @xcite . in this paper we present a theoretical analysis of the short - time bcs dynamics of a @xmath1li gas confined in a cigar - shaped laser trap . the excitation is modeled by a sudden change of the interaction strength which can be achieved through a feshbach resonance by an abrupt change of the external magnetic field @xcite . applying the well - known bcs theory in mean field approximation to ultracold fermi gases we show that the change of the coupling strength induces a collective oscillation of the bogoliubov quasiparticles close to the fermi level . this results in a damped amplitude oscillation of the bcs gap , which corresponds to the higgs mode . like in the case of confined bcs superconductors this oscillation breaks down after a certain time revealing rather chaotic dynamics afterwards . we explain these dynamics in terms of coupled harmonic oscillators which can be derived by linearizing the quasiparticle dynamics obtained from the heisenberg equation of motion . in doing so we first derive the quasiparticle equations of motion from the inhomogeneous bogoliubov - de gennes hamiltonian ( sec . [ sec : formalism ] ) which , because of using the standard contact - type interaction , requires a proper regularization of the gap equation . starting from the ground state calculated according to ref . @xcite we then calculate the dynamics of the superfluid gap after an instantaneous change of the coupling constant . the numerical results as well as their explanation follow in sec . [ sec : results ] , where we first discuss a rather small system and then proceed to a larger , experimentally more easily accessible system . finally , in sec . [ sec : conclusions ] we summarize our results and give some concluding remarks . our approach aims at describing the dynamics of the superfluid order parameter @xmath2 of an ultracold @xmath1li fermi gas , confined in a cigar - shaped , axial symmetric harmonic trapping potential @xmath3 here , @xmath4 is the mass of the @xmath1li atoms and @xmath5 ( @xmath6 ) is the confinement frequency in the @xmath7-@xmath8-plane ( @xmath9-direction ) , respectively . choosing @xmath10 yields an elongated cigar - shaped trap where the oscillator length @xmath11 provides a measure of the system length . the eigenvalues @xmath12 are measured with respect to the fermi energy @xmath13 . the index @xmath4 refers to the combination of quantum numbers @xmath14 , @xmath15 , and @xmath16 . for this geometry the one - particle states form one - dimensional subbands , labeled by @xmath17 [ cf . [ fig : states ] ] , while the states within each subband are labeled by @xmath16 . each subband has a constant one - particle density of states and thus the overall density of states exhibits finite jumps whenever a new subband appears . we consider the gas to be composed of two spin states , @xmath18 and @xmath19 , and start from the inhomogeneous bcs hamiltonian at @xmath20 . within the anderson approximation we then derive equations of motion for the corresponding bogoliubov quasiparticle excitations . the usual inhomogeneous bcs hamiltonian for an effective bcs - type contact interaction reads @xcite @xmath21 \ , d^3 r\notag \\ & - g \int \ , \psi_{\uparrow}^{\dagger}({\mathbf{r } } ) \psi_{\downarrow}^{\dagger}({\mathbf{r } } ) \psi_{\downarrow}^{}({\mathbf{r } } ) \psi_{\uparrow}^{}({\mathbf{r } } ) \ , d^3 r , \end{aligned}\ ] ] where @xmath22 and @xmath23 are the field operators for up and down spin , respectively , @xmath24 is the one - particle hamiltonian and @xmath25 is the coupling constant of the contact interaction @xmath26 . in the limit of low temperatures the main contribution to the interaction between two fermionic atoms in different internal spin states is given by scattering processes at low momentum . the description of those can be replaced by the widely known pseudopotential only depending on the scattering length @xmath27 @xcite , which yields @xmath28 . a bcs - like mean field expansion in terms of anomalous expectation values and a particle - hole transformation , leaving spin - up operators unchanged , @xmath29 , while interchanging spin - down ones , @xmath30 , leads to the bogoliubov - de gennes ( bdg ) hamiltonian @xcite @xmath31 where @xmath32 is the bcs order parameter . from eq . it becomes apparent that the corresponding eigenvalue equation can be written as the one - particle bogoliubov - de gennes equation @xmath33 @xmath34 can thus be diagonalized by bogoliubov s transformation , using the eigenfunctions @xmath35 and @xmath36 . this introduces non - interacting quasiparticles with energy @xmath37 obeying fermionic commutation relations with the corresponding creation operator @xmath38 \ , d^3 r.\ ] ] the spectrum of the bdg equation is symmetric with respect to the fermi energy and thus the eigenstates of the bdg equation occur in pairs . labeling @xmath39 for states @xmath40 and @xmath41 for @xmath42 , respectively , one finds the relations @xmath43 and @xmath44 for the eigenstates . therefore , all quantities can be expressed solely using the @xmath45 state wave functions @xcite . in the following we omit the @xmath45 , @xmath46 index of the eigenfunctions while they are still necessary for the quasiparticle operators . hereafter in the case of the eigenfunctions the index @xmath4 refers to the @xmath45 states . for our further calculations it is convenient to transform to the excitation picture ( @xmath47 , @xmath48 ) with @xmath49 and @xmath50 , where all quasiparticle excitations vanish in the ground state . we can rewrite the order parameter in the basis given by the eigenfunctions , where it reads : @xmath51 this yields the well - known result for the ground state order parameter @xmath52 which has to be solved self - consistently with eq . focusing on the underlying physics , we exploit the anderson approximation ( a.a . ) @xcite , choosing the bdg wave functions @xmath53 and @xmath54 proportional to the one - particle wave functions of the confinement potential @xmath55 , i.e. , @xmath56 here the amplitudes of the bdg wave functions @xmath57 , @xmath58 are obtained from the bdg eq . and read @xmath59 with the quasiparticle energies given by @xmath60 and the one - particle energies @xmath61 given by eq . . applying the anderson approximation to eq . additionally yields @xmath62 , where @xmath63 are the one - particle eigenfunctions . the anderson approximation has been tested in several nanostructured geometries and no qualitative deviations have been found @xcite . it is applied to all our calculations . from eqs . and we obtain the well - known bcs - like self - consistency equation , also referred to as gap equation . the ground state order parameter in the state @xmath64 is given by @xmath65 here @xmath66 is the interaction matrix element @xmath67 which exhibits maxima for states at the subband minimum ( i.e. , states with low @xmath16 ) . the contact interaction used here leads to a well - know ultraviolet divergence in the summation over all states , i.e. , in eq . , which can be regularized by applying a scattering length regularization @xcite . this has been established for the homogeneous gap equation and subsequently extended to the inhomogeneous gap equation , where ref . @xcite gives a careful derivation for confined systems . however , refs . @xcite state that a much simpler regularization is sufficient since the results are not sensitive to the details of the method used . the corresponding regularized gap equation reads @xmath68 which can be rewritten as a multiplication by a factor @xmath69 thus , each quasiparticle state @xmath70 is weighted by a factor @xmath71 . in order to extend this in a consistent manner to the nonequilibrium case , in which @xmath72 deviates from its ground state value @xmath73 and is determined by the full eq . , the same procedure has to be applied to the nonequilibrium version of eq . as will be discussed below [ see eq . ] . and thus the ground state gap @xmath74 in the dynamical equations . since in the dynamical case the time - dependent gap @xmath75 differs from the ground state value this may seem unusual . however , we have checked that the method is not sensitive on the detailed parameters used , e.g. , on using one constant value for all @xmath74 in order to omit the newly introduced dependence . thus the proposed regularization scheme is a suitable extension since the physical behavior of the system remains unchanged . ] figure [ fig : dispersion ] shows the dependence of the quasiparticle energies @xmath76 on the one - particle energies @xmath61 . for all subbands crossing the fermi energy finite minima of the quasiparticle energy evolve at @xmath77 . subbands with the minimum close to the fermi energy ( i.e. , @xmath78 ) exhibit a larger @xmath74 due to the larger interaction matrix elements @xmath66 . this leads to a rather shallow parabolic - like minimum in these subbands and , thus , to a maximum in the corresponding density of states ( see fig . [ fig : densityofstates ] ) . subbands showing this feature will be called resonant in the following and systems in which such a subband exists are referred to as resonant systems . the behavior can be compared to the well - known parabolic - like dispersion relation in the homogeneous bcs theory with @xmath79 . a similar behavior has also been obtained for nanostructured superconducting systems @xcite . overall the density of states combined with the interaction matrix @xmath66 leads to the quantum size oscillations of the order parameter upon changing the lateral size of the system ( i.e. , @xmath5 ) found in ref . @xcite . in this paper we consider an excitation of the fermi gas by a quantum quench , i.e. , a sudden change of the coupling constant @xmath80 . since in the region of a feshbach resonance the coupling constant strongly depends on an external magnetic field @xmath81 , this can be achieved experimentally by rapidly switching @xmath81 from an initial value @xmath82 to a final value @xmath83 . we assume that this switching process occurs on a time scale much faster than the characteristic time scale of the order parameter dynamics , such that the excitation can be taken to be instantaneous . this assumption is realistic since fast linear magnetic ramps with rates of @xmath84g / ms are experimentally available @xcite while the timescale of the gap dynamics is of the order of @xmath85ms @xcite . the excitations considered in this paper require a shift in the magnetic field of a few gauss , which indeed can be assumed to be instantaneous on the typical ms time scale in ultracold fermi gases . in this case during the switching of the magnetic field the state of the system remains unchanged . as usual , the dynamics of a quantum mechanical system can be described in different basis systems , which from a mathematical point of view are all equivalent . in our case , to calculate the dynamics of the order parameter after a quench from the _ initial _ system ( @xmath86 , @xmath87 , @xmath88 ) to the _ final _ system ( @xmath53 , @xmath54 , @xmath25 ) we choose a time - independent basis rather than remaining in the diagonal basis . for convenience we take the basis corresponding to the eigenstates of the system after the switching , i.e. , to the coupling constant @xmath25 . all our calculations are thus carried out in the basis @xmath53 , @xmath54 . the initial state , which is characterized by the ground state order parameter @xmath89 , corresponding to the coupling constant @xmath88 and the basis functions @xmath86 , @xmath87 , has to be expressed in terms of the basis @xmath53 , @xmath54 , which in particular gives rise to non - vanishing quasiparticle excitations in this basis . since the confinement potential is unchanged , also the corresponding one - particle wave functions @xmath55 remain unchanged . according to eq . , in the present case only the bdg amplitudes @xmath57 and @xmath58 change while the spatial shapes of @xmath53 and @xmath54 remain unchanged . therefore , all orthogonality relations are preserved and only diagonal quasiparticles are populated . for the initial values of the normal and anomalous expectation values , respectively , one finds @xmath90 in addition @xmath91 holds for all times @xmath92 . since the instantaneous order parameter @xmath93 deviates from the ground state value of the final system @xmath73 , the hamiltonian in the basis @xmath53 , @xmath54 becomes non - diagonal depending on the difference @xmath94 . it thus becomes explicitly time dependent according to @xmath95 \gamma_{ma}^{\dagger}\gamma_{na}^ { } \notag \\ & + \sum_{m , n } \left[\left(\delta-\delta_{gs}\right)_{u_m^ * u_n^ * } - \left(\delta^*-\delta_{gs}^*\right)_{v_m^ * v_n^ * } \right ] \gamma_{ma}^{\dagger}\gamma_{nb}^{\dagger } \notag \\ & - \sum_{m , n } \big[\left(\delta-\delta_{gs}\right)_{v_m v_n } - \left(\delta^*-\delta_{gs}^*\right)_{u_m u_n } \big ] \gamma_{mb}^{}\gamma_{na}^ { } \notag \\ & - \sum_{m , n } \left[\left(\delta-\delta_{gs}\right)_{v_m u_n^ * } + \left(\delta^*-\delta_{gs}^*\right)_{u_m v_n^ * } \right ] \left(1-\gamma_{mb}^{\dagger}\gamma_{nb}^{}\right ) . \end{aligned}\ ] ] with @xmath96 v_n({\mathbf{r } } ) d^3 r \notag \\ & \underbrace{=}_{\text{a.a . } } \left(\delta - \delta_{gs } \right)_{m } u_m^ { } v_m^ { } \delta_{mn}^{}. \end{aligned}\ ] ] here , the anderson approximation has been applied to the dynamical equations as proposed in ref . @xcite , yielding @xmath97 . the time evolution of the system is thus described by the time - dependent quasiparticle expectation values . the corresponding equations of motion can be obtained via heisenberg s equation of motion . for the considered instantaneous change of the coupling constant only diagonal expectation values are excited . the required equations of motion read @xmath98 where @xmath99 , \label{eq : e_ren}\ ] ] @xmath100 and @xmath101 v_{mk } \chi_k . \end{aligned}\ ] ] in eq . again the regularization factor @xmath102 has been introduced . equations - represent a finite set of coupled ordinary differential equations that we solve numerically . it is interesting to note that the evolution of the anomalous expectation values [ eq . ] corresponds to a set of harmonic oscillators with energies approximately given by @xmath103 [ first term in eq . ] while eq . as well as the second terms of eqs . and contain nonlinear couplings to all other oscillators via the factor @xmath104 , which vanishes when the order parameter agrees with its ground state value . we will come back to this separation into linear and nonlinear terms in sec . [ sec : linearized ] . in the following the temporal evolution of the amplitude of the spatially averaged gap @xmath105 will be shown and analyzed for different system parameters . here , the normalization volume @xmath106 is set to @xmath107 with @xmath108 being the oscillator length in @xmath45 direction . in order to concentrate on the physics we will start our analysis by investigating a very small system : all the main features occurring in the dynamics of larger , experimentally accessible confinements arise in small systems , too , but with a strongly reduced degree of numerical complexity . thus , the dynamics of the superfluid gap will at first be explained on the basis of small systems . the results for a larger system will be shown afterwards . an exemplary result for the gap dynamics after a quantum quench , obtained by changing the scattering length from @xmath109 nm to @xmath110 nm for a system with the confinement frequencies @xmath111khz and @xmath112hz , is shown in fig . [ fig : dynamik+ft ] . the fermi energy has been set to @xmath113 yielding @xmath114 according to ref . as can be seen the amplitude of the gap shows an initial drop corresponding to the decreased coupling and thus decreased ground state gap . afterwards a smoothly damped oscillation around the new ground state value of the gap occurs , which after a certain transition time @xmath115 turns into an irregular , rather chaotic oscillation . here @xmath115 is defined as the time of the first deviation is assumed to be reached if the distance between two adjacent maxima shows a deviation of more then 50@xmath116 compared to the averaged distance of all preceding maxima . ] from a smooth oscillation . nm to @xmath117 nm ; inset : fourier transform of the gap dynamics . the confinement frequencies are @xmath118khz and @xmath119hz . ] the inset of fig . [ fig : dynamik+ft ] suggests that this irregular oscillation after @xmath115 is the result of a superposition of several frequencies . here a segment of the fourier spectrum of the gap dynamics is shown . the spectrum is khz . the arrows mark the transition time @xmath115 . ] composed of a series of sharp peaks in the range of @xmath120pev to about @xmath121pev , which can each be assigned to a corresponding quasiparticle state , i.e. , @xmath122 . the main peaks at the lower end of this series correspond to the frequency of the initial damped oscillation and to the dominant frequencies of the irregular dynamics afterwards . their values are given by the quasiparticle energies closest to the fermi level . these lie in the vicinity of a quasiparticle subband minimum . the corresponding frequencies are thus given by @xmath123 , where @xmath124 is the @xmath9 quantum number referring to the state with minimal quasiparticle energy , i.e. , the state at the fermi energy . the other peaks belong to higher quasiparticle states and decrease continuously with increasing energy . while the qualitative picture of the gap dynamics is the same for all investigated systems , the quantitative values of the features mentioned above crucially depend on the system parameters . on the one hand the ground state gap and thus the mean value of the oscillation and its frequency contributions strongly depend on the perpendicular confinement @xmath125 ( due to the size - dependent superfluid resonances @xcite ) and on the scattering length @xmath27 . the transition time , on the other hand , increases with decreasing parallel confinement @xmath126 i.e . , with increasing system length as can be seen in fig . [ fig : uebergangszeit_entwicklung ] . here the gap dynamics is shown for the same perpendicular confinement and excitation as in fig . [ fig : dynamik+ft ] but for increasing system length , i.e. , decreasing @xmath127 ( from bottom to top ) . figure [ fig : uebergangszeit_entwicklung ] shows that the transition time @xmath115 moves to larger times as the length of the system increases . in addition a revival of the oscillation can be seen for the two largest systems with @xmath128hz and @xmath129hz , which for smaller systems would occur after the breakdown . a quantitative analysis of the transition time for different perpendicular confinements @xmath125 over a wide range of parallel confinements is shown in fig . [ fig : uebergangszeit ] . here the transition time is plotted against the inverse parallel confinement frequency @xmath130 . one can see that @xmath115 is independent of the size of the gas in the @xmath7-@xmath8-plane , since the values for every @xmath125 lie on the same curve . it is only influenced by the confinement in @xmath9-direction , where a linear dependence on @xmath131 can be observed . this is in full agreement with the behavior found for superconducting quantum wires @xcite . even a beating - like pattern was found for thin quantum wires which corresponds to the revivals seen in fig . [ fig : uebergangszeit_entwicklung ] . compared to the prediction from the linearized theory ( dashed curve ) . ] to investigate the smooth regime of the gap dynamics fig . [ fig : resonant_vergleich ] shows calculations for a rather large system length and two different perpendicular confinements . for this case of large lengths in ref . @xcite it was found that thick quantum wires exhibit a damping of the gap oscillation given by a power law @xmath132 with @xmath133 for resonant systems and @xmath134 for off - resonant ones . however , thin quantum wires were found to differ from this power law showing a more irregular oscillation but still a rather fast decay of the gap oscillation when resonant subbands are present . khz ; red curve ) and an off - resonant system ( @xmath135khz ; blue curve ) ; in both cases @xmath136hz . inset : fourier spectra . ] figure [ fig : resonant_vergleich ] shows that this situation applies to ultracold fermi gases as well . here a calculation of the gap dynamics is shown for a resonant system ( upper , red curve ) , which is again characterized by the same perpendicular confinement as in figs . [ fig : dynamik+ft ] and [ fig : uebergangszeit_entwicklung ] , as well as for a system far away from resonance ( lower , blue curve ; @xmath135khz ) . the excitation is the same as before and the parallel confinement frequency is chosen as @xmath137hz , which corresponds to a rather long cloud . it can be seen that both systems show an initial decay of the gap oscillation until a minimal amplitude is reached . in the resonant case this initial decay is rather strong and fast . here , the amplitude of the oscillation falls close to zero . afterwards it exhibits revivals until the smooth oscillation breaks down . in contrast , the off - resonant system shows an only moderate , comparatively slow decay of the oscillation , which after a short time exhibits a nearly constant amplitude . thus , on the one hand the decay of the oscillation is much stronger in the resonant than in the off - resonant case . on the other hand revivals and a beating like pattern occur for the resonant case before the breakdown while systems far away from resonance exhibit a nearly constant oscillation amplitude . the inset of fig . [ fig : resonant_vergleich ] shows that these different temporal evolutions correspond to different fourier spectra . here the fourier transforms of the resonant ( positive @xmath8-axis ) and the off - resonant ( negative @xmath8-axis ) system are shown . the resonant spectrum is composed of several strong components at the lower end and weaker peaks towards higher energies . in contrast , the off - resonant spectrum it is shifted due to the smaller gap contains only one dominant frequency part at low energies while the rest of the spectrum is strongly suppressed . thus , resonant systems on the one hand correspond to spectra with several strong modes and weaker high - frequency parts . off - resonant systems on the other hand are strongly dominated by one single mode with only minor contributions from other energies . the features described above the damped oscillation , the breakdown , and the irregular dynamics will be explained in the following section . as mentioned before , they also occur in bcs - superconductors in a very similar way , where cigar - shaped fermi gases correspond to thin , short quantum wires . the following explanations therefore apply to both ultracold fermi gases and confined bcs - superconductors thus showing the close relation between these systems . to analyze the mechanisms underlying the gap dynamics and its features we introduce a linearized set of equations of motion . this can be derived by neglecting all terms of second and higher order in the quasiparticle excitations ( this is valid due to the weak excitation investigated in this paper , which leads to @xmath138 ) . in doing so , eq . can be neglected since by inserting eqs . and into this equation only terms of at least second order in the normal excitations or products of anomalous and normal excitations contribute , which are to be neglected in the linearized case . khz and @xmath119hz . ] equation reduces to a closed set of equations for the anomalous excitations which , by performing the derivative in time of one equation and inserting the other , can be separated into its real and imaginary parts leading to @xmath139\chi_k , \hspace{0.01 cm } \label{eq : bwg - gl - lin } \end{aligned}\ ] ] where again all terms nonlinear in the quasiparticle expectation values have been neglected . this equation describes a set of linearly coupled harmonic oscillators with the uncoupled frequencies @xmath140 and @xmath141 the coupling strength of the oscillators @xmath142 with @xmath143 therein is weak due to the in this case mostly small matrix elements @xmath144 . the shift of the eigenfrequencies of the coupled system [ eq . ] with respect to the uncoupled frequencies [ eq . ] should therefore be small , as should be the shift of the uncoupled frequencies compared to the bare ones @xmath145 . figure [ fig : linear_vergleich ] shows the dynamics of the bcs gap obtained from eq . compared to the full dynamics . , in the dynamical calculations ( i.e. , in eqs . , and ) the coupling has to be slightly lowered by a small amount , i.e. , @xmath146 . it should be mentioned that the full ( nonlinear ) equations never lead to divergences . ] the parameters correspond to the system shown in fig . [ fig : dynamik+ft ] and are exemplary for all investigated systems . the linearized equations clearly reproduce the full dynamics and all its features in very good agreement . the inset shows that the positions as well as the strengths of most of the frequencies in the fourier spectra are well described by this approximation . the spectrum corresponding to the full equations of motion ( upper , red curve ) and the one corresponding to the linearized equation ( lower , black curve ) show only slight differences . only one low lying weak frequency component close to zero , which is present in the linearized version ( outside the range shown in the inset of fig . [ fig : linear_vergleich ] ) as well as weak side peaks that occur adjacent to every main peak in the full dynamics are not fully reproduced . the latter can be attributed to the nonlinear couplings . their influence on the dynamics , however , is obviously negligible . the main features of the gap dynamics can thus be explained on the basis of eq . , which can be solved analytically . the analytic solution can be expressed in terms of a linear superposition of simple ( co)sine - oscillators . the corresponding frequencies are determined by the uncoupled oscillator frequencies @xmath147 and the coupling @xmath142 . they are thus completely fixed by the the system parameters and the bcs gap , but they are independent of the initial conditions . in the considered case of weak coupling the coupled spectrum is only slightly shifted compared to the uncoupled frequencies . the eigenfrequencies of the coupled dynamics are thus approximately given by twice the quasiparticle energies , as observed above . the amplitudes of the different eigenmodes of the coupled system are determined by the initial values of the dynamics and thus depend on the details of the excitation . in general , one observes that each of the quasiparticle oscillators @xmath148 carries strong contributions from oscillators in the vicinity of its uncoupled frequency @xmath147 and in areas of a high density of states . these areas are located close to the minimum of a quasiparticle subband ( see section [ sec : formalism ] ) , i.e. , near @xmath149 . in addition one observes that the contributions from each oscillator to the lowest energies are mostly in phase while these to higher energies are more and more out of phase . a sum of all quasiparticle oscillators , which according to eq . yields the bcs gap , thus leads to dynamics dominated by low - lying frequencies at a quasiparticle band minimum . these can be understood as cumulative peaks created by collective oscillations of all quasiparticles in the system . all spectra obtained by the full equations indeed show such a spectrum with rather dense , strong frequency contributions near @xmath150 and relatively widespread suppressed frequency contributions at higher energies . the explanations given above have shown that the gap dynamics can be understood as a linear superposition of quasiparticle oscillations which themselves are given by a sum of simple oscillators . to finally explain the main features of the gap dynamics in the time domain the damping , the transition and the irregular oscillation one can therefore use an even simpler picture of a set of independent cosine - oscillators with the frequencies @xmath147 . due to the abrupt excitation all these oscillators start in phase at maximum deflection . starting to oscillate they will soon dephase . a sum of all oscillators ( i.e. , the gap ) thus performs a damped oscillation ( cf . @xcite ) . here systems with several strong frequency contributions i.e . , resonant systems show a rather fast and persistent damping since a large part of the cumulative amplitude is able to dephase . off - resonant systems in contrast exhibit only a slight decay of the oscillation since the main part of the cumulative oscillation is carried by one single mode . proceeding in time the damped oscillation continues until all oscillators are completely dephased and the amplitude of the oscillation is minimal . then the oscillators start to rephase , the amplitude grows and the oscillation reappears ( see fig . [ fig : uebergangszeit_entwicklung ] ; for off - resonant systems this effect is strongly suppressed since one single frequency dominates the spectrum ) . as soon as the first adjacent pair of oscillators goes back in phase again this beating - like pattern is interrupted . at this moment a spike in the cumulative oscillation indicates the breakdown of the regular oscillation and thus the transition time @xmath115 . afterwards all other frequencies rephase successively and create a rapid sequence of spikes which leaves a picture of an irregular oscillation . the preceding argumentation suggests that the time of this breakdown should be inversely proportional to the maximum spacing of adjacent quasiparticle energies . strictly speaking the transition times are found to be determined by adjacent quasiparticles from the same subband @xcite , i.e. , @xmath151 it thus increases @xmath152 with decreasing parallel confinement frequency since the energy spacing of the atomic spectrum then decreases . this is in full agreement with the relation found before . in fact , eq . gives exactly the linear curve shown in fig . [ fig : uebergangszeit ] . this indicates that the breakdown of the smooth initial oscillation of the bcs gap indeed is due to adjacent frequencies rephasing in time . as a final example the gap dynamics of a rather large system with @xmath153hz and @xmath154khz again after a sudden change of the scattering length from @xmath155 nm to @xmath117 nm is shown in fig . [ fig : dynamik_gross ] . this corresponds to a gas with @xmath156 and @xmath157 and is thus on an experimentally accessible length - scale @xcite . the fermi energy is chosen as @xmath158 which corresponds to @xmath159 atoms in the trap . nm to @xmath117 nm ; inset : fourier transform of the gap dynamics . the confinement frequencies are @xmath160khz and @xmath161hz . ] figure [ fig : dynamik_gross ] shows that the qualitative behavior of the gap dynamics is the same as for the smaller systems : a slowly decaying oscillation of the gap occurs . the transition time @xmath162 calculated from eq . is in good agreement with a small bump in the curve at @xmath163ms , the first deviation from a smooth oscillation . afterwards more and more deviations occur and the gap dynamics becomes successively irregular . when looking at the damping of the gap oscillation one can see that although the system is resonant one subband is close to the fermi energy the strength of the damping is situated somewhere between the resonant and the off - resonant case of fig . [ fig : resonant_vergleich ] . this is due to the weak perpendicular confinement and therefore large number of states contributing to the condensate : compared to the overall number of relevant states the resonant ones give only a small contribution to the gap . for larger systems the resonances are thus less pronounced @xcite . the fourier transform in the inset of fig . [ fig : dynamik_gross ] shows a familiar picture , too , with strong contributions at the lower end of the spectrum and successively decaying peaks towards higher energies . the difference with respect to the spectra shown before is on the one hand the high density of peaks for the large system . this is due to the weaker confinement and thus higher density of bare and quasiparticle states . on the other hand the gap in the fourier spectrum and thus the main frequency of the oscillation is comparatively small . this is because of the larger ratio of the perpendicular to the parallel length @xmath164 . with the fermi energy fixed at @xmath165 this leads to a comparatively low particle density of the trapped gas and thus a weaker condensate and a smaller gap . in conclusion , we have calculated the higgs amplitude dynamics in the bcs phase of an ultracold @xmath166li gas confined in a cigar - shaped trap . the dynamics is induced by a quantum quench resulting from a sudden change of an external magnetic field . we have shown that the amplitude of the spatially averaged gap performs a damped oscillation breaking down after a certain time @xmath115 , which is determined by the parallel confinement frequency @xmath127 , i.e. , by the length of the cloud . afterwards a rather irregular oscillation involving many different frequencies occurs . we have investigated the influence of the confinement on the gap dynamics and the impact of the size - dependent superfluid resonances on its qualitative behavior . it turned out that in the case of a resonant system , i.e. , a system where the fermi energy is close to a subband minimum , the dynamics of the order parameter exhibits a strong damping and , for sufficiently long systems , a revival before eventually the irregular regime is reached . in contrast , in an off - resonant system the damping is much less pronounced and the oscillation before the transition to the irregular regime is mainly determined by a single frequency . by analyzing the linearized version of the equations of motion for the quasiparticle excitations we were able to interpret the observed features of the dynamics . it turned out that for the excitations studied in this paper the linearized equations well reproduce the dynamical behavior of the gap , except for some slight details resulting from the nonlinearities in the full equations of motion . from the linearized model it becomes evident that the system approximately behaves like a set of weakly coupled harmonic oscillators . the frequencies as well as the couplings of these oscillators are completely determined by the system parameters after the excitation while the amplitudes of the different eigenmodes depend on the details of the excitation . the analysis revealed in particular that the transition time to the irregular dynamics is directly related to the energy separation of the one - particle energies while the differences between resonant and non - resonant systems is caused by the different densities of states and coupling efficiencies close to the subband minima .

the higgs amplitude mode of the order parameter of an ultracold confined fermi gas in the bcs regime after a quench of the coupling constant is analyzed theoretically . characteristic features are a damped oscillation which at a certain transition time changes into a rather irregular dynamics . we compare the numerical solution of the full set of nonlinear equations of motion for the normal and anomalous bogoliubov quasiparticle excitations with a linearized approximation . in doing so the transition time as well as the difference between resonant systems , i.e. , systems where the fermi energy is close to a subband minimum , and off - resonant systems can be well understood and traced back to the system and geometry parameters .

You are an expert at summarizing long articles. Proceed to summarize the following text: waves propagating in a curved spacetime develop `` tails '' . in particular , it is well established that the _ dominant _ late - time behaviour of massless fields propagating in black - hole spacetimes is a power - law tail . price @xcite was the first to analyze the mechanism by which the spacetime outside a ( nearly spherical ) collapsing star divests itself of all radiative multipole moments , and leaves behind a schwarzschild black hole ; it was demonstrated that all radiative perturbations decay asymptotically as an inverse power of time . physically , these inverse power - law tails are associated with the backscattering of waves off the effective curvature potential at asymptotically far regions @xcite . the analysis of price was extended by other authors . bik @xcite generalized the analysis and studied the dynamics of a scalar field in a _ charged _ reissner - nordstrm spacetime . he also found an asymptotic inverse power - law decay of the field , with the _ same _ power indices as in the schwarzschild spacetime ( with the exception of the _ extremal _ reissner - nordstrm black hole @xcite ) . in a brilliant work , leaver @xcite demonstrated that the late - time tail can be associated with the existence of a branch cut in the green s function for the wave propagation problem . gundlach , price , and pullin @xcite showed that these inverse power - law tails also characterize the late - time evolution of radiative fields at future null infinity , and at the black - hole outer horizon . furthermore , they showed that power - law tails are a genuine feature of gravitational collapse the existence of these tails was demonstrated in full non - linear numerical simulations of the spherically symmetric collapse of a self - gravitating scalar field @xcite ( this was later reproduced in @xcite ) . our current understanding of the late - time tail is , however , somewhat unsatisfactory . the ( _ leading order _ ) power - law tails in black - hole spacetimes are well established @xcite , but the resultant formulae are only truly useful at very _ late _ times . in a typical evolution scenario there is a considerable time window in which the signal is no longer dominated by the quasinormal modes @xcite , but the leading order power - law tail has not yet taken over @xcite . the purpose of this paper is to derive analytic expressions for the _ higher - order corrections _ which `` contaminate '' the well - known power - law tail in a spherically symmetric gravitational collapse . the determination of these higher - order terms is important from several points of view : the analyses of bik @xcite and gundlach et . @xcite established the fact that the leading - order power - law tail is _ universal _ in the sense that it is _ independent _ of the black - hole electric charge ( i.e. , the power index in a _ charged _ reissner - nordstrm spacetime was shown to be identical with the one found by price @xcite for the neutral schwarzschild black hole ) . this observation begs the question : what fingerprints ( if any ) does the black - hole electric charge leave on the field s decay ? moreover , the calculation of higher - order corrections to the leading order power - law tail is also of practical importance ; this is especially crucial for the determination of the power index from numerical simulations . the dominant inverse power - law tail is _ `` contaminated '' _ by higher - order terms , whose effect become larger as the aveliable time of integration decreases . the precise power index is expected only at infinitely - late time . thus , in practice , the _ limited _ time of integration introduces an inherent error in the determination of the power index . the only systematic approach to _ quantify _ the errors which are introduced by the finite integration time is to study _ higher - order corrections_. if one computes the contaminated part of the late - time tail , then the ratio of the corrections to the leading order term is a systematic , quantitative , indication of the error caused by the _ finite_-time numerical calculation . these questions and several others are addressed in the present paper . the plan of the paper is as follows . in sec . [ sec2 ] we give a short description of the physical system and formulate the evolution equation considered . in sec . [ sec3 ] we give an analytical description of the late - time evolution of scalar fields in black - hole spacetimes . in sec . [ sec4 ] we confirm our analytical results by numerical simulations . we conclude in sec . [ sec5 ] with a brief summary of our results and their implications . we consider the evolution of a spherically symmetric massless scalar field in a spherically symmetric charged background ( a collapsing star or a fixed black hole ) . the external gravitational field of a spherically symmetric charged object of mass @xmath4 and charge @xmath5 is given by the reissner - nordstrm metric @xmath6 using the tortoise radial coordinate @xmath7 , which is defined by @xmath8 , the line element becomes @xmath9 where @xmath10 . the wave equation @xmath11 for the scalar field in the black - hole background is @xmath12 where @xmath13 in terms of the tortoise coordinate @xmath7 and for @xmath14 the curvature potential eq . ( [ eq4 ] ) reads @xmath15 the general solution to the wave - equation ( [ eq3 ] ) can be written as a series depending on two arbitrary functions @xmath16 and @xmath17 @xcite @xmath18}\ .\end{aligned}\ ] ] here @xmath19 is a retarded time coordinate and @xmath20 is an advanced time coordinate . for any function @xmath21 , @xmath22 is the @xmath23th derivative of @xmath24 ; negative - order derivatives are to be interpreted as integrals . the first two terms in eq . ( [ eq6 ] ) represent the zeroth - order solution ( with @xmath25 ) . the star begins to collapse at a retarded time @xmath26 . the world line of the stellar surface is asymptotic to an ingoing null line @xmath27 , while the variation of the field on the stellar surface is asymptotically infinitely redshifted @xcite . this effect is caused by the time dilation between static frames and infalling frames . a static external observer sees all processes on the stellar surface become `` frozen '' as the star approaches the horizon . thus , he sees all physical quantities approach a constant . we therefore make the explicit assumption that after some retarded time @xmath28 on @xmath27 . this assumption has been proven to be very successful @xcite . we begin with the first stage of the evolution , i.e. , the scattering of the field in the region @xmath29 . the first two terms in eq . ( [ eq6 ] ) represent the primary waves in the wave front , while the sum represents backscattered waves . the interpretation of these integral terms as backscatter comes from the fact that they depend on data spread out over a _ section _ of the past light cone , while outgoing waves depend only on data at a fixed @xmath30 @xcite . after the passage of the primary waves there is no outgoing radiation for @xmath31 , aside from backscattered waves . this means that @xmath32 . hence , for a large @xmath7 at @xmath33 , the dominant term in eq . ( [ eq6 ] ) is @xmath34 . the functions @xmath35 satisfy the recursion relation @xmath36 for @xmath37 , where @xmath38 , and @xmath39 . thus , one finds @xmath40g^{(-1)}(u_1 ) \big[1+o(m / y ) \big]\ .\ ] ] this is the dominant backscatter of the primary waves . with this specification of characteristic data on @xmath41 , we shall next consider the asymptotic evolution of the field . we confine our attention to the region @xmath42 , @xmath43 . in this region the spacetime is approximated as flat @xcite . ( the validity of this approximation is ultimately justified by numerical simulations ) . thus , the solution for @xmath44 can be written as @xmath45 comparing eq . ( [ eq8 ] ) with the initial data on @xmath41 eq . ( [ eq7 ] ) , one finds @xmath46 where @xmath47 for late times @xmath48 we can expand @xmath49 and similarly for @xmath50 . using these expansions we can rewrite eq . ( [ eq8 ] ) as @xmath51}\ , \ ] ] where the coefficients @xmath52 are those given in @xcite . using the boundary conditions for small @xmath53 ( regularity as @xmath54 , at the horizon of a black hole , or at @xmath55 for a stellar model ) , one finds that at late times the terms @xmath56 and @xmath57 must be of the same order ( see @xcite for additional details ) . thus , we conclude that @xmath58 and @xmath59 we therefore find that the late - time behaviour of the field for @xmath60 is @xmath61 \big[1+o(m / t)\big ] \ . \end{aligned}\ ] ] this is the late - time behaviour of the field at a fixed radius . it is straightforward to integrate eq . ( [ eq3 ] ) using the method described in @xcite . we have used , however , a modified version of the numerical code used in @xcite , which is essential to achieve the extremely high accuracy needed for the computation ( see @xcite for additional details ) . the late - time evolution of the scalar field is independent of the form of the initial data used . the results presented here are for a gaussian pulse on @xmath62 @xmath63^{2 } \right \}\ , \ ] ] with a center at @xmath64 and a width @xmath65 . the black - hole mass is set equal to @xmath66 ; this corresponds to the freedom to rescale the coordinates by an overall length scale . the temporal evolution of the field at a fixed radius @xmath67 is shown in the top panel of fig . the dominant _ power - law _ fall off is manifest at asymptotic late times . in order to study the contamination effect of higher - order terms ( [ eq14 ] ) ] , we use the notion of a _ local power index _ @xmath68 , defined by @xmath69 @xcite . taking cognizance of eq . ( [ eq14 ] ) we find @xmath70 the approach of the local power index to its well - known asymptotic value @xmath71 is depicts in the bottom panel of fig . the plot shows that @xmath72 from above , with a qualitative agreement with eq . ( [ eq16 ] ) . in order to establish _ quantitatively _ the physical picture presented in sec . [ sec3 ] , we define the quantity @xmath73 . figure [ fig2 ] . depicts @xmath74 as a function of @xmath75 at three surfaces of constant radius @xmath76 , and @xmath77 ( from bottom to top ) . the numerical result @xmath78 ( independently of the value of @xmath7 ) is in excellent agreement with the _ analytically _ predicted behaviour @xmath79 [ see eq . ( [ eq16 ] ) ] . thus , fig . [ fig2 ] . establishes the existence of the contamination term of order @xmath0 . it should be noted that the value of @xmath80 used in this figure is @xmath81 rather than the theoretical value @xmath82 . this slight deviation from the theoretical value is expected due to the corrections of order @xmath83 in the expression for @xmath68 eq . ( [ eq16 ] ) . the purpose of this paper was to derive analytic expressions for the _ higher - order corrections _ which `` contaminate '' the well - known power - law tail in a spherically symmetric gravitational collapse . we have shown , both analytically and numerically , that the dominant correction dies off at late times as @xmath0 . this late - time decay of the contamination is much _ slower _ than has been considered so far @xcite ( see the discussion in appendix b ) . aside from being _ theoretically _ important , the result eq . ( [ eq14 ] ) is also of _ practical _ importance . it follows that an ` exact ' ( numerical ) determination of the power index demands extremely long integration times . the most accurate method for determining the power index experimentally applies to the concept of the _ local _ power index . in this way one discard the relatively large contamination which characterizes the early stages of the evolution . still , it follows from eq . ( [ eq16 ] ) that a determination of the power index to within @xmath1 requires an integration time of order @xmath84 . the dominant power - law tail is known to be universal in the sense that it is _ independent _ of the black - hole parameters @xcite . we have shown , however , that this universality is removed once we consider higher - order corrections terms the leading order fingerprint of the black - hole electric _ charge _ behaves as @xmath3 . * acknowledgments * i thank tsvi piran for discussions . this research was supported by a grant from the israel science foundation . it follows from eqs . ( [ eq8 ] ) , ( [ eq9 ] ) , and ( [ eq12 ] ) that the asymptotic behaviour of the field at future null infinity @xmath85 ( i.e. , at @xmath86 ) is @xmath87 we finally consider the behaviour of the field at the black - hole outer horizon @xmath88 . as @xmath89 the curvature potential eq . ( [ eq4 ] ) is exponentially small , and the general solution to eq . ( [ eq3 ] ) can be written as @xmath90 . on @xmath91 we take @xmath92 ( for @xmath93 ) . thus , @xmath94 must be a constant , and with no loss of generality we can choose it to be zero . we next expand @xmath95 for @xmath96 as @xmath97 in order to match the @xmath98 solution eq . ( [ eqa2 ] ) with the @xmath99 solution eq . ( [ eq14 ] ) , we make the ansatz @xmath100 $ ] for the solution in the region @xmath98 and @xmath96 . in other words , we assume that the solution in the @xmath98 region has the same late time @xmath75 dependence as the @xmath14 solution . this assumption has been proven to be very successful for the leading order behaviour of both neutral @xcite and charged @xcite fields . using this assumption , one finds @xmath101 $ ] , where @xmath102 is a constant . thus , the asymptotic behaviour of the field at the black - hole horizon is @xmath103\ .\ ] ] the first attempt to calculate higher - order corrections to the dominant power - law tail was made by andersson @xcite ( in the context of the schwarzschild spacetime ) . this analysis was based on an _ approximated _ curvature potential in the region far away from the black hole [ see eq . ( 26 ) of @xcite ] . while this approximated potential simplifies the analysis , it actually _ misses _ the genuine leading order corrections terms . the leading order correction in @xcite was found to be of order @xmath104 . this ( artificial ) result is caused by the fact that terms of order @xmath105 ( and smaller ) were totally neglected by the approximated curvature potential used in @xcite . moreover , the approximated approach presented in @xcite , _ if _ extended along the same lines to a reissner - nordstrm spacetime would imply that the influence of the black - hole electric charge on the late - time tail vanishes _ identically _ ( i.e. , it would vanish to _ any _ order in @xmath106 ) . this result is again a direct consequence of the fact that the approximated approach of @xcite does not take into account curvature terms of order @xmath107 ( and @xmath108 ) which appear in the ( exact ) curvature potential . temporal evolution of the scalar field , evaluated at @xmath67 in a schwarzschild spacetime with @xmath66 . the initial data is a gaussian distribution with @xmath109 and @xmath110 . asymptotic _ power - law fall off is manifest at late times ( top panel ) . the bottom panel depicts the evolution of the local power index @xmath111 . the power index approaches the well - known asymptotic value @xmath112 at late times.,width=642 ] the time evolution of the quantity @xmath73 , evaluated at @xmath113 , and @xmath77 ( from bottom to top ) . the asymptotic value @xmath78 is in excellent agreement with the _ analytically _ predicted behaviour @xmath79 at late times . this result establishes the existence of the contamination term of order @xmath0.,width=642 ]

it is well known that the late - time behaviour of gravitational collapse is _ dominated _ by an inverse power - law decaying tail . we calculate _ higher - order corrections _ to this power - law behaviour in a spherically symmetric gravitational collapse . the dominant `` contamination '' is shown to die off at late times as @xmath0 . this decay rate is much _ slower _ than has been considered so far . it implies , for instance , that an ` exact ' ( numerical ) determination of the power index to within @xmath1 requires extremely long integration times of order @xmath2 . we show that the leading order fingerprint of the black - hole electric _ charge _ is of order @xmath3 .

You are an expert at summarizing long articles. Proceed to summarize the following text: in ashtekar s theory of gravity , a @xmath2-connection captures the extrinsic curvature of a space - like leaf in a time - transversal foliation , and the intrinsic geometry of the leaf is given by a tetrad @xcite . it is shown that the einstein - hilbert functional and einstein equations can be written in terms of @xmath2-connections and tetrads , thus these variables together recast einstein s theory of gravity . by rewriting einstein s gravity with the connection variables and tetrad variables , one could attempt to quantise gravity via the hamiltonian formalism , and obtain a theory of quantum gravity @xcite . the reader may refer to thiemann s introductory @xcite . this article offers an alternative view of the space of connections to ones that appear in other loop quantum gravity literatures , and proposes a semi - classical limit using strict @xmath1-algebraic deformation quantisation formalism @xcite . to take the connections as dynamic variables in a quantum theory , one studies wave functions , also known as probability amplitudes , on the space of connections . unfortunately , the lack of a measure on such a space poses the first challenge , since in such case probabilities can not be defined . one typical solution to obtain a measure on the connection space comes from spaces of progressively refined cylindrical functions . in another description , one uses a finite set of curves to probe the space of @xmath0-connections to obtain a finite dimensional manifold that depend on the sets of curves . by successively refining the finite sets of curves , such as successive triangulation of the manifold , one obtains a pro - manifold that extends the original space of connections to the so - called space of generalised connections @xcite . as a step to quantising gravity in the ashtekar framework , there are recent developments of describing such an extended space of connections using a spectral triple in noncommutative geometry @xcite , which captures the geometry of the space of generalised connections as operators on a hilbert space . while the geometries of the base manifold and the space of @xmath0-connections on it are in theory retained , the construction of the spectral triple is considered too discrete to practically allow one to recapture the geometry of the base manifold and its @xmath0-connections . to rid the discrete description of using finite sets of embedded curves , this article proposes an alternative approach to smoothly probe the space of connections using the tangent groupoid . and then followed by an application of strict deformation quantisation in @xmath1-algebraic formalism , a possible semi - classical limit of can be obtained . the purposes of this article are to present an alternative idea to the studies of loop quantum gravity with the goal of obtaining a semi - classical limit , which is what loop quantum gravity still lacks today . this article contains six small sections , they are arranged as follows : section [ sec2 ] discusses the traditional way of putting a measure on the space of connections in loop quantum gravity literature , and the short coming of such a method . in section [ sec3 ] , we review the definitions of a lie groupoid , a lie algebroid , and tangent groupoid of a lie groupoid in an elementary way . section [ sec4 ] shows that how one uses the tangent groupoid as a tool to model the space of connections in a smooth way , and discuses the corresponding gauge action . section [ sec5 ] is the first attempt of deforming connections into noncommutating operators acting on a hilbert space . section [ sec6 ] introduces the notion of strict deformation quantisation in @xmath1-algebra formalism , and an important theorem by landsman @xcite , which states that a tangent groupoid defines a strict deformation quantisation . hence , using this formalism , we can deformation quantise @xmath0-connections . finally , section [ sec7 ] is an outlook that summarises the article and the current state of work toward obtaining a semi - classical limit in loop quantum gravity . we start by requiring that @xmath0 to be a semi - simple , simply connected , compact lie group , such as @xmath2 . and @xmath3 to be the corresponding lie algebra of @xmath0 . @xmath4 is the exponential map from @xmath3 to @xmath0 . the space of smooth @xmath0-connections @xmath5 over a manifold @xmath6 is the space of @xmath3-valued @xmath7-forms over @xmath6 . given a smooth manifold @xmath6 and a smooth ( compact ) curve @xmath8 in @xmath6 , one obtains a map @xmath9 from the space of smooth @xmath0-connections @xmath5 to @xmath0 @xmath10 given by taking the holonomy @xmath11 of each connection @xmath12 along the curve @xmath8 . here it is assumed that there is a fixed local trivialisation of the principal bundle , so that @xmath5 is realised as a vector space . suppose that there is a smoothly embedded finite graph @xmath13 in @xmath6 ( edges are smooth , and the number of them is finite ) , one can repeat the holonomy evaluation to obtain a map from @xmath5 to multiple copies of @xmath0 s , one for each edge . thus , one obtains @xmath14 where @xmath15 denotes the number of edges of the graph @xmath13 . [ agn : prop : denserange ] for any finite graph @xmath13 , the map @xmath16 is a surjection . while the space of smooth @xmath0-connections @xmath5 lacks structures , such as a measure , it surjects to @xmath17 , a compact measure space . it is done by forgetting the values of the connections outside the edges of the graph . hence , a lot of information is lost . however , one can imagine that there is a collection of finite graphs with one finer than the other , such that the collection of graphs is dense in the manifold @xmath6 in a certain sense . therefore , at every small neighbourhood in the manifold , there is an edge from the collection of graphs in the neighbourhood to probe the holonomies of the connections . we do not define the notion of a * directed system * of finite graphs , but to say that it is defined naturally by the associated groupoid of a directed finite graph @xcite . from which , there is associated a directed system of compact measure spaces @xmath18 . to put the above description in a mathematical context , we state denote by @xmath19 the projective limit @xmath20 . suppose that there is a system of of smoothly embedded finite graphs @xmath21 , such that the set of vertices is dense in @xmath6 and every neighbourhood of a point @xmath22 contains edges that span the vector space @xmath23 . then there exists an embedding @xmath24 from the property of the product ( tychonoff ) topology , one has that [ agn : prop : denserange ] let @xmath25 be a directed system of finite graphs in @xmath6 . then the image of @xmath5 under @xmath26 is dense in @xmath19 . that is @xmath27 this compactification procedure depends on the system of graphs used in probing the connection space . we give some examples of graph systems that provide good compactifications . 1 . [ agn : ex : triangulation ] let @xmath28 be a triangulation of @xmath6 and @xmath29 be the graph consisting of all the edges in this triangulation with any orientation . let @xmath30 denote the triangulation obtained by barycentric subdivision of each of the simplices in @xmath28 @xmath31 times . the graph @xmath32 is the graph consisting the edges of @xmath30 with consistent orientation . in this way @xmath33 is a directed system of finite graphs , and @xmath5 densely embeds into @xmath34 . [ agn : ex : dlattice ] let @xmath29 be a finite , @xmath35-dimensional lattice in @xmath6 and let @xmath36 denote the lattice obtained by subdividing each cell in @xmath29 into @xmath37 cells . correspondingly , let @xmath32 denote the lattice obtained by repeating @xmath31 such subdivisions of @xmath38 . in this way @xmath39 is a directed system of finite graphs , and @xmath5 densely embeds into @xmath40 . moreover , @xmath19 is a compact measure space . therefore , such a procedure of surjecting @xmath5 to a coarse approximation @xmath41 , then consider the limit of the approximations provide an extension of the space @xmath5 to a compact measure space @xmath19 . the space @xmath19 is called the space of generalized connections . now it is possible to consider probability amplitudes on the space of generalized connections , and proceed to canonical quantisation . the drawback of this compactification is that the original space of connections @xmath5 is forever lost in @xmath19 , consequently obtaining a semi - classical limit from quantisation on @xmath19 is impossible . the source of the problem comes from probing the space @xmath5 with finite graphs , which are very rigid objects that can not be perturbed . one can understand or abstract this process of compactifaction of @xmath5 as probing @xmath5 with a groupoid , where for the case of a graph @xmath13 , the groupoid is the associated fundamental groupoid . it is a finitely generated groupoid , which is very discrete and can not be perturbed . however , one can replace this discrete groupoid with a smooth groupoid , say a lie groupoid . this article proposes the use of the tangent groupoid . [ groupoid ] a * groupoid * @xmath42 is a ( small ) category in which every morphism is invertible . that is , a set of morphisms @xmath42 together with a set of objects @xmath43 such that * there exist surjective structure maps , called the source and range maps @xmath44 * there exists an injection , called the identity inclusion , @xmath45 * there exists a partially - defined associative composition @xmath46 with identities @xmath47 , and * there exists an inversion map @xmath48 with the usual properties . [ liegroupoid ] a * lie groupoid * is a groupoid @xmath49 with smooth manifold structures on @xmath42 and @xmath43 such that @xmath50 are submersions , the inclusion of @xmath43 in @xmath42 as the identity homophism and the composition @xmath51 are smooth . [ liegroupoidex ] 1 . any lie group @xmath0 is a groupoid @xmath52 over the identity @xmath53 . 2 . the tangent bundle @xmath54 of a manifold @xmath55 forms a lie groupoid @xmath56 with the _ source _ and _ range _ maps @xmath57 given by @xmath58 for @xmath59 , the inclusion @xmath60 given by the zero section , and composition @xmath61 given by @xmath62 . 3 . the product @xmath63 forms a lie groupoid @xmath64 with the _ source _ and _ range _ maps @xmath65 given by @xmath66 , @xmath67 , the inclusion @xmath68 is given by the diagonal embedding , and the composition @xmath69 is given by @xmath70 . given two lie groupoids @xmath49 and @xmath71 , their direct product @xmath72 is also a lie groupoid . a * lie algebroid * on a manifold @xmath55 is a vector bundle @xmath73 over @xmath55 , which is equipped with a vector bundle map @xmath74(called the anchor ) , as well as with a lie bracket @xmath75_e$ ] on the space @xmath76 of smooth sections of @xmath73 , satisfying @xmath77_e = [ \rho \circ x,\rho \circ y],\ ] ] where the right - hand side is the usual commutator of vector fields on @xmath78 , and @xmath79_e = f[x , y]_e + ( ( \rho \circ x)f)y\ ] ] for all @xmath80 and @xmath81 . a lie algebroid is also a lie groupoid with groupoid product given by fibre - wise addition . [ algebroidex ] 1 . a lie algebra @xmath3 with its lie bracket is a lie algebroid over a point . 2 . the tangent bundle @xmath54 of a manifold @xmath55 defines a lie algebroid under the lie bracket of vector fields , and the anchor map @xmath82 is the identity . [ algebroidex3 ] for a lie groupoid @xmath83 , let @xmath84 be the normal vector bundle defined by the embedding @xmath85 , with bundle projection given by @xmath86 . identify the normal bundle by @xmath87 , the anchor map is given by @xmath88 . finally , by identifying @xmath89 with @xmath90 , equip @xmath84 with the lie bracket coming from @xmath91 . @xmath84 is a lie algebroid . 4 . following the construction in the first example . @xmath92 is the lie algebroid of the lie groupoid @xmath93 . the anchor map @xmath94 consist of bundle projection on @xmath54 and zero on @xmath3 . in example [ algebroidex ] above , @xmath84 is called the associated lie algebroid of the lie groupoid @xmath83 . it is in itself a groupoid over @xmath6 , similar to @xmath95 . given the associated lie algebroid of @xmath84 of the lie groupoid @xmath83 . there exists a unique local diffeomorphism @xmath96 . let @xmath83 be a lie groupoid with lie algebroid @xmath84 . the * tangent groupoid * @xmath97 of @xmath42 is the lie groupoid @xmath97 over the base @xmath98 $ ] , such that * as a set , @xmath99 $ ] ; * and @xmath84 and @xmath42 are glued together by the local diffeomorphism @xmath100 . we do not elaborate the definition of @xmath100 here , but rather attempt to illustrate it with an example below . [ tangentgroupoid ] 1 . the tangent groupoid @xmath101 of a lie group @xmath0 is just @xmath102 $ ] glued together with the exponential map . the tangent groupoid @xmath103 of a manifold @xmath55 is @xmath104 $ ] as a set . and the groupoids @xmath54 and @xmath105 are glued together such that for @xmath106 , then @xmath107 $ ] . we think of an element @xmath108 of @xmath109 $ ] as a geodesic starting at @xmath110 and ending at @xmath111 , and @xmath112 is the time it takes to travel from @xmath110 to @xmath111 in a given velocity . hence , it is considered a one dimensional object . in this section , we propose using a smooth groupoid to probe the connection space with , and the holonomies will be encoded by the smooth @xmath0-valued functions over the smooth groupoid . denote by @xmath113 the space of smooth functions from @xmath63 to @xmath0 and @xmath114 the space of smooth functions from @xmath54 to @xmath3 such that @xmath115 is linear for @xmath116 and each @xmath117 . + here we think of an element @xmath118 of @xmath113 as a holonomy presentation of a connection for paths described by @xmath109 $ ] . [ qconnection ] define the space of * q - connections * @xmath119 to be @xmath120 $ ] as a set . and @xmath114 and @xmath113 are glued together as @xmath121 for all @xmath122 $ ] . the definition is inspired by the following intuition . the two points @xmath110 and @xmath123 are connected by the geodesic @xmath124 for @xmath125 $ ] . and the holonomy along @xmath8 is some group element . as @xmath112 approaches to zero and the end point of @xmath8 shrinks to its starting point @xmath110 , the holonomy contribution gets closer to the identity element in the group . therefore , the infinitesimal of the geodesic @xmath8 at @xmath110 gives the infinitesimal change in the group ; so for a point @xmath126 in @xmath54 , one associates it an element in @xmath3 . one observes the following remarks . the gluing condition of @xmath119 implies that @xmath127 and @xmath128 where @xmath129 is the identity of @xmath0 . each @xmath130 gives rise to a @xmath3-valued 1-form in a unique way . hence , @xmath114 is naturally identified as the space of @xmath0-connections @xmath5 . as a result , there is an embedding @xmath131 @xmath113 forms a group under point - wise multiplication of @xmath0 , and @xmath114 forms a group under point - wise addition of @xmath3 . the product @xmath119 inherits from @xmath114 and @xmath113 is smooth . the proof follows from @xmath132 the q - connection space @xmath119 is a package that captures information about probing the @xmath0-connection space with the tangent groupoid . that is , an element @xmath118 of @xmath113 evaluated at @xmath133 gives the holonomy of a connection along the geodesic from @xmath110 to @xmath111 . this holonomy presentation has a natural gauge action given by applying a symmetry at the starting point @xmath110 , then evaluate the holonomy along the geodesic from @xmath110 to @xmath111 , and finally apply a reverse symmetry at @xmath111 . we make it formal by the following definition . denote by @xmath134 the set of smooth functions from @xmath55 to @xmath0 . define the * gauge action * of @xmath134 on @xmath113 by @xmath135 for @xmath136 and @xmath137 . and define the gauge action of @xmath134 on @xmath114 by @xmath138 for @xmath136 and @xmath116 . equation is the usual gauge action on @xmath0-connections . the following proposition shows that the gauge actions defined on @xmath139 and @xmath114 are compatible . the @xmath134 action on the q - connection space @xmath119 induced from equations , is smooth . suppose that @xmath140 . then @xmath141 the proof is complete . let @xmath142 denote the diffeomorphism group of @xmath55 . @xmath143 $ ] carries a smooth @xmath142 action given by @xmath144 subsequently , @xmath142 acts on the q - connection space @xmath119 smoothly via the induced action @xmath145 for @xmath146 . suppose that @xmath147 unitarily represents on some finite dimensional vector space , say without loss of generality @xmath148 . then @xmath0 is included in the matrix algebra @xmath149 as unitary matrices . let us fix an orientation on @xmath55 , hence a volume form . then one obtains the hilbert space @xmath150 that @xmath113 acts on by convolution @xmath151 where @xmath137 , @xmath152 , the dot @xmath153 is the unitary representation of @xmath0 . denote by @xmath154 the space of @xmath149-valued smooth functions on @xmath63 , thus @xmath155 and it acts on @xmath156 . elements of @xmath154 will again be denoted by @xmath118 . @xmath154 comes equipped with an involution given by the point - wise conjugate transpose of @xmath149 . the action of @xmath154 on @xmath156 gives rise to a noncommutative product on @xmath157 given by the convolution @xmath158 for @xmath159 . the space @xmath154 forms a @xmath160-algebra , and it is identified with the ideal of trace - class operators on the hilbert space @xmath150 . let @xmath161 denote the operator trace on @xmath156 , and @xmath162 is explicitly given by @xmath163 where @xmath164 is the matrix trace of @xmath149 . the linear functional @xmath165 is invariant under the gauge action of @xmath134 , as @xmath166 such a property is called * gauge invariant*. the group element @xmath167 represents the holonomy of a connection around a loop with base point @xmath110 . the functional @xmath162 being gauge invariant is parallel to the fact that loop variables being gauge invariant in loop quantum gravity . the package of @xmath154 acting on @xmath156 with gauge group @xmath168 resembles the noncommutative standard model , where the algebra is given by the the gauge group , the hilbert space is unchanged , and the resolvent of the dirac operator gives rise to an element of @xmath154 . a * system of haar measures * for a groupoid @xmath83 is a family of measures @xmath169 , where each @xmath170 is a positive , regular , borel measure on @xmath171 . for every lie groupoid @xmath83 , there exists a smooth system of haar measures @xmath169 , so that the convolution product with respect to the haar system , @xmath172 together with the star structure , @xmath173 give rise to a @xmath1-algebra structure on the space of continuous functions @xmath174 on @xmath42 . here we are not being specific on the @xmath1-norm and its closure . for our case of a lie groupoid , the different closures coincide . 1 . @xmath175 of a lie group @xmath0 is the group @xmath1-algebra . 2 . given a smooth system of haar measures @xmath169 , where each @xmath176 is a haar measure on @xmath177 , and two smooth functions @xmath178 on @xmath54 . the product @xmath179 defines , by continuity , a @xmath1-algebra structure on the space of continuous functions on @xmath54 , denoted @xmath180 . 3 . given a smooth system of haar measures @xmath169 , where each @xmath176 is a haar measure on @xmath6 , and two smooth functions @xmath181 on @xmath182 . the product @xmath183 defines , by continuity , a @xmath1-algebra structure on the space of continuous functions on @xmath63 , denoted @xmath184 . similarly , there is associated a groupoid @xmath1-algebra @xmath185 to @xmath103 , which can also be seen as gluing the groupoid @xmath1-algebras @xmath180 and @xmath186 together . by fourier transform , the groupoid @xmath1-algebra @xmath180 of @xmath54 is isomorphism to the continuous function algebra @xmath187 on the cotangent bundle under point - wise multiplication . this algebra contains the poisson algebra @xmath188 . therefore , one has a poisson structure on ( a sub - algebra of ) @xmath180 . similarly , one has a poisson structure on ( a sub - algebra of ) @xmath189 . a * continuous field of @xmath1-algebras @xmath190 } \right)$ ] * over @xmath191 $ ] consists of a @xmath1-algebra @xmath192 , @xmath1-algebras @xmath193 , @xmath122 $ ] , with surjective homomorphisms @xmath194 and an action of @xmath195)$ ] on @xmath192 such that for all @xmath196 : 1 . the function @xmath197 is continuous ; 2 . @xmath198 } \lvert \varphi_\hbar(c ) \rvert$ ] ; 3 . for @xmath199)$ ] , @xmath200 . [ fieldalgebra ] for the tangent groupoid @xmath97 , we define @xmath201 for @xmath202 and @xmath203 . the pullback of the inclusion @xmath204 induces a map @xmath205 , which extends by continuity to a surjective @xmath206-homomorphism @xmath207 . the @xmath1-algebras @xmath208 and @xmath209 with the maps @xmath210 for a continuous field over @xmath191 $ ] . a strict deformation quantisation of a poisson manifold @xmath211 consists of 1 . a continuous field of @xmath1-algebras @xmath212})$ ] , with @xmath213 ; 2 . a dense poisson algebra @xmath214 under the given poisson bracket @xmath215 on @xmath211 ; 3 . a linear map @xmath216 that satisfies ( with @xmath217 ) @xmath218 for all @xmath219 and @xmath220 , and for all @xmath221 satisfies the dirac condition @xmath222 - { \mathcal{q}}_\hbar ( \{f , g\ } ) \right\rvert = 0 .\ ] ] [ liequantisation ] let @xmath42 be a lie groupoid and @xmath84 its associated lie algebroid . the continuous field of @xmath1-algebras @xmath223}\right)$ ] , as defined in example [ fieldalgebra ] , defines a strict deformation quantisation of the poisson manifold @xmath224 . the poisson structure on @xmath224 is induced dually by the lie bracket on @xmath84 @xmath225 is a poisson algebra with poisson bracket induced from the lie bracket of @xmath3 . take @xmath192 to be the @xmath1-algebra generated by the tangent groupoid @xmath226 , @xmath227 to be @xmath189 , and @xmath193 to be the group @xmath1-algebra @xmath175 for @xmath228 . the quantisation map @xmath229 is the inclusion of @xmath225 into @xmath230 , and @xmath231 is @xmath229 followed by map induced by the inclusion of @xmath3 or @xmath0 into @xmath226 . 2 . @xmath232 is a symplectic manifold . its symplectic structure defines the poisson algebra @xmath188 , which includes into continuous function algebra @xmath233 . by fourier transform , @xmath233 is isomorphic to @xmath180 . the groupoid @xmath54 exponentiates to @xmath63 , thus one obtains the inclusion of @xmath188 into the @xmath1-algebra @xmath185 . 3 . take the lie groupoid @xmath93 , it has the associated lie algebroid @xmath92 . @xmath234 is a dense poisson algebra in @xmath235 . the continuous field of @xmath1-algebras , @xmath236 is given by @xmath237 , @xmath238 for @xmath202 , and @xmath239 . the quantisation map @xmath240 is the inclusion , and @xmath241 is @xmath229 followed by the restriction map . recall that a @xmath242-connection @xmath243 is precisely a @xmath3-valued 1-form when @xmath244 . by considering ( the characteristic function supported on ) the graph of the function @xmath118 , one obtains a distribution on the tangent groupoid @xmath245 of the lie groupoid @xmath246 . therefore , one has an action of the space of connections @xmath5 on the @xmath1-algebra @xmath247 , for @xmath248 . by a smearing the characteristic function , that is integrating the distribution with some smooth function , one turns the characteristic function into an element in @xmath247 . therefore , a map from from @xmath119 , which includes the space of ordinary connections @xmath5 , to @xmath247 . then theorem [ liequantisation ] allows us to strictly deformation quantise the connections . therefore , what one has obtained in this procedure is a strict deformation quantising of the space of connections @xmath5 , which is obtained by mapping @xmath5 into a @xmath1-algebra that constitutes the properties of a strict deformation quantisation . the goal of this formalism is to provide a deforming parameter @xmath112 , so that when @xmath244 , ordinary connections are retrieved . this line of work is to provide loop quantum gravity a semi - classical limit , so that the theory of classical gravity , general relativity , returns as one takes the limit @xmath249 . the lack of a semi - classical limit in loop quantum gravity has been a long standing problem . the main obstacle of obtaining such a limit is by nature how one traditionally constructs a measure on the space of connections probing the connection space with a collection of finite graphs as described in section [ sec2 ] . finite graphs are very rigid and discrete , they do not provide a parameter that one usually encounters in quantum theory to adjust . as a result , obtaining a semi - classical limit simply becomes impossible if a `` smoother '' way of probing the connection space is not introduced . to come up with a smooth way of probing the connection space , we understand this probing procedure as evaluating some one - dimensional objects , which is a groupoid . thus , by using a smooth groupoid , or more specifically a lie groupoid , one has a hope of circumventing the discreteness problem . the proposal we give here is the tangent groupoid . as it turns out , when the right tangent groupoid is used , the space of connections includes into the groupoid . from there , one could deform the connections to convolution operators acting on a hilbert space . this formalism recreates some elements appear in the noncommutative standard model @xcite in the way that , the gauge group is the unitary part of the algebra in the spectral triple of noncommutative standard model , the hilbert space remains the same , and the connections are realized as trace - class operators , with the trace being a gauge invariant quantity that resembles the loop variables in loop quantum gravity literature . the tangent groupoid provides another important feature concerning deformation , which is the strict deformation quantisation result of landsman @xcite . deformation quantisation can naturally be formulated in terms of @xmath1-algebraic language , a theorem due to landsman shows that a tangent groupoid gives rise to a deformation quantisation in the strict sense . by combining landsman s result and our realization of connections using tangent groupoids , we obtain a deformation quantisation of @xmath0-connections . and this quantisation formalism permits the existence of a parameter @xmath112 , so that when @xmath244 , one retrieves the classical connections . however , this is not the end of the story . since connection variables are only half of the variables in gravity in arnowitt - deser - misner formulation , one still has to look into the other half of the variables that the connections are conjugate dual to , tetrads or metrics . it is known that tetrads are quantised to degree one differential operators @xcite , and its semi - classical limit can be obtained from the @xmath249 limit of integral kernel of the differential operator multiplied by @xmath112 , which gives nothing but the symbol of the differential operator @xcite . in the case of the manifold @xmath55 being three dimensional , the symbol is an @xmath250-valued function the poisson manifold @xmath232 . hence the symbol of the differential operator lives in the same space as a connection does . following up the work of deformation quantisation of connections here , the next step is to examine the interaction of the tetrads with the connections at both the classical level @xmath244 and the quantum level @xmath202 . at @xmath251 , one needs to examine the poisson bracket of a connection and a symbol ( of a differential operator ) , and determine which symbol is conjugate dual to a given connection . at @xmath202 , one repeats the same procedure except now that the poisson bracket is replaced with a commutator . we will leave those considerations to another article . finally , the author would like to stress that the line of work here is toward obtaining a semi - classical limit in loop quantum gravity , and this tangent groupoid application has not been seriously considered before , thus the work here may appear incomplete . nonetheless , the development so far shows that a semi - classical limit in loop quantum gravity is more within reach than before .

motivated by the compactification process of the space of connections in loop quantum gravity literature . a description of the space of @xmath0-connections using the tangent groupoid is given . as the tangent groupoid parameter is away from zero , the @xmath0-connections are ( strictly ) deformation quantised to noncommuting elements using @xmath1-algebraic formalism . the approach provides a mean to obtaining a semi - classical limit in loop quantum gravity .

You are an expert at summarizing long articles. Proceed to summarize the following text: many intriguing and fascinating results were shown during this conference , which certainly require some revision of our classical understanding of the evolution of the first stellar generations in the universe . let us briefly recall a few of them here : * the measured abundances of many elements at very low metallicity present a very small scatter ( cayrel et al . this might indicate that already at this low metallicity , stars are formed from a well mixed reservoir composed of ejecta from stars of different initial masses . * at least down to a metallicity of [ fe / h ] equal to -4 , there is no sign of enrichments by pair instability supernovae ( see the contribution by heger in this volume ) . * the n / o ratios observed at the surface of halo stars by israelian et al . ( @xcite ) and spite et al . ( @xcite ) indicate that important amounts of primary nitrogen should be produced by very metal - poor massive stars ( chiappini et al . @xcite ) . * below [ o / h ] @xmath8 the c / o ratio presents an upturn indicating either that very metal poor stars produce more carbon , or less oxygen or both ( akerman et al . @xcite ; spite et al . @xcite ) * if most stars , at a given [ fe / h ] , present a great homogeneity in composition , a small group , comprising about 20 - 25% of the stars with [ fe / h ] below -2.5 , show very large enrichments in carbon with a great scatter , these are the c - rich extremely metal poor stars ( cemp ) . * the surface abundances of stars in globular clusters show an anti correlation between the abundances of sodium and oxygen , those of magnesium and aluminum ( see the contributions by gratton and charbonnel in this volume ) . halo stars in the field do not show such anti correlations . * the zero age main sequence in the very massive globular cluster @xmath9 cen can be decomposed in a blue and a red component . the blue component is paradoxically more metal rich by about a factor two with respect to the red one . the only way , presently , to understand such a behavior , is to suppose that the stars of the blue component are very helium - rich ( see the contributions by gratton and maeder et al . in this volume ) . many more results could be added to the above list . such a large amount of unexplained features points toward the need of exploring new lines of research . some observational facts concern only a subsample of the stars ( for instance the c - rich stars , or the stars in globular clusters ) . in those cases some special circumstances can be invoked in order to reproduce the observed features . other observational facts as the c / o upturn or the primary nitrogen production point toward some general characteristics shared by a great part if not all the extremely metal poor stars . classically much more massive stars can be formed at very low metallicity as a result of the absence of dust which acts as an efficient cooling agent ( see e.g abel et al . the absence of cno elements modifies the way the h - ignition occurs in massive stars . the lower opacity of metal poor material implies more compact stars and very weak stellar winds . other characteristics might be different in pop iii stars , for instance the binary frequency , the amplitudes and the effects of magnetic fields . here we want to focus on the effects of rotation . as explained in the contribution by maeder et al . in this volume , at low metallicity , the present models predict that stars have more chance to reach break - up during the main - sequence phase , are more efficiently mixed and may undergo strong mass loss through stellar winds . here we want to present some new results illustrating this behavior and draw some consequences for nucleosynthesis at very low metallicity . the case of rotating pop iii models is briefly discussed elsewhere in the present volume ( see the contribution by ekstrm et al ) . in the present paper , we shall concentrate on stars with a very small initial amount of metals . it is interesting to point out that already a very small amount of metals makes a world of difference with respect to strictly pop iii stars . this is illustrated in fig . [ fig1 ] , presenting the chemical structure for two rotating 60m@xmath10 stars with @xmath11 = 800 km s@xmath12 . three differences between the pop iii and the @xmath0 model can be seen : in the @xmath0 stellar model 1 ) the co core is smaller ; 2 ) the quantity of primary nitrogen is much greater ; 3 ) an extended convective zone is associated to the h - burning shell ( from about 20 to 34 m@xmath10 ) . in the @xmath0 stellar model , the cno content in the region of the star where the h - shell ignites is already sufficient for the h - burning to occur through the cno cycle . this implies that from its birth the h - burning shell can compensate for a great part of the energy lost by the surface . typically , the h - shell luminosity is about one half the total one . the star then adjusts its structure so that the he - burning core has only to compensate for the other half of the total luminosity . instead in the pop iii model , the cno content in the region where the h - burning shell occurs is so low that the shell only succeeds in compensating for a very small fraction of the total luminosity ( typically a few percents ) , the rest having to be compensated by the he - burning core . this explains the much bigger helium core in the pop iii model and the very modest h - burning shell . as a consequence in the pop iii model , much less primary nitrogen is produced and the h - shell remains radiative for a much longer period . we do not know what was the initial rotation of the massive first stellar generations . on the other hand , we can observe the rotational velocity of solar metallicity massive stars , and suppose that the very metal poor stars had , at the beginning of their evolution , about the same angular momentum as their solar metallicity counterparts . for pop iii stars this implies an initial velocity of about 800 km s@xmath12 . the same initial velocity of 800 km s@xmath12 was considered for the very small metallicities @xmath0 and @xmath1 considered in the present paper . the evolution we shall describe is more dependent on the choice of the initial velocity than on the precise choice of the initial metallicity . both rotating models at @xmath0 and @xmath1 present the same qualitative behaviour , namely they both produce large amounts of primary nitrogen and lose a great part of their initial mass through stellar winds . [ fig2 ] shows the evolution of the ratio @xmath13 at the surface during the ms phase . at both @xmath0 and @xmath1 , the fast models reach the break - up limit , the `` metal - rich '' model at an earlier stage than the `` metal - poor '' one . this comes from the fact that when the metallicity increases , a given value of the initial velocity corresponds to a higher initial value of the @xmath14 ratio . the stars then remain at break - up for the rest of their ms lifetimes and undergo an enhancement of their mass loss rates . as a consequence , the @xmath0 model ends its ms life with 57.6 m@xmath10 , having lost 4% of its initial mass , while the @xmath15 model ends its ms life with 53.8 m@xmath10 , having lost 10% of its initial mass . despite the stars stay in the vicinity of the break - up limit during an important part of their ms lifetime , they do not lose very important amounts of mass . this is due to the fact that only the outermost layers of the stars are above the break up limit and are ejected . these layers have low density and thus contain little mass . the 60 m@xmath10 model at @xmath16 with @xmath11= 300 km s@xmath12 computed by meynet & maeder ( @xcite ) reaches the break - up velocity much later , only at the end of the ms phase . at @xmath15 , the velocity 300 km s@xmath12 appears thus as the lower limit for the initial rotation , allowing a 60 m@xmath10 star to reach the break - up limit during its ms phase . this 60 m@xmath10 star ends its ms life with 59.7 m@xmath10 , having lost only 0.5% of its initial mass . during the ms phase , the surface of the rotating stars is enriched in nitrogen and * depleted in carbon * as a result of rotational mixing . the n / c ratios are enhanced by more than two orders of magnitude at the end of the h - burning phase . the total amount of cno elements remains however constant . during the core he - burning phase , primary nitrogen is synthesized in the h - burning shell , due to the rotational diffusion of carbon and oxygen produced in the helium core into the h - burning shell ( meynet & maeder @xcite ) . this is well illustrated in fig . [ fig4 ] for the @xmath15 rotating model . in contrast to what happens during the ms phase , rotational mixing , during the core he - burning phase , induces strong changes of the surface metallicity . these changes only occur at the end of the core he - burning phase , when an outer convective zone appears and rapidly deepens in mass , dredging up newly synthesized elements to the surface . from this stage onwards the surface metallicity increases in a spectacular way . typically our 60 m@xmath10 model at @xmath0 ends its life with a surface metallicity one million times higher than its initial metallicity , _ i.e. _ with @xmath17 , a metallicity greater that the metallicity of the large magellanic cloud ! this increase of the surface metallicity is entirely due to the arrival in great quantities at the surface of primary cno elements , carbon and oxygen being produced in the he - core and the nitrogen in the h - shell . the consequence of such important surface enrichments on the mass loss rates remains to be studied in details using models of stellar winds with the appropriate physical characteristics ( position in the hr diagram and chemical composition ) . in absence of such sophisticated models , we applied here the usual rule , namely @xmath18 , where @xmath2 is the metallicity of the outer layers . with this prescription , the stars lose a great part of its initial mass . our 60 m@xmath10 at @xmath0 ends its life with only 24 m@xmath10 , the corresponding model at @xmath15 reaches a final mass of 37 m@xmath10 . we have also computed a 7 m@xmath10 stellar model at @xmath15 with @xmath19 km s@xmath12 . this model , in contrast with its more massive counterpart , never reaches the break - up limit during the ms phase . the 7 m@xmath10 is more compact than the 60m@xmath10 model , this makes the gratton - pick term in the expression for the velocity of the meridional circulation smaller ( see the contribution by maeder et al . in this volume ) . the angular momentum is thus less efficiently transported outwards than in the more massive model , making the reaching of the break - up limit more difficult . interestingly , the model present a higher degree of mixing than the 60 m@xmath10 stellar model , because , due to the relative inefficiency of the angular momentum transport , the gradients of @xmath20 remain steeper , making the shear diffusion stronger . this is in contrast to what happens at higher metallicity . indeed it was shown ( see maeder & meynet @xcite ) that at a given metallicity and for a given initial velocity , the mixing was more efficient in the more massive stars . this is correct as long as the gradients of @xmath20 do not depend too much on the initial mass . here at very low @xmath2 , the gradients are very sensitive to the initial mass , being steeper in smaller initial mass stars . the 7 m@xmath10 remains in the blue part of the hr diagram during the whole he - burning phase , preventing an outer convective zone to appear and to dredge - up the primary cno elements . only at the end of the core he - burning phase , the star evolves to the red and approach the base of the asymptotic giant branch . at this point , an outer convective zone appears and produces an enormous enhancement of the surface metallicity . this can be seen in fig . [ fig5 ] , which compares the chemical structure of the 7 m@xmath10 at the early agb phase , with that of the 60 m@xmath10 model at the end of the core c - burning phase . we see that in the outer layer of the 7 m@xmath10 the abundances of cno elements are at about the same level , while in the 60 m@xmath10 , the abundance of nitrogen is about at the same level as in the 7 m@xmath10 model , while those of carbon and oxygen are many orders of magnitude below . , @xmath21 and @xmath1 with and without rotation . the contribution of the stellar winds have been accounted for when present.,width=377 ] let us first discuss the cno yields of pop iii stellar models ( see ekstrm et al . in this volume ) . the yields of carbon and oxygen appear to be little affected by rotation , while those of nitrogen are enhanced by 3 to 4 orders of magnitude in rotating models . the yields for nitrogen are compared with those obtained in other models in fig . we see that the pop iii yields from rotating model with @xmath22 km s@xmath12 are in general higher than those for @xmath15 with @xmath23 km s@xmath12 computed by meynet & maeder ( @xcite ) . however , according to chiappini et al . ( @xcite ) , the yields from massive stars , required to fit the n / o ratios in the unmixed sample of halo stars observed by israelian et al . ( @xcite ) and spite et al . ( @xcite ) , should be enhanced by a factor 50 ( 1.7 dex ) , _ i.e. _ should be much higher than predicted by the present pop iii stars for stars in the mass range between 15 and 85 m@xmath10 . unless , only very massive stars are formed , it appears thus difficult to explain the high n / o ratios observed in halo stars . moreover it is likely that the production of primary nitrogen by massive stars extends over a range of metallicities and is not restricted to pop iii stars . as shown above , a tiny amount of metal can boost in a very important way the amount of primary nitrogen produced . in that case , starting with a velocity of @xmath22 km s@xmath12 would allow to reach levels well above those required by chiappini et al . ( @xcite ) to fit the observed n / o in halo stars . thus rotating massive star models have no difficulty in making the great amounts of primary nitrogen which seem to be required by the observations . more work is still needed in order to explore the range of initial conditions which would give the best agreement . spectroscopic surveys of very metal poor stars ( beers et al . @xcite ; beers @xcite ; christlieb @xcite ) have shown that carbon - rich extremely metal poor stars ( cemp stars ) account for up to about 25% of stars with metallicities lower than [ fe / h]@xmath24 . a star is said to be c - rich if [ c / fe]@xmath25 . the two most iron - deficient stars observed so far , he 0107 - 5240 , a giant halo star , and he 1327 - 2326 , a dwarf or subgiant halo star , are c - rich stars . this might indicate that the frequency of c - rich stars increases when the metallicity decreases . observations of c - rich non - evolved stars ( _ i.e. _ dwarf or subgiant ) indicate also that the pattern of abundances was already present in the cloud from which the star formed and is therefore not due to a mechanism occurring in the star itself . many scenarios have already been proposed to explain the very peculiar abundances at the surface of these stars ( supernova with mixing and strong fall back , umeda & nomoto @xcite ; mixing of the ejecta of two supernovae , limongi et al . @xcite ; mass transfer from an agb star , suda et al . @xcite ) . here we propose new scenarios based on rotating models . let us first see if the cemp stars could be formed from material made up of massive star wind ( or at least heavily enriched by winds of massive stars ) . at first sight such a model might appear quite unrealistic , since the period of strong stellar winds is rapidly followed by the supernova ejection , which would add to the wind ejecta the ejecta of the supernova itself . however , for massive stars , it might occur that at the end of their nuclear lifetime , a black hole , swallowing the whole final mass , is produced . in that case , the massive star would contribute to the local chemical enrichment of the interstellar medium only through its winds . let us suppose that such a situation has occurred and that the small halo star that we observe today formed from the shock induced by the stellar winds with the interstellar material . what would be its chemical composition ? its iron content would be the same as that of the massive star since the iron abundance in the interstellar medium had no time to change much in the brief massive star lifetime . also , the massive star wind ejecta are neither depleted nor enriched in iron . the abundances of the other elements in the stellar winds for our two rotating 60 m@xmath10 at @xmath0 and 10@xmath26 are shown in fig . [ fig8 ] . for the two metallicities considered here , the wind material of rotating models is characterized by n / c and n / o ratios between @xmath27 1 and 40 , and @xmath7c/@xmath5 c ratios around 4 - 5 . these values are compatible with the ratios observed at the surface of cs 22949 - 037 ( depagne et al . @xcite ) : n / c @xmath27 3 and @xmath7c/@xmath5c @xmath274 . the observed value for n / o ( @xmath270.2 ) is smaller than the range of theoretical values , but greater than the solar ratio ( @xmath27 0.03 ) . thus the observed n / o ratio also bears the mark of some cno processing , although slightly less developed than in our stellar wind models . on the whole , a stellar wind origin for the material composing this star does not appear out of order in view of the above comparisons , especially if one considers the fact that , in the present comparison , there is no fine tuning of some parameters in order to obtain the best agreement possible . the theoretical results are directly compared to the observations . moreover only a small subset of possible initial conditions has been explored . other cemp stars present however lower values for the n / c and n / o ratios and higher values for the @xmath7c/@xmath5c ratio . for these cases it appears that the winds of our rotating 60 m@xmath10 models appear to be too strongly cno processed ( too high n / c and n / o ratios and too low @xmath7c/@xmath5c ratios ) . better agreement would be obtained if the observed abundances also result from material from the co - core , ejected either by strong late stellar winds or in supernova explosion . to explore this possibility we have study the case where wind ejecta are mixed with supernova ejecta and interstellar material . at least two free parameters have to be introduced : 1 ) the dilution factor between the ejecta and the interstellar medium ; 2 ) the mass of iron in the supernova ejecta . it is possible to adopt values of these two parameters so that the observed values of [ fe / h ] , [ o / fe ] and [ c / fe ] can be reproduced by both our non - rotating and rotating 60 m@xmath10 model . however , the observed [ n / fe ] ratios can only be obtained using the rotating model . the wind and supernova ejecta model lowers the n / c and n / o ratios improving the agreement with respect to the pure wind model , however at the cost of two free parameters ! another model has been suggested by suda et al . ( @xcite ) : the small halo star , observed today , was the secondary in a binary system whose primary went through the agb stage . at this stage , part of the agb envelope has been accreted by the secondary greatly modifying its original surface abundances . as was the case for the massive star wind model seen above , the iron must come from a previous star generation , since no iron is produced by the agb star . using our models computed in meynet & maeder ( @xcite ) for initial masses between 2 and 7 m@xmath10 at @xmath15 and with @xmath28 and @xmath29 km s@xmath12 , we can estimate the chemical composition of the envelope of intermediate mass stars at the beginning of the thermal pulse asymptotic giant branch phase . the envelope is all the matter above the co - core . the range of values for the cno elements given by these models are shown in fig . [ fig9 ] ( vertical thin lines : no - rotation ; vertical thick lines , with rotation ) . we have also computed a new 7 m@xmath10 with @xmath22 km s@xmath12 at @xmath15 ( continuous line with solid circles ) . we see that the envelopes of agb stellar models with rotation show a chemical composition very similar to that observed at the surface of cemp stars . in particular , only rotating models are able to simultaneously explain the large abundances of c , n and o. it is however difficult to conclude that rotating intermediate mass star models are better than rotating massive star models in reproducing the abundance pattern of cemp stars . probably , some cemp stars are formed from massive star ejecta and others from agb star envelopes . interestingly at this stage , some possible ways to distinguish between massive star wind material and agb envelope do appear . indeed , massive star wind material is characterized by very low @xmath7c/@xmath5c ratio , while intermediate mass stars seem to present higher values for this ratio . agb envelopes would also present very high overabundances of @xmath6o , @xmath30o , @xmath31f and @xmath32ne , while wind of massive rotating stars present a weaker overabundance of @xmath6o and depletion of @xmath30o , @xmath31f and @xmath32ne . as discussed in frebel et al . ( @xcite ) , the ratio of heavy elements as the strontium to barium ratio can also give clues as to the origin of the material from which the star formed . in the case of he 1327 - 2326 , frebel et al . ( @xcite ) give a lower limit of [ sr / ba ] @xmath33 , which suggests that strontium was not produced in the main s - process occurring in agb stars , leaving thus the massive star hypothesis as the best option , in agreement with the result from @xmath7c/@xmath5c in g77 - 61 ( plez & cohen @xcite ) and cs 22949 - 037 ( depagne et al .

rotating massive stars at @xmath0 and @xmath1 lose a great part of their initial mass through stellar winds . the chemical composition of the rotationally enhanced winds of very low @xmath2 stars is very peculiar . the winds show large cno enhancements by factors of @xmath3 to @xmath4 , together with large excesses of @xmath5c and @xmath6o and moderate amounts of na and al . the excesses of primary n are particularly striking . when these ejecta from the rotationally enhanced winds are diluted with the supernova ejecta from the corresponding co cores , we find [ c / fe ] , [ n / fe],[o / fe ] abundance ratios very similar to those observed in the c rich extremely metal poor stars ( cemp ) . we show that rotating agb stars and rotating massive stars have about the same effects on the cno enhancements . abundances of s - process elements and the @xmath7c/@xmath5c ratio could help us to distinguish between contributions from agb and massive stars . on the whole , we emphasize the dominant effects of rotation for the chemical yields of extremely metal poor stars .

You are an expert at summarizing long articles. Proceed to summarize the following text: planetary nebulae ( pns ) have become increasingly important in extragalactic astronomy , for distance determinations via their luminosity function ( lf ) ( * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * and references therein ) , as kinematic tracers of the dark halos of galaxies @xcite , and as tracers for the distribution and kinematics of the diffuse stellar population in galaxy clusters @xcite . due to their strong narrow line emission at @xmath3 \lambda 5007 $ ] , pns can be easily detected out to distances beyond @xmath4 with narrow - band photometry and slitless spectroscopy @xcite , and to @xmath5 with multi - slit imaging spectroscopy @xcite . moreover , they are observed in elliptical and spiral galaxies , making them an indispensible tool to support distances obtained by other methods ( such as cepheids , surface brightness fluctuations , the tully - fisher relation , sne ia ) , and to measure the kinematics of stellar populations whose surface brightness is too faint for absorption line spectroscopy . for distance determination the planetary nebulae luminosity function ( pnlf ) is normally modeled as having a universal shape that depends only on the absolute bright magnitude cutoff @xmath6 : @xmath7 where @xmath8 is the number of pns with absolute magnitude @xmath9 @xcite . observationally , the cutoff magnitude @xmath6 has a quasi - universal value of @xmath10 with only a weak dependence on host galaxy metallicity expressed by the system s oxygen abundance , and which can be compensated for by a quadratic relation in @xmath11 $ ] @xcite . in practice , the pn magnitudes @xmath12 , after correcting for the interstellar reddening , are fitted to the model pnlf of eq . [ pnlfeqn ] convolved with the photometric error profile , yielding a value of the distance modulus @xcite . the absence of any systematic variations in @xmath6 and the pnlf shape has been verified in galaxies with significant population gradients , and among galaxies of different morphologies within galaxy clusters / groups up to virgo ( * ? ? ? * ; * ? ? ? * and references therein ) . this universality of the pnlf and the cutoff magnitude @xmath6 must be considered surprising , given that the pn luminosity in the @xmath3 \lambda 5007 $ ] line depends on the mass and metallicity of the central star , as well as on the electron gas temperature , optical thickness and dust extinction of the surrounding nebula . indeed , some current semi - analytic simulations of the pnlf seem to be at odds with the observational trends . @xcite indicate small possible dependencies of @xmath6 on the total size of the pn population , on the time elapsed since the last episode of star formation , and on how optically thin the pns are ; concluding , however , that only careful studies would detect such effects in the observed pnlf . in contrast , more recent pnlf simulations by @xcite contradict the observed narrow spread in @xmath6 and predict large variations of several magnitudes depending on a variety of realistic star formation and evolution scenarios . so is the pnlf truly quasi - universal and its cutoff magnitude nearly independent of population age and metallicity ? pns are also important as test particles to study the kinematics and dark matter distribution in the halos of elliptical galaxies . since the pn population is expected to arise from the underlying galactic stellar distribution , their radial velocities can be used as effective kinematic tracers of the mass distribution . however , the required pn sample sizes are many 100s @xcite , or at least 100 or more in conjunction with absorption line spectroscopy , which has limited this application to only a few nearby galaxies @xcite . in recent simulations of disk galaxy mergers involving dark matter , stars , and gas , @xcite predict that the young stars formed in the merger have steeper density profiles and larger radial anisotropy than the old stars from the progenitor galaxies , and they argue that if the pns observed in elliptical galaxies were to correspond to the young population rather than to all stars in the simulations , their velocity dispersion profile would match the measured dispersion profiles of @xcite . so do pns really trace the stars and their kinematics in elliptical galaxies ? different stellar populations may have , and in general would have , different phase - space distributions in the same galaxy potential . the simplest approach for dynamical modelling , taking the pn velocities as a random sampling of the stellar velocities , is however valid only when the pn population properties and their kinematics are uncorrelated . except in special cases this also requires that the pnlf is independent of the stellar population . vice - versa , if there existed differences in the pnlf or the bright cutoff magnitude for different stellar populations , they would best be identified by studying the correlations between pn magnitudes and kinematics or positions of these tracers , in a single galaxy where all pns are at the same distance . in this paper , we report on such a study in the elliptical galaxy ngc 4697 , an excellent target for this purpose because of the large sample of pn velocities known from @xcite . our analysis shows the existence of distinct pn populations which differ in their kinematics , brightnesses , and spatial distributions . this suggests that the answer to both the questions posed above may be no in general , different stellar populations may have slightly different pnlfs , and the observed pn population in elliptical galaxies may not be a fair tracer of their stars . the paper is organised as follows : in [ data ] we review the properties and pn data of this galaxy and discuss the magnitude and velocity completeness of our sample . our statistical analysis of these data is given in [ analysis ] where we demonstrate the inhomogeneity of the sample in the space of velocities , magnitudes , and positions . [ discn],[result ] conclude our work , giving also a brief discussion of its implications . ngc 4697 is a normal , almost edge - on e4 - 5 galaxy located along the virgo southern extension . from the multi - colour ccd photometry of @xcite , the effective radius is @xmath13 , the mean ellipticity is 0.45 , and the pa is constant , consistent with a near - axisymmetric luminosity distribution . isophotal analysis shows that this galaxy has a positive @xmath14 coefficient suggesting a disk - like structure within @xmath15 @xcite . from optical spectroscopy , its dominant stellar population has an age of @xmath16 gyr @xcite , consistent with the red mean b - v=0.91 colour . stellar absorption line kinematics along the major axis ( pa @xmath17 ) of ngc 4697 have been reported by @xcite and @xcite ; these velocity data can be well described by dynamical models based on the luminous mass distribution only . @xcite detected and measured radial velocities of 531 pns extending out to @xmath18 in this galaxy , with observational errors of @xmath19 . via dynamical analysis , they determined a constant mass to light ratio @xmath20 within @xmath21 , which is consistent with a @xmath22 gyr old stellar population with a salpeter mass function and slightly super solar metallicity . x - ray observations with rosat @xcite show only small amounts of hot gas in the halo of this galaxy . using more recent _ chandra _ data , @xcite could resolve most of the x - ray emission into nonuniformly distributed x - ray binary point sources ( xps ) , suggesting that ngc 4697 has lost most of its interstellar gas . though ngc 4697 does not show any signature of recent interactions , @xcite present evidence that the distribution of these x - ray sources is inconsistent with the optical morphology of ngc 4697 , and propose that these sources are mostly high mass x - ray binaries ( hmxbs ) associated with young stellar populations related to fallback of material in tidal tails onto a relaxed merger remnant , or to shock - induced star formation along tidal tails . they estimate typical fallback times of such tidal tails to be much longer than the settling timescale of the remnant and expect similar results for other elliptical galaxies with populations of @xmath23 gyr age . for the work in this paper we use the pn sample presented by @xcite . after removing the possible contaminants and unclear detections , they report unambigous detection of 535 pns . however only 526 out of 535 pns have confirmed measurements of velocity _ and _ magnitude , and we use these in our analysis . in order to determine the pnlf , it is crucial to estimate the magnitude where pn detection incompleteness sets in . detectability of a pn varies with the background galaxy surface brightness ; for a statistically complete sample the surface number density of pns should be directly proportional to it . @xcite show that their pn sample is statistically complete down to @xmath24 magnitudes outside an elliptical region of semi - major axis @xmath25 . in our analysis , we have thus defined two data sets : a _ complete sample _ ( with pns brighter than @xmath26 , outside the central ellipse of semi - major axis @xmath25 ) , and a _ total sample _ ( consisting of all pns with measured magnitude and radial velocity ) . the total number of pns in these data sets is 320 and 526 , respectively . the systemic velocity ( @xmath27 ) of ngc 4697 , obtained by averaging the observed velocity of all 526 pns is @xmath28 , which agrees with the values quoted in the literature ( * ? ? ? * and references therein ) . the on - band filter configuration used to detect and measure velocities of these pns has a peak wavelength of @xmath29 , peak transmission of @xmath30 , equivalent width of @xmath31 , and fwhm of @xmath32 @xcite . the fwhm corresponds to a velocity range of @xmath33 around the systemic velocity of ngc 4697 . thus the filter transmission width is large enough to facilitate observations of pns with all velocities bound to ngc 4697 , irrespective of their magnitude . indeed , even at magnitudes as faint as @xmath34 in the total sample , pns with large velocities @xmath35 are detected . thus the velocity coverage in both samples ( total and complete ) is not biased with respect to the pn magnitudes . the pn magnitudes were measured by @xcite from their undispersed images ; they are accurate to 0.1 and 0.2 mag for @xmath12 brighter and fainter than 26.5 , with systematic effects below 2% . as a further test relevant for the present work , @xcite used the redundancy provided between their e and w fields : plotting magnitude differences between the two measurements ( e and w ) of pn candidates as a function of difference in distance from the center of the ccd , they found a scatter diagram without any evidence of correlation . @xcite estimated the errors in the pn velocities from calibration , image registration , spectrograph deformation and guiding errors to be @xmath36 . the velocities of 165 pns were measured independently in the e and w field of @xcite ; these velocities agree within a standard deviation of @xmath37 . in order to check whether a systematic difference between the velocities of bright and faint pns could be introduced by an asymmetric psf ( a possibility suggested by k. freeman ) , we have superposed the psf s of three groups of 10 of the brightest pns , one selected at random , and two selected among those pns with the highest and lowest radial velocities . in the three cases we estimated the shift of the centroid of the entire psf with respect to the centroid of the upper part . the shifts were smaller than @xmath38 , and in some cases they were in the opposite sense compared to the results discussed below . in this section , we search for stellar population effects in the kinematics of the pns in ngc 4697 , by analysing the total and complete data sets with respect to their three observables : velocity , magnitude and position . for both data sets , we convert the observed pn radial velocities into co - rotating or counter - rotating velocities , as follows . with the galaxy center at the origin of the reference frame , and the x - axis oriented along the major axis ( pa@xmath39 deg ) , the absorption line stellar - kinematic data predict positive line - of - sight mean velocity with respect to the galaxy systemic velocity , at slit positions towards the south - west of the center with x coordinate @xmath40 , and vice versa . we denote this sense of rotation as _ co - rotating _ , and the opposite sense as _ counter - rotating_. by definition , the major axis absorption line data is _ co - rotating_. after subtracting the systemic velocity from the pn radial velocities , we define reduced velocities @xmath41 and denote the pns with @xmath42 @xmath43 ( @xmath44 ) 0 as co - rotating ( counter - rotating ) . by definition , the major axis absorption line data have @xmath45 . the resulting values of @xmath42 are displayed against the observed magnitudes in figure [ mvplot ] . even at magnitudes as faint as @xmath34 in the total sample , pns with large velocities @xmath35 are detected , showing that there is no kinematic bias at faint magnitudes . henceforth , unless stated otherwise , we will always use the complete sample for our analysis . figure [ mvplot ] shows that the complete pn sample appears to exhibit a correlation between magnitudes and kinematics , with faint pns showing more co - rotation than bright pns . we have performed several statistical tests to verify the significance and look for the origin of this correlation . table [ tabpear ] shows the results of pearson s r - test for correlated data . velocities of counter - rotating pns are strongly linearly correlated with their brightness , while those of co - rotating pns are almost independent of their magnitude distribution . further , we divided our sample into 3 equal magnitude bins each of size @xmath46 , hereafter referred to as _ faintest _ , _ intermediate _ , and _ brightest pns _ , and computed the mean reduced velocity and its dispersion in each of these bins along with the significance of their differences . as shown in table [ tabvtftest ] , the _ faint _ and _ bright pn subsamples _ defined through these bins have different mean reduced velocity and dispersions at @xmath47 and @xmath48 confidence , respectively . in figure [ 6bins ] we have plotted the cumulative velocity distribution of pns in the brightest and faintest magnitude bins . there is a visible excess of bright pns with counter rotating velocities . it is particularly evident from this figure that the brightest counter rotating pns display a velocity distribution that differs from the rest of the pns from the complete sample with high confidence . thus it is clear that the observed correlation between the pns kinematics and their magnitudes is compelling , and it arises because the faintest and brightest pns have significantly different velocity distributions . there appears to exist an additional component of bright , counter - rotating pns with respect to the overall sample . .pearson s r - test for linear correlation of pn magnitudes with reduced velocity @xmath49 , for the entire complete sample , the co - rotating , and the counter - rotating subsamples . values of r close to @xmath50 indicate a strong linear correlation ; values close to @xmath51 indicate little or no correlation . @xmath52 is the probability that two uncorrelated variables would give the r - coefficient as large as or larger than the measured one , for a normal distribution of r. small values of @xmath52 imply significant correlation . [ cols="^,^,^",options="header " , ] , orange dashed line ) , faintest pns ( @xmath53 , purple dashed - dotted line ) , and the entire complete sample ( black solid line ) . the entire velocity range is shown in the top panel . in the middle panel the velocities are divided according to their sense of rotation . the bottom panel shows the cumulative distribution ( shown from @xmath54 to @xmath55 values ) , normalised at @xmath56 . the kolmogorov smirnov probabilities show that the brightest counter rotating pns have significantly different velocity distribution from the rest of the pns.,title="fig : " ] , orange dashed line ) , faintest pns ( @xmath53 , purple dashed - dotted line ) , and the entire complete sample ( black solid line ) . the entire velocity range is shown in the top panel . in the middle panel the velocities are divided according to their sense of rotation . the bottom panel shows the cumulative distribution ( shown from @xmath54 to @xmath55 values ) , normalised at @xmath56 . the kolmogorov smirnov probabilities show that the brightest counter rotating pns have significantly different velocity distribution from the rest of the pns.,title="fig : " ] , orange dashed line ) , faintest pns ( @xmath53 , purple dashed - dotted line ) , and the entire complete sample ( black solid line ) . the entire velocity range is shown in the top panel . in the middle panel the velocities are divided according to their sense of rotation . the bottom panel shows the cumulative distribution ( shown from @xmath54 to @xmath55 values ) , normalised at @xmath56 . the kolmogorov smirnov probabilities show that the brightest counter rotating pns have significantly different velocity distribution from the rest of the pns.,title="fig : " ] if these correlations have a physical origin , they should also be manifest in the spatial distribution of these pns . thus we now enquire whether the pn kinematics and magnitudes depend on their spatial location in the galaxy . in figure [ tspace ] we plot the spatial locations of all the 526 pns in this galaxy . the central incompleteness ellipse is also displayed . pns brighter than @xmath57 ( which is @xmath58 magnitude deeper than the brightest pn ) are shown as filled blue squares ( co - rotating ) and filled red triangles ( counter - rotating ) . inside the incompleteness ellipse , the distribution of the bright pns appears to be concentrated around an elliptical annulus . however , we did not find any kinematic evidence ( like a rotation curve signature ) relating these bright pns to the central stellar disk : either they are not physically related to the disk , or the evidence from the data is inconclusive . outside the incompleteness ellipse , the distribution of bright pns does not follow the surface brightness of the host galaxy : there is a significant left - right asymmetry , with more bright pns to the right side of the galaxy minor - axis ( @xmath59 ) than to the left side ( @xmath60 ) . @xcite discuss at length the possible differences in their e ( east ) and w ( west ) fields , and are convinced that the maximum systematic errors in the measured photometry , positions and radial velocities are below @xmath61 , @xmath62 and @xmath63 , respectively . hence we conclude that the left - right asymmetry in number counts of bright pns is not affected by detection uncertainties . subsequently , we carried out several tests to check whether the brightest and faintest , or the co- and counter - rotating pns are distributed differently in the galaxy . it turns out that the radial pn distribution is independent of their sense of rotation . however , the distribution of pn distances from the galaxy mid - plane differs for co- and counter - rotating pns at a confidence level of @xmath64 . are shown by the orange dashed line , the red dash - triple dotted line , and the purple dash - dotted line , respectively . all pns in the complete sample are shown by the solid black line . note the large increase in the distribution of bright pns and the moderate increase in the distribution of intermediate pns , relative to the faint pns , at @xmath65 ( see text for details ) . the ks probability of the faintest and brightest pns being drawn from the same underlying distribution is only @xmath66 , while the intermediate pns are still compatible with the faint pns ( ks probability @xmath67).,title="fig : " ] are shown by the orange dashed line , the red dash - triple dotted line , and the purple dash - dotted line , respectively . all pns in the complete sample are shown by the solid black line . note the large increase in the distribution of bright pns and the moderate increase in the distribution of intermediate pns , relative to the faint pns , at @xmath65 ( see text for details ) . the ks probability of the faintest and brightest pns being drawn from the same underlying distribution is only @xmath66 , while the intermediate pns are still compatible with the faint pns ( ks probability @xmath67).,title="fig : " ] . solid ( dashed ) histograms show the mean velocity ( dispersion ) and their errors in different angular bins . the red ragged lines show running averages of mean velocity . _ top : _ velocities of pns brighter than @xmath68 in the complete sample . bin sizes are chosen so as to have approximately constant number of pns ( @xmath69 ) in each bin . the mean velocity along the major ( minor ) axis is positive ( negative ) on both sides of the center . the running average is also over @xmath70 pn velocities . the velocity dispersion is largest ( smallest ) on the minor ( major ) axis . _ middle : _ velocities of pns with @xmath71 in the same angular bins as in the top panel . the running average is over @xmath72 pns . _ bottom : _ velocities of pns with @xmath73 in the same angular bins as in the top panel . the faint pns show a rotation pattern as for the absorption line data but with a smaller peak velocity , and are consistent with a flat dispersion of @xmath74 . the running average is over @xmath75 pns . the blue dotted line shows a sinusoidal fit . the kinematics of the intermediate brightness pns is intermediate between the faint and bright pns.,title="fig : " ] . solid ( dashed ) histograms show the mean velocity ( dispersion ) and their errors in different angular bins . the red ragged lines show running averages of mean velocity . _ top : _ velocities of pns brighter than @xmath68 in the complete sample . bin sizes are chosen so as to have approximately constant number of pns ( @xmath69 ) in each bin . the mean velocity along the major ( minor ) axis is positive ( negative ) on both sides of the center . the running average is also over @xmath70 pn velocities . the velocity dispersion is largest ( smallest ) on the minor ( major ) axis . _ middle : _ velocities of pns with @xmath71 in the same angular bins as in the top panel . the running average is over @xmath72 pns . _ bottom : _ velocities of pns with @xmath73 in the same angular bins as in the top panel . the faint pns show a rotation pattern as for the absorption line data but with a smaller peak velocity , and are consistent with a flat dispersion of @xmath74 . the running average is over @xmath75 pns . the blue dotted line shows a sinusoidal fit . the kinematics of the intermediate brightness pns is intermediate between the faint and bright pns.,title="fig : " ] . solid ( dashed ) histograms show the mean velocity ( dispersion ) and their errors in different angular bins . the red ragged lines show running averages of mean velocity . _ top : _ velocities of pns brighter than @xmath68 in the complete sample . bin sizes are chosen so as to have approximately constant number of pns ( @xmath69 ) in each bin . the mean velocity along the major ( minor ) axis is positive ( negative ) on both sides of the center . the running average is also over @xmath70 pn velocities . the velocity dispersion is largest ( smallest ) on the minor ( major ) axis . _ middle : _ velocities of pns with @xmath71 in the same angular bins as in the top panel . the running average is over @xmath72 pns . _ bottom : _ velocities of pns with @xmath73 in the same angular bins as in the top panel . the faint pns show a rotation pattern as for the absorption line data but with a smaller peak velocity , and are consistent with a flat dispersion of @xmath74 . the running average is over @xmath75 pns . the blue dotted line shows a sinusoidal fit . the kinematics of the intermediate brightness pns is intermediate between the faint and bright pns.,title="fig : " ] the left - right asymmetry is confirmed by inspecting the azimuthal distribution of the faint and bright pns . in the literature we found a related analysis by @xcite who compared the azimuthal distribution of _ chandra _ x - ray point sources ( xps ) with the optical surface brightness of ngc 4697 . we follow their pa convention , and plot the cumulative angular distributions of the bright , intermediate , and faint pns in our complete sample in figure [ zezang ] . for comparison , the right panel of fig . 2 from @xcite is also shown . the angular distribution of all pns in the complete sample has a shape somewhere in between that of the xps and that of the optical light . the brightest pns are in complete disagreement with either of these distributions ; they seem to be more concentrated in a narrow angular sector between @xmath76 ( see fig . [ tspace ] ) , with only @xmath66 probability that the faint and bright pn subsamples are drawn from the same azimuthal distribution . at the same time , the radial distribution of the faint and bright subsamples are not significantly different ( figure [ radial ] ) . the velocity distribution of the brightest pns is also correlated with their azimuthal distribution . in figure [ vang ] we plot the mean radial velocity and its dispersion in angular sectors containing approximately constant number of pns from the bright subsample , as well as an angular running average of their mean radial velocity . along both sides of the major - axis of the galaxy , the brightest pns have positive velocity , while showing a relatively low velocity dispersion , perhaps due to infall as suggested for the xps by @xcite . along the minor - axis they have a large dispersion with a mean velocity that is negative . this kinematics is compatible with neither the faint pn velocities nor the stellar absorption line data . on the other hand , the fainter pns show a regular azimuthal distribution in the mean line - of - sight radial velocity and velocity dispersion . note that their velocity dispersion is in the mean smaller than that of the bright sample . the kinematics of the intermediate luminosity pns is intermediate between those of the bright and faint subsamples . the intermediate luminosity pns thus contain significant contributions from both of the different populations that dominate the bright and faint bins , respectively . of the major axis for bright and faint / intermediate pns separated at @xmath77 . blue circles show stellar absorption line spectroscopy ( als ) data from @xcite , red crosses are individual pn velocities , and grey diamonds are near neighbour running averages . the faint / intermediate sample is in approximate agreement with the als data , while the bright sample has a velocity consistent with zero within @xmath78 and positive velocities on both sides of the galaxy center . ] furthermore , including the bright pns in deriving kinematics for the whole pn sample introduces signficant contamination effects . in figure [ majkindb ] , we demonstrate this by plotting the mean pn velocities along a @xmath79 wide slit about the major - axis . the pns have been separated into faint / intermediate and bright sub - samples at @xmath77 . inside the @xmath78 , the mean velocities of the faint pns agree with stellar absorption line spectroscopic ( als ) data from @xcite , while the bright pns have much smaller mean velocities , consistent with @xmath80 . in the outer parts , the velocity asymmetry in the bright pns leads to a positive mean velocity on both sides of the galaxy center . both the faint / intermediate sample and the entire pn sample also show some asymmetry : the outer mean velocities on the @xmath81 side reach zero , but not the negative values expected from reflecting the positive values at @xmath82 . this confirms that also the intermediate / faint subsample contains a fraction of pns stemming from the out - of - equilibrium population traced by the bright pns . however , the streaming velocity of the majority of the faint pns does appear to decrease on both sides of the center . similarly contaminated results could be expected for the derived velocity dispersions . several important conclusions can be drawn from these figures . ( i ) the bright pns as defined in section [ subpop ] and fig . [ mvplot ] do not trace the azimuthal distribution of light in ngc 4697 . ( ii ) they do not trace the fainter pns in their azimuthal kinematics ; thus , a large fraction of them must belong to a separate pn subcomponent originating from a separate stellar population . ( iii ) third , they are not in dynamical equilibrium in the gravitational potential of ngc 4697 . ( iv ) because this subpopulation does not trace the stars , including its pn velocities into dynamical analysis of the galaxy will lead to significant errors in the results . probability . _ bottom : _ cumulative magnitude distributions of the main pn sample ( @xmath83 , blue dotted line ) and the extreme counter - rotating sample ( @xmath84 , red dash - dotted line ) defined in the text , along with that of the total sample ( black solid line ) . the first two luminosity functions are different with 99.7% confidence ; thus the pnlf can not be universal.,title="fig : " ] probability . _ bottom : _ cumulative magnitude distributions of the main pn sample ( @xmath83 , blue dotted line ) and the extreme counter - rotating sample ( @xmath84 , red dash - dotted line ) defined in the text , along with that of the total sample ( black solid line ) . the first two luminosity functions are different with 99.7% confidence ; thus the pnlf can not be universal.,title="fig : " ] what can we learn about the luminosity functions of the two kinematic components of the pn system in ngc 4697 ? the cumulative magnitude distributions of the co - rotating and counter - rotating subsamples in fig . [ mvplot ] have only @xmath67 ks probability of stemming from the same distribution ( top panel of figure [ ksmag ] ) . however , the fact that there are both co - rotating and counter - rotating bright pns in the bright sample whose azimuthal distribution is unmixed , shows that counter - rotation is not a _ clean _ discriminator after all for the secondary population of pns in ngc 4697 . thus the luminosity functions of the main and secondary pn populations in this galaxy may be a lot more different than this figure of @xmath67 would suggest . from fig . [ vang ] we estimate that the main population of pns in ngc 4697 has a mean radial velocity @xmath85 , with a dispersion of @xmath86 , so its reduced mean velocity is @xmath87 . thus in the bottom panel of fig . [ ksmag ] we show the cumulative magnitude distribution of the pns in the velocity range @xmath88 and compare it with the cumulative distribution of the pns with @xmath84 . these velocity ranges are dominated by the main and secondary pn populations in the sample , respectively . now the kolmogorov smirnov significance test shows that the magnitude distributions from these sections of fig . [ mvplot ] have only probability @xmath89 of being drawn from the same distribution . this result is strong enough to imply that the pnlf can not be universal the pnlf in ngc 4697 depends on a kinematic selection . in the following , we will use the velocity range @xmath90 as an approximate kinematic selection criterion for the main pn population in ngc 4697 . the luminosity function of the strongly counter - rotating pns in fig . [ mvplot ] a first approximation to the luminosity function of the secondary pn population in ngc 4697 differs from that of the main pn distribution so defined in the sense that it contains more bright pns near the cut - off and fewer faint pns than the main population ( see fig . [ ksmag ] ) . now an important question is : in what proportion do both populations contribute to the brightest pns , and do they have different cutoff magnitudes ? . circled symbols denote counter - rotating pns ; many of these are kinematic outliers . the two grey lines denote the position angle of the minor axis . the distribution of blue diamonds is approximatively uniform in pa except near pa@xmath91 where there are 12 pns ( crossed diamonds ) instead of the expected 2 pns . _ bottom : _ radial velocity magnitude plane . the symbols are as in the top panel . 6/8 of the brightest pns are either kinematic outliers or are found in the overdense angular region . , title="fig : " ] . circled symbols denote counter - rotating pns ; many of these are kinematic outliers . the two grey lines denote the position angle of the minor axis . the distribution of blue diamonds is approximatively uniform in pa except near pa@xmath91 where there are 12 pns ( crossed diamonds ) instead of the expected 2 pns . _ bottom : _ radial velocity magnitude plane . the symbols are as in the top panel . 6/8 of the brightest pns are either kinematic outliers or are found in the overdense angular region . , title="fig : " ] to investigate this , we show in figure [ vla ] the velocities , magnitudes , and position angles of the entire bright pn subsample ( see figs . [ mvplot ] and [ vang ] ) . the top panel shows ( i ) that 13/16 of the bright pns , whose radial velocities differ most from those of the faint population , are counterrotating . this explains the differences between the velocity distributions of co- and counter - rotating pns that first suggested more than one pn population in section [ subpop ] . ( ii ) also , even in the kinematically normal bright pns , there is a large overdensity ( 10/12 ) in the angular range pa@xmath91 . the bottom panel shows in addition that many of the brightest pns are either kinematic outlyers or found in the angular overdensity ( 6/8 ) . clearly , to arrive at a main population of ngc 4697 pns that is in dynamical equlibrium in the gravitational potential , we must remove the angular overdensity . then we are left with 16 pns in the range @xmath92 of a total bright subsample of 42 pns ; however , there is some freedom in the way the angular overdensity is removed . in the following , we explore two assumptions : ( i ) using all 12 pns in the angular overdensity , but weighting each one by @xmath93 , and ( ii ) removing the brightest 10 of the 12 pns in the overdensity . the resulting luminosity functions for both cases are plotted in figure [ clumfun ] ; they differ only slightly . thus in the following we use the kinematic condition @xmath94 together with assumption ( ii ) above to ensure azimuthal uniformity of the bright pns , as an improved selection criterion for pns in the main population in ngc 4697 . we keep all pns in the intermediate luminosity bin with @xmath92 , because fig . [ zezang ] showed that their azimuthal asymmetry is not large . also , we have checked that the mean angular velocity distribution in this magnitude bin after applying the kinematic selection follows approximately the sinusoidal variation of the faint pns , and the velocity disperion is approximately constant . the resulting main population sample is identified by their brightness , distribution and kinematic properties . comparing the luminosity function in figure [ clumfun ] of this main pn population , with the luminosity function of all pns in ngc 4697 , we see that the bright cutoff of the main population is shifted to fainter magnitudes . we note that the bright cutoff could be shifted further to fainter magnitudes if some of the kinematically normal and azimuthally uniform bright pns were also part of the secondary population , which we can not tell from the present data . we can ask now what is the effect , in practice , on the pnlf distance determination . the reduced sample for the main pn population in ngc 4697 has 214 objects . after binning these data into 0.2 mag intervals , we transform the apparent magnitudes @xmath12 into absolute magnitudes , adopting the extinction correction of 0.105 mag @xcite and assuming different distance moduli , and we compare the results with the pnlf simulations of @xcite . this is the same procedure used in @xcite for the pnlf distance determination . the comparison is shown in fig . we conclude that the pnlf distance modulus should be increased slightly from 30.1 ( the earlier determination based on the full sample ) to 30.2 or 30.25 . for comparison , the @xmath95-fit to the same data ( blue line in fig . [ clumfun ] ) gives 30.22 . this correction would bring the pnlf distance modulus in better ( but not perfect ) agreement with the surface brightness fluctuation ( sbf ) distance modulus ( @xmath96 ) reported by @xcite . ) of 214 objects , binned into 0.2 mag intervals . the apparent magnitudes @xmath12 have been transformed into absolute magnitudes @xmath97 by adopting an extinction correction of 0.105 mag and distance moduli indicated in each plot . the three lines are pnlf simulations @xcite . for a distance modulus 30.15 the brightest pns are a bit too weak , therefore the distance must be increased . but 30.35 is clearly excessive . the best fit is for 30.2 or 30.25 . , title="fig : " ] ) of 214 objects , binned into 0.2 mag intervals . the apparent magnitudes @xmath12 have been transformed into absolute magnitudes @xmath97 by adopting an extinction correction of 0.105 mag and distance moduli indicated in each plot . the three lines are pnlf simulations @xcite . for a distance modulus 30.15 the brightest pns are a bit too weak , therefore the distance must be increased . but 30.35 is clearly excessive . the best fit is for 30.2 or 30.25 . , title="fig : " ] bright pns play a significant role in the analysis of the last section . is it possible that the brightest pns in ngc 4697 are contaminated by compact hii regions such as those observed in @xcite ? the recent observations of @xcite have shown that this can not be so . these authors took spectra of 13/42 pns in our bright subsample in ngc 4697 with fors2@vlt ; these have no detectable continuum and the line ratios of metal - rich pns . the same argument also shows that these bright pns can not be background emission line galaxies . moreover , ly alpha emission galaxies would come in at [ oiii ] magnitudes of m@xmath98 ( see fig . 4 of * ? ? ? * ) , while the bright pns in ngc 4697 have m@xmath99 . furthermore , one may wonder whether the bright pns in ngc 4697 might simply be foreground objects which could be closer to us and hence brighter than the true ngc 4697 pns on which they would be superposed . given that ngc 4697 is located in the southern extension of the virgo cluster , known to contain an intracluster population of pns @xcite , this possibility deserves to be considered . however , the following observational facts show that the bright pns in ngc 4679 are not intracluster pns ( icpns ) . ( 1 ) ngc 4697 is in fact closer than the virgo cluster . the pnlf from @xcite places it at @xmath100 ( m - m@xmath101 ) , while the distance modulus from the pnlf of m87 is m - m@xmath102 @xcite . even if we discarded the entire brightest 0.3 mag of the ngc 4697 pns , ngc 4697 would still be at 75% of the distance of m87 . ( 2 ) the velocity dispersion of the bright and unrelaxed pns in ngc 4697 is @xmath103 , but varying azimuthally , while that of the fainter main population is @xmath104 ( table [ tabvtftest ] , figs . [ mvplot ] , [ zezang ] ) . both are much smaller than the velocity dispersion of icpns in virgo @xcite . the radial distribution of the bright population outside the incompleteness ellipse is concentrated towards the galaxy center and ks compatible ( 92% ) with the radial distribution of the fainter pns ( see fig . [ radial ] ) . thus also the bright pns in ngc 4697 are bound to the galaxy . ( 3 ) the surface density of pns with m@xmath99 is 0.58 pns / arcmin@xmath105 , while the mean surface density of virgo icpns is 0.02 pns / arcmin@xmath105 , much smaller @xcite . this last argument also rules out significant contamination by chance superpositions of pns even closer than ngc 4697 . we conclude that the bright pn population in ngc 4697 consists of genuine pns , and that it is dynamically bound to the ngc 4697 system . the irregular angular distribution and kinematics of these pns , by comparison with the fainter main population , show that they must belong to a separate stellar population not yet in dynamical equilibrium with ngc 4697 . pn samples as large as the one for ngc 4697 are still the exception . yet to undertake the analysis described in section [ analysis ] it was crucial to work with a complete sample of some 300 pns . the pn.s project @xcite may lead to complete samples of similar size but as of now the typical sample sizes are @xmath106 @xcite and their magnitude distributions and completeness have not been studied . the only other elliptical galaxy with a comparable ( even larger ) sample is cen a @xcite . a detailed analysis of the distribution of cen a pns in the magnitude velocity plane is still pending , but @xcite analysed the pnlf in cen a as a function of radius , based on narrow - band surveys . they concluded that no population effect on the pnlf bright cut - off could be seen , suggesting that the 0.3 mag difference between their main and outer halo samples was due to filter transmission uncertainties . analysis of the large sample from @xcite will clarify whether the pns in cen a are consistent with one or more subpopulations . this will be particularly interesting because cen a is believed to be the remnant of a galaxy merger , so one might expect pns from both the older stellar populations of the progenitors as well as from the stars formed in the subsequent interaction between them . without large kinematic samples , searches for pn subpopulations must be based on the pn luminosity distributions . the work of jacoby , ciardullo and collaborators cited in the introduction has shown that the pnlf is remarkably uniform . however , there are exceptions : we recall that in the halo of m84 there exists a small population of overluminous pns whose cutoff is @xmath107 mag brighter than that of the main m84 population , but which appear nonetheless bound to the halo of m84 @xcite . they must therefore be intrinsically bright , due to some stellar population difference . in m87 , the only overluminous pns projected onto the galaxy for which velocities have been measured @xcite , have very large relative velocities with respect to m87 . this is most naturally explained if these pns have fallen into the deep potential well of m87 from far out in the cluster ; this would again imply an intrinsic population difference . however , a larger kinematic sample in m87 is required to be certain . the existence of the bright pn subpopulation in ngc 4697 implies some uncertainty in the pnlf cut - off luminosity for this galaxy . depending on whether or not the azimuthally symmetric brightest pns are part of the main population , the cut - off luminosity of the main pn population in ngc 4697 is fainter than that of the whole population by @xmath108 mag ( figs . [ clumfun ] , [ rob ] ) . while this is consistent with the m84 result , is it also consistent with the systematic studies of pnlf distances ? @xcite have compared the pnlf and sbf distances , finding a distribution of residuals with a systematic offset by @xmath109 mag , which they suggested is due to internal extinction effects , and with a fwhm of @xmath110 mag . the offset has in the mean - time been reduced to @xmath111 mag , following a revision of the sbf distance scale by @xcite . the width of this distribution is consistent with their determination of the observational errors in both methods . however , the offset we have determined for ngc 4697 is also consistent with the distribution of residuals in @xcite . we have shown that a large fraction of the bright pns in ngc 4697 belong to a secondary , dynamically young stellar population that is not well - mixed in the gravitational potential of the galaxy . late infall of tidal structures @xcite or a merger with a smaller galaxy some time ago would be natural ways to add such an unmixed stellar component to ngc 4697 . what physical population difference is correlated with this dynamical youth ? @xcite show from their spectroscopic data for 13 bright pns that these have near - solar metallicities . of these 13 bright pns , 6 are inside the incompleteness ellipse , one has no measured velocity , and 6 belong to our secondary population . @xcite also use long - slit spectroscopy to show that the metallicity of the integrated stellar population within one effective radius has solar or higher metallicities . these observations make it unlikely that metallicity is the main factor responsible for the different magnitude distributions of the main and secondary pn populations in ngc 4697 . thus the more likely driver would appear to be an age difference , as suggested by @xcite and as might generally be expected in an accretion event . based on their result that the distribution of x - ray point sources in ngc 4697 does not follow the stellar light , @xcite have argued that this is because these sources were formed several @xmath112 years ago in tidal tails that are now falling back onto the galaxy . note , however , that the integrated light in ngc 4697 shows no evidence of young stars with mean age @xmath113 gyr @xcite , so this younger component could not be luminous enough to contaminate the integrated light to the level measured . also note that the observed increase of extinction in the pn envelope with pn core mass more than compensates for the increase of core luminosity with core mass , for bright pne in local group galaxies @xcite , so that stars with ages below @xmath114 gyr may not reach the [ oiii ] luminosity at the pnlf cutoff . a secondary stellar population younger than @xmath114 gyr is therefore unlikely as well . recently , @xcite have argued that the brightest pns in the pnlf must have core masses of @xmath115 , corresponding to main sequence masses of @xmath116 . they argue further that for such high - mass objects to occur in elliptical galaxies , these early - type galaxies would either have to contain a small , smoothly distributed component of young ( @xmath117 gyr age ) stars , or more likely , that the bright pns in these systems have evolved from blue straggler stars created through binary evolution . their blue straggler model , due to the assumption of a fixed distribution of primary - to - secondary mass ratios for the initial binaries , predicts that older stellar populations produce fewer bright pns per unit luminosity , as is observed , because the number of binary stars in a stellar population that can coalesce to @xmath116 blue stragglers decreases with time . if correct , this blue straggler model could also explain how the secondary population we found in ngc 4697 can contain a large fraction of the brightest pns in this galaxy , provided that the stellar population corresponding to this secondary pn population is appreciably younger than the main stellar population , whose age is @xmath16 gyr from optical spectroscopy @xcite . at the same time , this secondary stellar population must not be so young to violate either the constraints from the optical colours or from the envelope absorption - pn core mass correlation , i.e. , must be older than @xmath118 gyr . we can give an estimate for the effect of such an intermediate age population on the optical colours as follows . the secondary subpopulation traced by the bright and predominantly counterrotating pns may contain @xmath119 of all pns in the complete sample for ngc 4697 . a stellar population as blue as the bulge of m31 has a luminosity - specific pn density per unit @xmath120 up to 5 times higher than the populations characteristic for old elliptical galaxies @xcite . thus the subpopulation corresponding to the secondary pn population in ngc 4697 would be expected to contain @xmath121 of the blue luminosity of ngc 4697 , spread over a large fraction of at least the e image . to detect this we need deep and accurate photometry . the unmixed spatial and velocity distributions of the secondary pn population in ngc 4697 show that that this population is _ dynamically young _ , i.e , has not had time to phase - mix and come to dynamical equilibrium in the gravitational potential of ngc 4697 . it may well be associated with tidal structures that were formed in a merger or accretion event @xmath122 gyr ago , and that have now fallen back onto the galaxy , or be associated with a more recent merger / accretion with a red galaxy such as described in @xcite . in a universe in which structures form hierarchically , such secondary stellar populations might be quite common in ellipitical galaxies , but they would be difficult to see . the present work shows that studying their pn populations is one promising approach of looking for such secondary populations . however , large pn samples are required ; most existing pn studies of early - type galaxies do not have the statistics for such an investigation . moreover , in only a fraction of cases may there be enough asymmetry signal to detect with a few hundred pns . we have analysed the magnitudes , kinematics and positions of a complete sample of 320 pns in the elliptical galaxy ngc 4697 from @xcite . this data set is large enough for drawing statistically significant conclusions , and it does not suffer from detection incompletenesses in either magnitudes or radial velocities . we know of no systematic effects in the data that could explain our results . our main conclusions are : 1 . bright and faint pns in ngc 4697 have significantly different radial velocity distributions . the mean velocities of the faint and bright subsamples ( co - rotating and @xmath80 , respectively ) and their velocity dispersions are different , with 94% and 99.3% confidence . thus the pns in ngc 4697 do not constitute a single population that is a fair tracer of the distribution of all stars . 2 . the luminosity functions of the extreme counter - rotating subsample ( @xmath84 ) and of the main population ( defined by @xmath123 ) are different with 99.7% confidence . the pnlf is therefore not universal . 3 . based on this , we suggest that there exist ( at least ) two pn populations in this galaxy . the secondary pn population in ngc 4697 is prominent in a sub - sample of counter - rotating pns brighter than @xmath0 . the luminosity function of the entire extreme counterrotating sample may be a first approximation to the luminosity function of the secondary population . the spatial distribution of bright pns with @xmath124 is different from that of the faint pns . the bright pns do not follow the azimuthal distribution of the optical light , show a left - right asymmetry , and have a positive mean radial velocity on both sides of the galaxy major axis , but zero velocity and larger dispersion on the minor axis . they are not in dynamical equilibrium in the potential of the galaxy . the fainter population has rotation properties more similar to the absorption line velocities , with azimuthally constant dispersion . 5 . using both their kinematics and angular distribution , we can estimate a lower limit to the statistical fraction of bright pns in the secondary population . based on this we estimate that the bright cutoff of the main population is uncertain by @xmath125 mag . our results have two main implications for the use of pns in extragalactic astronomy . first , for distance determinations with the pnlf , it may be important to understand how uniform the pn populations in the target galaxies are . from our analysis in ngc 4697 we estimate that unrecognized subpopulations of pns in smaller samples than that in ngc 4697 may lead to variations of @xmath125 mag in the bright cutoff . this would correspond to distance errors of some @xmath126 , which , although a minor effect , could be significant in some cases . it will be necessary to verify how frequently such subpopulations occur in elliptical galaxies . we recall that also in the halo of m84 there exists a small population of overluminous pns whose cutoff is @xmath107 mag brighter than that of the main m84 population , but which appear nonetheless bound to the halo of m84 @xcite . the second implication concerns the use of pns as tracers for the angular momentum and gravitational potentials of elliptical galaxies . our analysis has shown that in ngc 4697 the bright pns do not trace the distribution and kinematics of stars and are not in dynamical equilibrium in the gravitational potential of the galaxy . the fainter pns look more regularly distributed but may also contain a fraction of pns that belongs to this out - of - equilibrium population . clearly therefore , mass determinations based on pn kinematics will in future require careful study of the pn samples being used , not only to verify that these pns are in dynamical equilibrium , but also to test for different dynamical components . even if in equilibrium , a younger population of stars may be more flattened or have a steeper fall - off than the main body of the elliptical galaxy . if the pns from this population are indeed somewhat brighter than the main population , one can recognize such differences from their lower velocity dispersion or different radial density profile . however , deep observations and large pn samples will be required . we are grateful to m. arnaboldi , r. ciardullo and r. saglia for helpful discussions , and to k. freeman , g. jacoby , e. peng and a. romanowsky for helpful comments on the manuscript . ns and og thank the swiss nationalfonds for financial support under grant 200020 - 101766 . rhm would like to acknowledge support by the u.s.national science foundation , under grant 0307489 .

we have analysed the magnitudes , kinematics and positions of a complete sample of 320 pns in the elliptical galaxy ngc 4697 . we show ( i ) that the pns in ngc 4697 do not constitute a single population that is a fair tracer of the distribution of all stars . the radial velocity distributions , mean velocities , and dispersions of bright and faint subsamples differ with high statistical confidence . ( ii ) using the combined data for pns brighter than @xmath0 , we have identified a subpopulation of pns which is azimuthally unmixed and kinematically peculiar , and which thus neither traces the distribution of all stars nor can it be in dynamical equilibrium with the galaxy potential . ( iii ) the planetary nebula luminosity functions ( pnlf ) of two kinematic subsamples in ngc 4697 differ with 99.7% confidence , ruling out a universal pnlf . we estimate that the inferred secondary pn population introduces an uncertainty in the bright cutoff magnitude of @xmath1 mag for this galaxy . we argue that this secondary pn distribution may be associated with a younger , @xmath2 gyr old stellar population , perhaps formed in tidal structures that have now fallen back onto the galaxy , as has previously been suggested for the x - ray point sources in this galaxy , or coming from a more recent merger / accretion with a red galaxy . the use of pns for extragalactic distance determinations is not necessarily compromised , but their use as dynamical tracers of dark halos will require deep observations and careful analysis of large pn samples .

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Proceed to summarize the following text: the energetic gamma - ray experiment telescope ( egret ) detected 66 active galactic nuclei ( agn ) with @xmath0-ray emission at energies greater than 100 mev @xcite . almost all of the agn identified were blazars , triggering immense interest and prompting a series of studies focused on the multi - wavelength properties of these agn . egret s successor , the large area telescope ( lat , onboard the _ fermi gamma - ray space telescope _ ) , has identified over 1000 @xmath0-ray sources with energies greater than 100 mev @xcite , revealing the @xmath0-ray sky in detail . the lat instrument presents the opportunity to revisit questions first examined in the egret era with far superior data . egret s discovery opened a new direction in investigating the nature and physical processes of blazars . blazars are radio sources characterized by a compact core and relativistic jets that are aligned at a small angle to our observational line - of - sight , i.e. flat - spectrum radio quasars and bl lac objects @xcite . it is supposed that @xmath0-ray emission from agn is produced from inverse - compton ( ic ) processes occuring within the jets and becomes apparently enhanced via relativistic beaming @xcite . although there is good support for this scenario , several questions remain , particularly with respect to the origin of @xmath0-ray production within the jets @xcite , as well as how ic processes act to produce @xmath0-ray emission @xcite . relativistic beaming ( or doppler boosting ) is evidenced by other associated properties that characterize blazars : high @xmath0-ray luminosity ( which comprises a large portion of the total radiative output ) @xcite ; apparent superluminal motion @xcite ; and rapid variability from radio to @xmath0-ray wavelengths @xcite . from the non - thermal nature of this emission , it is understood that it is produced from within the agn jets and is relativistically beamed along the line - of - sight . this could explain why @xmath0-ray - loud agn are also radio - loud but not all radio - loud agn are @xmath0-ray - loud . if @xmath0-ray emitting regions move at faster speeds than radio emitting regions , @xmath0-ray emission would be doppler boosted within a narrower cone @xcite . outside the @xmath0-ray beaming cone and within the radio beaming cone , @xmath0-ray radiation would be doppler dimmed , giving the appearance of a @xmath0-ray - quiet yet radio - loud agn . a second possibility suggests that the appearance of @xmath0-ray emission may be related to agn jet - bending @xcite . if a bend in the jet occurs downstream of the @xmath0-emitting region and upstream of regions of extended radio emission , the emissions would be doppler - boosted in alignment to their respective sections of jet . as such , it may be possible that @xmath0-ray emission is beamed away from us whilst radio emission is beamed towards us due to a better alignment of the radio - emitting jet section with our line - of - sight . several studies have compared @xmath0-ray - loud and @xmath0-ray - quiet agn populations with respect to characteristics of the agn core and jet components . statistically significant differences between these populations are found for core brightness temperature , jet opening angles and core polarization @xcite . the distinction between populations is also established for radio flux density variability between @xmath0-ray - loud and @xmath0-ray - quiet agn @xcite , suggesting that radio and @xmath0-ray emission are produced within the same region within agn jets @xcite . @xcite propose that radio - to-@xmath0-ray variability is caused by oblique shocks in the agn jets . the role of oblique shocks has also been explored in jets on parsec @xcite and kiloparsec scales @xcite . @xcite finds a correlation between @xmath0-ray and radio flux in agn after consideration of biases unaccounted for in previous studies @xcite . studies have also correlated superluminal ejections and @xmath0-ray flares in agn jets , suggesting that @xmath0-ray emission is produced within the parsec - scale region of the jets @xcite . furthermore , studies have related @xmath0-ray emission with jet kinematics @xcite , and there is evidence to suggest that fsrqs and bl lac objects may have intrinsically different mechanisms for @xmath0-ray emission @xcite . in examining the jet bend scenario of @xcite , ( * hereafter paper i ) used egret identifications to compare @xmath0-ray - loud and @xmath0-ray - quiet agn with respect to their jet - bending characteristics . their results showed a tendency in @xmath0-ray - quiet sources to both have more jet bends and more pronounced jet - bend angles at parsec scales . @xcite did not find evidence to support this scenario , but their study was limited in only examining parsec - to - kiloparsec jet misalignment angles . the jet bend proposition was revisited in @xcite and @xcite with inconclusive results ( limited to 4 egret candidates and 30 lat sources respectively ) . similarly , @xcite found that no significant difference in jet - bend angles between @xmath0-ray - loud and @xmath0-ray - quiet populations ( limited to 19 lat sources ) . utilizing lat s superior @xmath0-ray detection capabilities and a larger sample size , we re - examine the suggestion that jet - bending may be a significant factor for the detection of @xmath0-ray emission in agn . we follow the analysis of paper i , testing the statistical significance of jet - bend angles , as well as the number of bends between @xmath0-ray - loud and @xmath0-ray - quiet agn . we use the ` clean sample ' of the second lat agn catalog ( 2lac , * ? ? ? * ) and vlbi images accessible in the published literature or from the radio fundamental catalog ( rfc , version ` rfc2013d ' ) . we examine the statistics of two samples , a large but inhomogeneous sample and a smaller but homogeneous subsample based on ( * ? ? ? * hereafter mojave ) . in order to investigate the significance of jet - bending between @xmath0-ray - loud and @xmath0-ray - quiet agn populations , we surveyed the literature for vlbi images and measured source properties . the most recent version of the rfc contains 8310 compact radio sources studied in 5719 vlbi observing sessions , making it the most complete catalog of radio sources with available vlbi data . we have compared the rfc with the crates catalog @xcite , a nearly uniform survey of of flat - spectrum radio sources ( @xmath1 ) containing flux densities and spectral indices for over 11,000 objects . by cross - referencing data from the crates catalog to sources in the rfc , we obtain a large pool of sources to apply selection criteria and draw upon in order to define a ` rfc - based sample ' for the purpose of our investigation . there are a number of considerations in formulating our selection criteria for the rfc - based sample . a true cross - identification between radio / x - ray and @xmath0-ray counterparts can only be determined when there is a correlation in variability in both bands . in practice , cross - identifications were made only for 28 sources in the 2-year fermi - lat sources catalog ( 2fgl , * ? ? ? * ) . as such , most sources identified as @xmath0-ray - loud in 2lac are high - confidence statistical associations determined by three association methods using the detected positions and uncertainties of radio / x - ray and @xmath0-ray emissions within a source s 95% error ellipse @xcite , along with the physical properties of likely candidates . the determination of a high - confidence association in 2lac is also subject to a number of considerations @xcite . candidate gamma - ray sources are ` detected ' in 2fgl if a region of concentrated flux against its background obtains a test statistic ts @xmath2 25 ( corresponding to over 4 @xmath3 significance ) . there is low availability of spectroscopic information for radio sources in southern declinations . there are in some case multiple associations within a gamma - ray source s 95% error ellipse . a candidate s gamma - ray spectral intensity may not lie within lat s energy - dependent sensitivity curve . candidates with weak radio flux would also have reduced likelihoods of association as they become indistinguishable from other weak radio sources within the lat error ellipse . these factors would cause bias to higher unassociated agn despite possibly having detectable @xmath0-ray flux . to address these issues , we employ the ` clean sample ' subset of 2lac where sources are excluded if they presented difficulties in the analysis of making an association . we also apply a low threshold to flux densities to account for the possibility of false non - associations below the cutoff . there is also possible bias introduced by @xmath0-ray attenuation due to the extragalactic background light ( ebl ) , resulting in a lower rate of association at higher redshifts . predictions from ebl models show that @xmath0-rays with energies below @xmath4 10 gev from redshifts up to z @xmath4 3 do not undergo significant attenuation @xcite , and @xcite find no redshift dependence in the flux ratio of @xmath0-ray photons f(@xmath2 10 gev)/f(@xmath2 1 gev ) across blazar subclasses from z = 0 to above z = 2 . applying an upper limit to redshift can eliminate the possibility of this bias in our sample . furthermore , errors in modelling the diffuse @xmath0-ray background near the galactic ridge or in nearby interstellar cloud regions may introduce false associations between @xmath0-ray and radio emissions belonging to two distinct sources , identifying ` @xmath0-ray - loud ' agns that are actually @xmath0-ray - quiet . however , 2lac selection criteria include only sources with high galactic latitudes ( @xmath5 ) , eliminating the possibility of false or non - associations occuring near the galactic ridge . in review of these factors in @xmath0-ray detection and source association , our source selection criteria are therefore as follows : 1 . identified in rfc ; 2 . galactic latitude @xmath5 ; 3 . total 4.85 ghz flux density @xmath6 jy ; 4 . redshift z @xmath7 2 ; 5 . spectral index @xmath8 ; 6 . included in clean sample if included in 2lac ; 7 . associated with vlbi images in literature . these criteria define our rfc - based sample , consisting of 351 agn in total . the sample contains 151 @xmath0-ray - loud agn from the clean sample , a subset of sources from the second lat catalog that possess high - confidence associations between radio and @xmath0-ray counterparts @xcite ( see table [ tab : gl_list ] ) . a @xmath0-ray - quiet subsample was defined by the 200 agn not detected by lat ( see table [ tab : gq_list ] ) . we consider a subsample of the rfc - based sample defined by sources existing in mojave to investigate concerns of sample inhomogeneity @xcite . the ` mojave subsample ' is an unbiased subset since the sources included conform to our selection criteria for the rfc - based sample . the sample contains 65 @xmath0-ray - loud agn and 29 @xmath0-ray - quiet agn , giving a total sample of 94 agn . the sources were observed using the vlba and processed uniformly . the selection criteria used in mojave are also relatively homogeneous , only departing slightly from uniformity in the criterion for radio flux density between agn at northern ( 1.5 jy ) and southern ( 2.0 jy ) declinations to account for differences in instrument sensitivity . the high fidelity imaging of this sample makes it ideal for jet - bend measurements . the sample is also relatively large , which is advantageous from a statistical standpoint . the data from mojave were not available when paper i was published . all mojave agn were imaged at 15 ghz in the period august 1994 to september 2007 @xcite . @xcite suggest two scenarios where jet bending could influence @xmath0-ray detection in radio - loud agn . if @xmath0-ray emission is produced within an agn jet , and if the jet bends significantly downstream of the @xmath0-ray emitting region , there is a possibility that the @xmath0-ray emission is beamed away from our line of sight ( doppler - dimmed ) whilst radio emission is beamed towards us . in this case we would observe a @xmath0-ray - quiet but radio - loud agn . the other possibility is that there may be agn with jets either with no bends or with bends upstream of the @xmath0-ray - emitting region , producing aligned radio and @xmath0-ray beaming cones . in this case , we would observe a @xmath0-ray and radio - loud agn . such possibilities may account for @xmath0-ray observations not otherwise explained by relativistic beaming . two other simple scenarios may also be considered . first , objects for which we lie within the @xmath0-ray beaming cone but not the radio beaming cone . such sources would be observable to us as @xmath0-ray - loud , radio quiet agn . however , it is reasonable to assume that substantial radio emission occurs in regions of @xmath0-ray emission , making this scenario unlikely . second , agn jets for which both @xmath0-ray and radio beaming cones are misaligned with our line of sight . these sources will likely not be identified in @xmath0-rays and may appear as weak radio sources . in our approach , the number of parsec - scale jet bends and the maximum parsec - scale jet - bending angles are recorded for agn with vlbi images that present a discernible jet structure from which jet - bend data may be extracted . the hypothesis of @xcite allows the possibility that jet - bending could be significant on kiloparsec scales . however , neither @xcite nor paper i find a significant difference in the parsec - to - kiloparsec misalignment angle between @xmath0-ray - loud and @xmath0-ray - quiet populations . in addition , paper i does not find a statistically significant difference in the number of kiloparsec - scale jet - bends . more recent investigations assert that @xmath0-ray emission originates in the parsec - scale region , near the core , although the precise region continues to be discussed @xcite . in the scenario proposed in these studies , both @xmath0-ray and radio emission may be produced far upstream of any kiloparsec - scale bending , allowing the detection of both kinds of emission regardless of the existence of kiloparsec bending . kiloparsec - scale bending does not appear to impact @xmath0-ray detection in agn , consistent across a number of studies , and is not explored here . we applied the following method to our jet - bending measurements . for a given agn , we compared vlbi images available from the rfc and in other major surveys . considering the resolution and common features between epochs , we determined the agn s jet structure that could be discernable by eye . in this process , we assume that an abrupt change in the apparent direction of a jet , associated with a ` shock ' region of high flux density , corresponds to the bending between two jet sections . we selected the best vlbi image to derive its jet properties , with preference always given to available mojave images due to their superior image quality . the initial angle of the jet from the core was measured east of north in the equatorial coordinate system . the angles of any existing bends in the jet are then measured ( in the same sense ) with respect to the previous jet trajectory closer to the core . the number of bends and the maximum jet - bending angle were then recorded for each source , shown in tables [ tab : gl_list ] and [ tab : gq_list ] . there are a number of factors affecting an agn s derived jet properties . inhomogeneity is introduced from the variable resolution and sensitivity of vlbi instrumentation , the choice of radio frequency for imaging , and different image epochs for different sets of images associated with a particular agn . as such , the low fidelity of some vlbi images may affect the consistency of jet measurements across the whole sample . despite this , using the best available vlbi images for each agn will provide the most accurate jet - bending properties to retest the jet - bending hypothesis . the ability to resolve jet components also diminishes with decreasing radio flux density . in weak radio sources , this is particularly evident in the reduced ability to resolve low flux density jet components ( and therefore jet bending ) at large distances from the core . deflections in jets may not be obvious at small jet - bending angles , and where there are jet - bends , they may not occur at a single , well - defined point along the jet . the comparison between multiple vlbi images of agn across different imaging frequencies and epochs also allows us to obtain jet properties that reflect the jet structure from the observer s point of view as accurately as possible . furthermore , bright individual features with different trajectories within or around a jet may also give the false appearance of jet bending in jet structure in certain imaging epochs @xcite . comparing vlbi images at different epochs assists in identifying bright regions that may temporarily distort the appearance of the jet structure . measurements are subject to image fidelity and morphological interpretation , however , these considerations should provide a robust result when testing the significance of jet bending in @xmath0-ray detection compared to the findings of previous , more - limited studies . we employ a number of statistical tests to explore differences between @xmath0-ray - loud and @xmath0-ray - quiet populations in our sample . a two - sided kolmogorov - smirnov ( k - s ) two - sample test was used to determine the statistical significance of differences at 4.85 ghz flux density and maximum jet - bend angle distributions between @xmath0-ray - loud and @xmath0-ray - quiet agn populations . since the values for the number of jet bends are discrete , a @xmath9 test was applied to characterize differences between the distributions in this case . 13 @xmath0-ray - loud and 32 @xmath0-ray - quiet sources did not show discernible jets and thus were omitted from the statistical testing of jet properties . for all tests , the difference between sample populations is considered significant for a given property when , assuming the null hypothesis represents both samples being drawn from the same parent population , the probability of observing a test statistic at least as extreme as the one obtained ( @xmath10 ) is less than 0.05 ( 95% confidence that the null hypothesis can be rejected ) . these results are summarized in table [ tab : agn_stat ] . figures [ fig : lfx_agn_gl ] and [ fig : lfx_agn_gq ] show similar distributions for agn total flux densities at 4.85 ghz . a k - s test determines that the probability that the two samples have been drawn from the same parent population is @xmath11 , finding no statistically significant difference between the two distributions . the result gives confidence that there no biases in lat detections that may skew jet - bending statistics . we now consider the jet - bending statistics of @xmath0-ray - loud and @xmath0-ray - quiet samples . the distributions between samples for the number of jet bends ( figures [ fig : nb_agn_gl ] and [ fig : nb_agn_gq ] ) show no significant differences . a @xmath9 test finds a statistically insignificant probability of @xmath12 ( @xmath13 with 2 degrees of freedom ) . similarly , histograms for the parsec - scale maximum jet - bend angle ( figures [ fig : mb_agn_gl ] and [ fig : mb_agn_gq ] ) demonstrate no significant difference between the two populations . the probability that the two samples belong to the same parent population is @xmath14 . we find that there are no statistically significant differences between @xmath0-ray - loud and @xmath0-ray - quiet agn populations for any of the tested properties . the results of the statistical tests on the mojave subsample are given in table [ tab : agn_stat ] . the @xmath0-ray - loud and @xmath0-ray - quiet distributions for these properties are shown in figure [ fig : lmn_mjv ] . we have investigated the suggestion that @xmath0-ray - quiet agn have a larger tendency for jet bending than @xmath0-ray - loud agn @xcite . we present a statistical analysis of a large sample of agn that represents the population of radio - loud agn , using data from the fermi lat instrument . we also conducted an analysis of a homogeneous subsample ( mojave ) to address the inhomogeneity of selection criteria and vlbi image quality in the rfc - based sample . our analysis of the rfc - based sample shows that jet - bending does not play a significant role in predicting @xmath0-ray detectability in radio - loud agn . as such our results do not support the suggestion of @xcite . this conclusion is in contradiction to paper i , and supports the findings of @xcite . the contrast of results obtained by new data with those of paper i reveals the strong dependence on the fidelity and homogeneity of vlbi images in jet - bend measurements . the limited quality and homogeneity of vlbi images available at the time of study appears to be the reason for the reported conclusions in paper i. the results of the mojave subsample generally support the conclusions drawn from the rfc - based sample . statistically insignificant results were found for both the number of parsec - scale jet bends and maximum parsec - scale jet bend , indicating that the rfc - based sample is not adversely affected by vlbi image inhomogeneity and giving additional confidence to the null result . we thank the referee for providing helpful comments and suggestions . this research has made use of data from the radio fundamental catalog ( rfc ) maintained by l. petrov , solution rfc2013d ( unpublished , available on the web at http://asrtogeo.org/vlbi/solutions/rfc2013d ) ; the mojave database , maintained by the mojave team ( lister et al . 2009 , aj , 137 , 3718 ) ; the crates catalog ( healey et al . 2007 , apjs , 171 , 61 ) ; and the nasa / ipac extragalactic database ( ned ) , operated by the jet propulsion laboratory , california institute of technology , under contract with the national aeronautics and space administration . abdo , a. a. , ackermann , m. , ajello , m. , et al . 2010b , , 723 , 1082 ackermann , m. , ajello , m. , allafort , a. , et al . 2011a , 741 , 30 ackermann , m. , ajello , m. , allafort , a. , et al . 2011b , 743 , 171 aller , m. f. , aller , h. d. , & hughes , p. a. 1996 , in iau symposium 175 , extragallactic radio sources , ed . r. d. ekers , c. fanti , & l. padrielli ( dordrecht : kluwer ) , 283 balsara , d. s. , & norman , m. l. 1992 , , 393 , 631 beasley , a. j. , gordon , d. , peck , a. b. , et al . 2002 , , 141 , 13 ( rfc ) bondi , m. , padrielli , l. , fanti , r , et al . 1996 , , 308 , 415 bower , g. c. , backer , d. c. , wright , m. , et al . 1997 , , 484 , 118 dodson , r. , fomalont , e. b. , wiik , k. , et al . 2008 , , 175 , 314 fey , a. l. , & charlot , p. 1997 , , 111 , 95 fey , a. l. , & charlot , p. 2000 , , 128 , 17 fey , a. l. , clegg , a. w. , fomalont , e. b. 1996 , , 105 , 299 fichtel , c. e. , bertsch , d. l. , chiang , j. , et al . 1994 , , 94 , 551 fomalont , e. b. , frey , s. , paragi , z. , et al . 2000 , , 131 , 95 fomalont , e. b. , petrov , l. , macmillan , d. s. , gordon , d. , & ma , c. 2003 , , 126 , 2562 ( rfc ) franceschini , a. , rodighiero , g. , & vaccari , m. 2008 , , 487 , 837 hartman , r. c. , bertsch , d. l. , bloom , s. d. , et al . 1999 , , 123 , 79 healey , s. e. , romani , r. w. , taylor , g. b. , et al . 2007 , , 171 , 61 ( crates ) henstock , d. r. , browne , w. a. , wilkinson , p. n. , et al . 1995 , , 100 , 1 homan , d. c. 2012 , int . , 8 , 163 hughes , p. a. , aller , m. f. , & aller , h. d. 2011 , , 735 , 81 immer , k. , brunthaler , a. , reid , m. j. , et al . 2011 , , 194 , 25 ( rfc ) impey , c. 1996 , , 112 , 2667 jorstad , s. g. , marscher , a. p. , mattox , j. r. , et al . 2001 , , 556 , 738 kellermann , k. i. , lister , m. l. , homan , d. c. , et al . 2004 , , 609 , 539 kovalev , y. y. , aller , h. d. , aller , m. f. , et al . 2009 , , 696 , l17 kovalev , y. y. , petrov , l. , fomalont , e. b. , & gordon , d. 2007 , , 133 , 1236 ( rfc ) lhteenmki , a. , & valtaoja , e. 1999 , in asp conf . 159 , bl lac phenomenon , ed . l. o. takalo & sillanp ( san francisco : asp ) , 213 lhteenmki , a. , & valtaoja , e. 2003 , , 590 , 95 lebedev , s. v. , ampleford , d. , ciardi , a. , et al . 2004 , , 616 , 988 lee , s .- s . , lobanov , a. p. , krichbaum , t. p. , et al . 2008 , , 136 , 159 len - 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s & 0.121 & no + maximum pc - scale jet bend & k - s & 0.998 & no + number of pc - scale jet bends & @xmath9 & 0.945 & no + total 4.85 ghz flux density & k - s & 0.145 & no + maximum pc - scale jet bend & k - s & 0.948 & no + number of pc - scale jet bends & @xmath9 & 0.402 & no +

we investigate the hypothesis that @xmath0-ray - quiet agn have a larger tendency for jet bending than @xmath0-ray - loud agn , revisiting the analysis of @xcite . we perform a statistical analysis using a large sample of 351 radio - loud agn along with @xmath0-ray identifications from the fermi large area telescope ( lat ) . our results show no statistically significant differences in jet - bending properties between @xmath0-ray - loud and @xmath0-ray - quiet populations , indicating that jet bending is not a significant factor for @xmath0-ray detection in agn .

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Proceed to summarize the following text: the topic of parity - time ( @xmath9 ) symmetry and its relevance for physical applications on the one hand , as well as its mathematical structure on the other , have drawn considerable attention from both the physics and the mathematics community . originally the proposal of bender and his collaborators @xcite towards the study of such systems was made as an alternative to the postulate of hermiticity in quantum mechanics . in view of the formal similarity of the schrdinger equation with maxwell s equations in the paraxial approximation , it was realized that such @xmath9 invariant systems can in fact be experimentally realized in optics @xcite . subsequently , these efforts motivated experiments in several other areas including @xmath9 invariant electronic circuits @xcite , mechanical circuits @xcite , and whispering - gallery microcavities @xcite . concurrently , the notion of supersymmetry ( susy ) originally espoused in high - energy physics has also been realized in optics @xcite . the key idea is that from a given potential one can obtain a susy partner potential with both potentials possessing the same spectrum ( with exception of possibly one eigenvalue ) @xcite . an interplay of susy with @xmath9 symmetry is expected to be quite rich and is indeed useful in achieving transparent as well as one - way reflectionless complex optical potentials @xcite . in a previous paper @xcite we explored the interplay between @xmath9 symmetry , susy and nonlinearity . in ref . @xcite we derived the exact solutions of the general nonlinear schrdinger equation ( nlse ) with arbitrary nonlinearity in 1 + 1 dimensions when in an external potential given by a shape invariant @xcite supersymmetric and @xmath9 symmetric complex potential . in particular , we considered the nonlinear partial differential equation @xmath10 for arbitrary nonlinearity parameter @xmath3 , with @xmath11 and the partner potentials arise from the superpotential @xmath12 giving rise to [ vpvm ] @xmath13 for @xmath14 , the _ complex _ potential @xmath15 has the same spectrum , apart from the ground state , as the _ real _ potential @xmath16 and we used this fact in our numerical study of the stability of the bound state solutions of the nlse in the presence of @xmath15 ( see ref . @xcite ) . to complete our study of this system of nonlinear schrdinger equations in @xmath9 symmetric susy external potentials , we will study the stability properties of the bound state solutions of nlse in the presence of the external real susy partner potential @xmath16 , and compare the stability regime of these solutions , which depends on the parameters @xmath17 , to the stability regime of the related _ solitary _ wave solutions to the nlse in the absence of the external potential , which depend only on the parameter @xmath3 . because the nlse in the presence of @xmath16 is a hamiltonian dynamical system , we can use variational methods to study the stability of the solutions when they undergo certain small deformations . we will compare the results of this type of analysis with a linear stability analysis based on the v k stability criterion @xcite . in our previous paper @xcite we determined the exact solutions of the equation for @xmath14 and for @xmath15 , which was complex . we studied numerically the stability properties of these solutions using linear stability analysis . we found some unusual results for the stability which depended on the value of @xmath18 . in that paper , because of the complexity of the potential , the energy was not conserved and a hamiltonian formulation of the problem was not possible . however for the partner potential @xmath16 , when @xmath14 , the potential is real . we note that @xmath16 has the symmetry @xmath19 , so that we can obtain @xmath16 from two _ different _ superpotentials @xmath20 and @xmath21 that have the @xmath0 symmetric forms at arbitrary @xmath22 : [ ssw1w2 ] @xmath23 in particular , when @xmath14 we can determine the spectrum of bound states of the linear schrdinger equation with potential @xmath16 using the _ real _ superpotential @xmath24 and the real shape invariant sequence of partner potentials . when @xmath14 the first superpotential @xmath20 has the _ complex _ partner potential @xmath15 which we studied in @xcite , whereas @xmath25 is the usual _ real _ shape invariant susy potential . both superpotentials yield the same real potential @xmath26 . because this potential is real , one can use variational methods to study the stability of the exact solutions to the nlse in the potential @xmath16 . we will consider both derrick s theorem @xcite as well as a time dependent variational approximation @xcite to study the stability of the exact solutions . because of the similarity of the solutions to those of the nlse equation , we are able to use a variant of the v k stability criterion to study spectral stability of the solutions . the results of the v k analysis agree with the new regime of stability found from derrick s theorem and the time dependent variational approach . the latter approach allows us to obtain the frequency of small oscillations of the perturbed solutions . we are able to obtain exact analytic results because we can formulate the problem in terms of hamilton s action principle . the euler - lagrange equations lead to the dynamical equations which have a conserved hamiltonian . this is in sharp contrast with the stability analysis for the solutions in the presence of @xmath15 which had to be done numerically . the latter system is dissipative in nature . this paper is structured as follows . in sec . [ s : ssmodel ] we review the non - hermitian susy model that we studied in ref . @xcite . in sec . [ s : hamprinc ] we consider hamilton s principle for the nlse in the real external potential @xmath16 . section [ s : derrick ] describes the application of derrick s theorem for determining the domain of stability of the solutions . here we determine analytically a new domain of stability for the solutions , when compared to the solutions in the absence of an external potential . we find that for @xmath7 a new domain of stability exists . [ s : collective ] contains a collective coordinate approach which allows one to study dynamically the blowup or collapse of the solution as well as small oscillations around the exact solution when slightly perturbed . by setting the frequency to zero we obtain analytically a domain of stability that agrees with the result of derrick s theorem . in sec . [ ns.s : linearstability ] we provide the details of a linear stability analysis based on the v - k stability criterion , which leads to identical conclusions as derrick s theorem . section [ s : conclude ] contains a summary of our main results . we are motivated by the @xmath27 symmetric susy superpotential @xmath28 which gives rise to the supersymmetric partner potentials given by eqs . . in what follows we will specialize to the case @xmath14 . in that case , @xmath16 is the well known pschl - teller potential @xcite . the relevant bound state eigenvalues assume an extremely simple form as @xmath29 ^ 2 \>.\ ] ] such bound state eigenvalues only exist when @xmath30 . we notice that for the ground state ( n=0 ) to exist requires @xmath31 . the existence of a first excited state ( n=1 ) requires @xmath32 . we will find that the stability of the nlse solutions in this external potential will depend on the depth of the well , which will lead to a critical value of @xmath33 , above which the solutions are stable . in what follows we will be concerned with the properties of the nlse in the presence of the external potential centered at @xmath34 , @xmath35 in particular we are interested in the bound state solutions of @xmath36 if we assume a solution of the form @xmath37 it is easy to show that an exact solution is given by : @xmath38 where @xmath39 and @xmath40 for the complex potential @xmath15 , the amplitude of the solution is given by @xcite @xmath41 so that there are two separate regimes where @xmath42 is real . in contrast , for @xmath16 there is only one regime for an attractive @xmath16 where @xmath42 is real , namely @xmath43 the analysis of the stability of the solutions for @xmath15 in ref . @xcite showed a very complicated pattern . even for @xmath44 there is a regime of instability as a function of @xmath18 for the nodeless solution . for @xmath45 all solutions in that case found analytically and numerically were unstable . only for @xmath46 were the solutions stable . in contrast to that analysis where each value of @xmath3 had to be investigated separately , in the case of the real potential @xmath16 , we are able to address the stability question for all @xmath3 using derrick s theorem as well as the v k stability criterion . the mass @xmath47 of the bound state for the case @xmath16 is given by @xmath48^{1/\kappa } \ , \gamma[1/\kappa ] } { \gamma[1/2 + 1/\kappa ] } \>. \notag\end{aligned}\ ] ] if we turn off the external potential by setting @xmath49 , @xmath50 \to ( 1/\kappa + 1/2)$ ] and the mass @xmath47 of the bound state goes to the mass of the solitary wave solutions , @xmath51^{1/\kappa } \ , \gamma[1/\kappa ] } { \gamma[1/2 + 1/\kappa ] } \>.\ ] ] let us first discuss hamilton s principle of least action for the usual nlse without a confining potential . the nlse with arbitrary nonlinearity in 1 + 1 dimensions is given by @xmath52 the second term causes diffusion and the third term attraction and the competition allows for solitary wave blowup which depends on @xmath3 . here @xmath53 can be scaled out of the equation by letting @xmath54 so that the linear equation for the rescaled equation is obtained in the limit @xmath55 . while the solitary waves are stable for @xmath56 , for @xmath57 there is a critical mass @xmath47 necessary for blowup to occur , where the width of solitary wave goes to zero . for @xmath58 , blowup occurs in a finite amount of time . the classical action for the nlse is @xmath59 = \int l[\psi,\psi^{\ast } ] \dd{t}$ ] , where the lagrangian @xmath60 $ ] is given by @xmath61 & = \int \dd{x } \bigl \ { \ , i \ , \qty [ \psi^{\ast } ( \partial_t \psi ) - ( \partial_t \psi^{\ast } ) \psi ] / 2 \label{lagnlse } \\ & \qquad + ( \partial_x \psi^{\ast } ) ( \partial_x \psi ) - | \psi |^{2(\kappa+1 ) } / ( \kappa + 1 ) \ , \bigr \ } \>. \notag\end{aligned}\ ] ] the nlse follows from the hamilton s principle of least action , @xmath62 and @xmath63 , which leads to eq . with @xmath64 . multiplying this equation by @xmath65 and subtracting its complex conjugate , it is easy to prove that the mass @xmath47 , defined by @xmath66 , is conserved . we now want to add a _ real _ susy potential to the nlse . we will consider the addition of @xmath16 given in eq . so that the equation of motion is now given by @xmath67 the action which leads to eq . is given by @xmath59 = \int l[\psi,\psi^{\ast } ] \dd{t}$ ] where @xmath61 & = \int \dd{x } i \ , \qty [ \psi^{\ast } ( \partial_t \psi ) - ( \partial_t \psi^{\ast } ) \psi ] /2 - h[\psi,\psi^{\ast } ] \ > , \notag \\ h[\psi,\psi^{\ast } ] & = \int \dd{x } \bigl \ { \ , ( \partial_x \psi^{\ast } ) ( \partial_x \psi ) - | \psi |^{2(\kappa+1 ) } / ( \kappa + 1 ) \notag \\ & \qquad\qquad + \psi^{\ast } \ , v^{-}(x ) \ , \psi \ , \bigr \ } \>. \label{hdef}\end{aligned}\ ] ] derrick s theorem @xcite states that for a hamiltonian dynamical system , for a solitary wave solution to be stable it must be stable to changes in scale transformation @xmath68 when we keep the mass of the solitary wave fixed . that is the hamiltonian needs to be a minimum in @xmath69 space . first let us look at the case of the nlse without an external potential : derrick s method is based on whether a scale transformation which keeps the mass @xmath47 invariant , raises or lowers the energy of a solitary wave . for the nlse with hamiltonian @xmath70 \\ & \equiv h_1 - h_2 \ > , \notag\end{aligned}\ ] ] where both @xmath71 and @xmath72 are positive definite . a static solitary wave solution can be written as @xmath73 the exact solution has the property that it minimizes the hamiltonian subject to the constraint of fixed mass as a function of a stretching factor @xmath69 . this can be seen by studying a variational approach as done in @xcite , or by directly studying the effect of a scale transformation that respects conservation of mass . in the latter approach , which generalizes the method used by derrick @xcite , we let @xmath68 , and consider the stretched wave function , @xmath74 so that @xmath75 is preserved by the transformation . defining @xmath76 as the value of @xmath77 for the stretched solution @xmath78 , one finds that @xmath79 is consistent with the equations of motion . the stable solutions must then also satisfy : @xmath80{h_{\beta}}{\beta } \ge 0 \>.\ ] ] if we write @xmath77 in terms of the two positive definite pieces @xmath71 , @xmath72 , then @xmath81 we find @xmath82 so that @xmath83 . this result is consistent with the equations of motion . in fact for the nlse the exact solution has @xmath84 where @xmath85 . one finds then using @xmath86 } { \gamma[1/2+r/2 ] } \>,\ ] ] that [ h1andh2eval ] @xmath87^{1/\kappa } \gamma[1/\kappa ] } { 2 \kappa^2 \ , \gamma[3/2 + 1/\kappa ] } \ > , \label{h1-val } \\ h_2 & = \frac { \sqrt{\pi } \ , \qty [ ( \kappa + 1)/\kappa^2 ] ^{1/\kappa } \gamma[1/\kappa ] } { \kappa^3 \ , \gamma[3/2 + 1/\kappa ] } \ > , \label{h2-val } \end{aligned}\ ] ] so that the exact solution is indeed a minimum of the hamiltonian with respect to scale transformations , with @xmath88 . the second derivative is given by @xmath89{h_{\beta}}{\beta } = 2 \ , h_1 - \kappa(\kappa - 1 ) \ , \beta^{\kappa - 2 } \ , h_2 \>,\ ] ] which when evaluated at the stationary point yields @xmath90{h_{\beta}}{\beta } = 2 \ , ( 2 - \kappa ) \ , h_1 \ge 0 \>,\ ] ] for stability . this result indicates that solutions are unstable to changes in the width , compatible with the conserved mass , when @xmath58 . the case @xmath91 is a marginal case where it is known that blowup occurs at a critical mass ( see for example ref . the result found above for the nlse has also been found by various other methods such as linear stability analysis and using strict inequalities . numerical simulations ( see ref . @xcite ) have been done for the critical case @xmath91 showing that blowup ( self - focusing ) occurs when the mass @xmath92 . for @xmath93 a variety of analytic and numerical methods have been used to study the nature of the blowup at finite time @xcite . in the case of the nonlinear schrdinger equation , one can perform a linear stability analysis of the exact solutions . namely one lets @xmath94 \ , e^{-i \omega t } \>,\ ] ] and linearizes the nlse to find an equation for @xmath95 to first order , @xmath96 and studies the eigenvalues of the differential operator @xmath97 . if the spectrum of @xmath97 is imaginary , then the solutions are spectrally stable . k showed @xcite that when the spectrum is purely imaginary @xmath98 . also they showed that when @xmath99 , there is a real positive eigenvalue so that there is a linear instability . for the nlse , there is a class of solutions with arbitrary nonlinearity parameter @xmath3 . namely [ nlse.e:3 ] @xmath100 when we do not have an external potential , we know explicitly how the mass changes when we change @xmath101 at fixed @xmath3 . that is @xmath102^{1/\kappa } \gamma[1/\kappa ] } { \kappa \ , \sqrt{-\omega } \ , \gamma[1/2 + 1/\kappa ] } \ > , \notag \end{aligned}\ ] ] where @xmath103 we find @xmath104 thus for @xmath58 the solitary waves are unstable . this agrees with the result of derrick s theorem . when we have an external potential , we will need to determine the solutions numerically as we change @xmath101 . this will be accomplished in sec . [ ns.s : linearstability ] . so now let us look at our situation when we have in addition the real external potential : @xmath105 the exact solution to the nlse in the presence of @xmath16 is given by eq . . this solution is similar in form to the usual solution to the nlse except this nodeless solution is pinned to the potential so that there is no translational invariance . when @xmath106 this solution goes over to a particular solution of the nlse with width parameter @xmath107 . under the scale transformation @xmath108 , the stretched solution which preserves the mass @xmath47 is given by : [ dext.e:2 ] @xmath109 the stretched wave function @xmath110 is no longer an exact solution . the stretched hamiltonian for the external potential case is now given by @xmath111 where @xmath112 } { 2 \kappa^2 \ , \gamma[3/2 + 1/\kappa ] } \ > , \end{aligned}\ ] ] and @xmath113 } { ( \kappa + 1 ) \ , \gamma[3/2 + 1/\kappa ] } \ > , \end{aligned}\ ] ] with @xmath114 now given by , and @xmath115 thus , we find @xmath116 \dd{x } \>.\end{aligned}\ ] ] using the identity , @xmath117 we obtain @xmath118 } { ( \kappa + 1 ) \ , \gamma[3/2 + 1/\kappa ] } \>.\ ] ] as in the case when @xmath119 , we again find @xmath120 for our exact solution . so the stretched solution is again an extremum of @xmath76 with @xmath47 kept fixed . for the second derivative we have @xmath121{h_{\beta}}{\beta } } _ { \beta=1 } \!\!\ ! = 2 \ , h_1 - \kappa ( \kappa - 1 ) \eval { \pdv[2]{h_{3}}{\beta } } _ { \beta=1 } \>,\ ] ] where @xmath122{h_{3}}{\beta } } _ { \beta=1 } \!\!\ ! = ( 1/4 - b^2 ) \ , a^{2}(b,\kappa ) \ , \qty [ i_1 + i_2 + i_3 ] / \kappa \>,\ ] ] with @xmath123 we can again evaluate these integrals using the first identity eq . and the identity : @xmath124{\sech^{2 + 2/\kappa } ( \lambda x ) } { \lambda } \label{dext.e:9 } \\ & \quad = ( 2/\kappa + 2)(2/\kappa + 3 ) \ , x^2 \sinh^2(\lambda x ) \ , \sech^{2/\kappa + 4}(\lambda x ) \notag \\ & \qquad\qquad - ( 2/\kappa + 2 ) \ , x^2 \ , \sech^{2/\kappa + 2}(\lambda x ) \>. \notag\end{aligned}\ ] ] we will also need the following hypergeometric function : @xmath125 using these results , eq . gives @xmath126{h_{3}}{\beta } } _ { \beta=1 } \!\!\!\!\ ! & = ( 1/4 - b^2 ) \ , a^{2}(b,\kappa ) \ , f_3(\kappa ) \ > , \label{dext.e:11 } \\ f_3(\kappa ) & = - \qty [ \frac{4 \ , ug(\kappa)}{2 + 3\kappa } + \frac{4 \sqrt{\pi } \ , \kappa \ , \gamma[1 + 1/\kappa ] } { ( \kappa+1)(3\kappa+2 ) \ , \gamma[3/2 + 1/\kappa ] } ] . \notag\end{aligned}\ ] ] the critical value is determined from : @xmath126{h_{\beta}}{\beta } } _ { \beta=1 } \!\!\ ! & = a^{2}(b,\kappa ) \ , \bigl [ \ , 2 \ , f_1(\kappa ) \label{dext.e:12 } \\ & \hspace{-2em } - \kappa ( \kappa + 1 ) \ , a^{2\kappa}(b,\kappa ) \ , f_2(\kappa ) + ( 1/4 - b^2 ) \ , f_3(\kappa ) \ , \bigr ] \>. \notag\end{aligned}\ ] ] solving for the critical value of @xmath5 , we find @xmath127 the result of calculating the second derivative at @xmath107 and setting it equal to zero is that the domain of stability is now as follows : for @xmath56 and all @xmath18 in the range @xmath128 , the solution is stable , as it was for the solitary wave solutions of the nlse . here @xmath106 corresponds to no external potential . when @xmath58 the solitary wave solutions of the nlse were _ unstable_. instead , in the presence of the confining potential , a new domain of stability occurs when @xmath58 as long as @xmath129 , where @xmath8 is given by eq . . we see this in the result for @xmath130 shown in fig . [ ns.f : dhdbeta ] . in fig . [ ns.f : bvskappa ] , we show both @xmath131 and @xmath132 as a function of @xmath3 . the region between @xmath106 and @xmath131 is unstable . as we will show in sec . [ ns.s : linearstability ] , this analytic result for @xmath8 given by eq . is confirmed by our linear stability analysis . just as there is a critical mass for instability in the nlse at @xmath91 , for @xmath133 we can interpret the critical value of @xmath18 in terms of a critical mass which depends on @xmath3 above which the solution is unstable . since from eq . , we have @xmath134^{1/\kappa } \ , \gamma[1/\kappa ] } { \gamma[1/2 + 1/\kappa ] } \>,\ ] ] we see that the mass @xmath47 _ decreases _ as we increase @xmath18 at fixed @xmath3 . so as we go from the unstable case @xmath106 ( no potential ) and increase @xmath18 we decrease the mass until we reach an @xmath135 below which the solution is stable . finally we reach the curve @xmath136 which corresponds to @xmath137 = 1/2 + 1/\kappa$ ] . the different regimes are shown in fig . [ ns.f : mcrit ] . in the lightly shaded regime the solutions are unstable . the maximum value of the mass is given by the case @xmath106 , when there is no longer a stabilizing potential . the interval @xmath138 corresponds to the regime @xmath139 . this is the stable regime denoted by the darker shaded area in fig . [ ns.f : mcrit ] . in order to follow the time evolution of a slightly perturbed solitary wave or bound solution to a hamiltonian dynamical system , without solving numerically the time dependent partial differential equations for @xmath140 , one can introduce time - dependent collective coordinates assuming that the general shape of the original solution is maintained apart from the height , width , and position , etc . this will allow us to see whether these parameters just oscillate around the original values or whether the parameters grow or decrease in time . when instabilities are seen in the variational results , it suggests that the exact solutions are also unstable . unlike derrick s theorem when applied to the nlse , the collective coordinate method can be applied to the special case @xmath91 . it also gives an approximate description of wave function blow - up or collapse in the unstable regime , and oscillation of the perturbed solution in the stable regime . in the next section , we first apply this approach when there is no external potential . let us remind ourselves of the collective coordinate variational approach to blow - up for the nlse @xcite with no external potential . using this method , we found previously that when @xmath57 , there is a critical value of the mass required before blowup could take place . derrick s theorem has nothing to say about the stability of the solitary wave solution for this case . to make the collective coordinate approach concrete , we assume self - similar solutions of the form : @xmath141 \\ & \qquad \times \exp [ i \qty ( v \ , y(t ) / 2 + \lambda(t ) \ , y^2(t ) - \omega t ) ] \>. \notag\end{aligned}\ ] ] here @xmath142 , @xmath143 , and @xmath144 are arbitrary real functions of time alone , and @xmath145 . for no external potential translation invariance gives @xmath146 . in particular at @xmath147 and @xmath148 , we will start with the exact solution of the form @xmath149 and assume that this solution just changes during the time evolution in amplitude and width . with this assumption one can derive the dynamical equations for @xmath143 and @xmath144 from hamilton s principle of least action with the lagrangian given in eq . . noether s theorem yields three conservation laws : conservation of probability , conservation of momentum , and conservation of energy . conservation of probability gives `` mass '' conservation : @xmath150 and allows one to rewrite @xmath143 in terms of the conserved mass @xmath47 , the width parameter @xmath144 , and a constant @xmath151 whose value depends on @xmath152 . thus , @xmath153 we will therefore keep @xmath47 in our definition of @xmath143 since it will be a relevant parameter when @xmath91 . for @xmath154 , one obtains @xmath155}{\gamma[1/2 + 1/\kappa ] } \>.\ ] ] setting @xmath156 , in terms of the new collective coordinates @xmath157 $ ] , the lagrangian is given by @xmath158 = k[g,\lambda ] - h[g,\lambda ] \>,\ ] ] where [ var.e:6 ] @xmath159}{m } & = \frac{i}{2 m } \int \dd{x } \qty [ \psi^{\ast } ( \partial_t \psi ) - ( \partial_t \psi^{\ast } ) \psi ] \notag \\ & = \frac{1}{2 } \ , v^2 + \omega - \dot{\lambda } \ , g^2 \ , \frac{c_2}{c_1 } \ > , \label{var.e:6a } \\ \frac{h[g,\lambda]}{m } & = - \frac{1}{m } \int \dd{x } \qty [ ( \partial_x \psi^{\ast } ) ( \partial_x \psi ) + | \psi |^{2(\kappa+1 ) } / ( \kappa + 1 ) ] \notag \\ & \hspace{-3em } = \frac{v^2}{4 } + \frac{c_3}{c_1 } \ , \frac{1}{g^2 } + 4 \lambda^2 \ , \frac{c_2}{c_1 } \ , g^2 - \frac{1}{(\kappa + 1 ) } \ , \frac{c_4}{c_1 } \ , \qty ( \frac{m}{c_1 g } ) ^{\kappa } \ > , \label{var.e:6b}\end{aligned}\ ] ] where [ var.e:7 ] @xmath160}{2 \kappa \ , \gamma[3/2 + 1/\kappa ] } \ > , \notag \\ c_4 & = \int_{-\infty}^{\infty } \!\!\ ! f^{(2 \kappa + 2)}(z ) \label{var.e:7c } \\ & = \frac{\sqrt{\pi } \ , \gamma [ 1 + 1/\kappa ] } { 2 \kappa \ , \gamma[3/2 + 1/\kappa ] } = 2 \kappa \ , c_3 \>. \notag\end{aligned}\ ] ] collecting terms from and , the lagrangian is given by @xmath161}{m } & = \frac{1}{4 } \ , v^2 + \omega - \dot{\lambda } \ , g^2 \ , \frac{c_2}{c_1 } - \frac{c_3}{c_1 } \ , \frac{1}{g^2 } \label{var.e:8 } \\ & \qquad - 4 \lambda^2 \ , \frac{c_2}{c_1 } \ , g^2 + \frac{1}{(\kappa + 1 ) } \ , \frac{c_4}{c_1 } \ , \qty ( \frac{m}{c_1 g } ) ^{\kappa } \>. \notag\end{aligned}\ ] ] from the euler - lagrange equations we obtain the second order differential equation for @xmath162 , @xmath163 and the relation @xmath164 . in solving these equations , we will use for the mass @xmath47 , when we are not at the critical value @xmath91 , the expression for the mass for the solitary wave solution given by eq . . if we do this , we can rewrite eq . as @xmath165 one notices that for @xmath166=1 $ ] , @xmath167 , as it must for an exact solution . we see that to get @xmath168 when @xmath91 we need to have @xmath169 where @xmath170 is the value of the mass for the exact solution . so initial conditions with a mass greater than this are necessary to see blow up at @xmath91 . by multiplying both sides of by @xmath171 and integrating with respect to time we obtain a first integral of the second order differential equation , which up to a multiplicative factor is the same as setting the conserved hamiltonian divided by the mass @xmath47 to a constant @xmath172 . this gives @xmath173 we notice that at the critical value of @xmath174 , the last two terms both go like @xmath175 . self - focusing occurs when the width can go to zero . since @xmath176 needs to be positive , this means that at @xmath91 , the mass has to be greater than @xmath177 for @xmath162 to be able to go to zero . we find @xcite @xmath178 provided we use the exact solution ( which is a zero - energy solution ) for @xmath91 , namely @xmath179 . this agrees well with numerical estimates of the critical mass @xcite and is slightly lower than the variational estimate obtained earlier by cooper _ _ @xcite using a post - gaussian trial wave functions instead of a trial wave function based on the exact solution . for @xmath180 , if we use the mass of the exact solitary wave solution , the energy conservation equation simplifies to @xmath181 in the supercritical case when @xmath182 , we have @xmath183 this `` mean - field '' result was obtained earlier in refs . @xcite . to show the difference between the stability at @xmath184 and @xmath185 , we have solved eq . for the initial conditions @xmath186 , @xmath187 , with the results shown in fig . [ f : nlse-3 - 5 ] . for small oscillations we can assume @xmath188 from which we obtain the equation , @xmath189 \>. \notag\end{gathered}\ ] ] setting @xmath190 in eq . , leads to the same criterion for the critical mass when @xmath91 . the same equation gives the frequency of small oscillations when @xmath56 . for @xmath191 , the predicted period of oscillation is @xmath192 in good agreement with fig . [ f : nlse-3 - 5]a . @xmath193 from eq . for ( a ) @xmath191 and ( b ) @xmath194 . the latter case corresponds to blowup " . ] potential fit for @xmath191 . the solid line ( blue online ) is the exact derivative of the potential from eq . , the dashed line ( red online ) is the fitted function @xmath195 of eq . . ] now we would like to see how this argument is modified when we add the external potential @xmath16 . in this case the exact solution is `` pinned '' to the origin . the lagrangian is again given by eq . with the addition of the potential term : @xmath196 / m & = \int_{-\infty}^{+\infty } \!\!\ ! \dd{x } \psi^{\ast}(x , t ) \ , v^{-}(x ) \ , \psi(x , t ) / m \\ & = - ( b^2 - 1/4)\ , { \mathcal{v}}[g,\kappa ] / c_1 \ > , \notag\end{aligned}\ ] ] where @xmath197 & = \int_{-\infty}^{+\infty } \!\!\ ! \dd{y } \sech^{2/\kappa}(y ) \ , \sech^{2}(g y ) \ > , \\ & \xrightarrow{g \to 1 } \frac{\sqrt{\pi } \ , \gamma[1 + 1/\kappa]}{\gamma[3/2 + 1/\kappa ] } \>. \notag\end{aligned}\ ] ] the lagrangian now becomes : @xmath198}{m } = \frac{1}{4 } \ , v^2 + \omega - \dot{\lambda } \ , g^2 \ , \frac{c_2}{c_1 } - \frac{c_3}{c_1 } \ , \frac{1}{g^2 } - 4 \lambda^2 \ , \frac{c_2}{c_1 } \ , g^2 \notag \\ & \ > + \frac{1}{(\kappa + 1 ) } \ , \frac{c_4}{c_1 } \ , \qty ( \frac{m}{c_1 g } ) ^{\kappa } + \frac{(b^2 - 1/4)}{c_1 } \ , { \mathcal{v}}[g,\kappa ] \>. \label{var.e:17}\end{aligned}\ ] ] the euler - lagrange equations now give @xmath199}{g } \ > , \notag\end{aligned}\ ] ] where @xmath200}{g } & = - 2 \int_{-\infty}^{+\infty } \!\!\ ! \dd{y } y \ , \sech(gy ) \ , \sech^{3}(g y ) \ , \sech^{2/\kappa}(y ) \notag \\ & \xrightarrow{g \to 1 } - \frac{\kappa}{\kappa + 1 } \ , \frac{\sqrt{\pi } \ , \gamma[1 + 1/\kappa]}{\gamma[3/2 + 1/\kappa ] } \>. \label{var.e:19}\end{aligned}\ ] ] to solve this equation numerically we fit the numerical values of the integral in by a function of the form : @xmath201 using mathematica , one obtains an extremely accurate 4-parameter fit . for example , the result of this fit for @xmath191 is shown in fig . [ f : vfit ] for @xmath202 , @xmath203 , @xmath204 , and @xmath205 . different fit parameters are used for each value of @xmath3 . plots of the solutions @xmath193 of eq . for different values of @xmath18 and for @xmath191 , @xmath206 , @xmath207 , and @xmath208 are shown in fig . [ f : gt ] . the solid ( blue online ) , dotted ( green online ) , and dashed ( red online ) lines are the solutions @xmath193 of eq . for @xmath191 , @xmath206 , @xmath207 , and @xmath208 , and ( a ) @xmath209 , for @xmath191 , ( b ) @xmath210 , for @xmath57 , ( c ) @xmath211 , for @xmath130 , and ( d ) @xmath212 , for @xmath194 . ] we can study analytically the stability of the solutions in this variational approximation by linearizing eq . around the exact solution @xmath213 , @xmath214 to evaluate the effect of the external potential on the small oscillation equation we just need to know that : @xmath215 + \order { \epsilon^2 } \notag \>. \end{aligned}\ ] ] substitution of this expansion into gives @xmath216 \notag \\ \hspace{3em } - 4 \ , ( b^2 - 1/4 ) \ , \qty [ 2 \ , ug(\kappa ) - 3 \ , ug_2(\kappa ) ] / c_2 \notag \ > , \end{gathered}\ ] ] with @xmath217 , where @xmath218 is given by eq . and where @xmath219 the collective coordinate method allows one to approximately calculate the small oscillation frequency as well as the time evolution of the system using eq . . [ f : gt ] we assumed @xmath220 . the relevant values of @xmath8 are @xmath221 for @xmath222 . for @xmath184 we get oscillation for the entire range from @xmath106 to @xmath223 as seen in fig . [ f : gt]a . as predicted , for @xmath91 , once we get above @xmath224 , which is the case with no potential , then the solution is stable as seen in fig . [ f : gt]b . for @xmath225 , once we get above @xmath226 , then the solution is stable as seen in fig . [ f : gt]c . for @xmath185 we get similar results to @xmath227 , as seen in fig . [ f : gt]d . in the stable regime , the oscillation periods are accurately predicted by eq . . setting @xmath228 , determines the critical value of @xmath18 at a given @xmath3 below which the solutions are unstable for @xmath58 . the expression for @xmath8 obtained this way is identical to the expression for @xmath8 obtained from derrick s theorem in eq . and shown in fig . [ ns.f : bvskappa ] . in the domain of instability one finds that if we look at initial conditions where @xmath229 and @xmath230 , @xmath231 , then for the minus sign one gets `` blow up '' ( @xmath168 ) , and for the plus sign we get collapse of the solution ( @xmath232 ) . in fig . [ collapse ] , we give an example of collapse when @xmath233 and we are in the unstable regime . a first integral of the second order differential equation resulting from the lagrange s equation for @xmath162 can be obtained by setting the conserved hamiltonian to a constant @xmath172 . one then has @xmath234 \>. \label{econs}\end{aligned}\ ] ] from the energy conservation equation we can see immediately that at @xmath91 the exact solution we found does not blow up . this is for two reasons : first , when the width parameter @xmath235 , then @xmath236 $ ] becomes a constant independent of @xmath162 and therefore the potential does not affect the small @xmath162 behavior of the differential equation ; secondly the mass of the exact solution depends now on @xmath18 and @xmath3 and it is lower than the critical mass needed for blowup . that is , the mass of the bound solution is given by : @xmath237 \ , c_1[\kappa ] = \frac{\sqrt{\pi } \ , \qty ( b_{\text{max}}^2(\kappa ) - b^2 ) ^{1/\kappa } \ , \gamma[1/\kappa ] } { \gamma[1/2 + 1/\kappa ] } \notag \\ & \hspace{2em } \xrightarrow{\kappa \to 2 } \pi \ , \sqrt{1 - b^2 } \>. \label{mkappav } \end{aligned}\ ] ] the maximum value of this occurs when the external potential goes to zero at @xmath106 . when @xmath31 , the mass of the exact solution is _ always _ less than @xmath170 , so that these solutions are always stable when @xmath91 . for the nlse with no external potential , when the stability depends on the mass of the initial wave function at @xmath91 , the critical value is that of the exact solitary wave solution . see also fig . [ ns.f : mcrit ] and the discussion thereof . let us perform the linear stability analysis of the solitary wave solutions @xmath238 to the nonlinear schrdinger equation in the external potential . we take a perturbed solitary wave solution in the form @xmath239 \ , e^{-i\omega t}$ ] and consider the linearized equation on @xmath240 $ ] , @xmath241 if the spectrum of @xmath242 has eigenvalues with positive real part , then the corresponding solitary wave is called linearly unstable ; otherwise , it is called spectrally stable . in general , the spectral stability does not imply nonlinear stability , but for the nodeless solutions to the nonlinear schrdinger equation one can use the lyapunov - type approach to prove the orbital stability ; see e.g. ref . @xcite . the equation we are solving is @xmath243 with @xmath244 we are interested in the stability of the solitary wave solution @xmath245 to , with the amplitude @xmath246 satisfying @xmath247 \ , \phi_{\omega , b}(x)\>.\ ] ] for @xmath248 , one has the explicit expression @xmath249^{1/(2\kappa ) } \ , \sech^{1/\kappa}{x } \>,\ ] ] with @xmath217 . we will perform the spectral analysis of the linearization operator following the v k approach @xcite . we consider the perturbation of the solitary wave , @xmath250 \ , e^{-i\omega t}$ ] , with @xmath251 , and with @xmath252 and @xmath253 real . the linearized equation on @xmath252 and @xmath253 is given by @xmath254 where the self - adjoint operators @xmath255 are given by [ lpmdefs ] @xmath256 the stationary equation satisfied by @xmath257 and its derivative with respect to @xmath101 give the relations @xmath258 we need to perform the spectral analysis of @xmath259 and @xmath260 . we start with reviewing the v k approach from @xcite for the case @xmath106 , when @xmath261 . for a given value @xmath262 , let @xmath263 be the profile of a solitary wave for the case when @xmath264 ( when @xmath106 ) . by the v k theory , the linearization at @xmath265 is such that @xmath266 has a simple eigenvalue @xmath267 as its smallest eigenvalue , corresponding to the eigenfunction @xmath268 , while @xmath269 has one simple negative eigenvalue on the subspace of even functions , and a simple eigenvalue at @xmath270 on the subspace of odd functions corresponding to the eigenfunction @xmath271 . for any nonzero eigenvalue @xmath272 of the linearization operator from , one has the relation @xmath273 with nonzero @xmath274 . being in the range of @xmath275 , which is self - adjoint , @xmath274 is orthogonal to the null space of @xmath276 ; this allows us to arrive at @xmath277 hence @xmath278 . thus , the linear instability could only be caused by a positive eigenvalue of @xmath279 . from , one can see that one could have @xmath280 if the right - hand side of becomes positive for some @xmath274 orthogonal to the kernel of @xmath276 ; in other words , if the minimization problem @xmath281 gives a negative value of @xmath282 . by @xcite , finding the minimum of under constraints @xmath283 and @xmath284 leads to the relation @xmath285 with @xmath286 lagrange multipliers ; pairing the above with @xmath274 shows that @xmath282 in and is the same . writing @xmath287 and taking into account that @xmath284 , we see that we need to analyze the location of the first root of the v k function @xmath288 which is defined for @xmath289 in the resolvent set of the operator @xmath290 restricted onto the subspace of even functions . this domain includes the interval @xmath291 , where @xmath292 is the smallest negative eigenvalue of @xmath293 and @xmath294 is the next eigenvalue of @xmath293 , on the subspace of even functions . since clearly @xmath295 for @xmath296 , one has @xmath297 at some @xmath298 , @xmath299 ( hence stability ) if and only if @xmath300 , which leads to @xmath301 , and , using , we arrive at the v k stability condition @xmath302 an elementary computation based on shows that is satisfied ( for all @xmath303 ) if and only if @xmath304 . the left - hand side of becomes identically zero for @xmath57 and becomes positive for @xmath58 ( again , for all @xmath303 ) . now let us consider @xmath305 with @xmath31 and @xmath248 . as in the case of no potential , one has @xmath306 , with @xmath267 a simple eigenvalue corresponding to the eigenfunction @xmath307 . at @xmath308 , one has @xmath309 and @xmath310 we note that @xmath311 hence the smallest eigenvalue @xmath312 of @xmath313 ( assumed on the subspace of even functions ) is negative . at @xmath314 , one has @xmath315 hence for @xmath316 and @xmath317 , @xmath318 just as in the case @xmath106 which we considered above , the linear instability takes place when the minimization problem @xmath319 gives a negative value of @xmath282 . as we already pointed out in the case @xmath106 , one has @xmath320 for @xmath304 ( equivalently , @xmath321 are linearly stable ) , and @xmath322 for @xmath91 . due to , one then also has @xmath320 for @xmath323 , @xmath324 , @xmath325 and for @xmath91 , @xmath326 , @xmath327 . thus , for these values of @xmath3 and @xmath18 , the solitary waves @xmath328 are spectrally stable . for @xmath6 , the story is different : while @xmath282 in is negative for @xmath106 corresponding to the linear instability of @xmath329 , @xmath282 could become positive if @xmath18 exceeds some critical value @xmath131 : @xmath330 numerically , we proceed as follows . we pick @xmath6 and use the shooting method to construct a solitary wave @xmath257 and find the critical value @xmath331 above which @xmath332 becomes negative [ that is , when @xmath333 , a positive eigenvalue from the spectrum of @xmath334 collides with a negative eigenvalue , and they produce a pair of purely imaginary eigenvalues ; for @xmath335 , spectral stability takes place ] . this gives us the critical values @xmath8 vs. @xmath3 in agreement with fig . [ ns.f : bvskappa ] . we find remarkably that derrick s theorem and the v k spectral analysis of stability give identical results . the same result for the stability regime was also obtained by setting the oscillation frequency for small oscillations around the exact solution to zero using the time dependent variational method . in distinction with the case without a potential , in the presence of the external potential @xmath16 the results of the stability analysis are much more interesting because of the additional @xmath18 dependence of the exact solution . for @xmath93 it is possible to interpret the results of v - k and derrick s theorem in terms of a critical mass @xmath336 below which the solution is stable , or equally in terms of a critical depth @xmath18 for the confining potential above which the solution is stable . in this paper we studied the stability of the exact solution of the nlse in a real pschl - teller potential which is the susy partner of a _ complex _ @xmath27 symmetric potential studied previously @xcite . unlike the previous problem which required detailed numerical analysis for every value of the nonlinearity parameter @xmath3 , the real external potential problem here results in a hamiltonian dynamical system amenable to several variational approaches to the stability problem , such as derrick s theorem @xcite , v k theory @xcite , and a time dependent variational approach . using these methods we were able to show that for @xmath93 the pinned solution has a region of stability that was not available to the solitary wave solution of the nlse without an external potential . the latter solutions are known to blow up in a finite time interval when perturbed appropriately . the analytic result for the re - entry regime of stability found using derrick s theorem was corroborated by a numerical study of spectral stability based on the v k theory . this result is different from the result found numerically for the stability of the solution for the complex susy partner external potential @xmath15 . the analysis of the stability of the solutions for @xmath15 in @xcite showed a very complicated pattern . even for @xmath44 there is a regime of instability as a function of @xmath18 for the nodeless solution . at @xmath45 all the solutions found for @xmath15 , were unstable due to oscillatory instabilities . only for @xmath46 were the solutions stable . in contrast , for the @xmath16 potential we are able to address the stability question for all @xmath3 _ analytically _ and show that the effect of the external potential is to introduce a new domain of stability for all @xmath58 , when compared to the stability of the related solitary wave solutions in the absence of an external potential . the stability properties of the solutions of the nlse in the presence of the partner potentials @xmath337 are quite different from one another due to the dissipative versus conservative nature of these potentials . f.c . would like to thank the santa fe institute and the center for nonlinear studies at los alamos national laboratory for their hospitality . is grateful to indian national science academy ( insa ) for awarding him insa senior scientist position at savitribai phule pune university , pune , india . the research of a.c . was carried out at the institute for information transmission problems , russian academy of sciences at the expense of the russian foundation for sciences ( project 14 - 50 - 00150 ) . b.m . and j.f.d . would like to thank the santa fe institute for their hospitality . b.m . acknowledges support from the national science foundation through its employee ir / d program . the work of a.s . was supported by the u.s . department of energy . see special issues : h. geyer , d. heiss , and m. znojil , eds . , j. phys . a : math . gen . * 39 * , _ special issue dedicated to the physics of non - hermitian operators _ ( _ phhqp iv _ ) ( university of stellenbosch , south africa , 2005 ) ( 2006 ) ; 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we discuss the stability properties of the solutions of the general nonlinear schrdinger equation ( nlse ) in 1 + 1 dimensions in an external potential derivable from a parity - time ( @xmath0 ) symmetric superpotential @xmath1 that we considered earlier [ kevrekidis _ et al . _ phys . rev . e * 92 * , 042901 ( 2015 ) ] . in particular we consider the nonlinear partial differential equation @xmath2 for arbitrary nonlinearity parameter @xmath3 . we study the bound state solutions when @xmath4 , which can be derived from two different superpotentials @xmath1 , one of which is complex and @xmath0 symmetric . using derrick s theorem , as well as a time dependent variational approximation , we derive exact analytic results for the domain of stability of the trapped solution as a function of the depth @xmath5 of the external potential . we compare the regime of stability found from these analytic approaches with a numerical linear stability analysis using a variant of the vakhitov - kolokolov ( v k ) stability criterion . the numerical results of applying the v - k condition give the _ same _ answer for the domain of stability as the analytic result obtained from applying derrick s theorem . our main result is that for @xmath6 a _ new _ regime of stability for the exact solutions appears as long as @xmath7 , where @xmath8 is a function of the nonlinearity parameter @xmath3 . in the absence of the potential the related solitary wave solutions of the nlse are _ unstable _ for @xmath6 .

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Proceed to summarize the following text: the study of adjoint actions and variants thereof , and in particular the classification of orbits for such actions and the description of the orbit closures , are a common theme in lie representation theory . the archetypical example is the jordan - gerstenhaber theory for the conjugacy classes of complex @xmath2-matrices . + a more recent case is a. melnikov s study of the action of the borel subgroup @xmath0 acting on upper - triangular @xmath1-nilpotent matrices via conjugation @xcite . the orbits and their closures are described there combinatorially in terms of so - called link patterns , which we will recapitulate in section [ rom ] . + our aim in this paper is to generalize the work of a. melnikov by extending the variety of upper - triangular @xmath1-nilpotent matrices to all @xmath1-nilpotent matrices . the basic setup to reach this goal is a translation of the classification problem to a problem in representation theory of finite - dimensional algebras . more precisely , this translation yields a bijection between the orbits and the isomorphism classes of certain representations of a specific finite - dimensional algebra , see section [ trans ] . after a brief summary of methods from the representation theory of algebras ( see , for example , @xcite ) in section [ roa ] , we are able to calculate all indecomposable representations using auslander - reiten theory @xcite in section [ covsection ] and to classify the required representations . this gives a combinatorial classification in terms of oriented link patterns in section [ cob ] . + since several results on orbit closures for representations of finite - dimensional algebras are available through work of g. zwara @xcite , we can also characterize the orbit closures of @xmath1-nilpotent matrices in section [ close ] . + finally , we study the conjugation action of upper - triangular matrices on arbitrary nilpotent matrices . we provide a generic normal form for the orbits of this action in section [ gnf ] and construct a large class of semiinvariants in section [ si ] . + * acknowledgments : * the authors would like to thank k. bongartz and a. melnikov for valuable discussions concerning the methods and results of this work . in this section , we fix some notation and collect information about the aforementioned group action . in addition , we summarize material from the representation theory of finite - dimensional algebras . + let @xmath3 be the field of complex numbers . we denote by @xmath4 the borel subgroup of upper - triangular matrices , by @xmath5 the variety of nilpotent @xmath2-matrices , and by @xmath6 the closed subvariety of @xmath1-nilpotent such matrices . obviously , @xmath7 and @xmath0 act on @xmath8 and on @xmath6 via conjugation . + in case of the action of @xmath7 on @xmath8 , the classical jordan - gerstenhaber theory gives a complete classification of the orbits and their closures in terms of partitions ( or , equivalently , young diagrams ) . + our aim is to classify the orbits @xmath9 of @xmath1-nilpotent matrices @xmath10 under the action of @xmath0 . such a classification will be given in terms of _ oriented link patterns _ ; these are oriented graphs on the set of vertices @xmath11 such that every vertex is incident with at most one arrow . this is followed by a description of the orbit closures , by giving a necessary and sufficient condition to decide whether one orbit is contained in the closure of another , and by a method to construct all orbits contained in a given orbit closure . these descriptions are also given in terms of oriented link patterns . the group @xmath0 also acts on @xmath12 , the space of all upper - triangular matrices in @xmath8 , and on @xmath13 . the orbits and their closures for the latter action are described by a. melnikov in @xcite . since these results will be generalized in the following , we describe them in more detail . + let @xmath14 be the set of involutions in the symmetric group @xmath15 in @xmath16 letters . an element @xmath17 of @xmath18 is represented by a so - called link pattern , an unoriented graph with vertices @xmath11 and an edge between @xmath19 and @xmath20 if @xmath21 . for example , the involution @xmath22 corresponds to the link pattern ( 100,30 ) ( 10,10)(20,0)5 ( 10,-5)1 ( 30,-5)2 ( 50,-5)3 ( 70,-5)4 ( 90,-5)5 ( 10,10)(20,40)(30,10 ) ( 50,10)(70,40)(90,10 ) . for @xmath23 , define @xmath24 by @xmath25 and denote by @xmath26 the @xmath0-orbit of @xmath27 . [ mel1]@xcite every orbit of @xmath0 in @xmath28 is of the form @xmath29 for a unique @xmath23 . the next step is to look at the ( zariski-)closures of the orbits @xmath29 . + for @xmath30 , consider the canonical projection @xmath31 deleting the first @xmath32 and the last @xmath33 columns and rows of a matrix in @xmath28 . define the rank matrix @xmath34 of @xmath35 by @xmath36 the rank matrix @xmath34 is @xmath0-invariant , and we denote @xmath37 for @xmath23 . we define a partial ordering on the set of rank matrices by @xmath38 if @xmath39 for all @xmath19 and @xmath20 , inducing a partial ordering on @xmath14 by @xmath40 if @xmath38 . @xcite the orbit closure of @xmath41 is given by @xmath42 . moreover , the entry @xmath43 of the rank matrix equals the number of edges with end points @xmath44 and @xmath45 such that @xmath46 in the link pattern of @xmath17 . the theorem thus gives a combinatorial characterization of the @xmath0-orbits in @xmath28 and their orbit closures in terms of link patterns . as we make key use of results from the representation theory of finite - dimensional algebras for the study of the action of @xmath0 on @xmath6 , we now recall the basic setup of this theory and refer to @xcite and @xcite for a thorough treatment . let @xmath47 be a finite quiver , that is , a directed graph @xmath48 consisting of a finite set of vertices @xmath49 and a finite set of arrows @xmath50 , whose elements are written as @xmath51 ; the vertices @xmath52 and @xmath53 are called the source and the target of @xmath54 , respectively . a path in @xmath47 is a sequence of arrows @xmath55 such that @xmath56 for all @xmath57 ; we formally include a path @xmath58 of length zero for each @xmath59 starting and ending in @xmath19 . we have an obvious notion of concatenation @xmath60 of paths @xmath55 and @xmath61 such that @xmath62 . + the path algebra @xmath63 is defined as the @xmath64-vector space with basis consisting of all paths in @xmath47 , and with multiplication @xmath65 the radical @xmath66 is defined as the ( two - sided ) ideal generated by paths of positive length . an ideal @xmath67 of @xmath63 is called admissible if @xmath68 for some @xmath69 . + the key feature of such pairs @xmath70 consisting of a quiver @xmath47 and an admissible ideal @xmath71 is the following : every finite - dimensional @xmath64-algebra @xmath72 is morita - equivalent to an algebra of the form @xmath73 , in the sense that their categories of finite - dimensional @xmath64-representations are ( @xmath64-linearly ) equivalent . + a finite - dimensional @xmath64-representation @xmath74 of @xmath47 consists of a tuple of @xmath64-vector spaces @xmath75 for @xmath76 , and a tuple of @xmath64-linear maps @xmath77 indexed by the arrows @xmath78 in @xmath50 . a morphism of two such representations @xmath79 and @xmath80 consists of a tuple of @xmath64-linear maps @xmath81 such that @xmath82 for a representation @xmath74 and a path @xmath83 in @xmath47 as above , we denote @xmath84 . we call @xmath74 bound by @xmath67 if @xmath85 whenever @xmath86 . + the abelian @xmath64-linear category of all representations of @xmath47 bound by @xmath67 is denoted by @xmath87 ; it is equivalent to the category of finite - dimensional representations of the algebra @xmath73 . we have thus found a `` linear algebra model '' for the category of finite - dimensional representations of an arbitrary finite - dimensional @xmath64-algebra @xmath72 . + we define the dimension vector @xmath88 of @xmath74 by @xmath89 for @xmath76 . for a fixed dimension vector @xmath90 , we consider the affine space @xmath91 ; its points naturally correspond to representations @xmath74 of @xmath47 of dimension vector @xmath92 with @xmath93 for @xmath59 . via this correspondence , the set of such representations bound by @xmath67 corresponds to a closed subvariety @xmath94 . it is obvious that the algebraic group @xmath95 acts on @xmath96 and on @xmath97 via base change @xmath98 . by definition , the @xmath99-orbits @xmath100 of this action naturally correspond to the isomorphism classes of representations @xmath74 in @xmath87 of dimension vector @xmath92 . + by the krull - schmidt theorem , every representation in @xmath87 is isomorphic to a direct sum of indecomposables , unique up to isomorphisms and permutations . thus , knowing the isomorphism classes of indecomposable representations in @xmath87 and their dimension vectors , we can classify the orbits of @xmath101 in @xmath102 . + for certain classes of finite - dimensional algebras , a convenient tool for the classification of the indecomposable representations is the auslander - reiten quiver @xmath103 of @xmath73 . its vertices @xmath104 $ ] are given by the isomorphism classes of indecomposable representations of @xmath73 ; the arrows between two such vertices @xmath104 $ ] and @xmath105 $ ] are parametrized by a basis of the space of so - called irreducible maps @xmath106 . several standard techniques are available for the calculation of @xmath103 , see for example @xcite and @xcite . we will illustrate one of these techniques , namely the use of covering quivers , in subsection [ covsection ] in a situation relevant for our setup . our aim in this section is to translate the classification problem for the action of @xmath0 on @xmath6 into a representation - theoretic one . the following is a well - known fact on associated fibre bundles : [ basicthm ] let @xmath107 be an algebraic group , let @xmath108 and @xmath109 be @xmath110varieties , and let @xmath111 be a @xmath107-equivariant morphism . assume that @xmath109 is a single @xmath107-orbit , @xmath112 . define @xmath113 and @xmath114 . then @xmath108 is isomorphic to the associated fibre bundle @xmath115 , and the embedding @xmath116 induces a bijection between @xmath117-orbits in @xmath118 and @xmath119-orbits in @xmath108 preserving orbit closures . we consider the following quiver , denoted by @xmath47 from now on , \(m ) [ matrix of math nodes , row sep=0.05em , column sep=2em , text height=1.5ex , text depth=0.2ex ] : & & & & & & & + & 1 & 2 & 3 & & n-2 & n-1 & n + ; ( m-1 - 2 ) edge node[above=0.05 cm ] @xmath120 ( m-1 - 3 ) ( m-1 - 3 ) edge node[above=0.05 cm ] @xmath121(m-1 - 4 ) ( m-1 - 6 ) edge node[above=0.05 cm ] @xmath122(m-1 - 7 ) ( m-1 - 7 ) edge node[above=0.05 cm ] @xmath123 ( m-1 - 8 ) ( m-1 - 8 ) edge [ loop right ] node@xmath54 ( m-1 - 8 ) ; together with the ideal @xmath71 generated by the path @xmath124 . we consider the full subcategory @xmath125 of @xmath87 consisting of representations @xmath74 for which the linear maps @xmath126 are injective . corresponding to this subcategory , we have an open subset @xmath127 , which is stable under the @xmath101-action . we consider the dimension vector @xmath128 . [ thm ] there exists a closure - preserving bijection between the set of @xmath0-orbits in @xmath6 and the set of @xmath101-orbits in @xmath129 . consider the subquiver @xmath130 of @xmath47 with @xmath131 and @xmath132 . we have a natural @xmath101-equivariant projection @xmath133 . the variety @xmath134 consists of tuples of injective maps , thus the action of @xmath101 on @xmath135 is easily seen to be transitive . namely , @xmath134 is the orbit of the representation @xmath136 with @xmath137 being the canonical embedding from @xmath138 to @xmath139 . the stabilizer @xmath117 of @xmath140 is isomorphic to @xmath0 , and the fibre of @xmath141 over @xmath140 is isomorphic to @xmath6 . thus , @xmath142 is isomorphic to the associated fibre bundle @xmath143 , yielding the claimed bijection . by the results of the previous section , it suffices to classify the indecomposable representations in @xmath87 to obtain a classification of the orbits of @xmath0 in @xmath6 . we compute the auslander - reiten quiver @xmath144 of @xmath73 using covering theory , which is described in @xcite as mentioned before . we consider the ( infinite ) quiver @xmath145 given by \(m ) [ matrix of math nodes , row sep=0.85em , column sep=2em , text height=1.5ex , text depth=0.2ex ] & & & & & & & + & & & & & & & + : & & & & & & & + & & & & & & & + & & & & & & & + & 1 & 2 & 3 & & n-2 & n-1 & n + ; ( m-2 - 2 ) edge node[above=0.05 cm ] ( m-2 - 3 ) ( m-2 - 3 ) edge node[above=0.05 cm ] ( m-2 - 4 ) ( m-2 - 6 ) edge node[above=0.05 cm ] ( m-2 - 7 ) ( m-2 - 7 ) edge node[above=0.05 cm ] ( m-2 - 8 ) ( m-3 - 2 ) edge node[above=0.05 cm ] ( m-3 - 3 ) ( m-3 - 3 ) edge node[above=0.05 cm ] ( m-3 - 4 ) ( m-3 - 6 ) edge node[above=0.05 cm ] ( m-3 - 7 ) ( m-3 - 7 ) edge node[above=0.05 cm ] ( m-3 - 8 ) ( m-4 - 2 ) edge node[above=0.05 cm ] ( m-4 - 3 ) ( m-4 - 3 ) edge node[above=0.05 cm ] ( m-4 - 4 ) ( m-4 - 6 ) edge node[above=0.05 cm ] ( m-4 - 7 ) ( m-4 - 7 ) edge node[above=0.05 cm ] ( m-4 - 8 ) ( m-1 - 8 ) edge node[right=0.05 cm ] ( m-2 - 8 ) ( m-2 - 8 ) edge node[right=0.05 cm ] @xmath137 ( m-3 - 8 ) ( m-3 - 8 ) edge node[right=0.05 cm ] @xmath146 ( m-4 - 8 ) ( m-4 - 8 ) edge node[right=0.05 cm ] ( m-5 - 8 ) ; with the ideal @xmath147 generated by all paths @xmath148 , and the quiver @xmath149 given by \(m ) [ matrix of math nodes , row sep=0.85em , column sep=2em , text height=1.5ex , text depth=0.2ex ] : & & & & & & & + & & & & & & & + & 1 & 2 & 3 & & n-2 & n-1 & n + ; ( m-1 - 2 ) edge node[above=0.05 cm ] ( m-1 - 3 ) ( m-1 - 3 ) edge node[above=0.05 cm ] ( m-1 - 4 ) ( m-1 - 6 ) edge node[above=0.05 cm ] ( m-1 - 7 ) ( m-1 - 7 ) edge node[above=0.05 cm ] ( m-1 - 8 ) ( m-2 - 2 ) edge node[above=0.05 cm ] ( m-2 - 3 ) ( m-2 - 3 ) edge node[above=0.05 cm ] ( m-2 - 4 ) ( m-2 - 6 ) edge node[above=0.05 cm ] ( m-2 - 7 ) ( m-2 - 7 ) edge node[above=0.05 cm ] ( m-2 - 8 ) ( m-1 - 8 ) edge node[right=0.05 cm ] @xmath54 ( m-2 - 8 ) ; the quiver @xmath145 carries a natural action of the group @xmath150 by shifting the rows , such that @xmath151 . moreover , @xmath149 naturally embeds into @xmath145 , such that the composition of this inclusion with the projection @xmath152 is surjective . by results of covering theory @xcite , we have corresponding maps of the auslander - reiten quivers , namely an embedding @xmath153 and a quotient @xmath154 , such that the composition is surjective . since @xmath149 is nothing else than a dynkin quiver of type @xmath155 , it is routine to calculate its auslander - reiten quiver ( see @xcite ) , and we derive the auslander - reiten quiver @xmath156 just by making the identifications resulting from the action of @xmath150 , which can be read off from the dimension vectors of indecomposable representations . more examples and details concerning the calculation of auslander - reiten quivers using covering theory can also be found in @xcite . + we finally arrive at the picture ( the marked regions have to be identified ) given in figure [ ark ] . the auslander - reiten quiver of @xmath87,scaledwidth=100.0% ] we define the following representations @xmath157 for @xmath158 , @xmath159 for @xmath160 and @xmath161 for @xmath162 in @xmath87 ( graphically represented by dots for basis elements and arrows for a map sending one basis element to another one ) : + @xmath157 for @xmath163 : @xmath164 { 0 & { \xrightarrow}{0 } & \cdots & { \xrightarrow}{0 } & 0 & { \xrightarrow}{0 } & k & { \xrightarrow}{id } & \cdots & { \xrightarrow}{id } & k & { \xrightarrow}{e_1 } & k^{2 } & { \xrightarrow}{id } & \cdots & { \xrightarrow}{id } & k^{2 } \\ & & & & & & \bullet & { \rightarrow } & \cdots & { \rightarrow } & \bullet & { \rightarrow } & \bullet & { \rightarrow } & \cdots & { \rightarrow } & \bullet \\ & & & & & & j & & & & & & i & & & & n\\ & & & & & & & & & & & & \bullet & { \rightarrow } & \cdots & { \rightarrow } & \bullet \\ } ; \path[- > ] ( m-2 - 17 ) edge [ bend left=80 ] ( m-4 - 17 ) ( m-1 - 17 ) edge [ loop right ] node{$\alpha$ } ( m-1 - 17 ) ; \end{tikzpicture}\ ] ] @xmath157 for @xmath165 : @xmath164 { 0 & { \xrightarrow}{0 } & \cdots & { \xrightarrow}{0 } & 0 & { \xrightarrow}{0 } & k & { \xrightarrow}{id } & \cdots & { \xrightarrow}{id } & k & { \xrightarrow}{e_2 } & k^{2 } & { \xrightarrow}{id } & \cdots & { \xrightarrow}{id } & k^{2 } \\ & & & & & & \bullet & { \rightarrow } & \cdots & { \rightarrow } & \bullet & { \rightarrow } & \bullet & { \rightarrow } & \cdots & { \rightarrow } & \bullet \\ & & & & & & i & & & & & & j & & & & n\\ & & & & & & & & & & & & \bullet & { \rightarrow } & \cdots & { \rightarrow } & \bullet \\ } ; \path[- > ] ( m-4 - 17 ) edge [ bend right=80 ] ( m-2 - 17 ) ( m-1 - 17 ) edge [ loop right ] node{$\alpha$ } ( m-1 - 17 ) ; \end{tikzpicture}\ ] ] @xmath166 for @xmath160 : @xmath167 { 0 & { \xrightarrow}{0 } & \cdots & { \xrightarrow}{0 } & 0 & { \xrightarrow}{0 } & k & { \xrightarrow}{id } & \cdots & { \xrightarrow}{id } & k \\ & & & & & & i & & & & n\\ & & & & & & \bullet & { \rightarrow } & \cdots & { \rightarrow } & \bullet \\ } ; \path[- > ] ( m-1 - 11 ) edge [ loop right ] node{$0 $ } ( m-1 - 11 ) ; \end{tikzpicture}\ ] ] @xmath161 for @xmath168 : @xmath167 { 0 & { \xrightarrow}{0 } & \cdots & { \xrightarrow}{0 } & 0 & { \xrightarrow}{0 } & k & { \xrightarrow}{id } & \cdots & { \xrightarrow}{id } & k & { \xrightarrow}{0 } & 0 & { \xrightarrow}{0 } & \cdots & { \xrightarrow}{0 } & 0 \\ & & & & & & i & & & & j & & & & & & n\\ & & & & & & \bullet & { \rightarrow } & \cdots & { \rightarrow } & \bullet & & & & & & \\ } ; \path[- > ] ( m-1 - 17 ) edge [ loop right ] node{$0 $ } ( m-1 - 17 ) ; \end{tikzpicture}\ ] ] here we denote @xmath169 , @xmath170 and @xmath171 . [ uvw ] the representations @xmath157 , @xmath159 and @xmath161 form a system of representatives of the indecomposable objects in @xmath87 . the representations @xmath157 and @xmath159 form a system of representatives of the indecomposable objects in @xmath125 the endomorphism rings of these representations are easily computed to be @xmath172/(x^2)\mbox { for } i\leq j,\ ] ] @xmath173 thus they are indecomposable . their dimension vectors are @xmath174 respectively . these are precisely the dimension vectors appearing in @xmath103 , thus we have found all indecomposables . it is clear from the definition that the indecomposable representations belonging to @xmath125 are the @xmath157 and the @xmath159 . our next aim is to parametrize the isomorphism classes of representations in @xmath125 of dimension vector @xmath92 . as mentioned before , the krull - schmidt theorem states that every representation can be decomposed into a direct sum of indecomposables in an essentially unique way . [ classification ] the isomorphism classes @xmath74 in @xmath125 of dimension vector @xmath92 are in natural bijection to 1 . @xmath2-matrices @xmath175 with entries @xmath176 or @xmath177 , such that @xmath178 for all @xmath179 , 2 . oriented link patterns on @xmath11 , that is , oriented graphs on the set @xmath11 such that every vertex is incident with at most one arrow . moreover , if an isomorphism class @xmath74 corresponds to a matrix @xmath72 under this bijection , the orbit @xmath180 and the orbit @xmath181 correspond to each other via the bijection of lemma [ thm ] . let @xmath74 be a representation in @xmath125 of dimension vector @xmath92 , so @xmath182 for some multiplicities @xmath183 by theorem [ uvw ] . since @xmath184 , we simply need to calculate all tuples @xmath185 such that @xmath186 applying the automorphism @xmath187 of @xmath188 defined by @xmath189 this condition is equivalent to @xmath190 if we fix @xmath191 , this condition states that @xmath192 we can extract an oriented graph on the set of vertices @xmath11 from @xmath193 as follows : for all @xmath158 , we have an arrow from @xmath20 to @xmath19 if @xmath194 . the conditions on @xmath193 ensure that this graph is in fact an oriented link pattern . the matrix @xmath193 is obviously @xmath1-nilpotent . + the decomposition of @xmath74 into indecomposables can be visualized as follows . @xmath195 { & \bullet & \rightarrow & \bullet & \rightarrow & \bullet & \cdots & \bullet & \rightarrow & \bullet & \rightarrow & \bullet & ~~ & 1 \\ & & & \bullet & \rightarrow & \bullet & \cdots & \bullet & \rightarrow & \bullet & \rightarrow & \bullet & ~~ & 2\\ & & & & & \bullet & \cdots & \bullet & \rightarrow & \bullet & \rightarrow & \bullet & ~~ & 3\\ m : & & & & & & & & \vdots & & \vdots & & & \\ & & & & & & & \bullet & \rightarrow & \bullet & \rightarrow & \bullet & ~~ & n-2\\ & & & & & & & & & \bullet & \rightarrow & \bullet & ~~ & n-1\\ & & & & & & & & & & & \bullet & ~~ & n\\ } ; \path[- > ] ( m-1 - 12 ) edge [ bend left=80 ] ( m-3 - 12 ) ( m-7 - 12 ) edge [ bend right=50 ] ( m-2 - 12 ) ( m-6 - 12 ) edge [ bend right=120 ] ( m-5 - 12 ) ; \end{tikzpicture}\ ] ] the arrows in the rightmost column of the diagram allow us to read off the indecomposable direct summands of @xmath74 . namely , @xmath157 is a direct summand of @xmath74 if and only if there is an arrow @xmath196 . if there is no arrow at @xmath64 , the indecomposable @xmath197 is a direct summand of @xmath74 . + shortening the above picture to the rightmost column , @xmath74 corresponds to an oriented link pattern : @xmath198 for a given matrix @xmath10 , we would like to decide to which oriented link pattern it corresponds . define @xmath199 , the span of the first @xmath19 coordinate vectors in @xmath200 , and define a matrix @xmath201 by setting @xmath202 ( we formally define @xmath203 for @xmath204 or @xmath205 ) . the matrix @xmath206 is obviously an invariant for the @xmath0-action on @xmath6 . it is easy to extract an oriented link pattern from @xmath206 as follows : the matrix @xmath72 belongs to the orbit of a matrix @xmath193 as above if and only if @xmath207 or , conversely , @xmath208 for all @xmath158 . by @xmath0-invariance , we just have to compute @xmath206 for @xmath175 as in the previous theorem . we have @xmath209 if and only @xmath210 and there exists @xmath211 such that @xmath212 or , equivalently , such that there exists an arrow @xmath213 in the corresponding oriented link pattern . since both @xmath214 and @xmath215 are spanned by coordinate vectors @xmath216 , we thus have @xmath217 . the second formula follows . we can also rederive theorem [ mel1 ] of a. melnikov : every @xmath218orbit of an upper - triangular @xmath1-nilpotent matrix corresponds to the orbit of a representation in @xmath125 of dimension vector @xmath92 which does not contain @xmath157 for @xmath219 as a direct summand . in this case , the corresponding link pattern consists of arrows pointing in the same direction . we can thus delete the orientation and arrive at a link pattern as in @xcite . + * remark : * our method is easily generalized to obtain a classification of orbits for a more general group action : let @xmath220 be the parabolic subgroup consisting of block - upper triangular matrices with block - sizes @xmath221 . then @xmath222 acts on @xmath6 by conjugation , and the same reasoning as above yields a bijection between @xmath222-orbits in @xmath6 and isomorphism classes of representations in @xmath125 of dimension vector @xmath223 . using the analysis of this section , the @xmath222-orbits in @xmath6 correspond bijectively to matrices @xmath193 such that @xmath224 for all @xmath225 . consequently , they correspond bijectively to `` enhanced oriented link patterns of type @xmath221 '' , namely , to oriented graphs on the set @xmath226 such that the vertex @xmath19 is incident with at most @xmath227 arrows for all @xmath19 . after classifying the orbits via oriented link patterns , we describe the corresponding orbit closures . again , we will solve this problem using results about the geometry of representations of algebras . two theorems of g. zwara are the key to calculating these orbit closures , see @xcite and @xcite for more details . let @xmath74 and @xmath228 be two representations in @xmath87 of the same dimension vector @xmath92 . we say that @xmath74 degenerates to @xmath228 if @xmath229 in @xmath230 , which will be denoted by @xmath231 . since the correspondence of lemma [ thm ] preserves orbit closure relations , we know that @xmath231 if and only if the corresponding @xmath1-nilpotent matrices , denoted by @xmath175 and @xmath232 , respectively , fulfill @xmath233 in @xmath6 . [ zw](zwara ) suppose an algebra @xmath73 is representation - finite , that is , @xmath73 admits only finitely many isomorphism classes of indecomposable representations . let @xmath74 and @xmath228 be two finite - dimensional representations of @xmath73 of the same dimension vector . + then @xmath234 if and only if @xmath235 for every representation @xmath236 of @xmath73 . to simplify notation , we set @xmath237 : = \dim_k \operatorname{hom}(u , v ) $ ] for two representations @xmath236 and @xmath238 . since the dimension of a homomorphism space is additive with respect to direct sums , we only have to consider the inequality @xmath239\leq[u , m']$ ] for indecomposable representations @xmath236 to characterize a degeneration @xmath231 furthermore , since @xmath240=0 $ ] for all representations @xmath74 in @xmath125 by a direct calculation , we can restrict these indecomposables @xmath236 to those of type @xmath157 and @xmath159 of the previous section . + we can easily calculate the dimensions of homomorphism spaces between these indecomposable representations . [ hom ] for @xmath241 we have * @xmath242 = \delta_{i\leq k}$ ] , * @xmath243 = \delta_{i\leq k}$ ] , * @xmath244 = \delta_{i\leq l}$ ] , * @xmath245 = \delta_{i\leq l } + \delta_{j\leq l } \cdot \delta_{i\leq k}$ ] , where @xmath246 for a representation @xmath74 in @xmath125 of dimension vector @xmath92 ( or equivalently , for the corresponding @xmath1-nilpotent matrix @xmath72 ) , consider the corresponding oriented link pattern . define @xmath247 as the number of vertices to the left of @xmath64 which are not incident with an arrow , plus the number of arrows whose target vertex is to the left of @xmath64 . define @xmath248 as @xmath249 plus the number of arrows whose source vertex lies to the left of @xmath250 and whose target vertex lies to the left of @xmath64 . [ pq ] we have @xmath234 ( or equivalently , @xmath233 in the notation above ) if and only if @xmath251 and @xmath252 for all @xmath253 . given two representations @xmath74 and @xmath228 , we write @xmath254 for an indecomposable @xmath236 , the condition @xmath255 \leq [ u , m ' ] $ ] is then equivalent to @xmath256 + \sum_{i=1}^n n_i[u , { \mathcal{v}}_i ] \leq \sum_{i , j=1}^{n } m'_{i , j } [ u , { \mathcal{u}}_{i , j}]+ \sum_{i=1}^n n'_i[u , { \mathcal{v}}_i ] .\ ] ] using the dimensions of homomorphism spaces between indecomposable representations stated in the previous lemma , we calculate @xmath257 and @xmath258 and the condition @xmath239\leq [ u , m']$ ] is equivalent to the conditions @xmath251 and @xmath252 for all @xmath253 . the claimed interpretation of these values @xmath247 and @xmath259 in terms of oriented link patterns follows from theorem [ classification ] . as a next step , we develop a combinatorial method to produce all degenerations of a given representation @xmath74 in @xmath125 of dimension vector @xmath92 out of its oriented link pattern . it is sufficient to construct all minimal degenerations , that is , degenerations @xmath260 such that if @xmath261 , then @xmath262 or @xmath263 . minimal degenerations are denoted by @xmath264 . + in @xcite , g. zwara describes all minimal degenerations ; the result is stated here in a generality sufficient for our purposes . denote by @xmath265 the transitive closure of the relation on representations given by @xmath266 if there exists a short exact sequence @xmath267 such that @xmath268 . let @xmath74 and @xmath228 be representations in @xmath125 . + if @xmath269 , then one of the following holds : 1 . . there are representations @xmath271 , @xmath272 , @xmath273 in @xmath125 such that 1 . @xmath274 2 . @xmath275 3 . @xmath276 4 . @xmath277 is indecomposable . combining this theorem with the technique of ( * * theorem 4 ) , we obtain a characterization of minimal disjoint degenerations , that is , minimal degenerations @xmath264 such that @xmath74 and @xmath228 do not share a common direct summand : let @xmath264 be a minimal disjoint degeneration as before . then either @xmath228 is indecomposable or @xmath278 , where @xmath236 and @xmath238 are indecomposables and there exists an exact sequence @xmath279 or @xmath280 . thus we see that all minimal degenerations are of the form @xmath281 , where @xmath74 and @xmath228 are as in the corollary , thus @xmath228 involves at most two indecomposable direct summands . translating this to the language of oriented link patterns using theorem [ classification ] , we have `` localized '' the problem to the consideration of at most four vertices of an oriented link pattern . in this local case , we can apply theorem [ pq ] and easily work out all minimal degenerations . every minimal degeneration is of the form given in one of the following diagrams showing parts of the degeneration posets in terms of oriented link patterns . we assume that @xmath282 ( resp . @xmath283 , resp . @xmath284 ) are vertices of an oriented link pattern , and only indicate the changes to the arrows incident with one of these vertices ; all other arrows are left unchanged . @xmath285 @xmath286 * remark : * note that , although every minimal degeneration is of the form @xmath281 as above , the choice of @xmath271 is not arbitrary , that is , addition of an arbitrary @xmath271 might lead to a non - minimal degeneration . the precise conditions on @xmath271 neccessary for this degeneration to be minimal will be described in @xcite ; as a consequence , it will be shown in @xcite that all minimal degenerations are of codimension @xmath177 . + we have thus obtained a constructive way of describing an orbit closure @xmath287 of a @xmath1-nilpotent matrix @xmath72 in terms of its corresponding link pattern : by repeated application of the local changes to the arrows as in the theorem , we produce a list of all link patterns corresponding to matrices @xmath288 such that @xmath289 ( although this list will contain repetitions due to non - minimal degenerations ) . in this section , we consider the action of @xmath0 on @xmath8 by conjugation in general ( for the analogous problem of the action of @xmath0 on @xmath290 , see @xcite ) . + the starting point is the following observation ( see @xcite ) : + * example : * consider the action of @xmath291 on @xmath292 via conjugation . then the matrices @xmath293 $ ] for @xmath294 are pairwise non - conjugate . furthermore , on the open set @xmath295 of nilpotent matrices @xmath296 $ ] where @xmath297 or @xmath298 , the map @xmath299 given by @xmath300 is surjective and @xmath291-invariant . + we generalize some aspects of this example to arbitrary @xmath16 . it is appropriate to reformulate the problem as follows : we consider the action of @xmath301 on pairs @xmath302 consisting of a complete flag @xmath303 and a nilpotent operator @xmath304 of an @xmath16-dimensional @xmath64-vector space @xmath238 . then the orbits of this action are precisely the orbits of @xmath0 in @xmath8 since the variety of complete flags is isomorphic to the homogeneous space @xmath305 . 1 . @xmath306 for all @xmath307 , 2 . @xmath308 for all @xmath307 , or , equivalently , all induced maps @xmath309 are invertible , 3 . there exists a unique basis @xmath310 of @xmath238 such that 1 . @xmath311 for all @xmath312 , 2 . @xmath313 for all @xmath314 . obviously , the second property implies the first . we show that the third property implies the second one ; so assume there exists a basis @xmath310 with the properties ( a ) and ( b ) . by an easy induction , we have @xmath315 for all @xmath316 , and @xmath317 if @xmath318 . we thus have @xmath319 and the second property follows since @xmath320 . + conversely , assume that @xmath321 for all @xmath64 . in particular , we have @xmath322 , and thus @xmath323 and @xmath324 for all @xmath64 . we choose an arbitrary basis @xmath325 of @xmath238 which is adapted to @xmath326 , that is , such that @xmath327 for all @xmath64 . then , for all @xmath64 , the elements @xmath328 generate the @xmath64-dimensional space @xmath329 , thus they form a basis of this space . we can thus write the element @xmath330 uniquely as @xmath331 and we define @xmath332 for all @xmath64 . note that the elements @xmath333 do not depend on the choice of basis elements @xmath325 . we have @xmath334 : otherwise @xmath335 , and application of @xmath336 yields @xmath337 and thus @xmath338 for all @xmath19 by linear independence of the elements @xmath339 . then @xmath340 , a contradiction . since the elements @xmath341 form a basis and the @xmath342 are non - zero , the elements @xmath333 form a basis , too , which is again adapted to @xmath326 . + we have @xmath343 by definition . for @xmath318 , we thus have @xmath344 it follows that @xmath345 belong to @xmath346 , thus they form a basis of this space for dimension reasons . it also follows that @xmath347 form a basis of @xmath348 . + writing @xmath349 , we apply @xmath350 and calculate @xmath351 @xmath352 and thus @xmath353 and @xmath354 for all @xmath355 by linear independence of @xmath356 . we thus have @xmath357 for all @xmath316 , and , in particular , @xmath358 for all @xmath64 . the basis @xmath310 thus has the claimed properties . 1 . for @xmath307 , the first @xmath64 columns of @xmath365 are linearly independent , 2 . for @xmath307 , the minor @xmath366 is non - zero , 3 . @xmath72 is @xmath0-conjugate to a unique matrix @xmath117 such that @xmath367 for @xmath368 and @xmath369 for all @xmath370 . we apply the previous theorem to the vector space @xmath371 with coordinate basis @xmath372 , the standard flag defined by @xmath373 and the endomorphism @xmath336 given by multiplication by @xmath72 . the first property of the theorem immediately translates into linear independence of column vectors , whereas the second property translates to the non - vanishing of minors . the basis @xmath310 of the theorem yields an upper - triangular base change matrix , and representing @xmath72 with respect to this basis yields the desired @xmath0-conjugate @xmath117 . + we construct a class of determinantal @xmath0-semiinvariants on @xmath8 , that is , regular functions @xmath374 on @xmath8 such that @xmath375 for all @xmath376 and @xmath360 ; here @xmath377 is a character on @xmath0 called the weight of @xmath374 . for @xmath179 , we denote by @xmath378 the character defined by @xmath379 ; the @xmath380 form a basis for the group of characters of @xmath0 . + fix non - negative integers @xmath381 such that @xmath382 . moreover , fix polynomials @xmath383 $ ] for @xmath384 and @xmath385 , and denote the datum @xmath386 by @xmath387 . for all such @xmath19 and @xmath20 , consider the @xmath388-submatrices @xmath389 as defined in the previous section , and form the block matrix @xmath390 ; this is a @xmath391-matrix . for @xmath376 and @xmath395 , denote by @xmath396 ( resp . by @xmath397 ) the submatrix formed by the last @xmath362 rows and columns ( resp . by the first @xmath363 rows and columns ) of @xmath398 . with these definitions , it follows immediately that @xmath399 this yields the following equalities of block matrices @xmath400 @xmath401 and thus @xmath402 with the aid of these semiinvariants , we can see that the entries of the normal form @xmath117 associated to a matrix @xmath72 fulfilling the conditions of corollary [ hnf ] depend polynomially on @xmath72 , by describing them as the value of a special semiinvariant @xmath393 : for @xmath19 and @xmath20 such that @xmath403 and @xmath404 , consider the datum @xmath387 as above defined by @xmath405 , @xmath406 , @xmath407 , @xmath408 , @xmath409 , @xmath410 , @xmath411 , @xmath412 . then , for a matrix @xmath117 in the form of corollary [ hnf ] , we have @xmath413 . by a direct calculation , the matrix @xmath414 consists of the blocks @xmath415 @xmath416 thus , the matrix @xmath414 is lower triangular , all diagonal entries being @xmath177 except the @xmath417-entry , which equals @xmath418 . it seems likely that the semiinvariants @xmath393 generate the ring of all semiinvariants at least for a certain cone of weights . the generic normal form of corollary [ hnf ] allows to find identities between the @xmath393 by evaluation on matrices @xmath117 in normal form .

the orbits of the group @xmath0 of upper - triangular matrices acting on @xmath1-nilpotent complex matrices via conjugation are classified via oriented link patterns , generalizing a. melnikov s classification of the @xmath0-orbits on upper - triangular such matrices . the orbit closures as well as the `` building blocks '' of minimal degenerations of orbits are described . the classification uses the theory of representations of finite - dimensional algebras . furthermore , we initiate the study of the @xmath0-orbits on arbitrary nilpotent matrices .

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Proceed to summarize the following text: embedded control systems are ubiquitous and can be found in several applications including aircraft , automobiles , process control , and buildings . an embedded control system is one in which the computer system is designed to perform dedicated functions with real - time computational constraints @xcite . typical features of such embedded control systems are the control of multiple applications , the use of shared networks used by different components of the systems to communicate with each other for control , a large number of sensors as well as actuators , and their distributed presence in the overall system . the most common feature of such distributed embedded control systems ( des ) is shared resources . constrained by space , speed , and cost , often information has to be transmitted using a shared communication network . in order to manage the flow of information in the network , protocols that are time - triggered @xcite and event - triggered @xcite have been suggested over the years . associated with each of these communication protocols are different set of advantages and disadvantages . the assignment of time - triggered ( tt ) slots to all control - related signals has the advantage of high quality of control ( qoc ) due to the possibility of reduced or zero delays , but leads to poor utilization of the communication bandwidth , high cost , overall inflexibility , and infeasibility as the number of control applications increase . on the other hand , event - triggered ( et ) schedules often result in poor control performance due to the unpredictable temporal behavior of control messages and the related large delays which occurs due to the lack of availability of the bus . these imply that a hybrid protocol that suitably switches between these two schedules offers the possibility of exploiting their combined advantages of high qoc , efficient resource utilization , and low cost @xcite . such a hybrid protocol is the focus of this paper . to combine the advantage of tt and et policies , hybrid protocols are increasingly being studied in recent years . examples of such protocols are flexray and ttcan @xcite , used extensively in automotive systems . while several papers have considered control using tt protocols ( see for example , @xcite ) and et protocols ( see for example , @xcite ) , control using hybrid protocols has not been studied in the literature until recently . the co - design problem has begun to be addressed of late as well ( see for example , @xcite ) . in @xcite , the design of scheduling policies that ensure a good quality of control ( qoc ) is addressed . in @xcite , the schedulability analysis of real - time tasks with respect to the stability of control functions is discussed . in @xcite , modeling the real - time scheduling process as a dynamic system , an adaptive self - tuning regulator is proposed to adjust the bandwidth of each single task in order to achieve an efficient cps utilization . the focus of most of the papers above are either on a simple platform or on a single processor . a good survey paper on co - design can be found in @xcite . our focus in this paper is on the co - design of adaptive switching controllers and hybrid protocols so as to ensure good tracking in the presence of parametric uncertainties in the plant being controlled while utilizing minimal resources in the des . the hybrid protocol that is addressed in this paper switches between a tt and a et scheme . the tt scheme , which results in a negligible delay in the processing of the control messages , is employed when a control action is imperative and the et scheme , which typically results in a non - zero delay , is employed when the controlled system is well - behaved , with minimal tracking error . the latter is in contrast to papers such as @xcite and @xcite where the underlying _ event _ is associated with a system error exceeding a certain threshold , while here an _ event _ corresponds to the case when the system error is small . the controller is to be designed for multiple control applications , each of which is subjected to a parametric uncertainty . an adaptive switching methodology is introduced to accommodate these uncertainties and the hybrid nature of the protocol . switched control systems and related areas of hybrid systems and supervisory control have received increased attention in the last decade ( see e.g. , @xcite ) and used in several applications ( see e.g. @xcite ) . adaptive switched and tuned systems have been studied as well ( see @xcite ) . the combined presence of uncertainties and switching delays makes a direct application of these existing results to the current problem inadequate . the solution to the problem of co - design of an adaptive swtiched controller and switches in a hybrid protocol was partially considered in @xcite , where the control goal was one of stabilization . in this paper , we consider tracking , which is a non - trivial extension of @xcite . the main reason for this lies in the trigger for the switch , which corresponds to a system error becoming small . in order to ensure that this error continues to remain small even in the presence of a non - zero reference signal , we needed to utilize fundamental properties of the adaptive system with persistent excitation , and derive additional properties in the presence of reference signals with an invariant persistent excitation property . these properties in turn are suitably exploited and linked with the switching instants , and constitute the main contribution of this paper . in section [ sec : problem ] the problem is formulated , and preliminaries related to adaptive control and persistent excitation are presented . in section [ sec : switchingadaptivecontroller ] , the switching adaptive controller is described and the main result of global boundedness is proved . concluding remarks are presented in section [ sec : conclusion ] . the problem that we address in this paper is the simultaneous control of @xmath0 plants , @xmath1 , @xmath2 , in the presence of impulse disturbances that occur sporadically , using a hybrid communication protocol . we assume that each of these @xmath0 applications have the following problem statement . the plant to be controlled is assumed to have a discrete time model described by @xmath3 + b_0u(k - d)+\sum_{l=1}^{m_2}b_lu(k - l - d)+d(k - d)\end{gathered}\ ] ] where @xmath4 and @xmath5 are the input and output of the @xmath6-th control application , respectively , at the time - instant @xmath7 and @xmath8 is a time - delay . the disturbance @xmath9 are assumed to be impulses that can occur occasionally with their inter - arrival time lower - bounded by a finite constant . the parameters of the @xmath6-th plant are given by @xmath10 , @xmath11 , @xmath12,@xmath13 and are assumed to be unknown . it is further assumed that the sampling time of the controller is a constant @xmath14 , so that @xmath15 . the goal is to choose the control input @xmath16 such that @xmath5 tracks a desired signal @xmath17 , with all signals remaining bounded . the model in ( [ eq : model ] ) can be expressed as @xmath18 where @xmath19 is the backward shift operator and the polynomials @xmath20 and @xmath21 are given by @xmath22 the following assumptions are made regarding the plant poles and zeros : \1 ) an upper bound for the orders of the polynomials in ( [ eq : polyab ] ) is known and 2 ) all zeros of @xmath23 lie strictly inside the closed unit disk . [ ass : fixeddelay ] for any delay @xmath24 , eq . ( [ eq : model ] ) can be expressed in a _ predictor form _ as follows @xcite : @xmath25 with @xmath26 where @xmath27 and @xmath28 are the unique polynomials that satisfy the equation @xmath29 equation ( [ eq : predictorform ] ) can be expressed as @xmath30 where @xmath31 , @xmath32 , @xmath33 , and @xmath34 are defined as @xmath35 @xmath36 with @xmath37 , @xmath38 , @xmath39 , @xmath40 , and @xmath41 and @xmath42 the coefficients of the polynomials in ( [ eq : alphabeta ] ) with respect to the delay @xmath24 and finite initial conditions @xmath43 from eqs . ( [ eq : regressor])-([eq : phiandtheta ] ) , we observe that a feedback controller of the form @xmath44 realizes the objective of stability and follows the desired bounded trajectory @xmath17 in the absence of disturbances . designing a stabilizing controller @xmath4 essentially boils down to a problem of implementing ( [ eq : feedbackcontroller ] ) with the controller gain @xmath32 . two things should be noted : ( i ) controller ( [ eq : feedbackcontroller ] ) is not realizable as @xmath32 and @xmath45 are not known , and ( ii ) the dimension of @xmath31 , @xmath32 as well as the entries of @xmath32 depend on the delay @xmath24 . since @xmath32 and @xmath45 are unknown , we replace them with their parameter estimates and derive the following adaptive control input @xmath46 where @xmath47 denotes the @xmath48-th element of the parameter estimation @xmath49 and is the estimate of @xmath45 . @xmath49 is adjusted according to the adaptive update law @xcite : @xmath50 with @xmath51^t$ ] . equation ( [ eq : avoidzero ] ) is necessary to avoid division by zero in the control law ( [ eq : adaptivecontroller ] ) . theorem [ thm : fixeddelay ] addresses the stability of the adaptive system given by ( [ eq : predictorform ] ) , ( [ eq : adaptivecontroller ] ) , and ( [ eq : updatelaw])-([eq : updateepsilon ] ) . the reader is referred to theorem 6.3.1 in @xcite or theorem 5.1 in @xcite for the proof of theorem [ thm : fixeddelay ] . [ thm : fixeddelay ] let @xmath52 . subject to assumption [ ass : fixeddelay ] and given a fixed delay @xmath24 , the adaptive controller ( [ eq : adaptivecontroller ] ) with the update law ( [ eq : updatelaw ] ) guarantees that the plant given by ( [ eq : predictorform ] ) follows the reference @xmath53 , i.e. , @xmath54 , and that the sequences @xmath55 , @xmath56 and @xmath57 are bounded for all @xmath58 . the following definitions related to persistent excitation are needed to introduce our switching controller . we define the terms _ persistently exciting _ and _ sufficiently rich _ in the following way : [ def : pe ] a sequence @xmath59 is said to be _ persistently exciting ( pe ) ( in @xmath60 steps ) _ , if there exists @xmath61 such that @xmath62 uniformly in @xmath63 . [ def : sr ] a sequence @xmath59 is said to be _ sufficiently rich ( sr ) of order @xmath64 ( in @xmath60 steps ) _ , if there exists @xmath61 such that @xmath65 with @xmath66 uniformly for all @xmath63 . the following lemma is useful to prove theorem [ thm : persistentexcitation ] . suppose that @xmath67 and @xmath68 are two bounded sequences taking values in @xmath69 satisfying @xmath70 . then @xmath67 is sr of order @xmath71 if and only if @xmath68 is sr of order @xmath71 . the reader is referred to @xcite for the proof of theorem [ thm : persistentexcitation ] . [ lem : reachability ] consider the discrete time system @xmath72 with @xmath73 , @xmath74 , @xmath75 , and @xmath76 . assume that ( [ eq : reachable ] ) is completely reachable and that the input @xmath77 is sr of order @xmath78 . then , @xmath79 for all @xmath80 . we first rewrite ( [ eq : reachable ] ) as @xmath81 and define the characteristic polynomial of @xmath20 to be @xmath82 and let @xmath83 then , from the cayley - hamilton theorem it follows that @xmath84 where @xmath85 . let @xmath86 then , @xmath87 where `` @xmath88 '' denotes the direct sum , @xmath89 is the @xmath90 identity matrix , and @xmath91 is the minimal singular value of @xmath92 . @xmath93 can be also stated in terms of @xmath94 in the following way @xmath95 where @xmath96 is given by @xmath97 then , @xmath98 and hence using ( [ eq : kleinergamma ] ) , @xmath99 that is , @xmath100 from theorem 1 in @xcite , it follows that an input signal which is sr of order @xmath71 implies a persistent excitation of at most @xmath71 directions in a @xmath0-dimensional space . thus , @xmath101 _ remark : _ much of the existing results pertaining to persistent excitation pertain to the case when the external input @xmath77 is sr of order @xmath0 . lemma [ lem : reachability ] above as well as corollary [ cor : inomega ] stated below address the case when @xmath77 is sr of order @xmath71 , where @xmath102 , which to our knowledge has not been examined in the literature . as our goal is tracking of an arbitrary signal and not identification , we do not need the sr - order to be @xmath0 , but arbitrary and fixed at some @xmath71 . [ cor : inomega ] consider the discrete time system @xmath103 with @xmath73 , @xmath74 , @xmath75 , and @xmath76 . assume that ( [ eq : reachable2 ] ) is completely reachable and that the input @xmath77 is sr of a fixed order @xmath78 . then , there exists a subspace @xmath104 such that @xmath105 that is , the columns of @xmath94 span the subspace @xmath106 . this follows directly from lemma [ lem : reachability ] and the fact that for any complex matrix @xmath107 the following is true @xmath108 where @xmath109 denotes the image of the linear transformation @xmath107 . we make the following assumption which refers to an invariant property of persistent excitation . [ ass : pe ] @xmath17 is sufficiently rich of constant order @xmath110 for all @xmath58 . theorem [ thm : persistentexcitation ] connects the sufficient richness of @xmath53with the tracking error and the parameter convergence in an adaptive system . [ thm : persistentexcitation ] let @xmath52 . suppose the adaptive controller ( [ eq : adaptivecontroller])-([eq : updateepsilon ] ) is used to control the plant in ( [ eq : thetaregressor ] ) and let assumptions [ ass : fixeddelay ] and [ ass : pe ] hold . then 1 . @xmath111 , and 2 . @xmath112 as @xmath113 3 . @xmath114 converges to @xmath115 where @xmath115 is defined as @xmath116 where @xmath33 is given in ( [ eq : phiandtheta ] ) . item ( i ) follows directly from theorem [ thm : fixeddelay ] as it is independent of any persistent excitation of the reference signal @xmath53 . item ( ii ) follows by noting that the adaptive system in ( [ eq : model ] ) and ( [ eq : adaptivecontroller])-([eq : updateepsilon ] ) becomes asymptotically linear , and this linear system in turn has a state that satisfies ( [ eq : dimmx ] ) due to assumption [ ass : pe ] . item ( iii ) follows from ( i ) and the fact that @xmath117 . hybrid communication protocols such as flexray @xcite provide time - triggered and event - triggered bus schedules . time - triggered communication offers highly predictable temporal behavior , and event - triggered communication provides efficient bandwidth usage . to exploit their combined advantages , we propose the use of a hybrid communication protocol in this paper . to illustrate our proposed scheme , we use flexray as it has been established as the de - facto standard for future automotive in - vehicle networks . the flexray protocol is organized in a sequence of communication cycles of fixed length . further , every such cycle is subdivided into a _ static _ segment ( st ) and a _ dynamic _ segment ( dyn ) . the static segment is partitioned into time windows of fixed and equal length which are referred as _ slots_. each processing unit is assigned one or more slots indexed by a slot number @xmath118 that indicates available time windows for bus access in the static segment . due to the predictable temporal behavior we use the static segment schedules for communication in the time - triggered mode and dynamic segment schedules in the event - triggered mode . the dynamic segment is partitioned into _ minislots _ of much smaller duration than the static slots . similar to the static segment , the minislots are indexed by a slot number to indicate allowable message transmissions . however , dynamic slots are of varying size depending on the size of the message which is transmitted in a certain slot @xmath119 , where @xmath120 is the set of available slot numbers in the dynamic segment . if no message is ready for transmission in a particular slot only one small minislot is consumed and the slot number is incremented with the next minislot . however , if a message is transmitted in a slot @xmath119 then the slot number increments with the next minislot after which the message transmission has been completed . hence , bus resources are only utilized if messages are actually transmitted on the bus ; otherwise only one minislot is consumed . dynamic segment schedules are used for communication in the event - triggered mode . the focus of this problem is the simultaneous control of several applications for stabilization . that is , the goal is to choose @xmath16 , the input of the @xmath6th control application such that @xmath5 , its output , converges to @xmath17 which is zero . in the context of the problem under consideration , all control applications are partitioned into a sensor task @xmath121 , a controller task @xmath122 , and an actuator task @xmath123 ( figure [ fig : bus ] ) . we consider a communication protocol where each communication cycle is divided into time - triggered and event - triggered segments . using _ time - triggered _ communication schedules , denoted as @xmath124 , applications are allowed to send messages only at their assigned slots and the tasks are triggered synchronously with the bus , i.e. , we assume that the communication delay due to the finite speed of the bus is negligible and hence the delay @xmath24 in ( [ eq : predictorform ] ) is equal to @xmath125 . on the other hand , in an _ event - triggered _ schedule , denoted as @xmath126 , the tasks are assigned priorities in order to arbitrate for access to the bus . note that in our setup , multiple control applications share the same bus and hence multiple control messages have to be sent using a common bus and thus the messages might experience a communication delay @xmath127 when the higher priority tasks access the event - triggered segment . we choose the event - triggered communication schedules such that the sensor - to - actuator delay @xmath127 is within @xmath128 sample intervals , i.e. , @xmath129 for the control - related messages and hence the delay @xmath24 is at most equal to @xmath130 with @xmath131 . in summary , the delay @xmath132 if @xmath133 and @xmath134 if @xmath135 where @xmath136denotes the protocol used at time @xmath58 . the properties of the varying delay of the tt and et protocol are directly exploited in the control design in the following way . whenever the error between the plant output and its desired value is above some threshold @xmath137 , we send the control messages over the tt protocol , as this guarantees an aggressive control action with minimal communication delay . otherwise , the control messages are sent using the et protocol . that is , @xmath138 that is , the protocol switches depending on the state of the control application , as in ( [ eq : protocol ] ) . [ cc][cc]@xmath139 [ cc][cc]@xmath140 [ cc][cc]@xmath121 [ cc][cc]@xmath122 [ cc][cc]@xmath123 [ cc][][0.8]@xmath5 [ tc][cc][0.7]sensor [ cc][cc][0.7]controller [ cc][cc][0.7]actuator [ cc][cc]shared communication network commensurate with the switching protocol in ( [ eq : protocol ] ) , we propose a switch in the adaptive controller as well , and is defined below : @xmath141where @xmath142 is given in eq . ( [ eq : phiandthetaklein ] ) , @xmath143 is given in eq . ( [ eq : phiandtheta ] ) , @xmath144^t$ ] is the estimation of the controller gains @xmath145 ( eq . [ eq : phiandtheta ] ) , and @xmath146 . if @xmath147 , the adaptive controller is given by @xmath148where @xmath149 is given in eq . ( [ eq : phiandthetaklein ] ) , @xmath150 is given in eq . ( [ eq : phiandtheta ] ) , @xmath151^t$ ] is the estimation of the controller gains @xmath152 ( eq . [ eq : phiandtheta ] ) , and @xmath153 . the following definitions are useful for the rest of the paper . we denote the instants of time when the switch from tt to et occurs with @xmath154 , @xmath155 , and the instants of time when the switch from et to tt occurs with @xmath154 , @xmath156 . that is , the tt protocol is applied for @xmath157,p\in{\ensuremath{\mathbb{n}}}_0 $ ] and the et protocol is applied for @xmath158,p\in{\ensuremath{\mathbb{n}}}_0 $ ] with @xmath159 and switches occurring between @xmath160,p\in{\ensuremath{\mathbb{n}}}$ ] ( see figure [ fig : error ] ) . [ ass : dimpulse ] the disturbance @xmath9 in ( [ eq : predictorform ] ) is an impulse train , with the distance between any two consecutive impulses greater than a constant @xmath161 . this is the main result of the paper : [ thm : betaknown ] let the plant and disturbance @xmath162 in ( [ eq : predictorform ] ) satisfy assumptions [ ass : fixeddelay ] , [ ass : pe ] , and [ ass : dimpulse ] . consider the switching adaptive controller in ( [ eq : ttcontroller ] ) and ( [ eq : etcontroller ] ) with the hybrid protocol in ( [ eq : protocol ] ) and the following parameter estimate selections at the switching instants @xmath163 then there exists a positive constant @xmath164 such that for all @xmath165 , the closed loop system has globally bounded solutions . a qualitative proof of theorem [ thm : betaknown ] is as follows : + first , theorem [ thm : fixeddelay ] shows that if either of the individual control strategies ( [ eq : ttcontroller ] ) or ( [ eq : etcontroller ] ) is deployed , then boundedness is guaranteed . that is , for a sufficiently large dwell time @xmath161over which the controller stays in the tt protocol , with the controller in ( [ eq : ttcontroller ] ) , boundedness can be shown . after a finite number of switches , when the system switches to an et protocol , it is shown that the regressor vector remains in the same subspace as in the earlier switch to et and hence , the corresponding tracking error remains small even after the switch to et . hence the stay in et is ensured for a finite time , guaranteeing boundedness with the overall switching controller . _ proof of theorem [ thm : betaknown ] : _ we define an equivalent reference signal @xmath166that combines the effect of both @xmath53and the disturbance @xmath162 as @xmath167 where @xmath168 is given by @xmath169 and @xmath170 is the transfer function of the plant ( [ eq : model ] ) . also , we define a reference model signal @xmath171 given by @xmath172 where the transfer functions @xmath173 is given by @xmath174 and the optimal feedback gain @xmath175 is given by ( [ eq : phiandthetaklein ] ) . the overall ideal closed - loop system is given by the block diagram shown in figure [ fig : blockdiagram01 ] . we note that when there is no disturbance , the output @xmath171 corresponds to the desired regressor vector , and its first element of the vector corresponds to @xmath53 . ] when the algorithm is in mode @xmath124 , the underlying error equation is given by @xmath176 with @xmath177 . when the system is in mode @xmath126 , the error equation is given by @xmath178 with @xmath179 . define @xmath180 as @xmath181 choose lyapunov function @xmath182 where @xmath183 . let @xmath184 . the proof consists of the following four stages : * stage :* let there exist a sequence of finite switching times @xmath185 with the properties described above . then the errors @xmath186 and @xmath187 are bounded for all @xmath58 . + the proof of stage 1 is established using the following three steps : * step - * there exists a @xmath188 such that @xmath1890;{\ensuremath{e_{\text{th}}}\xspace}]:\;|e_1(k_1)|<\varepsilon\leqslant{\ensuremath{e_{\text{th}}}\xspace}$ ] where @xmath190 . during @xmath124(@xmath126 ) , the error @xmath186 ( @xmath187 ) is bounded . there exists a constant @xmath191 with @xmath192 , for @xmath156 . the length of the interval @xmath193 $ ] is greater than 2 , i.e. , @xmath194 + stage 2 is established using the following steps : * step - * if @xmath195 then @xmath196 if @xmath197 , then @xmath198 @xmath199 for @xmath200 @xmath201 for @xmath200 @xmath202 is bounded for all @xmath203 . + the following steps will be used to establish stage 3 : * step - * @xmath204 and @xmath205 , for all @xmath206 . @xmath207 during @xmath124 and during @xmath126 the control input is bounded for all @xmath58 and hence all signals are bounded . + the following two steps will be used to prove stage 4 : * step - * @xmath208 all signals are bounded . we note that the proofs of stages 1 , 3 , and 4 are identical to that in @xcite and are therefore omitted here . since stage 2 differs significantly from its counterpart in @xcite due to @xmath209 , we provide its proof in detail below . [ ] [ ] [ 0.8]time are assumed to occur at @xmath210.,title="fig : " ] @xmath211 in this step we show that if the tracking error @xmath212 is small the state signal error @xmath213 is also small . the signal @xmath214 is the output produced by the following transfer function @xmath215 with @xmath212 as the input : @xmath216 where @xmath217 is the inverse of the plant transfer function @xmath218 with the input signal @xmath219 and @xmath220 given in ( [ eq : phistern ] ) . from assumption [ ass : fixeddelay ] , it follows that @xmath217 is a stable transfer function . hence , as @xmath212 tends to zero , @xmath214 also tends to zero . if @xmath197 , then @xmath198 we first show that @xmath221 for @xmath222 and @xmath223 . we note that the reference model given in ( [ eq : phistern ] ) is a linear system and hence there exists a state space representation @xmath224 with @xmath225 being completely reachable . then it follows directly from lemma [ cor : inomega ] that @xmath226 for @xmath222 and @xmath223 . together with step 2 - 1 it follows that if @xmath197 , then @xmath198 . @xmath199 for @xmath200 first , we show that the error of the signal generated by the reference model signal @xmath227 together with the last parameter estimation value @xmath228 at the end of the previous et phase is small and therefore the output error @xmath229 is below the threshold @xmath137 . from step 2 - 2 we know that @xmath227 is in the same subspace @xmath230 as @xmath231 . from step 2 - 1 we know that @xmath231 is close to @xmath232 which in turn generates together with @xmath233 and @xmath234 an error which is @xmath235 according to theorem [ thm : fixeddelay ] . hence , @xmath236 from step 2 - 1 we know that @xmath237 is close to @xmath227 hence , according to step 2 - 4 we have @xmath238 . @xmath239 and @xmath240 this step shows that the error at the beginning of the et mode is below the threshold for at least @xmath241 steps . from step 2 - 3 we know that @xmath242 . according to the parameter choice in ( [ eq : thetachoice2 ] ) , the controller uses a constant initial value for the first @xmath243 steps . thus , the error @xmath244 because steps 2 - 1 to 2 - 5 can be applied . theorem [ thm : betaknown ] implies that the plant in ( [ eq : predictorform ] ) can be guaranteed to have bounded solutions with the proposed adaptive switching controller in ( [ eq : ttcontroller ] ) and ( [ eq : etcontroller ] ) and the hybrid protocol in ( [ eq : protocol ] ) , in the presence of disturbances . the latter is assumed to consist of impulse - trains , with their inter - arrival lower bounded . we note that if no disturbances occur , then the choice of the algorithm in ( [ eq : protocol ] ) implies that these switches cease to exist , and the event - triggered protocol continues to be applied . and switching continues to occur with the onset of disturbances , with theorem [ thm : betaknown ] guaranteeing that all signals remain bounded with the tracking errors @xmath245 converging to @xmath246 before the next disturbance occurs . the nature of the proof is similar to that of all switching systems , in some respects . a common lyapunov function @xmath202 was used to show the boundedness of parameter estimates , which are a part of the state of the overall system ( in stage 3 ) . the additional states were shown to be bounded using the boundedness of the tracking errors @xmath247 and @xmath248 ( in stage 1 ) and the control input using the method of induction ( in stage 4 ) . since the switching instants themselves were functions of the states of the closed - loop system , we needed to show that indeed these switching sequences exist , which was demonstrated in stage 2 . to this end , the sufficient richness properties of the reference signal were utilized to show that the signal vectors of a reference model and the system converge to the same subspace . next , it was shown that the error generated by the reference model is small and thus concluded that the tracking error at the switch from tt to et stays below the threshold @xmath137 . it is the latter that distinguishes the adaptive controller proposed in this paper , as well as the methodology used for the proof , from existing adaptive switching controllers and their proofs in the literature . in this work we considered the control of multiple control applications using a hybrid communication protocol for sending control - related messages . these protocols switch between time - triggered and event - triggered methods , with the switches dependent on the closed - loop performance , leading to a co - design of the controller and the communication architecture . in particular , this co - design consisted of switching between a tt and et protocol depending on the amplitude of the tracking error , and correspondingly between two different adaptive controllers that are predicated on the resident delay associated with each of these protocols . these delays were assumed to be fixed and equal to @xmath125 for the tt protocol and greater than @xmath249 for the et protocol . it was shown that for any reference input whose order of sufficient richness stays constant , the signal vector and the parameter error vector converge to subspaces which are orthogonal to each other . the overall adaptive switching system was shown to track such reference signals , with all solutions remaining globally bounded , in the presence of an impulse - train of disturbances with the inter - arrival time between any two impulses greater than a finite constant .

the focus of this paper is on the co - design of control and communication protocol for the control of multiple applications with unknown parameters using a distributed embedded system . the co - design consists of an adaptive switching controller and a hybrid communication architecture that switches between a time - triggered and event - triggered protocol . it is shown that the overall co - design leads to an overall switching adaptive system that has bounded solutions and ensures tracking in the presence of a class of disturbances .

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Proceed to summarize the following text: in recent years , the standard model of electroweak interactions has been confirmed experimentally with outstanding success . not only was almost all the particle content discovered at accelerator experiments , but their properties and interactions have been measured with high precision , in agreement with the model prediction . the only missing piece is the higgs boson , which is responsible for electroweak symmetry breaking . however , even today we can obtain meaningful constraints on the higgs boson mass from electroweak precision measurements . due to the impressive accuracy of some of these experimental results , they are sensitive to electroweak radiative corrections at the next - to - leading ( nlo ) and sometimes next - to - next - to - leading ( nnlo ) level , and thus depend on the impact of the higgs boson entering in the loops . two of the most important quantities in this respect are the mass of the @xmath0 boson , @xmath1 and the sine of the leptonic effective weak mixing angle @xmath2 . the @xmath0-boson mass can be inferred from the muon decay constant @xmath3 , which is generated through virtual w - boson exchange , so that @xmath4 . the effective weak mixing angle , on the other hand , reflects the ratio of the vector and axial - vector couplings , @xmath5 and @xmath6 , of the @xmath7 boson to fermions ( @xmath8 ) at the @xmath7 boson pole : @xmath9 since these couplings can be measured most precisely for leptons , the _ leptonic _ effective weak mixing angle @xmath2is usually taken as a reference . the current experimental world average for the @xmath0-boson mass is @xmath10 gev @xcite . recently , a lot of progress has been made towards establishing accurate theoretical prediction for @xmath1 . the best result @xcite includes the complete two - loop corrections @xcite and some three - loop contributions @xcite . the remaining theoretical error is estimated to be @xmath11 mev , which is well below the current experimental uncertainty . still , the electroweak two - loop corrections total to @xmath12 mev and are thus mandatory for electroweak precision analyses . the effective weak mixing angle @xmath2 is mainly derived from various asymmetries measured around the @xmath7 boson peak at @xmath13 colliders after subtraction of qed effects . the current experimental accuracy , @xmath14 @xcite implies strong indirect constraints on the allowed range for the higgs boson mass @xmath15 . therefore it is important to develop precise theoretical calculations for this quantity . usually , @xmath2 is computed as a function of the electromagnetic coupling @xmath16 , the muon constant @xmath3 and the masses of the @xmath7 boson , @xmath17 , and the top quark , @xmath18 ( other fermion masses are numerically irrelevant ) . as explained before , @xmath1 is calculated from @xmath3 , but in addition to these corrections , the computation of @xmath2 involves the corrections to the @xmath7 vertex form factors . the latter are expressed by the quantity @xmath19 , in such a way that the effective weak mixing angle can also be written as : @xmath20 and at tree - level , @xmath21 . higher - order corrections to @xmath2 have been under extensive theoretical study over the last two decades . besides the one - loop result @xcite , two- and three - loop qcd corrections are available @xcite , but for the electroweak two - loop contributions only partial results were known . by means of a large mass expansion in the heavy top quark mass , the formally leading @xmath22 @xcite and next - to - leading @xmath23 @xcite terms were computed . a part of the missing two - loop contributions was incorporated by the complete electroweak two - loop corrections to @xmath1 @xcite . while these corrections effected a shift in @xmath1 of 4 mev compared to the previously known @xmath24 contributions , the induced shift in @xmath2 was very sizable , @xmath25 , thus implying that the missing two - loop terms in the form factor @xmath26 can be of similar order . as a first step towards completing the electroweak two - loop corrections to @xmath2 , results for the fermionic ( i.e. diagrams with closed fermion loops ) two - loop corrections were presented recently @xcite . the genuine two - loop vertex diagrams are represented by the generic topologies in fig . [ fig : diags ] . higher - order corrections to the process @xmath27 near the @xmath7 pole can be consistently computed by performing an expansion of the amplitude around the complex pole @xmath28 , @xmath29 = \frac{r}{s-{\cal m}_{\rm z}^2 } + s + ( s-{\cal m}_{\rm z}^2 ) s ' + \dots \label{eq : polexp}\ ] ] here @xmath30 is the @xmath7 decay width . after subtracting contributions from s - channel photon exchange and @xmath31-@xmath7 interference , the vertex corrections form factor at nnlo is derived to be @xmath32 where the superscripts in parentheses indicate the loop order . in this quantity , ir - divergencies from qed contributions drop out , which involves a delicate interplay between one- and two - loop terms in the form factors . the uv - divergencies are cancelled by on - shell renormalization . the relevant counterterms are derived using the methods of ref . @xcite . the new part of this work is the computation of the two - loop @xmath33 vertex corrections , which are treated with two independent technical methods . the first method uses large mass expansions for the diagrams with internal top - quark lines and the differential equation method for diagrams with only light fermions @xmath34 , the masses of which are neglected . contrary to previous work @xcite , the expansion in @xmath35 is performed to high precision , by executing the series to the order @xmath36 , reaching an overall relative precision of @xmath37 of the final result . the coefficients of the expansion are 2-loop tadpole and 1-loop vertex diagrams , which can be evaluated efficiently with well - known analytical formulae . the contributions from diagrams without top - quark propagators involve only two independent scales , @xmath17 and @xmath1 , allowing a fully analytical treatment . even for the limited set of diagrams with closed fermions loops , a large number of scalar integrals with non - trivial structures in the numerator are involved . they can be reduced to a set of scalar master integrals by using integration - by - parts identities @xcite . owing to size of the linear equation system associated with this reduction procedure , the algorithm has been implemented in the dedicated c++ library diagen / idsolver @xcite , which performs the necessary steps in a highly automized way . analytical results for these master integrals are obtained by the differential equation method @xcite . this is illustrated by the following example , @xmath38 = \frac{p^2}{p^2+m^2 } \biggl ( \frac{4-d}{2}(4 + 5 \frac{m^2}{p^2 } ) \left [ \raisebox{-5.4mm}{\psfig{figure = v2p1.ps , width=1.6cm}}\;\right ] + \frac{10 - 3d}{2 } \left [ \raisebox{-5.4mm}{\psfig{figure = v2p0.ps , width=1.6cm}}\;\right ] - \frac{2-d}{2 } \left [ \raisebox{-3mm}{\psfig{figure = t134.ps , width=0.95cm}}\right ] \biggr).\ ] ] here the thick lines represent massive propagators with mass @xmath39 , the thin lines denote massless propagators and @xmath40 is the momentum flowing into the vertex . @xmath41 is the dimension of dimensional regularization . the momentum derivate of the scalar integrals on the left - hand side results in the same integral and simpler integral topologies on the right - hand side . feeding in analytical results for these simpler integrals , the differential equation can be solved in terms of generalized polylogarithms . all integrals were also checked by using low - momentum expansions . the second method makes use of numerical integrations based on dispersion relations . a scalar two - loop integral with a self - energy sub - loop as in fig . [ fig : disp ] ( a ) can be expressed as @xcite ' '' '' [ cols="^,^ " , ] @xmath42 where @xmath43 is the discontinuity of the scalar one - loop self - energy function . the second integral can be evaluated into a standard @xmath44-point one - loop function , leaving the integration over @xmath45 to be performed numerically . in general , one can also introduce dispersion relations for triangle sub - loops @xcite , but it is often technically easier to reduce them to self - energy sub - loops by introducing feynman parameters @xcite , @xmath46^{-1 } \ ; [ ( q+p_2)^2-m_2 ^ 2]^{-1 } = \int_0 ^ 1 { \rm d}x \ ; [ ( q+\bar{p})^2 - \overline{m}^2]^{-2 } \\ \bar{p } = x\,p_1 + ( 1-x)p_2 , \qquad \overline{m}^2 = x \ , m_1 ^ 2 + ( 1-x ) m_2 ^ 2 - x(1-x)(p_1-p_2)^2 . \end{array}\ ] ] this is indicated diagrammatically in fig . [ fig : disp ] ( b ) . the integration over the feynman parameters is also performed numerically . as a result , all master integrals for the vertex topologies can be evaluated by at most 3-dim . numerical integrations . before performing the numerical integrations , possible uv- and ir - divergencies need to be subtracted from the integrals . while this second method is applicable to two - loop vertex corrections with an arbitrary number of mass scales , it is slower and leads to much large expressions than the first method . nevertheless it provides an important check of the result . special caution is needed for the diagrams with a fermion triangle loop ( see the third diagram in fig . [ fig : diags ] ) , which involve the @xmath47 matrix . in dimensional regularization , it is not possible to fulfill the two relations @xmath48 and tr@xmath49 simultaneously . as in ref . @xcite , the contributions resulting in @xmath50-tensors were therefore evaluated in four dimensions , based on the observation that these terms are free of uv - divergencies . potential soft and collinear divergencies of single diagrams are regulated using a photon mass , with a subsequent careful expansion for zero photon masses . the new result for the fermionic two - loop corrections is combined with corrections of order @xmath51 , @xmath52 @xcite , @xmath53 @xcite and leading three - loop terms of @xmath54 and @xmath55 @xcite . reducible terms of the same order are taken into account , but no resummations are preformed . the most precise prediction of @xmath2 is obtained as a function of the muon decay constant @xmath3 , from which the @xmath0-boson mass is calculated by including the radiative corrections to @xmath1 as given in ref . @xcite . [ fig : res ] shows the final result for @xmath2 as a function of the higgs mass compared to the current experimental value . included in the plot are the error bands due to the uncertainties of the experimental input parameters entering into the theoretical prediction and of the direct measurement of @xmath2 . as evident from the figure , the identification of computation and measurement for @xmath2 favor relatively small values for @xmath15 . with the new two - loop corrections , the best - fit value for @xmath15 moved from 148 gev ( using formulae of ref . @xcite ) to 168 gev ( for @xmath56 gev ) . the numerical result for the leptonic effective weak mixing angle has been published in ref . @xcite as a parametric fitting formula that is accurate in the range 10 gev @xmath57 1 tev . it has also been implemented into the newest version 6.42 of the program zfitter @xcite , with some changes recently discussed in ref . @xcite , and was used in the latest release of electroweak precision global fits of the standard model . the impact of the new result for @xmath2 shifts the 95% confidence level upper bound on the higgs mass upwards by 23 gev to 260 gev @xcite . together with the inclusion of the new two - loop result in the prediction for @xmath2 , a assessment of the uncertainties from missing higher order contributions is required . since for practical purposes @xmath2 is given as a function of the muon decay constant @xmath3 , it is useful to evaluate the theoretical error for this parametrization , i.e. combining the radiative corrections to @xmath1 and the @xmath7 vertex . a simple method to estimate the higher order uncertainties assumes that the perturbation series follows roughly a geometric progression . this presumption implies relations like @xmath58 with this method one obtains the following errors for @xmath15 between 10 and 1000 gev in units of @xmath59 : between 2.3 and 2.0 for the @xmath60 contributions beyond the leading @xmath61 term , between 1.8 and 2.5 for @xmath62 , between 1.1 and 1.0 for @xmath63 and between 1.7 and 2.4 for @xmath64 . the missing bosonic @xmath65 corrections can not be appraised from geometric progression . however , considering they have a prefactor @xmath66 but no specific enhancement factor , they are estimated to be about @xmath67 . to account for possible deviations from the geometric series behavior , an overall factor @xmath68 was included to arrive at a total error of @xmath69 . alternatively , the error from a higher - order qcd loop can be assessed by varying the scale of the strong coupling constant @xmath70 or the top - quark mass @xmath18 in the @xmath71 scheme in the highest available perturbation order . the scale variation leads to an error estimate of 0.1 to @xmath72 for the @xmath73 corrections and of less than @xmath74 for the @xmath75 contributions . these numbers are of the same order as the estimated errors from the geometric progression method , so that the total error given above seems to be fairly reliable . the new error estimate was used in the latest electroweak global fits @xcite and lead to a reduction of the width of the well - known _ blue band _ , which indicates the theoretical error in the indirect determination of the higgs mass . in this contribution , recent progress in the calculation of higher - order corrections to the most important electroweak precision observables and their impact on the indirect determination of the higgs mass was reported . the complete fermionic @xmath76 corrections to the leptonic effective weak mixing angle @xmath2 have been calculated and numerical results were presented . as an additional check , the computation of the two - loop vertex integrals was performed with two independent methods . the new result , together with an estimate of the remaining theoretical error , was included in the latest version of the program zfitter and used for the latest global electroweak fits of the standard model . the calculation of the remaining bosonic electroweak two - loop corrections is currently in progress and will be available soon @xcite . furthermore , we are working on adapting the new results for quark final states , where particular attention has to be paid to the @xmath77 vertex , since it includes additional massive top - quark propagators @xcite . the authors are grateful to g. weiglein for valuable contributions , comments and communications concerning this project . m. c. was supported by the sofja kovalevskaja award of the alexander von humboldt foundation sponsored by the german federal ministry of education and research . m. a. and m. c. were supported by the polish state committee for scientific research ( kbn ) for the research project in years 2004 - 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analytical program for fermion pair production in @xmath78 annihilation , from version 6.21 to version 6.42 _ , hep - ph/0507146 , submitted to comp . + [ ` http://www-zeuthen.desy.de/theory/research/zfitter/ ` ] .

electroweak precision observables allow stringent tests of the standard model at the quantum level and imply interesting bounds on the mass of the higgs boson through higher - order loop effects . very significant constraints come especially from the determination of the mass of the @xmath0 boson and from the effective leptonic weak mixing angle . after shortly reviewing the status of theoretical computations of the @xmath0 mass , the new calculation of two - loop corrections with closed fermion loops to the effective leptonic weak mixing angle is discussed in detail . the phenomenological implications of the new result are analyzed including an estimate of remaining uncertainties .

You are an expert at summarizing long articles. Proceed to summarize the following text: the atom interferometry used in this proposal is similar to @xcite and uses many of the techniques in @xcite . light pulse atom interferometry uses two counter - propagating lasers that couple hyperfine degenerate ground states of alkali ( or alkali - like ) atoms through a near - resonance raman transition . while the lasers are on , the system undergoes rabi oscillations between two states having a relative momentum @xmath12 . by performing a @xmath13 series of raman pulses , the atom s wave packet is split into a slow and fast component , then after an interrogation time , @xmath14 , the states are reversed and the wave packet is brought back together for the final beam splitter that interferes the two halves of the wave packet . the maximal spatial separation of the wave packets in the interferometer is @xmath15 . there often is an initial velocity , @xmath16 , to the atom s wave packets that is used to doppler - select the desired atomic transitions . an initial velocity can also arise from the thermal velocities . atom interferometers are always run in pairs with the same lasers driving both interferometers in order to remove laser phase noise . the paired interferometers also reduce other common mode backgrounds such as the sagnac effect . the distance between these interferometers can be large and a benchmark value of several meters will be used . finally , interferometers are run with @xmath17 atoms simultaneously in a bunch and the full experiment is performed @xmath18 times . @xmath17 is determined by the rate for cooling atoms and @xmath19 is currently possible . the interrogation time for the experiment will be @xmath20 , so a benchmark values of @xmath21 is reasonable . in several days @xmath22 trials can be run and gives a shot - noise phase sensitivity of @xmath23 . atomic fountains are frequently used in ai and start with an ensemble of evaporatively cooled atoms ( @xmath24 ) in a magneto - optical trap ( mot ) that localizes that atoms to @xmath25 . the atoms are then launched with a velocity , @xmath26 , vertically through a series of multiple bragg or raman transitions . the atoms subsequently follow ballistic motion for an interrogation time , @xmath27 , defining @xmath26 at the initial @xmath28 pules and performing the @xmath29 pulse occurs at the apex of the trajectory . in atomic fountains , the atoms are instantaneously in free fall and decoupled from their environment except during periods when they are coupled to the lasers which last . the atoms are more isolated than meso- or macroscopic measurements that are vibrationally coupled to the environment and essentially removes the `` chopping '' used to isolate torsional pendulums or cantilevers from the environment . configuring the interferometer as a gyroscope , with @xmath26 perpendicular to @xmath30 and @xmath16 , allows the atom s ballistic trajectory to be parallel to a planar face of the mass . this maximizes the time that the atoms spend close to the source mass , thus making a signal from a new short distance force as large as possible . the interferometer is sensitive to the potential as a function of distance away from the surface of the source mass if the recoil velocity is perpendicular to this face . by mapping out the potential , it is possible to look for new contributions beyond the standard model . this configuration minimizes the earth s gravity ; however , the phase from the corriolis force ( sagnac effect ) is maximized and controlling this is an important background discussed below . and @xmath31 from the nearer of the paired interferometers . the plane in gold represents a thin casimir shield fitted with retroreflectors that allow optical access for the right moving laser.,width=288 ] the relative phase between the two paths arises from several different contributions and not many exact results are known . there is a semi - classical , perturbative method for computing the phase differences and at lowest order in the potential , @xmath32 , speed of light , @xmath33 , and the width of the wave packet , @xmath34 , the general result is that the phase difference is integral of the perturbing potential over the unperturbed paths . there are several phase difference results that will be useful in deriving the sensitivity . the first is if the potential only depends on the position in the direction of @xmath30 and where @xmath16 is not important @xmath35 where the initial position of the atom is taken to be @xmath36 . in the second half the potential has been taylor expanded , if applicable . the first term is proportional to the acceleration , and the second term is referred to as the recoil phase because it vanishes as @xmath37 becomes large , keeping @xmath38 fixed . finally , if @xmath16 is important , the above expression becomes @xmath39 where @xmath40 , @xmath41 , and @xmath42 . if the potential depends on the distance in the @xmath26 direction , no closed form is possible , but simple expressions can be obtained if @xmath43 is taylor expandable . newton s constant is not known to a precision better than @xmath44 , so it is necessary to perform a series of measurements to remove absolute sensitivity to @xmath45 . the most straightforward manner to remove the uncertainty in @xmath45 is to test the @xmath46 behavior of gravity by having a moveable source mass . the source mass will be taken to be a cylinder of radius , @xmath47 , and width , @xmath48 , and density , @xmath49 , with the circular face of the cylinder forming a vertical plane . this geometry is motivated by calculational simplicity and none of the results depend sensitively upon the geometry . the height of the atomic fountains , @xmath50 , will take place near the center of the cylinder s face . the newtonian potential near the center of the cylinder can be calculated in the far field , @xmath51 , and near field , @xmath52 , limits near the center of the cylinder . in the far field limit @xmath53 where @xmath54 is the distance to cylinder s face and @xmath55 is the distance from the cylinder s center . correction to the potential is due to the cylindrical symmetry and will not be important for the results . ] in the near field limit the newtonian potential is given by @xmath56 where @xmath57 is an unmeasurable function , independent of the distance from the source . there are several limits necessary for the yukawa potential . the first is @xmath58 where the potential from cylinder is @xmath59 for @xmath60 , the yukawa potential looks newtonian plus a correction arising from taylor expanding the exponential . in the far field limit ( @xmath61 ) the yukawa potential is @xmath62 notice the absence of an @xmath63 term because it is always an unmeasurable constant . in the near field limit ( @xmath64 ) , the yukawa potential becomes @xmath65 the physical size of the experiment ultimately limit the sensitivity . the size of the source mass sets the @xmath4 with maximum sensitivity and the benchmark value used is @xmath66 and @xmath67 . the other relevant physical constraint is how near the source mass can get to the interferometers , @xmath68 . the distance that the source mass can be moved from the interferometers does not limit the sensitivity so long as @xmath69 . the strategy to distinguish newtonian gravity from a new yukawa potential is to perform a near and a far measurement of the phase . the near measurement fixes what the newtonian prediction is for the far measurement . if the inclusion of a yukawa potential with strength @xmath3 and range @xmath4 causes a difference between the far prediction and the far measurement greater than the shot noise limit , then this yukawa potential is visible assuming that no backgrounds or uncertainties are larger . the scaling of the limits depend on @xmath4 and there two cases to be considered . if , @xmath70 , it is possible to get within the @xmath46 behavior of the yukawa potential and move outside its range ; furthermore , the radius of the cylinder is not determining the sensitivity . in the second case @xmath71 , it is possible to get within the @xmath46 and outside the range of the potential , but the size of the source mass is determining the sensitivity . for large @xmath4 , it may seem beneficial to move the source mass further from the source to make the difference between a yukawa potential and the newtonian potential more pronounced , growing as @xmath72 ; however , the size of the newtonian phase shift is falling as @xmath73 meaning that there is no parametric gain for moving the source mass a distance greater than @xmath47 from the interferometers for any value of @xmath4 . the biggest challenge of this experiment is the strategy of using a near measurement to extrapolate to a far prediction . the challenge is knowing both the source mass position and orientation precisely enough to make a @xmath74 or better prediction for the far newtonian prediction . examining the subleading terms in the newtonian potentials in ( [ eq : newton 1 ] ) and ( [ eq : newton 2 ] ) , if there is an uncertainty in the position of the cylinder of @xmath75 , then there is an uncertainty in the newtonian potential of @xmath76 or @xmath77 , respectively . by having @xmath78 reduces the uncertainty in the newtonian prediction from uncertainty in the initial height of the atom . it is challenging but achievable to know the position of the block to @xmath79 to be sensitive to @xmath80 ; however , future improvements in sensitivity will not be limited by this . a better solution to knowing the position and orientation of the block is to use multiple interferometers situated near @xmath81 to actively measure the position and orientation of the source . since the edge effects become @xmath82 near the edges , it is possible to use these additional measurements to locate the block and then use a central interferometer to use the near measurement to make the far prediction . this strategy will require six additional interferometers to over - determine the source s solid body coordinates . the remaining challenge is to keep these seven interferometers locked in place as the source mass moves , but this should be possible to a much greater accuracy . there is a residual uncertainty coming from `` jitter '' in the location of the mots . this motion is of the order of @xmath83 ; however , it is stochastic with respect to the number of bunches of atoms , @xmath18 , and therefore the uncertainty is @xmath84 and only limiting after several rounds of improvements . the sagnac effect , the phase induced by the corriolis force is large , @xmath85 , where @xmath86 is the angular velocity of the earth . the use of paired interferometers described above cancels the leading order sagnac effect . the novel method of actively reducing the sagnac effect is to rotate the lasers to compensate for the rotation of the earth . it is possible to rotate lasers with nanoradian precision which reduces the sagnac effect by a factor of @xmath44 . the dominant way that the sagnac effect contaminates the signal is from variation in relative @xmath26 of the two interferometers which is stochastic in the number of bunches , reducing the size by @xmath87 . the final reduction in sagnac contamination must come from vibration isolation and requires @xmath88 . the casimir potential between an atom and a conducting plate is given by @xmath89 , with @xmath90 being the polarizability of the atom and is @xmath91 for cs@xcite . the phase difference arising from the casimir potential arises from ( [ eq : pot phase ] ) or ( [ eq : phase vi ] ) rather than ( [ eq : acc phase ] ) because of its rapid fall off . these reduce the size of the casimir force relative to a macroscopic measurement device and it is not important for distances , ( and consequently @xmath4 ) greater than @xmath92 , but becomes important at shorter distances . in order to gain to reduce any residual phases , having a thin casimir shield ( @xmath93 in width ) that isolates the atom from the source mass position is helpful . while the the casimir potential from the shield will still be measurable , it will not be correlated with the mass position and therefore not directly a systematic effect . the primary way the casimir potential enters as a background is by having the source mass deflect the shield and additionally through the `` jitter '' of the mots . the deflection of the shield is sufficiently small so long as the source mass is @xmath94 from the shield . the jitter gives a phase uncertainty of @xmath95 for @xmath96 and is sufficiently small .. ] in addition to isolating the atoms , it will be necessary to attach retroreflectors on the shield to provide optical access for the outward propagating lasers necessary to drive the raman transitions . with the results of the previous section , the sensitivity to @xmath3 and @xmath4 can be computed . using the previously stated benchmarks for cs the phase sensitivity of @xmath97 from @xmath98 , @xmath99 , @xmath100 , @xmath101 , @xmath102 for pb and @xmath103 , the sensitivity curve is plotted in fig . [ fig : sensitivity ] relative in blue to current experimental limits shown black . the peak sensitivity is @xmath104 and occurs for @xmath105 which is determined by the size of the source mass . for @xmath106 , current ai is more sensitive than existing experiments . additionally , from @xmath107 current ai techniques matches the experimental limits . this motivates considering future experimental improvements to if sensitivity is possible down to @xmath108 . in the near future it will be possible to increase the magnitude of the momentum transferred to the atoms in the interferometer using a technique called large momentum transfer ( lmt ) . lmt uses repeated raman or bragg transitions during the laser pulses that impart on the atoms up to 100 times more momentum as a single raman transition . lmt significantly impacts sensitivity for larger @xmath4 , increasing sensitivity by a factor of 100 ; however , for shorter wavelengths , where @xmath109 , there is no significant gain to sensitivity because the atoms immediately leave the region being sourced by yukawa potential . , from either a need to doppler - select atomic transitions or thermal motion , there is a small gain at short distances proportional to @xmath110 shown in ( [ eq : phase vi ] ) . ] the sensitivity is shown in fig . [ fig : sensitivity ] in green . on the short distance front , increases in sensitivity must come through increases in phase sensitivity . one possibility is to increase the number of atoms being used in the experiment , which is limited by the length of the experiment and the rate for cooling atoms . in the future , a more effective way to increase sensitivity would be to use wave packets of entangled atoms where it is possible to have the sensitivity scale as @xmath111 ( in contrast to @xmath112 ) which is known as heisenberg limited statistics ( hs ) . hs potentially allows significant gains in sensitivity and the sensitivity curve is shown for a phase sensitivity of @xmath113 in red in fig . [ fig : sensitivity ] . in yellow , the sensitivity is shown for combining a lmt factor 100 and @xmath113 ; however , at this level the stochastic uncertainty in the newtonian prediction starts to become limiting . there are some caveats that may limit the sensitivity at shorter distances . the first is that all of the phase results are computed in the semi - classical limit ; however , when @xmath114 this approximation breaks down and a more complete quantum mechanical treatment is necessary . additionally , the interaction between the casimir shield and atoms must be considered in much more detail , particularly for the case of hs where preventing the decoherence of the wave packet is necessary . this letter has demonstrated that atom interferometry has the potential to improve the sensitivity to new forces in the @xmath115 to @xmath116 range using current technology or technologies that will be available in the near future . additionally , atom interferometry can be useful in testing other types of forces other than the standard @xmath117 yukawa potential such as spin , velocity and composition dependent forces . jw would like to thank m. kasevich , p. graham and s. dimopoulos for collaboration during early stages of this work and also g. biederman , b. dobrescu , and j. hogan for illuminating conversations . jw is supported by the doe under contract de - ac03 - 76sf00515 . jw is supported by the doe outstanding junior investigator award . e. g. adelberger , b. r. heckel and a. e. nelson , ann . sci . * 53 * , 77 ( 2003 ) [ arxiv : hep - ph/0307284 ] . n. arkani - hamed , s. dimopoulos and g. r. dvali , phys . d * 59 * , 086004 ( 1999 ) [ arxiv : hep - ph/9807344 ] . i. antoniadis , n. arkani - hamed , s. dimopoulos and g. r. dvali , phys . b * 436 * , 257 ( 1998 ) [ arxiv : hep - ph/9804398 ] . n. arkani - hamed , s. dimopoulos and g. r. dvali , phys . b * 429 * , 263 ( 1998 ) [ arxiv : hep - ph/9803315 ] . r. d. peccei and h. r. quinn , phys . * 38 * , 1440 ( 1977 ) . f. wilczek , phys . * 40 * , 279 ( 1978 ) . m. dine , w. fischler and m. srednicki , phys . b * 104 * , 199 ( 1981 ) . m. a. shifman , a. i. vainshtein and v. i. zakharov , nucl . b * 166 * , 493 ( 1980 ) . j. e. kim , phys . * 43 * , 103 ( 1979 ) . j. e. moody and f. wilczek , phys . d * 30 * , 130 ( 1984 ) . r. barbieri , a. romanino and a. strumia , phys . b * 387 * , 310 ( 1996 ) [ arxiv : hep - ph/9605368 ] . m. pospelov , phys . d * 58 * , 097703 ( 1998 ) [ arxiv : hep - ph/9707431 ] . b. a. dobrescu and i. mocioiu , jhep * 0611 * , 005 ( 2006 ) [ arxiv : hep - ph/0605342 ] . s. dimopoulos and a. a. geraci , phys . d * 68 * , 124021 ( 2003 ) [ arxiv : hep - ph/0306168 ] . q. g. bailey and v. a. kostelecky , phys . d * 74 * , 045001 ( 2006 ) [ arxiv : gr - qc/0603030 ] . g. biedermann , `` gravity tests , differential accelerometry and interleaved clocks with cold atom interferometers , '' ( 2008 ) . k. y. chung , s. w. chiow , s. herrmann , s. chu and h. muller , phys . d * 80 * , 016002 ( 2009 ) [ arxiv:0905.1929 [ gr - qc ] ] . h. muller , s. w. chiow , s. herrmann , s. chu and k. y. chung , phys . lett . * 100 * , 031101 ( 2008 ) [ arxiv:0710.3768 [ gr - qc ] ] . s. dimopoulos , p. w. graham , j. m. hogan and m. a. kasevich , arxiv:0802.4098 [ hep - ph ] . s. dimopoulos , p. w. graham , j. m. hogan and m. a. kasevich , phys . lett . * 98 * , 111102 ( 2007 ) [ arxiv : gr - qc/0610047 ] . a. arvanitaki , s. dimopoulos , a. a. geraci , j. hogan and m. kasevich , phys . lett . * 100 * , 120407 ( 2008 ) [ arxiv:0711.4636 [ hep - ph ] ] . s. dimopoulos , p. w. graham , j. m. hogan , m. a. kasevich and s. rajendran , phys . d * 78 * , 122002 ( 2008 ) [ arxiv:0806.2125 [ gr - qc ] ] . s. dimopoulos , p. w. graham , j. m. hogan , m. a. kasevich and s. rajendran , phys . b * 678 * , 37 ( 2009 ) [ arxiv:0712.1250 [ gr - qc ] ] . berman , ed . _ atom interferometry _ ( new york : acad . pr . , 1997 ) .

atom interferometry is a rapidly advancing field and this letter proposes an experiment based on existing technology that can search for new short distance forces . with current technology it is possible to improve the sensitivity by up to a factor of @xmath0 and near - future advances will be able to rewrite the limits for forces with ranges from 100 @xmath1 m to 1 km . new short distance forces are a frequent prediction of theories beyond the standard model and the search for these new forces is a promising channel for discovering new physics . over the past 15 years there has been rapid advances in light pulse atom interferometry ( ai ) and in a wide variety of settings , ai is the most sensitive measurement . this letter will explore the sensitivity of ai to new forces . ai holds great promise in improving currently sensitivity over a wide range of distances from roughly to . new forces can couple to matter in many different ways ; however , there is a benchmark parameterization that is frequently applicable to new forces where the potential between two particles is proportional to the mass of the particles @xmath2 where @xmath3 is a dimensionless number that characterizes the new force s strength relative to gravity and @xmath4 is the compton wavelength of the particle being exchanged . the coupling , @xmath3 , could be composition , spin or velocity dependent or have a power - law fall off rather than an exponential / yukawa behavior ; however , this parameterization is a standard benchmark and will be used in this letter . theories predict a wide range of @xmath4 and @xmath3 . some theories give @xmath5 such as gauge mediated supersymmetry theories that have moduli mediated forces @xcite , large extra dimensions @xcite or theories that have gravity shut off at the scale of the cosmological constant @xcite . alternatively , many theories also predict @xmath6@xcite . the most reknowned of these theories are peccei - quinn axions can mediate forces with @xmath7 @xcite . thus , while it is important to continue the search for @xmath8 to shorter distance forces , searching for sub -gravitational strength forces is also an important frontier to continue pursuing . finally , there are forces that are not yukawa forces of ( [ eq : yukawa ] ) and may intrinsically be stronger than gravity , but at long distances may show up as sub - gravitational strength forces @xcite . see @xcite for other applications of atom interferometry to modifications of gravity . atom interferometry uses cold atoms that have their quantum mechanical wave packets spatially split in two and recombined . the final interference pattern measures the phase difference between the two paths . the experiment described in this letter uses a source mass to create a potential that causes a relative phase between the two paths . by subtracting off the newtonian potential and other backgrounds , a new yukawa potential is visible . the ai experiment in this letter is effective at probing new forces in the to range with sensitivities down to @xmath9 with already proven techniques in contrast to current experimental limits that have sensitivities of @xmath10@xcite . future improvements can increase the range of sensitivity to to and with sensitivities down to @xmath11 .

You are an expert at summarizing long articles. Proceed to summarize the following text: a wide variety of mass transport systems ranging from traffic flow to polymer gels @xcite exhibit nonequilibrium condensation phenomena . these include basic microscopic dynamics ubiquitous in nature such as aggregation , fragmentation and diffusion . the nonequilibrium steady states of these systems are classified into two types of phases , so - called fluid phase and condensed phase . a finite fraction of total particles condenses on a single site in the condensed phase . in the fluid phase , the particle number on each site fluctuates around the density of total particles ( @xmath10 ) without the condensation . as the rates of these processes vary , the condensation phase transitions between the two phases may take place at a certain critical density @xmath22 . one of the simplest mass transport models exhibiting the condensation transitions is a conserved - mass aggregation ( ca ) model @xcite . ca model evolves via diffusion , chipping and aggregation upon contact which arise in a variety of phenomena such as polymer gels @xcite , the formation of colloidal suspensions @xcite , river networks @xcite and clouds @xcite . in one - dimensional ca model , the mass @xmath4 of site @xmath1 moves either to site @xmath23 or to site @xmath24 with unit rate , and then @xmath25 and @xmath26 . with rate @xmath11 , unit mass chips off from site @xmath1 and moves to one of the nearest neighboring sites ; @xmath27 and @xmath28 . the generalization to higher dimensions is straightforward . as total masses are conserved , the conserved density @xmath10 and the rate @xmath11 determine the phase of ca model . the @xmath29 case corresponds to the well - known zero range process ( zrp ) with a constant hopping rate @xcite . the existence of the condensation transitions in ca model depends on the symmetry of movement , the constraints of diffusion rate and the underlying network structure @xcite . in the symmetric ca ( sca ) model @xcite in which diffusion and chipping direction are unbiased , the condensation transitions take place at a certain @xmath22 . the steady state properties of sca model is exactly described by the mean field theory @xcite . the single site mass distribution @xmath30 was shown to undergo phase transitions on regular lattices @xcite . for a fixed @xmath11 , as @xmath10 is varied across the critical density @xmath31 , the behavior of @xmath30 for large @xmath32 was found to be @xcite @xmath33 mean field theory predicts @xmath34 and @xmath35 @xcite . recently , ca model on unweighted scale - free networks ( sfns ) with degree distribution @xmath36 was studied to investigate the effect of underlying network structure on the condensation transitions @xcite . we call networks with equal weight on all links unweighted networks . on unweighted sfns , the same type of the condensation transitions as those of sca in regular lattice take place for @xmath37 . however for @xmath38 , the condensation always occurs for any density @xmath10(@xmath39 ) . it was shown that the existence of the transitions is directly related to the diffusive capture process on unweighted sfns @xcite . on the other hand , most real - world networks exhibit not only a heterogeneous distribution of degree , but also heterogeneous distribution of weights @xcite . weights assigned on links characterize the interaction strengths between nodes . there have been various attempts to understand the underlying mechanism and scale - free behaviors of empirically observed weighted networks @xcite . also there have been attempts to understand the effect of heterogeneous weights on various dynamics such as synchronization , dynamics of random walks , transport and percolation , and condensation of zero - range process @xcite . these studies showed that dynamical properties are modified and exhibit non - trivial dependence on the strength of weight . in this paper , as the generalization of our study on ca model on complex networks , we investigate the effect of both heterogeneous degree and weight on the condensation phenomena of ca model on weighted networks . the weight @xmath40 represents the weight to a link from the node @xmath1 to @xmath2 . in general , the strength @xmath41 of the node @xmath1 scales with the degree @xmath42 as @xmath43 . the exponent @xmath44 varies with network structures @xcite . thus it is natural to take the weight @xmath40 as @xmath45 . in this paper , we study the condensation transitions of ca on the wsfns with degree distribution @xmath46 and the symmetric weight @xmath47 . as in one dimension , the diffusion of the whole masses and the fragmentation of unit mass occur with the unit rate and the rate @xmath11 , respectively . in addition , masses move from a node @xmath1 to @xmath2 with hopping rate proportional to @xmath48 . we found that a certain critical @xmath8 exists below which the condensation transitions take place . however for @xmath9 , the condensation always occurs for any density @xmath49 . to find @xmath8 as a function of the degree exponent @xmath13 , one needs the steady state distribution @xmath14 of finding a walker at nodes with degree @xmath15 on the wsfns . @xmath14 gives the capture probability @xmath16 with which two walkers meet at a node with degree @xmath15 . we analytically derived @xmath50 and @xmath16 , and finally obtained @xmath51 . this paper is organized as follows . in sec . ii , we discuss the condensation transitions of ca model on the wsfns . to verify the existence @xmath8 , we investigate the steady state property of a single walker and the diffusive capture process on the wsfns in sec . iii and iv . we discuss the behavior of an average mass @xmath19 of a node with degree @xmath15 in sec . v and finally summarize our results in sec . we consider ca model on wsfns with the weight @xmath0 from node @xmath2 to @xmath1 defined as @xmath52 . for the construction of wsfn , we first construct an unweighted static sfn with @xmath53 nodes and @xmath54 links @xcite . the degree @xmath42 of a node @xmath1 is defined as the number of its links connected to other nodes . the average degree of a node @xmath55 is given as @xmath56 . the degree distribution @xmath57 of sfns is a power - law distribution @xmath36 . in the static model @xcite , it is desired to use large @xmath55 to construct fully connected networks . in simulations , we use @xmath58 . after then , we assign a weight @xmath59 to the link between node @xmath1 and @xmath2 . thus the hopping probability of masses from node @xmath1 to an @xmath60s linked neighbor @xmath2 is @xmath61 . @xmath62 denotes the sum over the linked neighbors of node @xmath1 . each node has an integer number of particles , and the mass on a node is defined as the number of particles on the node . initially @xmath63 particles are randomly distributed on @xmath53 nodes with given conserved density @xmath64 . next a node @xmath1 is chosen at random and one of the following events occurs : \(i ) diffusion : with the unit rate , the whole mass @xmath4 of node @xmath65 moves to one of the linked neighbors @xmath2 with probability @xmath5 . then the aggregation takes place ; @xmath66 and @xmath67 . \(ii ) chipping : with the rate @xmath11 , unit mass moves to a linked neighbor @xmath2 with the probability @xmath5 , and then the aggregation takes place , i.e. @xmath68 , @xmath69 . the @xmath29 case corresponds to zrp with constant chipping rate on wsfns @xcite . we perform monte carlo simulations with random initial mass distribution on the wsfns with @xmath70 and @xmath71 . we set @xmath72 and the network size @xmath73 with @xmath58 . we measure the single node mass distribution @xmath30 in the steady states . in fig . 1 , we plot @xmath30 for @xmath74 with two different @xmath44 , @xmath75 and @xmath76 . @xmath30 exhibits quite different behavior according to the value of @xmath44 . for @xmath75 ( fig . 1(a ) ) , @xmath30 decays exponentially without aggregates for sufficiently low density @xmath77 . on the other hand , for sufficiently high density , @xmath78 , an aggregate forms with the power - law decaying background mass distribution . it means that the condensation transition takes place at a certain critical density @xmath79 . hence @xmath30 follows eq . ( [ p ] ) . since in unweighted sfns , i.e @xmath21 , the condensation phase transitions take place for @xmath80 @xcite , one may expect the condensation transitions for very small @xmath44 . based on the following steps , we estimate @xmath22 and the exponent @xmath81 . in the condensed phase , the total density @xmath10 is written as @xmath82 , where @xmath83 is the density of an aggregate . since @xmath10 is given as @xmath84 , one can estimate @xmath22 from @xmath85 , where the upper bound @xmath86 is the cut - off mass at which the background distribution terminates . using this method , we estimate @xmath87 . we estimate the exponent @xmath81 from the scaling plot @xmath88 using @xmath30 of @xmath78 ( inset of fig.1(a ) ) . since the background distribution does not change for @xmath89 , we use @xmath30 of @xmath90 for the scaling plot . we estimate @xmath91 . on the other hand , for @xmath92 ( fig.1(b ) ) , @xmath30 shows the complete different behavior . the condensation takes place with an exponentially decaying background distribution for both sufficiently low and high density , @xmath93 and @xmath94 . therefore we conclude that the condensation always occurs for any nonzero density so a system is always in the condensed phase without any transitions for @xmath92 . the two different behaviors of @xmath30 for @xmath75 and @xmath76 indicate that a crossover @xmath44 ( @xmath8 ) should exist in the range @xmath95 for @xmath74 . a system undergoes the condensation transition for @xmath96 , while the condensation always occurs without the transition for @xmath97 . for @xmath74 with @xmath75 ( a ) and @xmath92 ( b ) . the inset of ( a ) shows the scaling plot @xmath98 with @xmath99 when @xmath78.,title="fig:",width=302][fig1 ] for @xmath70 with @xmath100 ( a ) and @xmath101 ( b ) . the inset of ( a ) shows the scaling plot @xmath98 with @xmath102 when @xmath78 . , title="fig:",width=302][fig2 ] similarly , for @xmath70 , @xmath30 exhibits the same different behavior according to the value of @xmath44 . the difference from the @xmath74 case is that the condensation transitions are observed for a negative @xmath44 . we observe the condensation transitions for @xmath103 ( fig . 2(a ) ) . with the same method used in the @xmath74 case , we estimate @xmath104 and @xmath105 respectively . however , for @xmath101 , the condensation is observed even for very low density @xmath93 , which means that a system is always in the condensed phase for @xmath101 ( fig . therefore , the crossover @xmath8 also exists for @xmath70 , but its value is negative unlike the @xmath74 case . together with the results of @xmath74 , we conclude that the crossover @xmath8 exists for any @xmath106 and @xmath8 varies with @xmath13 . in what follows , we discuss the existence of @xmath8 and next the condensation nature for @xmath96 . first , the condensation phenomena of ca model on wsfns is similar to that on unweighted sfns . on unweighted sfns , the condensation transitions exist for @xmath37 , while the condensation always occurs for @xmath38 @xcite . hence the crossover @xmath13 is @xmath107 . intriguingly , it was shown that the existence of transitions is determined by the survival probability of a diffusing prey chased by a diffusing predator , so - called the lamb - lion problem @xcite . the reason is as what follows . in the limit @xmath108 , let us assume only an infinite aggregate exists . with the rate @xmath11 , the unit mass is chipped off and moves around with the unit rate . if the chipped mass meets again with the infinite aggregate within a finite time interval , then the infinite aggregate is stable against the chipping process . on the other hand , if the chipped mass and the infinite aggregate does not meet again within a finite time interval , then the infinite aggregate would disappear by repeated chipping processes . therefore , the stability of the infinite aggregate is physically related to the capture process in which a diffusing lion ( infinite aggregate ) chases a diffusing lamb ( chipped mass ) . for the unweighted sfns with @xmath38 , it was shown that the survival probability @xmath109 of a lamb decays exponentially with finite life time @xmath110 @xcite . however , for @xmath111 , @xmath109 is finite in the thermodynamic limit . the behavior of @xmath109 implies that the condensation transition exist for @xmath37 due to the stable fluid phase in the limit @xmath108 , but only the condensation exist for @xmath38 . as a result , the asymptotic behavior of the survival probability of a lamb in the lamb - lion capture process determines the existence of the condensation transitions on unweighted sfns . similarly , on the wsfns , the existence of the condensation transitions is also expected to depend on the survival probability @xmath109 of a lamb . to see this , let us consider two limits , @xmath112 and @xmath113 for a given @xmath13 . in the limit @xmath114 , a walker always moves to a node with the larger degree . once a walker reach the hub node with the maximal degree , the walker is trapped at the hub node forever . as a result , a lion always captures a lamb at the hub node within a finite time interval . hence , @xmath109 should decay exponentially with a finite life time . on the other hand , in the limit @xmath115 , a walker is forced to reach nodes with the minimal degree . due to the inhomogeneous structure , the nodes with the minial degree are connected by nodes with larger degree . hence , a walker can not escape from one of the nodes with the minimum degree in this limit . it means that a lion can not always capture a lamb at some other node , so that @xmath109 is finite . from the behavior of @xmath109 in the two opposite limits , there should be a crossover @xmath8 . @xmath109 is finite for @xmath116 and decays to zero for @xmath97 . for the condensation phenomena , one expects no condensation transitions ( @xmath117 ) for @xmath97 due to finite life time of a lamb . instead , the condensation always occurs . on the other hand , the condensation transitions occur for @xmath116 . we analytically find @xmath12 for a given @xmath13 in sec . iv . from @xmath12 , one reads @xmath118 for @xmath70 and @xmath119 for @xmath120 respectively . our simulation results for @xmath120 and @xmath121 confirm the existence of @xmath8 and also the sign of @xmath8 for each @xmath13 . next , we discuss the critical behavior of ca model on wsfns . the ca model on any dimensional regular lattice and unweighted sfns with @xmath37 is well described by mean - field theory @xcite . on wsfns , interestingly , the transitions take place even for @xmath122 , which means that the transition nature is not affected by the inhomogeneity of network structure . since @xmath8 diverges for @xmath123 , the critical behavior of ca model on sfns with @xmath116 should be the same as that on random networks where @xmath44 , i.e. weight , does not have no special meaning due to the uniform degree distribution . as a result , one expects the mean - field critical behavior of sca model in regular lattice . our numerical estimates of @xmath81 , @xmath91 for @xmath70 and @xmath105 for @xmath71 , well agree with the mean - field value @xmath35 . therefore , we conclude that the critical behavior of ca model for @xmath124 on the wsfns belongs to the universality class of sca model in regular lattice . in summary , for a fixed @xmath13 , there is a crossover weight exponent @xmath8 . ca model undergoes the same type of condensation transitions as those of sca model in regular lattice for @xmath96 , while the condensation always takes place for nonzero density for @xmath97 . to find @xmath8 as a function of the degree exponent @xmath13 , one needs the steady state distribution @xmath14 of finding a walker at nodes with degree @xmath15 on the wsfns . in the next section , we derive @xmath14 on the wsfns . in sec . iv , we study lamb - lion capture process on the wsfns and finally find @xmath8 using @xmath14 . we consider a single walker on weighted networks with the weight @xmath0 . the connectivity of the network is represented by the adjacency matrix @xmath125 whose element @xmath126 if there is a link from a node @xmath2 to @xmath1 . otherwise , @xmath127 . we set @xmath128 conventionally . the degree @xmath42 of a node @xmath1 is given as @xmath129 . since we consider weighted networks with weight @xmath0 , we define the weighted adjacency matrix @xmath130 as @xmath131 . the motion of a walker on the weighted networks defined by the matrix @xmath130 is a stochastic process in the discrete time . we derive the stationary distribution @xmath132 of a walker being at node @xmath1 following the method of ref . @xcite . to set up the equation , we define the transition probability as follows . a walker at node @xmath1 at time @xmath133 selects one of its @xmath42 linked nodes with hopping probability @xmath5 . then , at time @xmath134 , the walker moves to the selected node . the hopping probability @xmath5 from node @xmath1 to @xmath2 is then given as @xmath135 , where @xmath136 is the strength of node @xmath1 . as an initial condition , assume that the walker starts at the node @xmath137 at time @xmath138 . then the recurrence relation of the transition probability @xmath139 of finding the walker at node @xmath1 at time @xmath133 is @xmath140 then the transition probability @xmath141 is written by iterating as @xmath142 for a symmetric @xmath130 with @xmath143 , one finds @xmath144 by comparing @xmath145 and @xmath139 . in the stationary state , the probability @xmath132 of finding a walker at node @xmath1 should be independent of initial starting nodes , which gives @xmath146 . summing up over @xmath137 , one finds @xmath147 where @xmath148 . in weighted networks with symmetric weights , @xmath132 is proportional to the strength of node @xmath1 , i.e. the sum of the weights of the nearest neighboring nodes . the same result was found in the recent study on the dynamics of random walks on growing weighted networks @xcite . in this paper , we consider the symmetric weight @xmath0 , @xmath149 for the weight ( [ w ] ) , @xmath132 is not given as a simple form . hence it is better to handle the distribution @xmath14 of finding a walker at nodes with degree @xmath15 . using eq . ( [ pi ] ) , one can see that @xmath150 to express the sum in eq . ( [ pk1 ] ) in terms of degree @xmath15 , we arrange the sum as follows . only terms with @xmath151 contributes nontrivially to the sum @xmath152 and thus @xmath153 nodes with the degree @xmath15 in a network have the nontrivial contributions to the sum . the node with the degree @xmath15 has @xmath15 linked neighbors whose degree ranges from @xmath154 to the maximal degree of the network @xmath155 . hence , the number of nontrivial terms in the sum @xmath156 is @xmath157 , which can be arranged in the order of increasing degree . then , the double sum of eq . ( [ pk1 ] ) is written as @xmath158 , where @xmath159 is the degree distribution of the node involved in such @xmath157 terms . for large @xmath53 , we approximate @xmath159 to @xmath160 . then @xmath14 is approximately given as @xmath161 on sfns with degree distribution @xmath36 , the integral in the second line is finite for @xmath162 . hence we finally obtain @xmath14 on wsfns as @xmath163 the exponent @xmath164 varies with @xmath44 and @xmath13 , and also changes its sign . for @xmath165 , i.e. @xmath166 , @xmath167 is independent of degree @xmath15 so a walker does not feel the inhomogeneity of the underlying network structure . while a walker performs biased walks to nodes with the larger degree for @xmath168 , the direction of the bias is reversed for @xmath169 . since the exponent @xmath44 is a free parameter , one can controls the direction of the bias for a given @xmath13 . and @xmath164 for @xmath70 ( a ) and @xmath170 ( b ) . insets show the relation ( [ pk ] ) ( solid line ) and numerical estimates of @xmath164 ( symbols ) . , title="fig:",width=302][fig3 ] to check the scaling relation ( [ pk ] ) , we perform monte carlo simulations on the wsfns with @xmath73 and the average degree @xmath58 . in the steady states , we measure @xmath14 for various @xmath44 up to @xmath171 for @xmath70 and @xmath94 for @xmath170 . fig . 3 shows the plot of @xmath14 against @xmath15 for several values of @xmath44 . as shown , @xmath14 scales in power - law with @xmath15 . the inset in each panel shows the plot of @xmath164 against @xmath44 . the simulation results agree well with the analytical prediction ( [ pk ] ) . in this section , we consider the capture process or the lamb - lion problem on wsfns with the symmetric weights ( [ w ] ) . a lamb and a lion initially locate separately on randomly selected two nodes . then the probability @xmath16 of finding two walkers at the same node with degree @xmath15 at the same time is proportional to @xmath172 . from eq . ( [ pk ] ) , one gets @xmath173 with @xmath174 then the probability @xmath175 of finding two walkers on the same node with any degree is given as @xmath176 since the upper bound @xmath155 diverges with @xmath53 , the integral @xmath177 diverges for @xmath178 . hence there exists a crossover value @xmath8 given as @xmath179 for @xmath116 , the lamb survives indefinitely with a finite probability . however , for @xmath97 , the lion captures the lamb with the unit probability . to check the scaling relation ( [ dk ] ) , we measure @xmath16 on the wsfns with @xmath73 and @xmath58 . for @xmath180 trials , we count the number @xmath181 of capture events on nodes with the degree @xmath15 . we obtain @xmath16 to divide @xmath181 by total trials(@xmath180 ) . 4 shows the plot of @xmath16 for several values of @xmath44 , which scales well with @xmath15 in power - law . as shown in the insets of fig . 4 , numerical estimates for @xmath182 satisfy the relation ( [ dk ] ) very well . and @xmath182 for @xmath70 ( a ) and @xmath170 ( b ) . insets show the relation of ( [ dk ] ) ( solid line ) and numerical estimates of @xmath182 ( symbols ) . , title="fig:",width=302][fig4 ] of a lamb for @xmath170 ( a ) and @xmath70 ( b ) . the solid line is a guide to the eye . insets show the semi - logarithmic plots of @xmath109 for @xmath73 . from top to bottom , each line corresponds to the @xmath109 of @xmath96 , @xmath8 and @xmath183 respectively.,title="fig:",width=294][fig5 ] to verify the existence of @xmath8 by another method , we now consider the survival probability @xmath109 of a lamb . @xmath109 always satisfies @xmath184 on random and scale - free networks due to the small world nature @xcite . as @xmath184 in sfns with any @xmath13 , we are interested in the average life time @xmath185 of a lamb rather than @xmath109 itself . from @xmath186\ ; dt$ ] and @xmath187 , we have @xmath188 . hence @xmath189 is infinite for @xmath116 and finite for @xmath97 in the limit @xmath190 . however , for the finite - sized networks , a lamb is eventually captured within @xmath53 time steps for any @xmath44 . for @xmath96 , the maximum life time should be the order of @xmath53 to guarantee the finite survival probability in the limit @xmath190 . hence @xmath185 is expected to scale as @xmath191 for @xmath96 . we measure @xmath185 on wsfns of @xmath70 and @xmath192 with network size @xmath53 up to @xmath193 . from eq . ( [ ac ] ) , one reads @xmath194 for @xmath70 and @xmath195 for @xmath170 . in fig . 5 , we plot @xmath185 against @xmath53 . as shown in each inset , @xmath109 exponentially decays for any @xmath44 . for @xmath170(fig . 5(a ) ) , @xmath185 increases with @xmath53 as @xmath196 with @xmath197 for @xmath198 ( @xmath199 ) and @xmath200 for @xmath201 . we estimate @xmath202 by measuring successive slopes from the log - log data in fig . 5(a ) . for @xmath92 ( @xmath203 ) , @xmath185 tends to saturate to the asymptotic value @xmath204 with decreasing successive slopes . the exponent @xmath202 of @xmath75 is close to the expected value @xmath205 . for @xmath206 , @xmath185 seems to diverge with @xmath207 . however , since @xmath185 of @xmath92 already tends to saturate , it is expected that @xmath185 of @xmath97 would saturate to a finite value in the network with @xmath208 , where @xmath209 is the characteristic size for given @xmath44 . for example , for @xmath92 , @xmath185 does not get into the saturation region even after @xmath210 , which implies @xmath211 for @xmath92 . since @xmath212 should increase as @xmath213 , it is empirically impossible to see the saturation of @xmath185 via simulations . therefore , the initial slope @xmath202 at @xmath8 may have no special meaning as that of @xmath183 . the same behavior for @xmath185 was observed for @xmath21 case @xcite , where @xmath185 initially algebraically increases with continuously varying @xmath202(@xmath214 ) as @xmath215 from below . for @xmath70 ( fig . 5(b ) ) , we estimate @xmath216 for @xmath100 ( @xmath199 ) as expected . however , for @xmath101 ( @xmath217 ) , @xmath185 algebraically increases with @xmath218 . since for @xmath21 @xcite , @xmath212 is already larger than @xmath193 for @xmath219 , it is difficult to see the saturation of @xmath185 . for @xmath220 , we estimate @xmath200 . as in @xmath170 , the initial slope for @xmath97 has no special meaning . based on our numerical results , we are convinced that @xmath221 approaches a finite value @xmath110 for @xmath97 and becomes infinite for @xmath96 in the limit @xmath190 . hence in the limit @xmath190 , we have @xmath222 with the characteristic time @xmath223 . another interesting quantity in condensation phenomena on networks is the average mass @xmath19 at a node with degree @xmath15 in the steady state @xcite . in zrp with chipping rate @xmath224 , the complete condensation takes place for @xmath225 , where @xmath226 for unweighted sfns @xcite and @xmath227 for wsfns with the weight ( [ w ] ) @xcite . for @xmath228 , @xmath19 increases as @xmath20 for @xmath229 , and as @xmath230 for @xmath231 on the wsfns . especially for @xmath232 , @xmath19 increases @xmath20 until @xmath233 and jumps to the value @xmath234 at @xmath155 . hence the condensation takes place at the node with @xmath155 degree in zrp . for @xmath170 . the solid and the dashed line correspond to @xmath92 and @xmath235 respectively . the inset shows the scaling plot @xmath236 with @xmath237 for @xmath75 ( dashed line ) and @xmath238 for @xmath92 ( solid line).,title="fig:",width=340][fig6 ] the recent study on ca model on unweighted sfns showed that @xmath19 linearly increases up to @xmath155 without the jump at @xmath155 unlike in zrp with constant chipping rate @xcite . the linearity of @xmath19 comes from the fact that all masses can diffuse . the mass @xmath239 formed on the node with the degree @xmath155 can diffuse throughout network to make the steady sate distribution @xmath132 . by taking average over all nodes , the @xmath239 soaks into the average mass @xmath19 unlike in zrp where all samples have @xmath239 at @xmath155 . the linearity of @xmath19 on unweighted sfns results from @xmath240 @xcite . therefore , from @xmath241 on the wsfns , we expect @xmath242 up to @xmath243 . to see this explicitly , we derive the relation @xmath244 as follows . we consider the average total mass @xmath245 of nodes with degree @xmath15 defined as @xmath246 where @xmath247 is the probability of finding a walker with mass @xmath32 at nodes with degree @xmath15 in the steady state . since the mass distribution @xmath30 in the steady state is independent of @xmath15 , we write @xmath248 . from ( [ pk2 ] ) and ( [ mk ] ) , one reads @xmath249 where we drop the normalization constant of @xmath14 . since the number of nodes with degree @xmath15 is @xmath153 , @xmath19 is given as @xmath250 to confirm the scaling behavior of @xmath19 , we measure @xmath19 in the condensed phase on the wsfns of @xmath170 and @xmath73 . in fig . 6 , we plot @xmath19 against @xmath15 for @xmath75 and @xmath76 . with @xmath72 , we set @xmath78 which corresponds to the condensed phase for both @xmath44 values . assuming @xmath251 , one expects @xmath252 for @xmath75 and @xmath253 for @xmath92 respectively . from the scaling plot @xmath254 ( inset of fig . 6 ) , we estimate @xmath255 for @xmath75 and @xmath256 for @xmath92 which agree well with the predictions . in summary , we investigate the properties of conserved - mass aggregation ( ca ) model on weighted scale - free networks ( wsfns ) . in wsfns , the weight @xmath0 is assigned to the link between node @xmath1 and @xmath2 . we consider the symmetric weight given as @xmath3 . in ca model , masses diffuse with unit rate and unit mass chips off from mass with rate @xmath11 . in addition , the hopping probability @xmath5 from node @xmath1 to @xmath2 is given as @xmath257 . on the wsfns , a walker finally reaches the hub node with @xmath155 degree for @xmath258 , while it is trapped forever at nodes with the minimal degree for @xmath259 . in the lamb - lion capture process , it means that the lion captures the lamb at the hub node within finite time interval for @xmath260 . on the other hand , a lamb survives indefinitely with finite probability for @xmath259 , because the lion can not escape from a node with the minimal degree to capture a lamb at some other node . in - between the two limits , one expects a crossover @xmath8 below which the life time @xmath221 of a lamb is infinite . however , for @xmath97 , @xmath221 is finite . the dependence of @xmath261 on @xmath44 is similar to that on unweighted sfns of @xmath21 where @xmath221 is infinite for @xmath80 and finite otherwise @xcite . to verify the existence of @xmath8 , we need the stationary distribution @xmath14 of finding a walker at nodes with degree @xmath15 . from the equation for the transition probability @xmath262 to go from node @xmath1 to @xmath2 in @xmath133 time steps , we analytically find @xmath17 . next , we consider the so - called lamb - lion capture process . with @xmath14 , we find the probability @xmath16 of finding two walkers at the same node with degree @xmath15 at the same time to scale @xmath263 . finally , integrating out @xmath16 , we find the death probability @xmath175 of a lamb . a lamb survives indefinitely with the finite survival probability for @xmath116 , while it is eventually captured by a lion for @xmath97 . we analytically find @xmath264 . therefore , in the limit @xmath190 , the life time @xmath221 of a lamb is finite for @xmath9 , while it is infinite for @xmath265 . we numerically confirm the all analytical results . the existence of the condensation transitions is known to depend on @xmath221 of a lamb @xcite . for @xmath9 , @xmath221 is finite so the condensation always occurs for any nonzero density . on the other hand , for @xmath266 , the infinite @xmath221 ensures the condensation transitions at a certain critical density @xmath22 . for @xmath97 , we numerically confirm that the condensation always takes place at very low density . we also numerically confirm that for @xmath116 , ca model on the wsfns undergoes the same type of the condensation transitions as those of sca model in regular lattice . finally , we investigate the behavior of the average mass @xmath19 of a node with degree @xmath15 . in zrp with constant chopping rate on networks @xcite , @xmath19 increases as @xmath20 , and jumps to the total mass of the system at @xmath155 . however , in the sca model on unweighted sfns , it was shown that @xmath19 linearly increases with @xmath15 up to @xmath155 without any jumps @xcite . furthermore the linearity of @xmath19 is valid for any @xmath267 , which comes from the fact that the diffusion is only the relevant physical factor to decide the distribution @xmath19 . similarly , on the wsfns , we analytically find and numerically confirm that @xmath19 algebraically increases as @xmath20 for any @xmath268 without any jumps . this work was supported by the korea science and engineering foundation(kosef ) grant funded by the korea government(most ) ( no . r01 - 2007 - 000 - 10910 - 0 ) and by the korea research foundation grant funded by the korean government ( moehrd , basic research promotion fund ) ( krf-2007 - 313-c00279 ) . m. r. evans , europhys . lett . 36 , 13 ( 1996 ) . o. j. oloan , m. r. evans , 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we investigate the condensation phase transitions of conserved - mass aggregation ( ca ) model on weighted scale - free networks ( wsfns ) . in wsfns , the weight @xmath0 is assigned to the link between the nodes @xmath1 and @xmath2 . we consider the symmetric weight given as @xmath3 . in ca model , the mass @xmath4 on the randomly chosen node @xmath1 diffuses to a linked neighbor of @xmath1,@xmath2 , with the rate @xmath5 or an unit mass chips off from the node @xmath1 to @xmath2 with the rate @xmath6 . the hopping probability @xmath5 is given as @xmath7 , where the sum runs over the linked neighbors of the node @xmath1 . on the wsfns , we numerically show that a certain critical @xmath8 exists below which ca model undergoes the same type of the condensation transitions as those of ca model on regular lattices . however for @xmath9 , the condensation always occurs for any density @xmath10 and @xmath11 . we analytically find @xmath12 on the wsfn with the degree exponent @xmath13 . to obtain @xmath8 , we analytically derive the scaling behavior of the stationary distribution @xmath14 of finding a walker at nodes with degree @xmath15 , and the probability @xmath16 of finding two walkers simultaneously at the same node with degree @xmath15 . we find @xmath17 and @xmath18 respectively . with @xmath14 , we also show analytically and numerically that the average mass @xmath19 on a node with degree @xmath15 scales as @xmath20 without any jumps at the maximal degree of the network for any @xmath10 as in the sfns with @xmath21 .

You are an expert at summarizing long articles. Proceed to summarize the following text: stellar feedback the injection of energy and momentum by stars originates at the small scales of star clusters ( @xmath01 pc ) , yet it shapes the interstellar medium ( ism ) on large scales ( @xmath11 kpc ) . at large scales , stellar feedback is necessary in order to form realistic galaxies in simulations and to account for observed galaxy properties . in the absence of feedback , baryonic matter cools rapidly and efficiently forms stars , producing an order of magnitude too much stellar mass and consuming most available gas in the galaxy ( e.g. , @xcite ) . stellar feedback prevents this `` cooling catastrophe '' by heating gas as well as removing low angular momentum baryons from galactic centers , thereby allowing only a small fraction of the baryonic budget of dark matter halos to be converted to stars . the removal of baryons may also flatten the dark matter mass profile , critical to form bulgeless dwarf galaxies ( e.g. , @xcite ) . furthermore , stellar feedback possibly drives kpc - scale galactic winds and outflows ( see @xcite for a review ) which have been frequently observed in local galaxies ( e.g. , @xcite ) as well as in galaxies at moderate to high redshift ( e.g. , @xcite ) . at the smaller scales of star clusters and giant molecular clouds ( gmcs ) , newborn stars dramatically influence their environments . observational evidence suggests that only a small fraction ( @xmath212% ) of gmc mass is converted to stars per cloud free - fall time ( e.g. , @xcite ) . this inefficiency can be attributed to stellar feedback processes of h ii regions that act to disrupt and ultimately to destroy their host clouds ( e.g. , @xcite ) . in addition to the pressure of the warm ionized h ii region gas itself , there are several other forms of stellar feedback that can drive the dynamics of h ii regions and deposit energy and momentum in the surrounding ism : the direct radiation of stars ( e.g. , @xcite ) , the dust - processed infrared radiation ( e.g. , @xcite ) , stellar winds and supernovae ( sne ; e.g. , @xcite ) , and protostellar outflows / jets ( e.g. , @xcite ) . from a theoretical perspective , sne were the first feedback mechanism to be considered as a means to remove gas from low - mass galaxies ( e.g. , @xcite ) and to prevent the cooling catastrophe ( e.g. , @xcite ) . however , resolution limitations precluded the explicit modeling of individual sne in galaxy formation simulations , so phenomenological prescriptions were employed to account for `` sub - grid '' feedback ( e.g. , @xcite ) . since then , extensive work has been done to improve and to compare these sub - grid models ( e.g. , @xcite ) . furthermore , the use of `` zoom - in '' simulations ( which can model feedback physics down to @xmath11 pc scale ) has enabled the modeling of several modes of feedback simultaneously ( e.g. , @xcite ) . while simulations are beginning to incorporate many feedback mechanisms , most observational work focuses on the effects of the individual modes . consequently , the relative contribution of these components and which processes dominate in different conditions remains uncertain . to address this issue , we recently employed multiwavelength imaging of the giant h ii region n157 ( 30 doradus ; `` 30 dor '' hereafter ) to assess the dynamical role of several stellar feedback mechanisms in driving the shell expansion @xcite . in particular , we measured the pressures associated with the different feedback modes across 441 regions to map the pressure components as a function of position ; we considered the direct radiation pressure exerted by the light from massive stars , the dust - processed radiation pressure , the warm ionized ( @xmath3 k ) gas pressure , and the hot shocked ( @xmath4 k ) gas pressure from stellar winds and sne . we found that the direct radiation pressure from massive stars dominates at distances @xmath075 pc from the central star cluster r136 , while the warm ( @xmath5 k ) ionized gas pressure dominates at larger radii . by comparison , the dust - processed radiation pressure and the hot ( @xmath4 k ) gas pressure are weak and are not dynamically important on the large scale ( although small bubbles of the hot gas can have significant pressures @xcite ; see appendix [ app : hot gas ] of this paper for a discussion on how choice of hot gas filling factor is critical when evaluating the dynamical role of hot gas ) . in this paper , we extend the methodology applied to 30 dor to a larger sample of 32 h ii regions in the large and small magellanic clouds ( lmc and smc , respectively ) , with the aim of probing how stellar feedback properties vary between sources . the organization of this paper is as follows . section [ sec : sample ] describes our lmc and smc h ii region sample and the data we have employed for our analyses . section [ sec : method ] outlines the methods we have used to assess the dynamical role of several stellar feedback mechanisms in the 32 sources . section [ sec : results ] presents the results from these analyses , and section [ sec : discussion ] explores implications of our findings related to the importance of radiation pressure ( section [ sec : radpressure ] ) , the confinement of hot gas in the h ii regions ( section [ sec : leakage ] ) and the momentum deposition of the dust - processed radiation to the warm gas ( section [ sec : dusty ] ) . finally , we summarize this work in section [ sec : summary ] . for our feedback analyses , we selected the 16 lmc and 16 smc h ii regions of @xcite , who chose sources based on their bright 24@xmath6 m and h@xmath7 emission and which are distributed throughout these galaxies . we opted to include sources based on both ir and h@xmath7 , since bright h@xmath7 emission alone is not unique to h ii regions . for example , several of the emission nebulae identified by @xcite are now known to be supernova remnants . furthermore , bright 24@xmath6 m emission arises from stochastically heated small dust grains ( i.e. , dust is heated by collisions with starlight photons : e.g. , @xcite ) , so it is well - correlated with h ii regions within the milky way and other galaxies . lccccc n4 & dem l008 & 04:52:09 & @xmath866:55:13 & 0.7 & 10.2 + n11 & dem l034 , l041 & 04:56:41 & @xmath866:27:19 & 10.0 & 145 + n30 & dem l105 , l106 & 05:13:51 & @xmath867:27:22 & 3.1 & 45.1 + n44 & dem l150 & 05:22:16 & @xmath867:57:09 & 7.1 & 103 + n48 & dem l189 & 05:25:50 & @xmath866:15:03 & 5.2 & 75.6 + n55 & dem l227 , l228 & 05:32:33 & @xmath866:27:20 & 3.6 & 52.4 + n59 & dem l241 & 05:35:24 & @xmath867:33:22 & 3.9 & 56.7 + n79 & dem l010 & 04:52:04 & @xmath869:22:34 & 4.4 & 64.0 + n105 & dem l086 & 05:09:56 & @xmath868:54:03 & 2.9 & 42.2 + n119 & dem l132 & 05:18:45 & @xmath869:14:03 & 5.9 & 85.8 + n144 & dem l199 & 05:26:38 & @xmath868:49:55 & 4.9 & 71.3 + n157 & 30 dor & 05:38:36 & @xmath869:05:33 & 6.8 & 98.9 + n160 & & 05:40:22 & @xmath869:37:35 & 5.0 & 40.0 + n180 & dem l322 , l323 & 05:48:52 & @xmath870:03:51 & 2.7 & 39.3 + n191 & dem l064 & 05:04:35 & @xmath870:54:27 & 2.1 & 30.5 + n206 & dem l221 & 05:30:38 & @xmath871:03:53 & 7.7 & 112 + dem s74 & & 00:53:14 & @xmath873:12:18 & 2.7 & 47.9 + n13 & & 00:45:23 & @xmath873:22:38 & 0.5 & 8.87 + n17 & & 00:46:41 & @xmath873:31:38 & 1.5 & 26.6 + n19 & & 00:48:23 & @xmath873:05:54 & 0.7 & 12.4 + n22 & & 00:48:09 & @xmath873:14:56 & 0.9 & 16.0 + n36 & & 00:50:26 & @xmath872:52:59 & 2.5 & 44.4 + n50 & & 00:53:26 & @xmath872:42:56 & 4.3 & 76.3 + n51 & & 00:52:40 & @xmath873:26:29 & 1.9 & 33.7 + n63 & & 00:58:17 & @xmath872:38:57 & 1.3 & 23.1 + n66 & & 00:59:06 & @xmath872:10:44 & 3.6 & 63.9 + n71 & & 01:00:59 & @xmath871:35:30 & 0.2 & 3.55 + n76 & & 01:03:32 & @xmath872:03:16 & 3.1 & 55.0 + n78 & & 01:05:18 & @xmath871:59:53 & 2.6 & 46.1 + n80 & & 01:08:13 & @xmath872:00:06 & 2.2 & 39.0 + n84 & & 01:14:56 & @xmath873:17:51 & 5.7 & 101 + n90 & & 01:29:27 & @xmath873:33:10 & 1.7 & 30.2 + [ tab : sample ] our final sample of h ii regions are listed in table [ tab : sample ] , and figures [ fig : lmcthreecolor ] and [ fig : smcthreecolor ] shows the three - color images of the lmc and smc h ii regions , respectively . we note that although our sample spans a range of parameter space ( e.g. , two orders of magnitude in radius and in ionizing photon fluxes @xmath9 ) , the h ii regions we have selected represent the brightest in the magellanic clouds in h@xmath7 and at 24 @xmath6 m . we utilize published ubv photometry of 624 lmc star clusters @xcite to assess upper limits on the cluster ages and lower limits on star cluster masses powering our sample . within the radii of the lmc h ii regions , we found 18 star clusters from the bica sample . to estimate the cluster ages , we compare the extinction - corrected ubv colors of the enclosed star clusters to the colors output from starburst99 simulations @xcite of a star cluster of @xmath10 which underwent an instantaneous burst of star formation . for this analysis , we adopt a color excess @xmath11 , the foreground reddening in the direction of the lmc @xcite . this value is almost certainly an underestimate and represents the minimum reddening toward our clusters ( for example , the reddening in r136 is @xmath12 ) and neglects local extinction . based on the clusters ubv colors , we find upper limit ages of @xmath2315 myr ; greater extinction toward the clusters would yield younger ages . additionally , we estimate the lower limit of the star cluster masses by normalizing @xmath10 by the ratio of the v - band luminosities of our clusters with those of the simulated clusters at their respective ages . we find cluster masses of @xmath2300@xmath13 . as relatively bright and evolved sources , the dynamical properties of our sample may differ from more dim h ii regions ( those powered by smaller star clusters ) and h ii regions which are much younger or older . for our analyses , we employed data at several wavelengths ; a brief description of these data is given below . throughout this paper , we assume a distance @xmath14 of 50 kpc to the lmc @xcite and of 61 kpc to the smc @xcite . to illustrate the h ii regions structure , we show the h@xmath7 emission of the 32 sources in figures [ fig : lmcthreecolor ] and [ fig : smcthreecolor ] . we used the narrow - band image ( at 6563@xmath15 , with 30@xmath15 full - width half max ) that was taken with the university of michigan / ctio 61-cm curtis schmidt telescope at ctio as part of the magellanic cloud emission line survey ( mcels : @xcite ) . the total integration time was 600 s , and the reduced image has a resolution of 2pixel@xmath16 . to estimate the h@xmath7 luminosity of our smc sources within the radii given in table [ tab : sample ] , we used the flux - calibrated , continuum - subtracted mcels data . as the flux calibrated mcels data of the lmc is not yet available , we employed the southern h@xmath7 sky survey atlas ( shassa ) , a robotic wide - angle survey of declinations @xmath17 to @xmath18 @xcite , to measure h@xmath7 luminosities of our lmc h ii regions . shassa data were taken using a ccd with a 52-mm focal length canon lens at f/1.6 . this setup enabled a large field of view ( @xmath19 ) and a spatial resolution of 47.64 pixel@xmath16 . the total integration time for the lmc exposure was @xmath2021 minutes . infrared images of the lmc were obtained through the _ spitzer _ space telescope legacy program surveying the agents of galaxy evolution ( sage : @xcite ) . the survey covered an area of @xmath27 @xmath21 7 degrees of the lmc with the infrared array camera ( irac ; @xcite ) and the multiband imaging photometer ( mips ; @xcite ) . images were taken in all bands of irac ( 3.6 , 4.5 , 5.8 , and 7.9 @xmath6 m ) and of mips ( 24 , 70 , and 160 @xmath6 m ) at two epochs in 2005 . for our analyses , we used the combined mosaics of both epochs with 1.2pixel@xmath16 in the 3.6 and 7.9 @xmath6 m irac images and 2.49pixel@xmath16 and 4.8pixel@xmath16 in the mips 24 @xmath6 m and 70 @xmath6 m , respectively . the smc was also surveyed by _ spitzer _ with the legacy program surveying the agents of galaxy evolution in the tidally stripped , low metallicity small magellanic cloud ( sage - smc : @xcite ) . this project mapped the full smc ( @xmath230 deg@xmath22 ) with irac and mips and built on the pathfinder program , the spitzer survey of the small magellanic cloud ( s@xmath23mc : @xcite ) , which surveyed the inner @xmath23 deg@xmath24 of the smc . sage - smc observations were taken at two epochs in 20072008 , and we employed the combined mosaics from both epochs ( plus the s@xmath23mc data ) . the lmc and smc were observed with the australian telescope compact array ( atca ) as part of 4.8-ghz and 8.64-ghz surveys @xcite . these programs had identical observational setups , using two array configurations that provided 19 antenna spacings , and the atca observations were combined with the parkes 64-m telescope data of @xcite to account for extended structure missed by the interferometric observations . for our analyses , we utilized the resulting atca@xmath25parkes 8.64 ghz ( 3.5-cm ) images of the lmc and smc , which had gaussian beams of fwhm 22 and an average rms noise level of 0.5 mjy beam@xmath16 . given the large extent of the lmc , _ chandra _ and _ xmm - newton _ have not observed the majority of that galaxy . thus , for our x - ray analyses of the 16 lmc h ii regions , we use archival data from _ rosat _ , the rntgen satellite . the lmc was observed via pointed observations and the all - sky survey of the rosat position sensitive proportional counter ( pspc ) over its lifetime ( e.g. , @xcite ) . the rosat pspc had modest spectral resolution ( with @xmath26 ) and spatial resolution ( @xmath225 ) over the energy range of 0.12.4 kev , with @xmath27 field of view . table [ tab : xrayobslog ] lists the archival pspc observations we utilized in our analyses of our sample . all the lmc h ii regions except for n191 were observed in pointed observations from 19911993 with exposures ranging from @xmath2400045000 s. some of these observations were presented and discussed originally in @xcite . lccc n4 & july 1993 & rp500263n00 & 12.7 + n11 & november 1992 & rp900320n00 & 17.6 + n30 & february 1992 & rp500052a01 & 8.0 + n44 & march 1992 & rp500093n00 & 8.7 + n48 & october 1991 & rp200692n00 & 44.7 + n55 & october 1991 & rp200692n00 & 44.7 + n59 & december 1993 & rp900533n00 & 1.6 + n79 & october 1993 & rp500258n00 & 12.7 + n105 & april 1992 & rp500037n00 & 6.8 + n119 & june 1993 & rp500138a02 & 14.6 + n144 & june 1993 & rp500138a02 & 14.6 + 30 dor & april 1992 & rp500131n00 & 16.0 + n160 & april 1992 & rp500131n00 & 16.0 + n180 & october 1993 & rp500259n00 & 4.0 + n191 & & & + n206 & december 1993 & rp300335n00 & 11.3 + dem s74 & november 2009 & 0601211401 & 46.8 + n13 & october 2009 & 0601211301 & 32.7 + n17 & october 2009 & 0601211301 & 32.7 + n19 & march 2007 & 0403970301 & 39.1 + n22 & october 2000 & 0110000101 & 28.0 + n36 & march 2010 & 0656780201 & 12.8 + n50 & december 2003 & 0157960201 & 14.8 + n51 & april 2007 & 0404680301 & 20.4 + n63 & october 2009 & 0601211601 & 32.3 + n66 & may 2001 & 1881 & 99.9 + n71 & june 2007 & 0501470101 & 16.1 + n76 & march 2000jan 2009 & 52 observations & 471.0 + n78 & dec 2000feb 2009 & 36 observations & 297.6 + n80 & november 2009 & 0601211901 & 31.6 + n84 & march 2006 & 0311590601 & 11.3 + n90 & april 2010 & 0602520301 & 96.3 + [ tab : xrayobslog ] the smc was surveyed by _ xmm - newton _ between may 2009 and march 2010 @xcite . we exploit data from this campaign as well as from pointed _ xmm - newton _ observations for 13 of the 16 smc h ii regions . for the other three smc sources ( n66 , n76 , and n78 ) , we use deep _ chandra _ acis - i observations . n66 was targeted in a 99.9 ks acis - i observation @xcite . n76 and n78 are in the field of a _ chandra _ calibration source , the supernova remnant 1e 0102@xmath87219 , so they have been observed repeatedly since the launch of _ chandra _ in 1999 . we searched these calibration data and merged all the observations where the _ chandra _ chip array imaged the full diameter of the sources : 52 observations for n76 , and 36 observations for n78 . we follow the same methodology as in our 30 dor pressure analysis @xcite with only a few exceptions , described below . instead of calculating spatially - resolved pressure components for the sources , we determine the average pressures integrated over the radii listed in table [ tab : sample ] . thus , these pressure components are those `` felt '' within the h ii shells . we describe the uncertainties associated with the calculation of each term in section [ sec : uncertainty ] . to select the radius of each region , we produced surface brightness profiles of their h-@xmath7 emission , and we determined the apertures which contained 90% of the total h-@xmath7 fluxes . we opted for this phenomenological definition of the radii to reduce the systematic uncertainties between sources . as seen in figures [ fig : lmcthreecolor ] and [ fig : smcthreecolor ] , the h ii regions are quite complex , and the calculations below are simple and aimed to describe the general properties of these sources . the light output by stars produces a direct radiation pressure that is associated with the photons energy and momentum . the resulting radiation pressure @xmath28 at some position within the h ii region is related to the bolometric luminosity of each star @xmath29 and the distance @xmath30 the light traveled to reach that point : @xmath31 where the summation is over all the stars in the region . the volume - averaged direct radiation pressure @xmath32 is then @xmath33 where @xmath34 is the total volume within the h ii region shell and @xmath35 is the h ii region radius . the above equation is the formal definition of radiation pressure ( i.e. , it is the trace of the radiation pressure tensor ) . we note that radiation pressure and radiation force do not always follow the same simple relationship as e.g. , gas pressure and force , where the force is the negative gradient of pressure . in particular , @xcite point out that in a relatively transparent medium ( such as the interior of an h ii region ) , it is possible for the radiation pressure to exceed the gas pressure while the local force exerted on matter by the radiation is smaller than the force exerted by gas pressure . however , at the h ii shells where the gas is optically thick to stellar radiation , radiation force and pressure follow the same relationship as gas force and pressure , and the radiation pressure defined by equation [ eq : pdir ] is the relevant quantity to consider . to obtain @xmath29 of the stars in our 30 dor analyses , we employed ubv photometry of r136 from @xcite using hst planetary camera observations , and the ground - based data of @xcite and @xcite to account for stars outside r136 . while several large - scale optical surveys of the lmc have now been done and include ubv photometry ( e.g. , @xcite ) , these data do not resolve the crowded regions of young star clusters , and they focus principally on the ( uncrowded ) field population . an alternative means to estimate the bolometric luminosities of the star clusters is using the extinction - corrected h@xmath7 luminosities of the h ii regions . from @xcite , for a stellar population that fully samples the initial mass function ( imf ) and the stellar age distribution , the bolometric luminosity @xmath36 is related to the extinction - corrected h@xmath7 luminosity @xmath37 by the expression @xmath38 . we use the shassa and mcels data to estimate the observed h@xmath7 luminosities @xmath39 within the radii listed in table [ tab : sample ] . to correct for extinction , we employ the reddening maps of the lmc and smc presented in @xcite , from the third phase of the optical gravitational lensing experiment ( ogle iii ) . these authors used observations of red clump and rr lyrae stars to derive spatially - resolved extinction estimates ( with typical subfield sizes of [email protected] ) across the lmc and smc , and these data are publicly available through the german astrophysical virtual observatory ( gavo ) interface . using gavo , we obtained the mean extinction in the b- and v - bands , @xmath40 and @xmath41 , respectively . in the cases when the h ii region radii included multiple subfields of the ogle extinction measurements , we calculated the average @xmath40 and @xmath41 over that aperture . then , we use the color excess @xmath42 to compute @xmath43 , the extinction at the wavelength @xmath44 of the h@xmath7 line , given @xmath45 where @xmath46 ) is the value of the extinction curve at the wavelength of the h@xmath7 line . @xcite derive the best - fit expression for @xmath47 ) at optical wavelengths as @xmath48 where @xmath49 . we adopt the standard @xmath50 , which @xcite demonstrate to be valid in the optical for the lmc and smc , and we find @xmath51 2.362 . finally , the extinction - corrected h@xmath7 luminosity @xmath37 is @xmath52 the parameters associated with these calculations , including the intrinsic h@xmath7 luminosities and corresponding @xmath36 of the 32 h ii regions , are listed in table [ tab : extinction ] . the extinction - corrected h@xmath7 luminosities are typically 1020% greater than the observed h@xmath7 luminosities . we note that local reddening and extinction may be greater than the average values obtained in the ogle iii maps , and thus the bolometric luminosities of the star clusters may be greater . however , even if the local extinction is a factor of ten larger , the direct radiation pressure will still be dynamically insignificant , as seen in the results given in section [ sec : results ] . lccccccc n4 & 0.31 & 0.24 & 0.17 & 37.1 & 37.2 & 39.4 & 49.2 + n11 & 0.08 & 0.06 & 0.05 & 38.9 & 39.0 & 41.1 & 51.0 + n30 & 0.26 & 0.20 & 0.14 & 37.7 & 37.8 & 39.9 & 49.7 + n44 & 0.28 & 0.22 & 0.14 & 38.5 & 38.6 & 40.7 & 50.6 + n48 & 0.19 & 0.14 & 0.12 & 37.8 & 37.9 & 40.0 & 49.9 + n55 & 0.30 & 0.23 & 0.17 & 38.0 & 38.0 & 40.2 & 50.0 + n59 & 0.36 & 0.28 & 0.19 & 38.4 & 38.5 & 40.6 & 50.5 + n79 & 0.40 & 0.30 & 0.24 & 38.1 & 38.2 & 40.4 & 50.2 + n105 & 0.20 & 0.15 & 0.12 & 38.1 & 38.2 & 40.3 & 50.1 + n119 & 0.20 & 0.15 & 0.12 & 38.5 & 38.5 & 40.7 & 50.5 + n144 & 0.35 & 0.27 & 0.19 & 38.4 & 38.4 & 40.6 & 50.4 + n157 & 0.76 & 0.59 & 0.40 & 39.5 & 39.7 & 41.8 & 51.7 + n160 & 0.57 & 0.44 & 0.31 & 38.9 & 39.0 & 41.1 & 51.0 + n180 & 0.36 & 0.28 & 0.19 & 38.0 & 38.1 & 40.2 & 50.1 + n191 & 0.18 & 0.13 & 0.12 & 37.0 & 37.0 & 39.2 & 49.0 + n206 & 0.30 & 0.23 & 0.17 & 38.5 & 38.5 & 40.7 & 50.5 + dem s74 & 0.16 & 0.12 & 0.09 & 37.1 & 37.1 & 39.3 & 49.1 + n13 & 0.25 & 0.19 & 0.14 & 37.0 & 37.1 & 39.2 & 49.0 + n17 & 0.21 & 0.16 & 0.12 & 37.1 & 37.2 & 39.3 & 49.1 + n19 & 0.25 & 0.19 & 0.14 & 36.7 & 36.8 & 38.9 & 48.8 + n22 & 0.27 & 0.21 & 0.14 & 37.0 & 37.1 & 39.2 & 49.1 + n36 & 0.24 & 0.18 & 0.14 & 37.8 & 37.9 & 40.0 & 49.9 + n50 & 0.19 & 0.14 & 0.12 & 37.8 & 37.8 & 39.9 & 49.8 + n51 & 0.15 & 0.12 & 0.08 & 36.8 & 36.8 & 39.0 & 48.8 + n63 & 0.22 & 0.17 & 0.12 & 37.0 & 37.0 & 39.1 & 49.0 + n66 & 0.08 & 0.06 & 0.05 & 38.6 & 38.6 & 40.8 & 50.6 + n71 & 0.11 & 0.09 & 0.05 & 36.2 & 36.3 & 38.4 & 48.2 + n76 & 0.09 & 0.07 & 0.05 & 38.0 & 38.0 & 40.2 & 50.0 + n78 & 0.13 & 0.10 & 0.07 & 37.7 & 37.7 & 39.9 & 49.7 + n80 & 0.16 & 0.12 & 0.09 & 37.4 & 37.5 & 39.6 & 49.4 + n84 & 0.32 & 0.24 & 0.19 & 38.2 & 38.2 & 40.4 & 50.2 + n90 & 0.19 & 0.14 & 0.12 & 37.5 & 37.5 & 39.7 & 49.5 + [ tab : extinction ] one issue related to the above estimates of @xmath36 is the star formation history . while both the h@xmath7 and bolometric luminosity of an actively star - forming region are dominated by massive stars with lifetimes @xmath53 myr , the bolometric luminosity also contains a non - negligible contribution from longer - lived stars . the implication is that the ratio of h@xmath7 to bolometric luminosity of a stellar population evolves with time . the relation @xmath38 is appropriate for a population with a continuous star formation history over 100 myr , but for a nearly coeval stellar population as in our star clusters , the h@xmath7 to bolometric ratio will start out somewhat larger than kennicutt & evans value , then decline below it over a timescale of @xmath54 myr . thus , depending on the age of the stellar population , @xmath36 can be either an underestimate or an overestimate . given that our stellar sources are bright h ii regions and thus the stars are likely to be young , the latter seems more likely . we also note uncertainty related to imf sampling . stellar populations with masses below @xmath55 @xmath56 do not fully sample the imf , and this makes the h@xmath7 to bolometric luminosity ratio vary stochastically @xcite . most of our clusters are near the edge of the stochastic regime . for a randomly selected cluster , the most common effect is to lower the h@xmath7 luminosity relative to the bolometric luminosity ; the expected magnitude of the effect is a factor of @xmath57 ( e.g. , figure 7 of @xcite ) . this will tend to make our @xmath36 an underestimate by this amount . however , the real effect is likely to be smaller , because our sample is not randomly selected . for a rare subset of clusters stochasticity has no effect or actually raises the h@xmath7 to bolometric ratio compared to that of a fully - sampled imf , and since our sample is partly selected based on h@xmath7 luminosity , it is biased in favor of the inclusion of such clusters . it is not possible to model this effect quantitatively without knowing both the underlying distribution of cluster masses and the selection function used to construct the sample . thus we restrict ourselves to noting that this effect probably introduces a factor of @xmath58 level uncertainty into @xmath36 . in the remainder of this paper , we will use @xmath59 to calculate @xmath32 . the pressure of the dust - processed radiation field @xmath60 is related to the energy density of the radiation field absorbed by the dust , @xmath61 ( i.e. , assuming a steady state ) , @xmath62 we follow the same procedure of @xcite to estimate the energy density @xmath61 of the radiation absorbed by the dust in our sample . specifically , we measure the flux densities @xmath63 in the irac and mips bands and compare them to the predictions of the dust models of @xcite ( hereafter dl07 ) . the dl07 models determine the ir spectral energy distribution for a given dust content and radiation field intensity heating the dust . dl07 assume a mixture of carbonaceous grains and amorphous silicate grains that have a size distribution that reproduces the wavelength - dependent extinction in the local milky way ( see @xcite ) . in particular , polycyclic aromatic hydrocarbons ( pahs ) contribute substantial flux at @xmath2319 @xmath6 m and are observed in normal and star - forming galaxies ( e.g. , @xcite ) . to account for the different spatial resolutions of the ir images , we convolved the 3.6 , 8 , and 24 @xmath6 m images with kernels to match the point - spread function of the 70 @xmath6 m image using the convolution kernels of @xcite . then , we measured the average flux densities @xmath63 at 8 , 24 , and 70 @xmath6 m wavelengths in the apertures listed in column 5 of table [ tab : sample ] . we removed the contribution of starlight to the 8 and 24 @xmath6 m fluxes using the 3.6 @xmath6 m flux densities and the empirical relations @xmath64 where @xmath65 is the non - stellar flux at the respective wavelengths . the coefficients 0.232 and 0.032 are given in @xcite . in figure [ fig : lmc_models ] , we plot the resulting ratios @xmath66 versus @xmath67 measured for the 32 h ii regions . additionally , we plot the @xcite predictions for given values of @xmath68 , the fraction of dust mass in pahs , and @xmath69 , the dimensionless scale factor of energy density @xmath61 of radiation absorbed by the dust , where @xmath70 here , @xmath71 is the energy density of the @xmath72 ev photons in the local ism , 8.65 @xmath21 10@xmath73 erg @xmath74 @xcite . the 32 h ii regions span a factor of @xmath220 in @xmath67 , with the smc h ii regions having systematically lower @xmath67 than the lmc h ii regions . the lmc and smc sources have a similar range of a factor of @xmath26 in @xmath66 . broadly , the data follow a similar arc - like trend in the ratios as we found in the spatially - resolved regions of 30 dor @xcite . errors in our flux ratios are @xmath202.8% from a @xmath202% uncertainty in the _ spitzer _ photometry . and scaling @xmath69 of the energy density of the dust - processed radiation field ( equation [ eq : u ] ) from @xcite . the black star denotes the values for 30 dor . we interpolate the grid of predicted flux ratios to obtain @xmath75 and @xmath69 for each region listed in table [ table : pirresults ] . ] we interpolate the @xmath69-@xmath75 grid using delaunay triangulation , a technique appropriate for a non - uniform grid , to find the @xmath69 and @xmath68 values for our regions . for the points that lay outside the grid , we translated them to @xmath67 within the grid . since the y - axis ratio largely determines @xmath69 , this adjustment does not affect the pressure calculation for those sources . figure [ fig : lmc_uvspah ] plots the interpolated values of @xmath69 versus @xmath68 ; we also print the results in table [ table : pirresults ] so individual sources can be identified . we find that the @xmath69 values of the lmc and smc h ii regions span a large range , with @xmath76856 ( corresponding to @xmath777.4@xmath78 erg @xmath74 ) , and several of the smc sources have @xmath79 . the pah fractions of the smc h ii regions ( with @xmath801% ) are suppressed relative to those of the lmc h ii regions ( with @xmath811% ) . the smaller pah fractions in the low metallicity smc are consistent with the results of @xcite , who find a deficiency of pahs in the smc based on observations with the _ spitzer _ infrared spectrograph . based on pah band ratios in the irs data , these authors suggest that this deficiency arises because smc pahs are smaller and more neutral than pahs in higher metallicity galaxies . versus pah fraction @xmath75 for the 16 lmc h ii regions ( black circles ) and 16 smc h ii regions ( open squares ) , as given by the interpolation of the grid in figure [ fig : lmc_models ] . the numerical values for the two parameters are given in table [ table : pirresults ] , and the black star denotes the values for 30 dor . ] lrrr n4 & 740 & 2.1 & 500 + n11 & 230 & 3.2 & 50 + n30 & 250 & 3.4 & 60 + n44 & 230 & 2.8 & 60 + n48 & 140 & @xmath824.6 & 50 + n55 & 200 & 2.6 & 50 + n59 & 400 & 1.9 & 120 + n79 & 320 & 2.0 & 80 + n105 & 340 & 2.2 & 130 + n119 & 200 & 3.0 & 60 + n144 & 270 & 2.3 & 70 + 30 dor & 860 & 1.0 & 250 + n160 & 380 & 2.1 & 120 + n180 & 230 & 2.1 & 120 + n191 & 500 & 1.9 & 50 + n206 & 140 & 3.4 & 50 + dem s74 & 40 & 0.9 & 30 + n13 & 280 & 0.7 & 260 + n17 & 120 & 0.8 & 70 + n19 & 140 & @xmath830.5 & 160 + n22 & 740 & @xmath830.5 & 160 + n36 & 80 & @xmath830.5 & 60 + n50 & 50 & 0.7 & 20 + n51 & 140 & 0.7 & 30 + n63 & 90 & 0.7 & 60 + n66 & 380 & @xmath830.5 & 100 + n71 & 240 & @xmath830.5 & 330 + n76 & 130 & 0.6 & 70 + n78 & 570 & @xmath830.5 & 70 + n80 & 90 & 0.6 & 50 + n84 & 160 & 0.6 & 30 + n90 & 110 & 0.6 & 50 + [ table : pirresults ] finally , we employ the interpolated @xmath69 values and equations [ eq : pir ] and [ eq : u ] to estimate the dust - processed radiation pressure @xmath60 in our 32 sources . the warm ionized gas pressure is given by the ideal gas law , @xmath84 , where @xmath85 , @xmath86 , and @xmath87 are the electron , hydrogen , and helium number densities , respectively , and @xmath88 is temperature of the hii gas , which we assume to be the same for electrons and ions . if helium is singly ionized , then @xmath89 . if we adopt the temperature @xmath88 = 10@xmath90 k , then the warm gas pressure is determined by the electron number density @xmath91 . one way to estimate @xmath91 is via fine - structure line ratios in the ir ( e.g. , @xcite ) : since these lines have smaller excitation potentials than optical lines , they depend less on temperature and depend sensitively on the density @xcite . here , we estimate @xmath91 using an alternative means : by measuring the average flux density @xmath63 at 3.5 cm , where free - free emission dominates in h ii regions . for free - free emission , @xmath91 is given by eq . 5.14b of @xcite : @xmath92 where @xmath93 is the gaunt factor and @xmath14 is the distance to the sources , and @xmath34 is the volume of the sources . if we set the gaunt factor @xmath94 , we derive the densities @xmath91 listed in table [ table : pirresults ] . we find both the lmc and smc h ii regions have moderate densities , with @xmath95 22500 @xmath74 . the hot gas pressure is also given by an ideal gas law : @xmath96 , where @xmath97 is the electron density and @xmath98 is the temperature of the x - ray emitting gas . the factor of 1.9 is derived assuming that he is doubly ionized and the he mass fraction is 0.3 . furthermore , we assume that the electrons and ions have reached equipartition , so that a single temperature describes both populations . to estimate @xmath97 and @xmath98 , we model the bremsstrahlung emission at x - ray wavelengths of our sources using pointed _ rosat _ pspc observations ( for the lmc sources ) and _ chandra _ observations ( for n66 in the smc ) . the other h ii regions in the smc are undetected by _ xmm - newton _ and _ chandra _ , and we use these data to set upper limits on hot gas pressure in those targets . in the analyses described below , we assume a filling factor @xmath99 of the hot gas ( i.e. that the hot gas occupies the full volume of our sources ) . for the purposes of measuring the large - scale dynamical role of the hot gas , the appropriate quantity is the volume - averaged pressure . we explain in detail why this approach is critical when assessing global dynamics in appendix [ app : hot gas ] . for the _ rosat _ analyses of the lmc h ii regions , we used ftools , a software package for processing general and mission - specific fits data @xcite , and xselect , a command - line interface of ftools for analysis of x - ray astrophysical data . we produced x - ray images of the sources ( shown in blue in figure [ fig : lmcthreecolor ] ) , and we extracted spectra from within the radii given in table [ tab : sample ] as well as from background regions to subtract from the source spectra . appropriate response matrices ( files with probabilities that a photon of a given energy will produce an event in a given channel ) and ancillary response files ( which has information like effective area ) were downloaded for each observation s date and detector . resulting background - subtracted source spectra ( shown in figure [ fig : rosat15 ] ) were fit using xspec version 12.4.0 @xcite . spectra were modeled as an absorbed hot diffuse gas in collisional ionization equilibrium ( cie ) using the xspec components _ phabs _ and _ apec_. in these fits , we assumed a metallicity @xmath100 , the value measured in h ii regions in the lmc @xcite , and we adopted the solar abundances of @xcite . in some sources ( n11 , 30 dor , and n160 ) , we found the addition of a power - law component was necessary in order to account for excess flux at energies @xmath12 kev , a feature that is likely to be from non - thermal emission from supernova remnants or from point sources in the regions . llccll n4 & 1.6 & [email protected] & [email protected] & 34.1 & 13/9 + n11 & 1.9 & [email protected] & [email protected] & 36.3 & 100/99 + n30 & 1.9 & [email protected] & [email protected] & 34.6 & 20/52 + n44 & 6.0 & [email protected] & [email protected] & 37.0 & 156/107 + n48 & 4.7 & [email protected] & [email protected] & 35.6 & 135/123 + n55 & 1.2 & [email protected] & [email protected] & 34.4 & 34/53 + n59 & 1.6 & [email protected] & [email protected] & 35.6 & 19/54 + n79 & 1.6 & [email protected] & [email protected] & 35.1 & 47/47 + n105 & 2.1 & [email protected] & [email protected] & 35.6 & 68/74 + n119 & 2.1 & [email protected] & [email protected] & 35.9 & 181/109 + n144 & 2.0 & [email protected] & [email protected] & 36.0 & 166/115 + 30 dor & 3.0 & [email protected] & [email protected] & 36.8 & 204/165 + n160 & 8.1 & [email protected] & [email protected] & 34.8 & 62/40 + n180 & 2.5 & [email protected] & [email protected] & 35.2 & 11/31 + n191 & & & & & + n206 & 3.0 & [email protected] & [email protected] & 36.3 & 141/96 + n66 & 3.3 & [email protected] & [email protected] & 35.7 & 128/86 + [ table : pxresults ] lcccclc dem s74 & 5.06 & 0.0293 & 1.8@xmath102 & 4.6@xmath103 & 36.3 & 0.37 + n13 & 3.58 & 0.0013 & 8.7@xmath104 & 1.0@xmath105 & 33.6 & 0.69 + n17 & 3.33 & 0.0078 & 5.3@xmath105 & 5.2@xmath102 & 35.4 & 0.31 + n19 & 4.76 & 0.0026 & 1.6@xmath105 & 3.6@xmath102 & 35.2 & 0.76 + n22 & 4.44 & 0.0025 & 1.6@xmath105 & 3.0@xmath102 & 35.1 & 0.52 + n36 & 5.02 & 0.0241 & 1.5@xmath102 & 3.7@xmath103 & 36.2 & 0.41 + n50 & 4.86 & 0.0532 & 3.3@xmath102 & 7.7@xmath103 & 36.5 & 0.24 + n51 & 4.41 & 0.0137 & 8.7@xmath105 & 1.6@xmath103 & 35.9 & 0.39 + n63 & 4.60 & 0.0065 & 4.1@xmath105 & 8.3@xmath102 & 35.6 & 0.55 + n71 & 2.49 & 0.0002 & 1.1@xmath104 & 6.5@xmath104 & 33.5 & 0.70 + n76 & 3.45 & 0.1821 & 2.9@xmath103 & 3.1@xmath106 & 37.1 & 0.46 + n78 & 3.49 & 0.0853 & 1.3@xmath103 & 1.5@xmath106 & 36.8 & 0.41 + n80 & 3.48 & 0.0173 & 1.2@xmath102 & 1.3@xmath103 & 35.8 & 0.25 + n84 & 3.52 & 0.2549 & 1.7@xmath103 & 1.9@xmath106 & 36.9 & 0.23 + n90 & 2.10 & 0.0194 & 1.4@xmath102 & 6.4@xmath102 & 35.5 & 0.26 + [ table : pxupperlimits ] for the _ chandra _ analysis of n66 , we extracted a source spectrum using the ciao command _ specextract _ ; a background spectrum was obtained from a circular region of radius @xmath250 offset @xmath21 northeast of n66 . the resulting background - subtracted spectrum ( grouped to 25 counts per bin ) is shown in figure [ fig : n66spectrum ] . we first attempted to fit the spectrum with an absorbed hot diffuse gas in cie as above ( with xspec components _ phabs _ and _ apec _ ) assuming a @xmath107 metallicity plasma . the fit was statistically poor ( with reduced chi - squared values of @xmath108/d.o.f . @xmath109 317/90 ) , with the greatest residuals around emission line features . consequently , we considered an absorbed cie plasma with varying abundances ( with xspec components _ phabs _ and _ vapec _ ) . in this model , we let the abundances of elements in the spectrum ( o , ne , mg , si , and fe ) vary freely . the fit was dramatically improved ( with @xmath108/d.o.f . @xmath109 128/86 ) in this case . we found that the mg and fe abundances were consistent with those of the smc , while o , ne , and si had enhanced abundances of @xmath20.7 @xmath110 . the elevated metallicity of the hot plasma is suggestive that the x - ray emission is from a relatively young ( a few thousand years old ) supernova remnant ( snr ) , and the enhanced abundances are signatures of reverse shock - heated ejecta . a young snr in n66 has been identified previously as snr b0057@xmath8724 based on its non - thermal radio emission @xcite , its high - velocity h@xmath7 emission @xcite , and its far - ultraviolet absorption lines @xcite . the _ rosat _ and _ chandra _ x - ray spectral fit results are given in table [ table : pxresults ] , including the absorbing column density @xmath111 , the hot gas temperature @xmath112 , the hot gas electron density @xmath97 , their associated 90% confidence limits , and the reduced chi - squared for the fits , @xmath108/d.o.f . hot gas temperatures were generally low , with @xmath113 0.150.6 kev . comparing _ results for 30 dor to those from _ chandra _ in @xcite , we find that the integrated _ chandra _ spectral fits gave temperatures a factor of @xmath260% above those given by _ rosat_. this difference can be attributed to the fact that the _ rosat _ spectra were extracted from a much larger aperture than those from _ chandra_. broadly , the x - ray luminosity @xmath114 derived from our fits are consistent with previous x - ray studies of h ii regions in the lmc @xcite . for the smc h ii regions ( except n66 ) , we calculate upper limits on @xmath115 based on the non - detections of these sources in _ chandra _ ( for n76 and n78 ) and _ xmm - newton _ data . in particular , we measured the full - band count rates ( 0.58.0 kev ) within the aperture of our sources and converted these values to absorbed x - ray flux @xmath116 upper limits using webpimms , assuming the emission is from a @xmath107 metallicity plasma with @xmath117 kev . we then corrected for absorption to derive unabsorbed ( emitted ) x - ray fluxes @xmath118 , assuming an absorbing column equal to the weighted average @xmath111 in the source direction , given by the @xcite survey of galactic neutral hydrogen . finally , we simulated spectra of the @xmath107 , @xmath117 kev plasma to determine the emission measure @xmath119 ( and consequently , the electron density @xmath120 ) . the results of these analyses for the 15 smc h ii regions are listed in table [ table : pxupperlimits ] . each pressure term calculated using the methods described above will have an associated error , and there are many uncertainties which will contribute given the variety of data and analyses required . nonetheless , we attempt to assess these errors in the following ways . for the direct radiation pressure @xmath32 , the dominant uncertainty is the relation of @xmath37 to @xmath29 , as described in section [ sec : pdir ] . thus , for our error bars on @xmath32 have incorporated the factor of 2 uncertainty in the conversion of @xmath37 to @xmath29 . our calculation of @xmath60 is fairly robust , and the largest error comes from the 2% uncertainty in the _ spitzer _ photometry , which corresponds to a 2.8% error in the flux ratios of figure [ fig : lmc_models ] . therefore , we interpolated the @xmath69@xmath75 grid for @xmath1012.8% of our flux ratios to obtain a corresponding error in @xmath69 . these uncertainties lead to errors of the order 510% in @xmath60 . in the case of @xmath121 , we have uncertainty in the flux density @xmath63 over the radii of our h ii regions due to the low resolution of the radio data . therefore , we have measured @xmath63 for @xmath101one resolution element in our radio image and obtained the corresponding uncertainty in @xmath91 . this error is relatively small , @xmath21015% in @xmath91 and @xmath121 . finally , the range of @xmath115 is given by the uncertainty in the x - ray spectral fits of emission measure ( and correspondingly , the hot gas density @xmath97 ) and of the temperature @xmath112 . we employ these 90% confidence limits derived in our spectral fits , as listed in table [ table : pxresults ] . generally , the density @xmath97 was poorly constrained in lower signal sources ( e.g. , n4 , n30 , and n59 ) , as further evidenced by the poor reduced chi - squared values in those fits . therefore , in some cases , the error bars on @xmath115 can be relatively large , although the typical uncertainties were around @xmath23050% in @xmath97 . following the multi - wavelength analyses performed above , we calculate the pressure associated with the direct stellar radiation pressure @xmath32 , the dust - processed radiation pressure @xmath60 , the warm ionized gas pressure @xmath121 , and the hot x - ray gas pressure @xmath115 . table [ tab : presults ] gives the pressure components and associated errors measured for all the h ii regions , and figure [ fig : pdirvsp ] plots the pressure terms versus their sum , @xmath122 , to facilitate visual comparison of the parameters . as shown in figure [ fig : pvsr ] , we do not find any trends in the pressure terms versus size @xmath35 of the h ii regions . in all the targets except one , @xmath121 dominates over @xmath60 and @xmath115 . the exception is n191 , which has a @xmath60 roughly equal to its @xmath121 , although the errors on @xmath60 are quite large . for all sources detected in the x - rays except n30 , @xmath121 is a factor 27 above @xmath115 and @xmath123 in all sources . broadly , the relation between the terms is @xmath124 . in the entire sample , @xmath32 is 12 orders of magnitude smaller than the other pressure components . we note that while @xmath125 at distances @xmath075 pc from r136 in the giant h ii region 30 doradus @xcite , the warm ionized gas is what is driving the expansion currently and dominates the energetics when averaged over the entire source . lcccc n4 & 18.2@xmath126 & 2.13@xmath127 & 13.8@xmath128 & [email protected] + n11 & 5.08@xmath129 & 0.66@xmath130 & 1.38@xmath131 & [email protected] + n30 & 3.31@xmath132 & 0.72@xmath133 & 1.51@xmath134 & [email protected] + n44 & 4.21@xmath135 & 0.65@xmath136 & 1.69@xmath137 & [email protected] + n48 & 1.57@xmath138 & 0.40@xmath139 & 1.33@xmath137 & [email protected] + n55 & 4.41@xmath140 & 0.58@xmath141 & 1.28@xmath137 & [email protected] + n59 & 11.4@xmath142 & 1.15@xmath143 & 3.35@xmath144 & [email protected] + n79 & 4.96@xmath145 & 0.94@xmath146 & 2.25@xmath130 & [email protected] + n105 & 9.34@xmath147 & 0.99@xmath139 & 3.63@xmath148 & [email protected] + n119 & 5.24@xmath149 & 0.57@xmath137 & 1.62@xmath150 & [email protected] + n144 & 6.18@xmath151 & 0.78@xmath152 & 1.97@xmath153 & [email protected] + 30 dor & 55.7@xmath154 & 2.47@xmath155 & 6.99@xmath144 & [email protected] + n160 & 21.1@xmath156 & 1.10@xmath157 & 3.32@xmath158 & [email protected] + n180 & 9.03@xmath159 & 0.67@xmath152 & 3.21@xmath160 & [email protected] + n191 & 1.34@xmath161 & 1.43@xmath162 & 1.43@xmath131 & + n206 & 3.26@xmath163 & 0.41@xmath164 & 1.28@xmath137 & [email protected] + dem s74 & 0.67@xmath165 & 0.11@xmath131 & 0.69@xmath166 & @xmath830.88 + n13 & 16.9@xmath167 & 0.81@xmath134 & 7.28@xmath168 & @xmath831.65 + n17 & 2.37@xmath169 & 0.33@xmath150 & 2.00@xmath170 & @xmath830.75 + n19 & 4.67@xmath171 & 0.40@xmath130 & 4.40@xmath172 & @xmath831.82 + n22 & 5.78@xmath173 & 2.12@xmath174 & 4.31@xmath175 & @xmath831.25 + n36 & 4.34@xmath176 & 0.22@xmath150 & 1.63@xmath134 & @xmath830.99 + n50 & 1.25@xmath177 & 0.15@xmath131 & 0.63@xmath131 & @xmath830.58 + n51 & 0.71@xmath178 & 0.39@xmath179 & 0.87@xmath131 & @xmath830.94 + n63 & 2.20@xmath180 & 0.26@xmath137 & 1.57@xmath181 & @xmath831.31 + n66 & 12.1@xmath182 & 1.10@xmath183 & 2.92@xmath139 & [email protected] + n71 & 16.6@xmath184 & 0.68@xmath152 & 9.16@xmath185 & @xmath831.69 + n76 & 4.10@xmath186 & 0.38@xmath179 & 2.01@xmath143 & @xmath831.10 + n78 & 3.02@xmath187 & 1.66@xmath188 & 1.96@xmath152 & @xmath830.98 + n80 & 2.21@xmath189 & 0.26@xmath150 & 1.27@xmath179 & @xmath830.60 + n84 & 2.01@xmath190 & 0.47@xmath179 & 0.91@xmath131 & @xmath830.55 + n90 & 4.25@xmath191 & 0.33@xmath179 & 1.47@xmath192 & @xmath830.62 + [ tab : presults ] for the 32 h ii regions . dashed lines are meant to show how much each term contributes to the total pressure . the light blue arrows represent the @xmath115 upper limits of the 15 smc h ii regions that are not detected in archival _ xmm - newton _ and _ chandra _ data ; for our calculation of @xmath122 , we assume the smc @xmath115 upper limits are the pressures of the hot gas . section [ sec : uncertainty ] describes how error bars were calculated for each term . ] of the 32 h ii regions . the light blue arrows represent the @xmath115 upper limits of the 15 smc h ii regions that are not detected in archival _ xmm - newton _ and _ chandra _ data . see section [ sec : uncertainty ] for how error bars were assessed for each term . ] from section [ sec : results ] , it is evident that direct radiation pressure does not play a significant role in the dynamics of the regions . however , given the age and size of our sources , they are too large / evolved for the radiation pressure to be significant . the reason is that the pressure terms have a different radial dependence : @xmath193 , while @xmath194 , where @xmath195 is the shell radius . one can obtain a rough estimate of the characteristic radius @xmath196 where a given source transitions from radiation - pressure driven to gas - pressure driven by setting the total radiation pressure ( i.e. , the direct radiation as well as the dust - processed radiation ) equal to the warm gas pressure and solving for @xmath196 . in this case , we find @xmath197 where @xmath198 ev , the photon energy necessary to ionize hydrogen , @xmath199 is the case - b recombination coefficient , and @xmath200 is a dimensionless quantity which accounts for dust absorption of ionizing photons and for free electrons from elements besides hydrogen . in a gas - pressure dominated h ii region , @xmath200 = 0.73 if he is singly ionized and 27% of photons are absorbed by dust @xcite . the @xmath201 represents the factor by which radiation pressure is enhanced by trapping energy in the shell through several mechanisms , including trapping of stellar winds , infrared photons , and ly@xmath7 photons . here , we adopt @xmath202 , as in @xcite , although we note this factor is uncertain and debated , as discussed in section [ sec : dusty ] . lastly , @xmath203 is the ratio of bolometric power to the ionizing power in a cluster ; we set @xmath204 using the @xmath205 and the @xmath206 relations of @xcite . using these values , the above equation reduces to @xmath207 where @xmath9 is the ionizing photon rate , and @xmath208 s@xmath16 . we note that the derivation of equations [ eq : rch ] and [ eq : rch_s49 ] required several simplifying assumptions ( e.g. , regarding the coupling of the radiation to dust ) , and thus the estimate of @xmath196 should be viewed as a rough approximation of the true radius when an h ii region transitions from radiation- to gas - pressure dominated . we can estimate @xmath209 for our h ii regions based on their h@xmath7 luminosity @xcite : @xmath210 we list the resulting ionizing photon rates @xmath9 for our sample in table [ tab : extinction ] . given these values , we find a range @xmath211 0.017 pc for 31 h ii regions and @xmath212 33 pc for 30 dor . as our sample have radii @xmath210150 pc , the 32 h ii regions are much too large to be radiation - pressure dominated at this stage . this result demonstrates the need to investigate young , small h ii regions to probe radiation pressure dominated sources . the best candidates would be hypercompact ( hc ) h ii regions , which are characterized by their very small radii @xmath00.05 pc and high electron densities @xmath213 @xmath74 @xcite . hc h ii regions may represent the earliest evolutionary phase of massive stars when they first begin to emit lyman continuum radiation , and thus they offer the means to explore the dynamics before the thermal pressure of the ionized gas dominates . giant h ii regions which are powered by more massive star clusters may also be radiation pressure dominated . for example , @xcite showed that the super star clusters ( with masses @xmath214 ) in the starburst galaxy m82 are likely radiation pressure dominated . in section [ sec : results ] , we have demonstrated that the average x - ray gas pressure @xmath115 is below the @xmath215 k gas pressure @xmath121 . for the x - ray detected h ii regions , the median @xmath216 is 0.22 , with a range in @xmath217 0.130.50 ( excluding n30 , which has @xmath218 ) . for the 15 non - detected sources , we set upper limits on @xmath115 requiring at least 13 of the 15 h ii regions to have @xmath219 and nine to have @xmath220 . the low @xmath115 values we derive are likely due to the partial / incomplete confinement of the hot gas by the h ii shells ( e.g. , @xcite ) . if completely confined by an h ii shell expanding into a uniform density ism , the hot gas pressure @xmath115 would be large @xcite . conversely , a freely expanding wind would produce a negligible @xmath115 @xcite . in the intermediate case , a wind bubble expands into an inhomogeneous ism , creating holes in the shell where the hot gas can escape and generating a moderate @xmath115 . for example , @xcite argue the carina nebula is experiencing hot gas leakage based partly on its observed x - ray gas pressure of @xmath221 dyn @xmath222 , whereas the complete confinement model predicts @xmath223 dyn @xmath222 and the freely expanding wind model predicts @xmath224 dyn @xmath222 for carina . recent observational and theoretical evidence has emerged that hot gas leakage may be a common phenomenon . simulations have demonstrated that hot gas leakage can be significant through low - density pores in molecular material @xcite . observationally , signatures of hot gas leakage in individual h ii regions has been noted based on their x - ray luminosities and morphologies , such as in m17 and the rosette nebula @xcite , the carina nebula @xcite , and 30 dor @xcite . the results we have presented here on a large sample demonstrate that hot gas leakage may be typical among evolved h ii regions , implying that the mechanical energy injected by winds and sne can be lost easily without doing work on the shells . although we have found that the warm gas pressure @xmath121 dominates at the shells of our sources , a couple h ii regions ( n191 in the lmc and n78 in the smc , although we caution that the uncertainty in @xmath60 in n191 is large ) have nearly comparable @xmath60 and @xmath121 , and all 32 sources have @xmath225 . physically , this scenario can occur if the shell is optically thick to the dust - processed ir photons , amplifying the exerted force of those photons . in all 32 regions of our sample , the amplification factor caused by trapping the photons @xmath226 is quite large , with @xmath227 4100 and a median value of @xmath227 10 . from a theoretical perspective , it has been debated in the literature how much momentum can be deposited in matter by ir photons . @xcite argued that the imparted momentum would be limited to @xmath228 a few because holes in the shell would cause the radiation to leak out of those pores . conversely , if every photon is absorbed many times , then all the energy of the radiation field is converted to kinetic energy of the gas ; this scenario imparts the most momentum to the shell . an intermediate case is in optically thick systems , where photons are absorbed at least once , and the momentum deposition is dependent on the optical depth @xmath229 of the region @xcite . recent simulations by @xcite indicate that @xmath230 can be large as long as the radiation flux is below a critical value that depends on the dust optical depth . this critical value corresponds to the radiation flux being large enough so that the pressure of the dust - trapped radiation field is at the same order of magnitude as the gas pressure . at fluxes above the critical value , a radiation - driven rayleigh - taylor ( rrt ) instability develops and severely limits the value of @xmath230 by creating low - density channels through which radiation can escape . for example , in one case in @xcite where the rrt instability does not develop , they obtain @xmath231 , whereas when the radiation flux is increased so that radiation forces become significant and there is instability , @xmath230 drops to a few . clearly in the case of our sources , we are in the regime where the radiation pressure is not dominant compared to the warm gas pressure , and rrt instability is not expected ( though two of our sources are near the threshold of instability ) . thus , the high values of @xmath230 we obtain are consistent with these models . in this paper , we have performed a systematic , multi wavelength analysis of 32 h ii regions in the magellanic clouds to assess the role of stellar feedback in their dynamics . we have employed optical , ir , radio , and x - ray images to measure the pressures associated with direct stellar radiation , dust - processed radiation , warm ionized gas , and hot x - ray emitting plasma at the shells of these sources . we have found that the warm ionized gas dominates over the other terms in all sources , although two h ii regions have comparable dust - processed components . the hot gas pressures are relatively weaker , and the direct radiation pressures are 12 orders of magnitude below the other terms . we explore three implications to this work . first , we emphasize that younger , smaller h ii regions , such as hypercompact h ii regions , should be studied to probe the role of direct radiation pressure and the hot gas at early times . secondly , the low x - ray luminosities and pressures we derive indicate the hot gas is only partially confined in all of our sources , suggesting that hot gas leakage is a common phenomenon in evolved h ii regions . finally , we have demonstrated that the dust - processed component can be significant and comparable to warm gas pressure , even if the direct radiation pressure is comparatively less . these observational results are consistent with recent numerical work showing that the dust - processed component can be largely amplified as long as it does not drive winds . support for this work was provided by national aeronautics and space administration through chandra award number go213003a and through smithsonian astrophysical observatory contract sv373016 to mit and ucsc issued by the chandra x - ray observatory center , which is operated by the smithsonian astrophysical observatory for and on behalf of nasa under contract nas803060 . support for lal was provided by nasa through the einstein fellowship program , grant pf1120085 , and the mit pappalardo fellowship in physics . mrk acknowledges the alfred p. sloan foundation , nsf career grant ast0955300 , and nasa atp grant nnx13ab84 g . adb acknowledges partial support from a research corporation for science advancement cottrell scholar award and the nsf career grant ast0955836 . err acknowledges support from the david and lucile packard foundation and nsf grant ast0847563 . dc acknowledges support for this work provided by nasa through the smithsonian astrophysical observatory contract sv373016 to mit for support of the chandra x - ray center , which is operated by the smithsonian astrophysical observatory for and on behalf of nasa under contract nas803060 . the conversion of emission measure @xmath119 to hot gas electron density @xmath97 requires an assumption about the volume occupied by the hot gas , parametrized by a filling factor @xmath232 . for a fixed gas temperature @xmath112 ( which is determined from the spectral fitting and is independent of the assumed @xmath232 ) , the inferred density and pressure scale as @xmath233 . one can attempt to deduce @xmath232 from a combination of morphology and spectral modeling ( as in e.g. , @xcite ) . however , for the purposes of understanding the global dynamics , this approach can be misleading , as we demonstrate here . following the reasoning outlined below , we set @xmath234 . we are interested in the global dynamics of the regions , which are described by the virial theorem . neglecting magnetic fields ( which may not be negligible , but we lack an easy means to measure them ) , the eulerian form of the virial theorem is @xcite : here , @xmath34 is the volume , @xmath9 is the surface of this volume , @xmath237 , @xmath238 , and @xmath239 are the gas density , velocity , and pressure , @xmath240 is the fluid pressure tensor , @xmath241 is the frequency - integrated radiation energy density , @xmath242 is the radiation pressure tensor , @xmath200 is the gravitational potential , and @xmath243 is the identity tensor . the terms @xmath244 , @xmath245 , @xmath246 , and @xmath247 may be identified , respectively , as the moment of inertia , the total thermal plus kinetic energy , the total radiation energy , and the gravitational binding energy . the terms subscripted with @xmath248 represent external forces exerted at the surface of the volume , and are likely negligible in comparison with the internal terms for an h ii region with large energy input by massive stars . since manifestly @xmath249 either is very positive now , or was in the recent past ( otherwise the shell would not have expanded ) , the goal of this work is to understand the balance between the various positive terms on the right - hand side of the equation . the terms @xmath60 and @xmath32 are simply two different parts of @xmath246 , corresponding to energy stored in different parts of the electromagnetic spectrum , while @xmath121 and @xmath115 are part of @xmath245 . writing out the virial theorem in this manner makes the importance of the filling factor clear . the term we are interested in evaluating is the kinetic plus thermal energy of the x - ray emitting gas , where we have dropped the @xmath251 term on the assumption that the flow velocity is subsonic with respect to the hot gas sound speed , and in the second step we have defined the volume - averaged pressure @xmath252 , as distinct from the local pressure at a given point . the quantity @xmath252 can be understood as the partial pressure of the hot gas , including proper averaging down for whatever volumes it does not occupy . thus we see that the quantity of interest is _ not _ the local number density or pressure of the hot gas , it is the volume - averaged or partial pressure . now recall that , for fixed @xmath98 and fixed observed emission measure , local pressure scales with filling factor as @xmath253 , so a small volume filling factor increases @xmath115 . however , since the volume occupied by the hot gas scales as @xmath254 , it follows that @xmath255 , i.e. , a small volume filling factor implies that the hot gas is less , not more , important for the large - scale dynamics . this analysis has two important implications . first , the choice that makes the hot gas as dynamically - important as possible is to set @xmath256 , i.e. , to assume that the hot gas fills most of the available volume . in this case we simply have @xmath257 , and this is the choice we make in this work . a detailed assessment of @xmath232 that gives a value @xmath258 , as performed by @xcite , can imply an even smaller dynamical role for the hot gas , but not a larger one ( although understanding of filling factors is important for other considerations , such as the internal dynamics of h ii regions ) . the second implication is that it is inconsistent to treat @xmath115 as the quantity of interest for the global dynamics while simultaneously adopting @xmath259 . once can certainly attempt to measure @xmath254 and thus obtain a more accurate assessment of @xmath115 , but in this case the quantities that should be compared with other pressures is @xmath260 , _ not _ @xmath115 . the volume - averaged pressure is the relevant quantity for global dynamics , not the local pressure . we note that the above discussion of the filling factor applies to the warm gas as well , and we have also assumed a filling factor of order unity for the warm 10@xmath90 k gas . , c. l. , krumholz , m. r. , ballesteros - paredes , j. , et al.et al . 2013 , arxiv : 1312.3223 , protostars and planets vi , ed . h. beuther , r. s. klessen , c. p. dullemond , & t. henning ( university of arizona press ) , in press

stellar feedback is often cited as the biggest uncertainty in galaxy formation models today . this uncertainty stems from a dearth of observational constraints as well as the great dynamic range between the small scales ( @xmath01 pc ) where the feedback originates and the large scales of galaxies ( @xmath11 kpc ) that are shaped by this feedback . to bridge this divide , in this paper we aim to assess observationally the role of stellar feedback at the intermediate scales of h ii regions ( @xmath210100 pc ) . in particular , we employ multiwavelength data to examine several stellar feedback mechanisms in a sample of 32 h ii regions ( with ages @xmath2310 myr ) in the large and small magellanic clouds ( lmc and smc , respectively ) . using optical , infrared , radio , and x - ray images , we measure the pressures exerted on the shells from the direct stellar radiation , the dust - processed radiation , the warm ionized gas , and the hot x - ray emitting gas . we find that the warm ionized gas dominates over the other terms in all of the sources , although two have comparable dust - processed radiation pressures to their warm gas pressures . the hot gas pressures are comparatively weak , while the direct radiation pressures are 12 orders of magnitude below the other terms . we discuss the implications of these results , particularly highlighting evidence for hot gas leakage from the h ii shells and regarding the momentum deposition from the dust - processed radiation to the warm gas . furthermore , we emphasize that similar observational work should be done on very young h ii regions to test whether direct radiation pressure and hot gas can drive the dynamics at early times .

You are an expert at summarizing long articles. Proceed to summarize the following text: deploying more cellular base stations ( bs ) has been the main remedy to cope with relentless traffic growth . to reach the highest capacities , interest has lately been turning toward ultra - dense bs deployment @xcite , @xcite , where the density of bs may even exceed the typical number of users in a given area . the impact of such extreme bs densification , however , has still not been explicitly analyzed . to be more specific , in an engineering perspective , the preceding work @xcite provides an analytic average spectral efficiency ( se ) calculation that reveals the se is independent of bs density . the result is accurate only when bs density is low as it relies on an assumption that every bs has at least a single serving user . for densely deployed bs environment , the authors @xcite consider turned - off bss when having no serving users as in the third generation partnership project ( 3gpp ) release 12 specifications @xcite , and predict the se is a logarithmic function of bs density via its compatible performance metrics , service capacity , outage probability , and common ( worst user s ) rate respectively . nevertheless , they can not represent the explicit relationship between the se and bs density due to the intractable forms of results . motivated by these discussions on the bs densification , we derive an analytic se expression valid for general bs density by utilizing a stochastic geometric approach @xcite , and further provide its closed - form representation under ultra - dense bs environment . the result verifies _ se logarithmically increases with bs density _ in ultra - dense networks where the bs densification makes interference restricted by turning off empty bss . in an economic perspective , the authors @xcite consider a network operator s profit maximization based on user demand prediction in a single cell scenario neglecting the bs densification effect . for a multi - cell network , the work @xcite deals with cost minimization when bs density is low . regarding the relationship between user demand and network supply , the authors @xcite provide an analytic approach although it resorts to an iterated simulation . this study , leading from the preceding works , focuses on the question : how much amount of bs density and spectrum a network operator should invest in when user demand changes ? to capture the user demand variation , we consider two user demand characteristics : ( i ) user density and ( ii ) each user s sensitivity to his downloading rate . we thereby answer the question via solving a profit maximization problem while considering the user demand meeting network s rate supply . this paper examines the ramifications of bs density increase in downlink cellular networks from both engineering and economic points of view in terms of se and profit respectively . the main contributions are listed below . 1 . this paper derives the closed - form se in an ultra - dense downlink cellular network that is a logarithmic function of bs density . 2 . the paper provides network planning guidelines in terms of the profit optimal bs density and spectrum amount in closed forms . we consider a downlink cellular network . let @xmath0 denote bs coordinates in a two - dimensional euclidean plane , following homogeneous poisson point process with density @xmath1 . similarly , user coordinates @xmath2 follow homogeneous poisson point process with density @xmath3 , independent of @xmath0 . users are associated with their nearest bss , which correspondingly forms bs coverage regions whose boundaries comprise a two - dimensional voronoi tessellation @xcite . each bs is tuned off when its coverage , a voronoi cell , is empty of serving users , otherwise transmitting with unity power . for a transmitted signal , we consider path loss attenuation with the exponent @xmath4 and rayleigh fading . , @xmath5 , and @xmath6 where user density @xmath7 . compared to the exact se when @xmath8 , the analytic value achieves : 81.75% , 90.9% , and 97.96% for bs density @xmath9 , @xmath10 , and @xmath11 respectively.,width=328 ] this section aims at providing a closed - form se , to be utilized for profit maximization in section [ sect : profitmax ] . to this end , we consider interference - limited regime neglecting noise , and define se @xmath12 as ergodic rate per user for unity spectrum amount , @xmath13 $ ] in units of nats / sec / hz ( 1bit @xmath14 0.693 nats ) where @xmath15 denotes signal - to - interference ratio . for brevity , we neglect multiple user access , to be considered in section [ sect : cap_approx_ma ] . according to proposition 1 in @xcite , a bs s turned - on ( non - empty ) probability @xmath16 is given as @xmath17 . applying this to the equation ( 16 ) in @xcite provides @xmath12 as : @xmath18^{-1}\hspace{-7pt } dt \label{eq : exact}\ ] ] where @xmath19 . + for a sparse ( @xmath20 ) cellular network when @xmath21 , the equation is in accord with the result in @xcite . for explanatory convenience , let @xmath22 hereafter denote the se without multiple access in a sparse network , given as @xmath23^{-1 } dt . \label{eq : sparse}\ ] ] this shows @xmath22 is independent of @xmath1 . it implies , in other words , bs densification does not yield any gain in se under sparse environment since its desired signal power improvement is on the same order of amount as the aggregate interference increase , cancelled out each other at calculating @xmath15 . for an ultra - dense ( @xmath24 ) network when @xmath25 by using taylor expansion , the equation hardly captures the relationship between @xmath12 and @xmath1 due to the complicated double integrations therein . we take a detour this problem by deriving the closed - form approximation of @xmath12 in the following proposition . interestingly , this simple expression shows that @xmath12 is a logarithmic function of @xmath1 . in other words , _ bs densification does increase the se whilst yielding diminishing returns _ under ultra - dense environment . a stochastic geometric point of view interprets this phenomenon as follows . at a typical user , increasing @xmath1 shrinks each bs s coverage , simultaneously yielding : ( i ) the shortened distance to his associated bs and ( ii ) increased the number of empty cells ( or turned - off bss ) . focusing firstly on the former , the shortened distance in the order @xmath26 ( see the average distance is @xmath27 in @xcite ) yields the received signal power increase from the associated bs , in the order @xmath28 . for the latter in an ultra - dense scenario , it makes almost all bss turned - off except for the ones serving users in their infinitesimal coverage regions . consequently , the interfering bs locations ( or interferer density ) converge to the users ( or @xmath3 ) , delimiting the quantity of interference . combining these results leads to the ever - increasing @xmath15 in the order @xmath29 , resulting in the logarithmic se increase . [ fig : capapprox ] visually validates the tightness of the value in proposition 1 for different @xmath30 s . when @xmath8 , for instance , the difference between and is less than @xmath31 for @xmath32 . thus , we hereafter regard as the approximation of @xmath33 . additionally , it is worth mentioning that this result shows the bs ultra - densification gain in se . comparing to @xmath22 in a sparse scenario , ultra - densification provides the se gain : 162% , 250% , and 464% for @xmath9 , @xmath10 , and @xmath11 respectively when @xmath7 . now we turn our attention to multiple access of users with a fixed amount of spectrum . so far we have considered a bs serves all users in its coverage . instead , we henceforth consider each bs serves at most a single user at a given time , who is selected according to a uniformly random scheduler @xcite . according to proposition 2 in @xcite , a typical user s selection probability by the scheduler for a sparse network is @xmath34 , and for an ultra - dense network is @xmath11 . applying these results to the equations and yields the following corollary . _ _ ( se with multiple access ) _ se with a uniformly random scheduler in a sparse or ultra - dense downlink cellular network is given as follows . \label{eq : udnwma}\end{aligned}\ ] ] _ in a sparse scenario , increasing @xmath1 alleviates multiple access congestion , and thereby linearly increases se in spite of a constant @xmath22 independent of @xmath1 in . in an ultra - dense scenario , on the other hand , reducing the access congestion along with bs densification does not ameliorate se . that is because multiple access of users barely occurs in the ultra dense network where almost all bss have at most a single user within their coverages . this results in being equivalent to . the following section utilizes these results so as to explore the economic impact of bs densification . in this section , our objective is to maximize a network operator s profit when user demand varies , caused by the changes in the number of users @xmath3 and/or each user s rate sensitivity @xmath36 . the operator is able to cope with these demand changes by adjusting his operating bs density @xmath1 and spectrum amount @xmath37 as well as price per nats / sec ( or per nat for unity time ) @xmath38 . since the profit is a joint function of user demand , price , average rate , and its operating cost , the operator s profit maximizing decision problem to clarify the adjustments in @xmath1 , @xmath37 , and @xmath38 is not trivial , being of our interest . to be more specific , we firstly predict average per - user demand @xmath39 , and then determine optimal @xmath1 and @xmath37 so as to maximize the average profit per unit area , formulated as : @xmath40 where @xmath41 and @xmath42 respectively denote bs and unit spectrum operating costs per unit area . in an operator s perspective , we divide this profit maximization problem into three sequential stages : stage 1 . user demand prediction ; stage 2 . price decision ; and stage 3 . the decision on bs density and spectrum amount , resulting in average rate to be supplied . [ fig:3stages ] elucidates these subdivided problems and their solving direction as well as the direction of network operation for given solutions of the problem . for the purpose of predicting user demand at stage 1 , consider a typical user s payoff @xmath43 having the following characteristics : logarithmically increasing with downloading rate @xcite , and linearly decreasing with cost under usage - based pricing @xcite . let @xmath44 denote average rate per user , and define @xmath45 as a user s willingness - to - pay , assumed to be uniformly distributed from @xmath46 to @xmath36 , @xmath47 . we hereafter interpret the maximum willingness - to - pay , @xmath36 , as the user s rate sensitivity . correspondingly , we represent @xmath43 as : @xmath48^+.\ ] ] consider users try to maximize their payoffs . since @xmath43 is a concave function of @xmath44 , applying the first order necessary condition and taking average over @xmath45 yield the payoff maximizing average rate per user ( or average demand per user ) @xmath39 as follows . @xmath49 [ fig : approx_lowden ] [ fig : approx_highden ] applying the result to the profit function in ` ( p1 ) ` reveals the profit decreases with @xmath38 . this intuitively indicates attracting more users by reducing @xmath38 yields higher profit than increasing @xmath38 . since @xmath39 is also a decreasing function of @xmath38 , the profit maximizing price @xmath50 at stage 2 is the price when the equality holds at the constraint in ` ( p1 ) ` , resulting in @xmath51 \nonumber \\ & \overset{(a)}{\approx } & \frac{b}{2(1 + w \gamma ) } \label{eq : optprice}\end{aligned}\ ] ] where ( a ) follows from taylor expansion for large @xmath52 . this subsection aims at deriving optimal bs density @xmath53 and spectrum amount @xmath54 in closed forms so that their resultant average rate is provided to a typical user while maximizing profit . exploiting @xmath50 , the equation in section [ sect : demandmodel ] , modifies ` ( p1 ) ` for stage 3 as follows . @xmath55 for sparse cellular networks , applying to @xmath12 of ` ( p2 ) ` yields the following profit maximization problem . @xmath56 for ultra - dense networks , in the same manner , applying to @xmath12 in ` ( p2 ) ` leads to the following profit maximization problem . \r)^{-1 } \r\}^{-1}\\ & & \hspace{10pt } - \ ( c_b \lambda_b + c_w w \ ) \end{aligned}\ ] ] solving ` ( p3 ) ` and ` ( p4 ) ` provides profit maximizing @xmath53 and @xmath54 of sparse and ultra - dense networks respectively , as provided in the following proposition . ( optimal bs density and spectrum amount ) we interpret the above results in the following perspectives : 1 ) unit operating costs @xmath41 and @xmath42 and the resultant profit optimal operating costs @xmath58 and @xmath59 and 2 ) user demand comprising per - user rate sensitivity @xmath36 and user density @xmath3 . increasing unit bs operating cost @xmath41 leads to investing more in spectrum as the bs substitute , captured by increasing @xmath54 , and vice versa for unit spectrum operating cost @xmath42 increase . multiplying these unit operating costs by and yields the following profit optimal operating bs / spectrum cost ratio . _ _ ( optimal cost ratio ) _ profit maximizing ratio of bs and spectrum operating costs in a sparse or ultra - dense downlink cellular network is given as : @xmath60 _ in a sparse network , the operator should invest in bs operating cost as much as the spectrum cost since bs density and spectrum amount equally affect average rate ( see with the spectrum amount @xmath37 , shown in ` ( p3 ) ` ) . in an ultra - dense network , on the other hand , bs density less affects average rate than spectrum amount due to the densification s logarithmic impact on average rate . this leads to the investment strategy that bs operating cost should be less than the spectrum cost , depending on @xmath30 ( note @xmath61 ) . the straight lines on the bs density - and - spectrum planes ( bottom ) in fig . 3 illustrate such operating cost ratios . both profit optimal bs density and spectrum amount increase as user demand grows , but the optimal value increments incurred by user density are higher than the values by per - user rate sensitivity ( see the exponents of @xmath3 and b in and , visualized in fig . while rate sensitivity growth solely increases user demand , user density growth not only increases the demand but also decreases average rate due to : ( i ) incurring more multiple access congestion in a sparse network or ( ii ) generating more interference in an ultra - dense network ( see the discussion after proposition 1 in section [ sect : cap_approx ] ) . consequently , increasing user density requires more bs density and/or spectrum amount in order to keep up with the demand growth as well as to recover the average rate decline . such different impacts of user density and per - user rate sensitivity furthermore affect the network profitability . for user density growth , fig . 4(a ) shows the profit of an ultra - dense network increases less than that of a sparse network . an ultra - dense network requires much more bs density and spectrum amount to compensate the interference generation caused by user density growth , and it engenders too much cost increase worsening the network profitability . for this type user demand increase , bs ultra - densification is not preferable . for per - user rate sensitivity growth , in contrast , fig . 4(b ) depicts the profit of an ultra - dense network increases more than that of a sparse network thanks to the network s delimited interference , promoting ultra - dense bs deployment . in this paper we have derived a closed - form relationship between bs density and se in an ultra - dense cellular network . the se is shown to be a logarithmic function of bs density as the density grows . this closed - form se expression was used to derive closed - form solutions for the optimal operating bs density and spectrum amount , for the traditional spectrum licensing ( auction ) case . this expression could aid the operator in his decision if he should invest more in bs densification or bid for more spectrum for his network . our results reveal some fundamentally unique characteristics of the ultra - dense network deployment , e.g. that the number of users has a larger impact on the optimal network configuration than each user s sensitivity to his downloading rate . further , in our simplified model we see that the network operator should maintain a reasonable balance in investment , allocating about equal amounts to spectrum and bs deployment . to the best of our knowledge , this paper is the first to simultaneously specify not only random spatial locations of both users and bss but also each user s demand model , bridging the gap between stochastic geometric and network economic analysis . a weakness of the study is the very simple , homogeneous propagation model , i.e. using a constant path loss exponent . the bss are typically in the same room as the users in line - of - sight ( los ) conditions , whereas the interfering bss are behind walls , i.e. in non - los conditions , creating a better sir . further work should therefore involve a two - slope model , describing the los and non - los cases and giving a more direct relationship to the physical environment . further extension to this work could also include milimeter - wave systems where non - los signals are very weak . in addition , it does not seem likely that future short range systems will use traditionally licensed spectrum . extending the economic analysis to spectrum sharing paradigms is another interesting avenue for future research . this research was supported by the international research & development program of the national research foundation of korea ( nrf ) funded by the ministry of education , science and technology ( mest ) of korea ( grant number : 2012k1a3a1a26034281 ) . let @xmath62 denote @xmath63 . since @xmath64 , @xmath12 is lower bounded as : @xmath65^{-1 } dt \nonumber \\ & \overset{(a)}{\geq } & \int_{t>0 } \ [ 1-a ( e^t - 1)^{\frac{2}{\alpha } } \]^+ dt\\ & = & \log\ ( 1 + a^{-\frac{\alpha}{2}}\ ) + a \pi \csc\ ( \frac{2 \pi}{\alpha } \ ) \nonumber\\ & & \hspace{-5pt } -\frac{\alpha } { 2\ ( 1 + a^{\frac{\alpha}{2}}\ ) } \underbrace { \ , _ 2f_1\left(1,1;1-\frac{2}{\alpha};1-\frac{1}{a^{\frac{\alpha}{2}}+1}\right)}_{(b ) } \label{eq : prop1pf}\end{aligned}\ ] ] where @xmath66 follows from taylor expansion , gaussian hypergeometric function @xmath67 , and @xmath68 rising factorial . the function @xmath69 monotonically increases with @xmath62 for all @xmath30 , having unity minimum value at @xmath70 . for @xmath71 ( or @xmath24 ) , therefore all terms in except @xmath72 become negligible , completing the proof . @xmath73 consider a sparse network . for sufficiently large average rate ( or small @xmath74 ) , applying taylor expansion to the objective function in ` ( p3 ) ` leads to the following problem . @xmath75 the profit function is concave with respect to both @xmath1 and @xmath37 , so it has a unique maximum result . exploiting the first order necessary condition yields @xmath76^{\frac{1}{2 } } \lambda_u \quad \text{and } \label{eq : pfprop3densb}\\ w^*&=&\ [ \frac{b}{2 \gamma_\alpha c_w \lambda_b^ * } \]^{\frac{1}{2}}\lambda_u . \label{eq : pfprop3w}\end{aligned}\ ] ] applying to proves the result . next , consider an ultra - dense network . since the logarithmic function in ` ( p4 ) ` is not appropriate for deriving solutions in closed forms , we resort to considering its lower bound @xmath77 as the approximation that is tight for large @xmath78 ( or @xmath24 ) . applying taylor expansion provides the formulation as shown below . @xmath79 in the same way as the sparse network , exploiting the first order necessary condition leads to @xmath80^{\frac{1}{\frac{\alpha}{4}+1 } } \lambda_u \quad \text{and } \label{eq : pfprop3densbudn}\\ w^*&=&\ [ \frac{b { \lambda_u}^{\frac{\alpha}{4}+1 } } { 2 c_w}\ ( \frac{\rho_0}{{\lambda_b}^*}\)^{\frac{\alpha}{4 } } \]^{\frac{1}{2}}. \label{eq : pfprop3wudn}\end{aligned}\ ] ] applying to finalizes the proof . @xmath73 l. duan , j. huang , and b. shou , `` investment and pricing with spectrum uncertainty : a cognitive operator s perspective , '' _ ieee transactions on mobile computing _ , vol . 10 , no . 11 , pp . 1590 1604 , 2011 . s. sen , c. joe - wong , s. ha , and m. chiang , `` incentivizing time - shifting of data : a survey of time - dependent pricing for internet access , '' _ ieee communications magazine _ , vol . 50 , no . 11 , pp . 9199 , 2012 .

this paper investigates the relationship between base station ( bs ) density and average spectral efficiency ( se ) in the downlink of a cellular network . this relationship has been well known for sparse deployment , i.e. when the number of bss is small compared to the number of users . in this case the se is independent of bs density . as bs density grows , on the other hand , it has previously been shown that increasing the bs density increases the se , but no tractable form for the se - bs density relationship has yet been derived . in this paper we derive such a closed - form result that reveals the se is asymptotically a logarithmic function of bs density as the density grows . further , we study the impact of this result on the network operator s profit when user demand varies , and derive the profit maximizing bs density and the optimal amount of spectrum to be utilized in closed forms . in addition , we provide deployment planning guidelines that will aid the operator in his decision if he should invest in densifying his network or in acquiring more spectrum . ultra - dense cellular network , base station density , average spectral efficiency , spectrum amount , profit maximization , stochastic geometry , network economics .

You are an expert at summarizing long articles. Proceed to summarize the following text: it is currently believed that pulsars are among the few objects in our galaxy that are candidate sources of ultrarelativistic charged cosmic rays . relativistic particles within the magnetosphere emit @xmath0-rays at energies up to several gev in various processes such as curvature radiation , synchrotron radiation and inverse compton ( ic ) scattering . thus , observations in the multi - gev @xmath0-ray domain allow one to study the acceleration sites in the magnetosphere of a pulsar . predicted sites where particle acceleration can take place are , for example , above the polar cap of the neutron star ( e.g. * * ; * ? ? ? * ) and in the so - called outer gap of the magnetosphere ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? furthermore , particle acceleration can take place outside the magnetosphere in the region where the pulsar wind interacts with the interstellar medium . if electrons are accelerated in these shocks , they could give rise to ic - scattered photons from , for example , the cosmic microwave background , synchrotron radiation , or thermal origin . psr b1951 + 32 was detected first at radio frequencies by @xcite , and is one of the six rotation - powered high energy pulsars whose gev emission was detected by egret @xcite . among @xmath0-ray pulsars , psr b1951 + 32 is the only source observed to emit up to @xmath8gev with no cutoff being evident in the differential energy spectrum . the spectrum shows a hard spectral index of 1.8 between @xmath9mev and @xmath8gev . the pulsar has an apparent high efficiency ( @xmath10 ) of converting its rate of rotational energy loss , @xmath11ergs@xmath12s@xmath5 , into @xmath0-rays above @xmath9mev ( assuming a distance of @xmath13kpc to the pulsar ) . moreover , the @xmath0-ray luminosity at @xmath14gev is comparable to that of the crab pulsar @xcite . as inferred from its rotational parameters , the spin - down age of psr b1951 + 32 is @xmath15yr @xcite , that is , about 100 times older than the crab pulsar . the magnetic field strength of @xmath16 g @xcite is lower than that in most rotation - powered pulsars . because of the lower magnetic field , curvature @xmath0-rays emitted near the stellar surface , as predicted in polar - cap models , are less affected by magnetic pair production . compared with younger , more strongly magnetized pulsars , the spectral cutoff energy is thereby shifted to higher energies , up to a few tens of gev ( @xcite ; see also @xcite for a discussion of low - field millisecond pulsars ) . on the contrary , if the @xmath0-rays are emitted in the outer magnetosphere , as predicted in outer gap models , the potential drop in the outer gap of psr b1951 + 32 is expected to be comparable to that of young pulsars ( see eq . [ 12 ] of @xcite and eq . [ 2.1 ] of @xcite ) . therefore , the cutoff energy , which reflects the maximum lorentz factor of the electrons or positrons accelerated in the outer gap , is expected to be around 10gev @xcite . thus , features in the predicted spectral shape of weakly magnetized pulsars at energies above 10gev are strongly dependent on the emission altitudes . in order to discriminate between emission models , psr b1951 + 32 is a prime candidate for observation by ground - based @xmath0-ray detectors with low energy thresholds such as the imaging air cherenkov telescope magic . this pulsar is located in the core of the radio nebula ctb 80 , which is thought to be physically associated with the pulsar . in x - rays the nebula shows a cometary shape @xcite , being confined by a bow shock that is produced by the pulsar s high proper motion ( @xmath17 km s@xmath5 ) @xcite . @xcite predict an over-@xmath18gev flux from the nebula at a level of @xmath19 of the crab s flux , by assuming that high - energy leptons can accumulate inside the well - localized nebula for long periods of time , as observed in the case of the crab nebula . the current tightest constraint on the emission above @xmath9gev from the pulsar and its nebula , obtained by the whipple collaboration @xcite , puts an upper limit of @xmath2gev on the cutoff energy of the pulsed emission and an upper limit of @xmath20@xmath4s@xmath5 , on the steady emission above @xmath21gev . the latter is within a factor @xmath22 of the prediction of @xcite . in polar - cap models , the cutoff energy is determined by the attenuation of @xmath0-rays due to magnetic pair production and hence by the emission altitude of @xmath0-rays . as a consequence , the energy spectrum above the cutoff energy is superexponentially attenuated . if the emission altitude in the polar cap model shown in figure [ 1951_pulsar ] changes from 1 to 2 stellar radii , the cutoff energy will increase from 20gev to 60gev , which is , according to our observations , the maximum allowed cutoff energy for a superexponentially shaped cutoff . on the contrary , in outer - gap models , the cutoff is determined by the maximum lorentz factor of the accelerated positrons and electrons . as a consequence , the cutoff of the @xmath0-ray spectrum is smoother , resulting in an exponential cutoff . if the magnetic field lines near the light cylinder are straighter than assumed for the outer - gap spectrum in figure [ 1951_pulsar ] , the predicted flux below 60gev will increase . for more precise predictions of the cutoff energy in polar - cap models , multidimensional and self - consistent electrodynamics have to be examined from first principles , whereas a three - dimensional magnetic field configuration has to be investigated in the outer - gap model . assuming that these improvements in theory will be achieved in the near future , measurements with higher statistics around @xmath23gev , for example , by glast , or measurements by future ground - based experiments with lower thresholds than magic , for example , magic ii or the cherenkov telescope array , will be needed in order to distinguish between models . the predicted ic flux at tev energies in the outer - gap model ( figure [ 1951_pulsar ] _ solid black line _ ) appears to be inconsistent with our upper limits . nevertheless , it must be noted that the ic flux is obtained by assuming that all the magnetospheric soft photons illuminate the equatorial region of the magnetosphere in which the gap - accelerated positrons are migrating outwards . therefore , the predicted ic flux as a function of energy specifies an upper boundary to the possible pulsed tev emission . the open poloidal magnetic field lines could have a single - signed curvature within 1.8 light - cylinder radii , as the solution of the time - dependent force - free electrodynamics of an oblique rotator indicates @xcite . if this is the case , soft photons emitted inside the light cylinder along the convex magnetic field lines will not efficiently illuminate the magnetic field lines , which are slightly convex even outside the light cylinder . as a result , the predicted ic flux at tev energies will be significantly reduced . this problem will be solved in future when the self - consistent gap electrodynamics @xcite and the three - dimensional force - free electrodynamics are combined . we are grateful for the preparation of the ephemeris of psr b1951 + 32 by andrew lyne , thus enabling us to perform the pulsed analysis . alice harding was so kind as to provide us with her polar - cap predictions of psr b1951 + 32 . we also would like to thank the instituto de astrofisica de canarias for the excellent working conditions at the observatorio del roque de los muchachos , in la palma . the support of the german bundesministerium fr bildung und forschung and the max - planck - gesellschaft , the italian istituto nazionale de fisica nucleare , the spanish comisin interministerial de ciencias y tecnologa , eth research grant th 34/04 3 , and grant 1p03d01028 from the polish ministerstwo nauki i informatyzacji are gratefully acknowledged .

we report on very high energy @xmath0-observations with the magic telescope of the pulsar psr b1951 + 32 and its associated nebula , ctb 80 . our data constrain the cutoff energy of the pulsar to be less than @xmath1gev , assuming the pulsed @xmath0-ray emission to be exponentially cut off . the upper limit on the flux of pulsed @xmath0-ray emission above @xmath2gev is @xmath3photons @xmath4 sec@xmath5 , and the upper limit on the flux of steady emission above @xmath6gev is @xmath7photons @xmath4 sec@xmath5 . we discuss our results in the framework of recent model predictions and other studies .

You are an expert at summarizing long articles. Proceed to summarize the following text: the galaxy luminosity function ( hereafter lf ) provides a @xmath14 order description of the gross properties of galaxy populations . although it may be regarded as somewhat simplistic , in reducing galaxy properties to the two numbers ( @xmath15 and @xmath16 ) that describe the lf in the @xcite form , reproducing the observed lf is a fundamental test for any viable theory of galaxy formation , and one that has proven surprisingly complex to achieve until relatively recently ( e.g. , @xcite ) . clusters of galaxies are , in several ways , ideal laboratories to study the lf and its evolution . cluster galaxies may be considered as a volume limited sample of objects observed at the same cosmic epoch and occupy a relatively ` constant ' environment corresponding to the highest density peaks at the redshift of observation . operationally , cluster members have a much higher surface density on the sky than the surrounding field and often exhibit distinctive morphologies and colors ( see fig . 1 below ) . it is therefore possible to establish membership in a cluster ( at least in a statistical sense ) without resorting to observationally expensive redshift surveys . on the other hand , clusters are obviously special places , containing only a small percentage ( @xmath17@xmath18 ) of all galaxies in the universe . their galaxy populations clearly implicate the cluster environment in a variety of processes that affect the morphological evolution of galaxies ( e.g. , @xcite ) and suppress or modify their star formation history @xcite . in order to use clusters as probes of galaxy evolution it is therefore important to understand the effects of environment and carry out careful comparisons with field samples . for local samples , this is now possible thanks to large redshift surveys such as 2df @xcite and sdss @xcite . although bright galaxies appear to have formed the majority of their stellar populations and assembled their mass at least by @xmath19 ( e.g. , @xcite and references therein ) , there is considerable evidence that fainter galaxies undergo significant evolution since @xmath20 . the red sequence in clusters in the ediscs sample appears to be truncated at faint magnitudes @xcite , while @xcite show that the luminosity function of red cluster galaxies flattens at higher redshifts , as do @xcite for clusters in a deep _ spitzer _ field . on the other hand , there are clusters with well - formed red sequences at redshifts approaching @xmath21 as well @xcite . it may not be surprising that the faint end of the lf shows considerable variations from cluster to cluster , as one expects that dwarfs are more sensitive to environmental effects . although the bright end of the lf ( @xmath22 ) is broadly universal , the faint end varies between clusters and may even change radially within clusters @xcite . the evolution of the faint end slope of the total lf provides an interesting test of galaxy formation models . even if the red sequence is truncated at faint luminosities at earlier epochs , one expects a steepening faint end as established by the initial power spectrum of fluctuations , which is very steep @xcite . in higher redshift clusters , there should be a steeper lf and a larger fraction of bluer dwarfs . the purpose of this paper is to explore the evolution of the lf to very faint magnitudes in a sample of @xmath11 clusters observed with the hubble space telescope ( hst ) , reaching well below the @xmath10 point and into the domain of dwarf spheroidal galaxies , using a multicolor imaging sample . this will allow to test scenarios for the evolution of dwarf galaxy populations in @xmath23cdm hierarchical models . the structure of this paper is as follows : we describe the data and photometry in the next section . we derive the composite lfs and then discuss the results in the context of galaxy formation models . we adopt the ` consensus ' cosmology with @xmath24 , @xmath25 and use h@xmath26=100 km s@xmath27 mpc@xmath27 . all data are photometrically calibrated to the ab system , using the latest acs zeropoints published on the hst instrument web page . observations for this paper consist of deep hst imaging of six clusters at @xmath6 taken with the advanced camera for surveys ( acs ) in the @xmath0 ( f435w ) , @xmath1 ( f475w ) , @xmath2 ( f555w ) , @xmath3 ( f625w ) , @xmath4 ( f775w ) and @xmath5 ( f850lp ) bands with exposure times of 5,00010,000 seconds in each band . each cluster was observed in a single acs exposure , covering about @xmath28 on the sky . the data were originally taken for studies of gravitational lenses in these clusters . table 1 shows the clusters observed , their redshifts , exposure times in each band , and hst proposal ids . table 2 summarizes some essential physical characteristics for the observed clusters . most data are from the compilation of @xcite , from which @xmath29 can be estimated via the formula of @xcite , except for a1413 where @xmath29 was derived by @xcite from the x - ray profile . we were unable to find data in the literature for a1703 . the data were retrieved from the hst archive as fully reduced and flatfielded files ( * .flt ) and then processed through the multidrizzle algorithm @xcite to produce fully registered and co - added images for each band . 1 shows color composites for our data . full resolution jpegs will be made available on the @xmath30 web site . detection of objects and photometry were carried out using sextractor @xcite . we experimented with sextractor parameters to maximize our detections and minimize the number of spurious objects . all detections were visually examined to eliminate noise spikes , bleed trails from bright stars and other contaminants , especially the numerous arclets present in the images . the sextractor parameters eventually employed are shown in table 3 . we used the same parameters for all images . for each object we compute a total ( kron - like ) magnitude and an aperture magnitude of area equivalent to the minimum number of connected pixels needed for detection . this provides an estimate of the central surface brightness for each object . the motivation behind this procedure is as follows . detections of objects in deep images depends on their total magnitude but also on their central surface brightness . an object could be brighter than the magnitude threshold but be lost in the night sky because of low surface brightness ( see discussion in @xcite ) . for this reason we need to determine both a limiting total magnitude for completeness and a surface brightness threshold for detection . we do this by plotting central surface brightness vs. total magnitude for all bands in fig . 2 . to save space we only show the data for abell 1703 , but equivalent figures for all other clusters are available on the @xmath30 web site . in these figures the sequence at high surface brightness that provides an upper limit to the scatter plot is caused by stars . we can therefore discriminate against stellar contamination in this way ( cf . the surface brightness threshold is chosen empirically . the limit is selected to include as many objects as possible , but avoiding regions of the parameter space where the sample is obviously very incomplete . selection lines in surface brightness , apparent magnitude and the star - galaxy separation line are shown in fig . 2 . in order to determine a limiting magnitude , we plot the raw number counts for galaxies in the abell 1703 field in fig . 3 . as for fig . 2 , figures for all the other clusters are made available electronically . the completeness magnitude is chosen to lie about 0.5 mag . brighter than the luminosity at which counts start to decrease , in order to select a highly complete sample of objects . the only way to establish cluster membership for faint galaxies is by statistical background subtraction . in order to do so , we need to analyze ` blank ' ( cluster - less ) comparison fields of similar or greater photometric depth . we choose to use the two goods fields @xcite as these are the deepest and widest fields where hst multicolor photometry is available . we used the same sextractor parameters as we used for the cluster fields and impose the same selection limits on the goods fields as we did for the target fields . 4 shows the detection and selection plots for the goods fields ( similar to fig . 2 above ) . the fields are much deeper than the cluster fields we use . it must be noted that we plot only @xmath18 of the detections in the goods fields in fig . 4 to avoid saturating the figure . fig . 5 plots galaxy number counts for the goods fields and comparison galaxy number counts from the literature . the hst data were corrected to the sdss and johnson - cousins systems using the transformations tabulated in @xcite . goods number counts were fitted with a quadratic of the form @xmath31 . the fit was carried out over galaxies brighter than the completeness limit shown in fig . 4 and fainter than m@xmath32 mag . as goods counts for brighter galaxies are affected by small scale clustering . table 4 shows the values of the coefficients for the quadratic fits . we now subtract the scaled contribution from fore / background counts from galaxy counts . errors in galaxy counts for clusters and the goods fields are assumed to be poissonian . we used our smoothed quadratic for the background counts , extrapolating for galaxies brighter than @xmath33 mag . field counts at these bright limits and over the small ( @xmath34 arcmin@xmath35 ) covered by acs are very small in any case . clustering errors over the cluster and goods fields are estimated following the counts - in - cells approach and the fitting formula described by @xcite . it may be possible to further refine the selection of cluster members using photometric redshifts or color cuts . contamination of the lf by background galaxies has been suggested as a possible cause of the faint end upturn ( see below ) by @xcite . however , this would lead to loss of generality . the argument by @xcite may be correct for nearby clusters ( although @xcite point out that the simulations used by @xcite are unrealistic and that the background subtraction approach used here works as long as proper care is taken of statistical uncertainties , see their 3.3.6 ) , but it is not appropriate for these distant objects where the redshift is much larger than the typical scale of structure in the universe ( so that the counts should be smooth ) . there are some mismatches between the goods data ( used for background subtraction ) and the cluster data . for the @xmath1 ( f435w ) and @xmath3 ( f625w ) bands , we can use the goods f475w and f606w counts ( respectively ) , which are close matches to the bands used , however , for f555w ( @xmath2 ) we used the average of the f475w and f606w counts . the lf in this bands is more uncertain , because we do not have an appropriate background field . we @xmath36 and @xmath37 correct the lfs to @xmath38 for ease of comparison with local data . in order to do so we use a @xcite model for a solar metallicity elliptical formed at @xmath39 with an e - folding time of 1 gyr . this is appropriate to the red ellipticals that dominate the bright end of cluster lfs . in effect this approach allows us to consider the differential evolution , if any , between the dwarfs and the giants , who appear to consist mainly of old galaxies with little or no recent star formation . the lf of each individual cluster is relatively uncertain . we therefore build composite lfs in each band following the method by @xcite in order to average out errors . the method assumes that the lf is broadly universal , but this appears to be borne out by observations at low redshift , at least for relatively bright galaxies @xcite . composite lfs are built as follows : the number of cluster galaxies in the @xmath40 magnitude bin of the composite lf is given by : @xmath41 where @xmath42 is the ( background corrected ) number of galaxies in the @xmath40 magnitude bin of the lf of the @xmath43 cluster lf , @xmath44 is the normalization of the cluster lf ( to take care of the different richnesses of the clusters ) and is taken to be the background corrected number of cluster galaxies brighter than @xmath45 ( or its equivalent in the other bands , assuming the color of an old elliptical ) , @xmath46 is the number of @xmath4 clusters contributing to the @xmath40 magnitude bin of the composite lf and @xmath47 is the sum of all the @xmath4 normalizations . @xmath48 essentially , this carried out a weighted average of the individual cluster lfs , scaled by the number of galaxies ( in each cluster ) brighter than approximately the @xmath10 point . the formal errors on @xmath49 are computed as : @xmath50^{1/2}\ ] ] where @xmath51 are the errors on the number counts ( including poissonian and clustering contributions ) for the @xmath40 magnitude bin of the @xmath43 cluster lf . for each cluster we build the composite lf by summing up the individual lfs in absolute magnitude bins , after correcting for the distance modulus , extinction and @xmath52 correction to the common redshift of @xmath38 . the resulting lfs are presented in fig . 6 , together with the 1 , 2 and 3@xmath53 error countours . we fit the data to a single schechter function ( see discussion below , regarding the existence of a faint end upturn ) . in some cases we exclude the fainter and/or brighter magnitude bin from the fit , because these points are dominated by either the poorly sampled very massive galaxies or the fainter objects that suffer from complex incompleteness due to magnitude and surface brightness selection . the fit parameters are tabulated in table 5 , together with the @xmath54 errors for each parameter , marginalized over the remaining parameters . there are only a few deep cluster lfs having comparable depth to our own . in general , we are in good agreement with previous work , based on smaller and shallower samples @xcite . the @xmath15 point we measure gets steadily brighter with increasing wavelength , tracking the color of old , passively evolving stellar populations that are the main component of bright elliptical galaxies @xcite . the value we find is also consistent with the local sdss value by @xcite : this is not surprising , as it is well known that massive galaxies evolve passively since high redshift . the main interest of this investigation is in the evolution of the lf slope . the lf is well fitted by a single schechter function , with a slope @xmath55 which is essentially the same in all bands . this suggests that the cluster lf is dominated by galaxies on the red sequence to a luminosity of @xmath56 in all bands , and that therefore the red sequence continues at least to 6 magnitudes below the @xmath10 point even at @xmath11 . this can be clearly seen in fig . 7 where we show the @xmath57 vs. @xmath5 color - magnitude relation for abell 1703 . the relation can be clearly followed at least to @xmath58 with little scatter . a more detailed analysis of the color - magnitude relation is deferred to a future paper ( harsono & de propris 2008 , in preparation ) . the red sequence appears to be well established and contain the majority of cluster populations even at a lookback time of @xmath59 gyrs ( cf . @xcite for a similarly deep relation in local clusters ) . the slope is very similar to the local value for @xmath22 derived by @xcite and @xcite . this argues that even galaxies to @xmath56 have formed their stellar populations and assembled the majority of their mass at least at @xmath11 . we can therefore place a significant lower limit to the assembly epoch of galaxies down to 1/600@xmath60 of the mass of the milky way . given the evidence for truncation of the red sequence at @xmath61 @xcite , our data point to the @xmath62 interval as a crucial epoch to investigate the formation of the fainter galaxy populations . the other issue we wish to address is the faint end upturn . this has been a controversial subject , ever since its original discovery by @xcite and @xcite . there have been a series of claims and counterclaims regarding the faint end slope and the existence of an upward inflection , sometimes even in the same cluster . for instance , @xcite and @xcite observe a steep lf at the faint end for the coma cluster , while @xcite and @xcite quote much flatter slopes . in virgo , @xcite claim a slope as steep as @xmath63 , while @xcite derive @xmath64 . @xcite review the existence of the upturn and the shape of the faint end of the lf and conclude that there is a steep mass function . the observation of the upturn in two composite lfs , derived by two different groups @xcite , suggests that the upward inflection of the lf is real ( however , cf . , @xcite ) . unlike @xcite and @xcite we do not find a steepening of the lf at @xmath13 . however , we must consider that the lf upturn is more pronounced at large clustercentric radii , while our fields generally cover only the central region of each cluster . the size of the acs fields used in this study covers between 350 and 750 @xmath65 kpc on the side . in terms of @xmath66 ( the radius at which @xcite and @xcite normalize their lfs ) , the areas imaged by acs cover between 20% and 40% of the area of the cluster out to @xmath29 . we therefore only derive lfs for the cluster cores . this should not affect our comparisons with the bright end of the lf ( @xmath22 ) , as this does not seem to vary significantly with radius . however , the steep upturn claimed by @xcite and @xcite is much more pronounced at large clustercentric radii , i.e. , in the cluster outskirts . on the other hand it is possible to see the upturn even in the more central regions of the sample studied by @xcite , inside @xmath67 ( see fig . 10 of @xcite ) , so our data should exhibit an upturn at faint magnitudes , although of course more clusters would be welcome to bolster the argument . as we do not detect a steepening in the lf , we suggest that , if the steepening is real , the faint galaxies contributing to the upturn consist of a population of recently infalling objects ( e.g. , @xcite ) whose star formation is curtailed by the cluster environment . this would be consistent with the observation that about 1/2 of the fainter cluster dwarfs in virgo and elsewhere have been forming stars until relatively recently @xcite , until their star formation stopped and their colors moved to the red sequence . the newly infalling population may be the one now contributing to the steep upturn . in a ` downsizing ' model , it may be expected that objects undergoing star formation suppression and cluster infall at the present epoch would indeed tend to be among the fainter and less massive galaxies and to have the steep lf characteristic of ` pristine ' cdm power spectra . our observations imply that the passive evolution of bright galaxies can be extended to faint dwarfs , at least to @xmath11 and suggest that the majority of galaxy evolution may have taken place at surprisingly early epochs even for the least massive objects . all of the data presented in this paper were obtained from the multimission archive at the space telescope science institute ( mast ) . stsci is operated by the association of universities for research in astronomy , inc . , under nasa contract nas5 - 26555 . support for mast for non - hst data is provided by the nasa office of space science via grant nag5 - 7584 and by other grants and contracts . adami , c. , nichol , r. c. , mazure , a. , durret , f. , holden , b. & lobo , c. 1998 , , 334 , 765 adami , c. , picat , j. p. , durret , f. , mazure , a. , pell , r. & west , m. 2007 , , 472 , 749 andreon , s. , punzi , g. , grado , a. 2005 , , 360 , 727 andreon , s. 2006a , , 448 , 447 andreon , s. 2006b , , 369 , 969 andreon , s. , cuillandre , j .- c . , puddu , e. & mellier , y. 2006 , , 372 , 60 baldry , i. k. , glazebrook , k. & driver , s. p. 2008 , , 388 , 945 barkhouse , w. a. , yee , h. k. c. & lopez - cruz o. 2007 , , 671 , 1471 beijersbergen , m. , hoekstra , h. , van dokkum , p. g. & van der hulst , t. 2002 , , 329 , 385 bernstein , g. m. , nichol , r. c. , tyson , j. a. , ulmer , m. p. & wittman , d. 1995 , , 110 , 1507 bertin e. , arnouts s. 1996 , , 117 , 393 boyce , p. j. , phillipps , s. , jones , j. b. , driver , s. p. , smith , r. m. , & couch , w. j. 2001 , , 328 , 277 bruzual , g. & charlot , s. 2003 , , 344 , 1000 capak , p. et al . 2007 , , 172 , 99 carlberg , r. g. , yee , h. k. c. & ellingson , e. 1997 , , 478 , 462 crawford , s. m. , bershady , j. a. & hoessel , j. g. 2008 , astro.ph 0809.1661 colless , m. m. 1989 , , 237 , 799 colless , m. m. et al . 2001 , , 328 , 1039 conselice , c. j. , gallagher , j. s. & wyse , r. f. g. 2001 , , 559 . 791 conselice , c. j. , gallagher , j. s. & wyse , r. f. g. 2003 , , 125 , 66 cool , r. j. , eisenstein , d. j. , scranton , r. , brinkmann , j. , schneider , d. p. & zehavi , i. 2006 , , 131 , 736 cross , n. j. g. & driver , s. p. 2002 , , 329 , 579 croton , d. et al . 2006 , , 365 , 11 de lucia , g. et al . 2007 , , 374 , 809 de propris , r. , pritchet , c. j. , hartwick , f. d. a. & mcclure , r. d. 1995 , , 450 , 534 de propris , r. , eisenhardt , p. r. , stanford , s. a. & dickinson , m. 1998 , , 503 , l45 de propris , r. , stanford , s. a. , eisenhardt , p. r. , holden , b. p. & rosati , p. 2007 , , 133 , 2207 dressler , a. 1980 , , 236 , 351 driver , s. p. , phillipps , s. , davies , j. i. , morgan , i. , disney , m. j. 1994 , , 268 , 393 eisenhardt , p. r. , de propris , r. , gonzalez , a. h. , stanford , s. a. , wang , m. & dickinson , m. 2007 , , 169 , 225 garilli , b. , maccagni , d. & andreon , s. 1999 , , 353 , 479 giavalisco , m. et al . 2004 , , 600 , l93 gomez , p. l. et al . 2003 , , 584 , 210 harsono d. & de propris r. 2007 , , 380 , 1036 hilker , m. , mieske , s. & infante , l. 2003 , , 397 , l9 holberg , j. b. & bergeron p. 2006 , , 132 , 1221 huang j .- s . , cowie l. l. , gardner j. p. , hu e. m. , songaila a. , wainscoat r. j. 1997 , , 476 , 12 jenkins , l. p. , hornschemeier , a. e. , mobasher , b. , alexander , d. m. & bauer , f. e. 2007 , , 666 , 846 koekemoer , a. , fruchter , a. s. , hook , r. n. & hack , w. 2002 , in _ the 2002 hst calibration workshop : hubble after the installation of acs and the nicmos cooling system _ ed . by s. arribas , a. koekemoer , b. whitmore ( baltimore : space telescope science institute ) , p. 337 khochfar , s. , silk , j. , windhorst , r. a. & ryan , r. e. 2007 , , 668 , l115 krick , j. e. , surace , j. a. , thompson , d. , ashby , m. l. n. , hora , j. l. , gorjian , v. & yan , l. 2008 , astro - 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we present deep composite luminosity functions in @xmath0 , @xmath1 , @xmath2 , @xmath3 , @xmath4 and @xmath5 for six clusters at @xmath6 observed with the hubble space telescope advanced camera for surveys . the luminosity functions reach to absolute magnitude of @xmath7 mag . and are well fitted by a single schechter function with @xmath8 mag . and @xmath9 ( in all bands ) . the observations suggest that the galaxy luminosity function is dominated by objects on the red sequence to at least 6 mags . below the @xmath10 point . comparison with local data shows that the red sequence is well established at least at @xmath11 down to @xmath12 of the luminosity of the milky way and that galaxies down to the regime of dwarf spheroidals have been completely assembled in clusters at this redshift . we do not detect a steepening of the luminosity function at @xmath13 as is observed locally . if the faint end upturn is real , the steepening of the luminosity function must be due to a newly infalling population of faint dwarf galaxies .

You are an expert at summarizing long articles. Proceed to summarize the following text: the proximity effect between a ferromagnet and a superconductor has attracted much attention recently due to its potential applications in such areas of technology as magnetoelectronics @xcite or quantum computing @xcite . the proximity effect in ferromagnet - superconductor ( f - s ) structures is also important from the point of view of the scientific interest as it allows the study of the interplay between ferromagnetism and superconductivity @xcite . due to time reversal symmetry breaking the proximity effect in f - s structures leads to new phenomena , not observed in usual normal metal - superconductor ( n - s ) proximity systems . those are : oscillations of the superconducting transition temperature @xcite , density of states @xcite and superconducting pairing amplitude in f - s multilayers , josephson @xmath0-junction behavior in s - f - s heterostructures @xcite , a giant mutual proximity effect in s - f systems @xcite , spin valve @xcite or spontaneous currents in f - s bilayers @xcite . for review of the literature on those and related effects see @xcite . we have recently examined f - s proximity system and found that a spontaneously generated current flows on both sides of the interface @xcite . such a current , which flows in the ground state of the system is a hallmark of the fulde - ferrell - larkin - ovchinnikov ( fflo ) like state , originally predicted for a bulk superconductor in a magnetic exchange field acting on spins only @xcite . it is the purpose of the present paper to see what the effect on the fflo state will have a normal metal sandwiched between ferromagnet and superconductor ( f - n - s structure ) . will the spontaneous current still be generated ? if so , how will it be modified ? in the rest of the paper we show that the presence of the normal metal has nontrivial consequences on the generation of the spontaneous currents and on the fflo state in general . our model system is schematically shown in fig . [ fig1 ] . it consists of a ferromagnet ( layers @xmath1 ) , a normal ( paramagnetic ) region ( layers @xmath2 ) and a semi - infinite superconductor ( layers @xmath3 ) . the model hamiltonian is given by single orbital hubbard model @xmath4 c^+_{i\sigma } c_{j\sigma } + \frac{1}{2 } \sum_{i\sigma } u_i \hat n_{i\sigma } \hat n_{i -\sigma } , \label{hamilt}\end{aligned}\ ] ] where , in the presence of a vector potential , the nearest neighbor hopping integral is @xmath5 . the site energy levels @xmath6 are equal to @xmath7 in the ferromagnet and @xmath8 in the normal and superconducting region , and @xmath9 is the chemical potential . @xmath10 is the on - site electron - electron interaction , which is assumed to be negative on superconducting side and zero elsewhere , @xmath11 ( @xmath12 ) are the usual electron creation ( annihilation ) operators and @xmath13 is the electron number operator . we study the above model in the spin - polarized hartree - fock - gorkov ( sphfg ) approximation , assuming the landau gauge for the magnetic field @xmath14 , thus @xmath15 ( see fig . [ fig1 ] ) . in the following we assume periodicity of our model in the direction parallel to the interface and therefore we work in @xmath16 space in the @xmath17 direction but in real space in the @xmath18 direction . as usual , self - consistency is assured by the relations determining sc order parameter @xmath19 , current @xmath20 and the vector potential @xmath21 on each layer @xmath22 @xmath23 @xmath24 @xmath25 where @xmath26 is the @xmath27 nambu green function and @xmath28 the fermi distribution . equation ( [ vec_pot ] ) is a lattice version of the ampere s law @xmath29 . the above equations have to be solved self - consistently , using the method described in @xcite . figure [ fig2 ] shows the spatial dependence of the normalized superconducting pairing amplitude @xmath30 ( left panel ) and spontaneous current @xmath31 ( right panel ) for a number of normal metal @xmath32 layers ( @xmath33 ) . as in our previous work on f - s structures @xcite , the superconducting pairing amplitude shows clear oscillations in the ferromagnet . the period of the oscillations is related to the ferromagnetic coherence length @xmath34 , i.e. @xmath35 . no such effect is observed in the n region . one could expect that the inclusion of the normal metal suppresses the oscillations of @xmath36 in the f region . interestingly , the amplitude of the @xmath36 oscillations on the ferromagnetic side remains almost unchanged . only the phase of the oscillations can change . moreover , as for f - s bilayer @xcite , the vanishing of the pairing amplitude at @xmath37 is related to the crossing of the andreev bound states ( abs ) through the fermi energy of the system . furthermore , in such a situation , spontaneous current is generated @xcite . the current flowing produces a magnetic field , which splits this zero energy abs , thus lowering the total energy of the system . the energy of this splitting is given in our model by @xmath38 , where @xmath39 is the layer averaged vector potential . as we already mentioned , the zero energy abs is responsible for the generation of the spontaneous current . the typical distribution of such current , flowing parallel to the interface , is shown in the right panel of fig . the current flows in the negative @xmath17 direction on the superconducting side and in the positive direction in the whole @xmath40 region , giving a total current equal to zero , as should be in the true ground state . interestingly , the current in the ferromagnet shows also oscillatory behavior , however , with a different period than the pairing amplitude does . the oscillations of the current are related to the oscillations of the density of states at the fermi energy @xcite . on the other hand , in the n region , the distribution of the current is rather smooth , similarly to the pairing amplitude ( compare the solid and thin dotted lines in fig . [ fig2 ] ) . the state with spontaneous current is a true ground state , as it has lower energy than the state with no current . this can be seen in fig . [ fig3 ] , which shows the change in total energy of the system @xmath41 between the solution with spontaneous current and the one where the current is constrained to be zero is shown . another important finding is that the system can be periodically switched between the states with and without spontaneous current by increasing of the thickness of the normal metal region . for @xmath42 layers and @xmath43 the period is equal to six ( see fig . [ fig3 ] ) . all above the results show that inclusion of the normal metal slab between the ferromagnet and superconductor has non - trivial effects on the physics of the proximity induced fflo state . it can not be argued that the normal metal simply acts as interface transparency , as in our case the n slab is in the clean ( ballistic ) regime , and thus does not suppress the proximity effect . however , it leads to the suppression of the oscillatory behavior of the pairing amplitude , spontaneous current and the density of states at the fermi energy . moreover , it strongly modifies the positions of the abs , which leads to the periodical switching between the states with and without current . interface transparency also leads to such periodical switching of the current @xcite , however , at the same time it also kills the proximity effect , suppressing the current and changing the period of the pairing amplitude oscillations . no such effects are caused by the inclusion of the normal metal slab . perhaps the effect of the n slab would be more similar to the effect of reduced interface transparency if the normal metal was in the dirty regime . in conclusion we have studied the ground state properties of f - n - s proximity system . we have observed oscillatory behavior of the superconducting pairing amplitude in ferromagnet , but not in the normal region . similarly to f - s structures , we have found spontaneously generated currents flowing in the whole f and n regions and within a distance of a few @xmath44 on superconducting side . interestingly , the system can be switched periodically ( changing @xmath32 ) between the states with and without the spontaneous current . all this suggests possible realization of the proximity induced fflo state in f - n - s heterostructure . this work has been partially supported by the grant no . pbz - min-008/p03/2003 , and by the esf network programme aqd jj . g. e. w. bauer _ et al . _ , materials sci . b * 84 * , 31 ( 2001 ) ; l. r. tagirov , phys . rev . lett . * 83 * , 2058 ( 1999 ) . g. blatter _ et al . _ , b * 63 * , 174511 ( 2001 ) . n. f. berk , j. r. schrieffer , phys . * 17 * , 433 ( 1996 ) ; c. pfleiderer _ et al . _ , nature * 412 * , 58 ( 2001 ) . h. k. wong _ et al . _ , j. low temp . phys . * 63 * , 307 ( 1986 ) ; j. s. jiang _ et al . _ , . lett . * 74 * , 314 ( 1995 ) ; th . et al . _ , lett . * 77 * , 1857 ( 1996 ) . t. kontos _ et al . _ , * 86 * , 304 ( 2001 ) . a. i. buzdin _ et al . _ , jetp lett . * 35 * , 178 ( 1982 ) ; e. a. demler _ et al . _ , b * 55 * , 15174 ( 1997 ) ; e. vecino _ et al . _ , b * 64 * , 184502 ( 2001 ) ; k. halterman , o. t. valls , phys . rev . b * 65 * , 014509 ( 2002 ) . v. v. ryazanov _ et al . 86 * , 2427 ( 2001 ) ; s. m. frolov _ et al . _ , b * 70 * , 144505 ( 2001 ) . v. t. petrashov _ lett . * 83 * , 3281 ( 1999 ) . l. r. tagirov , phys . lett . * 83 * , 2058 ( 1999 ) . m. krawiec _ et al . _ , b * 66 * , 172505 ( 2002 ) ; physica c * 387 * , 7 ( 2003 ) ; eur . j. b * 32 * , 163 ( 2003 ) . m. krawiec _ et al . _ , b * 70 * , 134519 ( 2004 ) . b. l. gyrffy _ et al . _ , in _ physics of spin in solids : materials , methods and applications _ , s. halilov , kluwer academic publ . y. a. izyumov _ et al . _ , 45 * , 109 ( 2002 ) ; i. f. lyuksyutov , v. l. pokrovsky , adv . phys . * 54 * , 67 ( 2005 ) . a. i. buzdin , cond - mat/0505583 ; f. s. bergeret _ et al . _ , cond - mat/0506047 . p. fulde , r. a. ferrell , phys . rev . * 135 * , a550 ( 1964 ) ; a. i. larkin , y. n. ovchinnikov , sov . jetp * 20 * , 762 ( 1965 ) .

we discuss the ground state properties of the system composed of a normal metal sandwiched between ferromagnet and superconductor within a tight binding hubbard model . we have solved the spin - polarized hartree - fock - gorkov equations together with the maxwell s equation ( ampere s law ) and found a proximity induced fulde - ferrell - larkin - ovchinnikov ( fflo ) state in this system . here we show that the inclusion of the normal metal layer in between those subsystems does not necessarily lead to the suppression of the fflo phase . moreover , we have found that depending on the thickness of the normal metal slab the system can be switched periodically between the state with the spontaneous current flowing to that one with no current . all these effects can be explained in terms of the andreev bound states formed in such structures .

You are an expert at summarizing long articles. Proceed to summarize the following text: precise measurements of the lifetimes of charm meson weak decays are important for understanding qcd in both perturbative and nonperturbative regimes . for mesons a joint expansion in heavy quark effective theory and perturbative qcd parameters treated through the third order in the heavy quark mass shows a term including non - spectator w - annihilation as well as pauli interference . the resulting non - leptonic decay rate differences between w - exchange in @xmath9 and w - annihilation in @xmath8 produce lifetime differences of order 10 - 20 % @xcite . the @xmath8 lifetime @xcite was dominated by the measurements from e687 collaboration ( 0.475 @xmath11 0.020 @xmath11 0.007 ps ) @xcite . recently new precision measurements of the @xmath8 lifetime have been made by the e791 collaboration ( 0.518 @xmath11 0.014 @xmath11 0.007 ps ) @xcite and the cleo collaboration ( 486.3 @xmath11 15.0 @xmath12 fs ) @xcite . both groups have taken advantage of improved precision in the @xmath13 lifetime measurement to report new results for the @xmath14 lifetime ratio of @xmath15 @xcite and @xmath16 @xcite . their average is 7.4 @xmath17 from unity , emphasizing the large difference in w contributions to @xmath8 and @xmath9 decays . in this letter we report the results of a new measurement of the @xmath8 lifetime based on data from the hadroproduction experiment selex ( e781 ) at fermilab . the measurement is based on about 1000 fully reconstructed decays into @xmath18 from a sample of 15.3 @xmath19 hadronic triggers . the selex detector at fermilab is a 3-stage magnetic spectrometer . the negatively charged 600 gev/@xmath1 beam contains nearly equal fractions of @xmath20 and @xmath21 . the positive beam contains 92% protons . beam particles are identified by a transition radiation detector . the spectrometer was designed to study charm production in the forward hemisphere with good mass and decay vertex resolution for charm momenta in the range 100 - 500 gev/@xmath1 . five interaction targets ( 2 cu and 3 c ) had a total target thickess of 4.2% @xmath22 for protons . the targets are spaced by 1.5 cm . downstream of the targets are 20 silicon planes with a strip pitch of 20 - 25 @xmath23 m oriented in x , y , u and v views . the scattered - particle spectrometers have momentum cutoffs of 2.5 gev/@xmath1 and 15 gev/@xmath1 respectively . a ring - imaging cerenkov detector ( rich ) @xcite , filled with neon at room temperature and pressure , provides single track ring radius resolution of 1.4% and 2@xmath17 @xmath24 separation up to about 165 gev/@xmath1 . a layout of the spectrometer can be found elsewhere @xcite . the charm trigger is very loose . it requires a valid beam track , at least 4 charged secondaries in the forward 150 mrad cone , and two hodoscope hits after the second bending magnet from tracks of charge opposite to that of the beam . we triggered on about 1/3 of all inelastic interactions . a computational filter linked pwc tracks having momenta @xmath25 gev/@xmath1 to hits in the vertex silicon and made a full reconstruction of primary and secondary vertices in the event . events consistent with only a primary vertex are not saved . about 1/8 of all triggers are written to tape , for a final sample of about @xmath26 events . in the full analysis the vertex reconstruction was repeated with tracks of all momenta . again , only events inconsistent with having a single primary vertex were considered . the rich detector identified charged tracks above 25 gev/@xmath1 . results reported here come from a preliminary reconstruction through the data , using a production code optimized for speed , not ultimate efficiency . the simulated reconstruction efficiency of any charmed state is constant at about 40% for @xmath27 where @xmath28 of selex events lie . to separate the signal from the noncharm background we require that : ( i ) the spatial separation @xmath29 between the reconstructed production and decay vertices exceeds 8 times the combined error @xmath30 , ( ii ) each decay track , extrapolated to the primary vertex @xmath31 position , must miss by a transverse distance length @xmath32 2.5 times its error @xmath33 , ( iii ) the secondary vertex must lie outside any target by at least 0.05 cm and ( iv ) decays must occur within a fiducial region . there are @xmath34 events @xmath35 candidates , each having two rich - identified kaons and a pion , for which no particle identification is required . we divide them into three decay channels : @xmath6 , @xmath7 and other kk@xmath21 . the resonant mass window for the @xmath36 ( @xmath37 ) was @xmath38 mev/@xmath39 ( @xmath40 mev/@xmath39 ) . @xmath41 misidentification causes a reflection of @xmath42 under the @xmath8 peak . we limit the maximum kaon momentum to 160 gev/@xmath1 to reduce misidentification in the rich . to evaluate the shape of this background we use the @xmath43 sample that passes all the cuts listed above and lies within @xmath11 15 mev/@xmath44 of the @xmath45 mass . we formed the invariant mass distribution of these events when one pion is interpreted as a kaon . at most one of the two possible reflections per event falls into the @xmath35 mass window . the reflected mass distribution was fit by a polynomial function rising at 1925 mev/@xmath44 and decreasing to zero at large invariant mass . dividing this distribution by the number of @xmath42 events gives us the contribution per mass bin for each misidentified @xmath42 in the @xmath35 sample . we count the misidentified @xmath42 in the @xmath8 sample by fitting the @xmath35 mass distribution within @xmath11 20 mev/@xmath44 interval around the @xmath35 mass with the sum of a gaussian signal , a linear background shape estimated from the sidebands and the @xmath42 shape with variable normalization . the resultant misidentified @xmath42 contribution to the @xmath35 mass distribution is shown as the hatched areas in fig . 1(a ) , ( b ) . the fit gives @xmath46 and @xmath47 misidentified @xmath42 events in the @xmath48 and @xmath49 decay mode , respectively ( the error quoted is statistical only ) . the rich kaon identification is a very powerful tool for rejecting @xmath42 contamination ; @xmath37 decay kinematics further reduces particle identification confusion in the @xmath50 channel . to estimate yields , we subtracted the sideband background and @xmath42 contamination as evaluated above from the total number of events in the signal region . we find 430 @xmath11 24 @xmath51 and 330 @xmath11 19 @xmath49 events . a gaussian fit to the combined data shown in fig . 1(a ) , ( b ) gives a mass of @xmath52 mev@xmath53 and @xmath54 mev@xmath53 . note that these fitted yields are not used as constraints in the lifetime fit , as discussed below . the average longitudinal error @xmath55 on the primary and secondary vertices is 270 @xmath23 m and 500 @xmath23 m , which gives a combined error of 570 @xmath23 m . in the @xmath8 sample , the average momentum is 215 gev/@xmath1 , corresponding to a time resolution of 18 fs , about @xmath56 of @xmath57 . because bin - smearing effects are small , we used a binned maximum likelihood fitting technique to determine the @xmath8 lifetime . the fit was applied to a reduced proper time distribution , @xmath58 where m is the known charm mass @xcite , @xmath4 the reconstructed momentum , @xmath29 the measured vertex separation and @xmath59 the minimum @xmath29 for each event to pass all the imposed selection cuts . @xmath59 is determined event - by - event , along with the acceptance , by the procedure described below . we fitted all events with @xmath60 fs in the mass range @xmath61 gev@xmath53 , @xmath62 from the @xmath8 central mass value . to evaluate the mean lifetime we used a maximum likelihood method . the probability density was performed by the function : @xmath63 @xmath64 where @xmath65 the function is the sum of a term for the @xmath8 exponential decay corrected by the acceptance function @xmath66 plus a background function @xmath67 consisting of a single exponential plus a constant to account for a flat background extending to large proper time . its parameters were determined from the @xmath68 distribution from the @xmath8 sidebands . it also includes a term for the @xmath42 exponential decay normalized to the number of misidentified events in the signal region . the @xmath69 lifetime used in the fit is 1051 @xmath70 fs @xcite . the mass range of the sideband background windows , @xmath71 gev@xmath53 and @xmath72 gev@xmath53 was twice the signal mass window . we defined asymmetric sidebands to avoid the influence of @xmath73 , and we excluded the @xmath74 mass region . the four parameters are : @xmath57 ( @xmath8 lifetime ) , @xmath75 ( background lifetime ) , @xmath76 ( background fraction in the signal region ) and @xmath77 ( background splitting function ) . @xmath78 is the total number of events in the signal region after @xmath42 contamination subtraction . the proper - time - dependent acceptance @xmath66 is independent of spectrometer features after the first magnet , e.g. , rich efficiency and tracking efficiency . these efficiencies affect only the overall number of events detected . the proper time distribution of these events depends crucially on vertex reconstruction . to evaluate @xmath66 we reanalyze each @xmath79 @xmath35 event after moving it to a large set of different proper times @xmath80 . only the longitudinal position of the charm decay point and the axial orientation of the 3-body decay vectors are changed @xcite . a reanalyzed event is accepted if it passes the same cuts as those applied to the data . the minimum flight path after which an event is accepted defines @xmath59 for this event . this method is independent of details of the true @xmath81 or transverse momentum distributions @xcite . 2 shows the overall fits to the data distributions as a function of reduced proper time for @xmath82 and @xmath83 decay modes . it also shows the acceptance @xmath84 , which differs significantly from unity only after 4 lifetimes where statistics are limited . table 1 summarizes the lifetime results for the two modes analyzed : @xmath85 ; and @xmath86 . the uncertainties are statistical only , evaluated where @xmath87 increases by 0.5 . combining these results for the two resonant modes , we measure an average lifetime @xmath88 fs . .lifetime results and signal yields for the two @xmath35 modes analyzed . the last row is the weighted average of the two resonant channels @xmath89 and @xmath90 . the errors are statistical only . [ cols="<,^,^,^",options="header " , ] the systematic uncertainties for the @xmath8 lifetime analysis are listed in table 2 and described below . we group them in the following categories : lifetime shifts due to reconstruction errors have been well studied in our @xmath13 and @xmath91 work , with an order of magnitude higher statistics @xcite . because of the high redundancy and good precision of the silicon vertex detector , vertex mismeasurement effects are small at all momenta . proper time assignment depends on correct momentum determination . the selex momentum error is less than 0.5% in all cases . we assign a maximum systematic error from proper time mismeasurement of 1 fs . the effect of @xmath42 contamination under the @xmath92 peak was studied by changing the width of the exclusion window around the nominal @xmath42 mass for the k/@xmath21 interchange discussed above . effects on the @xmath49 mode are negligible . for the @xmath6 mode this gives a systematic error of 2 fs . the technique to determine the acceptance correction dependence on proper time is discussed extensively in ref . it has been verified with much larger statistics there . the maximum systematic error here is dominated by the @xmath6 correction , 3 fs . for the @xmath7 mode it is less than 2 fs . the fit was performed by the maximum likelihood method using a background parametrized by an exponential function plus a constant . we varied the width of the sidebands and the @xmath93 independently . the sytematic error due to the fit procedure is 5 fs and less than 0.5 fs for @xmath6 and @xmath7 decay modes respectively . that error is mainly dominated by the @xmath35 background parametrization . combining in quadrature all the sources of systematic errors listed in table 2 we obtain a total systematic error of 6.2 fs ( 2.2 fs ) for the @xmath94 ( @xmath7 ) mode . we have made a new measurement of the @xmath35 lifetime in two independent resonant decay channels , @xmath94 and @xmath7 using a maximum likelihood fit . selex measures the @xmath35 lifetime to be @xmath95 fs . using the result reported in the pdg @xcite @xmath96 fs we evaluate a ratio @xmath97 , 3@xmath17 from unity . the @xmath98 lifetime reported in this letter is comparable in precision with previous experiments @xcite . our result , combined with other world data , @xcite , lowers the overall @xmath35 lifetime somewhat . nevertheless , it is clear that the lifetime ratio @xmath99 is significantly larger than unity for all the precision measurements . the authors are indebted to the staff of fermi national accelerator laboratory and for invaluable technical support from the staffs of collaborating institutions . this project was supported in part by bundesministerium fr bildung , wissenchaft , forschung und tecnologie , consejo nacional de ciencia y tecnologia ( conacyt ) , conselho nacional de desenvolvimento cientfico e tecnlogico , fondo de apoyo a la investigaci ' on ( uaslp ) , funda~ ao de amparo pesquisa do estado de s~ ao paulo ( fapesp ) , the israel science fundation founded by the israel academy of sciences and humanities , istituto nazionale di fisica nucleare ( infn ) , the international science foundation ( isf ) , the national science foundation ( phy 9602178 ) , nato ( grant cr6.941058 - 1360 - 94 ) , the russian academy of science , the russian ministry of science and technological research board ( t " ubitak),the u.s . department of energy ( doe grant de - fg02 - 91er40664 and doe contract number de - ac02 - 76cho3000 ) , and the u.s .- israel binational science foundation ( bsf ) . s. frixione , m. mangano , p. nason and g. ridolfi , `` heavy quark production in heavy flavour ii '' , a. j. buras and m. lindner eds . ( world scientific publishing singapore 1997 ) ; i.i . bigi and n.g . uraltsev , z. phys . c62 , 623 ( 1994 ) . particle data group d.e . et al . _ , j. c15 , 1 ( 2000 ) . p. l. frabetti _ et . e687 collaboration , phys . 71 , 827 ( 1993 ) . et al . _ e791 collaboration , phys . b445 , 449 ( 1999 ) . g. bonvicini _ et al . _ , cleo collaboration , phys . 82 , 4586 ( 1999 ) . j. engelfried _ et al . methods a431 , 53 ( 1999 ) . et al . _ , in _ proceedings of the 29th international conference on high energy physics , _ 1998 , edited by a. astbury _ et al . _ ( word scientific , singapore , 1998 ) vol . ii , p. 1259 ; hep - ex/9812031 . kushnirenko , ph . d. thesis , carnegie mellon university , 2000 ( unpublished ) . a.y . kushnirenko _ et al . _ , accepted to be published .

we report a precise measurement of the @xmath0 meson lifetime . the data were taken by the selex experiment ( e781 ) spectrometer using 600 gev/@xmath1 @xmath2 , @xmath3 and @xmath4 beams . the measurement has been done using 918 reconstructed @xmath0 . the lifetime of the @xmath0 is measured to be @xmath5 fs , using @xmath6 and @xmath7 decay modes . the lifetime ratio of @xmath8 to @xmath9 is @xmath10 . ( selex collaboration ) , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,

You are an expert at summarizing long articles. Proceed to summarize the following text: during the past decades the development of the epitaxial crystal growth techniques such as molecular beam epitaxy and metal - organic chemical vapor deposition has made the growth of the quasi - two - dimensional ( quantum well ) or quasi - one - dimensional ( quantum wire)@xcite systems with controllable well thickness or wire radius became possible . these quantum structures have been applied to many semiconductor devices , such as high - electron - mobility transistors . recent progresses in growth and fabrication techniques have been able to fabricate the quantum wires with radii less than 100 @xmath0 . theoretically , the electronic properties of a hydrogenic impurity in the quantum well5,6,7,8 and the quantum wire @xcite have been studied by many authors . the impurity binding energies of a quantum wire with infinite or finite potential barrier @xcite and with different shapes of the cross - section@xcite have been discussed . the effect of location10,11 of impurities with respect to the wire axis was also studied previously . the emission line for quantum wires was observed@xcite to be two to three times broader than that of quantum wells and with 6 - 10 mev higher binding energy . it is expected that the same properties in quantum wells were further improved by the reduction of dimensionality to quasi - one - dimensional quantum wires . the physics of impurity states in quantum wire is very interesting because specific properties can be easily achieved by varying the wire radius . an electron bound to an impurity on the axis of the quantum wire behaves like a bounded three - dimensional electron when the boundary is far away . however , as the wire radius is reduced , the electron confinement due to the potential barrier becomes very important . especially in the quantum wire with infinitely high potential wall , the total energy of the electron may change from negative to positive at a certain radius and finally diverges to infinity as the radius approaches zero . furthermore , it is well known that the reduction of dimensionality increases the effective strength of the coulomb interaction . the binding energy @xmath1 of the ground state of a hydrogenic impurity in n - dimension is given by @xmath1=@xmath2 ^{2}r_{y}^{*}$ ] , where @xmath3 is the effective rydberg . hence the dramatic change in the binding energy may serve as a clear signal for variation in the effective dimension of the quantum wire . it is known an electron weakly bound to a hydrogen impurity in a polar semiconductor will interact with the phonons of the host semiconductor . in the past decade , many authors have studied the polaron effect on the binding energy of impurity or exciton in quantum well16,17,18,19,20,21,22,23,24 . recently , the electron - phonon effect on the binding energy of the donor impurity in a quantum wire with rectangular cross - section was reported@xcite . it was found the polaron effect on the binding energy becomes sizeable as the electron gets more deeply bound . the polaron shifts in donor energy levels are found to be of the order of 10% in a weakly polar system . in studying the polaron effect on the impurity binding energy , most of the previous works considered the interaction of the electron and bulk optical(bo ) phonon only . however , in ionic crystal , the motion of an electron near the surface may be affected very much by the surface longitudinal optical ( so ) phonon@xcite . an electron may be trapped at the surface by the electron - so phonon interaction . besides , the electron phonon interaction hamiltonian in the previous works was valid only for the bulk . therefore , we will choose the hamiltonian derived by li and chen@xcite , who considered the confined phonon modes in the cylindrical quantum dot . most of the previous approaches concentrating on the polaron effect on the ground state of an impurity in a quasi - one - dimensional wire employ the variational method or perturbation method . since the construction of variational trial wave functions bases entirely on physical intuition , and the estimation of the accuracy of the result obtained from variational approach is very difficult . furthermore , the perturbation method is only a good access to those systems with very small perturbation in most cases . therefore , it would be most desirable to have an alternative approach which is not only simple but also efficient to the quantum wire problem . in this work , we employ a simple approximation treatment which combines the spirit of both variational principle and perturbational approach to study the effect of electron - phonon interactions on the ground state binding energy of a hydrogenic impurity located inside a quantum wire . consider now a hydrogenic impurity located on the axis of a rigid wall cylindrical quantum wire with a radius @xmath4 . the hamiltonian of the impurity electron interacting with the phonon can be expressed as:@xmath5 here , @xmath6 is the electronic part of the hamiltonian @xmath7 @xmath8 is the confining potential which is assumed as : @xmath9 and @xmath10 and @xmath11 are the dielectric constant of the well and the effective mass of the electron . recently , li and chen@xcite has derived the confined the longitudinal - optical phonon and surface phonon modes of a free - standing cylindrical quantum dot of radius @xmath4 and height @xmath12 . we will follow their hamiltonian and let @xmath13 approach infinity , such that the dot system can become a quantum wire . therefore , @xmath14 is the bulk phonon hamiltonian which can be expressed as : @xmath15 where @xmath16 is the dispersionless bulk optical ( bo ) phonon energy , @xmath17 is the creation ( annihilation ) operator for bo phonon . @xmath18 is the interaction between the electron and bo phonon which can be expressed as : @xmath19\end{aligned}\ ] ] with @xmath20 } \left ( \frac{1}{\varepsilon _ { \infty } } -% \frac{1}{\varepsilon _ { 0}}\right ) , \ ] ] where @xmath21 is the mth - order bessel function , @xmath22 is the nth - root of @xmath23 , and @xmath24 is the crystal volume . @xmath25 is the surface optical phonon ( so ) phonon hamiltonian which can be expressed as : @xmath26 where @xmath27 is the surface optical ( so ) phonon energy , @xmath28 is the creation ( annihilation ) operator for so phonon . @xmath29 is the interaction between electron and so phonon : @xmath30 with @xmath31 } \notag \\ & & \cdot \left ( \frac{1}{\varepsilon \left ( \omega _ { sp}\right ) -\varepsilon _ { 0}}-\frac{1}{\varepsilon \left ( \omega _ { sp}\right ) -\varepsilon _ { \infty } % } \right ) , \end{aligned}\ ] ] @xmath32 , \ ] ] @xmath33 where @xmath34 , and @xmath35 . @xmath36 and @xmath37 are , respectively , the mth - order modified bessel function of the first and second kind . following landau and pekar s variational approach@xcite , the trial wavefunction can be written as : @xmath38 where @xmath39 depends only on the electron coordinate , and @xmath40 is the phonon vacuum state defined by @xmath41 , @xmath42 , and u is a unitary transformation given by : @xmath43 where @xmath44 and @xmath45 are the variational function and the unitary operators @xmath46 and @xmath47 transform the bulk phonon and surface phonon operators as follows : @xmath48 the parameters @xmath49 , @xmath50 , @xmath51 , @xmath52 can be obtained by minimizing the @xmath53 with respect to the parameters @xmath44 , @xmath54 , @xmath45 , @xmath55 . then the expectation value @xmath56 turns out to be @xmath57 the axis of the wire is assumed to be along the z direction . to solve the electronic part , one can employ the perturbative - variational approach as we did in the above subsection . two variational parameters @xmath58 and @xmath59 are introduced by adding and subtracting two terms @xmath60 and @xmath61 into the original hamiltonian @xmath6 and then regroup @xmath6 into three groups:@xmath62 where @xmath63 in the above equations , @xmath64 is treated as a perturbation , and @xmath58 and @xmath65 are treated as variational parameters which can be determined by requiring the perturbation term to be as small as possible . decomposing @xmath6 into two terms @xmath66 and @xmath67 is equivalent to dividing the space into a two - dimensional ( in xy plane ) and a one - dimensional ( in z - axis ) subspace . the unperturbed part of the hamiltonian @xmath6 contains two terms , i.e. @xmath66 and @xmath68 , where @xmath66 represents the one dimensional harmonic oscillator , and @xmath69 represents a two dimensional hydrogen atom located inside a quantum disk@xcite . both can be solved exactly . for illustration , the ground state energy and wavefunction of the unperturbed part can be expressed as : @xmath70 respectively , where @xmath71 is the ground state wavefunction of the 1d harmonic oscillator , and @xmath72 is the ground state wavefunction of the 2d hydrogen atom located at the center of an infinite circular well . the ground state eigenvalue and eigenfunction of the 1d harmonic oscillator can be expressed as : @xmath73 the ground state eigenvalue and eigenfunction of the 2d hydrogenic impurity located at the center of an infinite circular well can be obtained as14:(1 ) for @xmath74 , @xmath75 where @xmath76 , @xmath77 , @xmath78 , @xmath79 is the confluent hypergeometric function , and @xmath80 is the normalization constant.(2 ) for @xmath81 , @xmath82 where @xmath83 , @xmath84 , @xmath85 , @xmath86 is the irregular coulomb wave function , and @xmath87 is the normalization constant.(3 ) the turning point for energy changing from @xmath81 to @xmath74 in the quantum circle system may be determined by setting @xmath88 = 0\;\;for\;\;m=0,\ ] ] and@xmath89 = 0\;\;for\;\;m=1,\ ] ] the requirement of the continuity of the wavefunctions and its first derivative at boundary yields:(1 ) for @xmath74 , @xmath90 ( 2 ) for @xmath81 @xmath91 the eigenvalues are then given as : @xmath92 the first order energy correction can thus obtained as : @xmath93 the second term of the above equation can be integrated analytically and the result is : @xmath94 then the total energy up to the first order perturbation correction can then be obtained as : @xmath95 the variational parameters are then chosen by requiring the total energy @xmath96 to be minimized with respect to the variation of @xmath97 and @xmath65 . this is equivalent to requiring : @xmath98 @xmath99 for the excited states , the eigenvalues and eigenfunctions can be treated in the same way . we have calculated the effect of the confined the longitudinal - optical phonon and surface phonon interactions on the hydrogenic impurity located in a quantum wire . and the well potential is considered as infinite . figure 1 shows the ground state energy as a function of the wire radius . the binding energy @xmath1 of the hydrogenic impurity is defined as the energy difference between the ground state energy of the cylindrical wire system with and without the impurity , i.e.@xmath100where @xmath101 is the ground state energy of the quantum wire system without the impurity , while @xmath102 is the ground state energy of the quantum wire system with the impurity located on the axis of the cylindrical wire . one can see from fig.1 that the energy of the 1@xmath103 state becomes negative when the wire radius is larger than 1.65@xmath104 . it means that the confining energy is larger than the coulomb energy as the wire radius is smaller than 1.65@xmath104 . and one can also note that as the radius of the quantum wire is decreased , the ground state energy increases . as the wire radius @xmath4 becomes smaller , the electron is pushed toward the axis of the cylindrical wire . this makes the electron get close to the nucleus . as the electron gets close to the nucleus , both the ground state energy and the binding energy increase rapidly this is because the coulomb potential , which varies with @xmath105 ( @xmath4 is the wire radius ) , becomes more negative , while the kinetic energy of the electron , which varies with @xmath106 ( by the uncertainty relation ) , increases more rapidly . as a result , the ground state energy is increased as the electron gets close to the nucleus . the binding energy defined in eq . ( 39 ) is effectively the negative sign of the of the coulomb interaction energy between the electron and the nucleus , i.e. @xmath107 , therefore , the binding energy of the electron is also increased as the electron gets near to the nucleus .. as a result , the ground state energy is increased as the electron gets close to the nucleus . the binding energy defined in eq . ( 39 ) is effectively the negative sign of the of the coulomb interaction energy between the electron and the nucleus , i.e. @xmath107 , therefore , the binding energy of the electron is also increased as the electron gets near to the nucleus . our results show that for small wire radius , the binding energies are in good agreement with previous results@xcite . as the radius becomes very large , our result approaches the correct limit 1@xmath108 while the previous work @xcite can only yield a value of 0.22@xmath108 . the large discrepancy of the previous work may be due to the artificial dividing of the variational trial wavefunction into a one - dimensional hydrogen atom and a two - dimensional hydrogen atom and thus forces the creation of an additional node of the wavefunction at z=0 . in this work , the trial wavefunction is adopted to be the form of 1d harmonic oscillator wavefunction instead of one dimensional hydrogen atom . this prevents our wavefunction from introducing any additional node at z=0.figure 2 presents the 2s excited state binding energies as the functions of wire radius . one can note from the figure that as the wire radius increases , the binding energy approaches 0.25@xmath108 which gives correctly the limiting value of 3d hydrogen atom . figure 3 presents the confined bo phonon and so phonon effect as a function of wire radius . with increasing the wire radius , the magnitude of the confined bo phonon effect decreases from large value and then approaches to the bulk value . when the wire radius is less than 1.5 a * , the polaron effect increases rapidly . one might think as the radius becomes very small , the confined bo phonon effect should approach zero , like the case in quantum well @xcite . in fact , similar results were obtained by oshiro in a spherical quantum dot@xcite . they found the polaron energy shift is enhanced as the dot radius becomes small . this is due to the fact that the electron becomes complete localized ( e@xmath109 approaches infinity ) in small wire ( or dot ) radius while the binding energy approaches 4@xmath108 in small well width . in the case of quantum well , the confined so phonon effect plays the dominant role for small well width@xcite . but in quantum wire , the confined so phonon is less important , just like that in quantum dot system @xcite . this is because the surface area of a quantum wire ( or quantum dot ) decreases with the radius . thus the number of vibration modes of confined so phonon becomes fewer . in fig.4 , three curves are presented . the dotted curve represents the binding energy of the impurity without considering the interactions between the electron and phonon . the dashed curve represents the binding energy of the impurity with only confined bo phonon effect being taken into account . while the solid curve is the binding energy of the impurity including both confined bo phonon and so phonon effects in the calculation . comparing to the impurity binding energy , the confined so phonon is negligible in quantum wire . we then conclude that because of the similarity in geometry , the behavior of the polaron effect on the quantum wire system is like that on the quantum dot system . in this work , analytical solutions for the effects of the electron - phonon interaction on the binding energies of an impurity located inside a quantum wire are obtained by a simple but efficient perturbation - variation method . as the radius becomes very large , the correct limiting value can be obtained . we have also discussed both the confined bo and so phonon effects . we found the confined bo phonon effect is prominently for a quantum wire with small radius . we also found that the energy corrections of the polaron effects on the impurity binding energies increase rapidly when the wire radius is less than 1.5 a*.

the effect of electron - optical phonon interaction on the hydrogenic impurity binding energy in a cylindrical quantum wire is studied . by using landau and pekar variational method , the hamiltonian is separated into two parts which contain phonon variable and electron variable respectively . a perturbative - variational technique is then employed to construct the trial wavefunction for the electron part . the effect of confined electron - optical phonon interaction on the binding energies of the ground state and an excited state are calculated as a function of wire radius . both the electron - bulk optical phonon and electron - surface optical phonon coupling are considered . it is found that the energy corrections of the polaron effects on the impurity binding energies increase rapidliy as the wire radius is shrunk , and the bulk type optical phonon plays the dominant role for the polaron effects . pacs : 71.38+i;73.20.dx;63.20.kr

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Proceed to summarize the following text: recent and ongoing advances in technology have led to the discovery of extrasolar planets ( e.g. , mayor & queloz 1995 , marcy _ et el . _ 1997a , b , butler & marcy 1996 , marcy & butler 1996 , cochran _ et al . _ 1997 , noyes _ _ 1997 ; see also the references listed in the encyclopedia of extrasolar planets , www.obspm.fr/darc/planets/encycl.html ) , and promise the discovery and even the imaging of additional planets ( angel & woolf 1997 ; fraclas & shelton 1997 ; labeyrie 1996 ; brown 1996 ) . these developments excite the imagination because they seem to bring us closer to the possible discovery of extraterrestrial life . it is therefore interesting to ask whether microlensing searches are likely to find planets on which life could thrive . we do not yet have a clear enough understanding of the nature of life to definitively answer this question , because the range of physical conditions compatible with life may well be wider than our limited experience would at first suggest . it has been proposed , for example , that life may exist on the outer planets of our own solar system and/or on their moons . ( see , e.g. , reynolds et al . 1983 ; raulin _ et al . _ 1992 ; sagan , thompson , & khare 1992 ; williams , kasting , & wade 1997 , mccord _ et al . it has even been postulated that life may exist in non - planetary environments , including the interiors of stars and molecular clouds ( see , e.g. , feinberg & shapiro 1980 ) . it therefore makes sense to consider any planet , however close to or far from its star , and whatever the nature of the star , as a possible harbor for life . nevertheless , in the absence of real information on the existence of life away from our own planet , one question is clearly interesting : will microlensing find evidence of planets similar to earth ? we must of course define what we mean by similar to earth " . if we mean that there is a chance that chemical processes necessary for earth - like life could occur , then we want to consider planets which can have similar surface and atmospheric make - up , and similar amounts of energy available to fuel the necessary chemical processes . the range of planetary masses and distances from a solar - type star compatible with earth - like conditions , particularly the presence of liquid water , has been dubbed the goldilocks problem " , and has been studied by many researchers . ( see , e.g. , rampino & caldeira 1994 , for an overview . ) in linking the goldilocks problem to microlensing observations , we must focus on the properties of ps . s that determine their detectability when searched for by microlensing monitoring programs . 2 is therefore devoted to a brief overview of the detection of planets via microlensing . the question of whether planets discovered through their action as microlenses may be earth - like , is addressed in 3 . one of the primary findings of this study is that when a planet discovered via microlensingreceives from its central star a flux of radiation comparable to that received by the earth from the sun , the central star will generally be more luminous than our sun . this means that light from the lensed star will be blended with light from the ps . . 4 is devoted to studying how blending ( a ) influences the detection of , and ( b ) helps us develop strategies to study ps . s discovered through microlensing . in 5 , i summarize the conclusions . the detection of a planet via microlensing is possible if the separation between the star and planet is close to or larger than the einstein radius , @xmath1 of the star : @xmath2 where @xmath3 is the stellar mass , @xmath4 is the distance to the lensed source , and @xmath5 is the ratio of the distance to the lens to @xmath4 . consider lensing by a mass , @xmath6 . define @xmath7 to be the time taken for the track of a lensed star to cross a distance in the lens plane equal to the einstein diameter , @xmath8 . note that the duration of the observable event generally differs from @xmath7 for several reasons . first , if the photometric sensitivity is good , the event may be detectable well before the track of the source comes within @xmath9 of the lens position . for example , the deviation from baseline is at the @xmath10 level , when the projected separation between the source and lens is @xmath11 second , the track of the source will not generally intersect the lens position . finally , blending may decrease and/or finite - source size effects may increase the time during which the event is detectable . ( see distefano 1998 ; distefano & scalzo 1997 or 1998a for discussions relevant to planet lenses . ) express the separation between the planet and star as @xmath12 . the minimum possible value of @xmath13 for which planets can be discovered via microlensing is @xmath14 ( gould & loeb 1992 , but see wambsganss 1997 for a detailed discussion ) . for planets located between roughly @xmath15 and @xmath16 ( the _ zone for resonant lensing _ ) , the signature of a lensing planet is a short - lived perturbation superposed on a light curve whose underlying structure is due to lensing by the ps . the time duration of the underlying event may be weeks or months . all of the literature published to date on planet discovery via microlensing has been limited to the possible discovery of planets in the zone for resonant lensing , and observing programs designed to increase the detection efficiencies for these types of events are already underway . for a more detailed description of these short - duration , resonant " perturbations , and a discussion of the boundaries of zone for resonant lensing see , e.g. , mao & paczyski 1991 , gould & loeb 1992 , bennett & rhie 1996 , paczyski 1996 , wambsganss 1997 , peale 1997 . the benefits of systematically extending the search , to look for planets located farther from the central star , have recently begun to be explored ( distefano & scalzo 1997 , 1998a , b , distefano 1998 ) . for larger values of @xmath17 , the planet generally acts as an independent lens . when the planet is the only lens , then the event will be an isolated event of short - duration . ( for a jupiter - mass planet , e.g. , the time duration of the perturbation from baseline lasts @xmath18 of the time the deviation due to a solar - mass star would . ) when the track of the source passes through the lensing region of one planet and also that of another system mass ( typically the central star ) , the event will appear to repeat ( distefano & mao 1996 , distefano & scalzo 1997 , 1998a , b ) . there is no maximum value of @xmath13 , other than that dictated by the dynamics of the ps . itself i.e . , planets located too far from the central starmay be lost from the system . because the probability of a repeating event falls off as @xmath19 , where @xmath17 is the orbital separation , while the probability of isolated short - duration events is nearly independent of position ( and actually increases at the expense of repeating events as the orbital separation increases ) , isolated short - duration events become the dominant mode of detection for planets in wider orbits particularly since ps . s may have several planets in s ( distefano & scalzo 1997 , 1998a ) . calculations indicate that it is likely that microlensing by planets in s will provide an important channel and , for low - mass ( e.g. , earth - mass ) planets , possibly the dominant channel for planet detection via microlensing ( distefano & scalzo 1997 , 1998a , b distefano 1998 ) . the phrase earth - like conditions " does not have a unique meaning . two requirements seem natural , however . ( 1 ) the radiation flux received from the central star should be neither too large nor too small ; ( 2 ) the planet should have a rocky surface , water , and a gaseous atmosphere . for earth - like conditions to exist , the primary requirement on @xmath20 the orbital separation between the planet and the central star , is that the incident flux of radiation from the star should be comparable to the flux received by the earth from the sun . that is , @xmath21 , should not be too different from unity . @xmath22 the mass and luminosity of the central star are @xmath3 and @xmath23 , respectively . as usual , @xmath4 ( @xmath24 ) is the distance to the lensed source ( lens ) , and @xmath25 . given that ( 1 ) the conditions that lead to life may be flexible , ( 2 ) the effects of radiation incident from the star are likely to be strongly influenced ( either enhanced or diminished ) by the planet s atmosphere , and ( 3 ) internal heating from geological processes or radioactive materials may be important , it is not clear how large a range of values of @xmath21 may be compatible with the development of life . we will therefore simply use @xmath26 as a guideline , and emphasize that this should not be viewed as an absolute requirement . figure 1 shows the relationship between @xmath3 and @xmath5 for those systems that satisfy the relationship @xmath27 we have set @xmath28 and have assumed that @xmath29 in the upper plot , and @xmath30 in the bottom plot ; the former would be appropriate for main - sequence stars , while the latter would be appropriate for slightly evolved stars . as eq . 1 and figure 1 make clear , if the planets we will discover via microlensing are to have incident flux comparable to the flux incident on earth , their stars will generally ( although not necessarily ) be more luminous than our sun . this has two obvious implications . the first is that the length of time during which the planet would have this flux incident will be shorter than the time to date that the earth has had roughly this flux incident . this is because the system s star may need to be more massive than the sun , or even slightly evolved . the time elapsed from the formation of the star until the present time could range from less than @xmath31 the present age of the sun to times comparable to the sun s main - sequence lifetime . we do not know how long it takes for complex life forms to develop , but it may be that the process is fast enough that intelligent life can develop and thrive during a time significantly shorter than the present age of the sun . indeed , it is likely that a long sequence of independent processes must occur in order for intelligent life to develop ; thus , the probability distribution may well be log normal , and the likelihood of intelligent life developing in times much shorter than the time apparently taken on earth may be significant . the second implication is that , since the central star must be fairly luminous , it may contribute a non - negligible fraction of the light incident along the line of sight to the lensed source ; i.e. , the light we receive may be strongly blended . we may also argue , as follows , that planets likely to be deemed earth - like have masses within a factor of @xmath0 of the mass of the earth . assuming that we would like a rocky surface , we also assume that the planet s average density should be similar to that of earth . this means that the acceleration due to gravity , @xmath32 , scales as the cubed root of the planet s mass , @xmath33 . thus , @xmath32 will be within a factor of 2.5 of @xmath34 (= 9.8 m / s@xmath35 ) if @xmath33 is within a factor of 15 of @xmath36 . increasing or decreasing the value of @xmath32 will lead to different atmospheric contents ; for any given atmospheric temperature , there is a lower limit to @xmath33 ( hence @xmath32 ) , below which an atmosphere will not be retained . additional ( and more sophisticated ) considerations can influence these limits . for example , the level of geothermal ( planetary - thermal ) activity may be less for planets of smaller mass , influencing atmospheric chemistry . in fact , it has been conjectured that part of the difference between earth and mars , which is roughly @xmath37 times less massive than earth , could be related to differences in planetary - thermal activity related to their mass difference . but it is not clear that this is the crucial difference with regard to liquid water , for example , and other possibilities exist . ( see , e.g. , rampino & caldeira 1994 . ) in this paper , however , i will not highlight such complementary restrictions . this is because the main relevance of such considerations to microlensing observations is simply that there is likely to be a range of planet masses , possibly spanning two orders of magnitude , consistent with earth - like conditions . this means that event durations for earth - like planets are likely to also have a range . the factor of @xmath38 derived above corresponds to a range of event durations from less than an hour to more than a day . in addition , finite - source - size effects could increase the time duration of events by a factor of a few ( distefano & scalzo 1997 , 1998a , distefano 1998 ) . if the central star of a ps . that serves as a lens is fairly luminous , then its light will blend with that from the lensed star . this blending can have two consequences . first , if is is measurable and can be quantified , blending makes it possible to learn more about the lens system : the spectral type of the central star can be determined and , in some cases , even the mass of the planetmay be inferred ( distefano 1998 ) . unfortunately , however , the second consequence of blending is that the peak magnification , or in some cases , the time interval during which the event is observable , may be decreased to the point that the planet - lens event is not detected at all . we must distinguish between the detection of planets in the zone for resonant lensing , and the detection of planets in wider orbits . the discussion below applies to cases in which the orbital separation is outside the zone for resonant lensing , when the planet acts as a more - or - less independent lens . at the end of 4.1 we return to the case of planets in the zone for resonant lensing . let @xmath39 represent the fraction of the baseline flux contributed by the lensed star . when the central star of the ps . is luminous , particularly if its flux comes close to satisfying the criterion studied in the last section ( @xmath40 ) , then @xmath39 can indeed be small enough for the effects of blending to be measurable . if , for example , the apparent @xmath41 magnitude of the combined light coming along the line of sight from a lensing event is [ @xmath42 , then @xmath43 if the lens is located in the bulge and is a main - sequence star of mass roughly equal to [ @xmath44 @xmath45 . a small value of @xmath39 can allow us to reliably determine the effects of blending and to thereby learn more about the lens . the question we address below is whether values of @xmath39 small enough to be useful may prevent the event from being detected . ( woniak & paczyski 1997 ) . in the latter case , however , evidence of blending can be obtained through comparisons of spectra taken at peak with spectra taken at baseline ( unless the lens and source are of similar spectral type ) . astrometry can also be useful ( see , e.g. , goldberg 1988 , goldberg & wozniak 1998 ) . ] when light from the lensed source is blended with light from other sources , the observed magnification , @xmath46 is smaller than the true magnification , @xmath47 @xmath48 thus , in order for a light curve perturbation to be brought above the detection limit , @xmath49 must be larger than it would otherwise have to be , the projected distance between the source and lens must be smaller , and the event will consequently appear to have a shorter . to describe this effect systematically , distefano & esin ( 1995 ) introduced the blended einstein radius " , @xmath50 . let @xmath51 be the minimum peak magnification needed for event detection . define the of each event to be time during which the magnificationwas greater than @xmath52 the expression for the blended einstein radius is then @xmath53 figure 2 illustrates the influence of blending on event as a function of both @xmath51 and @xmath54 for a given value of @xmath55 the event is longer if @xmath51 is smaller . thus , increasing the photometric sensitivity to accomodate smaller values of @xmath51 should increase the detection rate . note that if @xmath51 is @xmath56 then even if @xmath57 the event ( which would have been @xmath58 ) , is reduced by only a factor of @xmath59 to @xmath60 thus , while blending does tend to decrease the event , making more frequent monitoring desirable , we can expect to be able to detect a large majority of events by using sensitive photometry . note , however , that , even with @xmath61 the time of an event with @xmath62 would be decreased by only a factor of @xmath63 . if @xmath51 could be reduced to @xmath64a formidable task for a large - scale monitoring program then for @xmath65 an event that would have lasted for @xmath66 will have an observed of @xmath60 there has not been a detailed study of the effects of blending on the detection of planets in the zone for resonant lensing . we can make some general observations , however . blending does decrease the duration of stellar - lens events . since we expect most such events to last for times on the order of weeks or months , the arguments above show that it is unlikely that blending will cause us to miss the stellar - lens event altogether . what blending can do , though , is to shorten the event in such a way that we become aware of it only after the planet - lens perturbation has occurred , if the perturbation takes place early in the event . perturbations can take place early in stellar - lens events ( paczyski 1996 , wambsganss 1997 to @xmath67 . ] ) . thus , one effect of blending is to decrease the rate of observable resonant planet - lens events . the effect should not be large , however , because ( 1 ) the events that occur early are not generally the most distinctive and readily observed planet - lens events , and ( 2 ) careful monitoring , once we know the stellar - lens event is underway , can help us to catch even perturbations that take place as the measured flux declines toward baseline , so that we suffer the worst losses only on the upswing of the magnification . the second effect of blendingis to alter the light curve shape . when , however , finite - source - size effectsare unimportant , the spike in magnification associated with resonant events is so distinctive that frequent monitoring of the light curve during the event should allow us to detect it . the remaining challenge is therefore to estimate the combined effects of finite - source size and blending on the overall efficiency of detecting planets in the zone for resonant lensing . the up - side of blending is that the quantity and color of light from the central star are themselves valuable pieces of information about the ps . . thus , if the blending is significant enough to make these features measurable , it allows us to learn something interesting about the ps . . we would like to do the following . \(1 ) definitively establish that there is blending . \(2 ) establish that the light not emanating from the lensed source is most likely emitted by the central star of the ps . . this establishes that , even if the event was an isolated event of short - duration , the lens was a ps . . \(3 ) determine the spectral type , and possibly the mass of the central star . in cases in which the mass ratio can be extracted from the light curve , this will establish the mass of the planet lens . if , in addition to blending , there are measurable finite - source - size effects , then the mass of any planets that served as lenses may be determined directly . these possibilities , together with the combined effects of blending and finite - source - size effects on event detection , are discussed in more detail elsewhere ( distefano 1998 ) , since they apply to all ps . s discovered via microlensing , not just those most likely to include an earth - like planet . here i simply note that a combination of observations would generally be needed to accomplish these goals . these include : light curve studies to begin to assess the role of blending and finite - source - size effects ; possible astrometric studies ( e.g. , goldberg 1988 , goldberg & wozniak 1998 , boden , shao , & van buren 1998 , dominik & sahu 1998 , mao & witt 1998 ) ; spectra taken near peak magnification , compared with spectra taken at baseline , to determine the spectral type and radius of the lensed star ; and high - spatial - resolution follow - up observations to determine whether any light not emanating from the lensed star can be explained by a chance superposition of light from other stars in the field . microlensing searches for planets complement other types of searches . they have the advantage of probing vast volumes of space , and of providing information about the existence of planets in diverse stellar environments . an additional advantage is that microlensingcan discover low - velocity planets . because they are far away , however , planets discovered via microlensing are less amenable to detailed follow - up observations . in this paper i have shown that microlensing is a tool that can discover distant planets with earth - like conditions , should such planets exist . in addition , i find that , among planets discovered by microlensing surveys , those likely to be experiencing earth - like conditions are particularly good targets for some follow - up studies . this is because the central star of these ps . s may be luminous enough that the blending of its light with light from the lensed source may be detectable and quantifiable . through a combination of spectra taken during the event , high resolution images and spectra taken after the event , and light curve fitting , we may be able to determine the spectral classification of the central star , the distance of the ps . from us , and even , in some cases , the mass of the planet lens . ( see also distefano 1998 . ) although there may be a subset of such events that are rendered undetectable by the very blending that helps to make them potentially so interesting , the results presented here show that in most cases , the planet - lens events should be detectable . until now , microlensing discussions of earth - like planets have intrinsically focused on earth - mass plants . in addition , because ( 1 ) low - mass stellar lenses are almost certainly significantly more numerous than higher - mass stars , and ( 2 ) a larger value of the mass ratio between the planet and star is preferred for detection , it has become common for calculations to focus on a low - mass ( @xmath68 ) star orbited by an earth - mass planet . figure 1 indicates , however , that it is unlikely that a low mass star will harbor a planet with earth - like conditions that can be discovered via microlensing . furthermore , the considerations in 3.2 indicate that there may be room for more flexibility in the mass of planets with earth - like conditions than has been assumed so far . simple arguments indicate that the range could extend as much as a factor of ten above , and a factor of 10 below the mass of the earth . finally , it is important to note that we should consider the possibility that microlensing ( whether by resonant or wide planets ) is most likely to discover the outer planets in a system that may contain closer planets experiencing earth - like conditions . in this case , the microlensing events serve as beacons directing us to the ps . . in the near - term , such discoveries can contribute to developing the statistics of distant ps . s , such as the frequency of planets as a function of spectral type and the spatial distribution of planets around stars . in the far ( and so far unforeseeable ) future , astronomers with instruments capable of measuring the spectra of distant stars and detecting small - amplitude doppler effects can perhaps check these ps . s for the presence of planets in orbits smaller than those occupied by the planets discovered via microlensing . it is a pleasure to thank the referee , scott gaudi , for suggested changes to an earlier manuscript ( distefano & scalzo 1998b ) that contained much of this work ; his suggestions have helped to improve the presentation . i would like to thank arlin crotts , andrew gould , jean kaplan , christopher kochanek , david w. latham , avi loeb , shude mao , robert w. noyes , bodhan paczyski , bill press , penny sackett , kailash sahu , michael m. shara , edwin l. turner , michael s. turner , and the participants in the 1997 aspen workshops , the formation and evolution of planets " and microlensing " for interesting discussions . it is also a pleasure to thank the aspen center for physics and the institute for theoretical physics at santa barbarba for their hospitality while the first version of this and related papers was being written , and the inter - university center for astronomy and astrophysics in pune , india for its hospitality while the paper was revised . this work was supported in part by nsf under ger-9450087 and ast-9619516 , as well as by funding from axaf .

are microlensing searches likely to discover planets that harbor life ? given our present state of knowledge , this is a difficult question to answer . we therefore begin by asking a more narrowly focused question : are conditions on planets discovered via microlensing likely to be similar to those we experience on earth ? in this paper i link the microlensing observations to the well - known goldilocks problem " ( conditions on the earth - like planets need to be _ just right " _ ) , to find that earth - like planets discovered via microlensing are likely to be orbiting stars more luminous than the sun . this means that light from the ps . s central star may contribute a significant fraction of the baseline flux relative to the star that is lensed . such blending of light from the lens with light from the lensed source can , in principle , limit our ability to detect these events . this turns out not to be a significant problem , however . a second consequence of blending is the opportunity to determine the spectral type of the lensed star . this circumstance , plus the possibility that finite - source - size effects are important , implies that some meaningful follow - up observations are likely to be possible for a subset earth - like planets discovered via microlensing . in addition , calculations indicate that reasonable requirements on the planet s density and surface gravity imply that the mass of earth - like planets is likely to be within a factor of @xmath0 of an earth mass . _ earth - mass _ # 1#2#3#4 # 1#2 -.4 true in -.2 true in

You are an expert at summarizing long articles. Proceed to summarize the following text: while a static magnetic field can not change the kinetic energy of charged particles , it affects the direction of their motion due to a lorentz force . this property is widely used in physics and technology . for example , magnetic fields play a crucial role in cyclotrons and particle accelerators , where they keep beams of charged particles on correct trajectories . another application is the mass spectrometry , which utilizes magnetic fields in order to sort charged particles based on their mass - to - charge ratio . magnetic fields can be also used as lenses for deflecting and focusing beams of charged particles . such lenses are utilized , for example , in cathode ray tubes and electron microscopes @xcite . the history of geometric electron optics started from a paper by hans busch in 1926 @xcite , where he suggested that the magnetic field of a short coil can act as a converging lens for electrons . this proposal was instrumental for the construction of the first electron microscope by knoll and ruska @xcite . owing to the fact that the wavelengths of electrons can be much smaller than the wavelengths of the visible light , the resolution of the electron microscope quickly surpassed that of the conventional optical microscope . modern transmission electron microscopes with aberrations correctors achieve a few orders of magnitude better resolution than the usual light microscopes and , therefore , are used to investigate the structure of metals , crystals , large molecules , as well as biological specimens including microorganisms and cells . in geometrical particle optics , the motion of charged particles in macroscopic electromagnetic fields is described analogously to the propagation of the light rays . mathematically , the possibility of such a description stems from the eikonal or wentzel kramers brillouin ( wkb ) approximation @xcite . the eikonal approximation is an appropriate and efficient means to solve wave equations in the case when the de broglie wavelengths of particles are small compared to the characteristic inhomogeneities of the system or external fields . in mathematical physics , the wkb approximation is a well - established method for finding approximate solutions to linear differential equations with coefficients varying slowly in space @xcite . recent experimental discovery of dirac and weyl materials made possible a condensed - matter realization of the systems whose low - energy quasiparticles are massless dirac or weyl fermions ( for reviews , see refs . the corresponding quasiparticles have a well - defined chirality in the vicinity of dirac points or weyl nodes . in this connection it should be noted that a dirac point in the brillouin zone could be viewed as an overlap of two weyl nodes of opposite chirality . according to the celebrated nielsen - ninomiya theorem @xcite , weyl nodes in condensed matter materials , in fact , always come in pairs of opposite chirality . in general , however , the nodes in each pair need not coincide . experimentally , the first three - dimensional dirac materials , i.e. , na@xmath0bi and cd@xmath0as@xmath1 , were discovered only a few years ago @xcite . since then a large number of weyl materials ( i.e. , @xmath2 , @xmath3 , @xmath4 , @xmath5 , @xmath6 , @xmath7 ) were discovered as well @xcite . these discoveries opened a new chapter in studies of numerous effects associated with quantum anomalies and chirality by using simple table - top experiments , rather than accelerator techniques of high - energy physics . in addition to quantum field theoretical methods , there exist a number of very powerful semiclassical techniques for studying condensed matter systems such as weyl and dirac materials in external electromagnetic fields . one of them is the kinetic theory . because of the nontrivial topological properties of weyl fermions @xcite , the corresponding kinetic theory should account for the berry curvature effects @xcite . amazingly , the resulting chiral kinetic theory @xcite can reproduce even the effects due to the quantum chiral anomaly @xcite . note that the latter implies the nonconservation of the chiral charge in the presence of parallel background electric and magnetic fields . the dirac and weyl materials make also possible the investigations of novel quantum phenomena that can not exist in high - energy physics . in particular , a number of such phenomena can be associated with the quasiparticle response to background pseudoelectromagnetic ( axial ) fields . unlike the ordinary electromagnetic fields @xmath8 and @xmath9 , their pseudoelectromagnetic counterparts @xmath10 and @xmath11 couple to the left - handed and right - handed particles with opposite signs . indeed , as shown in refs . @xcite , such fields can be induced by mechanical strains in dirac and weyl materials . a pseudoelectric field @xmath12 , for instance , can be created by dynamically stretching or compressing the sample . a nonzero pseudomagnetic field @xmath13 is generated , e.g. , by applying a static torsion @xcite or bending the sample @xcite . a typical magnitude of the pseudomagnetic field @xmath14 is estimated to be about @xmath15 in the former case and about @xmath16 in the latter case . for the purposes of this study , it will be sufficient to consider only static deformations so that @xmath17 . in addition , we will also assume that the ordinary electric field is also absent @xmath18 . in this study we suggest that the strain - induced pseudomagnetic fields can be utilized for creating pseudomagnetic lenses for deflecting and focusing beams of weyl quasiparticles , depending on the particle chiralities . a general experimental setup that allows a maximum control of chiral beams is given by a combination of the magnetic and pseudomagnetic lenses as shown schematically in fig . [ fig : illustration ] . the system consists of a thin long crystal ( wire ) of a weyl material placed inside a solenoid . in this case , magnetic and pseudomagnetic fields are directed along the @xmath19 axis and are present in the region @xmath20 . ( generically , the sample itself can be longer than @xmath21 . ) the magnetic field is generated by an electric current in the solenoid and the pseudomagnetic one is produced by the torsion of the crystal . since the characteristic scales of spatial variations of the background magnetic and pseudomagnetic fields in the ( pseudo-)magnetic lens are much larger than the de broglie wavelengths of weyl quasiparticles , it is justified to use the eikonal approximation for the description of their motion . because of a nontrivial topology of chiral particles , we will pay a special attention to the berry curvature effects on the corresponding chiral rays . is produced by an electric current @xmath22 in the solenoid , the pseudomagnetic field @xmath11 is created by twisting the crystal of a weyl material.,scaledwidth=50.0% ] the paper is organized as follows . in sec . [ sec : eikonal ] we describe the eikonal approximation in application to weyl quasiparticles . the properties of magnetic and pseudomagnetic lenses for weyl quasiparticles are considered in sec . [ sec : lens ] . the discussion and summary of the main results are given in sec . [ sec : summary - discussions ] . in this section , we formulate the eikonal approximation for the motion of weyl quasiparticles in the presence of both magnetic and pseudomagnetic fields . according to ref . @xcite , the berry curvature not only modifies the semiclassical equations of quasiparticle motion in background fields @xcite , but also changes their dispersion relations . thus , we will begin our analysis by presenting the corresponding relations for chiral quasiparticles . in the framework of the chiral kinetic theory @xcite , it is straightforward to include corrections linear in the background field @xmath23 , where @xmath9 and @xmath13 denote the ordinary magnetic and pseudomagentic fields , respectively , and @xmath24 is chirality of the left- ( @xmath25 ) and right - handed ( @xmath26 ) quasiparticles . however , by noting that the eikonal approximation for the conventional magnetic lens requires the inclusion of quadratic terms in the magnetic field @xcite , we need the dispersion relations for the weyl quasiparticles that are also valid to the second order in @xmath27 . the energy dispersion for a general band structure with nonzero berry curvature was obtained in ref . @xcite . the corresponding explicit expression for weyl quasiparticles was derived by the present authors in ref . @xcite . for the quasiparticles of positive energy ( electrons ) , the corresponding relation reads @xmath28 -\frac{(\mathbf{b}_{\lambda}\cdot[\mathbf{e}_{\lambda}\times\mathbf{p}])}{p}\right\ } , \label{second - energy-2}\end{aligned}\ ] ] where @xmath29 is the fermi velocity , @xmath30 is the speed of light , @xmath31 is the momentum of quasiparticles , and @xmath32 is the electron charge . note that the second and third terms in eq . ( [ second - energy-2 ] ) describe corrections due to the berry curvature . using the standard eikonal approximation @xcite for charged particles , we derive the equation for the abbreviated action @xmath33 of weyl quasiparticles by making the following replacement in the dispersion relation ( [ second - energy-2 ] ) : @xmath34 where @xmath35 is an effective vector potential that describes the background field @xmath27 . note that the abbreviated action @xmath36 is related to the full action via @xmath37 , where @xmath38 denotes the quasiparticle energy and @xmath39 is time . for a constant @xmath27 in the @xmath19 direction , the effective vector potential in the symmetric gauge is given by @xmath40 then , by substituting eq . ( [ lens - eikonal - p - replace ] ) into eq . ( [ second - energy-2 ] ) , we obtain the following eikonal equation for the abbreviated action @xmath36 of weyl quasiparticles with energy @xmath38 : @xmath41^{3/2 } } -\frac{e^2 \hbar^2v_fb_{\lambda}^2}{16c^2 } \frac{(\nabla_zs_0)^2}{\left[(\bm{\nabla}s_0)^2+\frac{e^2}{4c^2}r_{\perp}^2b_{\lambda}^2\right]^{5/2 } } , \label{lens - eikonal - epsilon}\end{aligned}\ ] ] where , in view of the symmetry in the problem , we assumed that @xmath42 depends on @xmath43 and @xmath44 , and used @xmath45 in order to obtain an analytical solution to eq . ( [ lens - eikonal - epsilon ] ) , we will use the paraxial approximation , which is well justified when the quasiparticles propagate sufficiently close to the optical axis . in other words , we will assume that @xmath46 is small and expand the solution in powers of @xmath46 . in the case @xmath47 , the action should describe a free quasiparticle moving with momentum @xmath48 , i.e. , @xmath49 in the case of a nonzero @xmath27 , we will seek @xmath36 in a similar form , i.e. , @xmath50 where constant @xmath51 and function @xmath52 are to be determined . indeed , by matching the actions at the boundaries of the ( pseudo-)magnetic lens , one can easily show that the eikonal of a quasiparticle moving in ( pseudo-)magnetic fields should also contain only even powers of @xmath46 . it is worth mentioning that the form of the abbreviated action in eq . ( [ lens - eikonal - s0 ] ) is standard in the paraxial optics @xcite . by moving the second term on the right - hand side of eq . ( [ lens - eikonal - epsilon ] ) to the left - hand side and squaring both sides of the equation , we find @xmath53 + \frac{e^2v_f^2}{4c^2}r_{\perp}^2b_{\lambda}^2 , \label{lens - eikonal - epsilon-1}\end{aligned}\ ] ] where terms of order @xmath54 were dropped . then , by keeping the terms up to quadratic order in @xmath55 and making use of the following relations : @xmath56 , \label{lens - eikonal - eqs - ee}\end{aligned}\ ] ] we rewrite eq . ( [ lens - eikonal - epsilon-1 ] ) in the form @xmath57 + \frac{2\lambda b_{\lambda}}{b^ { * } } \left\{c^3 + r_{\perp}^2c\left[\frac{3}{2}c a^{\prime}(z)+(a(z))^2\right]\right\ } \nonumber\\ & & + \frac{1}{2}\left(\frac{b_{\lambda}}{b^{*}}\right)^2 \left\{c^2 + r_{\perp}^2\left [ ca^{\prime}(z ) -2(a(z))^2\right]\right\ } -\frac{r_{\perp}^2 eb_{\lambda}^2}{2c \hbar b^ { * } } c^4=0 . \label{lens - eikonal - epsilon-2}\end{aligned}\ ] ] note that in the last equation , we introduced the reference value of the magnetic field @xmath58 associated with the quasiparticle energy @xmath38 . as is easy to check , this field is defined so that the corresponding magnetic length @xmath59 is comparable to the de broglie wavelength of the weyl quasiparticle @xmath60 . in fact , they are related as follows : @xmath61 . it should be noted that the validity of the quasiparticles energy ( [ second - energy-2 ] ) is restricted only to the case of sufficiently weak magnetic and pseudomagnetic fields , i.e. , @xmath62 . at the zeroth order in @xmath55 , eq . ( [ lens - eikonal - epsilon-2 ] ) reduces to the following equation : @xmath63 this equation has four nontrivial solutions , i.e. , @xmath64 where we used an expansion in powers of the small parameter @xmath65 . by taking into account that @xmath51 should be equal to one in the limit of vanishing fields , @xmath47 , we conclude that the physical solution is given by @xmath66 . now , by equating the terms quadratic in @xmath46 in eq . ( [ lens - eikonal - epsilon-2 ] ) , we obtain the following first - order differential equation for the function @xmath52 : @xmath67 where @xmath68 \left [ c^2-\frac{3}{2 } c^4 + \frac{\lambda b_{\lambda}}{b^ { * } } c-\frac{1}{2 } \left(\frac{b_{\lambda}}{b^{*}}\right)^2 \right]^{-1 } \simeq 1+\frac{5}{4 } \left(\frac{b_{\lambda}}{b^{*}}\right)^2 , \\ a_2 ^ 2&=&\frac { eb_{\lambda}^2}{4c \hbar b^ { * } } c^4 \left[\frac{3}{2 } c^4 -c^2 -\frac{\lambda b_{\lambda}}{b^ { * } } c+\frac{1}{2 } \left(\frac{b_{\lambda}}{b^{*}}\right)^2 \right]^{-1 } \simeq \frac { eb_{\lambda}^2}{2c \hbar b^ { * } } \left[1 - 2\lambda \frac{b_{\lambda}}{b^ { * } } \right ] . \label{lens - eikonal - epsilon - r-1-a2}\end{aligned}\ ] ] in the next section we will use eqs . ( [ lens - eikonal - epsilon - r-1-comp ] ) , ( [ lens - eikonal - epsilon - r-1-a1 ] ) , and ( [ lens - eikonal - epsilon - r-1-a2 ] ) to analyze the motion of weyl quasiparticles in magnetic and pseudomagnetic fields . let us begin the analysis from the simplest case of uniform magnetic and pseudomagnetic fields , @xmath69 . in the regions @xmath70 and @xmath71 , the fields are absent . therefore , @xmath72 and @xmath73 there and the solutions to eq . ( [ lens - eikonal - epsilon - r-1-comp ] ) are given by @xmath74 where @xmath75 and @xmath76 are the integration constants that will be fixed by the boundary conditions . on the other hand , by solving eq . ( [ lens - eikonal - epsilon - r-1-comp ] ) in the region with nonzero background fields , i.e. , @xmath20 , and using eqs . ( [ lens - eikonal - epsilon - r-1-a1 ] ) and ( [ lens - eikonal - epsilon - r-1-a2 ] ) , we obtain the following solution : @xmath77 where @xmath78 is another integration constant that should be also determined by matching the solutions for @xmath52 at @xmath79 and @xmath80 , i.e. , @xmath81 finally , after excluding @xmath78 from these equations , we obtain the following lens equation relating @xmath75 and @xmath76 for the quasiparticles of a given chirality @xmath82 : @xmath83 where @xmath75 and @xmath76 can be interpreted as the coordinates of the quasiparticle source and its image , respectively . indeed , we see that when the source is placed at the left focal point , i.e. , @xmath84 , the position of the image @xmath76 goes to infinity . similarly , when @xmath85 , the location of the image is near the right focal point , i.e. , @xmath86 . therefore , @xmath87 and @xmath88 are the locations of the principal foci and @xmath89 is the principal focal length . in the case under consideration , we find that @xmath90 and @xmath91 by making use of the small - field asymptotes for the functions @xmath92 and @xmath93 given by eqs . ( [ lens - eikonal - epsilon - r-1-a1 ] ) and ( [ lens - eikonal - epsilon - r-1-a2 ] ) , we then obtain the following explicit expressions for functions @xmath94 and @xmath89 : @xmath95 these analytical expressions are the key characteristics of the ( pseudo-)magnetic lens and are the main results of this paper . when the paraxial approximation is justified , these results should be valid for arbitrary weyl and dirac materials . as is easy to see from eq . ( [ lens - eikonal - lens - f - app-2 ] ) , the focal lengths for the quasiparticles of opposite chiralities are different in general . this remains true even in the limit of the vanishing pseudomagnetic field ( i.e. , @xmath96 but @xmath97 ) . in such a case a small quantitative difference between @xmath98 and @xmath99 is connected with the berry curvature effects . this is in contrast to the case of the vanishing magnetic field ( @xmath100 but @xmath101 ) , when the focal lengths for the quasiparticles of opposite chiralities are exactly the same in the formalism at hand . in order to get a better insight into the parametric dependencies of the principal focal length on the strength of the ( pseudo-)magnetic field , we will use the numerical value of the fermi velocity for cd@xmath0as@xmath1 @xcite , i.e. , @xmath102 . in this case , the characteristic magnetic field can be estimated as @xmath103 the dependence of the focal length @xmath89 on the pseudomagnetic field @xmath14 at two fixed values of the magnetic field @xmath100 and @xmath104 are shown in the left and right panels of fig . [ fig : focal - f - b5-b ] for @xmath105 , respectively . as expected , in the presence of both magnetic _ and _ pseudomagnetic fields , the focal lengths are different for the quasiparticles of opposite chirality . in practice , this property may provide an efficient means for a space separation of chiral charges inside a weyl material . as is easy to see from the analytical expression ( [ lens - eikonal - lens - f - app-2 ] ) , as well as from the numerical results in fig . [ fig : focal - f - b5-b ] , the dependence of the focal length on the ( pseudo-)magnetic field is quasi - periodic . also , focal length is formally divergent at the following discrete value of the background field : @xmath106 , where @xmath107 . the mathematical reason for the divergencies is clear from the analytical result in eq . ( [ lens - eikonal - lens - f - app-2 ] ) , which has the sine function in the denominator . it should be noted , however , that the paraxial approximation for the eikonal equation breaks down in the vicinity of such divergencies . this is clear from the fact that the expansion in powers of @xmath108 in eq . ( [ lens - eikonal - s0 ] ) becomes unreliable when @xmath52 is too large . in such a situation , a geometrical optics approach fails and one needs to use exact , rather than approximate solutions to the wave equation . given by eq . ( [ lens - eikonal - lens - f - app-2 ] ) for quasiparticles of chirality @xmath26 ( red solid line ) and @xmath25 ( blue dashed line ) as a function of pseudomagnetic field . while the left panel corresponds to zero magnetic field @xmath100 , the right one is plotted for @xmath109 . we set @xmath105 and @xmath110.,title="fig:",scaledwidth=45.0% ] given by eq . ( [ lens - eikonal - lens - f - app-2 ] ) for quasiparticles of chirality @xmath26 ( red solid line ) and @xmath25 ( blue dashed line ) as a function of pseudomagnetic field . while the left panel corresponds to zero magnetic field @xmath100 , the right one is plotted for @xmath109 . we set @xmath105 and @xmath110.,title="fig:",scaledwidth=45.0% ] in order to illuminate the effects of the berry curvature , it is instructive to compare the focal length in eq . ( [ lens - eikonal - lens - f - app-2 ] ) with its counterpart when such effects are neglected , i.e. , @xmath111 the corresponding relative difference @xmath112 is plotted in fig . [ fig : focal - f - compare ] . as we see , the effects of the berry curvature quantified by @xmath113 are about @xmath114 orders of magnitude smaller than the focal lengths . we checked , however , that the corresponding effects may become noticeable at sufficiently large magnetic and pseudomagnetic fields or near the divergencies . between the focal length @xmath115 given by eq . ( [ lens - eikonal - lens - f - app - landau ] ) and the focal length @xmath89 with the effects of the berry curvature ( [ lens - eikonal - lens - f - app-2 ] ) as a function of pseudomagnetic field . the red solid line denotes the right - handed quasiparticles and the blue dashed one corresponds to the left - handed ones . the left panel corresponds to zero magnetic field @xmath100 , while @xmath109 in the right one . we set @xmath105 and @xmath110.,title="fig:",scaledwidth=45.0% ] between the focal length @xmath115 given by eq . ( [ lens - eikonal - lens - f - app - landau ] ) and the focal length @xmath89 with the effects of the berry curvature ( [ lens - eikonal - lens - f - app-2 ] ) as a function of pseudomagnetic field . the red solid line denotes the right - handed quasiparticles and the blue dashed one corresponds to the left - handed ones . the left panel corresponds to zero magnetic field @xmath100 , while @xmath109 in the right one . we set @xmath105 and @xmath110.,title="fig:",scaledwidth=45.0% ] the model situation with a constant ( pseudo-)magnetic field @xmath116 inside a quasiparticle lens considered in the previous subsection is not very realistic . indeed , in the case of real solenoids and torsion - induced strains there are always fringing fields at the ends . in order to get a better insight into the underlying physics , in this subsection we study the case of a nonuniform ( pseudo-)magnetic field . in particular , we consider the effective field with the following spatial profile : @xmath117 such a configuration mimics well the fringing fields at the ends of the ( pseudo-)magnetic lens and , at the same time , allows one to obtain an analytical solution in the paraxial approximation . because of the the unit step functions @xmath118 in eq . ( [ inhom - b - profile ] ) , the background field @xmath119 is still effectively confined in the region where the background ( pseudo-)magnetic field is nonzero @xmath20 . by limiting ourselves to the case of sufficiently weak ( pseudo-)magnetic fields , we can neglect the effects of the berry curvature in the eikonal approximation . this is supported by our findings in the previous subsection , showing that the main contribution to the spatial separation of weyl quasiparticles with different chirality comes from the terms of the leading - order in the ( pseudo-)magnetic field . thus , by dropping the terms due to the berry curvature in the quasiparticle energy , we have @xmath120 instead of eq . ( [ second - energy-2 ] ) . in this case , the analog of the differential equation ( [ lens - eikonal - epsilon - r-1-comp ] ) reads @xmath121 it has the following analytical solution inside the solenoid : @xmath122 } , \label{inhom - b - eikonal}\end{aligned}\ ] ] where @xmath123 is an integration constant , and @xmath124 by matching the solutions outside the solenoid ( [ lens - eikonal - a2-sol-1 ] ) with that in eq . ( [ inhom - b - eikonal ] ) at @xmath79 and @xmath80 , we obtain the following lens equation : @xmath125 where @xmath126 , @xmath127 } + l \sin{\left[2f_{\lambda}\ , \mbox{arccot}\left(\frac{2\xi}{l}\right)\right]}}{2\left(4\xi^2f_{\lambda}^2-l^2\right ) \sin{\left[2f_{\lambda}\ , \mbox{arccot}\left(\frac{2\xi}{l}\right)\right]-8l\xi f_{\lambda } \cos{\left[2f_{\lambda}\ , \mbox{arccot}\left(\frac{2\xi}{l}\right)\right ] } } } , \label{inhom - b - g}\end{aligned}\ ] ] and @xmath128 } -4l\xi f_{\lambda } \cos{\left[2f_{\lambda}\ , \mbox{arccot}\left(\frac{2\xi}{l}\right)\right]}}. \label{inhom - b - f}\end{aligned}\ ] ] as is easy to check , @xmath129 , where @xmath115 is the solution in the case of the homogeneous field given by eq . ( [ lens - eikonal - lens - f - app - landau ] ) . the dependence of the focal length @xmath130 on the pseudomagnetic field strength @xmath14 is presented in fig . [ fig : focal - inhom - b5-b ] , where the left and right panels correspond to @xmath131 and @xmath132 , respectively . as we see from this figure , the effect of the inhomogeneity is weak at small values of the pseudomagnetic field , but it increases with @xmath14 . the case of moderately large @xmath133 , i.e. , @xmath134 , depicted in the right panel is almost indistinguishable from the case of uniform fields , as it should be for the weakly varying ( pseudo-)magnetic field . on the other hand , at small values of @xmath133 ( i.e. , @xmath135 ) , the dependence of @xmath130 on the ( pseudo-)magnetic field is very different . qualitatively , with decreasing @xmath133 , the period of the focal length oscillations increases . furthermore , we note that for @xmath136 , we can still use eq . ( [ lens - eikonal - lens - f - app - landau ] ) , but with the following replacement of the lens size : @xmath137 where we used the explicit form of @xmath138 in eq . ( [ inhom - b - profile ] ) in order to perform the integration . given by eq . ( [ inhom - b - f ] ) for quasiparticles of both chiralities at @xmath100 ( red solid line ) as well as for @xmath26 ( blue dashed lines ) and @xmath25 ( green dotted lines ) at @xmath139 . while the left panel represents the results at @xmath131 , the right one corresponds to @xmath132 . we set @xmath105 and @xmath110.,title="fig:",scaledwidth=45.0% ] given by eq . ( [ inhom - b - f ] ) for quasiparticles of both chiralities at @xmath100 ( red solid line ) as well as for @xmath26 ( blue dashed lines ) and @xmath25 ( green dotted lines ) at @xmath139 . while the left panel represents the results at @xmath131 , the right one corresponds to @xmath132 . we set @xmath105 and @xmath110.,title="fig:",scaledwidth=45.0% ] in this study we investigated the conceptual possibility of pseudomagnetic lenses that can be used to focus the beams of chiral quasiparticles in weyl materials . we found that the maximum flexibility in controlling the beams of weyl quasiparticles is achieved when one combines the magnetic and pseudomagnetic lenses , as shown schematically in fig . [ fig : illustration ] . indeed , the presence of both magnetic _ and _ pseudomagnetic fields allows one to achieve different magnitudes of the effective magnetic fields exerted on the left- and right - handed quasiparticles . this provides enough control to manipulate the focal lengths @xmath89 independently for each chirality . in order to obtain the geometric optics description for the beams of weyl quasiparticles in background magnetic and pseudomagentic fields , we used the conventional eikonal approximation for solving the semiclassical wave equation . such an approximation is well justified for weyl materials when the de broglie wavelengths of quasiparticles in the beam are small compared to the spatial scale of the background - field inhomogeneities , as well as the sample size . in this study we argued that , because of the nontrivial topology of weyl fermions , the corresponding eikonal equation is affected by the berry curvature in a nontrivial way . the corrections due to berry curvature effects are rather small when the magnetic and pseudomagnetic fields are much weaker than the reference value @xmath140 , defined in eq . ( [ lens - eikonal - bcrit ] ) , which is the scale set by the quasiparticle energy . this regime seems to be relevant for applications in dirac semimetals such as cd@xmath0as@xmath1 when the effective background field is weaker than about @xmath141 . for stronger fields , or for other types of dirac / weyl materials , it is possible that the berry curvature could lead to substantial corrections in the quantitative description of the ( pseudo-)magnetic lenses . in fact , these corrections may lead to a spatial separation of the beams of chiral quasiparticles even by the ordinary magnetic field alone . in such a case , a more refined analysis using quantum , rather than semiclassical equations would be required . in the framework of a simplified model with a spatially uniform ( pseudo-)magnetic field inside the lens , we derived the explicit general expressions for the principal foci and the focal length . the corresponding results include the effects due to the berry curvature and are reliable only in the case of not too strong background fields . in addition , in order to address the role of inhomogeneous fields , we also studied a simple model with a nonuniform pseudomagnetic field , see eq . ( [ inhom - b - profile ] ) . in this case , we found that the effect of a highly nonuniform field is similar to that of a uniform one , but with the effective length of the lens @xmath142 given by eq . ( [ inhom - b - leff ] ) . it is worth noting that in this study we solved the eikonal equation for weyl quasiparticles by using the paraxial approximation . clearly , this is adequate only when the beam of chiral quasiparticles remains close to the axis of the ( pseudo-)magnetic lens . in practice , of course , this condition may break down and , then , one would have to reanalyze the problem by using numerical methods . in such a regime , optical aberrations will appear and further complicate the situation . while all these issues may be of real importance for making pseudomagnetic lenses in practice , they are beyond the scope of the conceptual study presented here . it is intriguing to suggest that the pseudomagnetic lenses could allow for new and powerful ways of controlling and manipulating the chiral beams of quasiparticles inside dirac and weyl materials . for example , they may open an experimental possibility to control the spatial distributions of the electric current densities due to weyl quasiparticles , depending on their chirality . by focusing the beams of the left- and right - handed quasiparticles in different spatial regions , one could achieve a chiral distillation " and/or steady states of matter with a nonzero chiral asymmetry . this could allow various new applications that utilize the possibility of a _ chiral spectrometer _ , in which spatial separation of chiral charges may be detected via the local probes . the work of e.v.g . was partially supported by the program of fundamental research of the physics and astronomy division of the nas of ukraine . the work of v.a.m . and p.o.s . was supported by the natural sciences and engineering research council of canada . the work of i.a.s . was supported by the u.s . national science foundation under grant no . phy-1404232 . o. vafek and a. vishwanath , ann . condensed matter phys . * 5 * , 83 ( 2014 ) . a. a. burkov , j. phys . : condens . matter * 27 * , 113201 ( 2015 ) . h. b. nielsen and m. ninomiya , nucl . b * 185 * , 20 ( 1981 ) [ erratum : nucl . b * 195 * , 541 ( 1982 ) ] ; 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it is proposed that strain - induced pseudomagnetic fields in weyl materials could be used as chirality sensitive lenses for beams of weyl quasiparticles . the study of the ( pseudo-)magnetic lenses is performed by using the eikonal approximation for describing the weyl quasiparticles propagation in magnetic and strain - induced pseudomagnetic fields . analytical expressions for the locations of the principal foci and the focal length are obtained in the paraxial approximation in the models with uniform as well as nonuniform effective magnetic fields inside the lens . the results show that the left- and right - handed quasiparticles can be focused at different spatial locations when both magnetic and pseudomagnetic fields are applied . it is suggested that the use of the magnetic and pseudomagnetic lenses could open new ways of producing and manipulating beams of chiral weyl quasiparticles .

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Proceed to summarize the following text: granular matter is a generic name given to a system composed of macroscopic , athermal particles that have mutual repulsive , dissipative interactions @xcite . it is an intensely studied field in the physics community given the several distinct behaviors shown by such systems as a consequence of different external conditions imposed on them . one of such conditions is that which imposes a flow of particles , named granular flow @xcite . within the several granular flow examples , the flow around immersed obstacles has received some attention lately @xcite . one of the objectives of such investigations is to measure the force in the obstacle due to interactions with the flowing grains , the so called granular drag @xcite , analogously to the viscous flow force on an obstacle . on the other hand , one knows that a viscous fluid flow around an obstacle produces an additional force called lift , which is perpendicular to the flow . given the analogy between a viscous flow and the granular flow , a lift force should exist in an obstacle immersed in a granular flow under suitable conditions . however , most investigations focus only on the drag , while the lift studies are restricted only to a few experiments and simulations @xcite . soller and koehler @xcite showed that the lift on a rotating vane inserted in a granular packing scaled with geometric parameters of the system , such as an effective aspect ratio and immersion depth . the word effective means that the referred quantity , say the immersion depth , is considered taking into account the finite grain size . ding _ et al_. @xcite , dragging an intruder at constant velocity through a granular packing , showed that a lift force is induced on the intruder and it depends on its geometry . common to both investigations is the fact that the lift arises as a consequence of the hydrostatic nature of the stress in granular packings . given that these experiments were carried on such hydrostatic stress systems , variations of this condition might reveal different aspects of this force . for instance , is there a lift force in intruders immersed in flows where the stresses are not hydrostatic ? if so , in what conditions this force arises ? this paper is aimed at studying the lift force on an obstacle due to a dilute granular flow through numerical simulations . the approach will be identical to the one used in probing the drag on a cylinder due to a dilute granular flow @xcite . the argument drawn here to obtain an analytical expression for the lift shows that there is no net lift in such dilute conditions on a circular obstacle . hence , the obstacle chosen here is an elliptical one . the dependence of this force on the flow parameters will be obtained and studied numerically . section [ sec1 ] is reserved for developing the theoretical argument and presenting its predictions . in section [ sim ] , the simulation is described along with the numerical results for the lift . section [ analy ] holds analyses regarding the argument given in the previous section in order to understand the discrepancies between the theoretical and the numerical results . finally , section [ conclusions ] has the conclusions . in this section , the theoretical argument leading to an expression for the lift force on the ellipse is developed . also , some predictions are shown in order to be compared to the numerical results in the next section . in @xcite , the line of thought that led to an expression for the drag force on the circular obstacle was based on the dissipative collisions among grains and obstacle . the same line of thought can be drawn here in order to obtain an expression for the lift force on the ellipse due to collisions with the incoming stream . first of all , all disks are assumed to have only the horizontal velocity component , @xmath0 . therefore , for the ellipse given in fig . [ local_ellipse_frame ] , which is located at @xmath1,@xmath2 , a collision will occur only if the impact parameter ( the vertical distance between a disk s center and the horizontal line through the ellipse s center ) is in the range @xmath3,@xmath4 $ ] . this is so because @xmath5 and @xmath6 are , respectively , the lowest and the highest points in the ellipse , in the same way as @xmath7 and @xmath8 are the leftmost and the rightmost ones . any collisions that happen in the segment @xmath9 will exert a downward lift , while collisions that occur in the @xmath10 segment will produce an upward lift ( from now on , the @xmath9 segment , and all other segments in the text , will be referred to only as @xmath9 ) . moreover , since the lengths of both segments are , in general , unequal , the longest of them , in this case @xmath9 , will suffer more collisions than the other , which implies a net , negative , lift force . it is this mechanism that prevents any net lift to take place on a circular obstacle under these conditions ( in general , any body that is symmetrical with respect to the flow direction , and is fully immersed in the flow , does not suffer a net lift ) . therefore , the determination of the coordinates of these four points in the ellipse , @xmath7 , @xmath8 , @xmath5 and @xmath6 , is very important . in order to do this , one should notice that the tangent line to the ellipse is horizontal at @xmath6 and @xmath5 and vertical at @xmath7 and @xmath8 . therefore , starting from the tilted ellipse s equation in the global frame : @xmath11 where @xmath12 and @xmath13 , one can write for @xmath6 and @xmath5 : @xmath14 and @xmath15 where the plus(minus ) sign is for point @xmath6(@xmath5 ) . a similar calculation yields , for points @xmath7 and @xmath8 : @xmath16 and @xmath17 where in both eqs . the plus(minus ) sign is for point @xmath8(@xmath7 ) . the force due to a collision between a disk and the ellipse can be obtained by calculating the change in linear momentum of the disk . the total force is obtained by integrating the product of this individual momentum change and a suitable collision rate over the obstacle section facing the flow , i.e. , integrating over @xmath18 . suppose a disk hits the obstacle in @xmath9 . figure [ coll_geo ] presents schematically the geometry of the collision ( assumed frictionless ) . the grain velocity is @xmath19 , and the impact angle is @xmath20 , which is the one between the direction @xmath21 and the normal at the contact point . the post - collisional velocity is @xmath22 . in order to obtain the final components as a function of @xmath0 , one has two equations that relate the normal and tangential velocities before and after the collision : @xmath23 and @xmath24 , where @xmath25 , @xmath26 are the normal and tangential unit vectors at the collision point and @xmath27 is the normal restitution coefficient , assumed velocity independent . from these considerations , the components of the disk velocity after the collision are @xmath28 and @xmath29 . therefore , the vertical momentum change for a general collision is given by : @xmath30 the number of disks that strike the ellipse in a time @xmath31 is given by the number of particles within the area of the parallelogram with sides @xmath32 and @xmath33 , this last quantity being the arc length around the collision point . given the geometry of the collision ( figure [ coll_geo ] ) , the area of this parallelogram is @xmath34 . hence : @xmath35 where @xmath36 is the area fraction of the incoming flow . the collision frequency is simply this number divided by @xmath31 . the arc length @xmath33 is given by the product of the collision radius , @xmath37 , by the arc element , @xmath38 . this product is readily evaluated for a circle . however , in the ellipse case , this is not so simple , since this radius varies with @xmath39 . besides , the grain size introduces an additional complication for calculating @xmath37 , as seen in fig . [ coll_tri ] , which shows the triangle formed by the disk center , the ellipse center and the collision point . hence , the ellipse is assumed much larger than the disks and the disk size only enters the expression for @xmath37 as a minor correction . with this assumption , the arc length is given by @xmath40 . in parametric form , @xmath41 and @xmath42 , this length reads : @xmath43 finally , the effective collision radius is given by : @xmath44 another difficulty to obtain the expression for the lift force is that the momentum change ( i.e. , the force ) depends explicitly on @xmath20 , while the collision frequency depends on @xmath39 . therefore , any hope of integrating the lift force over all collisions should pass through obtaining the relationship between the angles @xmath20 and @xmath39 . this is done in the following paragraph . since @xmath45 gives the tangent of the angle between the normal line to a point in the ellipse and the @xmath46 axis , from the ellipse equation ( [ ellipse_eq_glob_frame ] ) , one has : @xmath47 where @xmath48 , @xmath49 and @xmath50 are the coefficients of the terms @xmath51 , @xmath52 and @xmath53 in eq . ( [ ellipse_eq_glob_frame ] ) . since @xmath54 and from eqs . ( [ x_dc ] ) , ( [ y_dc ] ) , ( [ x_ab ] ) and ( [ y_ab ] ) , the relation between @xmath20 and @xmath39 can be cast in terms of the coordinates of the points @xmath6 and @xmath7 , since : @xmath55 and @xmath56 hence : @xmath57 the lift force can now be evaluated as : @xmath58 where @xmath59 . by using eqs . ( [ py_change ] ) , ( [ coll_freq ] ) and ( [ radius_param ] ) , the total lift on the ellipse due to collisions is given by : @xmath60 where @xmath61 is the flow mass density and the arc length is to be calculated bearing in mind ( [ theta_alpha_rel ] ) . the lift force scales with the obstacle size @xmath62 , as expected , since it depends linearly on the arc length . also , it is seen that , for more inelastic grains the lift is smaller , since more inelasticity implies less intense change of momentum , see eq . ( [ py_change ] ) . the dependence of the lift on the tilt angle is hidden in the integral over @xmath18 , because the lengths of @xmath10 and @xmath9 , which contribute forces with distinct signs , are unequal for a general @xmath63 . apart from the dependence of @xmath64 in these parameters , it also depends on the ellipse eccentricity , i.e. , on the ratio of the ellipse s axes @xmath65 . for @xmath66 , the obstacle is a circle , and the lift vanishes by symmetry . in the other extreme value , @xmath67 , which corresponds to a flat plate , the lift vanishes only for @xmath68 ( horizontal plate ) and @xmath69 ( vertical plate ) , which are the only symmetrical orientations with respect to an horizontal flow . in fig . [ lift_theo_k ] , it is shown the predictions of eq . ( [ lift_ad ] ) for distinct values of @xmath70 ( the parameter values in plotting these curves were chosen to agree with those used in the simulations ) . one can see that the maximum ( absolute ) value of @xmath64 increases when @xmath70 decreases . this happens because when @xmath71 , the points @xmath6 and @xmath5 merge with points @xmath8 and @xmath7 , as can be inferred from eqs . ( [ x_dc ] ) , ( [ y_dc ] ) , ( [ x_ab ] ) , and ( [ y_ab ] ) . in this case , @xmath72 and @xmath73 . since the lift is the difference of the contributions from @xmath9 and @xmath10 , all collisions will give positive contributions to the net lift value , and it should reach its maximum for a particular @xmath63 . the objective of the simulations is to study the lift force on the obstacle as a function of four parameters that appear on eq . ( [ lift_ad ] ) , namely , the obstacle size , @xmath62 , the restitution coefficient , @xmath27 , the ellipse eccentricity , @xmath70 , and the tilt angle , @xmath63 . in this section , the simulation is detailed , along with the parameter values used in the computations , and the numerical results for the lift are shown . the system is composed of @xmath74 soft disks , with unity diameter @xmath75 and unity mass @xmath76 , located in a workspace of lengths @xmath77 and @xmath78 in the horizontal and vertical directions , respectively . they interact through normal and tangential forces , according to the model used in @xcite . the normal force between two disks is : @xmath79 where the first term is the conservative part , given by a simple harmonic spring force @xmath80 where @xmath81 is the spring constant , @xmath82 is the @xmath83-th disk diameter , @xmath84 is the distance between the two disks and @xmath85 is a unit vector along the normal between the disks centers . the second term is a dissipative , velocity dependent force , given by @xmath86 where @xmath87 is the normal damping coefficient and @xmath88 is the relative velocity between the contacting disks . the tangential force at the contact point is given by the following expression : @xmath89 where @xmath90 is the sliding friction constant and @xmath91 , the static friction coefficient , while @xmath92 is the unit vector along the direction of the relative velocity at the contact point . this vector is calculated as : @xmath93 where @xmath94 is the @xmath83-th disk angular velocity and @xmath95 . the total contact force @xmath96 vanishes if the disks are not in contact , i.e. , if @xmath97 . the values of the elastic parameters used were : @xmath98 , @xmath99 and @xmath100 , @xmath101 and @xmath102 . the restitution coefficient @xcite for these parameters were @xmath103 , for @xmath99 , and @xmath104 , for @xmath100 . the ellipse s major and minor half - axes are @xmath62 and @xmath105 , respectively . the parameter @xmath62 has the values @xmath106 , @xmath107 , @xmath108 , @xmath109 and @xmath110 , with @xmath111 . for the studies of the dependence on @xmath70 , size @xmath112 was used , with @xmath113 , @xmath114 and @xmath115 . all these cases were studied for both @xmath27 values . for the two smaller obstacle sizes , results were obtained with a @xmath116 and @xmath117 packing , while for the others , a @xmath118 and @xmath119 packing was used . in both cases , the packing fraction was about @xmath120 . the ellipse is located in @xmath121 and @xmath122 , and its major half - axis is tilted related to the horizontal axis ( flow direction ) by @xmath63 . the values of the tilt angle were in the range @xmath123,@xmath124 $ ] , divided in @xmath125 increments , which gives a total of @xmath126 distinct @xmath63 values . at the beginning , all disks are randomly generated without overlap among them and the ellipse . at this stage , the obstacle is modeled as a circle with diameter @xmath127 since this facilitates the overlap check . all disks have initial velocity @xmath19 , where @xmath21 is the unit vector in the horizontal ( flow ) direction and @xmath128 . the system has periodic boundaries perpendicular to the flow direction , while , along the flow , the conditions are the same as in @xcite : whenever a disk leaves the system through the right boundary , it is placed at the left one in a random vertical position . in fact , the code searches for a position where the incoming grain overlaps with no other disk , which is fairly easy , giving the low density that is used here . its velocity is set as the incoming flow velocity @xmath129 plus a small random vertical component chosen uniformly in the interval @xmath130 $ ] , where @xmath131 ( results obtained with @xmath132 differ little from those shown here ) . the interactions between the disks and the ellipse are the same as those given above for two disks . the main problem is to resolve particle - ellipse contacts . in order to do this , all disks are checked for overlap with a circle of diameter @xmath127 . if one disk overlaps with the circle , it is mapped in the ellipse s local frame of reference ( elfr ) . then , the contact point coordinates , in the elfr , are calculated using the algorithm proposed in @xcite . the direction of the normal force is along the line joining the disk center and the contact point , while the tangential force is perpendicular to this direction . lengths , forces and time are given in units of @xmath75 , @xmath133 , and @xmath134 , respectively . equations of motion are integrated with a leapfrog scheme @xcite , with a time step of @xmath135 . each simulation is performed during @xmath136 molecular dynamics ( md ) cycles for thermalization and @xmath137 md cycles for measurements . all results are averaged over @xmath108 and @xmath107 independent runs for the @xmath116 and @xmath118 packings , respectively . the main interest is to measure the force exerted on the obstacle by the stream of grains due to the collisions among them . therefore , the components of this force , drag and lift , are measured at all cycles after thermalization . at regular intervals , the forces are recorded as averages over this period . this accumulation interval is @xmath138 md cycles long ( results were also obtained with @xmath139 md cycles long accumulation periods and do not differ appreciably from those reported here ) . the flow velocity and particle number density fields , @xmath140 and @xmath141 , were measured as follows : the workspace is divided in square bins of side @xmath75 . at each cycle , all particles are mapped in an appropriate bin and its velocity components are added to the respective field element . similarly , angular profiles related to the contact angle between the grains and the ellipse , the collision angle @xmath39 , were measured . they are the collision number and the velocity components . each time there is a disk - ellipse contact , the angle formed by the line joining the collision point and the ellipse center with the flow direction is calculated . then , the above mentioned quantities are added to the appropriate profile bins . the field and the profiles are presented as averages over cycles and runs . in fig . [ lift_k ] , the numerical results for the lift as a function of the tilt angle and the eccentricity are shown . comparing figs . [ lift_theo_k ] and [ lift_k ] , it is clear that eq . ( [ lift_ad ] ) captures the qualitative features of the lift force , even though the correspondence becomes weak for @xmath142 and @xmath103 . also , the results agree with the prediction that the lift should be higher for less inelastic flows . finally , the theoretical result overestimates the numerical ones by , roughly , a factor of @xmath143 , for these cases . in fig . [ lift_phi ] , the numerical results for the lift force as a function of the tilt angle and obstacle size are shown . as expected , the lift increases with the obstacle size . however , the lift does not grow linearly with @xmath62 . plotting these data against the obstacle size shows that @xmath144 . since the results are not conclusive , it suffices to acknowledge that more simulations are needed in order to obtain a more reliable scaling with the obstacle size . also , this figure shows that the lift force for @xmath104 packings grows faster with @xmath62 than for those with @xmath103 . finally , the numerical results , as seen in fig . [ lift_k ] , also are smaller than their theoretical counterparts ( for @xmath145 , the factor is about @xmath146 ) . this shows that the difference between model and theory depends only a little on the tilt angle , while the obstacle size , inelasticity and eccentricity are the factors that accept the most on this difference . the reasons behind the failure to reproduce the numerical results will be analyzed in the next section , where the assumption that led to eq . ( [ lift_ad ] ) will be reviewed in detail . there are three basic hypotheses on which the argument for the lift force in section [ sec11 ] was built . the first one is that particle - obstacle interactions were frictionless , despite the force model , eq . ( [ tang_force ] ) , is not . second , the arc length expression , @xmath40 , does not take into account the fact that the grains have a finite diameter @xmath75 . finally , the collision frequency expression ( [ coll_freq ] ) was obtained under the hypothesis that only one particle at a time hits the obstacle , i.e. , a disk - ellipse collision does not affect the next collision , a condition met only in very dilute or in ideal gas flows ( which is not the case here ) . therefore , its applicability here is clearly questionable , as inferred from the system configuration shown in fig . [ config_4_6 ] . there is a dense region that begins above point @xmath7 and forms a separation boundary in front of the ellipse . this is the typical granular shock wave @xcite that forms around bodies immersed in fast granular flows . clearly , the dilute flow assumption is not valid . figure [ vel_fields ] shows the clear signature of this structure in the velocity field . also , this picture shows that the two branches of the shock wave meet behind the obstacle . this fact could , at first sight , invalidates the lift results because this clearly does not allow the use of periodic boundaries perpendicular to the stream . this is not the case , however , as seen in the results for the lift obtained from simulations that used rigid walls in the @xmath147 direction . these results agree with those shown in figs . [ lift_k ] and [ lift_phi ] within numerical error . nevertheless , periodic boundaries in a particular direction should only be used when the system is independent in that direction . c + + before discussing the influence of the shock wave in the results , a few brief comments will be made regarding the first two assumptions in the theoretical argument . to take into account the grain size in computing the collision radius is to consider a larger ( effective ) obstacle facing the flow , which would increase the theoretical prediction for the lift . since this quantity is already larger than the numerical results , it was safe to ignore it in the calculations . the influence of the friction force on the lift can be inferred from the velocity field results , fig . [ vel_fields ] . as seen in this figure , particles follow tangential trajectories along the ellipse @xcite , which means that they slide along the obstacle . since friction is a tangential force , one concludes that a grain sliding along @xmath9 exerts an upward lift . by symmetry , a disk exerts a downward lift while sliding along @xmath10 . therefore , the net effect of friction is to decrease the lift exerted by collisions in each segment . the combination of both effects might decrease or increase the net lift . the numerical results for the lift rising from friction show that it is opposite to the one resulting from the normal force . in other words , friction decreases the net lift . the effect is small , though , because the sliding friction is also small , @xmath101 . in most simulations of granular matter , regardless of the tangential force model , the sliding friction is only mildly lower than the normal interaction parameter . here , a much lower value was used , which could greatly affect the results . simulations performed with a distinct set of parameters , @xmath148 and @xmath149 , values which are more common to general simulations of granular matter , show that the results do not change significantly , and are still qualitatively the same as those in figs . [ lift_k ] and [ lift_phi ] . in fact , the lift data are reduced only by @xmath150 to @xmath151 . this happens due to the fact that the velocities involved in the simulations are high enough for the tangential force to reach the coulomb static friction criterion , which caps its maximum value and limits its influence on the lift . this conclusion is supported by the fact that the fraction of all disk - obstacle contacts , for @xmath148 , that reach the failure condition is about @xmath152 , with only a very small @xmath63 dependence . as stated earlier , the dilute flow assumption does not hold . a rigorous calculation of the force in the obstacle , in which all dense packing effects are taken into account , is not a simple matter . in @xcite an attempt was made in this direction . the formation of the shock wave produces a dense region that shields the obstacle from the incoming particles . since the interactions are dissipative , particles should reach the obstacle with reduced velocities . this clearly reduces the force exerted by the disks . also , given the large density within the shock wave , the collision rate might be affected . finally , as seen from the velocity field data , fig . [ vel_fields ] , particles should hit the obstacle , on average , with a non vanishing vertical velocity component , i.e. , particles suffer oblique inelastic collisions , which could affect the force on the obstacle . both effects will be discussed in more detail in the following two subsections . this discussion will by made only qualitatively , since more simulations are needed to determine the amount of influence each of them has on the results . one of the consequences of the existence of the shock wave is that the obstacle is shielded from the flow , in a way that the grains hit it with lower horizontal velocity than the upflow value . this fact alone indicates that the numerical net lift should be lower than its theoretical prediction . aside from reducing the incoming flow velocity , the shock wave forms a dense region around the obstacle through which grains should go through in order to reach it . this could affect the collision rate and , in turn , the net lift . both effects must play a role in order to explain the faster growth of the lift force with the obstacle size for stronger inelasticity flow compared to that of lower inelasticity one . the results for the collision profiles show that , for @xmath104 , the shock wave is more localized around point @xmath7 ( where , for the @xmath113 obstacles the peak of the profiles is located ) and that the overall collision number is greatly increased compared to the number when @xmath103 . this is a consequence of the aggregation that grains suffer due to the inelastic interactions and is common to all cases studied . this fact implies that the force on the obstacle should increase due to this increase in the collision rate and could compensate , in part , for the decrease in the force due to the reduced incoming velocity . the collision profiles are changed , in a very different way , for obstacles with distinct eccentricities , @xmath70 . [ coll_ang_prof_2 ] has results that illustrate this effect . the most striking feature of these curves is that the profiles for @xmath142 develop two peaks , instead of the more familiar one around point @xmath7 , which is still present , but it moves away from this point as the ellipse is oriented vertically . the other one develops around point @xmath6 and is seen at tilt angles as high as @xmath153 . such features are also seen in the @xmath154 results , although with smaller peaks . these results can be explained by the fact that in some cases , for stream velocities below a critical value , there is the appearance of a gap , filled with a hot granular gas , between the obstacle and the shock wave @xcite . in the present case , the appearance of the gap is a function of @xmath70 alone , since the stream velocity is the same for all simulations . figure [ rho_f ] has a result for the density field that indicates the presence of the gap . notice the denser arc formed right in front of the ellipse , the region limited by this arc and the obstacle is the gap . as a last note , the collision rate increases faster with @xmath63 for @xmath142 than for @xmath113 . [ configs ] has two configuration snapshots that illustrate the shock wave for two obstacles with distinct eccentricities . c + + one can infer from these pictures the reason behind the asymmetrical peaks in the collision profiles when @xmath155 and @xmath142 . there is a very small concentration of particles in front of @xmath10 , which certainly allows for more collisions to occur . these considerations still left unanswered the question about the faster growth of @xmath156 with @xmath62 for stronger inelasticity compared to those at lower inelasticity . since the net lift is the difference of the forces exerted in @xmath10 and @xmath9 , one should seek a relative measure of the speeds in each of them in order to account for the size of the contributions each has in the net value . this is done by calculating the ratio between the average horizontal disk velocities in @xmath9 and @xmath10 . if this ratio is greater than @xmath157 , the collisions on @xmath9 exert , on average , a stronger force than those on @xmath10 . these averages are evaluated directly from the simulations . the results , for all tilt angles and obstacle sizes , are given in fig . [ vx_ratio ] . first it can be seen that except in a few cases , markedly at small obstacles , all ratios are larger than @xmath157 , and those for @xmath104 cases are larger than those for @xmath158 . it is possible to identify a trend in these graphs , despite the noise : the ratio grows with obstacle size . this is consistent with the results in fig . [ lift_phi ] , where the net lift grows faster with @xmath62 for the @xmath104 flows compared to those with @xmath103 . [ vx_ratio_k ] has the data for the same quantity measured for obstacles with distinct eccentricities . it is seen that the ratio increases as @xmath70 decreases ( it reaches a factor of @xmath143 for @xmath142 ) . these data , together with those for the collision profiles , show that the large lift observed for low @xmath70 obstacles is a consequence not only of a larger number of collisions at @xmath9 , but also due to collisions with larger speeds than those at @xmath10 . the last piece of this analysis is the role of the vertical velocity component in the lift value . before proceeding , the collision argument of section [ sec1 ] is reviewed by allowing the incoming disks to have a vertical velocity , @xmath159 , and calculating the new momentum change in each collision . this will elucidate the effect of the vertical velocity on the lift value . the velocity of a disk which collides with the obstacle is @xmath160 . the final disk velocity for a inelastic , frictionless collision is obtained as : @xmath161 and @xmath162 the vertical momentum change due to this collision is given by : @xmath163 using the result for @xmath164 , ( [ new_vy ] ) , and performing some algebra , the new momentum change is : @xmath165,\ ] ] where @xmath166 . by looking at eq . ( [ new_p_change ] ) one can see that a collision that happens in the range @xmath167 , which corresponds to @xmath10 , will produce a lower momentum change if @xmath168 . similarly , for collisions that take place in the range @xmath169 , which is @xmath9 , a lower momentum change will occur if @xmath170 . from the velocity field data , fig . [ vel_fields ] , it is clear that the average vertical velocities in @xmath9 and @xmath10 are positive and negative , respectively . the oblique impacts decrease the positive lift exerted by the flow in @xmath10 , but also decrease the negative lift in @xmath9 . therefore , it is not obvious if both effects combined increase or decrease the net lift . in order for this combination of effects to increase the net lift , the vertical velocity in @xmath9 should be lower than the corresponding velocity in @xmath10 . in figs . [ vy_ratio ] and [ vy_ratio_k ] , the absolute value of the ratio of the average vertical velocities in both segments is shown for distinct obstacle sizes eccentricities , respectively . it is seen that the ratio , for @xmath104 , depends little on the obstacle size and restitution coefficient , and that those for @xmath103 grow with @xmath62 . it varies strongly with tile angle and eccentricity . despite a few cases , the ratio is larger than @xmath157 which implies that the net lift decreases due to oblique impacts , as provided by the existence of the shock wave a theoretical argument and numerical results on the net lift force exerted by a granular stream of equal disks on an ellipse were presented . the argument used to obtain a theoretical expression for the lift relied on inelastic , frictionless collisions of a very dilute granular flow , in the horizontal direction , and the ellipse . the expression obtained , eq . ( [ lift_ad ] ) , captures nicely the qualitative features of the numerical results . it does not , however , reproduce the results quantitatively , neither does it reproduce the difference in the lift force for @xmath103 and @xmath104 . the numerical data , shown in figs . [ lift_k ] and [ lift_phi ] , are lower than the theoretical prediction by a factor that depends mostly on the obstacle size ( for @xmath145 , eq . ( [ lift_ad ] ) is about @xmath146 times larger than the measured results ) . additional analyses of the flow properties showed that , from the assumptions drawn in the text , the dilute flow one is the most problematic . the existence of the granular shock wave @xcite invalidates this hypothesis . this structure affects directly the velocity components of the disks at contact with the obstacle as well as the collision rate . the horizontal velocity at contact decreases due to dissipative collisions that occur before the disks reach the shock wave . also , some of the horizontal momentum is deviated by the shock wave around the obstacle , introducing a vertical component whose net effect is to decrease the lift . perspectives to this work include a deeper analysis of dense packing effects on the lift . this would answer questions about the scaling of the lift with obstacle size , flow speed and density . such analysis would also allows for a better understanding of the mechanism behind the appearance of the net lift , since an obvious consequence of the presence of the shock wave is that any collisions that occur on its edge should transmit some linear momentum through the dense region until pushing the disks closets to the obstacle , and that is when the actual force is made . force transmission models , such as those in @xcite can be good starting points for this analysis . moreover , the effect of eccentricity on the shock wave is something worth investigating since , as seen here , for a constant flow speed , there should be some eccentricity value that allows for the appearance of the gap between the shock wave and the obstacle . finally , the hydrodynamical approach to granular flow could also be tested in such a situation . as argued in the text , ignoring the disk size does not affect the results for large obstacles . hence , if the obstacle is much larger than the disks , the predictions of such theory should be realized in simulations , since scale separation could be achieved , at least approximately . i thank a. p. f. atman and j. c. costa for a critical reading of this manuscript . this work is financially supported by cnpq and fapespa . 100 h. m. jaeger , s. r. nagel , r. p. behringer , rev . phys . * 68 * , 1259 ( 1996 ) . k. wieghardt , annu . fluid mech . * 7 * , 89 ( 1975 ) . c. s. campbell , annu . fluid mech . * 22 * , 57 ( 1990 ) . i. goldhirsch , annu . fluid mech . * 35 * , 267 ( 2003 ) . gdr midi , eur . j. e * 14 * , 41 ( 2004 ) . y. amarouchene , j. f. boudet , h. kellay , phys . lett . * 86 * , 4286 ( 2001 ) . e. c. rericha , c. bizon , m. d. shattuck , h. l. swinney , phys . lett . * 88 * , 014302 ( 2002 ) . j. f. boudet , y. amarouchene , h. kellay , phys . . lett . * 101 * , 254503 ( 2008 ) . r. albert , m. a. pfeifer , a .- barabasi , p. schiffer , phys . * 82 * , 205 ( 1999 ) . d. chehata , r. zenit , c. r. wassgren , phys . fluids * 15 * , 1622 ( 2003 ) . v. buchholtz , t. poeschel , granular matter * 1 * , 33 ( 1998 ) . c. r. wassgren , j. a. cordova , r. zenit , a. karion , phys . fluids * 15 * , 3318 ( 2003 ) . pica ciamarra m. _ et al_. , phys . lett . * 92 * , 194301 ( 2004 ) . r. soller , s. a. koehler , phys . e * 74 * , 021305 ( 2006 ) . y. ding , n. gravish , d. i. goldman , phys . * 106 * , 028001 ( 2011 ) . d. c. rapaport , _ the art of molecular dynamics simulation _ , 2nd ed . , cambridge university press ( cambridge , 2007 ) . l. e. silbert _ et al_. , phys . e * 64 * , 051302 ( 2001 ) . a. dziugys , b. peters , int . . anal . meth . geomech . * 25 * , 1487 ( 2001 ) . w. h. press , b. p. flannery , s. a. teukolsky , w. t. vetterling , _ numerical recipes in c _ , 2nd edition , cambridge university press ( cambridge , 1986 ) . y. h. deng , j. j. wylie , q. zhang , phys . e * 82 * , 011307 ( 2010 ) . et al_. , science * 269 * , 513 ( 1995 ) . s. ostojic , d. panja , europhys . lett . * 71 * , 70 ( 2005 ) .

this paper investigates the lift force exerted on an elliptical obstacle immersed in a granular flow through analytical calculations and computer simulations . the results are shown as a function of the obstacle size , orientation with respect to the flow direction ( tilt angle ) , the restitution coefficient and ellipse eccentricity . the theoretical argument , based on the force exerted on the obstacle due to inelastic , frictionless collisions of a very dilute flow , captures the qualitative features of the lift , but fails to reproduce the data quantitatively . the reason behind this disagreement is that the dilute flow assumption on which this argument is built breaks down as a granular shock wave forms in front of the obstacle . more specifically , the shock wave change the grains impact velocity at the obstacle , decreasing the overall net lift obtained from a very dilute flow .

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Proceed to summarize the following text: seeger s mass formula @xcite was given in 1961 , with its constants fitted to ground - state ( g.s ) binding energies of some 488 nuclei available at that time . the temperature t - dependence of these constants was later introduced by davidson _ @xcite on the basis of thermodynamical considerations of the nucleus . these constants , however , need be fitted again since a large amount of data on experimental g.s . binding energies @xcite , and their theoretically calculated values @xcite for , not - yet observed , neutron- and proton - rich nuclei have now become available . furthermore , the t - dependence of the constants , in particular the pairing constant @xmath9 , need be looked in to because of their recent un - successful use in calculating the decay properties of some excited compound nuclear systems @xcite-@xcite . note that our aim here is not to obtain a new set of constants for seeger s mass formula , but simply to include the t - dependence on experimental binding energies @xmath6 . for this purpose , a readjustment of only two of the four constants , the bulk constant @xmath0 and the neutron - proton asymmetry constant @xmath1 , are enough to obtain the @xmath6 within @xmath21.5 mev . a similar job was first done in @xcite for nuclei up to z=56 , and then in @xcite up to z=97 , but is redone here with an improved accuracy and up to z=118 . thus , the domain of the work is extended to neutron - deficient and neutron - excess nuclides where @xmath6 are not available , but theoretical binding energies @xmath7 are available @xcite . these re - fitted constants have been successfully used in the number of recent calculations @xcite-@xcite for studying the decay of hot and rotating compound nucleus ( cn ) formed in heavy ion reactions over a wide range of incident centre - of - mass ( c.m . ) energies . a brief outline of the seeger s mass formula , and the methodology used to workout the temperature - dependent binding energies , are presented in section ii . possible applications of the liquid drop energy in heavy ion reaction studies are also included in this section . the calculations and results are given in section iii , together with the table of fitted constants , which could be of huge importance for people working in the relevant area of nuclear physics . finally , the results are summarized in section iv . according to the strutinsky renormalization procedure , the binding energy @xmath10 of a nucleus at temperature t is the sum of liquid drop energy @xmath11 and shell corrections @xmath12 @xmath13 where @xmath3 is the semi - empirical mass formula of seeger @xcite , with t - dependence introduced by davidson _ @xcite , and @xmath5 taken as the `` empirical '' formula of myers and swiatecki @xcite , also made t - dependent to vanish exponentially , @xmath14 with @xmath15=1.5 mev @xcite . seeger s liquid drop energy @xmath3 , with its t - dependence due to davidson _ _ , is @xmath16{}\nonumber\\ & & + \bigl(\frac{z^{2}}{r_0(t)a^{\frac{1}{3}}}\bigr ) \bigl[1-\frac{0.7636}{z^{\frac{2}{3}}}-\frac{2.29}{[r_0(t)a^{\frac{1}{3}}]^{2}}\bigr]{}+\delta(t){\frac{f(z , a)}{a^{\frac{3}{4 } } } } , \label{eq:3}\end{aligned}\ ] ] with @xmath17 and , respectively , for even - even , even - odd and odd - odd nuclei , @xmath18 seeger s constants of 1961 are @xcite : @xmath19 and the pairing energy @xmath20=33.0 mev from @xcite . in the following , the bulk constant @xmath21(0 ) , and the neutron - proton asymmetry constant @xmath1 , are found enough to be readjusted/ refitted to obtain the @xmath6 . [ = @xmath22 in mev as a function of neutron number n for z=97 , calculated by using the experimental data ( solid circles ) @xcite , theoretical data ( open circles ) @xcite , with newly fitted constants ( crosses and down open triangles ) and with the 1961 seeger s constants @xcite ( hollow squares ) . ] ) ] calculated for the decay of @xmath23ni@xmath24 formed in @xmath25s+@xmath26 mg reaction at t=3.60 mev for @xmath27=0 and 36 @xmath28 , and also at t=0 for @xmath27=0 @xmath28 . ] the t - dependence of the constants in eq . ( [ eq:3 ] ) were obtained numerically by davidson _ @xcite from the available experimental information on excited states of 313 nuclei in the mass region 22@xmath29250 by determining the partition function @xmath30 of each nucleus in the canonical ensemble and making a least squares fit of the excitation energy @xmath31 to the ensemble average @xmath32 the constants @xmath33 , @xmath34 , @xmath35 , @xmath36 and @xmath9 are given in fig . @xcite for t@xmath374 mev , extrapolated linearly for higher temperatures . however , @xmath9 is constrained to be positive definite at all temperatures , and with @xmath9=0 for t@xmath382 mev . also , for the bulk constant @xmath33 , instead , an empirically fitted expression using fermi gas model is obtained , as @xmath39 for shell effects @xmath5 , the empirical formula of myers and swiatecki @xcite is @xmath40 \label{eq:7}\ ] ] where @xmath41 with @xmath42 or @xmath43 , and @xmath44 . @xmath45 are the magic numbers 2 , 8 , 14 ( or 20 ) , 28 , 50 , 82 , 126 and 184 for both neutrons and protons . the constants @xmath46=5.8 mev and @xmath47=0.26 mev . note that the above formula is for spherical shapes , but the missing deformation effects in @xmath5 are included here to some extent via the readjusted constants of @xmath3 since we essentially use the experimental binding energies split in to two contributions , @xmath3 and @xmath5 , for reasons of adding the t - dependence on it . finally , as an application of the two components [ @xmath11 and @xmath12 ] of the ( t - dependent ) experimental binding energy in the field of heavy - ion reactons , we define the collective fragmentation potential @xmath49+\sum_{i = 1}^{2}[\delta u_i]\exp(-t^2/t^2_0){}+v_c(r , z_i,\beta_{\lambda i},\theta_i , t)\nonumber\\ & & + v_p(r , a_i,\beta_{\lambda i},\theta_i , t){}+v_{\ell}(r , a_i,\beta_{\lambda i},\theta_i , t ) , \label{eq:8}\end{aligned}\ ] ] where the nuclear proximity @xmath50 , coulomb @xmath51 and the angular - momentum @xmath27-dependent @xmath52 potentials are for deformed and oriented nuclei and are also t - dependent . for details , see , e.g. , ref . @xcite . based on @xmath53 at fixed r and t , and the scattering potential @xmath54 at fixed @xmath55 and @xmath56 , we calculate the cn decay cross - section by using the dynamical cluster - decay model ( dcm ) of gupta and collaborators @xcite-@xcite , worked out in terms of the decoupled collective coordinates of mass ( and charge ) asymmetry @xmath57 [ @xmath58 and relative separation r. in terms of these coordinates , using @xmath27 partial waves , the cn decay cross section is defined as @xmath59 where the preformation probability @xmath60 , refering to @xmath55 motion , is the solution of stationary schrdinger equation in @xmath55 at a fixed r , and @xmath61 , the wkb penetrability refers to r motion , both quantities carrying the effects of angular momentum @xmath27 , temperature t , deformations @xmath62 and orientations @xmath63 degrees of freedom of colliding nuclei with c.m . energy @xmath64 . @xmath65m$ ] , is the reduced mass , with m as the nucleon mass . ( [ eq:9 ] ) is applicable to the decay of cn to light particles ( lps , a@xmath374 , z@xmath372 ) , intermediate mass fragments ( imfs , 2@xmath37z@xmath3710 ) , the fusion - fission fragments and the quasi - fission ( q.f . ) process where the incoming channel does not loose its identity , i.e. , @xmath66=1 for qf . the @xmath67 could be fixed for the vanishing of the fusion barrier of the incoming channel , or the light particle cross - section @xmath680 , or else defined as the critical @xmath69}/\hbar$ ] . , using eq . ( [ eq:9 ] ) , for the decay of compound system @xmath23ni@xmath24 formed in @xmath25s+@xmath26 mg reaction at t=3.60 mev , taking pairing constant @xmath70=0 and 9.5 mev , compared with experimental data @xcite . ] ( mev ) as function of temperature t ( mev ) , readjusted empirically for temperatures [email protected] mev ( solid line and solid dots ) , compared with the original curve ( dashed line ) due to davidson _ _ @xcite . ] table 1 gives the newly fitted constants of seeger s @xmath3 for the experimental binding energy @xmath6 @xcite , and the theoretical @xmath7 values @xcite where the experimental data were not available . interestingly , only the bulk constant @xmath0 , working as an overall scaling factor , and the asymmetry constant @xmath1 , controlling the curvature of the experimental parabola , are required to be re - adjusted . the role of these re - fitted constants is illustrated in fig . [ fig:1 ] for z=97 nuclides . we notice in fig . [ fig:1 ] an excellent agreement between the present fits ( crosses and down open triangles ) corresponding to experimental ( solid circles ) @xcite and theoretical data ( open circles ) @xcite , respectively . the fits are with in 0 - 1.5 mev of the available @xmath6 or @xmath7 data . also plotted in fig . [ fig:1 ] are the results of calculations using the old 1961 seeger s constants ( hollow squares ) , showing the requirement and extent to which the fitting can clearly improve upon the older results . next , we consider an application of the re - adjusted @xmath3 with an idea to impress upon the need and to propose here atleast a partially modified variation of the pairing constant @xmath70 with temperature t , as compared to that of davidson _ et al . [ fig:2 ] shows the fragmentation potential v(a ) for the decay of @xmath23ni@xmath24 ( a complete mass spectrum ) into light particles ( lps ) and intermediate mass fragments ( imfs ) at t=3.60 mev for two different @xmath27 values ( @xmath27=0 and 36 @xmath28 ) , compared with one at t = 0 for @xmath27=0 @xmath28 . we notice that at t = 0 for @xmath27=0 @xmath28 , the pairing effects are very strong since all the even - even fragments lie at potential energy minima . on the other hand , if we include temperature effects as per prescription of davidson _ et al . _ ( dashed line in fig . [ fig:4 ] ) , we find that @xmath70=0 mev in @xmath3 for t@xmath382 mev , and hence in fig . [ fig:2 ] for t=3.60 mev , @xmath70=0 mev , the odd - odd fragments like @xmath71b , @xmath72n , @xmath73f , etc . , become equally probable as the even - even fragments , since minima are now equally stronger . the same result was obtained earlier in @xcite for the decay of @xmath23ni@xmath24 at t=3.39 mev , since there too @xmath70=0 mev was used from davidson _ et al._. however , if we empirically choose @xmath70=9.5 mev for t=3.60 mev ( for the best fit to imfs data in fig . [ fig:3 ] ) , the situation becomes again favourable . in other words , fig . [ fig:2 ] for t=3.60 mev , @xmath70=9.5 mev shows once again that the even - even fragments , like @xmath74c , @xmath75o , etc . , are equally favoured as odd - odd @xmath72n , @xmath73f , etc . it is important to note that in this experiment @xcite on @xmath25s+@xmath26mg@xmath76ni@xmath77 , only the imfs are measured , and theoretically lps are more prominent at lower @xmath27-values whereas imfs seem to supersede them at higher @xmath27-values , as is also evident from fig . [ fig:2 ] . the calculated decay cross - sections @xmath78 for imfs at t=3.60 mev , for both @xmath70=0 and 9.5 mev cases are shown in fig . [ fig:3 ] , compared with experimental data @xcite . we notice in this figure that better comparisons are obtained for the case of @xmath790 calculations , contrary to earlier results in fig . 13 of @xcite for @xmath70=0 mev , but supporting the one in fig . 7 of @xcite for @xmath70=9.5 mev . similar calulations , supporting non - zero @xmath70 values at t@xmath382 mev , are also reported for @xmath23ni@xmath24 at t=3.39 mev in fig . 7 of @xcite , and for fusion - fission cross - section in @xmath80ba ( fig . 2(b ) in @xcite ) , and the possible @xmath72c clustering in @xmath81o and @xmath82ne nuclei @xcite . these calculations lead us to modify the variation of @xmath70 as function of t , as shown in fig . [ fig:4 ] ( solid line throgh solid dots ) . apparently , many more calculations are needed for fig . [ fig:4 ] to represent a true @xmath9 . 0.5 mm [ cols="^,^,^,^,^,^,^,^,^,^,^,^,^,^,^,^",options="header " , ] in view of the large data for ground state ( g.s . ) binding energies having become available and to be able to include the t - dependence on binding energies , we have re - fitted two of the constants , the bulk @xmath0 and neutron - proton asymmetry @xmath1 , of seeger s mass formula . the experimental g.s . binding energies or theoretical binding energies for neutron- and proton - rich nuclei , where data are not yet available , are fitted within @xmath21.5 mev , and up to z=118 nuclei . the method used is the strutinsky renormalization procedure to define the g.s . binding energy as a sum of the liquid drop energy and the shell correction . taking shell correction from the empirical formula of myers and swiatecki , the two constants of seeger s liquid drop energy are fitted to obtain the experimental or theoretical binding energy . the fitted constants of liquid drop energy have been used for understanding the dynamics of excited compound nuclear systems , which point out to the inadequacy of the variation of pairing energy constant @xmath70 with temperature t. as per the given @xmath8 variation of davidson _ et al . _ , @xmath70=0 mev for t@xmath382 mev . however , the recent compound nucleus decay calculations suggest that @xmath790 for t@xmath38 2 mev and hence clearly indicate the need for re - evaluation of the t - dependence of seeger s constants . a new dependence of @xmath8 is suggested on the basis of already published calculations for compound nucleus decay studies . need for further studies are clealy indicated . the authors are thankful to prof . s. k. patra for his interest in this work . m.k.s . is thankful to csir , new delhi , and r.k.g . to department of science and technology , govt . of india , for financial support for this work in the form of research projects . p. a. seeger , nucl . phys . * 25 * , ( 1961 ) 1 . n. j. davidson , s. s. hsiao , j. markram , h. g. miller and y. tzeng , nucl . a * 570 * , ( 1994 ) 61c . g. audi , a.h . wapstra and c. thiboult , nucl . a * 729 * , ( 2003 ) 337 . p. moller , j. r. nix , w. d. myers and w. j. swiatecki , at . . data tables * 59 * , ( 1995 ) 185 . r. k. gupta , s. k. arun , r. kumar and niyti , int . ( irephy ) * 2 * , ( 2008 ) 369 . r. kumar and r.k . gupta , phys . c * 79 * , ( 2009 ) 034602 . m. bansal , r. kumar and r. k. gupta , contri . efes - in2p3 conf . on `` many body correlations : from dilute to dense nuclear systems '' , feb . 15 - 18 , 2011 , m. balasubramaniam , r. kumar , r. k. gupta , c. beck and w. scheid , j. phys . * 29 * , ( 2003 ) 2703 . b. b. singh , m. k. sharma and r. k. gupta , phys . c * 77 * , ( 2008 ) 054613 . r. k. gupta , r. kumar , n. k. dhiman , m. balasubramaniam , w. scheid and c. beck , phys . c * 68 * , ( 2003 ) 014610 . r. k. gupta , m balasubramaniam , r. kumar , d. singh and c. beck , nucl . a*738 * , ( 2004 ) 479c . r. k. gupta , m. balasubramaniam , r. kumar , d. singh , c. beck and w. greiner , phys . c * 71 * , ( 2005 ) 014601 . r. k. gupta , m. balasubramaniam , r. kumar , d. singh , s. k. arun and w. greiner , j. phys . g : nucl . part . phys . * 32 * , ( 2006 ) 345 . b. b. singh , m. k. sharma , r. k. gupta and w. greiner , int . e * 15 * , ( 2006 ) 699 . r. k. gupta , s. k. arun , d. singh , r. kumar , niyti , s. k. patra , p. arumugam and b.k . sharma , int . e * 17 * , ( 2008 ) 2244 . r. k. gupta , niyti , m. manhas , s. hofmann and w. greiner , int . e * 18 * , ( 2009 ) 601 . s. k. arun , r. kumar and r.k . gupta , j. phys . phys . * 36 * , ( 2009 ) 085105 . s. kanwar , m. k. sharma , b. b. singh , r. k. gupta and w. greiner , int . e * 18 * , ( 2009 ) 1453 . r. k. gupta , niyti , m. manhas and w. greiner , j. phys . phys . * 36 * , ( 2009 ) 115105 . r. k. gupta , s. k. arun , r. kumar and m. bansal , nucl . a * 834 * , ( 2010 ) 176c . m. k. sharma , g. sawhney , s. kanwar and r. k. gupta , mod . a * 25 * , ( 2010 ) 2022 . r. k. gupta , _ clusters in nuclei _ , lecture notes in physics , * 818 * , ( 2010 ) 223 , ed . c. beck , springer - verlag berlin heidelberg .. m. k. sharma , s. kanwar , g. sawhney and r. k. gupta , aip conf . proc . * 1265 * , ( 2010 ) 37 . niyti , r. k. gupta and w. greiner , j. phys . phys . * 37 * , ( 2010 ) 115103 . w. myers and w. j. swiatecki , nucl . a * 81 * , ( 1966 ) 1 . a. s. jensen and j. damgaard , nucl . a * 203 * , ( 1973 ) 578 . s. debenedetti , _ nuclear interactions _ , new york , wiley , ( 1964 ) . s. j. sanders , d. g. kovar , b. b. back , c. beck , d. j. henderson , r. v. f. janssens , t. f. wang , and b. d. wilkins , phy . c * 40 * , ( 1989 ) 2091 ; phys . * 59 * , ( 1987 ) 2856 .

seeger s semi - empirical mass formula is revisited for two of its constants ( bulk constant @xmath0 and neutron - proton asymmetry constant @xmath1 ) readjusted to obtain the ground - state ( g.s . ) binding energies of nuclei within a precision of @xmath21.5 mev and for nuclei up to z=118 . the aim is to include the temperature t - dependence on experimental binding energies , and not to obtain the new parameter set of seeger s liquid drop energy @xmath3 . our proceedure is to define the g.s . binding energy @xmath4 , as per strutinsky renormalization procedure , and using the empirical shell corrections @xmath5 of myers and swiatecki , fit the constants of @xmath3 to obtain the experimental binding energy @xmath6 or theoretically calculated @xmath7 if data were not available . the t - dependence of the constants of @xmath3 , is introduced as per the work of davidson _ et al . _ , where the pairing energy @xmath8 is modified as per new calculations on compound nucleus decays . the newly fitted constants of @xmath3 at t=0 are made available here for use of other workers interested in nuclear dynamics of hot and rotating nuclei .

You are an expert at summarizing long articles. Proceed to summarize the following text: at optical and near infrared wavelengths , interstellar dust depresses the observed flux of galaxy disks by both scattering and absorption . several authors have proposed that , even in the face on perspective , normal , non starburst disks are optically thick at optical wavelengths , while others have argued for substantial transparency ( see the volume edited by davies & burstein 1995 and the review by calzetti 2001 for details ) . using a sample of spiral galaxies within @xmath6 , we statistically derived photometric solutions for the degree of internal extinction at i band as a function of disk inclination ( giovanelli 1994 ) , finding a difference of more than a magnitude of flux between face on and edge on systems . at the same time , we found evidence for transparency of the outer parts of disks , at radii @xmath7 disk scale lengths from the center . we later reported that the amount of internal extinction is luminosity dependent : more luminous disks being more opaque than less luminous ones ( giovanelli 1995 ) . these results have been confirmed by more recent analyses ( e.g. tully 1998 ; wang & heckman 1996 ) . purely photometric techniques are subject to a peculiar set of selection effects , that can severely affect quantitative conclusions on internal extinction , as witnessed by the liveliness of the debate over the last decade . in 1981 , goad & roberts noted a kinematic effect which can provide an independent test for disk extinction . let @xmath8 be the rotation curve of a disk ( @xmath9 being the radial coordinate along the disk s projected major axis ) , assumed to have axial symmetry . let @xmath10 be a set of cartesian coordinates _ in the plane of the disk_. if the disk is thin and it is observed at inclination @xmath11 ( @xmath12 for edge on ) , the component of velocity along the line of sight which intercepts the disk at @xmath10 is v_= v(r ) x i + v_turb , where @xmath13 accounts for turbulence and @xmath14 is oriented along the disk s apparent major axis . an observed rotation curve , as derived for example from a long slit spectrum positioned along @xmath14 , is smeared by seeing , instrumental resolution , averaging across the slit width , the finite thickness of the disk and extinction occurring within the disk itself . as realistically thick disks approach the edge on perspective , lines of sight along the major axis sample regions of increasingly broad range in @xmath15 , yielding a velocity distribution with a peak velocity contributed by parcels of gas at @xmath16 and a low velocity wing contributed by parcels at @xmath17 . if extinction is important , only foreground parts of the disk contribute to the emission and the factor @xmath18 depresses the velocity distribution observed at @xmath19 . goad & roberts noted how this tapering effect may , in opaque edge on disks , produce observed rotation curves resembling solid body behavior , independently of the true shape of @xmath8 . bosma ( 1992 ) applied this technique to two edge on systems , ngc 100 and ngc 891 , by comparing hi synthesis and observations . the evidence led them to conclude that the disk of ngc 100 is transparent , while in the case of ngc 891 they could not exclude the possibility of extinction in the inner parts of the disk . prada ( 1994 ) compared long slit spectra of ngc 2146 in the optical and near ir and reported evidence for extinction in the inner parts of the galaxy . here , we apply the goad & roberts test in a statistically convincing manner to a sample of more than 2000 / [ ] rotation curves . our results clearly indicate substantial opacity in the inner disks of spiral disks . they also show the effect to be luminosity dependent , in a manner very much in agreement with our previous photometric determination . in section 2 we present our data sample and describe our rotation curve fits . results on extinction are discussed in section 3 . in section 4 , we discuss an interesting serendipitous finding : the apparent constancy of the mean value of the kinematic scale length of / [ ] rotation curves , across populations of different luminosity class . throughout this paper , distances and luminosities are obtained from redshifts in the cosmic microwave background reference frame and scaled according to a hubble parameter @xmath20 mpc@xmath21 . as an effort to map the peculiar velocity field of the local universe , we have assembled a sample of several thousand observations of i band photometry and rotational width data of spiral galaxies . our observations , including those listed in dale & giovanelli ( 2000 and refs . therein ) as well as several hundred obtained at the hale 5 m telescope after that report and to be presented elsewhere , have been complemented by the samples observed in the southern hemisphere by mathewson and co workers ( mathewson 1992 ; mathewson & ford 1996 ) . the kinematic information for a substantial subset of these data is in the form of rotation curves in electronic form , derived from long slit / [ ] spectra . rotation curves can be fitted by a variety of parametric models , some of which are motivated by the physical expectation of contributions by a baryonic component with a mass distribution mimicking that of the light plus a dark matter spheroidal halo . other models rely purely on the versatility of a mathematical form in fitting effectively the observed rotation curves with a minimum of free parameters . here we report on fits to 2246 rotation curves with a parametric model of the form v_pe(r ) = v_(1-e^-r / r_pe)(1+r / r_pe ) where @xmath22 regulates the overall amplitude of the rotation curve , @xmath23 yields a scale length for the inner steep rise and @xmath24 sets the slope of the slowly changing outer part . this simple model , which we refer to as _ polyex _ , has been found to be very `` plastic '' . it fits effectively both the steeply rising inner parts of rotation curves , as well as varying outer slopes , and it can be used to advantage in estimating velocity widths at specific radial distances from the galactic center as required by applications of the luminosity linewidth relation . in particular , it provides a very useful estimate of the inner slope of the rotation curve as given by @xmath23 . of the 2246 rotation curves fitted by the model shown in equation ( 2 ) , 425 have been rejected due to poor quality of the data or of the fit or because @xmath23 projects to less than 3 " and may thus be affected by poor angular resolution . mean values of @xmath24 , the value of @xmath23 as seen in face on systems , @xmath25 , and of the ratio between @xmath23 and the scale length of the disk light for face on systems are given in table 1 , for different luminosity bins ; mean values of @xmath23 are shown in figure 1 and discussed in the next section , where they are used to probe disk extinction . contents of columns 5 and 6 of table 1 are discussed in section 4 . cccccc @xmath26 & @xmath27 & @xmath28 & @xmath29 & @xmath30 & @xmath31 @xmath32 & @xmath33 & @xmath34 & @xmath35 & @xmath36 & @xmath37 @xmath38 & @xmath39 & @xmath40 & @xmath41 & @xmath42 & @xmath31 @xmath43 & @xmath44 & @xmath45 & @xmath46 & @xmath47 & @xmath48 @xmath49 & @xmath50 & @xmath51 & @xmath52 & @xmath53 & @xmath54 @xmath55 & @xmath56 & @xmath57 & @xmath58 & @xmath59 & @xmath60 in figure 1 , the mean value of @xmath23 is shown separately for different luminosity classes , as a function of the disk inclination as expressed by @xmath61 , where @xmath62 is the apparent axial ratio . each symbol is the average within a bin including between 20 and 30 galaxies . the horizontal dotted line in each panel indicates an arbitrary scale of @xmath63 kpc . the total number of galaxies and the absolute magnitude range is indicated within each panel of the figure . the parameter @xmath23 gives an indication of the radial distance at which the rotational velocity is @xmath64 of @xmath65 ( the asymptotic velocity for flat rotation curves , for which @xmath66 ) . figure 1 shows that , as @xmath61 approaches 0.45 ( @xmath67 ) , the kinematic scale length @xmath23 of observed rotation curves starts increasing . the effect is more marked for more luminous disks , for which edge on systems exhibit an average value of @xmath23 nearly three times as large as that of systems with @xmath68 . the effect is not evident in the least luminous systems ( those with @xmath69 fainter than -19.5 ) . this result is in agreement with the expectations based on the photometric determination of giovanelli ( 1995 ) . no axial ratio dependence is observed for @xmath24 , the outer slope of the rotation curves . we have carried out preliminary modelling of the observations shown in figure 1 by simulating the dust and distributions as exponential disks both in @xmath9 and @xmath70 . assuming the same distribution for dust and tracers , with a ratio between scale height and scale length of 0.06 ( as found by xilouris 1997 ; 1998 ) , we obtain approximate match between models and figure 1 for values of the the face on ( @xmath71 ) , central ( @xmath72 ) optical depth of the model at the wavelegth of ( @xmath73 ) , @xmath74 , of about 4 . these modelling results are preliminary and depend very sensitively on the assumed thickness of disks and on the relative scales of dust and sources . we will report more extensively on the results of this effort at a later stage . figure 1 shows that , for low axial ratios ( where the effect of internal extinction is negligible ) , the average value of the kinematical scale length @xmath23 is approximately the same for disks of all luminosities . is this result to be expected ? first , we exclude that the angular resolution of spectroscopic observations has a significant impact on the reported findings . not using galaxies for which @xmath23 projects to an angular size @xmath75 rules out bias due to extreme cases of poor angular resolution . since we use rotation curves from different sources , which have been produced with different angular sampling intervals , we have also verified that there is no systematic difference in the fitted values of @xmath76 among different data sources , taking advantage of significant sample overlaps . we have done so for several dozen galaxies appearing in at least two samples among those : of our own , of mathewson et al . , of courteau ( 1997 ) and of rubin et al . ( 1999 and refs . therein ) . second , we explore the effect of sample selection criteria . the values of @xmath77 displayed in figure 1 are affected by malmquist bias , which does not impact on the trends produced by extinction but alters the relative values of @xmath77 among different luminosity classes : since more luminous populations have larger average distance , for them the angular size limit of 3 `` produces an @xmath77 biased high . in order to remove this bias , we draw a subsample which is volume limited , by including in it only systems with @xmath78 , @xmath79 kpc and within the distance range @xmath80 to @xmath81 mpc . at @xmath81 mpc , an angular size limit of 3 '' translates into a linear size limit of about 1 kpc ; the reason for a lower limit in distance is to reduce the uncertainty introduced by peculiar velocities . we further restrict the subsample to low inclination disks ( @xmath82 , @xmath83 , @xmath84 respectively for @xmath85 , @xmath86 and @xmath87 ) in order to get around the extinction effect discussed in the previous section and compute the mean values of @xmath23 and @xmath76 , now identified by a subscript `` 1 '' ; those values are listed in columns 5 and 6 of table 1 , for each luminosity class . the exclusion of galaxies with @xmath88 kpc biases high the values of @xmath89 ; it does so however equally for all luminosity classes , through which @xmath89 remains approximately constant . the mean values @xmath90 , which increase with decreasing luminosity , illustrate the fact that less luminous populations are on the average more nearby . note that the values of @xmath90 are fairly large in comparison with the typical seeing , excluding the likelihood of an angular resolution bias . next , we inquire on the implications of a constant @xmath23 on the total mass distribution . assume the total mass of the galaxy @xmath91 is that comprised within a limiting radius @xmath92 , defined as that within which the halo mean density is 200 times the critical density of the universe . then , m(r_pe)=m_200|(r_pe)|_200 ( r_per_200)^3 where @xmath93 is the mass within @xmath23 , and @xmath94 and @xmath95 are respectively the mean density within @xmath23 and within @xmath92 . since @xmath96 , we can also express the ratio between the rotational velocity at @xmath23 and the asymptotic value @xmath97 as ^2 = |(r_pe)|_200 ( r_per_200)^2 for a flat rotation curve , the definition of @xmath23 yields @xmath98 ^ 2\simeq 0.4 $ ] ; then m(r_pe ) 0.4 m_200r_per_200 it can be shown that @xmath99 ; thus , for a constant value of @xmath23 : m(r_pe ) m_200 ^ 2/3 since @xmath65 does not generally coincide with @xmath97 , the effect of the luminosity dependence of @xmath24 needs to be taken into account : it can be parametrized by writing @xmath100 , where @xmath101 is small and positive . in general , equation ( 6 ) can then be rewritten as m(r_pe ) m_200 ^ 2/3 + we now inquire on the dependence of the halo mass within @xmath23 on the total mass . we assume a halo density form as proposed by navarro , frenk & white ( 1997 ) : _ h(r ) = _ crit_(r / r_s)(1+r / r_s)^2 where @xmath102 is the cosmological critical density , and the scaling parameters @xmath103 and @xmath104 can be expressed in terms of @xmath92 and a concentration index @xmath105 , with @xmath106^{-1}$ ] . for small values of @xmath9 , the halo mass within radius @xmath9 is m_h(r ) = 400c^2 _ crit r_200(1+c)-c/(1+c ) r^2 where @xmath107 . for a constant @xmath23 , m_h(r_pe ) m_200 ^ 1/3 numerical simulations indicate that @xmath108 is mildly dependent on halo mass , _ decreasing _ approximately as @xmath109 with @xmath110 ( navarro , frenk & white 1997 ) . a comparison of equations ( 7 ) and ( 10 ) thus indicates that the mass in the halo can not account for the inner shape of the rotation curve , unless the concentration index _ increases _ with halo mass , opposite to what numerical simulations suggest . an alternative interpretation is that the contribution of disk and bulge to @xmath93 increases with galaxy luminosity ( and mass ) , a well exercised idea in the current literature ( e.g. burstein & rubin 1985 ; broeils 1992 ; moriondo , giovanardi & hunt 1998 ; sellwood 1999 and refs . therein ) . if we assume that @xmath111 , @xmath112 and neglect the dependence of @xmath108 on @xmath91 , 1+f_d m_200 ^ 1/3 , suggesting that as the galaxy mass increases , the fraction of baryonic material in the inner few kpc grows rather quickly . for these systems , the baryonic mass fraction may saturate at its cosmological level ( i.e. these systems retain all their baryons ) and their inner rotation curves may be completely determined by it . the spirals in our sample have small bulges ( they are mostly of type sbc and sc ) . within that context , note that in an exponential disk dominated rotation curve a rotational velocity @xmath64 of the maximum is obtained at a radial distance of about 0.6 disk scale lengths . for galaxies in our high luminosity bin , the mean ratio between @xmath23 and the scale length of the disk light is @xmath29 , as shown in table 1 , while it rises to twice that value for the less luminous galaxies in our sample . we conclude that the simplest and most likely explanation of the near constancy of @xmath77 among luminosity classes is that the baryonic mass fraction within the inner few kpc of spirals increases with total luminosity , and that within that region the most luminous spirals are entirely dominated by baryonic matter . we integrate photometric profiles to @xmath113 in order to obtain i band luminosities within that radius , assuming a solar absolute magnitude of + 3.94 ( livingston 2000 ) . estimating mass from @xmath114 , for galaxies brighter than @xmath115 we obtain a mean baryonic mass to light ratio @xmath116 @xmath3 within @xmath23 . individual colors for galaxies in our sample are not available ; however , if we assume that spiral galaxies radiate approximately like blackbodies with @xmath117 , then a mean bolometric @xmath118 @xmath119 can be estimated . the coarse analysis presented here does not take into consideration the dynamical effect of a central bulge with a low spin parameter and we have not attempted to separate disk and bulge contributions to the light . it is interesting to compare our result with that of bottema ( 1999 ) for the sb galaxy ngc 7331 : from measurements of stellar velocity dispersions he obtains @xmath120 and @xmath121 . we have shown that the / [ ] rotation curves of spiral galaxies exhibit a dependence of the inner slope of their rotation curves on the galaxy axial ratio . we have interpreted this result as induced by extinction occurring within disks . the effect is luminosity dependent , being stronger for the most luminous systems as previously indicated by giovanelli ( 1995 ) on photometric grounds . serendipitously , it was also found that rotation curves of galaxies of different luminosity classes exhibit approximately the same average exponential scale length in the inner regions , of about @xmath122 kpc ( the result has potential in the determination of redshift independent distances : we will report elsewhere on this application ) . this is in agreement with the idea that in the inner parts of spiral galaxies the baryonic component contributes a rapidly increasing fraction of the mass , with increasing total luminosity of systems . in fact , the most luminous galaxies in our sample appear to be completely baryon dominated within the inner few kpc . we use this inference and the combination of spectroscopic and photometric data to estimate a mean baryonic mass to light ratio for that region @xmath123 @xmath3 in the i band .

we use the tapering effect of / [ ] rotation curves of spiral galaxies first noted by goad & roberts ( 1981 ) to investigate the internal extinction in disks . the scale length of exponential fits to the inner part of rotation curves depends strongly on the disk axial ratio . preliminary modelling of the effect implies substantial opacity of the central parts of disks at a wavelength of 0.66 @xmath0 . in addition , the average kinematic scale length of rotation curves , when corrected to face on perspective , has a nearly constant value of about @xmath1 kpc , for all luminosity classes . the interpretation of that effect , as the result of the increasing dominance of the baryonic mass in the inner parts of galaxies , yields a mean baryonic mass to light ratio in the i band @xmath2 @xmath3 , within the inner @xmath1 kpc of disks . 83@xmath4 50w@xmath5 = cmr8 = cmbx8 = cmti8 = cmsl8 = cmmi8 + 584 _ use346 6.5 truein

You are an expert at summarizing long articles. Proceed to summarize the following text: 40004000 = 1000 # 1 40004000 = 1000 brown , bildsten & rutledge ( * ? ? ? * bbr98 hereafter ) argued that the core of a transiently accreting neutron star ( ns ) ( for reviews of transient neutron stars , see ) is heated by nuclear reactions deep in the crust during the accretion outbursts . the core is heated to a steady state in @xmath1 yr ( see for a detailed calculation ) , after which the ns emits a quiescent thermal luminosity ( bbr98 ) @xmath3 where @xmath4 is the time - averaged ( over the ns core thermal timescale ) mass - accretion rate onto the ns , and @xmath5 is the amount of heat deposited in the crust per accreted nucleon (; see for a review ) . for an accretion flux onto the ns ( @xmath6 ) , the `` rock bottom '' quiescent flux due to deep crustal heating is then @xmath7 where @xmath8 is the accretion flux averaged over the ns core thermal timescale , and the accretion efficiency @xmath9 for accretion luminosity of @xmath10 . since @xmath11 , where @xmath12 is the mean outburst duration and @xmath13 is the mean recurrence timescale , this scenario relates the quiescent luminosity to the outburst properties ; comparisons to observations of several quiescent neutron stars ( qnss ) match the predictions reasonably well @xcite . it also helps us to understand why the black hole systems in quiescence are so much fainter than ns systems @xcite , as they can not be thermally emitting . equation ( [ eq : fq ] ) is the minimum quiescent flux from the neutron star . should residual accretion occur in quiescence @xcite , the accretion luminosity would be in addition to the thermal emission already present . is a transient , type - i x - ray bursting neutron star approximately 3.8 degrees from the galactic center @xcite . it was discovered in outburst in 1989 , when its luminosity was 1.3 ( d/8 ) @xmath14 ; subsequent analysis found that it had been in outburst at least since oct 1988 ( ) , and has remained bright since then ( see for a complete description of observations ) . sunyaev et al . ( 1990 ) suggested that it was a transient , apparently because it had not previously been detected , although no strong upper limits on previous x - ray emission were given . ( this was noted later by , who suggested the object may not be a transient at all . we adopt an outburst timescale of @xmath15 13 yr , although the outburst may have been longer ) . type i x - ray bursts were observed , at the rate of @xmath110 per day , and studied in detail with rxte @xcite . nearly coherent oscillations ( 580 hz ) have been observed during the type - i bursts , and the frequency is interpreted as the spin frequency of the ns @xcite . the uncertainty in the localization of this source was initially 4.2 ( 90% confidence ) . an improved error circle ( 10 with _ rosat_/hri ; ) found 13 candidate ir counterparts down to @xmath16 ( 10 @xmath17 ) . the integrated hi column density in the direction of is = 0.35 ( @xmath18 cm ) . found = 16 ( @xmath19=534 ; ) , suggesting that is variable , and thus there must be significant contribution to from the system itself . an _ rxte_/pca scan of the galactic center found that had entered a low luminosity state @xcite in early 2001 . then observed the source with _ chandra_/acis - s in late march ( @xmath11 months after the end of the outburst ) and detected it at an unabsorbed flux of @xmath20 ( 0.510 kev ) . the scenario described by equation ( [ eq : fq ] ) applies when the neutron star crust and atmosphere resemble that of a cooling neutron star that is , the temperature decreases with increasing radius . for ns transients with short outbursts , such as aql x-1 ( @xmath21 ) , this is a good approximation , as the increase in crust s temperature from the heating during the outburst is small @xcite . this is not the case for . the duration of its most recent outburst is of the order of ( or longer than ) the thermal diffusion timescale in the crust ( bbr98 ) . in a sense , in quiescence resembles a neutron star that accreted steadily at the _ outburst _ accretion rate , except that the core temperature will be at the lower value set by the time - averaged accretion rate ( over the previous @xmath22 ) . to illustrate how the crust is so dramatically heated during a long outburst , consider the rise in temperature if no heat were conducted away from the crust during the outburst . as we describe in [ sec : crust ] , the total heat capacity of the region of the crust where most of the heat is deposited is @xmath23 erg k@xmath24 . during the outburst , the total amount of heat deposited is @xmath25 and so , _ if no heat is conducted away from the heating region _ , the temperature there can rise to @xmath26 k during the 13 year outburst . even when thermal conduction is taken into account , the rise in crust temperature can still be @xmath27 , which is the typical temperature the crust and core would have if accreted steadily at @xmath28 yr@xmath24 ( for @xmath29 yr ; see [ sec : core ] ) . if the crust , which is composed of the ashes of h / he burning , has a low thermal conductivity , then there is a substantial temperature gradient between crust and core . as discussed by for the case of steadily accreting neutron stars , when there is a substantial thermal gradient in the inner crust , the temperature of the crust reaction layers becomes decoupled from that of the core . for the ns in , this means that , until the crust has thermally relaxed , we can not directly infer its core temperature , as is possible for short - outburst transients such as . however , this also means that one can directly compute the current quiescent luminosity from using the fluence during the last outburst , as the thermal state of the crust is only weakly sensitive to the temperature of the core , and , hence , is nearly independent of the uncertainty in the accretion history over the past @xmath30 yrs ( the thermal time of the core ) , unlike in the case of short - outburst transients . our simulations , presented here , predict a quiescent luminosity which agrees with the observed value to within observational and theoretical uncertainties ( factor of a few ) for the fiducial value of @xmath5 and outburst fluence . moreover , this dependence of the thermal flux in the crust - dominated regime on @xmath31 , and not on @xmath32 , allows us to predict the evolution of @xmath33 , which can be directly confronted with observations . the domination of the quiescent luminosity by the cooling crust sets apart from the other neutron star transients with short duration outbursts and makes it an ideal laboratory for separately measuring the thermal properties of the crust . in this paper we report our analysis of the _ chandra _ observation @xcite of in quiescence and describe our simulations of the thermal state of its crust and core . we begin in [ sec : anal ] by describing the _ chandra _ observation and spectral analysis of . we include a description of possible ir counterparts ( [ sec : ir ] ) . after describing the observations , we then discuss the implications for the crust and core thermal structure as outlined above . we first apply ( [ sec : core ] ) the scenario in which the crust heating for any single outburst is negligible , i.e. , we treat as if it were a short - duration outburst , such as aql x-1 . this analysis has been done previously by . we then explain , in [ sec : crust ] , why this analysis is inapplicable to this source , and how the quiescent luminosity is determined by the thermal properties of the crust ( its thermal diffusion timescale and heat capacity ) for the next 130 yr . we present simulations of the quiescent lightcurve and demonstrate that monitoring observations can constrain these properties , similar to proposals for glitching pulsars @xcite . we conclude in [ sec : con ] with a summary and discussion of these results . the data were obtained from the _ chandra _ public archive . the discovery and observations have been previously analyzed by , and details of the trigger and history of are included there . the observation was triggered when an _ rxte_/pca scan of the galactic center found that had entered a low luminosity state @xcite , and was made 2001 mar 27 00:18 - 06:23 ( tt ) with the acis - s instrument @xcite for a total exposure time of 19401 seconds . the x - ray source was imaged on the s3 ( backside illuminated ) chip , which was operated as a 1/4 array with 0.8 sec exposures . we analyzed the data using the ciao v2.1 with caldb v2.6 and xspec v11 @xcite . two x - ray point sources separated by 31.30.1 are found with _ celldetect _ above a signal / noise ratio of 5 , listed in table [ tab : objs ] . we compared this image with an archived _ rosat_/hri image of this region ( rh400718 ) taken in 1997 when was in outburst . the position of had been previously determined this way @xcite and clearly is at the position of _ chandra _ source # 1 . x - ray source counts were extracted within an area 5 pixels in radius about the source position , for a total of 183 counts . at 0.0075 counts / frame , the pileup fraction is negligible . background was taken from an annulus centered on the source position , with radii of 10 and 80 pixels . the expected number of background counts in the source region is 3 . a ks test @xcite finds that the times of arrival ( toa ) of the 183 counts is consistent with a constant countrate . we place 3@xmath17 upper - limits on variability of @xmath34=22% rms . we also produced a power - density spectrum to search for a pulsed signal ( number of frequency bins : 12750 , with frequency resolution of 5 hz , and a nyquist frequency of 0.625 hz ; see ) , using barycentered toas ( tool _ axbary _ ) . no evidence for a pulsed signal is found : the largest leahy - normalized power @xcite was 20.37 ( with a probability of chance occurrence from a poisson - distributed countrate of 0.48 ) . the absence of a coherent signal is not surprising in light of the detection of pulsations during type - i x - ray bursts @xcite at @xmath1580 hz , well above our nyquist frequency . we binned the data into 10 pi bins ( 0.5 - 10.0 kev ) , and fit several single component spectral models ( powerlaw , h atmosphere , or raymond - smith , a multicolor disk and blackbody ) . galactic absorption is initially left as a free parameter . the best - fit models were all statistically acceptable ; the parameters are given in table [ tab : fits ] . while all models we investigated are statistically acceptable , some can be argued against on physical grounds . the pure power - law spectrum is unusually steep ( @xmath35=5.20.6 ) , typical more of a thermal spectrum than other processes . the emission measure @xmath36 from the raymond - smith spectrum is higher by 2 orders of magnitude than typical from active stellar coronae in the analogous rs cvn systems (; see for discussion on coronal emission from companions in x - ray transients ) . the disk black - body spectral model implies an inner disk radius of @xmath10.7 km considerably smaller than a neutron star . our best - fit blackbody spectrum is consistent with that found by . as pointed out by bbr98 , for accretion rates 2 @xmath37yr , gravity stratifies metals in the ns atmosphere faster than they can be provided by accretion ( bildsten , salpeter , & wasserman ) , making a pure h atmosphere the appropriate description of the ns photosphere in quiescence . due to the strongly energy - dependent opacity of free - free transitions , these spectra are significantly different from black - bodies @xcite . unlike the results from black - body fits , the inferred ns radii ( = @xmath38 where @xmath39 is the proper radius , and @xmath40 means the value as measured by a distant observer ) of the four observed field transients using h atmosphere spectra are consistent with what are expected theoretically : @xmath112 km ( 4 , ; , ; 1608 - 522 ; and 4u 2129 + 47 ) , thus supporting the notion that much of the emission originates from a neutron star surface . the quiescent spectrum for qnss is thus interpreted as thermal emission from a pure hydrogen atmosphere ns , possibly with an underlying power - law ( whose origin is not understood ) which dominates the spectrum at high ( @xmath413 kev ) energies ( see rutledge ) . the spectral fit for a h atmosphere spectrum alone ( no power - law ) gives @xmath42=12030 ev , an emission area radius of = 10 ( d/8 kpc ) km , and an unabsorbed flux of 1.8 ( 0.5 - 10 kev ; the absorbed flux is 3.8 ) , corresponding to a luminosity of @xmath431.4 ( d/ 8)@xmath14 ( 0.5 - 10 kev ) . the 90% confidence range in the unabsorbed flux is ( 1.14.6 ) . the best - fit spectrum is shown in fig . [ fig : ks1731 ] . the bolometric thermal luminosity ( as observed at infinity ) is @xmath442.7 ( d/8 ) @xmath14 ( uncertain by a factor of @xmath13 , due largely to spectral uncertainty ) . if 100% of the emergent luminosity were from accretion ( at an efficiency of 0.2 ) , this sets @xmath45 @xmath37yr , sufficiently low for the assumption of an h atmosphere . when we include an underlying power - law with all parameters free , the s / n of the spectrum does not constrain the model parameters to better than an order of magnitude . when we hold and @xmath35 fixed at typical values ( = 12.5 km for d=8 kpc ; @xmath46 ; cf . ) , the spectral model provides an acceptable fit to the data . this demonstrates that the spectrum of in quiescence is consistent with that observed from other qnss . a comparison between chandra astrometry and 2mass images finds a possible ir counterpart at the position of source # 2 ( see below ) but not of , with limits in j , h and k@xmath47 of 15.8 , 15.1 and 14.3 magnitudes respectively . comparing with ir sources previously examined as possible counterparts to @xcite , the closest object is source `` h '' , which is 1.51.1 away ( we adopt 1 positional uncertainty for _ chandra _ , and 0.5 uncertainty for the ir source ) . the second closest is `` g '' , which is 3.3 away . in addition , there is a fainter ir object present in the h - band image ( `` new '' object ) , but not in the j - band image taken by barret , which is not marked , but which is consistent with the _ chandra _ position for . further work in the ir for example , searching for ellipsoidal variations in quiescence is needed for a positive identification of the counterpart . source # 2 is consistent in position with 2mass j173412.7 - 260548 ( @xmath48 in distance , about the accuracy of the 2mass position reconstruction error ) . for an ir field source density of @xmath49=6.9 arcsec@xmath50 ( 217700 objects in the 2mass catalog within a 1000 radius ) the probability of a chance coincidence is prob=@xmath51=3.9% , which is the significance of the association of the x - ray and ir objects . the source s 2mass magnitudes and colors are j=12.2640.033 , @xmath52=0.948 and @xmath53=1.33 . [ [ core - dominated - emission - crust - dominated - emission - and - the - future - lightcurve - of ] ] core dominated emission , crust dominated emission , and the future lightcurve of ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ having described the observations and spectral analysis of , we now turn our attention to the thermal properties of the neutron star s crust and core . we begin by estimating the amount of mass deposited during the previous outburst ; we then describe what the thermal structure would be if the change in crust temperature during this outburst were small ( bbr98 ; ) . this is the case for transients with short - duration outbursts . because of the length of the outburst for this source , however , the crust is substantially heated ; [ sec : crust ] describes how the observed emission is determined by the heating of the crust during the past outburst , and not by the core equilibrium temperature . using the method of ur01 , we simulate the evolution of the quiescent lightcurve for different regimes of the crust and core microphysics . has been observed several times , always in outburst , since 1988 , and nearly continuously so since jan 1996 with the _ rxte_/asm . to estimate @xmath54 , we integrated the asm counts using only data in which there was a @xmath55 1-day detection and taking only 1 day detections in which the previous observation was made @xmath564 days ( to insure adequate sampling ) prior . periods in which the time to the previous observation was @xmath414 days were not included . the total fluence is 13660 asm c / s@xmath57days , over an integration of 1599 days ( including both detections and non - detections ) . if we assume an additional countrate at the 2@xmath17 level on those observations when the countrate was below 3@xmath17 detection , the total fluence would only increase by 7% , which is our 2@xmath17 upper - limit for the uncertainty in integrated counts . assuming a @xmath58=5 kev bremsstrahlung spectrum with = 1.0 ( which gives 6.5 per 1 asm c / s , corrected for absorption , 0.0120 kev , from w3pimms ; in the 210 kev range , the conversion factor is 3.0 per 1 asm c / s ) , we find a mean outburst flux , corrected for absorption , of @xmath59 ( 6.5 per 1 asm count ) = 5.6 . we conservatively estimate the spectral uncertainty to be at the 40% level ( for a change in @xmath58 between 2 kev and 8 kev , the asm counts / flux conversion changes by 40% ) . extrapolating this mean to the entire outburst , we find an outburst fluence of @xmath60 . this estimate is greater by a factor of @xmath16 than that of , who estimated @xmath61 using the minimum observed outburst flux and an outburst duration of 11.5 yr , rather than 13 yr . the quiescent bolometric thermal flux is @xmath62 . if the change in crust temperature during the outburst were small , then equation ( [ eq : fq ] ) would apply : the time - averaged accretion flux is @xmath63 , so that @xmath64 there is an uncertainty of about 0.5 dex in this value , mostly due to the uncertainty in the bolometric value of @xmath65 . this estimate is similar to the one arrived at independently by , although our estimated mean recurrence time is longer ( 1500 0.5 dex yr vs. 200 yr ) , since our time - averaged outburst flux is a factor of six greater than that assumed by . strictly speaking , this estimate of @xmath13 is the recurrence time to an outburst of comparable fluence . a month - long outburst would have just 1% of the fluence of the previous outburst and would not affect the estimate of eq . ( [ eq : rec ] ) . this estimate also neglects any neutrino emission , which reduces the effective value of @xmath5 and hence @xmath13 . more seriously , however , this analysis ignores the fact that the crust is strongly perturbed away from the core equilibrium temperature during such a long outburst . as a result , the quiescent emission currently observed is set by the thermal relaxation of the heated crust , and as we now demonstrate , is mostly decoupled from the thermal state of the core . during an outburst , the heating from the reactions raises the temperature in the crust around the reaction layers . after the outburst ends , the crust thermally relaxes and the thermal profile comes to resemble a cooling neutron star , i.e. , the temperature decreases with radius . for neutron star transients with `` typical '' outburst fluences and recurrence times ( that is , @xmath66 month , with an outburst luminosity @xmath67 eddington ) , the variations in the crust temperature and hence @xmath68 are small ( bbr98 ; ur01 ) . for example , ur01 found that @xmath68 varied by @xmath1 few% for @xmath69 yr and by @xmath1 30% for @xmath70 the variation is small because the amount of energy deposited in such short outbursts is not significant compared to the heat content of the crust , and so the crust is not heated substantially compared to its temperature when in thermal equilibrium with the core . to ascertain the magnitude of the deviation of @xmath33 from the value predicted by eq . ( [ eq : brown ] ) , we performed several simulations of thermal relaxation of the ns crust in a transient with observational properties ( @xmath12 , @xmath13 , @xmath72 ) inferred for using the methods and microphysics as described in ur01 . in summary , we solve the non - relativistic heat equation in the crust , from @xmath73 g @xmath74 to @xmath75 g @xmath74 . heating due to nuclear reactions in the deep crust is simulated by depositing energy at densities corresponding to the nuclear transitions computed by , with the amount of energy set by the instantaneous accretion rate . at the outer boundary , we use the flux - temperature relation for a fully accreted crust from potekhin @xcite . the core is taken to be isothermal ( a good assumption for the timescales of interest because of the large thermal conductivity ) , and its temperature evolves according to the mismatch between its neutrino emissivity and the flux from the crust . we start our simulations with the temperature profile corresponding to persistent accretion at the rate @xmath32 corresponding to the time average of @xmath31 over the recurrence time . we then evolve the model through several outburst / quiescence cycles , as necessary for the model to `` forget '' the initial conditions and reach a limit cycle . if the crust resembles that of a neutron star steadily accreting at @xmath31 , then the estimate of the recurrence time ( eq . [ [ eq : rec ] ] ) is inapplicable ( except as a lower limit ) . this is because the crust temperature is decoupled from that of the core . a full survey of parameter space should include simulations over a range of @xmath13 . this is beyond the scope of this initial paper ; rather , we presume that @xmath13 is given by eq . ( [ eq : rec ] ) in order to survey the influence of the crust and core microphysics . in these simulations , we examine the two main uncertainties in ns microphysics , ( 1 ) the impurity fraction , and , hence , the conductivity of the crust and ( 2 ) the possible presence of `` enhanced '' core neutrino cooling mechanisms , such as direct urca or pion condensation . first , the composition of accreting ns crusts is set by the nuclear processing of products of burning in the upper atmosphere . published calculations @xcite presume that burning proceeds to pure iron , and , hence , the crust has no impurities . however , recent work by schatz and collaborators @xcite shows that burning products are a mix of elements substantially heavier than iron , with the impurity parameter of the elements of the mix from the average charge @xmath76 , @xmath77 , where the average takes into account the relative abundance of the species . ] @xmath78 comparable to the square of the average nuclear charge @xmath79 . we therefore set bounds on the crustal conductivity by using electron - ion scattering @xcite , which has the same form as electron - impurity scattering with @xmath80 , and electron - phonon scattering @xcite , which is appropriate for a pure crystal . we refer to these two cases as `` low @xmath81 '' and `` high @xmath81 '' , respectively . second , the uncertainty in core cooling is covered by simulations with `` standard cooling '' and `` enhanced cooling '' ( modified urca appropriately suppressed by nucleon superfluidity , and neutrino emission as when the core is a pion condensate , respectively ; see ur01 for details ) . to illustrate the deviation of the crustal temperature from the equilibrium core temperature and the subsequent thermal relaxation of the crust , we show in fig . [ fig : tevol ] the time evolution of the crustal temperature in the low @xmath81 , standard cooling case for observational parameters inferred for ( @xmath82 yr , @xmath83 yr , and @xmath84 yr@xmath24 ) . vertical slices of the surface in the density - temperature plane show the instantaneous temperature distribution in the crust , while slices in the time - temperature plane show the time evolution of the temperature since the beginning of the outburst . clearly , the crust is heated to well above its pre - outburst temperature profile , which tracks the core temperature ( note that , because the crust is heated to well above its temperature in equilibrium with the core , this qualitative result is not very sensitive to the assumed @xmath13 ) . a steep temperature gradient carries most of the heat from the deep crustal heating region around @xmath85 g @xmath74 into the core , while a shallower temperature gradient carries a fraction of the heat towards the surface . the temperature near the top of the crust ( @xmath1 g cm ; and , hence , the x - ray luminosity , @xmath86 ; ) reaches a maximum at the end of the outburst ( time=13 years ) , and then decays back to the pre - outburst value on the thermal timescale of the crust ( @xmath87 yrs for this model ) . thermal evolution in other cases is similar ( but , of course , is different in the absolute magnitude of the temperature change ) , and the temperature changes are always much bigger than those that occur during short - duration outbursts ( cf.fig . 1 of ur01 ) . in fig . [ fig : greg ] , we show the long - term evolution of the quiescent surface luminosity from after accretion onto the compact object has ceased for the four cases discussed above , namely , low and high crustal conductivity @xmath81 , and standard and enhanced core neutrino emission ( see the legend on the plot ) . first consider fig . [ fig : greg]a , where we set @xmath88 . in all four cases , the transition from crust - dominated ( at early times ) to core - dominated cooling ( at late times ) is evident as a drop in the luminosity , ranging from 50% ( high @xmath81 , standard cooling case ) to a factor of @xmath89 ( low @xmath81 , enhanced cooling case ) . regardless of the core neutrino emissivity , this transition occurs at @xmath87 yrs for low conductivity crusts , and after @xmath90 yr for high conductivity crusts . these timescales are easy to understand , as they are just the thermal time to the appropriate depth in the crust ( cf . figure 3 of ur01 ) . since the low @xmath81 crust has a longer thermal time , it stays hot about 30 times longer than the high @xmath81 crust . after the crust thermally relaxes , @xmath33 is set by the emission from a hot core . if the only core neutrino emission mechanism is modified urca , suppressed by nucleon superfluidity , then the amount of heat lost by neutrinos from the core is negligible . therefore , all the heat deposited into the star comes out as thermal emission , and @xmath33 asymptotically approaches the value given eq . ( [ eq : brown ] ) , with @xmath91 ( see solid and dashed lines in fig . [ fig : greg ] ) . on the other hand , when enhanced neutrino cooling is allowed , @xmath92% of the deposited heat escapes as neutrinos (; ur01 ) , and only the remaining @xmath93% is radiated thermally from the surface . in this case , the `` effective '' value of @xmath5 in eq . ( [ eq : brown ] ) , in the sense of the energy retained in the star and re - radiated thermally , is @xmath94 . as shown above , the crust is heated to well above the core temperature during the long outburst . how does the temperature difference between the crust and the core depend on the crustal conductivity ? as an estimate , we can assume that , during the outburst , the crust comes to an equilibrium where the heat deposited around @xmath95 g @xmath74 at a rate @xmath96 is conducted away from this region . this is a very good assumption for the high conductivity case , and an acceptable one for the low conductivity case , where the crustal thermal time is comparable to the outburst duration . during the outburst , most of the heat is conducted into the core with a flux @xmath97 , where @xmath98 m is the distance between the crustal heating region and the core . thermal balance then requires @xmath99 , or @xmath100 k @xmath101 , where @xmath102 is the conductivity in units of @xmath103 erg cm@xmath24 s@xmath24k . in the low conductivity case , @xmath105 , so @xmath106 . we see from fig . [ fig : greg]a that @xmath33 at early times in the low @xmath81 case is the same for standard and enhanced cooling ( solid and dotted lines ) . this is now easy to understand , since , as we showed above , the crustal temperature is decoupled from the core temperature if the conductivity is low . this situation is exactly the same as for persistent accreters discussed by brown @xcite . on the other hand , for high conductivity crusts , @xmath107 , so @xmath108 can not significantly deviate from @xmath109 during the outburst . therefore , crust - dominated @xmath33 in the high @xmath81 case ( dashed and dash - dotted lines in fig . [ fig : greg]a ) more closely reflects the core temperature , which is quite different between the standard and enhanced cooling cases . from examination of lightcurves in figs . [ fig : greg]a , it is likely that was observed in quiescence during the crust - dominated phase , and is presently evolving through this phase . in three of four cases considered in fig . [ fig : greg]a , the predicted quiescent luminosity @xmath33 right after the outburst is a factor of 3 higher than the observed @xmath110 ( itself uncertain by a factor of 3 , due to spectral uncertainty ) . since @xmath33 depends directly on @xmath111 over the outburst duration , this difference could also be due to a systematic overestimate of @xmath31 during the @xmath112 yr outburst , only the last 5 years of which have been covered @xmath1 daily with rxte / asm . alternatively , this difference could be due to a value of @xmath5 different from the fiducial , since the exact value of @xmath5 is uncertain ( see schatz 1999 , 2001 for a discussion on different crustal compositions than assumed by ) . to provide lightcurves which extrapolate from the currently observed luminosity , in fig . [ fig : greg]b we adjusted @xmath5 ( or , equivalently , @xmath113 ) by factors of 0.3 to 2 , as noted in the figure caption , to obtain @xmath114 erg s@xmath24 at the start of quiescence . we stress that the overall normalization of these lightcurves is uncertain due to uncertainty in @xmath5 and outburst fluence , but the shape is determined by the microphysics of the crust and the temperature of the core . the time evolution of the luminosity will permit distinction between the four cases , and , hence , directly infer the integrated conductivity of the crust and possibly indicate whether enhanced neutrino emission mechanisms are operating in the core . in particular , observing the drop in the luminosity will permit the measurement of the thermal timescale of the crust . in addition , the relative magnitude of the drop will tell us about the presence or absence of enhanced neutrino emission from the core . such measurements ( accurate to 10% ) are well within the capabilities of present x - ray instrumentation , as the present detection demonstrates . observations over the next year will be able to exclude the case where the crust is a pure crystal and hence has very high conductivity . since the initial understanding of type i bursts as thermonuclear flashes , there has been an open question as to how much flux is rising up from deep parts of the star into the burning region ( see ) . large values of this flux ( @xmath115 mev per accreted nucleon ) can stabilize burning at accretion rates lower than otherwise expected . first showed that most of the crustal heating in a persistent source went into the ns core and that , at most , 100 kev per accreted nucleon came out into the upper parts of the star . the quiescent observations of shortly after outburst strongly support this picture and tell us that deep nuclear heating will not impact the type i x - ray burst behavior . assuming that the source has been transiently accreting for the past @xmath116 or so , with outburst fluences similar to that of the most recent observed outburst , we can constrain the core temperature to be less than the maximum crust temperature . to determine the most conservative ( i.e. , largest ) upper bound on this peak crust temperature , we use the following assumptions . we take the atmosphere to be composed of pure iron ( which has a higher opacity than hydrogen / helium ) and we use low thermal conductivity ( electron - ion scattering ) in the crust . these assumptions lead to the largest temperature rise between the photosphere and the crust . taking @xmath117 , we find a peak crust temperature , and hence a maximum core temperature , of @xmath118 . the large amount of heat deposited during the recent ( and the only observed ) outburst of , and the lengthy duration of the outburst compared to the thermal diffusion time of the ns crust , imply that the quiescent flux recently observed is dominated by emission from the cooling crust and not the core . the importance of this result is that the quiescent luminosity _ is calculated using the properties of only the most recent outburst _ , rather than by estimating the accretion history over the past yr , as is the case in the core - dominated transients such as and 4@xcite . our prediction of the quiescent luminosity from the cooling crust following the outburst agrees , to within observational and theoretical uncertainties ( a factor of a few ) , with the estimate of the observed bolometric luminosity . unlike the case of the short @xmath12 transients where we can directly infer the average core temperature , in the case of we can not measure the core temperature , and hence the recurrence time , until after the crust has thermally relaxed ( 130 yr after the end of the outburst ) . assuming that the source has been transiently accreting for the past @xmath116 or so , with outburst fluences comparable to that of the most recent outburst , we can constrain the core temperature to be less than the peak crust temperature , which , using the relation of for an iron crust ( low conductivity ) , gives @xmath119 . we predict future lightcurves for on the basis of 4 different scenario for crustal and core microphysics ( high / low conductivity , and enhanced / normal neutrino emissivity ) . these can be distinguished from each other observationally one in @xmath561 yr ( excluding or confirming the high @xmath81 crust with enhanced neutrino emissivity core ) , others over longer terms ( 10 - 100 yrs ) . this offers the unique opportunity to directly constrain the properties of the ns crust thermal conductivity , heat capacity , and depth to the ns core through monitoring observations that would measure the predicted decaying lightcurve . is now the fifth after 4 , , 4u 1608 - 522 , and 4u 2129 + 47field transient neutron star which displays in quiescence a spectrum consistent with a h atmosphere ns of radius @xmath1 12 km , and ( possibly ) a hard ( @xmath1200.85 ) power - law component which comprises 1040% of the quiescent ( 0.510 kev ) flux . shows no variability on short ( @xmath56hours ) timescales . these spectral and variability properties are identical to those of the well - studied and 4 @xcite , and the spectral properties are similar to those from lower s / n data of 4u 1608 - 522 and 4u 2129 + 47 @xcite . the derived radius is also comparable to that derived from this object using the independent method of a cooling radius expansion burst @xcite . it is possible that a persistent source can turn off after long periods of accretion ( here , a persistent source is one accreting for a period @xmath121 yr , the core heating timescale ) . while eq . 1 would seem to be applicable to predict the post turn - off quiescent thermal luminosity from the crust , the equation assumes that all heat deposited in the crust is re - radiated during quiescence . that will not be the case for sources approaching eddington luminosity , such as , where the effect of neutrino cooling in the crust and core can not be neglected . in such cases , @xmath122 . here @xmath123 is the energy per accreted baryon emitted through the atmosphere from the crust immediately following the outburst . a minimum value , @xmath124 ev / nucleon , applies when the source accretes persistently at the eddington limit , in which only 10% of the deposited heat flows towards the surface @xcite . at lower persistent luminosities , however , the crust and core temperatures are lower and neutrino cooling in each becomes less important , so @xmath123 increases . in the case of ( @xmath125 ) , we obtain @xmath126250 ev , if were a persistent accreter . however , the quiescent luminosity of gives instead @xmath127 ev , a factor of 7 below the lower limit , even for a persistent source accreting at the eddington rate . this demonstrates that can not be a persistent accreter , because its crust ( and by extension the core ) is too cool to have been accreting for @xmath121 yr . note , however , that we have neglected the effect of of neutrino emission due to cooper pairing of superfluid particles ( see for a review ) which may reduce the value of @xmath123 . as discussed by , the study of qnss in globular clusters has yielded a new way to accurately measure ns radii . the distances to some globular clusters have been determined to 2% post - hipparcos @xcite , and can be measured even more directly with sim @xcite , effectively removing distance as an uncertainty . at present , there is only one globular cluster qns in which the thermal component has been spectroscopically analyzed ( in @xmath128 cen ; ) and there are two more proposed ( in 47 tuc ; ) . measuring ns radii to an accuracy of 0.5 km even without knowing the mass can exclude @xmath150% of the proposed equations of state for the ns core matter @xcite . even though we presently can not make such accurate radius measurements in the field transients due to uncertain distances ( which can be overcome with sim ) , they remain important targets for detailed studies of the observational phenomena of qnss , such as we report here . the relative luminosity of the ( possible ) power - law component poses a puzzle . it has been suggested that this component is due to accretion onto the ns magnetosphere @xcite . however , such accretion would be unrelated to the quiescent luminosity of nss as predicted from deep crustal heating ( which is dominated by outburst accretion ) , and it can only be ascribed to coincidence that in the cases where adequate sensitivity is obtained , the power - law component produces a comparable fraction of the thermal flux in different sources . this suggests that the power - law component and h atmosphere spectral component are perhaps more closely related than previously thought . the authors are grateful to the _ chandra _ observatory team for producing this outstanding observatory , and to the _ chandra _ data processing team who pre - handle all the data and provide the calibrations which are used in this work . we also thank a. y. potekhin for reading and for comments on the text prior to submission . this research has made use of data obtained through the high energy astrophysics science archive research center online service , provided by the nasa / goddard space flight center ; and of the nasa / ipac infrared science archive , which is operated by the jet propulsion laboratory , california institute of technology , under contract with the national aeronautics and space administration . this research was partially supported by the national science foundation under grant no . phy99 - 07949 and by nasa through grants nag 5 - 8658 , nag 5 - 7017 , nag 5 - 10865 and the _ chandra _ guest observer program through grant nas go0 - 1112b . l. b. is a cottrell scholar of the research corporation . e. f. b. acknowledges support from an enrico fermi fellowship . g. u. is a lee a. dubridge fellow . lr + & 1.0 + @xmath42(ev ) & 12030 + ( km ) & 6.5 + total model flux & 1.8 + /dof ( prob ) & 1.06/7 ( 0.39 ) + + & 1.30.3 + @xmath42(ev ) & 90 + ( km ) & 23 + @xmath35 & -0.1 + @xmath129 & 0.2 + total model flux & 3.1 + /dof ( prob ) & 0.43/5 ( 0.83 ) + + & 1.060.08 + @xmath42(ev ) & 1113 + ( km ) & ( 12.5 ) + @xmath35 & ( 0.85 ) + @xmath129 & 0.2 + total model flux & 2.1 + /dof ( prob ) & 0.47/7 ( 0.86 ) + + & 1.70.4 + @xmath35 & 5.20.6 + total model flux & 20.0 + /dof ( prob ) & 0.94/7 ( 0.48 ) + + & 0.7 + @xmath130 @xmath131 ) & ( 1 ) + @xmath132 ( kev ) & 1.40.1 + @xmath133 & ( 6.21.5 ) + total model flux & 0.93 + /dof ( prob ) & 2.32/7 ( 0.02 ) + + & 1.10.2 + @xmath134 ( ev ) & 37050 + @xmath135 ( km ) & 0.7 + total model flux & 2.0 + /dof ( prob ) & 1.1/7 ( 0.37 ) + + & 0.900.2 + @xmath42 ( ev ) & 30040 + ( km ) & 1.3 + total model flux & 1.3 + /dof ( prob ) & 1.11/7 ( 0.35 ) +

the type - i x - ray bursting low mass x - ray binary was recently detected for the first time in quiescence by wijnands , following a @xmath013 yr outburst which ended in feb 2001 . we show that the emission area radius for a h atmosphere spectrum ( possibly with a hard power - law component that dominates the emission above 3.5 kev ) is consistent with that observed from other quiescent neutron star transients , = 23(d/8 ) km , and examine possible ir counterparts for . unlike all other known transient neutron stars ( ns ) , the duration of this recent ( and the only observed ) outburst is as long as the thermal diffusion time of the crust . the large amount of heat deposited by reactions in the crust will have heated the crust to temperatures much higher than the equilibrium core temperature . as a result , the thermal luminosity currently observed from the neutron star is dominated not by the core , but by the crust . this scenario implies that the mean outburst recurrence timescale found by wijnands ( @xmath1 200 yr ) is a lower limit . moreover , because the thermal emission is dominated by the hot crust , the level and the time evolution of quiescent luminosity is determined mostly by the amount of heat deposited in the crust during the most recent outburst ( for which reasonable constraints on the mass accretion rate exist ) , and is only weakly sensitive to the core temperature . using estimates of the outburst mass accretion rate , our calculations of the quiescent flux immediately following the end of the outburst agree with the observed quiescent flux to within a factor of a few . in this paper , we present simulations of the evolution of the quiescent lightcurve for different scenarios of the crust microphysics , and demonstrate that monitoring observations ( with currently flying instruments ) spanning from 130 yr can measure the crust cooling timescale and the total amount of heat stored in the crust . these quantities have not been directly measured for any neutron star . this makes a unique laboratory for studying the thermal properties of the crust by monitoring the luminosity over the next few years to decades . ifundefinedchapter # 1

You are an expert at summarizing long articles. Proceed to summarize the following text: a quick search through the nasa / ads abstract service indicates that more than @xmath0 5000 papers were published over the past five years on agns ! clearly , a comprehensive review of this literature is well beyond the scope of the present article . the present review highlights important results from recent large - scale ground - based surveys and space missions , and describes the consequences of these results on our understanding of radio - quiet agns . the standard paradigm for radio - quiet agns is described in 2 along with a few key supporting observations . important new constraints on this standard picture have been derived from recent uv / optical , infrared , and x - ray surveys ; these are discussed in 3 . in 4 , the nature of the connection between the supermassive black hole , host galaxy , circumnuclear starburst , and agn is briefly reviewed taking into account recent new data . the issue of agn fueling / triggering is addressed in 5 . a summary of the outstanding issues is given in 6 along with suggestions for future avenues of research . 2.6 in figure 1 is an idealized representation of the inner structure of radio - quiet agns . direct observational support for this picture comes from the detection in a few radio - quiet agns of h@xmath1o - masing disk - like structures in orbit around central masses of a few @xmath2 10@xmath3 10@xmath4 m@xmath5 on scales of 0.1 1.0 pc ( @xmath0 10@xmath6 @xmath7 , where @xmath8 is the gravitational radius ) . the most convincing cases so far are ngc 4258 ( miyoshi et al . 1995 ) , ngc 1068 ( greenhill et al . 1996 ; greenhill & gwinn 1997 ; gallimore et al . 1997 , 2001 ) , and ngc 4945 ( greenhill , moran , & herrnstein 1997 ) . a survey by braatz , wilson , & henkel ( 1997 ) failed to detect h@xmath1o masers in seyfert 1s , perhaps indicating a less favorable face - on disk geometry in these objects ( small @xmath9 along the line of sight ) , as expected in the unification model of seyfert galaxies ( fig . indirect evidence for accretion disk in radio - quiet agns comes from the presence of broad ( @xmath10 70,000 km s@xmath11 in mcg06 - 30 - 15 ) , skewed and redshifted fe k@xmath12 lines in seyfert 1s . this line probes the purported accretion disk within a few @xmath7 of the black hole ( e.g. , tanaka et al . 1995 ; iwasawa et al . 1996 , 1999 ; nandra et al . 1997 ; guainazzi et al . 1999 ; wilms et al . 2001 ; lee et al . 2002 ; fabian et al . 2002 ; turner et al . 2002 ; see contribution by fabian in this volume ) . the presence of large - scale ( photo-)ionization cones in radio - quiet agns is further indirect evidence for an inner disk structure in these objects ( e.g. , haniff , wilson , & ward 1988 ; pogge 1989 ; tadhunter & tzvetanov 1989 ; wilson & tzvetanov 1994 ; mulchaey et al . 1996a , 1996b ; pogge 2000 ; veilleux 2002 ; kinkhabwala et al . 2002 ) . historically , spectropolarimetric studies of radio - quiet agns have been the main driver behind the unification model of seyfert galaxies ( e.g. , antonucci & miller 1985 ; miller , goodrich , & matthews 1991 ) . but recent spectropolarimetric surveys indicate that only @xmath0 30 50% ( _ i.e. _ not all ) of seyfert 2 galaxies harbor hidden broad - line regions ( hblrs ; e.g. , tran 2001 ) . seyfert 2s with hblrs tend to have larger radio - to - fir flux ratios and warmer dust temperatures than those without ( e.g. , miller & goodrich 1990 ; kay 1994 ; heisler , lumsden , & bailey 1997 ; moran et al . 2000 ; tran 2001 ) . these trends have been interpreted by some to imply the existence of two separate classes of seyfert 2s . however , several authors have been quick to point out the importance of the orientation - dependent selection criteria used in many of these studies ( e.g. , antonucci 2001 ; alexander 2001 ; lumsden et al . 2001 ; gu & huang 2002 ) . biases in luminosity , starburst contribution , and covering factor are necessarily present in these studies , so caution should be used when interpreting the results . recent results from near- and mid - infrared spectroscopy of radio - quiet agns have provided further support for the unification model . obscured blrs have been detected at @xmath0 2 4 @xmath13 m in a number of uv- and infrared - selected seyfert 2 galaxies ( e.g. , veilleux , goodrich , & hill 1997a ; veilleux , sanders , & kim 1997b , 1999b ; lutz et al . 2002 ) . clavel et al . ( 2000 ) also note that the weaker 7-@xmath13 m continuum and larger equivalent widths of the pah feature in seyfert 2s relative to seyfert 1s can be explained if the continuum in the seyfert 2s is more strongly extinguished ( @xmath14 mag ) . studies at near- and mid - infrared wavelengths can only probe down to @xmath9 @xmath15 few @xmath2 10@xmath16 @xmath17 , _ i.e. _ considerably smaller than the expected column density of the dusty torus in agn . hard x - ray observations have extended the range of column densities up to @xmath0 10@xmath18 @xmath17 . maiolino et al . ( 1998 ) and risaliti , maiolino , & salvati ( 1999 ) have used bepposax , asca , and ginga data to show that seyfert 2 galaxies generally have larger obscuration than seyfert 1s , as expected in the standard unification model . however , these observations do not put any constraints on the exact geometry and location of the obscuring material in seyfert 2s ( e.g. , inner disk / torus _ vs. _ galaxy - scale circumnuclear material ? ) . optical , uv , and x - ray observations of agns indicate that the simplest form of the standard picture ( fig . 1 ) is not tenable and must be modified to account for dependences on agn luminosity . optical spectroscopy of infrared - selected agns ( in which orientation - dependent biases are less important ) indicates that the ratio of seyfert 2s to seyfert 1s decreases significantly with infrared luminosity ( e.g. , veilleux et al . 1995 ; veilleux , kim , & sanders 1999a ) . the well - known decrease of the c iv @xmath191549 equivalent widths with increasing luminosity ( _ i.e. _ the uv baldwin effect ; e.g. , osmer & shields 1999 ; espey & andreadis 1999 ) is another manifestation of this strong luminosity dependence . the fe k@xmath12 line and compton scattering hump also are weaker in qsos than in seyferts ( the x - ray baldwin effect ; e.g. , nandra et al . finally , uv and x - ray narrow absorption lines are rarely seen in the spectra of quasars but not so in seyferts ( e.g. , turner et al . 1993 ; mathur et al . 1999 ; nicastro et al . 1999 ; kaastra et al . 2000 , 2002 ; kaspi et al . all of these observations can be explained in the context of the unification model if the opening angle , or location of the inner edge , of the disk / torus / wind structure is allowed to vary with luminosity ( e.g. , lawrence 1991 ; hill et al . 1996 ; elvis 2000 ) . the discussion in this section focusses on moderate - to - high luminosity agns . the reader should refer to reviews by vron - cetty & vron 2000 , ho ( 2002 ) , barth ( 2002 ) and contributions in this volume by filippenko and ho for discussions of recent results on low - luminosity agns ( llagns ) . radio - loud agns are reviewed by urry in this volume so radio surveys are not discussed here . several new surveys have improved our knowledge of agn demography as a function of redshift ; three of them are discussed here . the results from the hamburg / eso bright qso survey ( he s ; e.g. , wisotzki et al . 2000 ) indicate that the local qso density is @xmath0 50% larger than previously thought ( e.g. , bright quasar survey of schmidt & green 1983 ) . the 2df qso redshift survey ( e.g. , boyle et al . 2000 ) and the sloan digital sky survey ( sdss ; e.g. , fan et al . 2001 ) have nicely confirmed that the qso density rises steeply up to @xmath20 2 and decreases beyond @xmath21 . at the time of writing , sdss has revealed four quasars at @xmath22 . this set of high-@xmath23 qsos should at least double in size by the time the survey is completed . these objects will provide important new constraints on the epoch of qso / galaxy formation ( see , e.g. , haiman & cen 2002 ) . the two micron all sky survey ( 2mass ) has uncovered a large population of red ( @xmath24 ) agns at relatively small redshifts ( median @xmath23 @xmath0 0.25 ; cutri et al . 2001 ; see also gregg et al . approximately 75% of these sources are previously unidentified agns whose space density ( @xmath0 0.5 deg@xmath25 ) is comparable to that of optical / uv selected agns of the same ir magnitudes . a large fraction ( @xmath0 80% ) of these objects are type 1 agns . chandra follow - up observations of some of these agns indicate that all of them are x - ray faint , with the reddest being the faintest in x - rays ( wilkes et al . interestingly , these broad - lined agns show significant absorption ( several of them have @xmath26 @xmath17 ) and may be important contributors to the cosmic x - ray background ( cxb ; cf . 3.3 ; see also webster et al . 1995 ; benn et al . 1998 ; whiting et al . 2001 for analyses of similar pks radio - selected red qsos ) . as mentioned in 2 , x - ray observations generally provide support for the unification model of seyfert galaxies . however , there are a number of important exceptions . these mismatches in the optical hard x - ray classification fall into three broad categories : ( 1 ) several broad - line agns show significant absorption in the x - rays ( e.g. , seyferts and qsos : fiore et al . 2001 , 2002 , wilkes et al . 2002 ; balqsos : mathur et al . 2001 , gallagher et al . ( 2 ) deep x - ray surveys have revealed a large population ( @xmath0 40 60% ) of x - ray bright sources with no obvious optical agns ( e.g. , barger et al . 2001 , 2002 ; tozzi et al . 2001 ; stern et al . 2002b ; fiore et al . ( 3 ) a number of seyfert 2 galaxies show no obvious x - ray absorption ( e.g. , ptak et al . 1996 ; bassani et al . 1999 ; pappa et al . 2001 ; panessa & bassani 2002 ) . the first category of objects may be explained in the context of the unification model if dust near the agn is different from galactic ( e.g. , different dust / gas ratio , metallicity , grain size ; veilleux et al . 1997a ; maiolino et al . 2001 ) or is not co - spatial with the ( neutral and ionized ) material producing the x - ray absorption ( e.g. , veilleux et al . 1997a ) . x - ray bright sources with no obvious optical agns may be powered through inefficient adaf - like accretion with @xmath27 , or perhaps the apparent lack of optical agns in these sources is simply due to strong dilution effects by the host galaxy light ( e.g. , moran , filippenko , & chornock 2002 ) . finally , narrow - line agns with no obvious x - ray absorption have often been shown to be ( nearly ) compton thick when observed in the hard x - rays . half of all nearby optically selected seyfert 2s may fall in this category [ e.g. , risaliti et al . 1999 ; well - known examples include ngc 1068 , ngc 4945 ( done , madejski , & smith 1996 ) , and circinus ( matt et al . luminous examples also exist , although they may be less common : e.g. , qso-2s ( norman et al . 2002 ; crawford et al . 2002 ; stern et al . 2002a ) , x - ray detected extremely red objects ( eros ) with @xmath28 ( e.g. , alexander et al . 2002 ; mainieri et al . 2002 ) , and ultraluminous infrared galaxies ( e.g. , ngc 6240 ; iwasawa & comastri 1998 ; vignati et al . 1999 ) . the improved sensitivity of the current x - ray facilities has allowed to search for possible evolutionary effects in the x - ray spectra of quasars and agns over a broad redshift range . brandt et al . ( 2002 ) and mathur , wilkes , & ghosh ( 2002 ) recently failed to find convincing evidence for a redshift dependence of the optical x - ray slope in qsos out to @xmath29 . bechtold et al . ( 2002 ) come to a different conclusion using a larger comparison sample . a significant population of buried type 2 agns with peak emissivity around @xmath30 ( rather than @xmath31 for type 1 agns ; 3.1 ) appears to be needed to explain the properties of the cxb and the redshift dependence of the number density of x - ray sources in the deep chandra and xmm surveys ( e.g. , rosati et al . 2002 ; franceschini , braito , & fadda 2002 ) . these results may imply that type 1 and 2 agns follow different evolutionary paths , in disagreement with the agn unification model . note , however , that the recent discovery of a large population of strongly absorbed type 1 agns by wilkes et al . ( 2002 ) may affect these conclusions . ground - based and @xmath32 observations of the stellar and gas kinematics near the center of `` normal '' ( inactive ) galaxies have revealed a close connection between the mass of the supermassive black hole at the center of each of these objects and the mass of the spheroidal component ( e.g. , kormendy & richstone 1995 ; faber et al . 1997 ; magorrian et al . 1998 ; gebhardt et al . 2000 ; kormendy & gebhardt 2001 ; merritt & ferrarese 2001 ; tremaine et al . 2002 ) and perhaps also the mass of the dark matter halo ( ferrarese 2002 ) . the results of reverberation mapping in agns suggest that the @xmath33 relations found in inactive galaxies also apply to active galaxies ( e.g. , wandel , peterson , & malkan 1999 ; kaspi et al . 2000 ; peterson & wandel 2000 ; laor 2001 ; onken & peterson 2001 ; mclure & dunlop 2002 ; ferrarese et al . 2001 ; although see krolik 2001 ) . these results suggest a causal connection between spheroid / galaxy formation , black hole growth and agn activity ( e.g. , efstathiou & rees 1988 ; small & blandford 1992 ; haehnelt & rees 1993 ; chokshi 1997 ; silk & rees 1998 ; haiman & loeb 1998 ; fabian 1999 ; kauffmann & haehnelt 2000 ; burkert & silk 2001 ; adams , graff , & richstone 2001 ) . this tight smbh host galaxy connection is discussed in more detail by urry , peterson , and merritt in this volume . a strong connection also exists between starbursts and agns ( veilleux 2001 ; also ward this volume ) . nuclear starbursts appear to be present in several seyfert galaxies , based on the strength of uv and optical absorption and emission features from young / intermediate - age stars in the nuclear spectra of these objects ( e.g. , terlevich , diaz , & terlevich 1990 ; heckman et al . 1997 ; gonzalez delgado et al . 1998 , 2001 ) . the presence of extended soft thermal x - ray emission in seyferts also supports this scenario ( e.g. , levenson , weaver , & heckman 2001 ) . claims that starbursts are more common in seyfert 2s than in seyfert 1s have been made for many years ( e.g. , rodriguez - espinosa , rudy , & jones 1986 ; pier & krolik 1993 ; maiolino et al . 1995 ; nelson & whittle 1996 ; gonzalez delgado et al . 2001 ; gu , dultzin - hacyan , & de diego 2001 ) , but orientation - dependent selection criteria may seriously bias some of these samples and cause the apparent seyfert 1 / seyfert 2 dichotomy . the nuclear starbursts in active galaxies may have a very strong impact on the evolution of the host galaxy and the agn itself . the stellar winds and supernovae associated with the nuclear starburst may deposit sufficient amounts of mechanical energy at the centers of these objects to severely disrupt the gas phase of these systems and result in large - scale galactic winds . a well - known example of starburst - driven wind in an active galaxy is ngc 3079 . detailed optical , radio , and x - ray studies of this object have revealed the presence of a powerful galactic wind that is strongly interacting with the ambient material of the host galaxy ( e.g. , veilleux et al . 1994 ; cecil et al . 2001 ; cecil , bland - hawthorn , & veilleux 2002 ) . the wind event is clearly disturbing the distribution of gas within the central kpc of this object and may therefore also affect the feeding of the agn . the frequency of starburst - driven winds in agns is poorly constrained ( see veilleux & rupke 2002 for a discussion of a promising search technique ) , but if common these winds may provide negative feedback on the fueling mechanisms of agns ( e.g. , rupke , veilleux , & sanders 2002 ) . the broad range in luminosity of agn ( @xmath34 10@xmath35 10@xmath36 l@xmath5 ) implies accretion rates of order @xmath37 0.001 100 m@xmath5 yr@xmath11 ( assuming a radiative efficiency of 10% in rest mass units ) ; the required accretion rates are very small for llagns but quite substantial for qsos . the requirements on the fueling / triggering processes for agns are therefore highly dependent on the agn luminosity . local processes are sufficient to power llagns and seyferts , but galactic - scale phenomena may be required to explain powerful qsos . a broad range of processes including galaxy interactions and mergers , large - scale and nuclear bars , and nuclear gaseous spirals have been proposed to account for the fueling of seyferts and llagns ( see combes 2000 and the contribution by combes in this volume ) . the lack of obvious excess of companions and mergers seems to rule out the possibility that galaxy interactions and mergers are solely responsible for triggering these objects ( e.g. , fuentes - williams & stocke 1988 ; dultzin - hacyan et al . 1999 ; de robertis , yee , & hayhoe 1998 ; virani , de robertis , & vandalfsen 2000 ) . a slight ( 2.5-@xmath38 ) excess of bars appear to be present among seyferts ( e.g. , knapen , shlosman , & peletier 2000 ; laine et al . 2002 ; knapen this volume ; although see mulchaey & regan 1997 ; regan & mulchaey 1999 ; martini et al . 2001 ) , but this result can not explain seyfert activity in galaxies without bars . nuclear ( 0.1 1 kpc ) gaseous spirals have been detected in most seyferts but they also appear to be present in several normal galaxies and therefore can not be the only reason for the seyfert activity ( e.g. , ford et al . 1994 ; dopita et al . 1997 ; laine et al . 1999 , 2001 ; martini & pogge 1999 ; regan & mulchaey 1999 ; pogge & martini 2002 ; emsellen this volume ) . most likely , these various processes help replenish a reservoir of fuel on scales of @xmath0 100 pc or larger , but other processes are needed to carry the material down to sub - pc scales ( e.g. , gas instabilities , stellar ejecta , magnetic fields ) . unfortunately , very little is known observationally on scales @xmath37 10 pc . the origin of the activity in high - luminosity objects is perhaps less ambiguous than in low - luminosity objects . hosts of luminous qsos generally appear to be elliptical galaxies ( e.g. , dunlop et al . 2002 ; although see mcleod & mcleod 2001 ) , but signs of interaction are seen in several qsos , especially in those with infrared excess ( e.g. , surace , sanders , & evans 2001 ; canalizo & stockton 2000 , 2001 ) . several of these objects contain large quantities of molecular gas ( e.g. , evans et al . 2001 ) , suggestive of a gas - rich merger origin . ultraluminous infrared galaxies ( uligs ) have long been suspected to be the progenitors of optical quasars based on the fact that nearly all uligs show signs of recent galaxy mergers and starburst / agn activity , but only recently has there been a large enough homogeneous sample of uligs to look carefully at this question . recent optical , near - infrared , and mid - infrared spectroscopic surveys of the 1-jy sample of 118 uligs indicate that @xmath0 30% ( 50% ) of the objects with log[l@xmath39/l@xmath5 ] @xmath41 12.0 ( 12.3 ) are powered predominantly by a quasar rather than a starburst ( e.g. , kim , veilleux , & sanders 1998 ; genzel et al . 1998 ; veilleux , kim , & veilleux 1999a ; lutz , veilleux , & genzel 1999 ) . a recent morphological analysis of the 1-jy sample indicates strong trends between infrared luminosity and colors , quasar activity , and merger phase ( veilleux , kim , & sanders 2002 ) . all uligs in the 1-jy sample are found to be in the pre - merger or final merger phase . about @xmath0 70% of the extreme uligs with log[l@xmath39/l@xmath5 ] @xmath42 12.5 are single - nucleus advanced mergers . all ( most ) seyfert 1s ( 2s ) in the sample are also advanced mergers . the @xmath43- and @xmath44-band profiles in 73% of the single - nucleus advanced mergers are well fitted over @xmath43 = 4 12 kpc by an elliptical - like @xmath45 law ( see also scoville et al . 2000 ; cui et al . these elliptical - like hosts follow the same @xmath43-band @xmath46 @xmath47 relation as normal ellipticals , suggesting that these objects may eventually become intermediate - luminosity ( 1 3 l@xmath48 ) elliptical galaxies if they get rid of their excess gas or transform the gas into stars . the hosts of uligs show a broad range in luminosity ( mean @xmath0 2 l@xmath48 ) which overlaps with that of qso hosts . the @xmath43 @xmath44 colors of ulig hosts ( mean @xmath0 3.0 ) are also similar to those of qso hosts and normal ellipticals . the average half - light radius of uligs is 4.8 @xmath49 1.4 kpc at @xmath43 and 3.5 @xmath49 1.4 kpc at @xmath44 , similar to the qso host sizes measured at @xmath50 by mcleod & mcleod ( 2001 ) but slightly smaller than those measured at @xmath43 by dunlop et al . the reason for this apparent discrepancy between the two qso datasets is not known . overall , these results support the scenario in which the ulig is the result of the merger of two @xmath0 l@xmath48 disk galaxies which eventually evolves into an elliptical - like galaxy with a powerful agn . there are obviously exceptions to this scenario and this is why a large sample of objects like the 1-jy sample is needed to draw statistically meaningful conclusions . the following is a list of outstanding issues and unanswered questions : * there is strong general support for the agn unification model ( fig . 1 ) , but there are important exceptions : ( a ) not all seyfert 2s show hblrs . this may be a real effect or it could be due to instrument sensitivity ( _ i.e. _ not enough `` mirrors '' to scatter the blr emission back into our line of sight ) . ( b ) optical and x - ray classifications are not always consistent with each other . these classification mismatches may be explained by non - standard dust near agns , inefficient adaf - like accretion , dilution effects by galaxy light , or compton thickness . * the luminosity dependence of the type 2 / type 1 agn ratio and the uv and x - ray baldwin effects of the emission and absorption lines in agns require that the geometry of the disk / torus / wind structure depends on agn luminosity . the exact dependence relies on knowing the fraction of obscured type 2 agns , which is still subject to large uncertainties . * recent chandra investigations of high - redshift quasars come to conflicting conclusions regarding possible evolutionary effects of the sources of energy and absorption . the situation at low redshift is less ambiguous : the properties of the cxb and results from deep chandra and xmm surveys appear to require a significant population of buried type 2 agns with peak emissivity around @xmath30 , rather than @xmath31 for unobscured type 1 agns . these results may imply that type 1 and 2 agns follow different evolutionary paths , in contradiction with the agn unification model . the recent discovery of a large population of partially obscured type 1 agns at low redshifts may affect these conclusions . * there is now strong evidence that the host galaxy smbh connection originally found in inactive galaxies also applies to active galaxies . emission - line methods to test this connection at high redshifts are promising but will undoubtedly be less accurate ( e.g. , nelson 2000 ; mclure & jarvis 2002 ; vestergaard 2002 ) . * there appears to be a symbiotic relation between starbursts and agns . the origin of this relation is not known . the surveys suggesting a greater occurrence of ( circumnuclear ) starbursts in seyfert 2s than in seyfert 1s are often strongly affected by selection biases . feedback from starburst - driven galactic winds may be one of many side effects of this tight starburst agn connection . this wind phenomenon is particularly important in understanding agn activity at high redshift . * minor mergers , bars , and nuclear spirals may all combine to bring the fuel down to @xmath0 100 pc in seyferts and llagns , but it is not clear what happens next . there is now strong evidence that gas - rich mergers are able to trigger some qsos at low redshifts after undergoing a ulig phase , but there is very few observational constraints on the formation process for the bulk of qsos that were formed at @xmath51 . the author congratulates the conference organizers for a very successful meeting . some of the work discussed in this review was done in collaboration with d .- c . kim , d. b. sanders , j. bland - 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the current paradigm for radio - quiet agns is reviewed , taking into account new results from recent large - scale surveys carried out from the ground and from space . topics include structure of the central engine , agn demography , fueling / triggering processes , and connection between the supermassive black hole , host galaxy , circumnuclear starburst and agn . dependences on agn power and lookback time are pointed out in the discussion . suggestions for future avenues of research are mentioned in the last section .

You are an expert at summarizing long articles. Proceed to summarize the following text: a _ wavelet set _ relative to dilation by an expansive ( all eigenvalues greater than 1 in absolute value ) real @xmath3 matrix @xmath4 is a set @xmath5 whose characteristic function @xmath6 is the fourier transform of an orthonormal wavelet . that is , if @xmath7 then @xmath8 is an orthonormal basis for @xmath9 . this definition is equivalent to the requirement that the set @xmath10 tiles @xmath11dimensional space ( almost everywhere ) both under translation by @xmath12 and under dilation by the transpose @xmath13 , so that @xmath14 @xmath15 while wavelet set wavelets are not well - localized , and thus not directly useful for applications , they have proven to be an essential tool in developing wavelet theory . in particular , wavelet set examples established that not all wavelets have an associated mra @xcite , and that single wavelets exist for an arbitrary expansive matrix in any dimension @xcite . smoothing and interpolation techniques have also used wavelet set wavelets to produce more well - localized examples . ( see e.g. @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite . ) all of the early examples of wavelet sets for dilation by non - determinant 2 matrices in dimension greater than 1 were geometrically complicated , showing the fingerprints of the infinite iterated process used to construct them . ( see e.g. figure [ dim2](a ) ) . many early researchers , e.g.@xcite , @xcite , conjectured that a wavelet set for dilation by 2 in dimension greater than 1 could not be written as a finite union of convex sets . in support of this conjecture , benedetto and sumetkijakan @xcite showed that a wavelet set for dilation by 2 in @xmath0 can not be the union of @xmath16 or fewer convex sets . however , in 2004 , gabardo and yu @xcite used self - affine tiles to produce a wavelet set for dilation by 2 in @xmath17 that is a finite union of polygons ( figure [ dim2](b ) ) . in 2008 @xcite we used a technique based on generalized multiresolution analyses @xcite to construct such wavelet sets for arbitrary real ( @xmath18 ) scalar dilations in @xmath17 . figure [ dim2](c ) shows one of the wavelet sets for dilation by 2 from @xcite . although they were developed independently , and using very different techniques , these two examples are remarkably similar . in fact , the wavelet sets in figure [ dim2](b ) and [ dim2](c ) are equivalent in the sense that one can be transformed into the other under multiplication by a determinant 1 integer matrix . the similar shape of these two wavelet sets suggests the general @xmath16-dimensional result produced in this paper . [ dim2 ] @xmath19{holec.pdf } } \end{picture } & \setlength{\unitlength}{100bp } \begin{picture}(1,1)(.0,-.2 ) \put(0,0){\includegraphics[width=\unitlength]{gabyu.pdf } } \end{picture } & \setlength{\unitlength}{120bp } \begin{picture}(1,0.0)(-.2,-.2 ) \put(0,0){\includegraphics[width=\unitlength]{unsymw.pdf } } \end{picture}\\ \mbox{soardi / wieland 1998\quad\quad\quad}&\mbox{gabardo / yu 2004}&\mbox{\quad\quad merrill 2008}\\ \end{array}$ ] we call wavelet sets that are finite unions of convex sets _ simple wavelet sets_. in 2012 @xcite , we expanded the results in @xcite to produce simple wavelet sets for dilation by any @xmath20 matrix that has a positive integer power equal to a scalar times the identity , as long as its singular values are all greater than @xmath21 . in that paper , we also found examples of expansive @xmath20 matrices that can not have simple wavelet sets . it is our conjecture that , in any dimension , an expansive matrix whose determinant does not have absolute value equal to 2 can have a simple wavelet set if and only if it has a positive integer power equal to a scalar times the identity . in this paper , we generalize the 2-dimensional examples in @xcite to @xmath16-dimensional space , @xmath1 . we do this using neither the generalized multi - resolution analysis techniques of @xcite , nor the self - affine techniques of @xcite . rather , we use a remarkable result by sherman stein @xcite on tiling @xmath0 with notched cubes , together with the tiling conditions that are equivalent to the definition of a wavelet set . section 2 presents stein s result , and then skews and translates the notched n - cubes to produce notched parallelotopes that are simple wavelet sets for dilation by negative scalars . section 3 further modifies these notched parallelotopes by translating out a central parallelotope ( as in figure [ dim2](b ) and [ dim2](c ) ) . using this technique , theorem [ main ] creates simple wavelet sets for dilation by any scalar @xmath22 . this result establishes counterexamples , in every dimension greater than 1 , to the conjecture that wavelet sets for dilation by 2 can not be finite unions of convex sets . these counterexamples are composed of @xmath23 convex sets for dimension @xmath16 , as compared to the lower bound of @xmath24 given in the benedetto / sumetkijakan result mentioned above . theorem [ matrix ] generalizes theorem [ main ] to dilation by matrices that have a positive integer power equal to a scalar , as long as their singular values are not too small . one consequence of this theorem is to create simple wavelet sets for dilation by a scalar @xmath25 with @xmath26 , thus completing the scalar dilation case of the existence question for simple wavelet sets . for non - scalar dilations in dimension 3 and higher , theorem [ matrix ] offers support to the sufficiency direction of the conjecture above concerning exactly which matrices have associated simple wavelet sets . the examples that end section 3 further support this conjecture by showing that the theorem s additional condition on singular values need not always hold for matrices that have simple wavelet sets . we begin by establishing some notation . write @xmath27 for the standard basis of @xmath0 , and @xmath28 for the cyclic permutation matrix with columns @xmath29 . let @xmath30 stand for the vector @xmath31 , and write @xmath32 for translation by @xmath33 . given a vector @xmath34 in @xmath0 that is not a multiple of @xmath30 , let @xmath35=\{x_0v+x_1c(v)+\dots x_{n-1}c^{n-1}v\;:\;0\leq x_i\leq 1\}\ ] ] be the parallelotope spanned by the vectors @xmath36 . note that the spanning vectors of @xmath37 $ ] have equal length , since they are permutations of the same vector @xmath38 . in particular , when @xmath39 , @xmath37 $ ] is a rhombus . note also that @xmath37 $ ] has two vertices on the line determined by @xmath30 : one at the origin , and the other determined by the sum of the coordinates of @xmath38 , at @xmath40 , where the vector given an @xmath41 , @xmath42 , write @xmath43 $ ] for the notched parallelotope that results from deleting a subparallelotope scaled by @xmath41 from the vertex @xmath44 of @xmath37 $ ] . that is , let @xmath45=\mathcal p[v]\setminus\;\tau_{(1-\alpha)(\sum v_i)\vec 1}\;\alpha \mathcal p[v],\ ] ] or equivalently , @xmath46=\mathcal p[v]\setminus\;\tau_{(\sum v_i)\vec 1}\;\left(-\alpha \mathcal p[v]\right).\ ] ] we will need the following result about translation tilings by notched cubes due to sherman stein . [ notchedcube ] given a real number @xmath42 , let @xmath47 be the lattice spanned by the columns of @xmath48 , where @xmath28 is the cyclic permutation matrix . then the translates of the notched unit cube @xmath49 $ ] by the vectors in @xmath47 tile @xmath0 . see @xcite we use this result to produce a notched parallelotope that tiles @xmath0 under translation by the lattice @xmath12 : [ notchedpar ] for a fixed real number @xmath41 , @xmath42 , let @xmath50 . then the translates of @xmath51 $ ] by @xmath12 tile @xmath0 . by lemma [ notchedcube ] we know that @xmath49 $ ] tiles @xmath0 under translation by @xmath52 the lattice spanned by the columns of @xmath48 . if we define @xmath4 to be the linear transformation that maps @xmath47 to @xmath12 , we thus have that that @xmath53)$ ] tiles @xmath0 under translation by @xmath12 . note that @xmath54 so that @xmath55 . thus , @xmath56\right)&=&a\left(\mathcal p[e_1]\setminus\;\tau_{(1-\alpha)\vec 1}\;\alpha \mathcal p[e_1]\right)\\ & = & \mathcal p[a(e_1)]\setminus\;\tau_{(1-\alpha)a\vec 1}\;\alpha\mathcal p[a(e_1)]\\ & = & \mathcal p[w(\alpha)]\setminus\tau_{\vec 1}\;\alpha \mathcal p[w(\alpha)]\\ & = & \mathcal { n}[w(\alpha ) , \alpha]\end{aligned}\ ] ] to be a wavelet set , a notched parallelotope would have to also tile under dilation . for a scalar dilation @xmath25 , this is clearly impossible , since a notched parallelotope is defined to have an extreme point at the origin . thus , we consider instead the translated notched parallelotope @xmath57 $ ] , for @xmath33 . for such a set to tile under dilation by @xmath25 would require that the dilated outer parallelotope , @xmath58 $ ] , fit perfectly into the notch , which is of the form @xmath59 $ ] . first note that this would force the scale of the dilated outer parallelotope to match the scale of the notch , so that @xmath60 . for positive @xmath18 , this perfect fit also would require that @xmath61 so that @xmath62 . however , the outer parallelotope @xmath63 $ ] has one of its extreme points at @xmath64_j = t+\frac{d}{d-1}\vec 1 $ ] . thus , using the required value for the translation @xmath65 would again cause the outer parallelotope to have an extreme point at the origin . hence a set of the form @xmath57 $ ] can not tile by a positive scalar dilation . however , for negative scalar dilations , a wavelet set that is just a notched parallelotope is possible , as the following theorem shows . the examples produced by theorem [ negdilate ] generalize the wavelet sets for negative scalar dilations in @xmath17 found in @xcite and @xcite . [ negdilate ] for @xmath66 , @xmath18 , let @xmath67 , and @xmath68 . then @xmath69\ ] ] is a wavelet set for dilation by @xmath70 in @xmath0 . we know from lemma [ notchedpar ] that the notched parallelotope @xmath71 $ ] tiles @xmath0 under translation by @xmath12 , and thus that its translate @xmath72 does as well . it remains to show that @xmath72 tiles under dilation by @xmath70 . the proposed wavelet set @xmath73 $ ] has its vertices on the line determined by @xmath30 at @xmath74 and @xmath75 , while its outer parallelotope @xmath76 $ ] has its vertices at @xmath74 and @xmath77 . ( see figure [ neg]a . ) if we consider now the dilate by @xmath78 of these two polytopes , we see that @xmath79 has its vertices on the line determined by @xmath30 at @xmath80 and @xmath81 , while @xmath82 $ ] has its vertices at @xmath80 and @xmath83 . ( see figure [ neg]b . ) [ neg ] @xmath84{negdil.pdf } } \end{picture } & \setlength{\unitlength}{200bp } \begin{picture}(1.0,1)(0,0 ) \put(0,0){\includegraphics[width=\unitlength]{negdilb.pdf } } \end{picture}\\ \hskip.5in\mbox{(a)}\quad\mathcal w & \hskip.5 in \mbox{(b)}\quad -\frac 1d \mathcal w \end{array}$ ] using the value @xmath85 given in the statement of the theorem , we see that @xmath86 and @xmath87 , so that the dilation by @xmath78 of the outer parallelotope used to form @xmath72 exactly fits into the notch of @xmath72 . thus , @xmath88\setminus\left(\tau_{(t+1)\vec 1}\;\frac{1}{d^2}\mathcal p\left[w\left(\frac 1d\right)\right]\right)\\ & = & \tau_{t\vec 1}\mathcal p\left[w\left(\frac 1d\right)\right]\setminus\frac{1}{d^2}\left(\tau_{t\vec 1}\mathcal p\left[w\left(\frac 1d\right)\right]\right),\end{aligned}\ ] ] which tiles under dilation by @xmath89 . therefore , @xmath72 tiles under dilation by @xmath70 . figure [ neg]a shows the general shape of a wavelet set @xmath72 for dilation by @xmath70 in 3 dimensions . the size of the notch shown is for @xmath90 ; in general the size of the notch will be @xmath91 times the size of the whole . for dilation by @xmath70 , the lower left hand vertex of @xmath72 will be at @xmath92 , the inside corner of the notch at @xmath93 , and the outer corner of the notch at @xmath94 even though we have seen that a translate of a notched parallelotope @xmath95 $ ] can not itself tile under dilation by a positive scalar , we will make a simple alteration to such a set that retains the property of tiling under translation , and makes the set tile under dilation as well . we use the idea behind the wavelet set construction technique in @xcite . that is , we eliminate the overlap between our proposed wavelet set and its dilate , by translating the dilate out by an integer vector , creating a satellite . in order to avoid the iterated process required in @xcite , we translate out a little bigger piece than the dilate of @xmath95 $ ] ; that is , we translate out the dilate of the whole outer parallelotope @xmath96 $ ] . we choose the translation amount such that a dilate of the satellite exactly fills the notch . the details of this construction are carried out in the following theorem . [ main ] for @xmath66 , @xmath22 , let @xmath97 . suppose @xmath98 satisfies @xmath99 , and let @xmath100 then @xmath101\right)\;\setminus\;\left(\frac 1d\tau_{t\vec 1}\ ; \mathcal p\left[w\left(\frac1{d^2}\right)\right]\right)\right)\;\bigcup\;\tau_{k\vec 1}\left(\frac1d\tau_{t\vec 1}\;\mathcal p\left[w\left(\frac1{d^2}\right)\right]\right)\ ] ] is a wavelet set for dilation by @xmath25 in @xmath102 we claim that @xmath103\subset\tau_{t\vec 1 } \;\mathcal n\left[w(\frac1{d^2 } ) , \frac 1{d^2}\right]$ ] . to see this , first note that @xmath104 and @xmath105_i,$ ] so that @xmath106\subset\tau_{t\vec 1}\ ; \mathcal p\left[w\left(\frac1{d^2}\right)\right]$ ] . thus , to establish the claim , it will suffice to show that the vertex of the notch that is closest to the origin , namely @xmath107_i)\vec 1 $ ] , lies outside of @xmath108 $ ] . that is , we must show that @xmath109 . substituting @xmath110 we see that this is equivalent to @xmath111 which follows from the given conditions @xmath112 and @xmath22 . lemma [ notchedpar ] impies that @xmath113 $ ] tiles under translation by the integer lattice , and thus that @xmath114 $ ] does as well . by the claim , we have that @xmath72 is formed by removing @xmath115 $ ] from inside @xmath114 $ ] , and shifting it by a vector of the integer lattice . thus @xmath72 tiles @xmath0 under translation by @xmath12 . to establish tiling under dilation by @xmath25 , we first show that @xmath116 $ ] is disjoint from @xmath117\right)$ ] . that is , we must show that @xmath118 , which follows from the definition of @xmath65 together with the condition @xmath112 . now , note that @xmath119\setminus\frac1d\tau_{t\vec 1 } \;\mathcal p\left[w\left(\frac1{d^2}\right)\right]$ ] tiles @xmath0 under dilation by @xmath25 . the definition of @xmath65 together with the disjointness of the pieces of @xmath72 shows that @xmath72 is formed from this set by dilating the notch @xmath120\right)$ ] by @xmath25 . thus , @xmath72 also tiles @xmath0 under dilation by @xmath25 . see figure 9.7 of @xcite for illustrations of these tilings under both translation and dilation in the case @xmath39 . [ var]the wavelet sets produced by theorem [ main ] are a natural generalization of the 2 dimensional simple wavelet sets produced for scalar dilations in @xcite . we can also alter these examples to produce natural generalizations of the 2-dimensional example for dilation by 2 that appears in @xcite . note that if @xmath72 is a wavelet set for dilation by the scalar @xmath25 , then so is @xmath121 for any integer matrix @xmath122 of determinant @xmath123 . if we take @xmath122 to be the @xmath3 matrix that has 1 s on the diagonal , -1 s on the subdiagonal and 0 s elsewhere , then @xmath121 is a simple wavelet set for dilation by @xmath25 in dimension @xmath16 that is centered on the @xmath124 axis rather than the line @xmath125 . other variations are easily produced using other choices for the matrix @xmath122 . figure [ 3d ] shows one of the wavelet sets produced by theorem [ main ] for dilation by 2 in @xmath126 , as well as the variation described in remark [ var ] . [ 3d ] @xmath127{dil2wav3d.pdf } } \end{picture } & \setlength{\unitlength}{150bp } \begin{picture}(1.0,1)(-.2,0 ) \put(0,0){\includegraphics[width=\unitlength]{dil2wav3dx.pdf } } \end{picture } \end{array}$ ] the final theorem uses the same technique as theorem [ main ] to produce simple wavelet sets for dilation by an expansive matrix @xmath4 that has an integer power @xmath128 equal to a scalar multiple @xmath25 of the identity . in this case , the notched parallelotope @xmath129 $ ] fails to tile under dilation by @xmath13 because it is formed from the outer parallelotope @xmath130 $ ] by removing its dilate by @xmath131 instead of its dilate by the matrix @xmath132 . we would like to remedy the resulting overlap of @xmath129 $ ] and all of its negative dilates @xmath133 $ ] , by translating @xmath134 $ ] out by an appropriately chosen integer vector , to form a satellite that will be clear of @xmath129 $ ] and will perfectly dilate into the notch . this is possible only if @xmath135 $ ] is completely contained in @xmath136 $ ] . ( for an example where this containment does not hold , see figure 5(a ) . ) to overcome this difficulty , we put a restriction on @xmath4 , and sometimes replace @xmath131 with a higher power @xmath137 . [ matrix ] let @xmath4 be an @xmath3 integer matrix such that @xmath138)$ ] , where @xmath18 , @xmath139 , @xmath140 , and @xmath141 $ ] is the @xmath3 identity matrix . suppose further that all of the singular values of @xmath4 are greater than @xmath2 . let @xmath142 . let @xmath143 be the closest vector in @xmath12 to @xmath144 , and let @xmath145 then for @xmath146 sufficiently large , @xmath147\right)\subset\tau_t\mathcal n\left[w(\frac 1{d^q}),\frac 1{d^q}\right].$ ] for such @xmath148 @xmath149\right)\;\setminus\;\left(a^{*-1}\tau_{t}\ ; \mathcal p\left[w\left(\frac 1{d^q}\right)\right]\right)\right)\;\bigcup\;\left(\tau_{k}\left(a^{*-1}\tau_{t}\;\mathcal p\left[w\left(\frac 1{d^q}\right)\right]\right)\right)\ ] ] is a wavelet set for dilation by @xmath4 in @xmath102 by lemma [ notchedpar ] , @xmath150 $ ] , tiles under translation by @xmath12 , and thus its translate by @xmath65 does as well . thus , @xmath72 will also tile by translation as long as the set @xmath151 $ ] , which is translated out by the integer vector @xmath143 , is a subset of @xmath152 $ ] , and moves to a position disjoint from @xmath152 $ ] . as @xmath153 , the vector @xmath154 and @xmath155 , so that the set @xmath156 $ ] approaches the unit @xmath16-cube centered at the origin . thus , by taking @xmath146 sufficiently large , we can make the longest vector in @xmath156 $ ] to be arbitrarily close to @xmath2 times as long as the shortest vector . then , since the singular values of @xmath4 are greater than @xmath2 , we will have @xmath147\right)\subset\tau_t \mathcal p\left[w(\frac 1{d^q})\right]$ ] . the size of the notch also shrinks to 0 as @xmath153 , so that by taking @xmath146 larger if necessary , we will also have @xmath147\right)\subset\tau_t\mathcal n\left[w(\frac 1{d^q}),\frac 1{d^q}\right]$ ] . by the definition of @xmath143 , @xmath157 $ ] and @xmath158\right)$ ] are clearly disjoint for large @xmath146 . to establish tiling under dilation , note that for @xmath146 large enough that @xmath147\right)\subset\tau_t\mathcal n\left[w(\frac 1{d^q}),\frac 1{d^q}\right]$ ] , we have that @xmath159\setminus\left(a^{*-1}\tau_{t}\ ; \mathcal p[w(\frac 1{d^q})]\right)$ ] tiles @xmath0 under dilation by @xmath160 . to show that @xmath72 tiles under dilation as well , we must show that the outlier piece of @xmath72 exactly fits into the notch of @xmath161 $ ] under dilation by some integer power of @xmath13 . we have @xmath162\right)\right)&=&a^{*(-pq)}\left(\tau_{a^*k+t}\;\mathcal p\left[w\left(\frac 1{d^q}\right)\right]\right)\\ & = & \frac 1{d^q}\left(\tau_{a^*k-(d^q-1)t+d^qt}\;\mathcal p\left[w\left(\frac 1{d^q}\right)\right]\right)\\ & = & \frac 1{d^q}\left(\tau_{d^q\vec1+d^qt}\;\mathcal p\left[w\left(\frac 1{d^q}\right)\right]\right)\\ & = & \tau_{t+\vec 1}\left(\frac1{d^q}p\left[w\left(\frac 1{d^q}\right)\right]\right ) , \end{aligned}\ ] ] which is exactly the notch of @xmath163 $ ] . thus we have that @xmath72 also tiles @xmath0 under dilation by @xmath13 . theorem [ matrix ] also produces simple wavelet set for scalar dilations @xmath26 , which were not covered by theorem [ main ] . for scalar dilations by @xmath22 , theorem [ matrix ] produces a series of alternative wavelet sets to those of theorem [ main ] . as @xmath146 increases in this series , the parallelotope becomes closer to cubic , the notch becomes smaller , and the satellite becomes farther removed . [ mat1 ] let @xmath164 . then @xmath165 and @xmath166 , so theorem [ matrix ] applies . with @xmath167 , @xmath168 and @xmath169 , we have @xmath170\subset\tau_t \mathcal p\left[w(\frac 19)\right]$ ] . thus , we have a simple wavelet set @xmath171\right)\;\setminus\;\left(a^{*-1}\tau_{t}\;\mathcal p\left[w\left(\frac 19\right)\right]\right)\;\bigcup\;\left(\tau_{(1,1,-1)}\left(a^{*-1}\tau_{t}\;\mathcal p\left[w\left(\frac 19\right)\right]\right)\right),\ ] ] which is pictured in figure 4 . [ matrix1 ] ( 1,1)(.1,0 ) ( 0,0 ) of example [ mat1].,title="fig : " ] the hypothesis in theorem [ matrix ] that the singular values of @xmath4 be greater than @xmath2 is sufficient but not necessary , as the next example shows . [ mat ] let @xmath172 . then @xmath173 and @xmath174 , so theorem [ matrix ] does not apply . with @xmath175 , @xmath176 and @xmath177 , we do not have @xmath178\subset\tau_t \mathcal p\left[w(\frac 14)\right]$ ] . ( see figure 5(a ) . ) however , using @xmath179 , with @xmath180 , @xmath181 , and @xmath182 , the required containment does hold , yielding the wavelet set @xmath183\right)\;\setminus\;\left(b^{*-1}\tau_{t}\ ; \mathcal p\left[w\left(\frac 1{16}\right)\right]\right)\;\bigcup\;\left(\tau_{k}\left(b^{*-1}\tau_{t}\;\mathcal p\left[w\left(\frac 1{16}\right)\right]\right)\right).\ ] ] figure 5(b ) shows the central part of the wavelet set . ( the complete wavelet set also includes a translation of the missing inner parallelotope by @xmath184 . ) [ mat ] @xmath185{mat1.pdf } } \end{picture } & \setlength{\unitlength}{130bp } \begin{picture}(1.0,1)(0,0 ) \put(0,0){\includegraphics[width=\unitlength]{mat2.pdf } } \end{picture}\\ \mbox{(a ) } b^{*-1}\mathcal p\left[w\left(\frac14\right)\right]\not\subset\mathcal p\left[w\left(\frac14\right)\right]\qquad\qquad&\mbox{(b ) } \mathcal n\left[w\left(\frac1{16}\right),\frac1{16}\right]\setminus b^{*-1}\mathcal p\left[w\left(\frac1{16}\right)\right ] \end{array}$ ] the author wishes to thank the referee for several suggestions that clarified the presentation . k. d. merrill ( 2008 ) . simple wavelet sets for scalar dilations in @xmath190 . in , _ wavelets and frames : a celebration of the mathematical work of lawrence baggett _ ( p. jorgensen , k. merrill and j. packer eds . ) , birkhauser , boston , pp.177 - 192 .

wavelet sets that are finite unions of convex sets are constructed in @xmath0 , @xmath1 , for dilation by any expansive matrix that has a power equal to a scalar times the identity and also has all singular values greater than @xmath2 . in particular , we produce simple wavelet sets in every dimension for dilation by any real scalar greater than 1 .

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Proceed to summarize the following text: topological phase transitions in two dimensions , which are induced by the proliferation of topological defects , are naturally described via coulomb gas formulations . in the description of the kosterlitz - thouless transition of the xy model , the corresponding coulomb gas ( cg ) with integer charges is obtained as an effective theory for the topological defects of the model : the vortices @xcite . similarly , a collection of dislocations whose proliferation induce the melting transition in a two dimensional elastic lattice can be described by a coulomb gas with vector charges which belong to the reciprocal lattice@xcite . finally , most of two dimensional statistical models , such as the ising , potts and askin - teller models , can be transformed into coulomb gases @xcite ( see also the review @xcite ) . in all these cases , the transitions can be studied by renormalization in considering the screened interaction between two test coulomb charges . this interaction is logarithmic at large distance @xmath0 , and the renormalization procedure consists in neglecting the higher terms in an expansion in @xmath1 as being irrelevant @xcite . this renormalization is thus valid for a dilute gas of charges , and is usually implemented by an expansion of the coulomb gas partition function in powers of the _ charge fugacity _ @xmath2 . this fugacity @xmath3 is related to the _ core energy _ @xmath4 of the charges ( i.e the local energy to create a defect or their chemical potential ) . thus this coulomb gas renormalization procedure is perturbatively controlled in the limit of large @xmath4 , or equivalently at finite temperature in the limit of small fugacity . several topological transitions in two dimensions have been successfully described using this cg technique @xcite . soon afterwards , several authors attempted to extend these techniques to models with quenched disorder @xcite . the randomness in the original statistical models translates into random fields in the coulomb gas formulation . the averaged free energy ( instead of the partition function ) of this coulomb gas is then expanded , as in the pure case , in powers of the fugacity @xmath2 of the charges . usual scaling techniques then allow to study the topological transitions in these disordered systems . it does not seem to have been realized at that time that these approaches rely on the crucial assumption of a _ uniform fugacity _ for the charges in the sample , or equivalently of a core energy _ spatially uniform over the sample_. although this assumption is natural for pure models , it is at least questionable in the presence of a random potential @xcite . the randomness may favor the appearance of topological defects ( the charges of the cg ) in some sites where the core energy will be effectively lower than on other sites . at high temperature , the randomness of the core energy @xmath4 is irrelevant as it is averaged out by thermal phase fluctuations . in that case the large scale behaviour of the model can indeed be described by simply considering the averaged fugacity over the sample @xmath5 , as in @xcite . however , as we show here , when the temperature is lowered , the spatial inhomogeneities in the local core energy become more and more relevant . as a result , approaches based on a single uniform fugacity are doomed to fail . a correct detailed description of the scaling behaviour of the site - dependent fugacities ( or core energies ) becomes then necessary to determine the phase diagram and describe quantitatively the transitions in the random model . it is the purpose of this work to define a novel renormalization method which allows this description in a consistent and perturbatively controlled manner . we present here a detailed analysis , a shorter account of the method and results has appeared in @xcite . although we focus here on the random phase xy model , these techniques can be applied to a much wider class of systems which can be formulated as disordered coulomb gases ( e.g. see section [ part : sinegordon ] for a random sine - gordon formulation ) . in particular , extensions to the melting of two dimensional crystals in presence of disorder are studied in @xcite and applications to the statistics of localized wave functions of electrons in random magnetic fields as well as entropic transitions in liouville theory and other models are studied in @xcite . in this paper , we focus on the 2d xy model with randomness in the phase ( see eq . ( [ xy ] ) ) , originally studied in @xcite in the context of xy magnets with random dzyaloshinskii - moriya interactions . the topological defects ( vortices ) of this model are represented by integer charges @xmath6 at sites @xmath0 . two charges @xmath7 interact for a large separation via the usual coulomb interaction of strength @xmath8 , with the corresponding energy @xmath9 where @xmath10 is the typical size of the core region of the vortices , _ i.e _ the short distance cut - off ( as discussed in section [ part : model ] ) . these charges also couple to a random potential @xmath11 , which arises from the randomness of the xy model , via a term @xmath12 which depends on the sign of the charge . the crucial property of this random potential is that it has long range logarithmic correlations : @xmath13 the on - site variance being @xmath14 where @xmath15 is the system size . rubinstein , schraiman and nelson @xcite identified the most relevant vortices as the @xmath16 charges . they derived scaling equations for the stiffness @xmath17 , the disorder strength @xmath18 and a _ single fugacity _ for these charges @xmath19 . as in the pure case , the nature of the phase is determined by the behaviour of the fugacity : in a quasi - ordered phase , @xmath20 decreases with the scale while in a disordered phase it increases . in this last case , vortices appear at the scale @xmath21 at which their renormalized core energy @xmath22 is about zero , which corresponds to a fugacity of order @xmath23 : @xmath24 ( see _ e.g _ @xcite ) . interestingly the divergence of the correlation length at criticality found in @xcite , @xmath25 , is identical to the result of kosterlitz - thouless for the pure system . the phase diagram was found to be reentrant at low temperature , and is shown as a dashed line in fig . [ fig : phasediag - intro ] as a function of the renormalized value of @xmath26 and @xmath8 . recently , several authors @xcite have proposed a modified phase diagram for the same model , where the reentrance of the phase transition of fig . [ fig : phasediag - intro ] disappears . the main point , further developed in @xcite , is an energy argument for a single vortex at zero temperature in a finite size sample , in the spirit of the kosterlitz - thouless argument for the transition in the pure model @xcite . the energy to create a single defect has two main components ( neglecting the bare core energy ) : the elastic energy @xmath27 and the disorder energy . indeed the defect can take advantage of the spatial variations of @xmath11 and choose the minimal value of @xmath11 in the @xmath28 sites of a sample of size @xmath15 . to estimate this minimum @xmath29 of @xmath11 over @xmath30 sites , the above authors neglected the spatial correlations of @xmath11 . under that hypothesis , the problem of this single particle in the random potential @xmath11 becomes _ identical _ to the random energy model ( rem ) defined and solved in the seminal work of derrida @xcite . in particular , the averaged @xmath31 of the minimum behaves for large @xmath30 as @xmath32 , where @xmath33 is the onsite variance ( see above ) . adding the elastic contribution @xmath34 yields a creation energy @xmath35 . hence this simple single vortex argument points towards the existence of a topological phase transition at @xmath36 where vortices proliferate . the modified phase diagram is shown as a solid line on figure [ fig : phasediag - intro ] . although appealing , this single vortex argument is not sufficient by itself to prove the existence of a phase transition in the system of many interacting charges , and even less so to describe the critical behaviour . screening by mutual interactions usually plays an important role in coulomb systems , and screening of the disorder could also in principle modify the results . the single vortex argument can only become valid asymptotically ( at infinite scaling length ) if there indeed exists a phase with finite renormalized values for @xmath8 and @xmath26 and vanishing fugacity at zero temperature . the very existence of the xy phase in the present problem has been questionned in recent works @xcite and a further analysis is thus necessary . thus , we can not avoid the use of a renormalization approach to determine the true phase diagram . such a controlled rg description should correctly take into account the correlations of the random potential , neglected in the above argument . it should allow to precisely caracterize the universal features of the critical behaviour at the disorder driven transition . several attempts to construct a rg method were proposed @xcite prior to this work , but no agreement was found between the results of these different approaches . as we discuss below , this is largely because none of these approaches was internally consistent . in particular , they were all based on expansions in _ dipole _ fugacities while considering single charge fugacities not only appears more natural but is essential in constructing a consistent renormalization procedure . as a result these approaches missed the very important contribution of the fusion of environments ( see below ) . korshunov and nattermann started by approximating the free energy of a given disordered coulomb gas by the sum of the free energies of _ independent dipoles _ ] of charges @xcite . they found no renormalization of the disorder strength @xmath26 and a modified phase diagram in agreement with the figure [ fig : phasediag - intro ] . an interesting replica approach was developed by scheidl @xcite , with the use of a coulomb gas of @xmath37-component charges @xmath38 , where @xmath37 is the number of replica . scheidl noticed that , in the renormalization procedure , one must a priori take into account charges with a number @xmath39 of non zero components , contrarily to @xcite where only single component charges were considered . his rg equations are not compatible with the one of @xcite : since the disorder strength @xmath26 is renormalized , it leads to a different phase diagram when expressed in the bare constants . upon closer examination in section [ part : other - approaches ] and interpretation within our formalism , the assumption implicitly underlying scheidl s work @xcite appears to be the gaussian nature of the renormalized local disorder , while we are able to prove that the local disorder does not remain gaussian upon coarse - graining . although we are also able to show a posteriori that in the xy phase some results are compatible between the two approaches , the non gaussian nature of the local disorder becomes crucial to determine the critical behaviour at the transition . finally tang @xcite developed an appealing physical picture for the single vortex freezing phenomena in this model beyond the rem approximation mentionned above , and correctly foresaw the existence of a connection @xcite with the problem of the directed polymer on the cayley tree ( dpct ) . however , no precise coulomb gas renormalization procedure was developed following these ideas . finding a controlled rg procedure which allows to describe this physics of freezing in a collection of interacting coulomb charges in a random environments is thus a non trivial and challenging problem . as we will see ( and as can be expected from the above single charge argument ) , it requires a quantitative description of the scale dependence of the probability distribution of the fugacities associated with the rare sites were the charges are frozen . before embarking in the ( sometimes technical ) remainder of the paper , it is useful to depict the spirit of the method and summarize the main results . to capture the physics of freezing in a gas of coulomb charges we find it crucial to start by realizing that upon increasing the size of the charges ( cutoff @xmath10 ) , their core energy aquires a random component from the potential @xmath11 . increasing @xmath10 to @xmath40 , this potential @xmath11 with logarithmic correlation splits into the rescaled logarithmically correlated potential @xmath41 at scale @xmath42 , and a random part @xmath43 uncorrelated at scales larger than @xmath42 . this local potential is naturally incorporated in the ( renormalized ) core energy @xmath4 , which thus becomes random and site dependent : @xmath44 . thus , upon coarse graining , the fugacities of the @xmath45 charges of the coulomb gas become random ( and dependent on the sign of the charge ) : @xmath46 . let us stress that the definition of @xmath43 ( and thus of @xmath47 ) depends on details of the cutoff procedure . however the rg procedure developed here depends only on the logarithmic behaviour of the correlator of @xmath41 at large distances , which is cut - off independent . as a result , a remarkable universality will emerge at the end of the procedure , which happens to be related to universality of front solutions in non - linear equations ( see below ) . we are now faced with the description of the scale - dependence of a coulomb gas with random fugacities , and thus _ a priori _ of the full corresponding fugacity distribution . since we find that , at low temperature , the distribution of the local core energy does not remain gaussian under coarse graining we can not restrict a priori the renormalization study to follow a small number of variables ( such as the mean and the variance ) as illustrated in figure [ fig : distribution ] . instead , we must study the scale dependence of the full distribution @xmath48 and especially the precise form of its tails around @xmath49 ( i.e @xmath50 ) which control the low temperature physics ( at @xmath51 defects appear where the local core energy is negative ) . this requires new techniques in two dimensions . before developing a new rg procedure , we have to find a correct perturbative parameter to study the disorder driven transitions at low temperature . around the pure topological transition , this parameter corresponds to the charge fugacity @xmath2 ( for the most relevant charges @xmath16 ) . however , we expect that , upon lowering the temperature , a freezing of the defects occurs . in that case , most sites have a large core energy ( @xmath52 ) while only a few sites with @xmath49 are favorable to the defects . a natural choice for a perturbative parameter at low temperature in this highly non - homogeneous coulomb gas is thus the _ density of favorable sites _ depends on the sign of the charge : @xmath53 , and the perturbative parameter is the density @xmath54 of sites favorable either to @xmath55 charges or to @xmath56 charges . one must thus follow the rg flow of the full probability distribution @xmath57 . ] which we note @xmath58 . @xmath59 plays an analogous role here than @xmath20 in the pure case : the quasi - ordered phase is caracterized by a density @xmath59 decreasing with the scale @xmath60 , while a disordered phase , where disorder induced topological defects proliferate , corresponds to an increasing @xmath59 . this new small parameter allows us to study in a controlled perturbative expansion the physics of the vortices at low temperature and around the transitions . an essential contribution to the renormalization of @xmath57 , absent in previous approaches , originates from what we call the _ fusion of random environments_. the main idea is the followoing . each random fugacities @xmath61 is associated with a region of size @xmath10 around @xmath0 . upon increasing the cut - off , the size of these regions increases , and we thus have to merge two regions distant from less than the new cutoff @xmath42 ( see fig . [ fig : fusion - intro ] ) . the probability distribution of the fugacities in the new region can be deduced from the one in the two merged regions by a _ fusion rule _ ( see fig . [ fig : fusion - intro ] ) . the final renormalization of @xmath62 can be formulated into a single differential equation for the distribution function @xmath57 ( given in section [ part : replica ] ) . most interestingly we find that the universal features at the transition of the model ( @xmath63 ) do not depend on the precise definition of these regions . the fusion of environments corresponds to a non - linear term in the differential equation , which can thus be affected by a change of cut - off procedure . quite remarkably , the universality class at the new transition is determined by the properties of the front solutions of this differential equation ( i.e their velocity ) , that _ do not depend _ on the precise form of this non - linear term . hence universality in the usual rg sense finally appears from non trivial results of front propagation in non - linear equations . this differential rg equation for @xmath62 , together with the screening equations for @xmath8 and @xmath64 , can be obtained by a systematic and perturbatively controlled rg procedure . one formulation consists in using the replica trick to average the free energy over disorder , leading to a coulomb gas with @xmath37-component charges . in this formulation , the infinite set of fugacities , associated with the vector charges @xmath65 $ ] , encode the distribution function @xmath62 of the random fugacities @xmath47 . hence , since this last distribution can not be _ a priori _ described by a finite number of moments at low temperature , we have to consider the scaling behaviour of the fugacities of all the replica charges with components @xmath66 . the @xmath67 limit of this infinite set of rg equations is then taken by using an appropriate parametrization in terms of @xmath68 which yields the non linear rg equation for @xmath68 . the second , equivalent formulation , does not rely on replica . it is constructed using a new expansion of physical quantities in the `` number of independent regions '' , introduced in this paper . after some transformations , we reduce the non - linear rg equation that governs the scale dependence of the distribution @xmath62 to the celebrated kolmogorov - petrovskii - piscounov ( kpp ) equation ( also named the fisher equation ) . the known results on this equation allow us to derive the form of the tails of the distribution and thus to obtain the phase diagram . in particular , the velocity of the front solutions of the kpp equation directly determines whether the small parameter @xmath69 increases or decreases . hence using known results on the selection of this velocity , we obtain the phase diagram of the figure [ fig : phasediag - intro ] expressed in renormalized variables . the critical behaviour at the transition @xmath70 , follows from the finite size corrections to the velocity corrections of the kpp front and defines a new universality class . in particular , the correlation length @xmath71 diverges at the @xmath72 transition as @xmath73 in contrast with the kt behaviour found in @xcite . note that recent numerical simulations @xcite of this model seem to agree with the phase diagram of fig . [ fig : phasediag - intro ] and it would be interesting to also determine numerically the precise critical behaviour at low temperature . besides the caracterization of the new critical behaviour our work also enables to study the freezing of vortices , which occurs below a temperature @xmath74 ( see figure [ fig : phasediag - intro ] ) . this freezing corresponds to a transition for the single charge problem , whose study with our new rg technique is presented in @xcite . in particular , this renormalization approach draws a precise connection between the physics of freezing of the xy defects and the problem of branched processes ( or of random directed polymers on cayley trees ( dpct ) for a discrete version ) studied by derrida and spohn @xcite . this connection naturally emerges from our coulomb gas rg equations via the kpp equation which has also appeared in the exact solution of the dpct problem @xcite . it does not rely on any ad - hoc construction . the paper is organized as follows : in section [ part : model ] , the random xy model is defined and its cg formulation is carefully derived . in particular , the relation between the continuum limit and the decomposition of the random potential @xmath11 in to a local part ( random core energy ) and a long - range potential is discussed in section [ part : disdec ] . part [ part : replica ] describes the renormalization method of the replicated coulomb gas , and while a direct method ( without replica ) , which consists in expanding the free energy into the number of independent regions ( random environments ) is presented in section [ part : direct ] . the rg equations are analyzed in section [ part : analysis ] using results on the propagation of kpp - like fronts . the consequences for the determination of the phase diagram and the critical behaviour is detailed in section [ part : xyphase ] . a formulation in terms of a sine - gordon model is given in section [ part : sinegordon ] . the comparison with previous approaches is postponed to section [ part : other - approaches ] , and most of the technical details can be found in the appendices . in this section and in the following we study the xy model with random phases @xcite . we start with the model defined on the 2d square lattice by its partition function : @xmath75=\prod_{i}\int_{-\pi}^{\pi } d \theta_{i}~ e^{- \beta h[\theta , a ] } \quad \text { with } \quad h[\theta , a]=\sum_{\langle i , j \rangle } v(\theta_{i}-\theta_{j}-a_{ij})\ ] ] where the sum is over pairs of nearest neighbors ( i.e over bonds ) on the lattice and @xmath76 is the inverse temperature . the @xmath77 are random gauge fields , independent from bond to bond , each with gaussian distribution of variance @xmath78 . the periodic potential @xmath79 is defined for the xy model by @xmath80 where @xmath8 is the stiffness . in the limit @xmath81 this model corresponds to the 2d `` gauge glass model '' @xcite . for finite @xmath26 it was studied in ref . @xcite . the standard way to study this model is to decompose it into spin waves and vortex degrees of freedom @xmath82 = z_{sw } z_{cg}$ ] . this decomposition can be performed exactly ( see appendix [ part : cg ] ) for the corresponding villain model defined by the potential @xmath83 technically this is the model which we study here . it is reasonable however to expect that this villain model and the xy model should be in the same universality class ( as is the case without disorder ) . the rg analysis contained in the following sections is consistent with this assumption . the vortex part is described in terms of a coulomb gas with integer charges @xmath84 defined on the sites @xmath0 of the dual infinite ( square ) lattice : [ square ] @xmath85 where @xmath86 is the 2d coulomb potential with @xmath87 at large distance . $ ] is the lattice laplacian and we denote @xmath89 where @xmath90 is the lattice spacing . note that the energy associated with a dipole of unit charge of size @xmath91 is @xmath92 . as discussed in the appendix [ part : cg ] we can consider a neutral cg ( since @xmath93 ) with @xmath94 , and neutrality has already been used to arrive at ( [ square ] ) . in the vortex representation the random gauge fields @xmath95 translate into _ random dipoles _ ( along @xmath96 ) @xmath97 which couple to the vortex charges via the coulomb potential . as detailed in appendix [ part : cg ] this results in a gaussian bare disorder potential @xmath98 . an important feature of this problem is that the disorder potential seen by the charges ( the vortices ) has logarithmic _ long range correlations _ @xmath99 for @xmath100 , since @xmath101 . note that at a given point @xmath102 where @xmath15 is the system size and that up to now , these are exact transformations . before defining the continuum limit of this coulomb gas , we note that an alternative definition to this lattice coulomb gas consists in labelling the _ non zero charges _ in ( [ square ] ) by their positions @xmath103 and their corresponding charge @xmath104 . instead of the integer field @xmath84 of ( [ square ] ) defined at each site of the lattice , a configuration is represented by the set @xmath105 . we obviously have @xmath106 . with this representation , the partition function of ( [ square ] ) reads @xmath107 where @xmath108 and @xmath109 have been defined after eq . ( [ square ] ) and the sum over the charge configurations ( primed sum ) counts each distinct neutral configuration of non zero charges only once in the definition of the configuration sum of the cg , where @xmath110 is the total number of particles of charge @xmath111 . in order to implement a renormalization procedure one first needs to introduce a continuum version of this model . in the usual approach to 2d coulomb gas the continuum limit is obtained by replacing the lattice coulomb interaction by the approximation @xcite @xmath112 where @xmath113 for @xmath114 and @xmath115 otherwise , and @xmath116 with @xmath117 is the euler constant . this approximation is excellent @xcite on the lattice for @xmath118 . in the standard method the continuum cg model is then defined by considering a gas of integer hard core charges @xmath119 at point @xmath120 of diameter @xmath90 which interact with @xmath121 . using neutrality @xmath122 and ( [ approx ] ) , allows to rewrite @xcite the hamiltonian ( [ square - bis ] ) as a sum of a simple logarithmic interaction @xmath123 between the hard core charges and a _ fugacity term _ @xmath124 of bare value @xmath125 where @xmath126 can be interpreted as the bare core energy for the defects of the model ( [ square ] ) . in presence of disorder , special care has to be taken to define properly the continuum limit for the random potential , since its correlator is logarithmic . since on the lattice the correlator of the disorder @xmath127 is the same as the coulomb interaction , it is consistent to use the same @xmath128 for the disorder in the continuum model . this immediately leads to the fact that the disorder @xmath11 must be separated in two parts , using ( [ approx ] ) : a _ long range correlated gaussian _ part @xmath41 and a _ local _ part @xmath129 : [ decomposition ] @xmath130 with no cross correlation . using this decomposition we can now write the partition function of the continuum model : @xmath131 } \\ & & h[n , r ] = - j \sum_{i \neq j } n_i \ln \left(\frac{|{\bf r}_i-{\bf r}_j|}{a_{o}}\right ) n_j - \sum_{i } n_i v^{>}_{{\bf r}_i } - \sum_{i } \ln y[n_i,{\bf r}_i]\end{aligned}\ ] ] where the primed configuration sum counts only once each distinct neutral charge configuration . we have introduced the _ spatially dependent fugacity _ , of bare value , from ( [ approx],[decomposition ] ) : @xmath132 = - \gamma \beta j n^2 + \beta n v_{\bf r } \end{aligned}\ ] ] thus we find that disorder favors some regions resulting in a local fugacity for @xmath45 charges @xmath133 with a core energy @xmath134 which now varies from point to point . thus one anticipates that problems will arise in the conventional fugacity expansion if @xmath135 varies _ strongly _ from point to point . in addition , note that the local fugacities are different for @xmath136 charges and @xmath137 charges ( although there is still a statistical @xmath138 symmetry ) . we have thus defined a continuum model with a particular cutoff procedure . this is apparent , e.g. in the decomposition of the disorder that we have used . this decomposition is cutoff dependent and we have taken care to chose the same cutoff procedure for the disorder term and the interaction . we have chosen here a real space hard cutoff procedure , which is often used in cg studies . other cutoff procedures can be used . it is natural to expect , and we will partially verify that the large scale results will not depend on the particular procedure chosen . the alert reader will have already noticed that the disorders @xmath41 and @xmath43 as defined above in ( [ decomposition ] ) are not strictly speaking physical , since the fourier transform of their respective correlators is not positive . this is an artefact of this particular choice of a real space hard cutoff . it is not a serious problem , and is easily cured by choosing instead a cutoff in fourier space . the resulting decomposition of disorder is then completely legitimate . this is further explained in appendix [ part : cutoff ] . for simplicity , we will however proceed using the above cutoff choice , which has illustrative value , keeping in mind that the more legitimate choice detailed in appendix [ part : cutoff ] can be used instead , completely equivalently at all stages , for technical rigor . we now turn to the renormalization of the model . in this section we study , using replica , the renormalization group properties of the disordered coulomb gas defined by ( [ square ] ) . in the present case the replica method is particularly convenient in order to perform the combinatorics necessary to renormalize consistently the model . the strategy is first in ( [ part : replicacg ] ) to transform the model ( [ square ] ) into a vector coulomb gas with @xmath37-replica charges . the fugacities of these @xmath37-vector charges will then naturally encode the distribution of the spatially dependent fugacities defined above ( see ( [ locfug ] ) ) . the renormalization group ( rg ) equations for these fugacities are derived in ( [ part : replicarg ] ) for fixed @xmath37 . by a suitable parametrization , in ( [ part : replicalimit ] ) we then extract in the @xmath139 limit the rg equations which describe the scale dependence of the full distribution @xmath140 of the local fugacities of the topological defects in the original disordered model ( [ square ] ) , as well as the scale dependence of the stiffness @xmath8 and long wavelength disorder @xmath26 . we start again from the model ( [ square ] ) on a square lattice . as is well known , disordered averaged correlation functions and free energy can be obtained by studying the replicated partition function ( generating function ) @xmath141 in the limit @xmath139 . for integer @xmath37 , @xmath141 can then be written exactly as a cg with m - component _ vector _ charges @xmath142 , @xmath143 ( each @xmath142 is integer ) , living on the sites of the lattice ( note that @xmath10 denotes a replica index while @xmath90 denotes the cutoff ) . averaging over the bare disorder one obtains the partition sum of a fully coupled , translationally invariant , vector coulomb gas on a square lattice : @xmath144 with @xmath145 where @xmath146 and summation over repeated replica indices is assumed unless otherwise specified . the next step is to approximate the lattice replica model ( [ latticereplica ] ) by a continuum coulomb gas with @xmath37-component vector charges . in the following we consider a hard core cutoff in real space . the problem of the choice and consequences of the cutoff procedure is rather subtle here and will be discussed below . using the approximate propagator ( [ approx ] ) we obtain the continuum hamiltonian [ replicah ] @xmath147 & = & - \sum_{i \neq j } k_{ab } n^a_i g_{{\bf r}_{i}-{\bf r}_{j}}^{\text{app } } n^b_j\\ & = & - \sum_{i \neq j } k_{ab } n^a_i \ln\left(\frac{|{\bf r}_i-{\bf r}_j|}{a_{o } } \right ) n^b_j - \sum_{i } \ln y[{\bf n}_i]\end{aligned}\ ] ] where the charge @xmath148 is located in @xmath103 . in the second equality we have used the neutrality of the coulomb gas , i.e. @xmath149 for each @xmath143 , to introduce the local fugacity @xmath65 $ ] which is a function of the whole set of components of the _ vector _ charge @xmath150 . its bare value from ( [ approx ] ) is a simple quadratic function @xcite : @xmath151_{\text{bare } } = e^{- n_a \gamma k^{ab } n_b}\end{aligned}\ ] ] this quadratic form results from the _ gaussian nature of the bare local disorder _ and corresponds to ( [ locfug ] ) in the unreplicated version . if this form was preserved by the rg , as was implicitly _ assumed _ in @xcite , one would be able to study the model using only two coupling constants . however this is not the case . as shown below , the vector charge fugacity @xmath65 $ ] has a non trivial flow under rg and does not remain purely quadratic . the local disorder does not remain gaussian and we will have to follow its full probability distribution . we now study the scale dependent properties of the @xmath37-component vector coulomb gas using the _ expansion in the vector charge fugacity _ @xmath65 $ ] . although it is the natural way to study the renormalization of a vector coulomb gas , it may seem at this stage somewhat formal . this is not so however since , as will become clear below , it turns out to correspond exactly to the expansion in the number of rare favorable regions of local disorder , which is the physically relevant ( and novel ) expansion for this model . for the replicated partition function in the continuum model this vector fugacity expansion reads : @xmath152}\end{aligned}\ ] ] which contains fugacities via ( [ replicah ] ) and the primed sum over charge configurations counts each distinct neutral configuration only once . in presence of disorder , infrared divergences appear everywhere in the low temperature xy phase @xcite . to treat these divergences we now turn to the rg method . we perform the rg analysis of the present @xmath37-component vector coulomb gas on the partition function @xmath153 . it is a simple extension of the analysis for the scalar coulomb gas @xcite . details are presented in the appendix [ part : rgreplica ] and we only sketch the method here . for any fixed @xmath37 it is possible to leave the form of the expansion ( [ expansion ] ) unchanged under the increase of the hard core cut - off @xmath154 provided one defines scale dependent coupling constants @xmath155 and fugacities @xmath156 $ ] . this corresponds to the ( one loop ) renormalizability of the @xmath37-component vector model , which we checked here to order @xmath65 ^ 2 $ ] . the rg flow equations which determine these couplings are found as ( see appendix [ part : rgreplica ] ) : [ rgrep ] @xmath157 y[-{\bf n } ] \\ \label{rgrep2 } & & \partial_l y[{\bf n } ] = ( 2 - n^a k_{ab } n^b ) y[{\bf n } ] + c_2 \sum _ { \genfrac{}{}{0pt } { } { { \bf n}'+ { \bf n } '' = { \bf n } } { { \bf n}',{\bf n } '' \neq 0 } } y[{\bf n } ' ] y[{\bf n } '' ] \end{aligned}\ ] ] with @xmath158 , @xmath159 for our hard cut - off procedure , and the second equation ( [ rgrep2 ] ) is defined only for @xmath160 . the first equation ( [ rgrep1 ] ) comes from the annihilation of dipoles of opposite replica vector charges separated by @xmath161 . it gives the renormalization of the interaction ( screening by small dipoles ) and of the disorder . simple rescaling gives the first term of the second equation ( [ rgrep2 ] ) , i.e. the naive scaling dimension of @xmath65 $ ] . the second contribution in ( [ rgrep2 ] ) comes from the possibility of _ fusion of two replica vector charges _ upon coarse graining ( see fig . [ fusioncg2 ] ) . while such term can be neglected in the ( pure ) scalar coulomb gas ( as it yields less relevant operators ) it is usually crucial when studying most vector cg models , as e.g. in the analysis of two dimensional melting transition @xcite . indeed following too closely the analysis for the pure xy model has led previous studies @xcite to miss the possibility of fusion and thus such a contribution . as we see in the following it has important and non trivial consequences for the physics of the low t phase and the transition . for the present disordered coulomb gas , contrarily to the conventional analysis @xcite , one can not hope to capture the most relevant operators by restricting to single component charges ( e.g. @xmath162 ) . this was recently emphasized by scheidl @xcite . however , since this leads to considering multicomponent vector charges , it is thus crucial to treat properly this fusion term , which was not done previously ( e.g. in @xcite ) . moreover , discarding this term in ( [ rgrep2 ] ) leads to a set of rg equations which is _ not consistent _ to their lowest order @xmath163 ^ 2)$ ] . this term may _ a priori _ modify the scaling dimensions in a non trivial way in the @xmath139 limit , and it is thus crucial to study carefully its effect . before doing so in the next section , let us give for completeness the renormalization of the free energy density per replica , defined as @xmath164 . it reads : @xmath165 y[-{\bf n}']\end{aligned}\ ] ] from which the flow of the free energy density can be obtained as @xmath166 . we now have to find an analytical continuation to @xmath139 of the whole set of rg equations ( [ rgrep ] ) . this is a priori a formidable task because ( [ rgrep ] ) are in fact , for arbitrary @xmath37 , an infinite set of coupled equations . remarkably , in the process of performing this analytical continuation , an appealing physical interpretation in terms of probability distributions of local fugacities ( local disorder : see section [ part : disdec ] ) will emerge naturally and be our guide in the following . as a first step , we will consider only charges with components @xmath167 in each replica . how to incorporate higher charges ( e.g. @xmath168 ) is discussed in appendix [ highercharges ] , where it is shown that they are less relevant in the region of the phase diagram studied here . we first remark that the possible forms of the @xmath65 $ ] are severely constrained . replica permutation symmetry , which we will assume here and is preserved by the rg , together with @xmath167 implies that @xmath65 $ ] depends only on the number @xmath169 and @xmath170 of @xmath171 components of @xmath38 . a natural possible parametrisation of @xmath65 \equiv y[n_{+},n_{-}]$ ] consists in introducing a function of two arguments @xmath172 such that : @xmath173 = \langle z_{+}^{n_{+ } } z_{-}^{n_{- } } \rangle_\phi = \langle \prod_{a}\left[\delta_{n^{a},0}+z_{+}\delta_{n^{a},+1}+z_{-}\delta_{n^{a},-1 } \right ] \rangle_\phi \end{aligned}\ ] ] where we denote @xmath174 . our strategy is to establish an rg equation for @xmath175 ( in the limit @xmath139 ) whose solutions @xmath176 will parametrize solutions @xmath177 $ ] of ( [ rgrep1 ] , [ rgrep2 ] ) . let us now examine how ( [ rgrep2 ] ) can be transformed in an integro - differential equation for @xmath175 . the technical details are given in appendix [ part : replimit ] . the first terms in the r.h.s . of ( [ rgrep2 ] ) translate into a differential operator @xmath178 where : @xmath179 using new `` core energy '' variables @xmath180,@xmath181 such that @xmath182 , and the corresponding function @xmath183 such that @xmath184 , it can be interpreted as a _ diffusion process _ since the first term of ( [ rgrep2 ] ) now translates into @xmath185 with @xmath186 . to deal with the second term we first extend the rg equation ( [ rgrep2 ] ) so as to allow for zero charge @xmath187 , since it is easier to continue analytically unrestricted sums . after some combinatorics ( see appendix [ part : replimit ] ) , we find that using the representation ( [ eq : ydef ] ) , ( [ rgrep2 ] ) can be rewritten completely equivalently in terms of @xmath188 as : @xmath189 where @xmath190 . this equation describes the scale dependence ( @xmath60 ) of the function @xmath176 which parametrizes the whole set of scale dependent fugacities @xmath177 $ ] in the limit @xmath139 . up to now the function @xmath191 has been introduced as a generating function to parametrize the fugacities @xmath65 $ ] . it is a priori an arbitrary function and in particular @xmath192 is still undetermined . in this paragraph we will exchange @xmath172 for a _ physical _ function @xmath193 of norm unity , which will be interpreted in the following as the probability distribution for the local fugacities @xmath194 of @xmath45 charges . we start from the above equation ( [ loose ] ) for @xmath175 which can be simply interpreted as describing the sum of two processes . defining from the random fugacities @xmath195 the random core energy variables @xmath196 , the first process in ( [ loose ] ) corresponds to a brownian diffusion for @xmath181 ( i.e the local disorder potential as in ( [ locfug ] ) ) together with a convection for @xmath180 . the second process involves a fusion , with a rate @xmath197 upon increase of the cutoff , of two sets of random variables @xmath198 , @xmath199 into a single one @xmath200 according to the transformation law : @xmath201 as in a @xmath202 reaction . the term @xmath203 in ( [ loose ] ) corresponds to a loss of two charges , while the last term corresponds to a creation of the fused one . the term @xmath204 keeps track of the `` dead charges '' which disappear by setting them to @xmath115 ( since they decouple from the system ) . it is in a sense only a counting device , since by construction @xmath192 is unchanged upon fusion . we thus introduce @xmath205 restricted to @xmath206 , such that @xmath207 where @xmath208 is the total weight of non zero charges . integrating ( [ loose ] ) over @xmath209 we obtain that : @xmath210 thus in the presence of fusion it converges quickly towards @xmath211 . since @xmath212 converges to a constant , this suggests to introduce a normalized function @xmath213 . as shown below it is natural to interpret @xmath140 as a _ probability distribution_. from ( [ loose ] ) we find that it obeys the following rg equation : @xmath214 where @xmath215 denotes @xmath216 and the probability conserving diffusion operator @xmath217 has been defined in ( [ oo ] ) . the limit @xmath139 of the other rg equations ( [ rgrep1 ] ) which give the renormalization of the stiffness @xmath8 and disorder strength @xmath26 is performed in appendix [ part : replimit ] using @xmath188 . reexpressed in terms of @xmath218 they read : [ screening ] @xmath219 where we have chosen ( arbitrarily ) to keep @xmath220 fixed and renormalize @xmath8 ( only the combination @xmath221 flows ) . the above formulae ( [ rgeqp],[screening ] ) forms our complete set of rg equations . as will become clear in the following sections , @xmath140 represents the distribution of the fugacities @xmath194 of local environments and the last term in ( [ rgeqp ] ) corresponds to fusion of environments upon coarse graining . remarkably , once expressed in terms of @xmath218 the coefficients of the above rg equations exhibit some _ universality_. the factor of 2 in the last term in ( [ rgeqp ] ) arises from the fraction of environments @xmath222 which are fused when increasing the cutoff . note also that the coefficient @xmath223 which naturally appears in ( [ screening ] ) is not affected by a uniform rescaling of the fugacities . note that some features of these equations are cutoff dependent , as will be discussed in a following section . finally , the flow of the free energy density is found to be : @xmath224 these rg equations will be studied in section [ part : analysis ] . first we will present another renormalisation procedure , without replicas . although it is technically more difficult to implement , it allows for a more direct physical interpretation , which in turns sheds some light on the more systematic replica method presented above . in addition it may be more appealing to the replicaphobic section of the community . the above method which relies on an expansion in the vector fugacities @xmath65 $ ] can be justified provided there are _ few vector charges _ in the system i.e in the dilute limit . even though the @xmath65 $ ] may appear as formal fugacities , the above rg equations can be justified in an expansion of the exact renormalised potential @xmath225 seen by two test charges distant of @xmath91 in @xmath226 , as was emphasized by nienhuis @xcite . it is even claimed to be exact @xcite in that limit \sum_{{\bf n } ' \neq 0 } y[{\bf n } ' ] y[-{\bf n}']$ ] with @xmath227 in the second equation . we omitted this term as we found that it vanishes in the limit @xmath139 ] . as will become apparent in the following section the physical meaning of this diluted limit of _ vector charges _ exactly corresponds to the limit of a small density of regions favorable to the creation of frozen defects which is the physically relevant limit in the regimes studied in this paper . to understand how cutoff dependence comes in the method used here , it is instructive to study the limit of zero disorder . one can indeed check that one recover the usual results in the limit of the pure case @xmath229 . this however , requires some careful consideration of the cutoff procedure for the vector cg representation . as discussed in the appendix f , even in the pure case the distribution @xmath191 which parametrizes the vector fugacities solution of the cg rg equations can be non trivial . it does satisfy @xmath230 but the fugacity @xmath231 still has a non trivial distribution @xmath232 for a generic choice of cutoff . there is no paradox there and it is compatible with the standard korsterlitz - thouless rg equation of the pure case , as is explained in appendix f , universality being recovered at small fugacity . in this section we introduce a method to study the model ( [ square ] ) and more generally coulomb gas with disorder without using replicas . we start in section [ part : physics ] from the general motivation and rederive the rg equation for @xmath140 in ( [ rgeqp ] ) in a more physical way . we also identify the small parameter which allows to study perturbatively the present problem . in section [ part : exp ] we introduce a quantitative and systematic method to expand in this small parameter . the direct renormalization approach using this expansion is performed in section [ part : directrg ] . the connection with the replica method of section [ part : replica ] is presented in section [ part : connect ] . let us first explain the spirit of the direct method and illustrate how one is led to the rg equation ( [ rgeqp ] ) , derived more quantitatively in the next sections . we have seen in section [ part : disdec ] that the local disorder @xmath129 defines the site dependent fugacities . we concentrate on @xmath45 charges for which these fugacity variables read ( see ( [ locfug ] ) ) @xmath233 and are quenched random variables with only short range spatial correlations . one now studies the system under a change of cutoff @xmath234 ( coarse graining ) which includes an integration over the corresponding degrees of freedom . we find that the coarse grained model remains of the same form as the original one , with a renormalized stiffness @xmath235 , a renormalized gaussian long range disorder strength @xmath236 and a local disorder distribution @xmath68 . note that , although the bare local disorder @xmath129 is gaussian , it becomes _ non gaussian _ under coarse graining . this is a novel feature of the present approach , at variance with previous attempts at renormalizing the model @xcite . it complicates the analysis but is necessary to capture correctly the physics of the model which is driven by the rare events . the rg equations ( [ rgeqp ] , [ screening ] ) for the fugacity distribution @xmath68 of local environments ( higher charges are less important and considered later ) , for the stifness @xmath235 and for the correlated disorder strength @xmath237 can be understood from the following considerations . the correction to the fugacity distribution @xmath68 is the sum of two contributions : \(i ) _ rescaling _ : the first observation is that upon changing the cutoff , as can be seen from its correlator ( [ decomposition ] ) and is detailed below , the long range disorder @xmath238 produces an additional local disorder contribution which can be written as a renormalisation of the local charge fugacity : @xmath239 where @xmath240 is a gaussian random variable , uncorrelated from site to site and with @xmath241 . this contribution leads to an effective diffusion and drift in the random core energy variables @xmath242 as @xmath243 and thus produces the first terms in ( [ rgeqp ] ) . \(ii ) _ fusion of environments_. the second contribution comes from the fusion of environments . upon a change of cutoff , any two regions located around @xmath244 and @xmath245 with @xmath246 have to be considered as a single region in the system with the rescaled cutoff . as a consequence the two corresponding pairs of fugacities @xmath247 and @xmath248 must be combined and replaced by a single pair of effective fugacity variables @xmath249 associated with the new region at @xmath250 , as illustrated in fig . [ fig : fusion ] . @xmath251 can be determined by estimating the relative boltzman weight @xmath252 to have a configuration with charge 1 ( which lies either in @xmath244 or @xmath245 ) versus a neutral one ( either no charges or a dipole in @xmath244,@xmath245 ) , and similarly for @xmath253 . this gives the fusion rule : @xmath254 the corresponding correction to the distribution @xmath68 produces the last two terms of ( [ rgeqp ] ) . finally , the rg equation for @xmath235 and @xmath237 can be obtained from the screening by small dipoles of the effective interaction and disorder between two infinitesimal test charges as described in section [ part : directrg2 ] . several comments are in order concerning this rg procedure . first we note that in defining local fugacity variables ( [ locz ] ) we have added an explicit spatial dependence to the part @xmath255 of the fugacity which does not distinguish between a @xmath55 and a @xmath56 charge . this dependence is not explicitly present in the bare model formula ( [ locfug ] ) ( although it is present if an additional small disorder in the local stiffness @xmath256 is included ) but , as we can see from ( [ fusion1 ] ) it appears as soon as fusion takes place ( the fusion rule is not compatible with a uniform @xmath257 ) . second , there are some assumptions underlying the rg procedure : technically we treat the local regions as independent from point to point , we restrict @xmath41 to be strictly gaussian , together with the usual assumptions ( e.g. short distance expansions ) of the cg renormalization . these assumptions are consistent and amount to discard less relevant operators . these irrelevant operators can be identified within the method using replica of section [ part : replica ] where the above assumptions appear as standard in the rg of the @xmath37-component vector cg . for instance , the separation of the disorder into the two components @xmath258 and @xmath129 corresponds in the replica method to the natural splitting in ( [ replicah ] ) between the vector fugacity local operator @xmath65 $ ] ( originating from @xmath129 ) and the off - diagonal replica coulomb interaction @xmath259 ( from @xmath258 ) . this will be further apparent on the equivalent sine gordon representation of the problem presented in section [ part : sinegordon ] . accordingly , the definition of the independent local regions ( and thus of the local disorder environments and of the detailed form of the distribution @xmath260 ) is clearly cutoff dependent . so is the detailed form of the fusion rule ( [ fusion1 ] ) . however , universality of the physical results will be recovered in a remarkable way in section [ part : analysis ] , independently of the details of the cutoff procedure . as we will see , this is because , in the low temperature limit , the above definitions and fusion rules capture correctly ( to order @xmath261 ) the rare events which dominates the physics . they correctly evaluate the universal part of @xmath260 ( its tails in the low temperature region where they dominate the physics ) while they also correctly describe the weak disorder regime at higher temperatures . finally , we note that usual charge fusion between certain types of replica charges , represented on figure [ fusioncg2 ] corresponds , in the method without replica , to fusion of environments . to renormalize consistently the present model we need a method which can handle in a systematic way broad distributions of local fugacities . we have found such a method , which we now introduce . it is based on a _ systematic expansion of physical quantities in the number of independent points_. it generalises the conventional fugacity expansion in @xmath2 of the pure case , but is more powerful . in effect , it amounts to a partial resummation of the conventional expansion . it is versatile since , as we will see , it yields back the conventional expansion in the pure case or at high temperature , but is also able to handle the broad distributions which arise at low temperature . the idea there is that only few rare regions ( favorable to the charges ) in each environment will dominate the observables and thus it becomes possible again to expand in the density of such rare regions . the idea underlying this expansion in the number of `` independent points '' is that the site - dependent fugacities associated with distinct `` points '' can be considered as statistically independent . on a lattice , these `` independent points '' naturally correspond to the sites of the lattice , while in the continuum model , their definition is necessarily cut - off dependent . upon coarse graining , we will be able to use this systematic expansion to renormalize consistently the fugacity distribution associated with each point . for definiteness , we will show how to construct the expansion in number of points on the free energy @xmath262=-\beta^{-1 } \ln z[v]$ ] . it can be performed in a given environment , keeping the full functional dependence in the set of potentials @xmath263 . the construction can then be easily generalized to any physical quantity , such as arbitrary powers of the free energy @xmath264^p}$ ] ( which yield an expansion of all moments averaging term by term over disorder ) or all correlation functions of the field @xmath6 which can be obtained from products of free energies @xmath265\dots f^{a_{k}}[v_{k}]}$ ] by differentiation with respect to the potentials @xmath266 . in a second stage ( next section ) we will use this expansion to justify the rg equation for the disorder distribution . we recall that the coulomb gas model we consider is defined by its partition function @xmath267 = 1 + \sum_{p>0 } \sum'_{\{n_{1},\dots n_{p } \ } } \sum_{{\bf r}_{1}\neq \dots \neq { \bf r}_{p } } e^{\beta j \sum_{{\bf r}_{i}\neq { \bf r}_{j } } n_{i}g_{{\bf r}_{i}-{\bf r}_{j } } n_{j } + \beta \sum_{i } n_{i}v_{{\bf r}_{i}}}\ ] ] here and below , as in ( [ zcont ] ) , all formulae can be extended to the continuum model by replacing discrete sums over distincts sites @xmath268 by integrals with , _ e.g _ hard core conditions @xmath269 and introducing the uniform fugacity @xmath2 as was done in section [ part : model ] . note that in the above expression ( [ free ] ) we do not make use of the decomposition ( [ decomposition ] ) and @xmath270 denotes the original disorder . the expansion in the number of points of the free energy has the form : @xmath271&=&f^{(0 ) } + \sum_{{\bf r}_{1}\neq { \bf r}_{2 } } f^{(2)}_{{\bf r}_{1},{\bf r}_{2}}[v ] + \sum_{{\bf r}_{1}\neq { \bf r}_{2}\neq { \bf r}_{3 } } f^{(3)}_{{\bf r}_{1},{\bf r}_{2},{\bf r}_{3}}[v ] + \dots\end{aligned}\ ] ] where , in the disordered case , each term @xmath272 depends on the values taken by the disorder potential @xmath11 exactly and only at points @xmath273 . the explicit construction of these terms is given both for the disordered and pure case ( @xmath274 ) in appendix [ part : appexp ] . from a practical point of view , the explicit expression of @xmath272 is in all cases : @xmath275 } f_{{\bf r}_{i_{1}},\dots { \bf r}_{i_{l}}}[v]\end{aligned}\ ] ] where @xmath276 $ ] is the free energy of the coulomb gas defined only on the set of @xmath60 points @xmath277 ( instead of the full lattice ) and the summation is over all distinct subsets of the set @xmath278 . the definition ( [ free ] ) is unambiguous , though subtle . looking at the explicit expression ( [ fk ] ) as a sum over smaller subset of points one could imagine adding other terms to the @xmath272 depending on less than @xmath279 points . this is not possible in a global way , as the whole series must add up to the free energy , and the formula ( [ fk ] ) enforces it order by order . we refer to the appendix [ part : appexp ] for further details about the precise definition and construction . note that the term @xmath280 vanishes here because due to neutrality one can not define a cg on a single site and that the expansion in the number of points of powers @xmath281 $ ] involves a rearrangment of the expansion of @xmath262 $ ] . it is important to stress that each term @xmath282 of the above expansion corresponds to an infinite sum over terms of _ arbitrary high order in the conventional fugacity @xmath2 _ ( as defined in ( [ locfug ] ) ) . this can also be seen when setting disorder to zero , where we find that each @xmath283 ( @xmath284 ) and @xmath285 ( @xmath286 ) starts as @xmath287 plus an infinite number of additional higher order terms in @xmath2 ( see below ) . indeed the expansion ( [ free ] ) corresponds to a complete resummation of the conventional fugacity expansion usually performed in coulomb gas studies @xcite except that it is usually performed on the partition function @xmath288 while here we perform it on the free energy . the expansion ( [ free ] ) using the free energy is the appropriate expansion when the fugacities ( or the core energies ) are random and strongly site dependent with a broad distribution ( e.g. most of the sites have @xmath289 except for a few which have @xmath290 ) . indeed , in this case the only small parameter is the probability @xmath291 that a given point is favorable for a charge . thus the @xmath279-th term of the expansion ( [ free ] ) is of order @xmath292 , since it is associated with @xmath279 independent points . we can thus consider ( [ free ] ) as a perturbative expansion in the small parameter @xmath291 , valid in the @xmath293 phase , which replaces the conventional expansion in @xmath2 . to compute the @xmath272 we thus have to consider a coulomb gas defined by the partition function ( [ square - recall ] ) ( in the continuum limit ) , restricted to the system of points @xmath273 . we will in addition restrict ourselves to charges @xmath294 , as higher charges , examined later will be less relevant . let us closely examine the lowest order terms @xmath295 and @xmath296 . for @xmath295 we need only two points and the partition function ( [ square - recall ] ) reads simply @xmath297 the three terms of the partition function ( [ z-2 ] ) corresponds respectively to no charges or a dipole in @xmath298 , and the two possible positions for the dipole . this results in the boltzmann weigth for a dipole @xmath299 . the first terms of ( [ free ] ) thus reads @xmath300 restriction of the expansion ( [ free ] ) to its first nonvanishing term , _ i.e _ @xmath301 corresponds to the so - called _ independent dipole approximation_. this is the approximation on which the analysis of @xcite was based . this approximation , which neglects all interactions between dipoles , may seem at first sight to be a good enough approximation in the xy phase and at the transition . although this is reasonable in the pure xy model , this turns out to be incorrect here : by discarding the next term @xmath302 one throws away crucial statistical correlations . indeed , when renormalizing the distribution of local fugacities , we have to take into account correlations between dipoles induced by the disorder , which arise as follows . suppose that the site @xmath303 is favorable to the creation of a @xmath136 defect , _ i.e _ if @xmath304 and @xmath305 and @xmath306 are both favorable to creation of a @xmath137 defect ( while other neigbouring sites are unfavorable ) . within the independent dipole approximation the dominant configuration would be to put both a dipole in @xmath307 and one on @xmath308 to take advantage of these three favorable sites . these are the configurations ( 4 - 6 ) in fig . [ fig : expansion ] . however this configuration is forbidden because of the hard core constraint ( and we have restricted to @xmath45 charges , higher ones being less favorable energetically ) . thus we need to take into account the effective correlations between dipoles which arise because of rare favorable sites . this is done by considering the second term of the expansion @xmath302 . furthermore , as we will see below , consistent one loop renormalization requires to examine all terms in the expansion ( [ free ] ) and how they change under coarse graining , and thus to go well beyond the independent dipole approximation . let us derive the explicit formula for @xmath302 . the partition function with three sites reads : @xmath309= -\beta^{-1}\ln \left ( 1+w_{{\bf r}_{1}{\bf r}_{2 } } + w_{{\bf r}_{1}{\bf r}_{3}}+w_{{\bf r}_{2}{\bf r}_{3}}\right)\ ] ] however , since _ all terms of the expansion _ of @xmath310 in terms of @xmath311 depend exactly and only on @xmath312 , we have to substract to this free energy the terms depending on less than three sites , which can be identified as @xmath313 . the final expression for the second term of the expansion of the free energy is thus @xmath314\\ \nonumber & = & -\beta^{-1}\ln \left ( \frac{1+w_{{\bf r}_{1}{\bf r}_{2 } } + w_{{\bf r}_{1}{\bf r}_{3}}+w_{{\bf r}_{2}{\bf r}_{3 } } } { ( 1+w_{{\bf r}_{1}{\bf r}_{2}})(1+w_{{\bf r}_{1}{\bf r}_{3 } } ) ( 1+w_{{\bf r}_{2}{\bf r}_{3 } } ) } \right)\end{aligned}\ ] ] let us first notice that now the three sites component of the expansion ( [ free ] ) restricted to the first two terms is exactly @xmath315 . thus adding this second term has cured the problem coming from the configurations ( 4,5,6 ) of figure ( [ fig : expansion ] ) which , as discussed above are not allowed . it is interesting to note that the term @xmath310 is present ( and important ) in the expansion of @xmath316 , even though there are no actual configuration with three charges in a given environment ( from neutrality ) . indeed each term @xmath282 in the expansion ( [ free ] ) contains contributions from every even number ( less or equal to @xmath279 ) of charges . as can be seen from above ( fig [ fig : expansion ] ) it takes into account effective _ correlations _ between the distributions of fugacities of three differents sites @xmath312 induced by the hard core constraints . by contrast , there would be no such term involving three sites ( because of neutrality ) in the similar expansion carried out on @xmath288 ( conventional fugacity expansion ) . it was thus missed in previous studies , while in fact we show in the next section that it gives rise to a crucial contribution to the renormalization of the distribution of disordered fugacities : the _ fusion of environments_. the above defined expansion in number of independent points ( a type of cluster - virial expansion ) which we have illustrated on the first few terms , can be performed systematically to all orders . by coarse graining all terms of the expansion self - consistently , we will now obtain the renormalisation of the disorder distribution . we now propose a renormalisation scheme based on the above expansion ( [ free ] ) . from ( [ part : physics ] ) one captures the relevant physics by defining random local charge fugacities @xmath317 at each point and splitting the disorder distribution as @xmath318=\prod_{\bf r } p(z^{+}_{\bf r } , z^{-}_{\bf r } ) p[v^>_{\bf r}]$ ] as in ( [ decomposition ] ) . we know from the results of section [ part : replica ] that one can renormalize the model by defining only three scale dependent quantities : the full local disorder distribution @xmath319 , the stiffness @xmath235 and the second moment @xmath320 of the long range disorder @xmath321 . we now separately obtain the renormalization of each of these quantities , in a systematic perturbative expansion in the small parameter @xmath291 deduced from the expansion in the number of points . there are three types of contributions as follows . as can be seen from ( [ zcont ] , [ decomposition ] ) there is an explicit dependence in the cutoff @xmath322 in the expression of the interaction energy between charges and of the correlator of the long range part of the disorder @xmath321 . this dependence will appear in each term @xmath282 of the expansion ( [ free ] ) . upon changing the cutoff this will results in contributions of order @xmath323 which can be absorbed by appropriate redefinitions as follows . the interaction term changes as : @xmath324 where we have used the neutrality condition @xmath325 . the first term is the interaction term with the rescaled cutoff while the additional term produces an additive contribution to the @xmath45 charge fugacity @xmath326 . similarly from ( [ decomposition ] ) the correlator of @xmath238 can be rewritten as @xmath327 explicitly as the sum of a new long range disorder correlator with cutoff @xmath328 and a short range disorder correlator ( we have discarded terms of order @xmath329 ) . thus the original problem with cutoff @xmath322 can be rewritten as one with cutoff @xmath330 with ( i ) a new gaussian long range disorder with identical form of the correlator ( [ decomposition ] ) with @xmath322 replaced by @xmath330 ( ii ) a new local ( short range ) disorder @xmath331 with @xmath332 since it is clear from ( [ change ] ) that when @xmath333 the lr disorder produces an additive _ gaussian _ contribution @xmath334 to the sr disorder . adding the two contributions we find that the change of cutoff produces the total rescaling contribution ( [ rgrescaling ] ) . having introduced the expansion ( [ free ] ) one can now coarse grain its continuum version by increasing the cutoff @xmath335 and integrating over the corresponding degrees of freedom . upon this increase two points @xmath336 from the @xmath279 points integral ( @xmath282 ) will be fused if they satisfy @xmath337 . this will produce a contribution which is an integral at scale @xmath338 depending only on @xmath339 _ independent points _ and thus , by definition of the @xmath282 , produces a correction of order @xmath323 to the @xmath339 terms @xmath340 . all these corrections can be reabsorbed into a correction to the fugacity distribution ( together with an additive change to @xmath341 the free energy contribution of all degrees of freedom which have been eliminated in the change of cutoff ) . we now illustrate how this works on the case @xmath296 , and indicate in the appendix [ app : kpoint ] how it works for arbitrary @xmath279 . to lowest order in @xmath342 , this correction is independent of @xmath279 and can be easily performed by considering the three points integral , using ( [ free3 ] ) with _ @xmath343 : @xmath344\end{gathered}\ ] ] where we assume latexmath:[$|{\bf r}_{1}-{\bf r}_{3}|\geq \tilde{a}_{0 } , two points @xmath307 are fused to a single point @xmath346 and one obtains a correction to @xmath347 . using the decomposition ( [ decomposition ] ) of the disorder potential into a correlated component and a local part : @xmath348 , we can rewrite the boltzmann weigth for a dipole @xmath299 , defined in ( [ z-2 ] ) , as : @xmath349 where we have used the definition @xmath350 for the local charge fugacities . several simplifications now occur in evaluating ( [ correct1 ] ) . we first note that in ( [ correct1 ] ) the integral is of order @xmath342 . thus to find the leading correction of @xmath351 to order @xmath323 , we only have to consider the integrand expanded in order @xmath115 in @xmath342 . to this order , from the correlator ( [ decomposition ] ) of the disorder potential @xmath41 and using @xmath352 , we find that @xmath353 thus the weight of the fused pair can be simplified as @xmath354 . similarly , @xmath355 and @xmath356 simplify to order @xmath115 in @xmath342 using ( [ correct2 ] ) : @xmath357 using these simplifications , we can now rewrite the first term of ( [ correct1 ] ) : we obtain for the corresponding dipole weigth : @xmath358 it is now simple to verify that the change in the disorder averaged @xmath359 > _ v$ ] originating from the @xmath360 term is correctly accounted for by following rules for the random fugacity variables . the last two terms of ( [ correct1 ] ) produce , with the simplification ( [ correct1 - 3],[correct1 - 2 ] ) , the rules @xmath361 from ( [ correct1 - 1 ] ) , we find the rule corresponding to the first term of ( [ correct1 ] ) : @xmath362 this is true for any point @xmath306 _ different _ from @xmath303 and @xmath305 . one can thus rewrite these fusion corrections as a correction to the ( unnormalised ) distribution of local fugacities @xmath175 : @xmath363\end{aligned}\ ] ] the coefficient @xmath364 comes from the integration over the relative position @xmath365 . thus from ( [ fusion - direct ] ) and ( [ rgrescaling ] ) , we recover exactly the rg equation ( [ loose ] ) for the function @xmath188 . one can similarly check that the above rule correctly account for the corrections to @xmath339 term from the @xmath279 term upon change of cutoff . to be exhaustive , we must also consider a constant term in the expansion ( [ free ] ) of the free energy : @xmath366 which satisfies @xmath367 . this term corresponds to the free energy sum of all degrees of freedom which have been eliminated up to scale @xmath60 . indeed , @xmath140 ( or @xmath368 ) contains an average over all disorder configurations at smaller scales ( all environments which have been eliminated ) . a full and systematic proof can in principle be made by considering all averages of powers @xmath281 $ ] expanded in number of points and all fusions to order @xmath279 giving corrections to order @xmath339 . we will not attempt it here , as the systematic study to all orders is much easy to perform within the ( equivalent ) replica formalism . to derive the scaling behaviour of both the stiffness @xmath8 and the strength of the correlated disorder @xmath41 , we consider the screening of the interaction and the correlation of the disorder between two infinitesimal test charges @xmath369 and @xmath370 in the sample . to implement the study of this screening within our expansion in number of independentpoints , we first define @xmath371 $ ] as the free energy of the disordered coulomb gas defined in ( [ zcont ] ) with the two additional test charges @xmath372 fixed in @xmath307 . these test charges interact with the other integer charges of the coulomb gas , which screen the interaction between them , and the correlator of the disorder @xmath41 . from @xmath371 $ ] , one can define the screened interaction and disorder by [ def - screened ] @xmath373 } { \partial e_{1 } \partial e_{2 } } \right|_{e_{1}=e_{2}=0}\\ -\beta v^{r}({\bf r}_{1 } ) & = & \left . \frac { \partial f[v , e_{1},e_{2 } ] } { \partial e_{1 } } \right|_{e_{1}=e_{2}=0}\end{aligned}\ ] ] at large distance we expect from renormalisability @xmath374 , which imply the definitions for the renormalized coupling constant and disorder strength @xmath375 where the fourier transform of the 2d laplacian @xmath376 has been defined after equation ( [ square ] ) . these definitions can be transformed exactly using ( [ zcont ] ) into the relation @xmath377 up to now , these are only standard definitions of the renormalised stiffness @xmath378 and disorder strength @xmath379 . from this one deduces the rg equations for @xmath8 and @xmath26 since by coarse graining ( [ jr],[sr ] ) we obtain @xcite @xmath380 where the @xmath381 factor arises because we are dealing with correlations of the charge _ density_. the novel and most tricky part is how to evaluate the right hand side in a systematic way . our method is to expand these correlation functions in the number of independent points . this can be done in a systematic way using ( [ free ] ) and the definitions @xmath382 } { \partial v_{{\bf 0 } } \partial v_{{\bf r } } } \\ \label{def - correl2 } \langle n_{\bf 0}\rangle \langle n_{\bf r } \rangle & = & \frac { \partial f[v ] } { \partial v_{{\bf 0 } } } \frac { \partial f[v ] } { \partial v_{{\bf r}}}\end{aligned}\ ] ] note that while the expansion of the first term ( connected correlation ) in the number of points is readily obtained from the expansion ( [ free ] ) , the expansion of the second term ( disconnected correlation ) is more subtle since we need to square the derivatives of ( [ free ] ) and rearrange it again in an expansion in the number of points . fortunately , to the order we are working , we need only the two point term in the expansion . thus we can restrict ourselves to terms with two points on both expansions . the expansion of the first term ( [ def - correl1 ] ) follows straightforwardly from ( [ free ] ) : @xmath383 } { \partial v_{{\bf 0 } } \partial v_{{\bf r } } } = \sum_{{\bf r}_{1}\neq { \bf r}_{2 } } \frac { \partial^2 f^{(2)}_{{\bf r}_{1},{\bf r}_{2 } } } { \partial v_{{\bf 0 } } \partial v_{{\bf r } } } + \sum_{{\bf r}_{1}\neq { \bf r}_{2}\neq { \bf r}_{3 } } \frac { \partial^2 f^{(2)}_{{\bf r}_{1},{\bf r}_{2},{\bf r}_{3 } } } { \partial v_{{\bf 0 } } \partial v_{{\bf r}}}+ \dots \end{aligned}\ ] ] and thus from ( [ f-2 ] ) one finds : @xmath384 } { \partial v_{{\bf 0 } } \partial v_{{\bf r}}}|_{2 points } & & = \frac { \partial^2 f^{(2)}_{{\bf 0},{\bf r } } } { \partial v_{{\bf 0 } } \partial v_{{\bf r } } } \\ & & = \frac{-\beta^{-1}}{1+w_{{\bf 0},{\bf r } } } \frac { \partial^2 w_{{\bf 0},{\bf r } } } { \partial v_{{\bf 0 } } \partial v_{{\bf r } } } + \frac{\beta^{-1}}{(1+w_{{\bf 0},{\bf r}})^{2 } } \frac { \partial w_{{\bf 0},{\bf r } } } { \partial v_{{\bf 0 } } } \frac { \partial w_{{\bf 0},{\bf r } } } { \partial v_{{\bf r}}}\end{aligned}\ ] ] since we can keep only terms depending of the two points ( @xmath385 and @xmath386 ) in the last equality ( the other contributions vanish ) . similarly one finds that : @xmath387 } { \partial v_{{\bf 0 } } } \frac { \partial f[v ] } { \partial v_{{\bf r}}}|_{2 points } & & = \left(\sum_{{\bf r}_{1}\neq { \bf r}_{2 } } \frac { \partial f^{(2)}_{{\bf r}_{1},{\bf r}_{2 } } } { \partial v_{{\bf 0 } } } + \dots \right ) \left . \left(\sum_{{\bf r}_{3}\neq { \bf r}_{4 } } \frac { \partial f^{(2)}_{{\bf r}_{3},{\bf r}_{4 } } } { \partial v_{{\bf r } } } + \dots \right)\right|_{2 points } \\ \nonumber & & = \frac { \partial f^{(2)}_{{\bf 0},{\bf r } } } { \partial v_{{\bf 0 } } } \frac { \partial f^{(2)}_{{\bf 0},{\bf r } } } { \partial v_{{\bf r } } } = \frac{\beta^2}{(1+w_{{\bf 0},{\bf r}})^{2 } } \frac { \partial w_{{\bf 0},{\bf r } } } { \partial v_{{\bf 0 } } } \frac { \partial w_{{\bf 0},{\bf r } } } { \partial v_{{\bf r } } } \label{expcorr}\end{aligned}\ ] ] where the dipole weights @xmath388 have been defined in ( [ z-2 ] ) these expressions further simplify since it is sufficient to work to lowest order in @xmath342 , i.e to order @xmath115 in @xmath342 . we can use that @xmath389 since we need only to evaluate the correlation at @xmath390 . this yields that @xmath391 . one can also replace the derivatives with respect to the full disorder @xmath392 by derivatives with respect to the local disorder @xmath393 ( see ( [ decomposition ] ) ) and thus we find : @xmath394 by using this expansion into ( [ def - correl1],[def - correl2 ] ) we find the screening rg equations ( [ screening ] ) . thus the expansion in the number of points has allowed us to derive consistent rg equations _ without using replicas_. note the difference with the previous unsuccesful attempt in @xcite to derive rg equations for the stiffness and disorder without replicas . the authors of @xcite were working in the independent dipole approximation . thus they computed the correlation functions as in the first line of ( [ expcorr ] ) except that they kept the full expression @xmath395 . this led to intractable expressions involving up to _ four _ independent points . the beauty of the present method is that we know that we can keep , in a consistent way to this order , only the terms in this expression which involve two points . we have thus developed two rg procedures , one using replicas based on an expansion in the _ vector fugacity _ @xmath65 $ ] which may have seemed somewhat formal , the other one , direct without replicas , as an expansion in the number of favorable regions @xmath291 ( at low temperature , while at higher temperature it smoothly crosses over into the usual expansion in the uniform fugacity @xmath396 ) . in fact these two approaches are _ equivalent_. indeed , we have checked using some combinatorics detailed in appendix [ part : connection - replica ] , that the fugacity expansion in @xmath65 $ ] ( [ expansion ] , [ replicah ] ) is _ identical _ term by term , in the limit @xmath139 to the expansion in number of points of the free energy ( [ free ] ) . thus , working with one or the other is equivalent .. ] it also shows that the limit of diluted _ vector charges _ physically corresponds to the limit of rare favorable regions . [ part : kpp ] in this section we analyse the rg equations derived in the previous sections and we obtain the phase diagram and the critical behaviour of the xy model with random phases . our task now is to study the closed set of rg equations ( [ rgeqp],[screening ] ) for the scale dependent distribution of fugacities @xmath397 , the long range disorder strength @xmath237 and the stiffness @xmath235 . since we study below the zero temperature limit , we have chosen to keep @xmath220 fixed and renormalize @xmath8 ( only the combination @xmath221 flows ) . we started by studying the set ( [ rgeqp],[screening ] ) numerically . although we will not reproduce here the details we describe the main results . they confirmed the overall physical picture and led us to introduce an approximation , described below , which allows for an analytical solution . we found that at low @xmath220 and for @xmath26 smaller than some critical value @xmath398 an xy phase exists ( as in fig . [ fig : phasediag - intro ] ) , where both the stiffness @xmath8 and @xmath26 converge to finite non zero values at large @xmath60 : @xmath399 in general , at low @xmath220 we found that , starting at scale @xmath400 from ( [ locfug ] ) with a small initial averaged fugacity ( and small local disorder ) the distribution @xmath397 becomes broad and develops power law tails in the variables @xmath401 , @xmath402 which quickly extend up to fugacities @xmath403 . however , in the xy phase ( [ xydef ] ) we found that the typical @xmath404 goes to zero and that the concentration of rare favorable regions , after an initial increase , ends up decreasing towards zero at large @xmath60 . it is useful to define the _ single fugacity distribution _ : @xmath405 which does not satisfy a closed rg equation . then we find in the xy phase that @xmath406 decreases at large @xmath60 ( or equivalently @xmath407 where @xmath408 is some arbitrary threshold ) . thus the probability that either @xmath401 or @xmath402 is of order @xmath23 decreases . on the other hand , for @xmath409 we find that @xmath406 ends up increasing at large scale and the whole distribution @xmath410 ends up drifting towards increasing @xmath96 values . this corresponds to the disordered phase where @xmath235 vanishes at large scale ( @xmath411 ) . the most interesting flow occurs in the critical regime near the transition . there , if one interprets the fraction of favorable regions @xmath406 as the perturbative parameter , the structure of the flow near @xmath412 is reminiscent of the rg flow a la kosterliz - thouless with a separatrix , in the plane @xmath413 between the xy and the disordered phase . accordingly , exactly at the transition , i.e on the separatrix ( which , to be accurate , is a critical manifold in the space of distributions and couplings @xmath414 ) the distribution @xmath140 becomes very broad and develops a fixed shape , with @xmath406 slowly decreasing to zero ( which makes the critical regime perturbative in @xmath415 ) . a schematic representation of the distribution is given in fig . [ fig : distrib-2d ] . let us close this paragraph by noting that at higher @xmath220 , by contrast , we found that the distribution remains peaked and that the overall picture becomes more consistent with considering a uniform average fugacity ( as in @xcite ) . the xy transition then occurs when this uniform average fugacity ceases to flow to zero at large scale . to quantify these ( mostly numerical ) observations , we now turn to the single fugacity approximation . we now argue that , given the structure of the rg flow observed numerically , we can , with no loss of accuracy in all the regimes of interest ( i.e within and near the boundaries of the xy phase ) approximate the full rg equations ( [ rgeqp ] ) for @xmath68 by a simpler equation for the single fugacity distribution @xmath410 , and similarly for ( [ screening ] ) . let us first focus on the low @xmath220 regime , where the distribution @xmath397 is broad and the physics is dominated by rare favorable regions . for such a broad distribution the parameter which allows to organize perturbation theory is : @xmath416 where we distinguish symbolically the rare configurations @xmath417 from the typical ones where @xmath418 . the first observation is that @xmath401 and @xmath402 are even more rarely simultaneously @xmath419 . we note symbolically : @xmath420 indeed such configurations , absent from the start , are generated from the fusion term in ( [ rgeqp ] ) . to generate from the fusion rule @xmath421 a configuration where both @xmath422 one clearly needs at least that either @xmath423 or @xmath424 , which is of probability of order @xmath425 ( symbolically one can write @xmath426 ) . thus , and this can be further checked , the estimate ( [ p11 ] ) is valid ( the diffusion term does not change it ) . one can now inspect the corrections to the stiffness @xmath8 and to the long range disorder @xmath26 in ( [ screening ] ) which come mainly from configurations where either @xmath423 or @xmath424 . this leading contribution is thus of order @xmath427 . other fugacity configurations , e.g. with both @xmath428 contribute in ( [ screening ] ) but , from the above estimate ( [ p11 ] ) , their probability is @xmath429 and are thus subleading in an expansion in @xmath415 . to summarize , one can see schematically the rg equations ( [ screening ] ) as a correction of leading order @xmath430 to @xmath235 and @xmath237 and the rg equation ( [ rgeqp ] ) as a correction to @xmath415 of order @xmath415 ( diffusion term ) and @xmath430 ( fusion term ) . guided by these observations , we now approximate the full equation ( [ rgeqp ] ) by a rg equation for a function of a single variable which should correctly describe the behaviour of @xmath431 around @xmath432 . this approximation amounts to neglect the denominators in the fusion rule of ( [ rgeqp ] ) . indeed , in the spirit of our previous estimates , this denominator contributes only when @xmath423 ( or @xmath424 ) and then yields a contribution of order @xmath430 but which corrects only @xmath433 since this fusion produces both a @xmath434 . this corresponds to a correction of order @xmath435 to @xmath415 ( which we neglect , since the leading order of the correction to @xmath415 coming from the fusion term is @xmath425 ) . this can be seen e.g. since @xmath433 itself in turns contributes to @xmath415 to order @xmath425 ( through integration over one of the two variables ) , or by more detailed arguments , not reproduced here . neglecting the denominators gives the equation for the probability distribution @xmath436 this equation can now be explicitly integrated over one variable , say @xmath253 and yields a closed equation ) admits solutions of the form @xmath437 . ] for the single fugacity distribution @xmath438 defined in ( [ singledef ] ) : @xmath439p(z ) - 2 p(z ) \\ \nonumber & & + 2 \int_{z',z''>0}dz'dz '' \delta ( z - ( z ' + z '' ) ) p ( z ' ) p ( z'')\end{aligned}\ ] ] we can now obtain the corresponding equations for the corrections to @xmath8 and @xmath26 . in ( [ screening ] ) one keeps only the configurations corresponding to either @xmath440 or @xmath441 but discard the much less probable configurations where all four fugacities are @xmath419 . the expressions then nicely factor out in terms of @xmath410 and one obtains : [ screening2 ] @xmath442 where the additional factor of @xmath443 counts the equivalent integral with @xmath136 and @xmath137 exchanged . we now summarize the obtained closed set of rg equations in terms of the variable @xmath180 defined as : @xmath444 of distribution @xmath445 . physically the random variable @xmath180 can be interpreted as the random local core energy @xmath446 . one has : [ rg - eq-1d - u ] @xmath447 and thus the corrections to @xmath8 and @xmath26 involve only the convolution @xmath448 . we have thus justified , in the low temperature regime , that these approximate rg equations should describe the tails of the fugacity distribution exactly to the order @xmath430 in the expansion in @xmath415 to which we are working . indeed , since we have studied the rg in the previous sections only to order @xmath65 ^ 2 $ ] ( in the replica formulation , corresponding to @xmath430 in the rare events formulation ) it probably does not make sense at this stage to try to be more accurate .. are correct , only an exhaustive analysis of the two dimensional front solutions of ( [ rgeqp ] ) , clearly a complex task which goes beyond this paper , could confirm the exactness of the approximation and the validity of the arguments based on organizing the perturbation theory using a single small parameter @xmath415 . ] finally , let us note that although we have focused until now on the low @xmath220 regime it is rather obvious that the approximation of discarding the denominators will be even more valid at higher temperature since in the conventional fugacity expansion ( in a uniform averaged fugacity ) these terms corresponds to @xmath449 terms while we work to @xmath450 . so these new rg equations interpolate all limits correctly . one can now easily see on the form ( [ rg - eq-1d - u ] ) that the rg equations admit a well defined @xmath451 limit . in the equation ( [ toy ] ) , the fusion rule @xmath452 now becomes a max rule @xmath453 and ( [ toy ] ) transforms as @xmath454 similarly , the function of @xmath455 in ( [ toy2],[toy3 ] ) become respectively a delta and a theta function when @xmath451 , so that in that limit one has : [ rg1d-2 ] @xmath456 note that the last integral exactly evaluates the probability to find two local regions with a total negative core energy , energetically favorable for a dipole , in agreement with the physical picture valid at low @xmath220 . we thus obtain a close set of equations ( [ rg1d-1],[rg1d-2 ] ) which describes the scaling behaviour of the system at zero temperature . the equation for the distribution ( [ rg1d-1 ] ) can be conveniently simplified using the parametrization @xmath457 with @xmath458 . the function @xmath459 then satisfies @xmath460 in the case where @xmath26 and @xmath8 are @xmath60-independent , this equation is known as the kolmogorov - fisher equation and we will recall some results on this equation in the next section . before turning to its study , let us notice that the screening equations ( [ rg1d-2 ] ) can be rewritten using this new parametrization simply substituting @xmath461 . although ( [ rg - eq-1d - u ] ) at finite temperature @xmath463 could be in principle studied directly , it is much more convenient to introduce the generalization of the @xmath51 parametrization ( [ g - t0 ] ) as @xmath464 with @xmath465 . in the limit @xmath466 , this function reduces to the previous one ( [ g - t0 ] ) . using this new function and variable the integral equation ( [ toy ] ) can again be transformed exactly into a simpler differential equation which , interestingly , remains exactly the same as in the @xmath51 case : @xmath467 only the initial condition explicitly depends on temperature ( see below ) . thus solving the kpp equation ( [ kolmogorov ] ) allows us in principle to obtain the scale dependent fugacity distribution @xmath468 at any @xmath220 . however the relation between this function @xmath459 and the distribution @xmath469 becomes much more involved at @xmath463 . as a result , and contrarily to the @xmath51 case , the screening equations do not admit a simple expression in term of @xmath459 . for any fixed @xmath60 one can reconstruct the integer moments @xmath470 of the distribution @xmath469 by simply expanding the generating function in powers of @xmath471 since : @xmath472 taking back this expansion in the kpp equation ( [ kolmogorov ] ) and identifying the coefficient of the exponentials @xmath473 , we find the exact rg equations for the moments @xmath474 of the distribution @xmath469 : @xmath475p_{l } ( z')p_{l } ( z'')\end{aligned}\ ] ] starting from a reasonable initial distribution with finite moments , the moments remain finite for finite @xmath60 , but , as we now discuss , increase quickly as @xmath476 . let us examine the scaling dimension of the moments , neglecting for now the bilinear ( fusion ) term . we find that for fixed @xmath220 and @xmath8 , each successive moment @xmath474 diverges with the scale @xmath60 when the long range disorder @xmath26 becomes larger than a critical disorder @xmath477 ( see fig . [ fig : y - moments ] ) ( these values will be slightly renormalized by the screening equations but the conclusion will be similar ) . putting back the fusion term simply implies that the moments with @xmath478 also diverge when the first one does , as indicated in the fig . [ fig : y - moments ] . the divergence of the first moment @xmath479 which can be identified with the uniform fugacity @xmath2 of rubinstein et al . yields the ( incorrect ) reentrant phase diagram of @xcite where the xy phase is destroyed above the line @xmath480 . indeed it is already clear that the uniform fugacity approximation can not work since we now see that higher moments diverge for even smaller disorder strengths . thus even within the xy phase this result for the moments show that the distribution @xmath469 rapidly becomes broader and broader . atypical sites where @xmath96 is large appear and dominate the behaviour of the higher moments . thus it becomes meaningless to study the scale dependence of the integer moments , but rather we must now consider the whole probability distribution @xmath469 and in particular understand its tail . this can be achieved with the help of the known solutions of the kpp equation which we briefly review in the next section , before coming back to describing the scaling behaviour of @xmath468 . it is interesting to note at this stage that the above kpp equation also arises in the problem of the directed polymer with quenched disorder on the cayley tree ( dpct ) @xcite . there the variable @xmath60 corresponds to the number of generations and @xmath481 to the distribution of free energy @xmath482 . thus we have demonstrated in an explicit and non trivial way , i.e at the level of the rg equations , that there are close connections between the two problems , which both exhibit a similar freezing transition . there are also notable differences between the two problems . for instance , while the diffusion coefficient @xmath483 is constant in the dpct studied in @xcite , in the disordered cg , @xmath484 depends on the scale and , in a self consistent manner , on the solution of the kpp equation itself , via the equations ( [ screening2 ] ) which describes the physics of screening , absent in the dpct . as a result , additional phase transitions exist here as will be detailed in section [ part : xyphase ] . we recall in this section some known facts on the kolmogorov - petrovskii - piscounov ( kpp ) equation ( also known as the kolmogorov - fisher equation ) which we will need in the following sections . the equation reads , in a general form : @xmath485 where the diffusion coefficient is constant , the function @xmath486 satisfies @xmath487 , @xmath488 positive between @xmath115 and @xmath23 and @xmath489 between @xmath115 and @xmath23 . the usual case corresponds to @xmath490 . this equation has been applied to a wide range of problems , from chemistry to hydrodynamic instabilities or to the propagation of the meissner phase into the normal phase in a superconductor @xcite . it is the prototype of equations describing the diffusive invasion of an unstable state by a stable one . this can be seen by writing it as a landau equation @xmath491 whose free energy @xmath492 takes by construction its local maximum in @xmath493 ( the unstable state ) and its minimum in @xmath494 ( stable state ) . one usually chooses an initial condition @xmath495 monotonously decreasing from @xmath496 to @xmath497 . for a large class of initial conditions , the solutions of the kpp equations are known to converge uniformly towards traveling waves solutions of the form : @xmath498 with @xmath499 and @xmath500 . however the question of the determination , given an initial condition , of the asymptotic traveling wave @xmath501 and its velocity @xmath502 has been largely debated for kpp equations or for similar more complex non linear equations , and is still of current interest . it is known as the front _ velocity selection problem_. it can be illustrated as follows . a family of possible front solutions exists , parametrized by the velocity @xmath503 and noted @xmath504 , as can be seen by substituting ( [ front ] ) in ( [ kpp - general ] ) . constraints exist for the velocity . indeed , one can linearize the kpp equation in the region ahead of the front for large positive @xmath505 where @xmath501 is very small . as discussed below it is in fact this region which determines the velocity and is universal . in this region one has : @xmath506 and thus @xmath501 is a superposition of two exponentials @xmath507 with ( see figure [ fig : vitesse ] ) @xmath508 with @xmath509 . the large @xmath510 behaviour is dominated by the smaller @xmath511 and thus correspond to the left branch of the curve @xmath512 in figure [ fig : vitesse ] when @xmath513 . in the marginal case where @xmath514 and @xmath515 , the two eigenvalues are degenerate and the front thus has the asymptotic behaviour for @xmath516 : @xmath517 let us now give the known results for the selection of the asymptotic front among the family of possible @xmath518 . \(i ) _ velocity selection . _ although for more complex equations one relies on stability analysis and a marginal stability criterion @xcite , in the case of the kpp equation a rigorous result is available . a theorem due to bramson @xcite shows that the asymptotic traveling wave is determined by the behaviour at @xmath519 of the initial condition @xmath495 in the following manner . if @xmath495 decays fast enough , as @xmath520 for @xmath516 with @xmath521 ( or faster ) theorem b of @xcite states that @xmath522 uniformly converges towards the traveling wave solution @xmath523 of velocity @xmath524 . if @xmath495 decays slower , as @xmath520 with @xmath525 then @xmath522 uniformly converges towards @xmath526 of velocity @xmath512 continuously depending on @xmath511 and given by ( [ velocity ] ) . the asymptotic velocity is thus given by @xmath527 for steep enough initial condition @xmath528 and @xmath529 otherwise . moreover , the leading corrections to the velocity are also given by the theorem of bramson and are _ independent of the function _ @xmath530 in ( [ kpp - general ] ) . the corresponding position of the front @xmath531 is given by @xcite [ velocity - corrections ] @xmath532 note that cases ( [ 3demi ] ) and ( [ 1demi ] ) differ only by the velocity corrections but not by the asymptotic front shape which is @xmath533 in both case . let us emphasize again the remarkable universality which arises in this problem . clearly the detailed shape of the asymptotic front depends on the detailed form of the non linear term @xmath530 in the kpp equation . however , the selection itself , the selected velocity , its corrections , and the tail of the selected front function @xmath501 are all independent of @xmath530 . it is also natural physically that it is the region in which the front penetrates ( region ahead of the front @xmath516 ) which determines the selection . this universality has been explored further @xcite and it has even been shown that the next leading corrections to @xmath534 are also universal . finally the marginal case which interpolates between ( [ 3demi ] ) and ( [ 1demi ] ) has been also explored with @xmath535 one has : @xmath536 . the asymptotic front shape is the same for all these cases , but not the detailed finite @xmath60 shape as we now discuss . \(ii ) _ shape of the front for finite @xmath60 _ the problem has also been studied for finite but large @xmath60 @xcite . we start with the case where @xmath495 decays fast for @xmath516 as @xmath520 with @xmath537 , most relevant for the following sections . then one must distinguish two regions in the traveling wave solution as illustrated in figure [ fig : front - shape ] . the central region is the `` bulk '' or `` interior front '' for @xmath538 fixed and finite . there , the shape of the front corresponds to its asymptotic form @xmath539 and the center moves as @xmath540 with @xmath541 . far ahead of this `` interior front '' , @xmath542 , @xmath522 still decays faster that in the asymptotic solution @xmath543 . thus there exist an intermediate region ( which we call the `` intermediate front region '' ) which matches between the interior region and the region at infinity . in this intermediate region which corresponds to @xmath544 one must take into account diffusion , and one can solve the linearized kpp equation without assuming a front solution , but assuming a scaling form . it is found @xcite that in this scaling `` far front '' region @xmath544 , the solution @xmath522 behaves as with @xmath535 with a scaling function noted @xmath545 in @xcite ( for @xmath546 one gets the above result , but the form for general @xmath10 is more complicated ) . ] : @xmath547 note that later on we will need to distinguish a region even further ahead of the front for @xmath548 . let us close this section by returning to the question of the dependence of the rg equation of the random xy model in the cutoff procedure . we can show that the cutoff procedure will determine the function @xmath530 . let us consider for instance again the limit of zero disorder , discussed in appendix f. since in that case @xmath549 one should have : @xmath550 with in that case @xmath551 and @xmath218 characterizes the `` disorder '' associated with the choice of cutoff procedure . an interesting choice corresponds to @xmath552 where @xmath553 satisfies the kosterlitz thouless rg equation @xmath554 . it is easy to see that in that case @xmath522 satisfies ( [ zerodis ] ) with @xmath555 . this is further discussed in @xcite we now use the solutions of the kpp equation discussed in the previous section to determine the phase diagram of the xy model with random phase shift . let us first look for the xy phase where we expect that @xmath235 and @xmath556 reach limits , respectively @xmath557 and @xmath558 at large @xmath60 . thus in the xy phase at large @xmath60 the kpp equation with constant @xmath559 can be used . precise behaviour near the phase transition away from the xy phase , as well as intermediate scale dependence inside the xy phase _ a priori _ requires taking into account the @xmath60 dependence of @xmath484 which will be done in the following sections . note that @xmath60 dependence of @xmath235 itself results only in a shift that can always be trivially taken into account . first we note that the phase diagram will be entirely determined by the velocity selected in the kpp equation . indeed , we know from the previous section that @xmath522 converges to a traveling front solution @xmath560 of velocity @xmath503 . the parametrization ( [ g - tfinie ] ) then implies that the distribution @xmath410 of vortex fugacity for the random xy model , itself converges to a traveling front solution , more conveniently expressed in the random core energy variable @xmath561 as : @xmath562 where @xmath531 is given in ( [ velocity - corrections ] ) and @xmath563 . the center of the front of @xmath564 , located in @xmath565 corresponds to the maximum of the distribution @xmath566 ( as can be easily seen on ( [ g - t0 ] ) ) and to the typical values of the random variable @xmath180 . the front shape of @xmath564 is simply related to the kpp front solution @xmath518 through @xmath567 . the asymptotic velocity of the front of @xmath564 is thus : @xmath568 the total velocity in the @xmath180 variable is thus the kpp velocity minus the stiffness . the former comes from the spread of the distribution due to disorder , while the latter from the effect of interactions . in previous sections we have explained that the xy phase corresponds to a decrease of the density of favorable regions @xmath569 ( i.e @xmath570 ) and to the absence of topological defects at large scale . in particular , for @xmath8 and @xmath483 to reach finite asymptotic values @xmath557 and @xmath571 , it is necessary that @xmath572 decreases fast enough . @xmath415 can be estimated crudely as @xmath573 . thus , as shown in fig . [ fig : relative - velocity ] , when the total velocity @xmath574 is negative ( xy phase ) , the front moves to the left ( large @xmath96 or large @xmath180 ) and thus the probability of events @xmath575 decreases while if @xmath574 is positive , @xmath576 increases asymptotically ( disordered phase ) . the xy phase thus corresponds to the region where the total velocity is negative and thus to : @xmath577 in which case the whole probability distribution of the core energy @xmath578 , its typical and average values drift to @xmath579 at larger scale . the transition line between the disordered and the xy phase can be located , in the plane @xmath557 , @xmath580 , by finding the line where this relative velocity @xmath581 vanishes . the phase diagram can now be obtained by determining the velocity @xmath503 as a function of @xmath220 and @xmath26 . the crucial observation is that by construction ( as is the case in @xcite ) the initial condition @xmath495 decays for large @xmath510 as : @xmath582 thus , since @xmath583 is finite , it decays exponentially and one can apply bramson s results ( [ velocity - corrections ] ) detailed in the previous sections with the identification : @xmath584 thus in effect it is the temperature which selects the velocity @xmath503 . we find that , as depicted in fig [ fig : phase - diag ] : \(i ) for @xmath585 , i.e at high enough temperature @xmath586 , the velocity continuously depends on temperature : @xmath587 and thus , in that regime , the xy phase exists for : @xmath588 which is exactly the condition which would be obtained from the averaged fugacity ( at least when expressed in the renormalized parameters ) and leads to the transition line of rubinstein et al . as we have discussed earlier ( see eq.([rg - moments ] ) ) . \(ii ) the velocity of the front _ freezes _ at @xmath589 and for @xmath590 , i.e at low temperature @xmath591 , where it becomes temperature independent : @xmath592 and the total velocity of the front @xmath481 is now @xmath593 thus we obtain that below this freezing temperature at : @xmath594 the transition between the xy phase and the disordered phases occurs at : @xmath595 . for @xmath596 the system is unstable to the proliferation of topological defects induced by disorder . to describe the xy phase it is important to understand the various regions of the scale dependent fugacity distribution @xmath410 . indeed the fugacity distribution also gives the distribution of the charge correlation functions in the coulomb gas from the relations ( [ exp : correl1 ] ) . in the xy phase one can describe the large scale behaviour of the fugacity distribution by neglecting the slow scale dependence of @xmath26 and @xmath8 ( adiabatic approximation ) . the behaviour of @xmath481 can be obtained by studying @xmath597 since one has @xmath598 with @xmath599 . at @xmath51 this simplifies into @xmath600 . it is also useful to note that the laplace transform of @xmath410 is @xmath601 . at low @xmath220 in the xy phase @xmath410 becomes very broad and one must distinguish for large but finite @xmath60 several different regions represented in fig . [ fig : front - shape ] . using the results of the preceding sections on the kpp equation we now describe these regions in details : _ interior front _ : the first region corresponds to the bulk of the probability distribution centered around the typical value of the fugacity @xmath602 . from the results for the front velocity ( [ xlowt ] , [ xhight ] ) one thus obtains that in the xy phase at high temperature @xmath603 the _ typical _ renormalized fugacity decays with the scale as : @xmath604 which in that case is also the scaling of the average fugacity @xmath605 . on the other hand , below the freezing temperature @xmath606 it decays as : @xmath607 the bulk of the distribution is thus well described by the asymptotic front of the kpp equation as @xmath608 and @xmath609 . its precise shape is of course non universal as it depends on the details of the definition of the fugacity at the level of the cutoff ( e.g. the function @xmath530 in ( [ kpp - general ] ) ) . however , since the physically interesting region @xmath610 corresponds to the region ahead of the kpp front ( even though the _ total _ velocity @xmath611 is negative , see fig . [ fig : front - shape ] ) universal results about the kpp equation can be used . in particular the _ near tail _ ( i.e the scaling region @xmath612 fixed but large ) is universal and can be obtained since @xmath613 . through laplace inversion one gets that the fugacity distribution in that near tail region behaves as a power law : @xmath614 which is valid for @xmath615 and in the regime @xmath616 fixed ( as @xmath60 grows ) but large . the freezing which occurs at @xmath617 thus concerns _ typical regions _ which , deep in the xy phase have a very small fugacity @xmath618 . it corresponds to the temperature at which the first moment of the distribution on the r.h.s . of ( [ typical ] ) becomes infinite . thus for @xmath606 the true average fugacity @xmath619 and becomes dominated by rare local environments corresponding to values of @xmath96 outside of the bulk of @xmath410 ( where ( [ typical ] ) is invalid ) . _ intermediate front _ : the second region is the `` intermediate front region '' ( see fig . [ fig : front - shape ] ) and corresponds to @xmath620 . there we know from ( [ front - intermediate ] ) that @xmath621 with @xmath622 . this region will be crucial to describe the critical behaviour in the following section . _ far tail region _ : finally the most important region to describe the xy phase is the `` far tail region '' far ahead of the front with @xmath623 ( see fig . [ fig : front - shape ] ) . indeed , to obtain the renormalization of @xmath8 and @xmath26 ( via the screening equations ( [ screening2 ] ) ) we are interested in the rare events @xmath624 of small probability @xmath415 , but which dominate the _ average _ correlations . these events correspond to the region of fixed @xmath180 and thus to @xmath625 . fortunately , this region is so far ahead of the front that @xmath522 can be obtained with excellent accuracy by solving the linearized kpp equation : @xmath626 by straightforward integration from from @xmath400 to @xmath60 . this leads to : @xmath627 we also notice that in this regime @xmath481 can also be obtained explicitly . indeed , since the relation between @xmath481 and @xmath522 is a simple linear ( @xmath60-dependent ) convolution it is straighforward to show ( e.g. via @xmath510 fourier space ) that @xmath522 obeying the linearized kpp equation ( [ linkpp ] ) is equivalent to @xmath481 satisfying : @xmath628 this linearized equation was not entirely obvious to guess directly from ( [ rg - eq-1d - u ] ) and justifies , even in this simplest regime , the detour through the rigorous results for the kpp equation . using @xmath599 it yields : @xmath629 it is interesting to note that for _ fixed _ @xmath510 and @xmath180 , @xmath630 and @xmath481 can be estimated in the large @xmath60 limit as simple exponentials : @xmath631 with @xmath632 and an identical expression for @xmath633 with a different prefactor @xmath634 . these forms , which are valid in the region of fixed fixed @xmath635 ( and fixed @xmath636 respectively ) will be useful to estimate the renormalization of @xmath8 and @xmath26 below ) and ( [ kpplinearu ] ) by approximating @xmath637 , and similarly for @xmath180 ( to be precise the @xmath483 and @xmath8 which appear here are @xmath638 and @xmath639 respectively ) . they are valid only as long as the integrals which defines @xmath640 and @xmath641 are convergent . since @xmath642 for large positive @xmath643 the estimate for @xmath644 is valid only as long as @xmath645 . for @xmath646 one must instead perform saddle point estimate , see below ] . thus at low temperature we find the following decay of the probability of rare favorable local environments : @xmath647 with @xmath648 . we also find that the distribution @xmath469 has an algebraic tail at low temperature in the region @xmath624 ( i.e. @xmath96 fixed as @xmath649 ) as : @xmath650 with is a different power law behaviour the averaged fugacity @xmath651 is controlled by @xmath652 , while for @xmath653 is controlled by @xmath654 than the one which characterizes typical fugacities ( [ typical ] ) . these two different power law behaviours are represented in figure [ fig : front - shape ] . finally , one can check that the three regions match properly and thus we have a fairly complete description of @xmath655 . for instance , using the expression valid in the region ( ii ) for _ fixed _ @xmath510 at large @xmath60 one gets : @xmath656 which always gives the result ( [ papp ] ) up to the logarithmic corrections @xmath60 prefactors factors , and reproduces even the @xmath60 prefactors correctly for @xmath657 , which is when we expect the matching to become exact coefficient in the logarithmic correction to the velocity @xcite . ] . to study the screening equations ( [ screening2 ] ) in the xy phase we need the distribution @xmath658 of dipole core energy @xmath659 . fortunately in this regime it can be computed simply from ( [ kpplinearu ] ) and is given simply by : @xmath660 the last equality being valid for fixed @xmath180 and @xmath476 , and @xmath661 in terms of the initial dipole core energy distribution . substituting this last form in ( [ screening2 ] ) we obtain , in the large @xmath60 limit : [ screeningc ] @xmath662 with @xmath663 a constant roughly independent of the temperature . the two above integrals @xmath664 are convergent respectively only for @xmath665 and @xmath666 , in which cases it is indeed legitimate to replace @xmath667 by its above asymptotic form . note that the @xmath51 limit is well defined since in that case @xmath668 and @xmath669 . at higher temperature @xmath670 the full front solution controls the renormalization of @xmath8 . using the simple above gaussian form yields that the @xmath180 integral is dominated by the saddle point @xmath671 which gives : @xmath672 which corresponds for @xmath670 , to the behaviour of @xmath673 ( and for @xmath674 of @xmath675 ) , as expected . a similar , though more involved , analysis can be performed for @xmath26 . thus we have obtained the equations ( [ screeningc ] ) for the renormalization of @xmath8 and @xmath26 in the xy phase . since we have shown that @xmath415 decreases exponentially as ( [ pl1 ] ) we conclude that @xmath8 and @xmath26 reach their finite limits @xmath676 and @xmath677 as power laws of the systems size . this analysis also yields the full distribution of the correlation function of the charges in the xy phase . indeed , let us recall that : @xmath678 thus by following the rg up to the scale @xmath679 we can obtain from @xmath680 the distribution of the charge correlations at large @xmath681 . for instance , from the above and ( [ typ2 ] ) one finds the following decay of the _ typical _ thermal and disorder correlations : @xmath682 for @xmath683 and @xmath606 . for @xmath674 one has instead @xmath684 . the averaged moments can be obtained as above by substituting the exponential form of the distribution @xmath685 for @xmath686 . performing the corresponding integrals , one gets : @xmath687 with @xmath688 \gamma\left[p -\frac{t}{t^*}\right]}{\gamma[2 p ] } \\ & & b_p(t ) = \frac{c''}{\sqrt{8 \pi \sigma j^2 } } t ~\frac{\gamma\left[\frac{t}{t^*}\right ] \gamma\left[2 p - \frac{t}{t^*}\right]}{\gamma[2 p ] } \end{aligned}\ ] ] these formulae are valid only at low enough temperature @xmath689 . finally , although we have not attempted a precise rg calculation of the xy order correlation functions , the following behaviour should hold in the quasi ordered xy phase : @xmath690 with @xmath691 and @xmath692 . at the zero temperature transition point @xmath693 , @xmath51 , the value of these exponents are universal @xmath694 . we now study the transition from the xy to the disordered phases at low temperature at @xmath695 . from the previous sections we know that it occurs when the total velocity of the front of the distribution @xmath481 vanishes , i.e the critical region is defined by @xmath696 in a large @xmath60 regime . while in the xy phase the physics was dominated by the _ far tail _ of the traveling front @xmath623 in the critical regime it is the _ interior front _ @xmath697 , as well as the intermediate front region @xmath620 ( see fig . [ fig : front - shape ] ) which controls the transition . thus the correct description of the transition _ requires _ the knowledge of the kpp physics in an essential way , and is thus entirely novel . for simplicity , we will only present the analysis at @xmath51 . we will also work to first order in @xmath698 . we first assume that the coefficient @xmath699 varies sufficiently slowly near the transition so that the results from the kpp equation with a constant @xmath483 can be used . this assumption will be self consistently verified at the end . near the transition the center of the front is located at @xmath700 with @xmath701 as indicated by ( [ velocity - corrections ] ) . thus in the critical regime one still has @xmath702 , although logarithmically in @xmath60 exactly at the transition . thus this critical regime can still be studied perturbatively , as @xmath415 remains very small . the front velocity has the following scale dependence : @xmath703 where the h.o.t . contains the universal @xcite @xmath704 subdominant corrections to velocity in the kpp equation as well as additional subdominant corrections originating from the slowly varying @xmath484 . in the critical region at @xmath51 we can use the kpp front solution @xmath705 with , from ( [ front - intermediate ] ) : @xmath706 an expression valid as long as @xmath707 . the rg equations for @xmath8 and @xmath26 in the critical region thus read : [ sct0 ] @xmath708 we need to evaluate these expressions for @xmath709 large and negative ( typically either as @xmath710 on the critical manifold or as @xmath711 very close to it ) . at criticality they behave approximately as @xmath712 , which can be guessed by setting @xmath686 in the first expression . a more accurate estimate of the above integrals is performed in the appendix [ part : integrals ] . the end result is that , in the regime of interest ( where we can discard terms of order @xmath713 ) one has : @xmath714 where @xmath640 is a constant . introducing the small parameter : @xmath715 the density of favorable regions reads : @xmath716 since we can discard terms of order @xmath713 . from ( [ velcritical ] ) one finds that @xmath717 satifies precisely the rg flow equation : @xmath718 to lowest order in @xmath719 . note the @xmath720 _ universal _ factor which arises from the universal velocity corrections in the kpp equation ( [ velocity - corrections ] ) . a natural choice of parameters to describe the universal behaviour around the transition is thus @xmath721 ( which up to logarithmic corrections is equal to the density of favorable sites @xmath722 ) and @xmath26 which satisfies : @xmath723 these two equations ( [ eq - g ] , [ eq - sigma - g ] ) form our complete set of rg equations projected on the plane @xmath724 . they are somewhat analogous to the one describing a kosterlitz - thouless type transition , with an important difference . here one readily finds that the separatrix is _ vertical_. introducing the deviation from criticality , @xmath725 ( where @xmath726 ) one has for @xmath727 the flow : @xmath728 and thus satisfy @xmath729 . thus our rg equations yields @xmath730 at criticality @xmath731 at criticality ] and a correlation length : @xmath732 since starting away from criticality @xmath733 one finds read @xmath734 $ ] with @xmath735 = \int_x^{+\infty } dx x^{-3 } e^{-32 x}$ ] up to logarithmic corrections . ] this critical behaviour correspond to a new universality class which is different from kosterlitz - thouless and from the prediction of @xcite . note the crucial role of the @xmath720 universal factor . replacing it by any number less than @xmath23 would have led to usual kt behaviour . we can now check that the variations of @xmath483 should be unimportant at the transition . indeed one finds . writing @xmath737 $ ] as a function of @xmath709 , we get that @xmath738 $ ] since @xmath709 appears only explicitly in the bound of the integral in ( [ sct0 ] ) . thus one finds @xmath739 ( 1 - j \sigma \frac{d}{d x_l } \ln a[x_l])$ ] . using that using that @xmath740 = 2/\sqrt{d } + o(1)$ ] one gets that the leading variations cancel at the transition point @xmath741 . ] that @xmath742 at most . on the other hand we can estimate that if @xmath743 varies faster than @xmath744 the kpp results should not be affected ( see the scaling with @xmath60 of all terms in e.g. eq ( a7 ) in @xcite ) let us close by noting again how universality appears in the derivation of the critical behaviour . although most details of the fugacity distribution @xmath745 ( e.g. its bulk , the fusion rule .. ) depend on the cutoff procedure the universality in the xy transition appears in a remarkable way . it arises from the independence @xcite of the velocity , the velocity corrections , and front tail on the precise form @xmath746 $ ] of the non linear term in ( [ kpp - general ] ) . this gives us confidence in the method developped here . as is well known the coulomb gas can also be equivalently formulated as a sine gordon model ( see _ e.g _ @xcite ) . in this section we identify the random version of the sine gordon model to which our analysis applies . the scalar sine gordon model , of partition function @xmath747 is defined , in the absence of disorder by the action : @xmath748 where @xmath749 $ ] . as is well known it is equivalent to a coulomb gas since , expanding in @xmath721 one has : @xmath750 where @xmath751 denotes averages with respect only to the quadratic gradient part . the interaction @xmath752 has been defined in ( [ part : model ] ) . the above partition sums involve only @xmath753 neutral pairs of dipoles since , in the large size limit @xmath754 . finally @xmath755 in the last line involves a summation over all distinct charges with @xmath756 for @xmath757 and @xmath758 for @xmath759 . let us turn to the disordered version of the model . to reproduce the bare version ( [ zcont ] ) of the model , one must first add a short range correlated random _ imaginary _ field as follows : @xmath760 with correlations @xmath761 . since it imposes now that @xmath762 , each factor @xmath763 in the @xmath721 expansion in ( [ sgexp ] ) yields an additional @xmath764 where @xmath765 and thus reproduces the bare cg version ( [ zcont ] ) of our model . note that the above model still contains a uniform `` fugacity '' @xmath721 . it thus corresponds to the version studied in @xcite by rubinstein et al . however we know that this can not be the correct form under renormalization . the first obvious idea is to generalize @xmath766 , i.e a disordered fugacity ( the above expansion can be immediately generalized to this case ) . this term will be generated , but is not the end of the story . indeed let us consider the symmetries of the above action @xmath767 ( even in the presence of a @xmath768 ) . when @xmath769 the action is real and invariant through @xmath770 . in the presence of the random field this symmetry is broken ( as @xmath771 acquires a non zero , disorder environment dependent average ) and the action is complex . thus nothing prevents that under coarse graining the action will become : @xmath772 and this is indeed precisely what we have found in the cg formulation , i.e each sign of charge acquires a _ different _ local random fugacity . there is however a symmetry constraint on the distribution of the local random fugacities . indeed in the above bare model ( [ sgaction ] ) the full partition sum is real , as there is still a _ statistical symmetry _ : @xmath773 for any configuration @xmath771 and environment @xmath774 . since the probabilities of @xmath774 and @xmath775 are the same , all physical averages will be real . in the coarse grained model ( [ sgcoarse ] ) , since we have that @xmath776 the probability ( over environments ) of the random fugacity disorder configuration @xmath777 should be equal to the probability of @xmath778 . we can now check that this random sine gordon model ( [ sgcoarse ] ) is indeed equivalent to the coarse grained disordered cg considered in this paper by expanding its the partition sum in a given environment , which yields : @xmath779 and thus the @xmath95 random field disorder is associated to the long range part @xmath321 of the disorder in the cg ( see ( [ decomposition ] ) ) while the random couplings constants @xmath780 in ( [ sgcoarse ] ) are associated to the local random fugacities in the cg . similarly one can establish a correspondence directly on the replicated versions of both models . replicating the above sg model and averaging over disorder yields ( the limit @xmath139 is implicit ) : @xmath781 expanded to first order and treating the @xmath782 and @xmath783 as uncorrelated in space and reexponentiating yields the replicated sg model defined as : @xmath784 e^{i { \bf n } \cdot { \bf \phi } } \label{replicatedsg}\end{aligned}\ ] ] where @xmath785 is the scalar product in replica space and @xmath65 $ ] are analogous to the vector fugacities introduced previously in the replicated vector cg model . although the bare model obtained from ( [ averagesg ] ) contains only single component replicated charges , all multicomponent charges will be generated upon coarse graining , as in the replicated vector cg . as we have seen all these charges should be taken into account and thus the generic replicated sg model should contain all possible @xmath65 $ ] with @xmath786 . this sg model can also be studied using rg , either in its replicated form ( [ replicatedsg ] ) , or directly ( [ sgcoarse ] ) , very similarly to the vector cg studied in this paper . although variations in definitions of the fugacities @xmath65 $ ] and different cutoff procedures can induce some irrelevant differences in the details of the renormalization ( and different rg equations ) , the two models have the same physics . the sine gordon formulation has several advantages , such as exhibiting by construction the decomposition of the disorder in two physically very different components ( see [ decomposition ] ) . it is also more amenable to replica variational methods than the cg , left for future study . let us compare our method and results with the work of scheidl @xcite . in this work the multicomponent charges ( restricted to @xmath66 in each replica ) were considered , but the fusion was not taken into account . also only dipole fugacities were introduced . we can recover the rg equations and results of scheidl within our approach by ( i ) artificially setting the fusion coefficient @xmath197 to @xmath115 in ( [ loose ] ) ( ii ) assuming a log - normal distribution @xmath787 for the fugacities ( which is consistent only when fusion is neglected ) ( iii ) defining random `` dipole fugacities '' as @xmath788 and @xmath789 ( or equivalently @xmath790 ) identical to the `` dummy gaussian variables '' introduced by scheidl . within our diffusion formalism ( see section([part : replicalimit ] ) ) we find that the norm @xmath791 diverges exponentially , which allows to recovers the extra factors @xmath792 appearing in scheidl s screening equations . going from our diffusion formalism back to the replica formulation yields back the rg equations of @xcite . in presence of a fusion term @xmath793 , the equations become a priori very different . the fugacity distribution does not remain log - normal as we do not assume a priori the form of the distribution . interestingly , although the log - normal does not reproduce correctly the true distribution of fugacities ( and misses connections such as the one with the freezing of the dpct via the kpp equation ) , some of our results deep in the xy phase , such as the renormalization of @xmath794 and @xmath26 ( see ( [ screeningc ] ) ) , agree with the one of scheidl . it was not obvious a priori that the approach without fusion in the xy phase did not miss extra relevant physics in this model , and indeed near the transition fusion appears to be crucial and must be taken into account . to conclude , we have constructed in this paper a novel renormalization group method which allows to study perturbatively , and in a consistent way , a large class of disordered models which can be formulated as two dimensional coulomb gases with quenched disorder . we have applied it specifically to the xy model with quenched random phases . we have obtained the phase diagram for this model , confirmed the existence of a low temperature xy phase , and elucidated the critical behaviour at the transition where topological defects proliferate . it would be interesting to check our predictions in numerical simulations the present rg method is not based on the conventional perturbative expansion in a vortex fugacity @xmath2 spatially uniform over the system , which , as we have shown , is only justified for pure models or for disordered models at high enough temperature . instead , it is constructed by first defining the local random vortex fugacity ( or core energy ) and then following its full probability distribution under coarse graining . below a freezing temperature this distribution becomes very broad and can not be followed by conventional cg methods . our renormalization procedure allows to follow this broad distribution in a controlled way , by defining the new perturbative parameter as the concentration @xmath406 of rare regions favorable to vortices . this running parameter flows to zero in the xy phase and at the transition ( marginal flow ) . hence both can be studied perturbatively . the underlying physical picture obtained here is that the transition is controlled , as the scale increases , by the proliferation of vortices in less and less rare favorable regions . we find that it has some features reminiscent of a kosterlitz - thouless transition , with important differences , such as the scaling of the correlation length . to derive the rg equation for the distribution of vortex fugacities , we have introduced two equivalent methods . one is based on the replicated vector coulomb gas version of the model , and in an expansion in the vector fugacities . the second one is direct , with no use of replicas , and is based on a systematic expansion of all physical quantities in the number of points , i.e in independent local regions and thus , in the end , in powers of the concentration @xmath406 of rare favorable regions . as we have shown these two methods are fully equivalent : the first one being more systematic and the second one allowing for a clear physical understanding of the problem in terms of probability distributions . these two expansion methods are highly non perturbative in the original uniform fugacity variable @xmath2 . since they are constructed from charge fugacities they can be made fully consistent ( by contrast with previous approaches based on dipole fugacities ) . our method sheds light on the broader issue of universality in random systems . the spirit of the rg method is that at large scale most information about the system is irrelevant and can be discarded . in constructing our rg procedure we have first shown , using the very special properties of logarithmic interactions , that it is enough to follow , in addition to the two parameters @xmath794 and @xmath26 , the distribution of only one or two local ( i.e uncorrelated ) random variables @xmath795 . at this stage it is clear that we still keep too much information . indeed we have found that the precise form the non linear rg equation obeyed by @xmath795 as well as the detailed shape of this distribution are largely cutoff dependent . however , this complicated looking rg equation can be generally recast , up to irrelevant terms , as a well known non linear front propagation of the kpp type . using the known remarkable universality property of this type of non linear equations , we found that all the information needed to describe the universal properties of the xy phase and of the transition is indeed independent of the detailed shape of the fugacity distribution , or of the precise form of its rg equation . only its tails , and the finite size corrections to the front velocity seem to be needed to determine the physical quantities and the critical behaviour . since we are following a full distribution of local disorder , the present method could be termed a functional rg . we can indeed draw some parallel between this functional rg procedure for the disordered cg and two other known examples where the universal behaviour of a disordered finite - range system is extracted from a functional rg equation . the first one is the asymptotically exact real space rg in @xmath796 @xcite well suited to `` infinite disorder '' fixed points . as emphasized in @xcite there are indeed similarities when treating the single vortex , @xmath796 version of the present model . the method of @xcite can be applied for the sinai potential which has correlations growing as a power of distance . for the present case of weaker , logarithmic correlations , these methods can also be applied in principle ( at least around zero temperature where disorder is still very broad ) but become very hard to work out analytically . the other example of functional rg appears in a dimensional expansion for the problem of an interface in a random potential @xcite , which has infinite number of marginal directions at the upper critical dimension @xmath797 . an important question is whether in the problem studied in this paper there is also an infinite number of marginal directions . as discussed above , the results extracted from the kpp equation seem to suggest that a smaller amount of information than the exact full distribution may be needed here . thus one can speculate that the critical theory studied here could be equivalently formulated as a more conventional field theory , yet to be identified , with a small number of marginal or relevant operators . in any case the rg method developed here should provide a physically transparent guide to study the system . given the wide applications of coulomb gases in two dimensions it is likely that other two dimensional disordered models can be studied using methods similar to the one introduced here . finally , another outcome of the present work has been to unveil some interesting connections between the renormalization of a disordered system and the universal features of the propagation of invading fronts in non linear systems . the existence of intriguing relations between freezing transitions and velocity selection in non linear fronts was noticed previously by derrida and spohn @xcite in their study of the dpct . here , we found an even deeper and puzzling connection , betwen the `` universality '' in these selection mecanisms and the universality in the critical phenomena captured by the renormalization group . since attempts have been made to construct renormalization methods in order to extract the universal features of such non linear equations @xcite , this suggests that a common framework could be developed in connexion with two dimensional disordered models . * acknowledgements * it is a pleasure to thank b. derrida , v. hakim and w. van saarloos for useful discussions . in this appendix we derive , by exact transformations , a lattice disordered coulomb gas formulation of the villain form of ( [ xy ] ) , extending to this disordered case the approach of kadanoff @xcite . an alternative route to @xcite in the pure case in the method used in @xcite . for simplicity we firt turn to the villain version of ( [ xy ] ) before making the duality transformations , although the inverse procedure could be used ( see @xcite ) . this villain model corresponding to ( [ square ] ) is @xcite : @xmath798\in \mathbb{z } } \prod_{i}\int_{-\pi}^{+\pi } \frac{d \theta_{i}}{2 \pi } \exp \left [ - \frac{k}{2\pi } \sum_{\langle i , j \rangle}(\theta_{i}-\theta_{j}-2\pi l_{ij}-a_{ij } ) ^{2}\right]\end{aligned}\ ] ] where @xmath221 and we used the notation @xmath799 ( see fig . [ reseau ] ) . this villain partition function indeed corresponds to a @xmath800-gauge theory , since the action is invariant under @xmath801 with @xmath802 . choosing the _ gauge field _ @xmath803 such that for each horizontal link @xmath804 ( _ i.e _ @xmath805 ) ( free boundary conditions along @xmath510 ) we obtain a partition function of a gaussian field @xmath806 ( @xmath807 ) : @xmath808 } \prod_{i}\int_{-\infty}^{+\infty } d \phi_{i } e^{- 2\pi\beta j \sum_{i } \left ( \left ( \nabla_{i , y}\phi + l_{i , y } -\frac{1}{2\pi } { \bf a}_{i , y } \right)^{2 } + \left ( \nabla_{i , x}\phi -\frac{1}{2\pi } { \bf a}_{i , x } \right)^{2 } \right ) } \end{aligned}\ ] ] with the notation for the discrete gradient @xmath809 . integrating over this gaussian field yields the coulomb gas action @xmath810 } e^ { - \beta j \sum_{i , j } \left[(l_{i-\hat{e}_{y},y}-l_{i , y})- \frac{1}{2\pi}{\bf \nabla}_i.{\bf a}\right ] g_{ij } \left[(l_{j-\hat{e}_{y},y}-l_{j , y})- \frac{1}{2\pi}{\bf \nabla}_j.{\bf a}\right ] - \sum_i 2 \pi k~ l_{i , y}^2}\nonumber \\ & = & \label{latticecg } \sum_{n_{\alpha } } e^ { - \beta j \sum_{\alpha,\beta } ( n_{\alpha}-q_{\alpha } ) \gamma_{\alpha \beta } ( n_{\beta}-q_{\beta } ) } \end{aligned}\ ] ] where we have discarded the @xmath811 self interaction and we defined the dual lattice charges @xmath812 with @xmath813 ( see fig . [ reseau ] ) : @xmath814 and the quenched random dipoles @xmath815 note that the neutrality of disorder charges @xmath816 follows directly from the definition of @xmath817 ( e.g. taking @xmath818 to vanish at the boundary ) and implies neutrality for the integer charges @xmath819 ( from the divergence of @xmath820 ) . the _ coulomb potential _ @xmath821 is defined on the lattice by @xmath822 which gives the expression @xmath823 where the @xmath824 is the euler constant . using the neutrality of the charges and the definition @xmath825 , and the definition of the disorder potential @xmath826 , the lattice disordered cg action reads @xmath827 the coulomb gas , or its equivalent sine gordon version , can also be renormalized using smooth cutoff procedures . we will not give details here but refer the reader to @xcite ( see also @xcite ) . let us simply point out how , in that case , the full disorder @xmath270 can indeed be decomposed , as in ( [ decomposition ] ) , into two bona fide disorders . in the case of a soft cutoff , the continuum approximation of the lattice coulomb interaction reads ( instead of ( [ approx ] ) ) : @xmath828 where @xmath829 and we will choose @xmath830 a positive monotonously decreasing function of @xmath510 . one has the asymptotic large @xmath91 behaviour @xmath831 ( see e.g. @xcite ) . one also has : @xmath832 where @xmath833 . thus one can write the cutoff dependent decomposition @xmath834 and one gets upon increase of cutoff : @xmath835 and @xmath836 with : @xmath837 thus both @xmath41 and @xmath838 are well defined physical gaussian disorders since their correlators have positive fourier transform . in addition @xmath838 is short range correlated . for instance , taking @xmath839 one finds : @xmath840 in this appendix , for completeness , we explicitly renormalize the vector coulomb gas defined by ( [ expansion ] ) . this amounts to extend to @xmath37-component vector charges the renormalisation of scalar cg @xcite . the partition function reads : @xmath841 e^{\mathcal{a}_{a_{o } } [ { \bf n}_{1}\dots { \bf n}_{n}]}\end{aligned}\ ] ] where as usual the primed sum is over all distinct neutral charge configurations and the notation h.c . stand for all the hard core constraints @xmath842 for all pairs @xmath843 , implicit in the following . the action @xmath844 is defined by @xmath845=\frac{1}{2 } \sum_{i\neq j } 2k_{ab}~ n_{i}^{a}\ln \left(\frac { |{\bf r}_{i}-{\bf r}_{j}|}{a_{o } } \right ) n_{j}^{b}\ ] ] a common way to renormalise usual coulomb gas consists in coarse graining the partition function , leaving the expansion in number of fugacity ( here @xmath846 ) unchanged . this amounts to define scale dependent replica stiffness @xmath847 and fugacities @xmath848 $ ] . we will follow this scheme in this appendix . note that another equivalent way would be to renormalise the correlation function directly , following _ e.g _ @xcite . as in the scalar case @xcite , renormalisation of the generalised vector coulomb gas @xcite proceeds in the same three steps : rescaling , fusion and annihilation ( screening ) of small dipoles . we now turn to the description of these three contributions : first we increase the hard - core cut - off @xmath849 with @xmath850 . this increase of cut - off produces a naive rescaling : @xmath851 e^{\mathcal{a}_{a_{o}}[{\bf n}_{1}\dots { \bf n}_{n } ] } = \prod_{i=1}^{n}\int \frac{d^{2}{\bf r}_{i}}{\tilde{a}_{o}^2 } y[{\bf n}_{i } ] e^{\mathcal{a}_{\tilde{a}_{o}}[{\bf n}_{1}\dots { \bf n}_{n } ] } e^{dl ( 2-k_{ab}n_{i}^{a } n_{i}^{b})}\ ] ] where we used the neutrality @xmath852 to express the correction coming from the action . we can absorb the extra factor in ( [ app : rescaling ] ) to all order in @xmath846 by the change of fugacities corresponding to the equation @xmath853= ( 2-k_{ab}n^{a}n^{b } ) y[{\bf n}]\ ] ] upon the increase of cut - off , two charges @xmath854 and @xmath855 have to be coarse grained if they are located in @xmath856 and @xmath857 with @xmath858 . within the small charge density hypothesis , we consider only one such pair . for a dipole , these two charges have to be integrated out at scale @xmath859 : this corresponds to the annihilation , while for a non neutral pair , the coarse grained charge is simply the sum of the two charges at scale @xmath90 ( fusion ) . in both cases the partition function splits into @xmath860 where @xmath861 involves configurations with one pair of charges @xmath862 distant of less than @xmath330 while @xmath863 does nt . @xmath864 can be written as @xmath865 \\ \times \sum_{{\bf n}_{p},{\bf n}_{q } } \int_{a_{o}\leq |{\bf r}_{p}-{\bf r}_{q}|\leq a_{o}e^{dl } } \frac{d^{2}{\bf r}_{p}}{\tilde{a}_{o}^2 } \frac{d^{2}{\bf r}_{q}}{\tilde{a}_{o}^2 } y[{\bf n}_{p}]y[{\bf n}_{q } ] e^{\mathcal{a}_{p , q}[{\bf n}_{1}\dots { \bf n}_{n}]}\end{gathered}\ ] ] where , with the notation @xmath866 , the action reads @xmath867 we must now distinguish between a neutral pair and a non neutral one . in this case the small pair of charges @xmath868 at scale @xmath90 gives an effective charge @xmath869 at scale @xmath859 , located in @xmath870 . thus we must integrate over the relative position of the two charges @xmath871 when coarse graining the coulomb gas . to obtain the corresponding correction to @xmath863 ( of order one in @xmath342 ) , it is enough to expand ( [ expans ] ) to order 0 in @xmath872 . this expansion reads @xmath873 \sum_{{\bf n}_{p},{\bf n}_{q}}\int \frac{d^{2}{\bf r}}{a_{o}^2}~ y[{\bf n}_{p}]y[{\bf n}_{q } ] \left(\int_{a_{o}\leq \rho\leq a_{o } e^{dl } } \frac{d^{2}{\bf \rho}}{a_{o}^2 } \right ) \\ \times \prod_{i \neq p , q } \left ( \frac{|{\bf r}_{i}-{\bf r}|}{a_{o}}\right)^{\alpha_{ip}+\alpha_{iq } } \prod_{i < j \neq p , q } \left ( \frac{|{\bf r}_{i}-{\bf r}_{j}|}{a_{o}}\right)^{\alpha_{ij}}\end{gathered}\ ] ] using @xmath874 , we can rewrite this correction to @xmath863 as a single contribution to the fugacity @xmath875 $ ] of the _ non zero _ charge @xmath869 : @xmath876= 2 \pi \sum_{{\bf n}_{p},{\bf n}_{q } } y[{\bf n}_{p}]y[{\bf n}_{q}]\ ] ] note that in this expression @xmath854 and @xmath855 are distinguishable charges : this explains the factor 2 between ( [ rg - fusion ] ) and ( [ rgrep2 ] ) . in this case the small dipole of size @xmath877 is integrated out at scale @xmath859 . when coarse graining , we sum over both @xmath872 and @xmath386 , yielding a factor @xmath878 from the integration measure . thus diverging contributions correspond to the expansion of ( [ expans ] ) to order 2 in @xmath872 . using @xmath879 , this expansion of @xmath864 can be expressed as @xmath880 \prod_{i < j \neq p , q } \left ( \frac{|{\bf r}_{i}-{\bf r}_{j}|}{a_{o } } \right)^{\alpha_{ij } } \sum_{{\bf n}_{p}}y[{\bf n}_{p}]y[-{\bf n}_{p}]\\ \times \int_{a_{o}\leq \rho\leq a_{o } e^{dl } } \frac{d^{2}{\bf \rho}}{a_{o}^2 } \int \frac{d^{2}{\bf r } } { a_{o}^2 } \left ( 1+\sum_{i , j \neq p , q}^{n } \alpha_{i , p } \alpha_{j , p } \frac{1}{4 } { \bf \rho}.{\bf \nabla}\ln ( { \bf r}_{i}-{\bf r } ) { \bf \rho}.{\bf \nabla}\ln ( { \bf r}_{j}-{\bf r } ) \right)\end{gathered}\ ] ] performing the integral and reexponentiating the last term using the neutrality of the configuration @xmath881 , we get the correction to @xmath863 coming from @xmath882 : @xmath883 \\ \times \prod_{i < j \neq p , q } \left ( \frac{|{\bf r}_{i}-{\bf r}_{j}|}{a_{o}}\right)^{\alpha_{ij}-\pi^{2}dl \sum_{{\bf n}_{p } } \alpha_{i , p}\alpha_{j , p}~y[n_{p}]y[-n_{p } ] } \end{gathered}\ ] ] the constant @xmath884 corrects the intensive free energy while the second term can be absorbed in a correction to the coupling constant @xmath885 } n^c n^d y[{\bf n } ] y[- { \bf n}]\ ] ] the three above contributions can be summarized into the set of rg equations given in the text ( [ rgrep ] ) valid for _ all non zero vector charge _ , and for all @xmath37 . the coefficient @xmath886 and @xmath364 depends on the ir regularisation . for our hard cut - off , we find within our procedure that @xmath887 and @xmath888 . note that the ratio @xmath889 is independent of a uniform rescaling of the fugacities and is known , in the case of single component charges to be universal at a transition @xcite . for a discussion of regularisation of replicated coulomb gas , see discussion in appendix [ app - cutoff ] . in this appendix we explicitly perfom the @xmath67 limit of the whole set of rg equations ( [ rgrep1],[rgrep2 ] ) with the restriction that the vector charges have only @xmath890 components . this limit is taken using the parametrisation ( [ eq : ydef ] ) of the fugacities @xmath891 $ ] in terms of the function @xmath175 . in the three different terms of ( [ rgrep1],[rgrep2 ] ) , corresponding to rescaling , annihilation and fusion contributions ( see appendix [ part : rgreplica ] ) , we first modify sums to include fugacities for null charge , translate the expression in terms of @xmath188 and naturally take the @xmath139 limit . we will use the notation @xmath892 . the term corresponding to rescaling in ( [ rgrep2 ] ) is @xmath893 = ( 2 - n_a k^{ab } n_b ) y[{\bf n } ] \end{aligned}\ ] ] which holds for any vector charge @xmath38 . the second term can be expressed in terms of @xmath188 using ( [ eq : ydef ] ) and @xmath894 with @xmath221 . however the expression is much simpler if one uses , instead of ( [ eq : ydef ] ) , the equivalent parametrisation @xmath65=\int_{u , v } e^{\beta pu}e^{\beta qv } \tilde{\phi } ( u , v)$ ] with @xmath895 , @xmath896 and @xmath897 . with @xmath898 , this yields @xmath899&= & ( \beta j p - \sigma \beta^2 j^2 q^2 ) \int_{u , v } \tilde{\phi}(u , v ) e^{\beta pu } e^{\beta qv } \\ & = & \int_{u , v } \tilde{\phi}(u , v ) \left ( j \frac{\partial}{\partial u } - \sigma j^2 \frac{\partial^2}{\partial v^2 } \right ) e^{\beta pu } e^{\beta qv } \\ & = & - \int_{u , v } e^{\beta pu } e^{\beta qv } \left ( j \frac{\partial}{\partial u } + \sigma j^2 \frac{\partial^2 } { \partial v^2 } \right)\tilde{\phi}(u , v ) \label{lim - repar}\end{aligned}\ ] ] as this is true for any vector charge @xmath900 satisfying @xmath901 , _ i.e _ for any @xmath902 or equivalently @xmath903 , to satisfy ( [ dim - y ] ) we can search for a function @xmath898 such that : @xmath904 in terms of @xmath175 , this corresponds to the equation : @xmath905 where we have used that @xmath906 . note that for this process , the integral of @xmath188 , @xmath192 satisfies simply @xmath907 . in ( [ rgrep2 ] ) , the fusion term of two charges @xmath908 is restricted to @xmath909 and @xmath910 . furthermore since @xmath911 corresponds to the annihilation which is treated separately in ( [ rgrep1 ] ) ( see below ) and must not be counted twice , the equation ( [ rgrep2 ] ) can be used only for @xmath912 . it is convenient to extend the equation ( [ rgrep2 ] ) to include @xmath913 = \int \phi = \mathcal{n}$ ] for which the fusion contribution should be absent . thus the extended equation corresponding to fusion reads : @xmath914= c_{2 } \left ( \sum _ { \genfrac{}{}{0pt}{}{{\bf n}'+ { \bf n } '' = { \bf n}}{{\bf n}',{\bf n}''\neq 0 } } y[{\bf n } ' ] y[{\bf n } '' ] - \delta_{{\bf n},0 } \sum_{{\bf n } ' \neq 0 } y[{\bf n } ' ] y[-{\bf n } ' ] \right ) \\ & & = c_{2 } \left ( \sum _ { { \bf n}'+ { \bf n } '' = { \bf n } } y[{\bf n } ' ] y[{\bf n } '' ] - 2 y[{\bf 0 } ] y[{\bf n } ] - \delta_{{\bf n},0 } \left ( \sum_{{\bf n } ' } y[{\bf n } ' ] y[-{\bf n } ' ] - 2 y[0]^2 \right ) \right ) \nonumber\end{aligned}\ ] ] where in the second equality we allow for @xmath915 or @xmath916 . turning to the representation in terms of @xmath903 we have : @xmath917 y[{\bf n}'']\\ & & = \left < \prod_a \bigl [ ( 1 + z'_- z''_+ + z'_+ z''_- ) \delta_{n_a,0 } + ( z'_+ + z''_+ ) \delta_{n_a,+1 } + ( z'_- + z''_- ) \delta_{n_a,-1 } \bigr ] \right>_{\phi,\phi}\end{aligned}\ ] ] using permutation symmetry we find that ( [ eqphi1 ] ) can be written as : @xmath918 = \\ & & c_2 \left < ( 1 + z'_- z''_+ + z'_+ z''_- ) ^{m } \left(\frac{z'_+ + z''_+}{1 + z'_- z''_+ + z'_+ z''_-}\right)^{n_{+ } } \left(\frac{z'_- + z''_-}{1 + z'_- z''_+ + z'_+ z''_-}\right)^{n_{- } } \right>_{\phi \phi } \\ & & - 2 c_2 \mathcal{n } \left < z_+^{n_+ } z_-^{n_- } \right>_{\phi } - c_2 \delta_{n_+,0 } \delta_{n_-,0 } \left ( \left < ( 1 + z'_- z''_+ + z'_+ z''_- ) ^{m } -1 \right>_{\phi \phi } - \mathcal{n}^2 \right)\end{aligned}\ ] ] the following choice for @xmath919 allows to satisfy the above equation for all @xmath38 ( up to now @xmath37 is still arbitrary ) : @xmath920 we can now take the limit @xmath139 _ explicitly _ on this equation , which yields : @xmath921 putting all these contributions together , we obtain the scaling equations for the coupling constant @xmath8 , the correlated disorder strength @xmath26 and the distribution of local disorder @xmath175 : @xmath929 where the diffusion operator has been defined in ( [ def - o ] ) . in this appendix we explore the connections between the expansion in number of sites of the free energy , and the expansion of the replicated partition function in power of composite charge fugacities @xmath65 $ ] . as we will see , both expansions coincide exactly , and we can thus consider the expansion of the replicated partition function as a generating functional of the site - expansions . fist we consider the first term @xmath930 given by ( [ z-2 ] ) of the expansion in number of independent sites ( [ free ] ) . this term corresponds exactly to the approximation of independent dipoles considered in @xcite : dipoles do not interact with each other , and the free energy is thus the sum over all positions of the pairs of the free energy of a pair : @xmath931 , and the free energy of a dipole is simply @xmath932 using the decomposition of the disorder ( section [ part : disdec ] ) @xmath348 , and averaging over the correlated disorder @xmath41 using replica , we obtain @xmath933 where we used the non local correlation of the disorder @xmath934 and the definition @xmath935 . with the bare definition of the replica charge fugacities , this finally gives after average over @xmath43 , the expected first term of the expansion of @xmath936 : @xmath937 } y[{\bf n } ] y[-{\bf n } ] \left(\frac{|{\bf r}-{\bf r}'|}{a_{o}}\right)^{-2 n^{a}k^{ab}n^b}\end{aligned}\ ] ] the methods is the same as in previous section , but the calculations are slightly more tedious . we consider the second term of the free energy expansion : @xmath938 with the help of the previous section , it is enough to consider only @xmath939 using the decomposition of figure ( [ fig : secondorder ] ) , we obtain the sum @xmath940 by averaging the second term over @xmath41 , and using @xmath935 , we get @xmath941 inserting this result in the above expression averaged over @xmath41 , the sum over the partitions of the interval @xmath942 $ ] in @xmath943 can be exactly rewritten as a sum over all _ neutral _ triplets @xmath944 of @xmath37 component vector charges with components @xmath66 ( see figure [ fig : secondorder ] ) . thus one recovers the expression @xmath945(r ) y[n'](r ' ) y[n''](r'')e^{-s_{n , n',n '' } } -1\ ] ] we now note that the last three logarithmic terms in ( [ logs ] ) just give additional restrictions on this sum ( as they correspond respectively to @xmath946 ) . thus , averaging over the local fugacities , we end up with the expected second term ( see [ replicah ] ) of the expansion of @xmath153 in power of the vector fugacities @xmath65 $ ] : @xmath947 y[{\bf n ' } ] y[{\bf n '' } ] e^{- \beta h_{{\bf n , n',n''}}}\end{aligned}\ ] ] these simple combinatorics can be done on higher order terms : it gives the equivalence term by term between the @xmath67 limit of the expansion in @xmath65 $ ] and the expansion in the number of independent sites of the moments of the free energy . in this appendix we discuss the consequences on the rg equations of the choice of cutoff made in this paper for the replicated coulomb gas . we illustrate for simplicity only the case of zero disorder . although we will be mainly concerned with coulomb gas , most of this discussion can be applied to other general replica field theory . let us first recall the results for a single component cg ( @xmath948 ) . we restrict to the most relevant charges @xmath45 for simplicity . it is defined by the action @xmath949 the corresponding rg equation was derived by kosterlitz : @xmath950 we now consider @xmath37 copies of this model in the absence of disorder , as illustrated in fig . ( [ fig : cutoff ] ) . they are a priori physically completely uncoupled . the most natural cutoff procedure in that case would be independent cutoffs ( e.g. hard core for each ) ( left figure ) . another procedure , which becomes much more convenient in the presence of disorder , is to reformulate the @xmath37 copies as a single coulomb gas of vector charges with @xmath37 components . however in that case the cutoff is by definition columnar ( right figure ) ( e.g hard core vector charge are a hard columnar disk ) and in a sense the copies are coupled , via the cutoff . we now check that in the pure case the ensuing vector cg rg equations are still perfectly compatible with ( [ cutoff : rg1 ] ) as they should . they read : @xmath951 = \left ( 2 - k { \bf n } \cdot { \bf n } \right ) y[{\bf n } ] + c_{2 } \sum_{{\bf n}'\neq 0,{\bf n } } y[{\bf n } ' ] y[{\bf n}-{\bf n } ' ] + \text{higher order terms } \nonumber\ ] ] let us first illustrate the case of two copies @xmath952 . we can choose @xmath953=y[-1,0]=y[0,1]=y[0,-1 ] \equiv y_1 $ ] and @xmath954=y[-1,1]=y[1,-1]=y[-1,-1 ] \equiv y_2 $ ] . then considering all the possible fusions within this set the rg equations read : @xmath955 a solution of these equations , to the order @xmath956 at which we are working , is : @xmath957 = \lambda_1 y_l \quad y_l[1,1 ] = \lambda_2 y_l^2\end{aligned}\ ] ] with @xmath958 , where @xmath553 satisfies the single copy equation ( [ cutoff : rg1 ] ) . this can be generalized to higher charges so that in general one can find solutions of the type : @xmath959 } = \lambda_{i } y_l^{i}\ ] ] with coefficients @xmath960 which can be determined for a given regularisation procedure . they depend on both the initial ( @xmath948 ) and the replica regularisation ( see fig [ fig : cutoff ] ) . this example illustrates how a generic cutoff procedure will produce non trivial coefficients @xmath961 . thus if one reinterprets formally @xmath65 = < y^{n_+ + n_- } > _ { \phi(y)}$ ] in terms of a `` disorder '' @xmath232 as done in this paper , ] one must keep in mind that even the pure system corresponds to a non trivial `` bare disorder '' in the fugacities , which solely originates from the ( convenient ) choice of a _ columnar _ cutoff for a particular choice which does not introduce additional disorder ] . it is thus clear that the definition of the local fugacity distribution is strongly cutoff dependent . in this appendix we derive formula ( [ fk ] ) for the expansion of the free energy in the number of points . the same method can be applied to other physical observables . to organize this expansion we start by introducing _ fictitious site dependent fugacities _ for the charges : @xmath962 . these are introduced only as a trick in this appendix and should not be confused with the real disordered fugacities of ( [ decomposition ] ) . indeed , to recover the original model , we will set them back at the end either to @xmath963 ( for the lattice model ( [ square - bis ] ) ) or @xmath964 for the continuum model , where @xmath2 is the corresponding fugacity for the charge in the pure case . these fugacities are introduced by writing a more general form of the the partition function ( [ square - bis ] ) as @xmath965 = \sum_{p } \sum'_{\{n_{1},\dots n_{p } \ } } \sum_{{\bf r}_{1}\neq \dots \neq { \bf r}_{p } } \left(\prod_{i } \zeta_{{\bf r}_{i}}^{n_i^2 } \right ) e^{\beta j \sum_{{\bf r}_{i}\neq { \bf r}_{j } } n_{i}g_{{\bf r}_{i}-{\bf r}_{j } } n_{j } + \beta \sum_{i } n_{i}v_{{\bf r}_{i}}}\ ] ] here and below , as in ( [ zcont ] ) , all formulaes can be extended to the continuum model by replacing discrete sums over distincts sites @xmath268 by integrals with , _ e.g _ hard core conditions @xmath269 . note that in the above expression ( [ square - fict ] ) we do not make use of the decomposition ( [ decomposition ] ) and @xmath966 denotes the original disorder . let us consider for simplicity a system of @xmath30 distinct points @xmath967 . the free energy functional @xmath968 = - t \ln z[v , \zeta ] $ ] is a function of the @xmath30 variables @xmath969 , .. , @xmath970 . let us write the conventional taylor expansion of the free energy around @xmath971 : @xmath972 & = & \sum_{p_1,p_2, .. p_n = 0}^{+\infty } \frac{1}{p_1 ! \dots p_n ! } \frac{\partial^{p_1+ .. p_n } f[v,\zeta ] } { \partial \zeta_{{\bf r}_1}^{p_1 } \dots \partial \zeta_{{\bf r}_n}^{p_n}}|_{\zeta=0 } \times \zeta_{{\bf r}_1}^{p_1 } ... \zeta_{{\bf r}_n}^{p_n}\end{aligned}\ ] ] we now separate in this sum the terms which have only one non zero @xmath973 , then only two non zero @xmath973 , etc .. thus we can rewrite : @xmath974 & = & \sum_{k=1}^{+\infty } \sum_{\ { r_{i_1}, .. ,r_{i_k } \ } } \sum_{q_1, .. q_k = 1}^{+\infty } \frac{1}{q_1 ! \dots q_k ! } \frac{\partial^{q_1+ .. q_k } f[v,\zeta ] } { \partial \zeta_{{\bf r}_{i_1}}^{q_1 } \dots \partial \zeta_{{\bf r}_{i_k}}^{q_k}}|_{\zeta=0 } \times \zeta_{{\bf r}_{i_1}}^{q_1 } ... \zeta_{{\bf r}_{i_k}}^{q_k } \\ & = & \nonumber \sum_{k=1}^{+\infty } \sum_{\ { r_{i_1}, .. ,r_{i_k } \ } } f^{(k)}_{r_{i_1}, .. ,r_{i_k}}[v,\zeta]\end{aligned}\ ] ] where the sum is over all distincts _ sets _ @xmath975 of @xmath279 distincts points ( among the @xmath30 points ) and the sum over each @xmath803 goes from @xmath23 to @xmath579 . we have thus obtained an expansion of @xmath976 $ ] as a sum of terms of the form @xmath977 which depends exactly and only on the variables @xmath978 ( and thus also only on the variables @xmath966 ) evaluated at the @xmath279 _ distinct points _ @xmath979 . for a neutral coulomb gas it starts with @xmath295 and reads : @xmath980&= & \sum_{\ { { \bf r}_{i_1 } \neq { \bf r}_{i_2 } \ } } f^{(2)}_{{\bf r}_{i_1},{\bf r}_{i_2}}[v,\zeta ] + \sum_{\ { { \bf r}_{i_1 } \neq { \bf r}_{i_2 } \neq { \bf r}_{i_3 } \ } } f^{(3)}_{{\bf r}_{i_1},{\bf r}_{i_2},{\bf r}_{i_3}}[v,\zeta ] + \dots\end{aligned}\ ] ] with the definition @xmath981 & = & \sum_{q_1, .. q_k = 1}^{+\infty } \frac{1}{q_1 ! \dots q_k ! } \frac{\partial^{q_1+ .. q_k } f[v,\zeta ] } { \partial \zeta_{{\bf r}_{i_1}}^{q_1 } \dots \partial \zeta_{{\bf r}_{i_k}}^{q_k}}|_{\zeta=0 } \times \zeta_{{\bf r}_{i_1}}^{q_1 } ... \zeta_{{\bf r}_{i_k}}^{q_k}\end{aligned}\ ] ] note that in this last expression we can drop out the dependence of @xmath976 $ ] on the fugacities @xmath982 and the potential @xmath11 at points different from @xmath983 . a more explicit expression can be obtained by summing over the @xmath984 in ( [ def - fk ] ) : @xmath985 & = & \sum_{l=0}^{k } ( -1)^{k - l}\sum_{i_{1},\dots i_{l}\in [ 1,\dots k ] } f_{{\bf r}_{i_{1}},\dots { \bf r}_{i_{l}}}[v,\zeta]\end{aligned}\ ] ] where @xmath986 $ ] is the free energy associated with the system of sites @xmath277 ( instead of the full lattice ) . equivalently , it does depend only on the fugacities @xmath982 ( or potential @xmath11 ) at points @xmath277 . after setting @xmath987 ( lattice model ) or @xmath988 ( continuum model ) this gives the definition of the @xmath989 $ ] that we use throughout this paper , namely the equation ( [ fk ] ) . this last expression ( [ fk - expl ] ) allows to explicitly compute the coefficient @xmath989 $ ] of the expansion ( [ free - zeta ] ) for arbitrary order @xmath279 . in the section ( [ part : exp ] ) we use it to give explicit expressions for the first few terms of the expansion of the free energy . as shown in the previous section the free energy expansion involves the partition function @xmath990 of a system of finite number of sites . we illustrate here how the rule of fusion of environments described in section [ part : directrg ] works more generally . let us consider a system of @xmath991 sites , with charges restricted to @xmath992 and of energy and fugacities given by ( [ zcont ] ) . upon increase of the cutoff one must take into account fusion of the sites @xmath244 and @xmath245 into the site @xmath993 , one finds that : @xmath994 with : @xmath995 } ( w^{(q)}_{1,{\bf r}_{i_2}, .. {\bf r}_{i_q } } + w^{(q)}_{2,{\bf r}_{i_2}, .. {\bf r}_{i_q } } ) \end{gathered}\ ] ] where @xmath996 and @xmath997 } \end{aligned}\ ] ] where @xmath998 has been defined in ( [ zcont ] ) and the charge @xmath999 is located in @xmath1000 . note that @xmath1001 the dipole weight defined in section [ part : directrg1 ] . in ( [ sump ] ) the first term corresponds to the total weight of configurations with no charge in @xmath244 and @xmath245 . the second term correspond to configurations with a dipole in @xmath1002 . its original expression is complicated but simplifies when @xmath244 and @xmath245 are fused into @xmath1000 ( the interaction energy of any given other charge @xmath1003 in @xmath1004 with @xmath1005 becomes opposite to the one with @xmath1006 ( up to higher order terms ) and @xmath1007 simplifies into @xmath1008 to lowest order in @xmath342 as explained in section [ part : directrg ] ) . the last two terms correspond to all configurations with one charge either in @xmath244 or in @xmath245 . the important property with respect to the free energy expansion is that one can factor the term @xmath1009 and rewrite : @xmath1010 } \tilde{w}^{(q)}_{\tilde{{\bf r}},{\bf r}_{i_2}, .. {\bf r}_{i_q } } ) \nonumber\end{aligned}\ ] ] where @xmath1011 as the same definition as @xmath1012 except that the fugacity at @xmath1000 has been modified according to the fusion rule for fugacities ( [ rule2 ] ) . the total factor inside the last logarithm is exactly the partition function @xmath1013 of a system of @xmath1014 sites with the new fugacity given by ( [ rule2 ] ) on the site @xmath1000 . in this appendix we briefly indicate how higher charges can be included in the same ( fusion - diffusion ) formalism and why including them does not affect any of our results . let us consider for instance the charges @xmath1015 and define @xmath1016 and @xmath1017 respectively as the number of component charges @xmath1018 and @xmath1019 in the vector charge @xmath38 . the parametrization ( [ eq : ydef ] ) can be readily extended to encode also for the charges @xmath1015 simply by writing the vector fugacity @xmath65 $ ] as an average over a function @xmath1020 as @xmath1021 = \int_{z_{+},z_{-},z_{++},z_{-- } } \phi(z_{+},z_{-},z_{++},z_{-- } ) z_{+}^{n_{+ } } z_{-}^{n_{- } } z_{++}^{n_{++ } } z_{--}^{n_{-- } } \nonumber\end{aligned}\ ] ] where the random variables @xmath1022 and @xmath1023 represent the random local fugacities for the charges @xmath1015 . similar manipulations as in appendix ( [ part : replimit ] ) lead to a rg equation for a normalized @xmath1024 which we will not write explicitly . it does contain a diffusion operator as well as a fusion term . for illustration we simply give the diffusion operator , expressed using more convenient variables @xmath1025 and @xmath1026 and @xmath1027 . it reads , for the corresponding probability distribution @xmath1028 : @xmath1029 the detailed study of the corresponding rg equation , together with the fusion term will not be reported here . instead let us indicate how the irrelevance of the higher charges can be justified . since one has : @xmath1030 it is clear that the reduced probability @xmath1031 satisfies the same diffusion equation as @xmath193 in ( [ rgeqp ] ) but with the change @xmath1032 and @xmath1033 . it becomes relevant by power counting only at @xmath1034 . thus it is less relevant than @xmath193 in the region of the phase diagram of interest and we can rightly neglect the new fugacities @xmath1022 and @xmath1023 . one can also check that the fusion terms imply that @xmath1035 is of order @xmath1036 when this parameter is small ( since at lowest order the rg equation contains a term proportional to @xmath1037 ) . the rare events which involve the charges @xmath1015 are thus subdominant . to evaluate the integral ( [ sct0.1 ] ) in the screening equation for @xmath8 we consider separately three intervals @xmath1038 , @xmath1039 and @xmath1040 , where @xmath1041 is a number such that @xmath1042 can be replaced by its asymptotic expression for @xmath1043 with good accuracy . only in the middle interval can we use the universal tail expressions for both factors @xmath1044 . thus we have , using symetries : since , remembering that the integral @xmath1047 , the above integral is at most a finite number . the second integral in ( [ cutintwo ] ) is estimated as : @xmath1048 and gives the leading contribution . for @xmath657 it behaves as @xmath1049 and gives the estimate in the text . it is encouraging to note that the leading behaviour precisely originates from the interval where @xmath1050 can be replaced everywhere by its universal asymptotic form . the integral ( [ sct0.2 ] ) in the screening equation for @xmath26 can also be estimated by considering for each variable @xmath180 , @xmath1051 two intervals @xmath1052 and @xmath1053 . the end result is that we find to leading order :

we study the two dimensional xy model with quenched random phases and its coulomb gas formulation . a novel renormalization group ( rg ) method is developed which allows to study perturbatively the glassy low temperature xy phase and the transition at which frozen topological defects ( vortices ) proliferate . this rg approach is constructed both from the replicated coulomb gas and , equivalently without the use of replicas , using the probability distribution of the local disorder ( random defect core energy ) . by taking into account the fusion of environments ( i.e charge fusion in the replicated coulomb gas ) this distribution is shown to obey a kolmogorov s type ( kpp ) non linear rg equation which admits travelling wave solutions and exhibits a freezing phenomenon analogous to glassy freezing in derrida s random energy models . the resulting physical picture is that the distribution of local disorder becomes broad below a freezing temperature and that the transition is controlled by rare favorable regions for the defects , the density of which can be used as the new perturbative parameter . the determination of marginal directions at the disorder induced transition is shown to be related to the well studied front velocity selection problem in the kpp equation and the universality of the novel critical behaviour obtained here to the known universality of the corrections to the front velocity . applications to other two dimensional problems are mentionned at the end .

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Proceed to summarize the following text: the central regions of the dwarf spiral galaxy ngc 4395 show most of the common signatures of nuclear activity . the optical and ultraviolet ( uv ) spectra reveal prominent high - ionization forbidden lines on top of a nearly featureless continuum , and broad wings corresponding to gas velocities in excess of @xmath010@xmath7 km s@xmath8 are detected in the permitted lines ( filippenko & sargent 1989 ) . contrary to the objects of the same kind , the emission - line properties , the optical to x - ray variability pattern and the inferred accretion rate of ngc 4395 are those typical of the seyfert class , of which this source is usually considered to represent the least luminous member . different methods have been employed in the last years to derive the mass of its central black hole : the estimate obtained through reverberation mapping is @xmath9 ( peterson et al . 2005 ) , but the lack in this galaxy of a significant bulge and the stringent upper limit of 30 km s@xmath8 on its velocity dispersion suggest an even lower value , of the order of @xmath010@xmath1010@xmath1 ( filippenko & ho 2003 ) . anyhow , the engine of ngc 4395 falls somewhere between the stellar - mass black holes found in galactic x - ray binaries and the supermassive black holes residing inside active galactic nuclei ( agn ) . as such , it can provide critical information about the relationship between these two populations and the physics of accretion systems in general . + in the light of all these pieces of observational evidence , ngc 4395 is a true scaled - down version of an ordinary seyfert galaxy , the only difference with respect to its high - luminosity counterparts being the much smaller mass of the central black hole . on the other hand , the x - ray observations of this source indicate an unusual spectral hardness at @xmath0210 kev , and suggest a wide range of variations for the intrinsic photon index . the most extreme states ( @xmath11 ; moran et al . 2005 ) are even difficult to interpret within the standard two - phase model ( haardt & maraschi 1991 ) , posing serious questions on the production mechanism of the x - ray emission itself . the presence of undetected absorption effects has been frequently invoked as a possible explanation . indeed , the stronger flux variability characterizing systematically the soft x - ray bands can be attributed to a complex , multi - zone warm absorber , whose properties have been discussed in detail in several works ( iwasawa et al . 2000 ; shih , iwasawa & fabian 2003 ; dewangan et al . here we review the two highest - quality observations of ngc 4395 , performing in both cases an accurate time - resolved analysis mainly focused on neutral absorption , in order to test whether changes of its column density and/or covering factor play a role in the apparent x - ray spectral hardness of this source . the longest _ xmm - newton _ monitoring of ngc 4395 started on 2003 november 30 , for a total duration of @xmath0113 ks . after the subtraction of high - background periods , the useful exposure declines to 91.4 ks . we also take into account the deep _ suzaku _ observation , which was carried out on 2007 june 25 over a span of @xmath0230 ks , corresponding to a net integration time of 101.3 ks . we have followed the standard procedures for the reduction of the event files , and extracted the source and background spectra from circular regions with radii of 30(_xmm - newton _ ) and 2(_suzaku _ ) . in the first case , for the sake of clarity only epic - pn data are presented here and plotted in the figures , even though the mos spectra have been checked throughout and give fully consistent results ; in the second one , instead , only the data from the front - illuminated detectors of the x - ray imaging spectrometer ( xis ) have been examined , after merging the xis0 and xis3 spectra . the spectral analysis has been performed using the xspec v12.6 fitting package . all the uncertainties are given at the 90 per cent confidence level ( @xmath12 ) for the single parameter of interest . we first restricted our analysis to the energies above 1 kev , following the customary approach of fitting the spectrum averaged over the whole observation to obtain a benchmark model . we achieved a fully acceptable fit ( @xmath13 for 1390 degrees of freedom , with no obvious structure in the residuals ) through a simple model consisting of a power law with photon index @xmath14 , a narrow iron emission line with equivalent width @xmath15 ev , and a partial neutral absorber whose column density and covering fraction are @xmath16 @xmath5 and @xmath17 , respectively . @xmath5 ( kalberla et al . 2005 ) . ] no reflection component is strictly required for the continuum : however , by adding a pexrav model ( magdziarz & zdziarski 1995 ) for physical consistency and forcing its strength to match the width of the narrow iron line , the fit is slightly improved ( @xmath18 with the loss of one d.o.f . ) and the power - law photon index steepens to @xmath19 . both cases above yield a formally satisfactory fit , yet the flat slope of the intrinsic continuum deserves a thorough investigation . this result would actually confirm one of the most remarkable features of ngc 4395 , which had been previously caught by _ chandra _ even in harder states ( see moran et al . 2005 ) . in any case , the typical value of @xmath201.5 is still unusually low , compared with both the distribution of photon indices found in local agn ( e.g. bianchi et al . 2009 ) and the average 15150 kev spectrum of the source measured by the burst alert telescope ( bat ) onboard _ swift _ ( @xmath21 ; fig . [ bm ] ) . + a viable explanation for such a discrepancy is the presence of a complex ( variable ) absorber , strongly modifying the observed spectral shape of ngc 4395 below @xmath010 kev . the extrapolation of the basic model down to 0.5 kev supports this working assumption : the apparent extra emission at 0.50.7 kev , in fact , hints at absorption effects in the @xmath00.71.5 kev range rather than at a genuine soft excess like the one detected in a large fraction of seyfert galaxies ( e.g. boller , brandt & fink 1996 ; porquet et al . 2004 ) . moreover , a clearly time - dependent behaviour of the x - ray source is revealed not only by the overall flux light curve but also by the hardness ratio ( hr ) evolution . it is well established that ngc 4395 is characterized by strong variations of the x - ray flux on time - scales as short as a few hundreds of seconds ( fig . [ lc ] , upper panel ) : the fractional rms variability amplitude during the _ xmm - newton _ observation under review is exceptionally high ( @xmath090 per cent at 0.52 kev ; see vaughan et al . 2005 ) . .,width=321 ] on top of these fluctuations of the intrinsic x - ray brightness , mainly related to the workings of the primary source , significant _ spectral _ variations are also evident from the visual inspection of the hr light curve ( fig . [ lc ] , lower panel ) . the observed hr pattern can be accounted for through either changes of the column density or , alternatively , oscillations of the continuum slope : this proves that a time - averaged spectral analysis is not sufficient to fully understand the nature of the x - ray emission of ngc 4395 . the hardness ratio is defined here as the ratio between the 510 kev and the 25 kev flux , thus it is especially sensitive to values of @xmath22 of the order of 10@xmath310@xmath4 @xmath5 . as already mentioned , the spectral complexity and the larger fractional rms variability below @xmath02 kev are likely due instead to a multi - zone warm absorber , whose presence has been revealed since the early _ asca _ observations of the source ( iwasawa et al . 2000 ; shih et al . we anyway note that this is not a critical point , since the hr evolution is only used to select the appropriate intervals for the subsequent time - resolved spectral analysis , which is crucial to validate or dismiss the suggested interpretation . + the shape of the hr light curve reveals two periods of sudden spectral hardening around @xmath025 and 90 ks ; this hints at increased opacity , which would mainly affect the 25 kev band . according to fig . [ lc ] , six different regimes of the hardness ratio can be roughly defined . in this framework , the minor hr fluctuations on very short time - scales can be due to a flickering of the photon index in response to variations of the physical conditions in the disc / corona system , but the statistics is not enough for a complete investigation of these aspects . regardless of this , our aim is to check whether a reasonable configuration of the neutral absorber allows us to recover a steeper intrinsic photon index for the x - ray emission of ngc 4395 . + in order to obtain an adequate description of the entire 0.510 kev spectral range , we included in the reference model above also an apec component for the soft thermal emission ( smith et al . 2001 ) and a two - zone warm absorber , whose complex effects have been already pointed out in several previous works ( see also dewangan et al . 2008 ) . in our analysis we have assumed that the soft emission , warm absorption and reflection features do not vary in the course of a @xmath0100 ks long observation : since the light crossing time over a distance of a gravitational radius ( @xmath23 ) is of the order of one second , the reflection component is expected to change significantly in response to the primary continuum only if the scattering material is located well within @xmath010@xmath24 from the centre ( as in the disc reflection scenarios ; e.g. nardini et al . 2011 ) . similarly , the soft emission likely arises from a very extended region , while the warm absorber , in principle , could lie much closer to the x - ray source ( but see blustin et al . 2005 ) : if this is the case , we are just sampling its average properties . on the other hand , @xmath25 , @xmath22 and @xmath26 were initially left free to evolve among the different intervals . ( 1 ) or @xmath22(2 ) , and plotted respectively in red , green and blue.,width=321 ] in spite of the possible degeneracies , it turns out that the photon index and the column density are subject to very limited variations : in particular , @xmath27 . this suggests that @xmath25 and @xmath22 can be roughly treated as constant parameters as well , hence their values have been tied in all the six spectra : only the covering fraction of the cold absorber is allowed to vary with time . + on sheer statistical grounds , this model already gives a very good fit ( @xmath28 , with @xmath29 and @xmath30 @xmath5 ) , proving that , in first approximation , @xmath26 alone would be able to account for all the observed spectral variability : its qualitative trend across the six intervals , in fact , fairly correlates with the shape of the hr light curve . there are some problems with the reflection efficiency , though . it is not obvious how to assess the latter quantity in the time - resolved analysis , anyway by comparing the pexrav amplitude to the highest flux level of the power - law continuum we obtain a reflection strength of @xmath31 , which is completely inconsistent with the moderate equivalent width of the companion iron line ( george & fabian 1991 ) . moreover , when extrapolated at higher energies , such component would give rise to a very flat spectrum with a prominent compton hump at @xmath030 kev , which is not detected by _ swift_/bat . given that the time - scales are extremely different , it is still possible that the source is caught in a state of large reflection during the _ xmm - newton _ observation , but in this case the fluorescent iron line should be much stronger as well . forcing @xmath32 to have a standard value delivers a poor spectral description , and no improvement is achieved through the self - consistent reflionx table models ( ross & fabian 2005 ) . + as a consequence , we attempted to introduce a second partial covering cold absorber , assuming again that @xmath25 and @xmath22(1,2 ) are constant , and @xmath26(1,2 ) variable . the best fit quality is significantly improved ( @xmath33 ) , while the photon index @xmath34 is still in excellent agreement with the average high - energy slope of the source . the presence of an additional absorber with column density of @xmath010@xmath4 @xmath5 removes all the previous limitations , reducing substantially the required strength of the reflection continuum ( now @xmath35 ) . this is also clear from fig . [ sc ] , where the different model components are disentangled to display their relative contribution in a single time segment . the basic parameters of this final model are listed in table [ t1 ] , while the six individual spectra are shown in fig . [ xmm ] . . the data are rebinned for plotting purposes only.,width=321 ] .main parameters of the _ xmm - newton _ and _ suzaku _ best - fitting models , common to all the intervals on which our time - resolved spectral analysis has been performed . @xmath25 : photon index ; @xmath36 : temperature of the soft emission in ev ; @xmath37 , @xmath38 : column density in @xmath5 and ionization parameter in erg cm s@xmath8 of the warm absorption components ; ew@xmath39 : time - averaged equivalent width of the 6.4-kev iron line in ev ; @xmath32 : strength of the reflected continuum ; @xmath22 : column density of the neutral absorbers in @xmath5 ; @xmath40 , @xmath41 : average observed and intrinsic 0.510 kev flux in erg @xmath5 s@xmath8 . [ cols="<,^,^ " , ] the cold absorption model provides a good interpretation of the spectral variability of the x - ray source during both the _ xmm - newton _ and _ suzaku _ observations . moreover , considering the entries of table [ t1 ] , the basic physical quantities appear to be in fair agreement . this is a strong confirmation of the validity of the scenario explored in this work , where the changes are driven by the evolution of the covering fractions . the @xmath26(1 ) and @xmath26(2 ) progression is listed in table [ t2 ] . due to the smaller uncertainties , we focus our discussion on the _ xmm - newton _ case . we first point out that our model is defined in such a way that mutually exclusive regions of the x - ray source are affected by the two cold absorbers : in other words , we only consider a single - layer configuration of the neutral gas along the line of sight . all the @xmath26 sequences should be taken with some caution , though . the low ionization stage in one of the warm absorption component impairs to some extent the determination of @xmath26(1 ) , while possible changes in the reflection strength on that of @xmath26(2 ) . however , even from the face values obtained under our assumptions , some interesting considerations can be drawn . it is now @xmath42 that shows a tighter correlation with the hr light curve : the peaks during periods 2 and 5 might be explained in terms of _ eclipses _ from individual _ clouds _ ( e.g. lamer , uttley & mchardy 2003 ; risaliti et al . given that the duration of such events is @xmath010 ks , a typical dimension of the x - ray source of @xmath010@xmath43 corresponds to a transverse velocity of the putative clouds of @xmath010@xmath7 km s@xmath8 , placing the obscuring gas at the broad - line region scale . the size and the shape of these blobs are not precisely known . yet , our time - resolved analysis suggests that different column densities are present along the line of sight at the same time . the range of variations in @xmath26 implies that the number of intervening clouds is very limited ( a few at most , see fig [ blr ] ) , and that their size is comparable to the dimensions of the x - ray source . the extreme case entails a single irregular and inhomogeneous absorber . + the physical situation is then expected to be rather complex . in this view , the x - ray absorbers that we have included should be only regarded as a linear combination of the _ real _ ones , whose exact geometrical structure can not be probed with the present data quality . the large disparity ( roughly an order of magnitude ) between the column densities involved , and specifically the fact that @xmath44 , hints at a multi - layer configuration of the cold absorber , where the two phases have a different location and are superimposed in part on one another when seen in projection . such a scheme can be envisaged as in fig . [ abs ] : only the broad - line region component , with @xmath45 @xmath5 , has a time - dependent covering fraction ( @xmath460.8 ) . conversely , the absorption system with lower column density is constant ( or at least variable over much longer scales ) , and can be associated with a more distant gaseous component , within a narrow - line or torus - like region . indeed , this would be consistent with both the optical classification of ngc 4395 as a type 1.51.8 seyfert galaxy ( ho et al . 1997 ; panessa et al . 2006 ) and the large covering factor of the emission - line regions with respect to the central source ( kraemer et al . the broad- and narrow - line regions are expected to be responsible for some of the uv to x - ray absorption detected in ngc 4395 ( crenshaw et al . 2004 ) , and evidence for the identification of the rapidly variable x - ray absorber with the broad - line emitting clouds have been recently found in other seyfert galaxies ( e.g. maiolino et al . 2010 ; risaliti et al . 2011 ) . in this perspective , the frequency and amplitude of the variations in @xmath22 and/or @xmath26 would be linked to the degree of clumpiness of the circumnuclear environment at the different scales . + once the _ suzaku _ data are taken into account , all the considerations made above still hold , even though the uncertainties on the key parameters are quite large , and the smaller difference between @xmath22(1 ) and @xmath22(2 ) makes the covering fraction patterns less meaningful . apart from the speculations on the physical and geometrical structure of the cold absorption system , the scenario outlined in this work has the other great advantage of strengthening the correlation between ngc 4395 and the high - luminosity agn population . first , it gives reasons for the x - ray spectral hardness of this source , reconciling the estimate of its photon index with the typical values found among agn ; secondly , neutral absorption variability systematically occurs in a significant fraction of active galaxies , the prototypical case being another type 1.8 seyfert , ngc 1365 ( e.g. risaliti et al . our findings then represent a further point of contact between ngc 4395 and its more massive and luminous counterparts . incidentally , it is also worth noting that the intrinsic 0.510 kev emission of ngc 4395 implied in this cold absorption scheme is larger than the observed one by just a factor of @xmath02 ( table [ t1 ] ) . taking advantage of the _ swift_/bat spectral constraints , the resulting 0.5100 kev luminosity is @xmath06@xmath47 erg s@xmath8 , which exceeds the usual estimates of the bolometric luminosity ( see also moran et al . 2005 ; iwasawa et al . depending of the exact value of the black hole mass , the eddington ratio of ngc 4395 could be much closer to @xmath00.01 than previously thought . . the number of clouds simultaneously crossing the line of sight and their size are limited by the rapid changes of @xmath26 . even a single blob with irregular shape and non - uniform physical properties is consistent with the observed variability pattern . ( the observer is located towards the bottom of the page).,width=321 ] , but an alternative configuration is assumed : the absorption component with lower column density has a constant covering fraction of @xmath48 and is external to the system of clouds responsible of the x - ray spectral variability ( see the discussion in the text).,width=321 ] we have discussed a possible interpretation of the x - ray spectral hardness usually observed in the low - luminosity active galaxy ngc 4395 . this source harbours in its centre a black hole with estimated mass of @xmath010@xmath1 , and it is one of the few known objects whose study can shed light on the links between the physics of accretion processes in galactic black hole binaries and agn . in spite of being a genuine seyfert galaxy in many respects , ngc 4395 remains a somewhat puzzling source because of the inferred flatness of its primary x - ray continuum . the existence of complex absorption effects has often been proposed as a likely explanation . here we have provided for the first time an example of these effects based on observational evidence , by reviewing the two highest - quality looks of ngc 4395 taken by _ xmm - newton _ and _ suzaku _ : in both cases , a time - resolved analysis shows that the spectral evolution of the source can be interpreted by means of a two - phase neutral absorber with variable covering factor . as a first approximation , this cold absorber is identified with the system of broad - line clouds , allowing for a double - peaked @xmath22 distribution . the low column density component can otherwise be attributed to an external narrow - line or torus - like region , with nearly constant @xmath26 . this is presumably an oversimplification of the physical and geometrical structure of the circumnuclear environment ( which is known to comprise also a complex , multi - zone warm absorber ) , but even the existence of a single partial - covering cold screen can not be completely ruled out on statistical grounds . + we stress that the absorption variability scenario presented here is not unique , and different models based on intrinsically flat x - ray continua ( with significant changes of either the photon index or the reflection strength ) can describe the observed behaviour of this source equally well . this interpretation , however , fits into the analogy between the properties of ngc 4395 and those of _ standard _ high - luminosity seyfert galaxies , many of which are systematically affected by x - ray absorption variability due to the clumpiness of their circumnuclear regions . moreover , it allows us to retrieve in a natural way a @xmath49 photon index below 10 kev , in perfect agreement with the _ swift_/bat spectral slope and the usual values measured among agn . no alternative explanation would then be required for the intrinsic 210 kev flatness of ngc 4395 . this work has been partly supported by nasa grant nnx08an48 g . we thank the anonymous referee for constructive and useful comments which significantly improved the content of this paper . bianchi s. , guainazzi m. , matt g. , fonseca bonilla n. , ponti g. , 2009 , a&a , 495 , 421 blustin a. j. , page m. j. , fuerst s. v. , branduardi - 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ciamponi g. , 2011 , mnras , 410 , 1027 ross r. r. , fabian a. c. , 2005 , mnras , 358 , 211 shih d. c. , iwasawa k. , fabian a. c. , 2003 , mnras , 341 , 973 smith r. k. , brickhouse n. s. , liedahl d. a. , raymond j. c. , 2001 , apj , 556 , l91 vaughan s. , iwasawa k. , fabian a. c. , hayashida k. , 2005 , mnras , 356 , 524

we present a new x - ray analysis of the dwarf seyfert galaxy ngc 4395 , based on two archival _ xmm - newton _ and _ suzaku _ observations . this source is well known for a series of remarkable properties : one of the smallest estimated black hole masses among active galactic nuclei ( of the order of @xmath010@xmath1 ) , intense flux variability on very short time - scales ( a few tens of seconds ) , an unusually flat x - ray continuum ( @xmath2 over the 210 kev energy range ) . ngc 4395 is also characterized by significant variations of the x - ray spectral shape , and here we show that such behaviour can be explained through the partial occultation by circumnuclear cold absorbers with column densities of @xmath010@xmath310@xmath4 @xmath5 . in this scenario , the primary x - ray emission is best reproduced by means of a power law with a standard @xmath6 photon index , consistent with both the spectral slope observed at higher energies and the values typical of local agn . [ firstpage ] galaxies : active galaxies : individual : ngc 4395 x - rays : galaxies .

You are an expert at summarizing long articles. Proceed to summarize the following text: service - oriented architecture in wsns@xcite makes it possible to rapidly develop new applications . in a service - oriented wsn , a typical application requires several different services , e.g. , data aggregation , data processing , decoding , which are provided by service providers that are also sensors . the task of service composition is to assign each required service to an appropriate service provider according to certain criteria . service composition with various performance metrics @xcite , e.g. , load balance , end - to - end delay and resource , have been well studied . service composition in wsns has also recently been studied in @xcite . @xcite studies the minimum - cost service placement based on service composition graphs with a tree structure . @xcite considers the optimal placement of filters ( services ) with different selectivity rates . habitat and environmental monitoring represent a class of wsn applications . the queries in such applications in general are _ persistent _ ( or _ recurrent _ ) queries which need to be processed repetitively with a given frequency for a given duration @xcite , e.g. , an application requests receiving images in which the monitored area is dimly lit from 9:00am to 5:00pm@xcite . three services are required in such a persistent query : service @xmath0 checking for dim images , service @xmath1 checking for `` sufficient '' motion between successive frames , and service @xmath2 fusing the identified motions(e.g . , the appearance of a suspect ) . in a service - oriented wsn , such services are provided by sensor nodes in the network , thus , in - network processing is feasible , to reduce the possibly massive amounts of raw data . in wsns , energy consumption is a critical issue and sleep scheduling has been well studied as a conservative approach to save energy @xcite . when a node is in sleep mode , all its provided services are not available , which may cause disruption to service composition . @xcite studied a cross - layer sleep scheduling design in a service - oriented wsn which considers system requirement on the number of active service providers for each service at any time interval . in a service - oriented wsn , a query routing procedure which routes requesting services towards service providers is necessary . for a persistent query , the query routing procedure might need to be conducted many times during its lifetime due to the sleep schedule in the mac layer , which might introduce more energy consumption . take the query that starts at 09:00am and ends at 5:00pm with a frequency of 100s as an example . in fig . [ example1](a ) , at 09:00am , a path is chosen to provide the requested services , while after 100s one of the sensors in this path switches into sleep mode , which results in unavailability of the service composition path . it is necessary to conduct the query routing procedure again to find a new service composition path as shown in fig . [ example1](b ) . in this paper , we aim to use the minimum number of service composition solutions during a persistent query s lifetime such that the energy consumption caused by repetitively conducting query routing procedure is minimized . once the minimum number of required service composition solutions is derived , we then select the service composition solutions with minimum transmission cost . the contribution of the proposed work is summarized as follows : * we propose a service - oriented query routing protocol . traditional routing in wsns only involves finding a path from source sensors to a sink . service - oriented query routing protocol needs to ensure that the path from source sensors to the sink includes service providers , which imposes new challenges to routing in wsns . * we propose an optimal greedy algorithm to minimize the number of required service composition solutions during a persistent query s lifetime , which can minimize the energy consumption caused by conducting the service - oriented query routing protocols . * we propose a dynamic programming algorithm to minimize the total service composition cost which aims to reduce the transmission cost in executing a query . in a service - oriented wsn , width=240 ] the rest of the paper is organized as follows . the network architecture and problem definition are given in section [ network ] and [ prob ] respectively . the algorithms and simulation results are presented in section [ algorithm ] and [ simulation ] respectively . we conclude the paper in section [ conclusion ] . in our network architecture , the service providers form a service provider overlay network as shown in fig . [ model ] . two service providers in the service provider overlay network may be multiple hops away from each other and the communication between them can be a multi - hop communication in the same wsn or through existing 802.11 wlan . the service - oriented architecture at the sink has the following three layers : * _ service composition query layer_. this layer maps an application s query into a _ service composition query _ which specifies required services and their invocation order . for example , the aforementioned query will be converted to a service composition query with services @xmath0 , @xmath1 and @xmath2 . * _ service layer_. this layer has the service information provided by the sensors in service provider overlay network . we also assume that service layer has the sleep schedule information of service providers in service provider overlay network . * _ service composition layer_. this layer finds the service composition solutions for service composition queries , which is the problem to be studied in this paper . for a persistent query , the service composition layer may find several service composition solutions during its lifetime since some service composition solutions may not always be feasible due to sleep schedule . the service composition solutions are maintained in a _ service composition table _ as shown in fig . [ model ] . the service composition solution only specifies a service provider for each required service in a service composition query . once the service composition solutions are identified , a routing protocol is invoked to find paths from source sensors to the first service provider in the service composition solution and find paths between any two adjacent service providers . in this paper , we propose the following service - oriented query routing protocol : * the sink broadcasts a _ service composition query routing _ message which includes service composition solution , duration , and interest . such a message will reach all service providers in service provider overlay network . * upon receiving a _ service composition query routing _ message , if a service provider is the first service provider in the service composition solution , it will broadcast the interest to the sensor network . source sensors can then send the data to the first service provider using any data - driven routing protocol in wsns . thus , service composition is transparent to source sensors . * upon receiving a _ service composition query routing _ message , if a service provider is in the service composition solution but not the first service provider , it needs to find a path to its upstream service provider in the service composition solution . this can be done by any routing protocol in wsns . during the lifetime of a persistent query , it may be necessary to switch the service composition solutions due to the sleep schedule of service providers . the service - oriented query routing protocol needs to be conducted again when the service composition solution changes , which consumes more energy . the rest of the paper focuses on the service composition with minimum cost to avoid the frequent change of service composition solutions during a persistent query s lifetime . notice that the service - oriented query routing protocol is a distributed routing protocol . the sink only generates the service composition solutions which determines an appropriate service provider for each required service . to make such a decision , the sink only needs to maintain the services availability and the sleep schedule information of each service provider . in a large - scale wsn , service providers are only a small portion of the whole network . we believe that maintaining such information at the sink is well - paid when the duration of a persistent query is long . let @xmath3 be a persistent service composition query and @xmath4 be service providers . let @xmath5 be the set of services that sensor @xmath6 can provide and @xmath7 be the set of sensors that can provide service @xmath8 . fig [ example2](a ) shows the service availability at the service layer . given the duration @xmath9 and the frequency @xmath10 of a persistent query , the query should be executed for @xmath11 times during its duration @xmath9 and we assume that @xmath11 is an integer . let @xmath12 be the start time of @xmath13-th execution of the persistent query where @xmath14 . given the sleep schedule information of the service providers at the service layer , the sink can derive each service provider s availability at @xmath12 . let @xmath15 be 1 if service provider @xmath6 is active at @xmath12 , otherwise , set @xmath15 be 0 . figure [ example2](b ) gives the service provider availability at the service layer . with the service availability and the service provider availability information , the service composition layer can derive a service composition solution at @xmath12 for @xmath16 . as shown in fig . [ example2](c ) , the service composition solution @xmath17 is valid at @xmath18 and @xmath19 , @xmath20 is valid at @xmath21 and @xmath22 and so on . during this persistent query s lifetime , 4 service composition solutions are required and thus the service - oriented query routing protocol needs to be conducted 4 times , which consumes energy . this paper aims to minimize the number of service composition solutions for a persistent query . in a service - oriented wsn , width=268 ] let @xmath23 be 1 if the service composition solution at @xmath12 is different from that at @xmath24 , otherwise let @xmath23 be 0 . then @xmath25 represents total number of service composition solutions during a persistent query s lifetime , which needs to be minimized . under such an objective , a service composition solution may be used continuously in order to reduce the energy consumed by frequently invoking service - oriented query routing protocol . although some service providers may be used continuously , this will not decrease the longevity of network . since if a service provider is to be active , it has to provide services for the system according to sleep scheduling . though the service - oriented query routing procedure is the major source of energy consumption , the transmission of the data from the source sensor to the sink also consumes energy . two service providers in the service provider overlay network may be multiple hops away and if the communication between them is through the same service - oriented wsn , relay sensors may also be in sleep mode . thus , even a service composition solution can be used continuously over multiple executions , a local routing discovery procedure may be invoked between two service providers due to the sleep scheduling . we use average transmission cost between two service providers to characterize such energy consumption caused by the local routing discovery between two service providers . besides minimizing the number of service composition solutions during a persistent query s lifetime , it is also important to minimize total transmission cost . in fig . [ compositioncost ] , there are two sets of service composition solutions for a persistent query and both include 4 service composition solutions during the persistent query s lifetime . thus , these two sets of service composition solutions consumes the same energy caused by service - oriented query routing procedure . in the first set , @xmath26 will be used for 3 times , twice , 3 times and twice respectively with a total cost of 184 . in the second set , @xmath26 will be used for twice , 3 times , 3 times and twice respectively with a total cost of 174 . thus , the second set of service composition solutions will be more energy efficient . in this paper , firstly , we aim to minimize the number of service composition solutions during a persistent query s lifetime . such a problem is referred to as problem * p1*. secondly , we need to minimize the total cost of the service composition solutions . such a problem is referred to as problem * p2*. [ algorithm ] in this section , we first approach problem * p1*. then based on the result of * p1 * , we approach the second problem * p2*. let @xmath27 be the number of executions that service provider @xmath6 can be continuously available from @xmath13-th execution ( including at @xmath13-th execution ) . for example , if @xmath6 s availability at all execution instances of a persistent query is given as @xmath28 , @xmath29 is 2 since @xmath6 can be available at @xmath30st and @xmath31nd execution , @xmath32 is 0 as @xmath6 is not available at @xmath33rd execution . the greedy algorithm which is shown in algorithm . [ greedy ] is always to select the service provider with maximum @xmath27 for each @xmath8 in @xmath13-th execution such that the solution can be continuously used for the maximum number of times . after the service composition solution is determined for @xmath13-th execution , the number of times that this solution can be used is determined by the minimum @xmath27 among all selected service providers . let @xmath34 be the set of selected service providers for @xmath13-th execution and @xmath35 be the number of times that @xmath36-th service composition solution can be continuously used . the worst case running time of this greedy algorithm is o(@xmath37 ) . @xmath38 gives the minimum number of service composition solutions during a persistent query s lifetime . we now prove the optimality of the greedy algorithm . let @xmath25 be the solution obtained from the greedy algorithm where @xmath39 . let @xmath40 be an optimal solution where @xmath41 . [ lemma ] for any sequence @xmath42 and @xmath43 where @xmath44 , there must always exists @xmath45 . we use induction to prove this lemma . firstly , for @xmath46 , it is obvious that @xmath47 . when @xmath48 , as greedy algorithm always selects the provider with maximum @xmath27 for each service , the value of @xmath49 must be no less than 0 , so @xmath50 . assume that when @xmath51 we have @xmath52 . for @xmath53 , in the given optimal solution , there is a service composition solution which can be continuously used from @xmath54 to @xmath55 . if @xmath56 , then we have @xmath57 ; if @xmath58 , then we must have @xmath59 since the greedy algorithm always selects the service providers which can provide longest continuous services . in both cases , we have @xmath60 . thus , lemma holds when @xmath53 . [ theorem1]@xmath61 , the solution obtained from the greedy algorithm , must be optimal . we prove it by contradiction . assume that there exists @xmath62 , then @xmath63 . according to lemma 1 , we also have @xmath64 . the relationship among @xmath65 , @xmath66 , and @xmath67 is shown in fig . [ compare ] . and @xmath68,height=57 ] [ compare ] fig . [ compare ] denotes that there exists a service composition solution which can cover executions from @xmath67 to @xmath65 . according to our greedy algorithm , we can find a solution which can be continuously used from @xmath67-th execution to the last execution . thus @xmath69 , which conflicts the assumption . in the following , we approach problem * p2 * which minimizes the total routing cost based on the result of problem * p1*. let @xmath70 be the set of feasible service composition solutions at @xmath13-th execution which can be continuously used for the following @xmath71 executions . for any service composition solution @xmath72 , let @xmath73 be transmission cost if @xmath74 is selected to be executed once . let @xmath75 . @xmath76 can be obtained by finding a shortest path in an auxiliary graph @xmath77 which is constructed as follows : * @xmath78 is the set of nodes consisting of @xmath79 layers @xmath80 and the @xmath81 layer @xmath82 contains all service providers which can continuously provide @xmath8 from @xmath13-th execution to @xmath83-th execution , e.g. , if @xmath6 can provide @xmath8 and it is available from @xmath13-th execution to @xmath83-th execution , node @xmath84 . * let @xmath85 be the link set such that there is a direct link @xmath86 whenever @xmath87 and @xmath88 for @xmath89 . the cost of @xmath90 is the shortest path cost from @xmath6 to @xmath91 in the physical network . * add two special nodes @xmath92 and @xmath93 such that @xmath94 is the @xmath95 layer and @xmath96 is the @xmath97 layer . link @xmath92 to each node in @xmath98 and link each node in @xmath99 to @xmath93 with cost 0 . let @xmath100 be the minimum total cost from @xmath13-th execution to the last execution if @xmath36-th service composition solution starts at @xmath13-th execution . then we have the following recursion : @xmath101 where @xmath102 . we have the following boundary condition : @xmath103 for @xmath104 . the dynamic programming is given in algorithm . [ dynamic ] in which @xmath105 is the minimum total cost for the persistent query and @xmath106 $ ] stores @xmath36-th service composition solution . the time complexity of the algorithm is @xmath107 . in this section , we first introduce the design of our simulation . the number of service providers @xmath108 of each service is randomly generated between @xmath109 $ ] . we then randomly generate @xmath108 service providers for @xmath8 from @xmath110 . for each @xmath6 , we also randomly generate its availability at each execution . then we validate whether each @xmath8 can be provided by at least one active service provider at each execution . if infeasible , the instance is dropped from our simulation . to compare the performance of our algorithms , we also introduce a baseline algorithm named _ min - cost - based _ algorithm which aims to select the service composition solution with minimum transmission cost for each execution . we compare the number of service composition solutions during a persistent query s lifetime and the total transmission cost of our algorithms with _ min - cost - based _ algorithm respectively . [ simulation ] , height=153 ] in the first set of experiments , we evaluate the performance of algorithms by varying @xmath11 in @xmath111 for @xmath112 . the effectiveness of greedy algorithm for * p1 * is tested by comparing with _ min - cost - based _ algorithm . as shown in fig . [ solutions ] , the number of service composition solutions during persistent query s lifetime in greedy algorithm is much less than that in _ min - cost - based _ algorithm . for example , with @xmath113 , the number of service composition solutions in our greedy algorithm is only 21 while it is 32 in _ min - cost - based _ algorithm . the difference between two algorithms increases with the number of query s executions , which demonstrates the effectiveness and scalability of our work . [ cost1 ] illustrates that total service composition cost obtained from dynamic programming based on the result of the greedy algorithm is higher than that obtained from _ min - cost - based _ algorithm . as we explained in section [ prob ] , the energy consumed in service - oriented query routing protocol is much higher than that in conducting service composition . thus , though the solution obtained from our algorithms may consume more energy in the service composition phase , it consumes much less energy in service - oriented query routing phase which is the major energy consumption source in a persistent query . in the second set of our experiments , we study in detail the impact of the number of required services on the total service composition cost and the impact of the number of service providers on the total service composition cost . we have selected three scenarios @xmath114 , @xmath115 , @xmath116 by varying @xmath79 in @xmath117 $ ] . as shown in fig . [ cost2 ] , the total service composition cost increases with the number of required services @xmath79 since more service providers may be involved in a service composition . given @xmath79 , the service composition cost is lower in a network with more service providers . in a network with more service providers , more feasible service composition solutions are possible and our dynamic programming algorithm can find the service composition solution with minimum cost . , [ conclusion ] this paper studies service composition in service - oriented wsns with persistent queries . we aim to provide service composition solutions during a persistent query s lifetime such that the involved routing update cost and transmission cost is minimized . the optimality of greedy algorithm and dynamic programming provides the service composition solutions for persistent queries with the minimum energy consumption . r. ha , p .- h . ho and x.s . shen , cross - layer application - specific wireless sensor network design with single - channel csma mac over sense - sleep trees " , accepted by elsevier journal : computer communications .

service - oriented wireless sensor network(wsn ) has been recently proposed as an architecture to rapidly develop applications in wsns . in wsns , a query task may require a set of services and may be carried out repetitively with a given frequency during its lifetime . a service composition solution shall be provided for each execution of such a persistent query task . due to the energy saving strategy , some sensors may be scheduled to be in sleep mode periodically . thus , a service composition solution may not always be valid during the lifetime of a persistent query . when a query task needs to be conducted over a new service composition solution , a routing update procedure is involved which consumes energy . in this paper , we study service composition design which minimizes the number of service composition solutions during the lifetime of a persistent query . we also aim to minimize the total service composition cost when the minimum number of required service composition solutions is derived . a greedy algorithm and a dynamic programming algorithm are proposed to complete these two objectives respectively . the optimality of both algorithms provides the service composition solutions for a persistent query with minimum energy consumption . * keywords : * service composition , wireless sensor network , routing , query .

You are an expert at summarizing long articles. Proceed to summarize the following text: cold dark matter ( cdm ) theory successfully describes many aspects of the formation of large - scale structure in the universe @xcite . however , mismatches do exist between its predictions and observations , such as the cusp - core controversy , missing satellites @xcite and the angular momentum problem @xcite . in particular , the cusp - core issue has provoked much debate . cdm simulations consistently yield density profiles with steeper inner slopes ( power - law exponent between @xmath71 and @xmath71.5 ) than observational studies which have found a range of slopes , including constant density cores in dark matter dominated low surface brightness galaxies @xcite and shallow slopes in clusters with gravitationally lensed arcs @xcite . these results , among others , have initiated discussion about the role baryons play in softening simulated cores @xcite and the observational effects that may mask cusps in low surface brightness galaxies @xcite . recent @xmath0-body results demonstrate that , at the current resolution of simulations , the central power - law exponent does not converge to a universal value @xcite . in numerical simulations , a softened gravitational potential is used to prevent the macro - particles ( 10@xmath810@xmath9 ) from experiencing artificially strong two - body interactions ( * ? ? ? * for example ) . the softening length , @xmath2 , is chosen to maximise the resolution while suppressing two - body effects over the simulation running time in view of the ongoing disagreement regarding the form of @xmath3 , it is important to clarify _ analytically _ , by a code - independent argument , whether the choice of @xmath2 affects the physics of the system and hence @xmath3 . in this paper , we employ the framework of statistical mechanics @xcite , drawing upon recent results on phase transitions in @xmath0-body systems with attractive power - law potentials @xcite . self - gravitating particles behave qualitatively differently to many other statistical systems because gravity is an unscreened , long - range force . they are best examined within the microcanonical ensemble , where the energy and number of particles are fixed and phases with negative specific heat are allowed . we apply the results of the classical theory of self - gravitating , @xmath0-body systems to demonstrate the effects of introducing a short distance cutoff in numerical simulations , in particular the effect on stability . the study of the thermal stability of self - gravitating systems has a long history . @xcite showed that spherical systems of point particles in a box with reflecting walls are gravitationally unstable below a critical temperature , collapsing catastrophically to a central point . @xcite generalised this work to a spherical system of @xmath0 classical hard spheres in contact with a heat bath , showing the gravothermal instability to be a general feature of self - gravitating systems held at a constant temperature . @xcite , investigating point fermions obeying the pauli exclusion principle , showed that a _ stable _ low temperature phase can exist if the gravitational potential is softened , transforming the gravothermal instability to a phase transition to the low temperature phase . in this paper , we extend these results and use them to reinterpret some of the ambiguous results of numerical simulations of cdm haloes discussed above . a detailed comparison with preceding analytic work is presented in section [ comparison ] . section 2 briefly reviews the formalism for treating @xmath0 self - gravitating , collisionless particles statistically . in section 3 , we apply the formalism to compute @xmath3 analytically as a function of @xmath1 , the total energy , and @xmath2 . the result is a thermodynamic phase diagram that contains both collapsed and extended haloes . in section 4 , we locate published @xmath0-body simulations on the phase diagram and show they are biased towards the collapsed phase . this phase is unstable for @xmath10 , suggesting that collapsed haloes are an artificial by - product of the softened potential ; there is no immediate reason to expect agreement between simulated and observed profiles unless the gravitational potential is appreciably softened in nature . we emphasise at the outset that it is not our intention to reproduce realistic cdm haloes with non - zero angular momentum and hierarchical clustering ; rather , we demonstrate in a code - independent manner how the softening used in @xmath0-body simulations may artificially alter the density profiles found . the properties ( e.g. energy , entropy ) and collective behaviour ( e.g. gravothermal catastrophe ) of a self - gravitating gas of cdm particles in thermodynamic equilibrium take different values when computed in different statistical ensembles because the long - range nature of the gravitational potential renders the system inseparable from its environment @xcite . in this paper , we follow previous studies by considering the self - gravitating gas in the microcanonical ensemble ( mce ) , whose features are constant energy , volume and particle number . particles do not evaporate from the system over time and the walls of the container are perfectly reflecting . the mce is more appropriate than the canonical ensemble ( ce ) for three reasons : ( i ) it is unclear how to construct an external heat bath ( required by the ce ) for a long - range potential , because the system interferes with the environment @xcite ; ( ii ) states with negative specific heat are inaccessible in the ce @xcite ; and ( iii ) the equilibrium density profile in the violently relaxed ( smoluchowski ) limit is the singular isothermal sphere in the ce , contrary to observations @xcite . the density of states , @xmath11 , is the volume of the ( 6@xmath121)-dimensional surface of constant energy @xmath1 in phase space ( @xmath13 ) , where ( @xmath14 ) are the co - ordinates and momenta of the @xmath15-th particle . at any one moment , the system occupies one point in the 6@xmath0-dimensional phase space . for particles of equal mass @xmath16 , one has @xmath17d^{3n}pd^{3n}x,\ ] ] where the first and second sums give the kinetic and potential energy , and the integral is over phase space volume . the gravitational potential , @xmath18 , is given by @xmath19 or , if the potential is artificially softened over a characteristic length @xmath20 , by @xmath21^{-1/2}$ ] . the thermodynamic entropy @xmath22 ( up to a constant ) and the temperature @xmath23 of the system are defined in terms of @xmath11 : @xmath24 @xmath25 these quantities are hard to interpret when assigned to a system far from equilibrium . note that @xmath11 diverges for @xmath26 and @xmath27 ; any two particles can be brought arbitrarily close together , liberating an infinite amount of potential energy , so that the co - ordinate space integral diverges @xcite . this is a serious problem because it is impossible to achieve thermodynamic equilibrium if @xmath11 diverges ; the system does not have time to sample the infinite number of possible microstates with equal probability @xcite . if the dark matter particles are fermions , the pauli exclusion principle does prevent this problem . however , the fraction of the phase space volume sampled by @xmath0 mildly relativistic cdm particles in a time @xmath28 , given by @xmath29 , is exceedingly small for most proposed cdm particles , e.g. @xmath30m@xmath31/gev for self - interacting dark matter @xcite . the density of states is evaluated in the continuum ( mean - field ) limit by integrating over momentum and then expressing the remaining configurations as a functional integral over possible density profiles @xmath32 @xcite , @xmath33,\ ] ] where the effective dimensionless action @xmath34 and dimensionless energy @xmath35 are quantities defined by @xcite . in ( [ functional ] ) and ( [ action ] ) , and throughout the remainder of this paper , the density profile and position co - ordinates @xmath36 are written as dimensionless quantities , relative to the total mass @xmath37 and outer radius @xmath38 of the system , with @xmath39 and @xmath40 . upon evaluating the functional integral by a saddle point method ( which involves extremising the action ) , ( [ functional ] ) reduces to three coupled integral equations describing the density profile @xmath32 , the central density @xmath41 and the inverse temperature @xmath42 . for a newtonian potential , one has @xmath43 , \\\label{coupled2 } & & \frac{1}{\rho_0}=\int^{1}_{0 } 4{\pi}{x^2_2 } \exp\left[\frac{2\pi\beta}{x_2}\int^{1}_{0}\rho(x_1)x_1(|x_2+x_1|-|x_2-x_1|){dx_1}\right ] dx_2 , \\\label{coupled3 } & & \frac{3}{2\beta}=\xi + 4\pi^2\int^{1}_{0}\int^{1}_{0}\rho(x_1)\rho(x_2)x_1x_2(|x_1+x_2|-|x_1-x_2|)dx_1dx_2.\end{aligned}\ ] ] note that the factor @xmath44 in ( [ coupled2 ] ) was omitted due to a typographical error by @xcite . the solutions to these equations describe the density profile , entropy and temperature of an equilibrium system for a given energy . to obtain analogous equations for the softened potential , we replace @xmath45 everywhere with @xmath46^{1/2}$ ] . this extension is valid in the mean - field formalism for @xmath2 small : correction terms are @xmath47 @xcite . the mean - field limit is only meaningful physically for @xmath48 , otherwise @xmath11 diverges . @xcite verified the mean - field results against monte - carlo simulations and an alternative analytic method known as the mayer cluster expansion , where the density of states is expanded as a combinatorial series in the dilute limit ( @xmath49 ) . the different approaches are in accord in the dilute limit . in order to verify the mean - field approach in the high - density limit ( @xmath50 ) , relevant to the extended and collapsed phases studied here , we need to calculate the correction terms in this limit , following @xcite a project outside the scope of this paper . nevertheless , to give a rough idea of these corrections , we note ( by analogy ) that they are of order @xmath51 in the ce , where @xmath52 in the ce is a proxy for @xmath53 in the mce . note that the classical thermodynamic limit ( @xmath54 constant as @xmath55 ) does not apply for gravitating systems ; thermodynamic quantities are finite if proportional to @xmath56 as @xmath55 . we solve ( [ coupled])([coupled3 ] ) for the radial density profile @xmath32 by the following iterative relaxation scheme @xcite : given the current iterate of the profile , @xmath57 , apply ( [ coupled3 ] ) , ( [ coupled2 ] ) and ( [ coupled ] ) to compute @xmath58 , @xmath59 and @xmath60 in that order , then apply @xmath61 until the convergence criterion @xmath62 is satisfied . we typically adopt @xmath63 and @xmath64 ( @xmath65 ) in this work . the softening can be introduced into this scheme in two ways : ( i ) as a nonzero lower limit of integration in the integrals in ( [ coupled])([coupled3 ] ) ; and ( ii ) in the potential , @xmath66^{-1/2}$ ] . both approaches were tested and found to produce qualitatively similar behaviour ; we concentrate on the latter in this work as it is more closely allied to @xmath0-body simulations . with no softening present in the gravitational potential , a stable solution of ( [ coupled])([coupled3 ] ) formally exists above a cutoff energy @xmath67 . the density profile of the halo exhibits a flat central core , with @xmath68 as @xmath69 , and near - isothermal wings , with @xmath32 @xmath70 @xmath71 ( @xmath722.2 ) as @xmath73 , as illustrated in figure [ nice_density](a ) . we refer to it as the _ extended _ phase . it agrees with the solution for secondary infall onto a spherical perturbation @xcite and behaves asymptotically like the spherical , thermally conducting polytrope @xcite and infinite - dimensional brownian gas @xcite . for @xmath74 and @xmath10 , a formal solution of ( [ coupled])([coupled3 ] ) does not exist . the entropy and temperature are discontinuous at this cutoff energy as shown in figure [ nice_density](b ) . this is the well - known gravothermal catastrophe @xcite . note that , for @xmath75 , the singular isothermal sphere @xmath76 , @xmath77 and @xmath78 is always a solution of ( [ coupled])([coupled3 ] ) , as can be verified analytically , but it is not stable and so the iterative procedure never converges to it , but rather to figure [ nice_density](a ) . if the gravitational potential is softened , a stable phase exists for all values of @xmath80 . for @xmath81 , the halo is extended as in figure [ nice_density](a ) , with a flat core and near - isothermal envelope , @xmath82 . however , for @xmath74 , the halo is collapsed . figure [ collapse_density](a ) displays the density profile of such a collapsed halo for @xmath83 ( note that both axes are logarithmic ) . the halo has a steep dirac peak ( ` cusp ' ) at @xmath84 : @xmath32 is flat for @xmath85 , decreases as a large inverse power of @xmath86 for @xmath87 and flattens for @xmath88 . for intermediate energies in the range @xmath89 , the system is bistable : the halo can be either extended or collapsed depending on the initial conditions and the route to equilibrium . figure [ collapse_density](b ) , a plot of entropy and temperature as a function of energy , illustrates this bistability and the hysteresis to which it can lead . @xmath90 and @xmath91 jump discontinuously at both @xmath92 and @xmath93 . if @xmath80 enters the intermediate range from below , the halo remains collapsed until @xmath80 exceeds @xmath92 . alternatively , if @xmath80 enters from above , the halo remains extended until @xmath80 is reduced below @xmath93 @xcite . the critical energy @xmath93 , and the collapsed and extended profiles we obtain , are consistent with previous analyses @xcite . for example , @xcite find a phase transition at @xmath94 , consistent with our value @xmath95 , and a critical reciprocal temperature at the transition in the range @xmath96 for @xmath97 , consistent with @xmath98 in this paper ( @xmath83 ) . a more precise comparison is prohibited by the adoption of the ce rather than the mce in most previous work . the van der waals model proposed by @xcite is an exception ; it is examined in detail in the following section . the phenomenon of bistability was overlooked until the work of @xcite . the behaviour of the system depends somewhat on the choice of the relaxation parameter @xmath99 , defined at the start of this section . phase_table ] displays the minimum softening length for which the system makes the transition to a stable collapsed phase , for a given value of @xmath99 . the phase transition from the extended to the collapsed phase is increasingly delicate as @xmath2 decreases : for larger values of @xmath99 , the system is less likely to reach the critical point where the phase transition occurs and thus remains in the extended phase . although the effect is numerical , it potentially reflects the relative likelihoods of the possible routes that the real system can take to equilibrium . .minimum softening , @xmath100 , for which a stable collapsed solution is found for a given relaxation parameter @xmath99 . [ phase_table ] [ cols="<,<",options="header " , ] we have investigated the equilibrium configurations of @xmath0 self - gravitating collisionless particles , interacting via a softened gravitational potential , in the mce and mean - field limit . below a critical energy , @xmath101 , a system with @xmath102 exists in a stable , collapsed phase . this phase is unstable for pure gravity ( @xmath10 ) . above another critical energy @xmath103 , both softened and unsoftened systems exist in an stable , extended phase . in the intermediate region @xmath104 , both the collapsed and extended phases are accessible ; the detailed route to equilibrium determines which one is picked out . the density profiles for the extended and collapsed phases are qualitatively different . the extended profile has a flat core and near - isothermal outer envelope . the collapsed profile is a centrally condensed dirac peak , whose logarithmic slope depends on @xmath2 . we compare our results with published @xmath0-body simulations by using the softening parameter , @xmath2 , size , @xmath105 , and concentration parameter , @xmath106 , to place simulated haloes on the @xmath80-@xmath2 phase diagram . we find that many published simulations inadvertently sample the collapsed phase only , even though this phase is unstable for pure gravity and arguably irrelevant astrophysically . we remind the reader that we neglect several effects that are important in real cdm haloes , such as hierarchical clustering , nonzero angular momentum , and cosmological expansion . our results elucidate some of the artificial behaviour that a softened potential can introduce ; they are not a substitute for a full @xmath0-body calculation . we are grateful to bruce mckellar for extensive discussions on the theoretical basis of the mean - field equations , and the anonymous referee for useful comments that improved the manuscript . this work was supported by the australian research council discovery project grant 0208618 . cmt acknowledges the funding provided by an australian postgraduate award . antonov , v. a. 1962 , vest . leningrad univ . , 7 , 135 aronson , e. b. & hansen , c. j. 1972 , , 177 , 145 athanassoula , e. 2004 , in iau symposium 220 , dark matter in galaxies , ed . s. d. ryder , d. j. pisano , m. a. walker , and k. c. freeman , ( san francisco : asp ) , 255 athanassoula , e. , fady , e. , lambert , j. c. & bosma , a. 2000 , , 314 , 475 bertschinger , e. 1985 , , 58 , 39 binney , j. & tremaine , s. 1987 , galactic dynamics , ( princeton u. p. ) chabanol , m .- corson , f. & pomeau , y. 2000 , europhys . lett . , 50 , 148 davis , m. , efstathiou , g. , frenk , c. s. & white , s. d. m. 1985 , , 292 , 371 de blok , w. j. g. , mcgaugh , s. s. & rubin , v. c. 2001 , , 122 , 2396 de vega , h. j. & sanchez , n. 2002 , nuclear phys . b , 625 , 409 ghigna , s. , moore , b. , governato , f. , lake , g. , quinn , t. & stadel , j. 2000 , , 544 , 616 hamilton , a. j. s. , kumar , p. , lu , e. & matthews , a. , 1991 , , 374 , 1 hayashi , e. , navarro , j.f . , taylor , j.e . , stadel , j. & quinn , t. 2003 , , 584 , 541 huang , k. 1987 , statistical mechanics , ( 2nd ed . ; 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the statistical mechanics of @xmath0 cold dark matter ( cdm ) particles interacting via a softened gravitational potential is reviewed in the microcanonical ensemble and mean - field limit . a phase diagram for the system is computed as a function of the total energy @xmath1 and gravitational softening length @xmath2 . for softened systems , two stable phases exist : a collapsed phase , whose radial density profile @xmath3 is a central dirac cusp , and an extended phase , for which @xmath3 has a central core and @xmath3 @xmath4 @xmath5 at large @xmath6 . it is shown that many @xmath0-body simulations of cdm haloes in the literature inadvertently sample the collapsed phase only , even though this phase is unstable when there is zero softening . consequently , there is no immediate reason to expect agreement between simulated and observed profiles unless the gravitational potential is appreciably softened in nature .

You are an expert at summarizing long articles. Proceed to summarize the following text: reactor neutrino experiments have played a critical role in the history of neutrinos . among them , savannah river experiment @xcite by reines and cowan in 1956 observed the first neutrino . chooz @xcite determined the most stringent upper limit of the last unknown neutrino mixing angle @xmath1 in 1998 . kamland @xcite observed the first reactor neutrino disappearance in 2003 . flux is an important source of uncertainties for a reactor neutrino experiment . it is determined from thermal power measurements , reactor core simulation , and knowledge of neutrino spectra of fuel isotopes . past reactor neutrino experiments have determined the flux to ( 2 - 3)% precision . in the coming years , three precision experiments on neutrino mixing angle @xmath0 using reactor neutrinos , daya bay @xcite , double chooz @xcite , and reno @xcite , will start operation . all these experiments use near - far detector configurations . most uncertainties from the reactor will cancel out . the residual error will range from 0.1% to 0.45% . however , correlation among reactor cores need better understanding of the error sources . recently there are increasing interests on non - proliferation monitoring @xcite using ton - level neutrino detectors . a reactor neutrino experiment at an intermediate baseline @xmath260 km with a giant detector @xcite will have rich physics content . precise determination of neutrino flux will greatly improve the sensitivity of these experiments . another category of reactor neutrino experiments is @xmath3-electron or @xmath3-nucleus scattering experiments , such as texono @xcite , munu @xcite , gemma @xcite , etc . normally they wo nt rely on precise neutrino flux . in this note , we will review the calculation of the reactor neutrino flux and recent progresses on the error analysis . most commercial reactors are pressurized water reactor ( pwr ) or boiling water reactor ( bwr ) . they are very similar in neutrino flux calculation . we will use pwr as examples in the following . the @xmath4u enrichment in fresh fuel of a pwr is normally ( 3 - 4)% , and more than 95% is @xmath5u . electron antineutrinos are emitted from subsequent @xmath6-decays of fission fragments . they are dominated by 4 isotopes , @xmath4u , @xmath7pu , @xmath8pu , and @xmath5u . other isotopes contribute only at 0.1% level . one can calculate the neutrino energy spectrum of each isotope by summing all fission fragment @xmath6-decay branches . there are a lot of efforts on such studies . however , the fission products are very complex . due to lack of accurate nuclear data , such calculations carry large uncertainties at the 10% level . the most accurate neutrino spectra of the first 3 isotopes were determined at ill @xcite by measuring the @xmath6 spectra of fissioning . the @xmath6 spectra are then converted to neutrino spectra , with an average uncertainty 1.9% . the @xmath5u spectrum are calculated theoretically @xcite . they are shown in fig . [ fig : spectra ] . , scaledwidth=40.0% ] the isotope concentration in fuel will evolve during reactor operation as @xmath4u depletes and @xmath7pu and @xmath8pu breed . the @xmath5u concentration is relatively stable . such evolution can be obtained by core simulation . a typical isotope evolution as a function of operation time , in terms of fission rates of the reactor , is shown in fig . [ fig : evol ] . , scaledwidth=40.0% ] when we know the fission rates of each isotope from core simulation , and neutrino energy spectrum of each isotope , we can easily get the neutrino flux @xmath9 , where @xmath10 is the fission rate of isotope @xmath11 and @xmath12 is its neutrino spectrum . however , the fission rates are proportional to the thermal power of the core , which is fluctuating . it is unrealistic to repeat core simulation to reflect the power fluctuation . normally we scale the neutrino flux to the measured thermal power . @xmath13 where @xmath14 is the thermal power , @xmath15 is the energy release per fission for isotope @xmath11 , and @xmath16 is the sum of @xmath10 , thus @xmath17 is the fission fraction of each isotope . among the inputs , the thermal power data is provided by the nuclear power plant . the uncertainty is generally estimated to be ( 0.6 - 0.7)% @xcite . fission fractions are obtained by core simulation as a function of burn - up . burn - up is the amount of energy in mega watt days ( mwd ) released per unit initial mass ( ton ) of uranium ( tu ) . the simulated fission fraction carries @xmath25% uncertainties from statistics of hundreds of analyses for various codes and various reactors @xcite . the 5% fission fraction uncertainties corresponds to @xmath20.5% uncertainty in neutrino yield . energy release per fission varies slightly for different cores at different time due to neutron capture and non - equilibrium products . average numbers @xcite can be used , with uncertainties of ( 0.30 - 0.47)% . alternatively we can extract them from the core simulation to accurately reflect the core differences and burn - up effects . recently there are studies to include contributions from non - equilibrium isotopes in the core @xcite as well as that from spent fuel which is temporarily stored adjacent to the core @xcite . these are at sub - percent level and only contribute to the low energy region . besides fission products , @xmath5u(n,@xmath18)@xmath7u reaction also contributes to the neutrino yield . it is below inverse @xmath6-decay threshold ( 1.8 mev ) but will contribute significantly to low energy @xmath3-electron scattering experiments @xcite . the most accurate thermal power measurement is the secondary heat balance method . detailed description of this measurement can be found , for example , in @xcite . this is an offline measurement , normally done weekly or monthly . the uncertainty is cited as 0.7% by chooz and palo verde . primary heat balance tests are online thermal power measurement . normally it is calibrated to the secondary heat balance measurement weekly . daya bay power plants control the difference of these two measurements to less than 0.1% of the full power . these data are good for neutrino flux analysis . to 0.1% level , it can be taken as the secondary heat balance measurement . the power plants also monitor the ex - core neutron flux , which gives the nuclear power . this monitoring is online , for safety and reactor operation control . it is normally calibrated to the primary heat balance measurement daily . this measurement is less accurate , controlled to be less than 1.5% of the full power by daya bay power plant . recently there are a lot of studies on the power uncertainties and instrumentation improvements by the power plants @xcite , with the motivation of power uprates . the power measurements can be more accurate than what were cited in the past reactor neutrino experiments . the uncertainties of the secondary heat balance is dominated by the flow rate measurement . in the past , there are two kinds of widely used flow meters , venturi type flow meters and orifice plate flow meters . venturi flow meters are used by most us and japan reactors . the uncertainty is often 1.4% . it can be as low as 0.7% if properly calibrated and maintained . but they suffer from fouling effects , which could grow as high as 3% in a few years . to improve the measurement , ultrasonic flow meters have begun to be in use in some us and japan reactors . they have uncertainties 0.45% for type i and 0.2% for type ii @xcite . the orifice plate flow meters are used by french reactors . they have no fouling effects . typically they have an uncertainty of 0.72% and could be improved to 0.4% with laboratory tests . it should be noted that the uncertainties of above flow meters are at the 95% c.l . ( confidence level ) , as defined in iso-5167 . unless specified , the thermal power uncertainty given by the power plant is also at 95% c.l . an edf ( electricite de france ) n4 reactor with four parallel steam generators , which is the chooz type , is analyzed in @xcite . main components of the uncertainties are shown in table [ tab : n4 ] . it is dominated by the discharge coefficient , which is an empirical formula in the flow rate measurement and its uncertainty is specified in iso 5167 - 1 - 2003 . the final uncertainty of the thermal power is 0.40% at 95% c.l . in this evaluation , it is assumed that the discharge coefficients of the orifice plates in four coolant loops are independent , thus the final uncertainty is statistically reduced . if the discharge coefficients are fully correlated for all four orifice plates , then there is no statistical reduction . the power uncertainty will be 0.37% at 1@xmath19 level , which is still significantly smaller than 0.7% that chooz used . .error table for the thermal power measurements for n4 reactor @xcite.[tab : n4 ] [ cols="^,^,^ " , ] electron antineutrinos are emitted from the subsequent @xmath6-decays of fission fragments . due to the lack of data for the @xmath6-decays of the complex fission products , theoretical calculations of the neutrino spectra of isotopes carry large uncertainties . ill @xcite measured the @xmath6 spectra of fissioning of @xmath4u , @xmath7pu , and @xmath8pu by thermal neutrons , and converted them to neutrino spectra . the normalization error is estimated to be 1.9% . spectrum shape error is from 1.34% at 3 mev to 9.2% at 8 mev , as shown in fig . [ fig : spectra ] . @xmath5u can not fission with thermal neutrons . its spectrum relies on theoretical calculation . the uncertainty is estimated to be 10% @xcite . normally @xmath5u contributes ( 7 - 10)% of fissions in a pwr . the calculated neutrino counting rate and spectra were verified by bugey and bugey-3 @xcite . the normalization error is further lowered to 1.6% , which was used by chooz . the ill spectra are derived after 1.5 days exposure time with thermal neutron . thus , long - lived fission fragments have not reached equilibrium . in a real reactor , these fission products will accumulate and contribute to the neutrino flux . chooz estimated this contribution to be @xmath20.3% on the average and ignored it in the detailed analysis due to its small size comparing to other errors . six chains have been identified in @xcite , with half lives from 10 hours to 28 years . they only contribute to the low energy region . further studies show that for a typical pwr , on average these contributions are @xmath20.2% of total neutrino detection rate via inverse @xmath6-decay @xcite . in the 2 - 4 mev region , it increases to 0.8% after one year s accumulation , as shown in fig . [ fig : noneq ] -decay cross section @xcite.[fig : noneq ] , scaledwidth=40.0% ] spent fuel is normally stored temporarily adjacent to the core . the storage could be as long as 10 years . similar to the non - equilibrium contributions , the long - lived fission fragments in the spent fuel will contribute to the neutrino flux . a pwr is normally refueled very 12 - 18 months . the spent fuel from one refueling will contribute @xmath20.2% of the total neutrino rate after first several days . before the 1980 s , the reactor neutrino flux was determined to an uncertainty of 10% . with a lot of efforts , especially by ill , bugey , chooz , palo verde , etc . , the uncertainty has improved to ( 2 - 3)% . motivated by high precision neutrino measurements by scientific collaborations and power uprates by the power plants , we have more accurate thermal power . the uncertainty could be lowered from 0.7% to 0.4% . small corrections from non - equilibrium isotopes and spent fuel , as well as energy release per fission are studied in detail . we also have a global picture of uncertainties of fission rate simulations . however , there is no new data on neutrino spectra of fuel isotopes . for single - detector experiments , the neutrino spectrum uncertainty of about 2% will dominate . the next @xmath0 experiments with near - far relative measurements will suffer little from reactor flux uncertainties , which is estimated to be 0.1% to 0.45% , depending on the experiment layout . the correlation among reactor uncertainties is important for @xmath0 experiments since correlated errors will cancel out . meanwhile , high precision detector and high statistics at the near detector of these experiments may help to improve the knowledge of neutrino spectra . since the time that this manuscript was written , it has been suggested @xcite that the antineutrino spectra for all of the relevant fission isotopes ( @xmath4u , @xmath5u , @xmath7pu , and @xmath8pu ) are several percent larger than those of ref . @xcite . this issue requires further attention .

flux is an important source of uncertainties for a reactor neutrino experiment . it is determined from thermal power measurements , reactor core simulation , and knowledge of neutrino spectra of fuel isotopes . past reactor neutrino experiments have determined the flux to ( 2 - 3)% precision . precision measurements of mixing angle @xmath0 by reactor neutrino experiments in the coming years will use near - far detector configurations . most uncertainties from reactor will be canceled out . understanding of the correlation of uncertainties is required for @xmath0 experiments . precise determination of reactor neutrino flux will also improve the sensitivity of the non - proliferation monitoring and future reactor experiments . we will discuss the flux calculation and recent progresses . neutrino flux , reactor , fission rate

You are an expert at summarizing long articles. Proceed to summarize the following text: star formation occurs across the galactic disc with most molecular material concentrated at low latitudes @xmath1 . both in the inner and in the outer galaxy , young stars still partly embedded in the dense gas and dust in molecular clouds have been found ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? they represent current active star formation sites . the study and census of star formation sites in regions of higher galactic latitudes have received comparatively little attention and coverage . however , the detection and characterisation of star formation sites across the whole galaxy have strong implications on its structure and evolution . in our previous studies of star formation in the outer galaxy , we have reported discovery of what can be designated as star formation sites in extreme environments , namely near the distant `` edge '' of the galactic disc ( at galactocentric distances of about 16.5 kpc @xcite ) , or far below the midplane of the galactic disc ( at vertical distances of about 500 pc @xcite ) . in both cases , the presence of molecular material and of clusters of embedded stars in these environments posit that the physical conditions required to form a cluster of stars are met in locations at distances away from the galactic plane ( or away from the galactic centre ) larger than reasonably expected . iras 06345 - 3023 is an iras psc source in the outer galaxy that currently appears classified in the simbad data base as `` 6dfgs gj063630.0 - 302542 galaxy '' . this means that it is listed in the 6dfgs , the six - degree field galaxy survey catalogue as being an extragalactic source . however , a search in the 6dfgs catalogue , at the iras coordinates , does not return this source . instead , a different source ( g0636403 - 302842 ) is presented , which is located about 4 arcminutes away . in addition , ned ( the nasa extragalactic database ) does not contain this source . @xcite first called attention on the presence of two nebulous `` stars '' on eso / src j plates at this location . @xcite used the 4 metre nanten milimetre telescope to study a sample of intermediate - to - high latitude iras sources and reported detection of co emission from iras 06345 - 3023 . their co maps , although at a coarse angular resolution , show a clear peak at the position of the source . as part of our study of young embedded clusters in the outer galaxy ( e.g. * ? ? ? * ; * ? ? ? * ) , we have conducted observations ( near - infrared @xmath0 imaging , and millimetre co line ) towards iras 06345 - 3023 . these observations revealed the presence of young stars embedded in a molecular cloud core . we report here our near - infrared discovery of an aggregate of young stars seen towards iras 06345 - 3023 . in addition , we use new co data , to characterise the molecular environment and derive its kinematic distance . section 2 describes the observations and data reduction . in section 3 , we present and discuss the results . a summary is given in section 4 . near - infrared ( @xmath2 , @xmath3 and @xmath4 ) images were obtained on 2000 november 12 and on 2002 january 8 using the eso antu ( vlt unit 1 ) telescope equipped with the short - wavelength arm ( hawaii rockwell ) of the isaac instrument . the isaac camera @xcite contains a 1024 @xmath5 1024 pixel near - infrared array and was used with a plate scale of 0.147 arcsec / pixel resulting in a field of view of 2.5 @xmath5 2.5 arcmin@xmath6 on the sky . for each filter , dithered sky positions were observed . a series of 12 images with individual on - source integration time of 6 and 3.55 seconds was taken in the @xmath2 and in the @xmath3 bands , respectively . similarly , series of 6 images , each of 2 second integration time , were obtained in the @xmath4 band . the images were reduced with the image reduction and analysis facility ( iraf ) , using a set of our own scripts to correct for bad pixels , subtract the sky background , and flatfield the images . dome flats were used to correct for the pixel - to - pixel variations of the response . the selected images were then aligned , shifted , trimmed , and co - added to produce a final mosaic image for each band @xmath0 . correction for bad pixels was made while constructing the final mosaics that cover about @xmath7 arcmin@xmath6 on the sky . point sources were extracted using daofind with a detection threshold of 5@xmath8 . the images were inspected to look for false detections that had been included by daofind in the list of detected sources . these sources were eliminated from the source list . aperture photometry was made with a small aperture ( radius = 0.5@xmath9 ) and aperture corrections , found from bright and isolated stars in each image , were used to correct for the flux lost in the wings of the psf . the error in determining the aperture correction was @xmath10 0.03 mag in all cases . we used the 2mass all - sky release point source catalogue @xcite to calibrate our observations . the @xmath0 zeropoints were determined using 2mass stars brighter than @xmath4 = 14.7 mag . the standard deviations of the offsets between isaac and 2mass photometry are 0.1 mag , in all bands . given the relatively small number of stars in these images , it is not possible to make a robust estimate of the completeness limit of the observations based on the statistics of the stars detected . however , a relatively good determination of the completeness limit can be obtained calculating _ e.g. _ the @xmath11-limit in each band . those are 20.6 in @xmath2 , 20.0 in @xmath3 , and 18.9 mag in @xmath4 . as part of a survey of molecular gas in the third galactic quadrant , the region around the position of the iras source was mapped using the single - dish 15-m sest radiotelescope in 1999 september . the map was obtained in the rotational line of @xmath12co(1 - 0 ) ( 115.271 ghz ) and consisted of @xmath13 positions , in full - beam spacing ( @xmath14 ) , and centred on the iras coordinates . the integration time was 60 seconds . the spectra were taken in frequency switching mode , recommended to save observational time when mapping extended sources . one additional central spectrum was taken in beam - switching mode , with integration time of 120 seconds , in order to obtain improved baselines to search for the presence of molecular outflows . a high - resolution 2000 channel acousto - optical spectrometer was used as a back end , with a total bandwidth of 86 mhz and a channel width of 43 khz , which at the frequency of 115 ghz corresponds to approximately 0.11 km s@xmath15 . the antenna temperature was calibrated with the standard chopper wheel method . pointing was checked regularly towards known circumstellar sio masers , and pointing accuracy was estimated to be better than @xmath16 . the data reduction pipeline was composed of the following steps : _ i ) _ folding the frequency - switched spectrum ; _ ii ) _ fitting the baseline by a polynomial and subtracting it ; _ iii ) _ obtaining the main - beam temperature @xmath17 by dividing the antenna temperature @xmath18 by the @xmath19 factor , equal to 0.7 . the spectrum baseline rms noise ( in @xmath17 ) , averaged over all map positions , has been found to be 0.15 k. ( blue ) , @xmath3 ( green ) , and @xmath4 ( red ) colour composite image towards iras 06345 - 3023 covering @xmath20 . north is up and east to the left . notice the nebular emission around a small concentration of red stars towards the centre of the image , and the very small number of field stars.,width=291 ] figure [ rgb995 ] presents the vlt / isaac @xmath0 near - infrared colour composite image obtained towards iras 06345 - 3023 . centred on the image , extended nebular emission involving a few bright stars is displayed . in addition , when compared to other fields , usually at low galactic latitudes , we notice a very small number of field stars , either background or foreground , that populate the galactic disc and frequently contaminate deep near - infrared images . a close look at the image ( see the zoomed - in view of the same region shown in figure [ rgb995 - 1 ] ) reveals the presence of a butterfly - shaped bipolar dark cloud , to the north - northeast and to the south - southwest . notice also the bright rims of the dark `` butterfly wings '' illuminated by the central stars . the richness of the structures seen includes a quasi - circular bright ring and bright wisps or rays of light against the black opaque northern wing ( see also fig . [ rgb999 ] for the nomenclature used here ) . in fig . [ rgb999 ] we reproduced the colour composite image using a colour coding emphasizing the bright stars as opposed to the diffuse nebular emission . a few important points can be derived as follows . the bright source labeled `` wispy neb '' is not a point source , it corresponds to pure diffuse emission . several wisps in this nebula ( better distiguished in fig . [ rgb995 - 1 ] ) seem to point away from a common point , the location of one of the reddest sources , that labeled `` irs 4 '' . the bluer colour of the wispy nebula supports the idea that it corresponds to scattered light from a cavity located close to the centre of the butterfly - shaped core where opacity is lower . in fact , the position of one of the two `` nebulous stars '' that were noticed on the optical eso / src images @xcite coincides with the position of this wispy nebula . the other optical `` nebulous star '' corresponds to the blue emission seen to the west including the circular arc ( the `` arc '' ) . this arc - like structure appears to be associated with the source labeled `` irs 3 '' which is the brightest red embedded source present in the images . the arc is quasi - circular , very well delineated , and constitutes an engimatic structure , similar to the structure seen around a few other young stellar objects . examples of these are the diamond - ring object of @xcite , which was conjectured to represent a circumbinary structure existing during a brief stage in the formation of a binary star ; and gg tau , a multiple young source , which exhibits a circumbinary ring ( e.g. * ? ? ? * ; * ? ? ? * ) , and that has recently been suggested to be forming planets @xcite . irs 3 could be another case of a multiple young object denouncing its multiplicity through the presence of circumbinary arcs or rings , seen in the near - infrared or submillimetre . interestingly , neither of the two optical `` nebulous stars '' is a real star ( point source ) but instead corresponds to diffuse scattered emission escaping through lines - of - sight of lower optical depth . several cavities excavated by winds from young stars are now known and have been modelled ( e.g. * ? ? ? * ; * ? ? ? our images seem to confirm the rule that young stars disrupt the parent cloud core reducing the opacity from specific lines - of - sight and resulting in very non - homogenous distributions of extinction across cloud cores . also remarkable is the detection of wisps of bluer light escaping from this star formation site , similar to sunlight under low - lying clouds on a partly cloudy day on earth . in addition , despite its faintness , a careful look reveals the presence of a bright rim of this molecular cloud core specially in the southwest part of the dark nebula ( fig . [ rgb995 - 1 ] ) . but with a colour coding emphasizing the emission from bright stars . notice the nomenclature of the structures labeled here , some of which can be seen more clearly in fig.[rgb995 - 1],width=302 ] we have detected co emission towards the positions of the nebular emission and of this group of stars . moreover , the location of the nebula coincides with the position where the molecular gas peaks . [ co1 ] shows the beam switching spectrum from the iras 063453023 revealing a clear detection of a co line . a gaussian fit yields a @xmath21 peak value of 8.1 k at the @xmath22 km s@xmath15 in good agreement with @xcite . the line presents some deviation from a gaussian , exhibiting moderate wings that could be due to a molecular outflow . our @xmath23 co map ( fig . [ comap ] ) shows relatively constant co emission , but peaking at the centre position where the iras source is located . because the map was obtained in frequency - switching mode , we did not try to look for the presence of outflows in the beams of this map . either adopting a flat rotation curve in the outer galaxy or using a circular rotation model , a heliocentric distance @xmath24 kpc , and a galactocentric distance @xmath25 kpc are found , respectively . this is in good agreement with the distance quoted by @xcite . thus , the projected size of the region occupied by the near - infrared nebula and butterfly - wings turns out to be @xmath26 pc , a typical size for a molecular cloud core , and coincident with the c@xmath27o - derived size @xcite . however , as expected , the size of the molecular cloud is much larger as can be seen by the large scale map of @xcite which covers more than @xmath28 on the sky . interestingly , due to its relatively high galactic latitude ( @xmath29 ) , this source is located about 450 pc below the mid - plane of the galactic disc . this value is among the largest known for molecular clouds at this distance in the outer galaxy @xcite , placing this source at the edge of the galactic ( thin ) disc @xcite . a similar large vertical distance was found for the star formation site reported by @xcite . in that case , flaring and the location of the galactic warp could account for the presence of the star formation site . however , this is not the case for iras 06345 - 3023 . in fact , flaring of the gas component ( both hi and h@xmath30 ) in the galactic disc is known to occur but is significant only at larger galactocentric distances @xcite . thus , the present day existence of molecular gas and dust at this relatively large vertical distance raises new questions about the evolution of the galactic disc , as well as the ability to form stars in this extreme environment . in order to determine the nature of the group of stars seen in the near - infrared images , and specially those seen towards the nebula , photometry of the stars in the @xmath0 images was performed . not many stars are seen , neither field stars ( background or foreground ) nor embedded young stars . the scarcity of field stars is due to the location of this star formation site in the outer galaxy , and about 450 pc below the galactic disc . on the other hand , given the good resolution and depth of the images , and considering the distance , the scarcity of young stars forming in this cloud core could indicate a lack of ability to form rich clusters at this large distance below the galactic disc . table 1 presents the results of our photometry for the sources detected in the @xmath0-bands in the region shown in fig.[rgb995 - 1 ] . columns ( 1 ) and ( 2 ) give their coordinates . the @xmath0 magnitudes ( in the 2mass system ) are presented in columns ( 3 ) , ( 4 ) and ( 5 ) , where errors are of the order of 0.10 mag . in column ( 6 ) we indicate the sources that are likely to be young stellar objects according to their near - infrared @xmath31 and @xmath32 colours @xcite . the sources labeled in fig . [ rgb999 ] are indicated by their names . only nine stars are detected as young stellar object candidates and likely to be forming in this region . their near - infrared colours are compatible with class i ysos , that is optically invisible objects still surrounded by cirumstellar discs and envelopes and frequently exhibiting modest outflows . in fact , these young stars are not optically visible ( absent in the digitized sky survey ) even though they have created regions of lower extinction through the cloud that manifest as optical nebulae . we estimate the luminosity of the brightest one ( irs 3 ) by integrating the near - infrared fluxes and the iras fluxes . the result is 41@xmath33 and it represents an upper limit for the luminosity of the brightest object in the @xmath4-band . this low value of the luminosity for a pre - main - sequence object implies that no high - mass star is present in this star formation site . .vlt / isaac photometry of sources towards iras 063453023 [ cols="^,^,^,^,^,^,^ " , ] high - resolution , deep , near - infrared ( @xmath0 ) images of the region towards iras 06345 - 3023 reveals a small group of stars compatible with the presence of a small aggregate of young stars . 2 . the stars are located in a region exhibiting nebular emission and are embedded in a dense molecular cloud core detected through co line emission . the nebular light is rich in morphological structures , including arcs in the vicinity of embedded stars , wisps , and bright rims of a butterfly - shaped dark cloud . in particular , the circular arc may represent a circumbinary ring . 4 . at about 1,5 kpc ( heliocentric distance ) in the outer galaxy , and at a galactocentric distance of 10 kpc , the relatively large value of its galactic latitude places this group of young stars at 450 pc below the galactic plane , at the edge of the molecular disc . thus , active star formation is taking place at vertical distances larger than those typical of the ( thin ) disc . this may raise questions on the origin of the gas forming this cloud and on the sharpness of the stellar and of the gaseous galactic thin disc . the group of young stars is likely to be composed of low - mass stars , mostly class i and flat spectrum young stellar objects . 6 . inspection of wise images and catalogues confirms the yso nature of the sources and the absence of additional young stars associated to this star formation site . this research was based on observations collected at the eso 8.2-m vlt - ut1 antu telescope ( program 68.c-0214a ) . jy acknowledges support from fct ( sfrh / bsab/1423/2014 ) . this research made use of the nasa/ ipac infrared science archive , which is operated by the jet propulsion laboratory , california institute of technology , under contract with the national aeronautics and space administration . this research also made use of the simbad database , operated at cds , strasbourg , france , as well as saoimage ds9 , developed by the smithsonian astrophysical observatory . this publication makes use of data products from the wide - field infrared survey explorer , which is a joint project of the university of california , los angeles , and the jet propulsion laboratory / california institute of technology , funded by the national aeronautics and space administration .

we report the discovery of a small aggregate of young stars seen in high - resolution , deep near - infrared ( @xmath0 ) images towards iras 06345 - 3023 in the outer galaxy and well below the mid - plane of the galactic disc . the group of young stars is likely to be composed of low - mass stars , mostly class i young stellar objects . the stars are seen towards a molecular cloud whose co map peaks at the location of the iras source . the near - infrared images reveal , additionally , the presence of nebular emission with rich morphological features , including arcs in the vicinity of embedded stars , wisps and bright rims of a butterfly - shaped dark cloud . the location of this molecular cloud as a new star formation site well below the galactic plane in the outer galaxy indicates that active star formation is taking place at vertical distances larger than those typical of the ( thin ) disc . [ firstpage ] stars : formation infrared : stars ism : clouds ism : individual objects : iras 06345 - 3023 dust , extinction galaxy : stellar content .

You are an expert at summarizing long articles. Proceed to summarize the following text: the topology or inherent shape and form of an object is important . in data analysis , the inherent form and structure of data clouds are important . quite a few models of data form and structure are used in data analysis . one of them is a hierarchically embedded set of clusters , a hierarchy . it is traditional ( since at least the 1960s ) to impose such a form on data , and if useful to assess the goodness of fit . rather than fitting a hierarchical structure to data ( e.g. , @xcite ) , our recent work has taken a different orientation : we seek to find ( partial or global ) inherent hierarchical structure in data . as we will describe in this article , there are interesting findings that result from this , and some very interesting perspectives are opened up for data analysis and , potentially , perspectives also on the physics ( or causal or generative mechanisms ) underlying the data . a formal definition of hierarchical structure is provided by ultrametric topology ( in turn , related closely to p - adic number theory ) . we will return to this in section [ sect23 ] below . first , though , we will summarize some of our findings . ultrametricity is a pervasive property of observational data . it arises as a limit case when data dimensionality or sparsity grows . more strictly such a limit case is a regular lattice structure and ultrametricity is one possible representation for it . notwithstanding alternative representations , ultrametricity offers computational efficiency ( related to tree depth / height being logarithmic in number of terminal nodes ) , linkage with dynamical or related functional properties ( phylogenetic interpretation ) , and processing tools based on well known p - adic or ultrametric theory ( examples : deriving a partition , or applying an ultrametric wavelet transform ) . in @xcite and other works , khrennikov has pointed to the importance of ultrametric topological analysis . local ultrametricity is also of importance . this can be used for forensic data exploration ( fingerprinting data sets ) : see @xcite and @xcite ; and to expedite search and discovery in information spaces : see @xcite as discussed by us in @xcite , @xcite , and @xcite . in section [ sect23 ] we show how extent of ultrametricity is measured . section [ sect3 ] presents our main results on the remarkable properties of very high dimensional , or very sparse , spaces . as dimensionality or sparsity grow , so does the inherent hierarchical nature of the data in the space . in section [ hfda ] we then discuss application to very high frequency time series modeling . summarizing a full description in murtagh @xcite we explored two measures quantifying how ultrametric a data set is , lerman s and a new approach based on triangle invariance ( respectively , the second and third approaches described in this section ) . the triangular inequality holds for a metric space : @xmath0 for any triplet of points @xmath1 . in addition the properties of symmetry and positive definiteness are respected . the `` strong triangular inequality '' or ultrametric inequality is : @xmath2 for any triplet @xmath1 . an ultrametric space implies respect for a range of stringent properties . for example , the triangle formed by any triplet is necessarily isosceles , with the two large sides equal ; or is equilateral . * firstly , rammal et al . @xcite used discrepancy between each pairwise distance and the corresponding subdominant ultrametric . now , the subdominant ultrametric is also known as the ultrametric distance resulting from the single linkage agglomerative hierarchical clustering method . closely related graph structures include the minimal spanning tree , and graph ( connected ) components . while the subdominant provides a good fit to the given distance ( or indeed dissimilarity ) , it suffers from the `` friends of friends '' or chaining effect . * secondly , lerman @xcite developed a measure of ultrametricity , termed h - classifiability , using ranks of all pairwise given distances ( or dissimilarities ) . the isosceles ( with small base ) or equilateral requirements of the ultrametric inequality impose constraints on the ranks . the interval between median and maximum rank of every set of triplets must be empty for ultrametricity . we have used extensively lerman s measure of degree of ultrametricity in a data set . taking ranks provides scale invariance . but the limitation of lerman s approach , we find , is that it is not reasonable to study ranks of real - valued ( values in non - negative reals ) distances defined on a large set of points . * thirdly , our own measure of extent of ultrametricity @xcite can be described algorithmically . we examine triplets of points ( exhaustively if possible , or otherwise through sampling ) , and determine the three angles formed by the associated triangle . we select the smallest angle formed by the triplet points . then we check if the other two remaining angles are approximately equal . if they are equal then our triangle is isosceles with small base , or equilateral ( when all triangles are equal ) . the approximation to equality is given by 2 degrees ( 0.0349 radians ) . our motivation for the approximate ( `` fuzzy '' ) equality is that it makes our approach robust and independent of measurement precision . a supposition for use of our measure of ultrametricity is that we can can define angles ( and hence triangle properties ) . this in turn presupposes a scalar product . thus we presuppose a complete normed vector space with a scalar product a hilbert space to provide our needed environment . quite a general way to embed data , to be analyzed , in a euclidean space , is to use correspondence analysis @xcite . this explains our interest in using correspondence analysis : it provides a convenient and versatile way to take input data in many varied formats ( e.g. , ranks or scores , presence / absence , frequency of occurrence , and many other forms of data ) and map them into a euclidean , factor space . murtagh @xcite , and earlier work by rammal et al . @xcite , has demonstrated the pervasiveness of ultrametricity , by focusing on the fact that sparse high - dimensional data tend to be ultrametric . in such work it is shown how numbers of points in our clouds of data points are irrelevant ; but what counts is the ambient spatial dimensionality . among cases looked at are statistically uniformly ( hence `` unclustered '' , or without structure in a certain sense ) distributed points , and statistically uniformly distributed hypercube vertices ( so the latter are random 0/1 valued vectors ) . using our ultrametricity measure , there is a clear tendency to ultrametricity as the spatial dimensionality ( hence spatial sparseness ) increases . as @xcite also show , gaussian data behave in the same way and a demonstration of this is seen in table [ tabunifgauss ] . to provide an idea of consensus of these results , the 200,000-dimensional gaussian was repeated and yielded on successive runs values of the ultrametricity measure of : 0.96 , 0.98 , 0.96 . .typical results , based on 300 sampled triangles from triplets of points . for uniform , the data are generated on [ 0 , 1 ] ; hypercube vertices are in @xmath3 , and for gaussian , the data are of mean 0 , and variance 1 . dimen . is the ambient dimensionality . isosc . is the number of isosceles triangles with small base , as a proportion of all triangles sampled . equil . is the number of equilateral triangles as a proportion of triangles sampled . um is the proportion of ultrametricity - respecting triangles (= 1 for all ultrametric ) . [ cols="<,<,<,<,<",options="header " , ] we find clearly distinguishable peaks in figure [ fig10 ] . the lower and the higher peaks belong to the two arima components . the central peak belongs to the inter - cluster distances . we have shown that our methodology can be of use for time series segmentation and for model identifiability . we will assess this further in future work . given the use of a hilbert space as the essential springboard of all aspects of this work , it would appear that generalization of this work to multivariate time series analysis is straightforward . what remains important , however , is the availability of very large embedding dimensionalities , i.e. very high frequency data streams . what we have observed in all of this work is that in the limit of high dimensionality a hilbert space becomes ultrametric . it has been our aim in this work to link observed data with an ultrametric topology for such data . the traditional approach in data analysis , of course , is to impose structure on the data . this is done , for example , by using some agglomerative hierarchical clustering algorithm . we can always do this ( modulo distance or other ties in the data ) . then we can assess the degree of fit of such a ( tree or other ) structure to our data . for our purposes , here , this is unsatisfactory . * firstly , our aim was to show that ultrametricity can be naturally present in our data , globally or locally . we did not want any `` measuring tool '' such as an agglomerative hierarchical clustering algorithm to overly influence this finding . ( unfortunately @xcite suffers from precisely this unhelpful influence of the `` measuring tool '' of the subdominant ultrametric . in other respects , @xcite is a seminal paper . ) * secondly , let us assume that we did use hierarchical clustering , and then based our discussion around the goodness of fit . this again is a traditional approach used in data analysis , and in statistical data modeling . but such a discussion would have been unnecessary and futile . for , after all , if we have ultrametric properties in our data then many of the widely used hierarchical clustering algorithms will give precisely the same outcome , and furthermore the fit is by definition optimal . we have described an application of this work to very high frequency signal processing . the twin objectives are signal segmentation , and model identification . we have noted that a considerable amount of this work is model - based : we require assumptions ( on clusters , and on model(s ) ) for identifiability . motivation for this work includes the availability of very high frequency data streams in various fields ( physics , engineering , finance , meteorology , bio - engineering , and bio - medicine ) . by using a very large embedding dimensionality , we are approaching the data analysis on a very gross scale , and hence furnishing a particular type of multiresolution analysis . that this is worthwhile has been shown in our case studies . f. murtagh , from data to the physics using ultrametrics : new results in high dimensional data analysis , in a.yu . khrennikov , z. raki and i.v . volovich , eds . , p - adic mathematical physics , american institute of physics conf . proc . 826 , 151161 , 2006 . r. rammal , g. toulouse and m.a . virasoro , ultrametricity for physicists , reviews of modern physics , 58 , 765788 , 1986 . rohlf and d.r . fisher , tests for hierarchical structure in random data sets , systematic zoology , 17 , 407412 , 1968 .

an ultrametric topology formalizes the notion of hierarchical structure . an ultrametric embedding , referred to here as ultrametricity , is implied by a natural hierarchical embedding . such hierarchical structure can be global in the data set , or local . by quantifying extent or degree of ultrametricity in a data set , we show that ultrametricity becomes pervasive as dimensionality and/or spatial sparsity increases . this leads us to assert that very high dimensional data are of simple structure . we exemplify this finding through a range of simulated data cases . we discuss also application to very high frequency time series segmentation and modeling . pacs : 02.50.-r , 05.45.tp , 89.65.gh , 89.20.-a

You are an expert at summarizing long articles. Proceed to summarize the following text: bose - einstein condensates ( becs ) in optical lattices ( ols ) are presently attracting a great deal of interest @xcite due both to their flexibility in parameter design and to the possibility they offer to observe interesting phenomena such as superfluid to mott - insulator transition @xcite , bloch oscillations @xcite , landau - zener tunneling @xcite , generation of coherent atomic pulses ( atom laser ) @xcite , atom interferometry @xcite , etc . in this respect , ols allow to control important properties of bec by means of their periodic structure , this allowing , for example , the existence and stability of localized nonlinear excitations with chemical potentials inside band - gaps ( so called gap - solitons ( gss ) ) even in the presence of repulsive interactions ( positive scattering lengths ) . this fact , that would be obviously impossible in absence of the ol @xcite , has been experimentally demonstrated in @xcite . modulations of the ol can be used to accelerate , decelerate or to scatter gss as well as to control their velocities @xcite . uniform accelerations of the ol combined with periodic modulations of the scattering length , either in space or in time , were shown to be effective tools to induce long lived bloch oscillations in the nonlinear regime @xcite , as well as band - gap tunneling phenomena such as the landau - zener tunneling @xcite . periodic time dependent ol accelerations were also used to achieve the dynamical localization of nonlinear matter waves @xcite and the rabi - oscillations of gs states across a band - gaps @xcite which survive on a long time scale in the presence of nonlinearity . moreover , the combination of the above phenomena permits the stirring of gss both in the reciprocal and in the direct lattice space as recently demonstrated in @xcite . besides the one - dimensional contexts in which these effects have been investigated , ols also play an important role for stabilizing gss against collapse or decay in higher dimensions @xcite . all what said above refers to the case of perfect ols , e.g. ols without distortions or defects which compromise the periodicity structure . in this case the interplay between periodicity and nonlinearity is the only source for the localization of matter waves in the system . ol defects , however , introduce additional bound states in the band - gaps ( so called impurity modes ) @xcite providing an alternate source of localization in the system . opposite to gss , impurity modes exist both in absence and in presence of nonlinearity and can interfere in the scattering process of gss by ol defects . scattering properties of solitons with single ( non periodic ) potential wells have been extensively investigated during the past years . for the case of the nonlinear schrdinger equation ( nlse ) scattering of solitons by extended defects were numerically investigated in @xcite where the occurrence of a series of reflection , transmission , and trapping regions as a function of the defect strength was reported . a similar problem has been recently investigated for solitons of the gross - pitaevskii equation ( gpe ) with rectangular potential wells and for attractive interatomic interactions @xcite . similar studies were also done for point defects of the discrete nlse equation @xcite which corresponds , under suitable conditions , to a tight binding model of bec in a deep ol @xcite . the resonant transmission , reflection and trapping of discrete breathers by point - defects was investigated in @xcite . in contrast to these studies , however , the scattering of continuous gpe gss by ol defects have been scarcely investigated . in this context , we mention the numerical study performed in @xcite where the scattering properties of gss were suggested to be useful to construct quantum switches and quantum memories . to our knowledge , however , the mechanism underlying resonant transmissions of gpe gss by ol defects and the possibility of multiple defects resonant scattering have not yet been discussed . in the present paper we provide an extensive numerical investigation of the scattering properties of gss by ol defects and identify the physical mechanism underlying the phenomenon of the resonant transmission . in particular , we show that the presence of repeated reflections transmission and trapping regions observed for increasing strengths of an ol defect is associated to the impurity modes inside the defect potential with energies ( chemical potentials ) and numbers of atoms matching corresponding quantities of the incoming gs . as the ol defect strength is increased , new impurity modes enter from the gap edges , moving toward the center of the gap ( bottom of the potential ) . this implies that for incoming gss with chemical potentials very close to band edges , the number of resonances observed in the scattering coincides with the number of bound states which can exist in the defect potential for the given defect strength . this fact is demonstrated both by studying stationary states inside the defect potential and by direct numerical integrations of the gpe . an excellent agreement between the two approaches is found , this confirming the correctness of our interpretation . the dependence of the resonant transmission on the incoming velocity of the gs is also investigated both for fundamental gss in the semi - infinite gap and for gss in the first gap zone . as a result we show that the transmission resonant peaks become wider as the incoming velocity is increased , with very sharp peaks at small velocities . the multiple resonant transmission through a series of ( two and three ) ol defects is also demonstrated . we show in this case that for equally spaced identical ol defects the widths of the full transmission resonances in general decreases as the number of defects is increased . we demonstrate , however , that the resonant transmission through a series of defects can be achieved if defects are designed so to compensate off - resonance detunings introduced by velocity changes . this fact gives rise to the possibility of using arrays of ol defects as very precise filters for matter wave dynamics . the paper is organized as follows . in sec . ii we present the model equation and discuss the basic properties of gss of the gpe . in the sec . iii we consider the interaction of small amplitude gss with a localized gaussian impurity in the ol . the problem is investigated by means of direct numerical integrations of the gpe for both attractive and repulsive interactions as well as for attractive and repulsive defects . in section iv we use a stationary defect mode analysis to show that the repeated resonant transmission , reflection and trapping regions occurs in correspondence of resonances with impurity modes inside the defect potential and investigate their dependence on gs incoming velocities . in section v the multiple resonant transmission across a series of equidistant ( equal and unequal ) ol defects , is investigated . in sect . vi the main results of the paper are shortly resumed . let us consider a cigar - shaped bec described by the following normalized one - dimensional gpe @xcite @xmath0 where @xmath1 denotes an external potential of the form : @xmath2 , with @xmath3 a perfect ol of period @xmath4 : @xmath5 , and @xmath6 a defect potential consisting of a sum of @xmath7 single wells potentials localized on a distance of several lattice period around the ol sites @xmath8 , @xmath9 . in the following we assume @xmath10 and @xmath11 to have the form @xmath12 \label{defect}\end{aligned}\ ] ] where without loss of generality the period of the ol is taken @xmath13 . we remark that results of this paper will not qualitatively dependent on type of defect ( we used gaussian defects just for numerical convenience ) and similar results can be obtained for other shapes of the defects , like square well defects , for example ) . in the following we will mainly restrict to the cases of few ol defects : @xmath14 only . we also remark that in eq.([gp ] ) the normalization has been made by measuring the energy in units of recoil energy @xmath15 , where @xmath16 and @xmath17 is the lattice constant , the space coordinate and time in units of @xmath18 and @xmath19 , respectively . the dimensionless macroscopic wave function is also normalized as @xmath20 , where @xmath21 is the s - wave scattering length . it is well known that in the absence of the defect potential @xmath22 , eq.([gp ] ) posses families of exact gs solutions with energy located in the band - gaps of the linear eigenvalue problem @xmath23\varphi_{\alpha k}=0,\ ] ] where @xmath24 are orthonormal set of bloch functions with @xmath25 denoting the band index and @xmath26 the crystal - momentum inside the first brillouin zone ( bz ) : @xmath27 $ ] . it is also known that small - amplitude gss with chemical potentials @xmath28 very close to band edges are of the form @xmath29 with the envelope function @xmath30 obeying the following nlse @xmath31 where @xmath32 and @xmath33 are slow temporal and spatial variables , @xmath34 denotes the soliton effective mass and @xmath35 the effective nonlinearity @xcite . the condition for the existence of such solitons is @xmath36 and coincides with the condition for the modulational instability of the bloch wavefunctions at the edges of the bz @xcite . examples of small amplitude gss with chemical potential inside the band - gap structure are depicted in fig . [ fig_e(k ) ] . in the presence of very diluted ol defects , gss will continue to exist and away from defects they practically coincide with gss of the undistorted ol . an attractive ( local potential well ) or repulsive ( local potential barrier ) ol defect will be seen by the gss differently , depending on the sign of their effective mass . thus , for example , in the case of repulsive interactions and a negative effective mass , a gs approaching a repulsive defect will see it as a trapping potential ( rather than as a potential barrier ) , thus besides being totally or partially transmitted / reflected , it can also be trapped at the defect site , a fact which would be impossible in absence of the ol . in all the numerical simulations presented in this paper we have used stationary gss of eqs . ( [ gp])([defect ] ) , exactly determined by the shooting method @xcite or by self - consistent calculations @xcite , and have put them in action by means of phase imprinting ( e.g. we multiply the state by the phase factor @xmath37 , with @xmath38 being the gs velocity ) . this provides initial condition for the gpe numerical time integration of the form : @xmath39 . for possible experimental implementations of our results , an alternate method to use to put the gs in action could be the acceleration of the ol for a short time interval to move the stationary state away from the bz edges ( center ) so that it can acquire a small bloch velocity @xmath40 ( for details on how this can be done see @xcite ) . we consider first the scattering of a gs by a single localized defect ( @xmath41 in eq . ( [ defect ] ) ) . in order not to perturb the soliton initially , the distance between the soliton center and the ol defect is taken much larger than the width of the gs ( @xmath42 ) . in the following we compute the trapping , transmission and reflection coefficients defined as @xmath43 , @xmath44 and @xmath45 , with @xmath46 , @xmath47 and @xmath48 denoting the numbers of atoms trapped , transmitted and in the initial state , respectively . the final time @xmath49 depends on the initial velocity of the gs and in the numerical experiment is determined as the time necessary for the coefficients @xmath50 to become stationary after the scattering process has occurred . the trapping region @xmath51 $ ] has the size of the initial soliton and in all our calculations we fix @xmath52 . for the scattering of a gs in the semi - infinite gap to occur , the existence criterion for a gs near the bottom edge of the first band ( where the effective mass is positive ) implies that the nonlinear coefficient must be negative sign@xmath53 . by applying a small initial velocity to the gs in the defect direction , depending on the amplitude @xmath54 and width @xmath55 of the defect , three possible scenarios can occur : i ) complete reflection , @xmath56 ; ii ) complete transmission , @xmath57 ; iii ) partial trapping , @xmath58 . the regions of the parameter space @xmath59 where these different regimes occur , as obtained from direct numerical integration of eq.([gp ] ) , are reported in fig . [ eta(d)_w-0125_v005 ] . the dependence of @xmath60 on the defect strength @xmath54 for two different incoming gs velocities and for a fixed value of the defect width @xmath61 is depicted in fig . we see that by changing the strength of the defect it is possible to achieve complete reflections ( @xmath62 ) or transmission ( @xmath63 ) as well as partial trappings ( @xmath58 ) . the profiles of the initial , reflected and transmitted gs are depicted in the top panels of fig.[fig3 ] for defect strengths corresponding to points labeled in the middle panel by letters a , b , c. by comparing the middle and the bottom panels of fig.[fig3 ] it is clear that the sharp peaks at small incoming velocities ( @xmath64 ) for which @xmath63 ( see points a , c , e , f , g in the middle panel ) become wider as the velocity is increased while the regions for which @xmath62 are a bit reduced ( also see fig.[fig3_transm_0.475 ] for the case of first band - gap gss ) . the first four impurity modes corresponding to the transmission peaks @xmath65 in the middle panel of fig.[fig3 ] have been depicted in fig . [ modes_fig3 ] . notice the alternating odd - even symmetry of these modes ( modes d and f being odd and modes e and g being even with respect to the center @xmath66 of the defect potential ) as usual for eigenstates of one - dimensional trapping potentials . we remark that the existence of four resonant transmission peaks ( and reflection regions ) seen in fig.[fig3 ] for @xmath67 correlates with the existence of four impurity modes for the given defect strength region ( see stationary defect mode analysis below ) . similar results as those of the previous section can be found for gss inside the first band - gap , with the only difference that now there are two possibilities for the existence of small - amplitude solitons : i ) in the vicinity of the top edge of the first band where the effective mass is negative and therefore gss can exist only for repulsive interactions @xmath68 ; ii ) and in the vicinity of the bottom edge of the second band where effective mass is positive and gs exist only for attractive interactions @xmath69 . the small - amplitude gs near the first band - gap edges b , c , in fig.[fig_e(k ) ] are shown by the corresponding profiles @xmath70 and @xmath71 depicted in the figure . as remarked before , the sign of the effective mass determines the type of the interaction of the gs has with the defect potential and for negative gs effective mass ( repulsive interatomic interactions ) the defect will be seen as a defect trapping potential ( supporting therefore bound states ) if the defect strength @xmath54 is positive rather than negative ( as seen for the case of a positive effective mass ) . except for this , results go in parallel with those of the previous section and have been collected in figs.[fig3_transm_0.475 ] , [ fig3_transm_0.475_v ] for the case of an initial gs close the top edge of the first band ( negative effective mass ) . in particular , from the bottom panel of fig . [ fig3_transm_0.475 ] and fig . [ fig3_transm_0.475_v ] we clearly see that for a fixed defect width the region of the resonant transmission ( corresponding to red color ) becomes wider as the incoming gs velocity is increased , while the full reflected regions are achieved mainly for small velocities , as one could have expected . we remark that also in this case the number of resonant transmission reflection and trapping regions are found to correlate with the number of impurity modes present in the defect potential for the given range of the defect strength ( first four modes corresponding to the maxima of the trapping coefficient c in fig.[fig3_transm_0.475 ] are depicted in fig.[fig3_transm_0.475_modes ] , [ modes_fig4_0475 ] ) . as remarked before , the presence of a repulsive ( attractive ) ol defect in the gpe affects the existing gs states for @xmath72 and introduces additional localized states inside the gap in presence of repulsive ( attractive ) nonlinearities . numerical calculations show that the band structure is only slightly affected by an ol impurity , the main effect being the introduction of bound states spatially localized at the impurity sites and with chemical potentials inside the band - gap . note that repulsive ( attractive ) ol defects in the presence of an attractive ( repulsive ) nonlinearity can not introduce additional bound states because the corresponding impurity potentials correspond to barriers rather than potential wells , due to the positive ( negative ) effective mass . recalling that the opposite signs of nonlinearity and effective mass is a necessary condition for the gs existence , one has that the effective impurity potential acts as a trapping potential when @xmath54 and @xmath73 have equal signs . away from the ol impurity localized states are practically the same as for @xmath72 case . at the defect site , however , gss levels get slightly shifted by the impurity potential and additional impurity modes enter the gap . a gs moving through the impurity will have , in general , a mismatch in energy and in number of atoms with the impurity modes , this giving a partial reflection / transmission of the incoming matter wave . a total transparency of the ol defect is expected for incoming gs energies and number of atoms exactly matching those of an impurity modes inside the defect potential ( notice that the energy of impurity modes depend on @xmath54 and on the number of atoms ) . by increasing the strength of the impurity , the depth of the defect potential increases and more impurity modes enter the gap . as @xmath74 is increased , the energies of these modes enter the gap from the top ( bottom ) of a band for impurity strength and the nonlinearity both positive ( negative ) . this implies that the transparency ( complete transmission ) of the impurity occurs in correspondence of each impurity mode entering the gap and matching the energy of the incoming gs given by @xmath75 where the first term @xmath76 represents the energy of the stationary gs state at @xmath77 ( eg . with @xmath78 ) , while the second one is the contribution due to the kinetic energy ( here @xmath79 is the bloch velocity and @xmath80 the effective mass ) . for small velocities the energy ( [ energyshift ] ) practically coincides with @xmath76 but in general the kinetic term should be accounted in the matching in energy with the impurity levels ( see below ) . notice that eq . ( [ energyshift ] ) is only valid near stationary points @xmath81 ( bottoms or tops of a band ) where @xmath82 and @xmath83 can be written as @xmath84 in the range of initial velocities @xmath85 $ ] we have considered , the energy curves @xmath86 in vicinity of @xmath87 and @xmath88 are very well approximated by eq . ( [ energyshiftapprox ] ) with @xmath89 , @xmath90 , and @xmath91 and @xmath92 for bottom and top edges of the band , respectively . by knowing @xmath93 ( e.g. @xmath26 ) and @xmath94 one can compute the energy shift due to the nonzero velocity to be accounted in the matching between the gs and the impurity levels ( see lower panels of figs . [ modes_fig4_-0125 ] , [ modes_fig4_0475 ] ) . notice that the kinetic energy has the sign of the effective mass so that @xmath83 is pushed forward the corresponding band edge for finite @xmath93 , meaning that inside the impurity potential the gs matching condition with an impurity mode can be achieved for a lower values of @xmath74 . from this we expect the resonance transmission peaks to be shifted away from the @xmath95 resonance toward lower values of @xmath74 as @xmath93 is increased . to confirm these prediction with gpe calculations we have solved the stationary problem @xmath96u-\sigma u^3=0\end{aligned}\ ] ] searching for bound states @xmath97 localized at the ol defect using both shooting method and self - consistent calculations . using these approaches we found that the values of @xmath54 for which the gaussian defect becomes transparent to the gs dynamics , correspond to the values for which a localized mode inside the defect potential has energy and number of atoms matching the corresponding quantities of the incoming gs . since the discussion is very similar ( a part implications due to signs of the effective masses ) for gs of the semi - infinite and for the ones inside finite - gaps , we refer to the case of an initial gs inside the semi - infinite gap , with a positive effective mass with chemical potential close to the bottom of the first band . in the top panel of fig.[modes_fig4_-0125 ] we have shown the dependence of the number of atoms @xmath98 in a defect mode of a given symmetry on @xmath54 , for an energy @xmath99 corresponding to a gs with zero incoming velocity ( point a of figs . [ fig_e(k ) ] , [ fig3 ] ) . the horizontal dotted line refers to the numbers of atoms in the initial gs . in the bottom panel of fig . [ modes_fig4_-0125 ] we show the energy mismatch @xmath100 between the energy of a defect modes @xmath83 and the one of an incoming gs given by eq . [ energyshiftapprox ] for different incoming velocities . notice that the intersection points @xmath101 of the dotted line @xmath102 with the curves @xmath103 are in coincidence with the zeros of the function @xmath100 displayed in the bottom panel for @xmath64 and correspond to the maxima of the transmission coefficient @xmath104 in fig.[fig3 ] ( see curves @xmath64 in the middle panel ) . [ modes_fig4_-0125 ] ( see also [ modes_fig4_0475 ] for repulsive case ) also explains the decay of the reflection coefficient and the rising of the trapping coefficient coinciding up to the maxima observed ( just before the resonance ) for increasing values of @xmath74 away from the resonance points @xmath101 . to the right of these points in fig.[modes_fig4_-0125 ] , the number of particles in the defect mode is higher then the number of particles in the incoming soliton @xmath105 . in this case the gs can not `` use '' a defect mode to pass the ol impurity and will be fully reflected . on the left of the resonance points , the number of atoms in the gs is higher than the one in the defect mode so that the gs can be captured by the ol impurity by releasing the excess number of atoms into the reflection and transmission channels . it is clear that the peak of the capture coefficient occurs just before the resonant transmission ( e.g. for @xmath106 ) of the gs through the defect ( achieved when the condition @xmath107 is exactly fulfilled ) . notice from fig.[modes_fig3 ] that at the resonant transmission , the profiles of the stationary impurity modes obtained by solving eq.([matheiu_def ] ) exactly coincide with the solution of the gpe during the passage trough the defect . in fig.[-0125_odd_even_eta(d ) ] we have also depicted , similarly to fig.[eta(d)_w-0125_v005 ] , the level curves @xmath102 as a function of the amplitude @xmath54 and the width @xmath55 of the defect . from the bottom panel of fig . [ modes_fig4_-0125 ] it is also clear that for @xmath108 the resonances shift in the direction of lower values of @xmath74 ( as expected from our analysis ) . this well correlates with the gpe calculations reported in the bottom panel of fig.[fig3 ] . similar results are obtained for a gs near the bottom of the first band gap ( point b in fig . [ fig_e(k ) ] ) for the case of repulsive interactions . this is shown in fig.[modes_fig4_0475 ] . from which we see that the shift of the resonance due to the finite bloch incoming velocity is always in the direction of lower values of @xmath54 and are in good agreement with the gpe numerical results reported in the bottom panel of fig . [ fig3_transm_0.475 ] . in particular , for the velocity @xmath109 and the resonance near @xmath110 we obtain from fig . [ modes_fig4_0475 ] that the resonance peak is at @xmath111 while from the gpe result in fig . [ fig3_transm_0.475 ] we obtain the value @xmath112 . it is also worth to note from the bottom panels of figs . [ modes_fig4_-0125 ] , [ modes_fig4_0475 ] , that for a fixed energy mismatch the widths of the curves increase as the incoming velocity is increased , a fact which correlates with the broadening of the transmission peaks observed in the gpe calculations ( compare bottom panels in fig . [ fig3 ] and in fig [ fig3_transm_0.475 ] , respectively ) . from this we conclude that the above impurity mode analysis fully confirms the interpretation of the transmission peaks as resonances between incoming gss and impurity modes . an interesting question to ask is whether the soliton resonant transmission could survive multiple impurity scatterings . in fig . [ fig10 ] we show the transmission , reflection and trapping regions of an attractive gs of the gpe with two identical gaussian ol impurities , placed at @xmath66 and @xmath113 , respectively . the tcr curves depend also on the distance between the two impurities and this could be varied so to find optimal values for double resonant transmission to occur . the phenomenon , however , is more sensitive to variations of the incoming gs velocity as one can see by comparing fig.[fig10 ] with the case of single impurity in fig . [ fig3 ] . notice that while the transmission resonance peaks at small velocities are practically unaffected by the presence of the second impurity , they become more narrow at larger velocities for the resonant scattering on two impurities . the shrinking of the resonances at larger velocities can be understood by the fact that for higher incoming velocities the velocity of the gs after the first impurity is slightly reduced and the variation introduces a detuning from resonance in the scattering with the second impurity which in turn reduces the double resonance width . from this one can expect that in the presence of more lattice defects the multiple resonance transmission peaks become very narrow and only solitons with very precise initial matching velocities will be able to pass . in general the gs may be able to pass only a finite number of impurities before remaining trapped at one impurity or becoming scattered back and forth between them . this is shown in fig.[fig11 ] where contour plots of the gs space - time dynamics in presence of two identical gaussian defects are reported for different strengths @xmath54 and for the same parameters as for the bottom panel of fig.[fig10 ] . in the left top and bottom panels we see the occurrence of total reflection and transmission for a resonant value of @xmath54 while in the corresponding right panels we show the back and forth dynamics of the soliton between two impurities ( top panel ) and the trapping by the second impurity ( top panel ) for a non resonant values of @xmath54 . notice the small change of the soliton velocity after the passage through the first defect . to overcome the detuning from resonance induced by the change of velocity one could design the strength of the second impurity so to match the value of the intermediate velocity and still achieving @xmath63 . in this way one can realize a double ( or multiple ) filtering of the soliton motion so that only gss having very precise initial velocity can overcome the defect series . this possibility is illustrated in fig . [ fig12 ] where the resonant transmission of a gs though a series of three gaussian defects of the ol , with defect strengths designed so to compensate the detunings introduced by the velocity changes , is shown . notice from the left panel of fig . [ fig12 ] that in the case of equal defects with the initial velocity matching the resonance transmission peak of the first defect , the gs can not be transmitted through all the three defects but it is stopped at the second defect . the dynamics of the soliton in presence of multiple defects may be quite complicated and is beyond the aim of this work ( an investigation in the parameter space of the possible scenarios for the gs time evolution will be discussed elsewhere ) . in this paper we have investigated the scattering properties of matter wave gap - solitons with optical lattice defects in the framework of the mean - field gross - pitaevskii equation . we have shown that the occurrence of repeated reflection , transmission and trapping regions are in correspondence of the defects strengths for which the number of atoms and energies of additional bound states created by the optical lattice defect , match the ones of the incoming gap - soliton . this has been demonstrated by a study of the stationary defect modes energies ( chemical potentials ) and number of atoms as a function of the defect strength . a very good agreement between the predicted values of the resonant transmission peaks by means of impurity modes and the ones found by direct time integrations of the gross - pitaevskii equation , is found . the behavior of the reflection and trapping curves have also been explained by impurity mode analysis . these investigations have been performed both for attractive and repulsive interactions and for localized states both in the semi - infinite gap ( attractive case ) and in the first gap zone ( attractive and repulsive cases ) . the dependence of the resonant transmission on the initial gap - soliton velocity has been also investigated . we have shown that the positions of resonant transmission peaks shift toward lower values @xmath74 as the initial gap - soliton velocity is increased , while the widths of resonances shrinks to zero at very small gap - soliton velocities . the possibility of multiple resonant transmission through an arrays of defects was also demonstrated . in particular , we have shown that for an optical lattice with two equally spaced identical gaussian defects the widths of the full transmission resonances for larger incoming velocities , decreases as the number of defects is increased . the sharpening of the transmission peaks has been explained in terms of the detuning from resonance introduced by the small velocity change after the passage of an optical lattice defect . finally , the resonant transmission of a gap - soliton though a series of gaussian defects with unequal strengths designed so to compensate detunings introduced by the velocity changes , was demonstrated . these results give the possibility to construct very precise filters for the matter wave gap - soliton dynamics by means of properly designed arrays of ol defects . v.a.b . acknowledges support from the fct grant , ptdc / fis/64647/2006 . ms acknowledges the miur ( prin-2008 initiative ) for partial financial support . see review papers o. morsch and m. oberthaler , rev . phys . * 78 * , 179 ( 2006 ) and references therein ; i. bloch , j. phys . b * 38 * , s629 . ( 2005 ) ; d. jaksch , and p. zoller , ann . phys . n.y . * 315 * , 52 ( 2005 ) ; v.a . brazhnyi and v. v. konotop , mod . . lett . * 18 * 627 ( 2004 ) . o. morsch , j. h. mller , m. cristiani , d. ciampini , and e. arimondo , phys . lett . * 87 * , 140402 ( 2001 ) ; i. carusotto , l. pitaevskii , s. stringari , g. modugno , and m. inguscio , phys . rev . lett . * 95 * , 093202 ( 2005 ) . m. jona - lasinio , o. morsch , m. cristiani , n. malossi , j. h. mller , e. courtade , m. anderlini , and e. arimondo , phys . lett . * 91 * , 230406 ( 2003 ) ; s. wimberger , r. mannella , o. morsch , e. arimondo , a. r. kolovsky , and a. buchleitner , phys . a * 72 * , 063610 ( 2005 ) . baizakov , v.v . konotop , and m. salerno , j. phys . b * 35 * 5105 ( 2002 ) ; e.a . ostrovskaya , yu.s . kivshar , phys . lett . * 90 * , 160407 ( 2003 ) ; j. yang and z. musslimani , opt . lett . * 23 * , 2094 ( 2003 ) ; b.b . baizakov , b.a . malomed , and m. salerno , europhys . lett . * 63 * , 642 ( 2003 ) .

the physical mechanism underlying scattering properties of matter wave gap - solitons by linear optical lattice defects is investigated . the occurrence of repeated reflection , transmission and trapping regions for increasing strengths of an optical lattice defect are shown to be due to impurity modes inside the defect potential with chemical potentials and numbers of atoms matching corresponding quantities of an incoming gap - soliton . for gap - solitons with chemical potentials very close to band edges , the number of resonances observed in the scattering coincides with the number of bound states which can exist in the defect potential for the given defect strength . the dependence of the positions and widths of the transmission resonant on the incoming gap - soliton velocities are investigated by means of a defect mode analysis and effective mass theory . the comparisons with direct integrations of the gross - pitaevskii equation provide a very good agreement confirming the correctness of our interpretation . the possibility of multiple resonant transmission through arrays of optical lattice defects is also demonstrated . in particular , we show that it is possible to design the strength of the defects so to balance the velocity detunings and to allow the resonant transmission through a larger number of defects . the possibility of using these results for very precise gap - soliton dynamical filters is suggested .

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Proceed to summarize the following text: type ii superconductors @xcite allow for a partial penetration of magnetic field into the bulk of the superconducting ( sc ) material when the applied field @xmath4 satisfies @xmath5 . in a seminal work abrikosov @xcite showed that when the ratio @xmath6 , where @xmath7 is the magnetic field penetration depth and @xmath8 is the coherence length , is greater than @xmath9 the magnetic field penetrates the sc material in the form of flux - lines ( fls ) . these fls are also called vortices , since they are surrounded by circular currents . each fl carries a quantized unit of flux @xmath10 called the fluxoid . the fls have cylindrical shape of radius @xmath11 ( the radius is not sharp since the magnetic field decays exponentially like @xmath12 , where r is the distance from the axis ) and a non - sc core of radius @xmath13 . due to a repulsive interaction among the fls , they arrange themselves in a triangular lattice referred to as the vortex solid ( vs ) . this result follows from mean - field theory . after high - temperature superconductors were discovered in the 1980 s , it became apparent that thermal fluctuations , not included in the mean - field theory @xcite , play an important role at relatively high temperatures and fields , still below @xmath14 and @xmath15 . these fluctuations can cause the abrikosov lattice to melt into a disordered liquid via a first order transition ( fot ) , which can be roughly estimated using the lindemann criterion known from solid state physics @xcite . technologically , the melting of the fl lattice is important since the vortex liquid ( vl ) is not actually sc due to the dissipation caused by the fl motion when an electric current passes through the system . pinning of fls by naturally occurring defects in the form of vacancies , interstitials , twin and grain boundaries etc . , is effective to impede fl motion in the vs phase , where the fls form a rigid correlated network . the effectiveness of the pinning manifests itself by leading to high critical currents . in the vl phase pinning of a few vortices does not inhibit others from moving when a current is applied . thus for practical purposes the sudden increase in resistivity occurs at the melting transition rather than when @xmath16 for any reasonably non - vanishing currents . the existence of the melting transition in high-@xmath14 pristine materials has been established through numerous experimental @xcite and numerical @xcite studies . as was mentioned above , disorder in the form of points defects and sometimes more extended defects can and does occur naturally in laboratory samples . in addition artificial point defects can be induced by bombarding the sample with electrons originating from particle accelerators . extended columnar defects in the form of linear damaged tracks piercing through the sample can be induced by heavy ion irradiation . both naturally occurring and artificially induced defects are situated at random positions in the sample and their effective pinning strength ( i.e. their interaction with fls ) can also vary from defect to defect . thus defects play the role of quenched disorder . the adjective `` quenched '' refers to the immobility of these defects during experimental time scales . introduction of disorder in terms of point defects or columnar pins affects both the properties of the solid and liquid phases and might also shift the location of the melting transition in the h - t plane @xcite . in the case of point pins , the vs phase is replaced with a bragg glass phase @xcite , characterized by quasi - long - range order . the melting transition is predicted to shift towards lower temperatures @xcite . in the case of columnar pins the vs phase is replaced with a so called pinned bose glass @xcite where fls are trapped by the columnar defects and the whole lattice becomes immobile . the bose glass phase is similar to the localized phase of a two dimensional repulsive bose gas in the presence of quenched disorder , as will be explained in more detail in the next section . the effect of both kinds of disorders on the fls melting has been studied experimentally in various high - temperature superconductors . two common materials that have been extensively investigated are yba@xmath0cu@xmath1o@xmath2 ( ybco ) and bi@xmath0sr@xmath0cacu@xmath0o@xmath3 ( bscco ) , both having critical temperatures ranging between 90 - 120 k at @xmath17 . the main difference between these materials is their anisotropy parameter @xmath18 , where @xmath19 and @xmath20 denote the effective masses of electrons moving along the @xmath21-axis and the @xmath22-plane respectively . bscco is much more anisotropic : its anisotropy parameter @xmath23 lies in the range of 50 - 200 compared to the range of 5 - 7 for ybco @xcite . this fact causes the fls to be much `` softer '' or elastic . thus in the case of bscco the fls are sometimes described as a collection of loosely connected `` pancakes '' residing in adjacent cu - o planes . experimental studies on ybco have shown a marked shift in the irreversibility line in the presence of the columnar disorder @xcite . the irreversibility line in the h - t plane marks the onset of hysteresis effects and is located close to the melting transition on the solid phase side . for bscco , many experimental studies have been conducted @xcite . the more recent ones have shown @xcite that the melting line is not shifted when the density of columnar defects is relatively low , @xmath24 , but for @xmath25 a shift in the position of the melting transition is observed . here the matching field @xmath26 is defined as @xmath27 where @xmath28 is the density of the columnar defects and @xmath29 is the flux quantum . theoretical work on columnar disorder includes bose glass theory @xcite . radzihovsky @xcite considered the possibility of two kinds of bose glass phases ( strongly or weakly pinned ) depending on whether @xmath30 or @xmath31 . more recently , columnar as well as point disorder were investigated by goldschmidt @xcite using replica field theory . he showed that the melting line shifts to lower temperature in the case of point disorder and to higher temperature in the case of columnar disorder . due to the complexity of the problem , especially in the presence of disorder , simulations have been very useful in studying the fl system . there have been many simulation studies of the vortex system in the presence of disorder . however , most simulation work has been confined to the addition of point disorder only . in particular , there is little work done on the effects of columnar disorder on the fl melting . recent work by wengel and tuber @xcite concentrated on the case of high defect density region @xmath32 . in contrast , a recent simulation study by sen _ et al . _ @xcite uses a small density of columnar defects , each of infinite strength , but considers an extremely small magnetic field . similarly , nandgaonkar _ et al . _ @xcite also investigate the case of a very small magnetic field . in this paper we consider columnar defects of a finite strength ( @xmath33 ) with a relatively low defect density , @xmath34 , and with realistic magnetic fields as used in the experiments . we first discuss the method implemented for ybco : following nelson @xcite , we map the system of @xmath35 vortices ( fls ) in a high-@xmath14 superconductor onto @xmath35 interacting bosons in 2-space + 1-time dimensions . then we do a path integral monte carlo ( pimc ) simulation on this system . the partition function of @xmath35 vortices can be expressed as : @xmath36 where @xmath37 denotes the 2d position vector of the ith vortex at a height @xmath38 along the @xmath21-axis , @xmath39 is the free energy as a function of temperature @xmath40 , @xmath41 is the length of the sample along the @xmath38-direction . the london free - energy functional @xmath42 is given by @xmath43 where @xmath44 @xmath45 is the vortex line energy per unit length , the line tension is @xmath46 and @xmath47 . here @xmath48 is the lattice spacing , @xmath49 is the magnetic field along the @xmath38-direction and @xmath23 is the anisotropy parameter . @xmath50 is the in - plane interaction potential between two fls a distance @xmath51 apart . @xmath52 denotes the modified bessel function of the first kind . this expression for the london free - energy is an approximation that neglects the non - local interaction of real vortices and replaces it by in - plane interactions only , which is really justified if the fls do not deviate too much from straight lines along the @xmath38-axis @xcite . with this approximation , the system of @xmath35 interacting fls is equivalent to a system of @xmath35 bosons in @xmath53 dimension interacting with a pairwise potential @xmath50 . the path integral representation of a system of @xmath35 bosons of mass @xmath54 each in two dimensions , interacting through a potential @xmath55 , with @xmath56 being the strength and @xmath7 being the range of the repulsive interaction , is given at finite temperature @xmath57 in terms of the imaginary time action @xmath58 here @xmath59 is the imaginary time and @xmath57 is the temperature of the bose system . we see that there is a one to one parameter mapping between the boson system and the vortex system @xcite : @xmath60 where @xmath61 is the average density of bosons ( and fls ) and @xmath62 is the area of the sample . we can write the london free - energy functional in a dimensionless form as follows : @xmath63 where @xmath64 all lengths are measured in units of @xmath65 and energies in units of @xmath66 for bosons and @xmath67 for fls . we next discretize the integral along the @xmath38-axis by dividing it into @xmath68 segments : @xmath69/kt}\label{xi}\ ] ] where @xmath70 , @xmath54 labels the planes and @xmath71}{kt}=\sum _ { i , m } \frac{\left(\mathbf{r}_{i , m+1}-\mathbf{r}_{i , m}\right)^{2 } } { 2\lambda^{2}\tau } + \sum _ { i < j}\tau k_{0}\left(r_{ij , m}/\tilde{\lambda } \right).\ ] ] we work only in the limit where @xmath72 is large , which amounts to taking @xmath41 very large and in the mapping onto @xmath73 bosons corresponds to the ground state of the bosons at the absolute zero temperature ( @xmath74 ) . we used @xmath75 and discretized the @xmath38-axis into @xmath76 planes . we used the matrix - squaring method @xcite to calculate the right action so that we can work with a small number of planes along the @xmath38-direction . working with the primitive action requires the use of a large number of slicing of the @xmath38-direction and is very time consuming @xcite . the boundary conditions in the @xmath38-direction for a system of bosons are @xmath77 , with all permutations @xmath78 of the indices being allowed . this is what is meant by the summation over @xmath79 in eq . ( [ xi ] ) . for small @xmath56 ( a small repulsive interaction ) , which corresponds to large @xmath80 , we expect a bose - einstein condensed phase where permutations are important . this is the superfluid phase . for small @xmath80 , repulsion is large and permutations are rare as the bose system is in its classical phase which is a wigner crystal . these two phases correspond respectively to the fl liquid and solid phases . the melting line represented by @xmath81 is given approximately by the expression @xmath82 . near @xmath83 this behavior is more complicated ( see below ) . we now discuss the method of simulations of bscco : because of the high anisotropy of the bscco system one can not use the simple picture given above . here , instead , we follow a different model which can be cast in a form analogous to ybco . we take the lawrence - doniach model @xcite as our starting point for bscco . it leads to the following form of the london free - energy for inter - layer ( il ) josephson coupling @xcite : @xmath84 , \label{quadjos}\ ] ] for @xmath85 , and @xmath86 , \label{linjos}\ ] ] for @xmath87 , where @xmath88 is the inter - layer spacing and @xmath89 is the healing length defined by @xmath90 . for the in - plane ( ip ) coupling we use : @xmath91 in principle one should add to the above interaction an electromagnetic interaction among the pancake vortices @xcite . the electromagnetic interaction becomes dominant in the limit of infinite anisotropy ( @xmath92 ) . @xcite argue , ( see also ref . ) that for the value @xmath93 for bscco the josephson coupling still dominates by one order of magnitude over the electromagnetic coupling . their argument goes as follows : clem @xcite shows that if one has a straight array of pancake vortices along the @xmath38-axis , and one pancake is displaced a distance @xmath94 in the @xmath95-direction than the magnetic energy of the configuration increases by an amount @xmath96 where @xmath97 is euler s constant ( = 0.5772 ... ) . for large @xmath94 ( @xmath98 ) , the modified bessel function @xmath99 decays exponentially and thus the energy increases like @xmath100 . for small @xmath94 the bessel function can be expanded in a power series in @xmath101 @xmath102 and thus the magnetic energy behaves like @xmath103 to leading order in @xmath94 which is the same as the quadratic behavior of the josephson energy in eq . ( [ quadjos ] ) above . the ratio of the coefficients of the quadratic terms in the magnetic and josephson energies goes roughly like @xmath104 ( where @xmath105 for bscco ) . thus for anisotropy @xmath93 we get a factor of 0.25 ( a somewhat more precise estimate @xcite gives a ratio of about 0.1 ) . thus the magnetic interaction is negligible compared to the josephson interaction for @xmath93 . for samples with @xmath106 these interactions are already comparable . for large values of @xmath94 the magnetic interaction increases logarithmically and the josephson interaction increases linearly so the magnetic interaction is always negligible . the key to the estimate given above is to consider not just two pancake vortices but a whole line with one displaced pancake . this argument is valid if the deviations of the vortices from straight lines are not too large . as for the in - plane interaction clem showed that a linear array of pancake vortices gives rise to exactly the same magnetic field at a distance @xmath94 away from it as produced by an abrikosov vortex line . ( [ logjos ] ) is consistent with the magnetic interaction of pancake vortices , again when the fls do not deviate too much from straight lines . equations ( [ quadjos])-([logjos ] ) for bscco can be cast in a form similar to that for the ybco with the following substitutions : @xmath107 with these changes the london free - energy functional would look like@xmath108,\ ] ] for @xmath109 , and @xmath110- \frac{(2r_{g}-(|\mathbf{r}_{i , m}-\mathbf{r}_{i , m+1}|))^{2 } } { 2\lambda ^{2}\tau } , \label{second}\ ] ] for @xmath111 , when now again all lengths are measured in units of @xmath65 . while doing simulations at a fixed magnetic field @xmath49 and temperature @xmath40 , the term @xmath112 will remain constant and would drop out of @xmath113 term in the boltzmann factor . it however needs be considered during the measurement of the energy . the second term in eq . ( [ second ] ) can be easily handled at the last stage of the bisection method ( see the next section for a discussion of the bisection method ) . we can make use of a reduced temperature variable to make some expressions look simpler . first , using the fact that the temperature dependence of @xmath114 arises mainly through @xmath115 and neglecting the logarithmic corrections , one gets @xcite @xmath116 and hence@xmath117 defining reduced temperature as : @xmath118 we obtain@xmath119 this shows that the equation for the melting line is approximately@xmath120 note that some authors @xcite use a temperature dependence of @xmath121 in @xmath7 , or even @xcite @xmath122 . all these choices coincide near @xmath14 . the choice of temperature dependence of @xmath7 is not expected to have a significant effect on the results . the technique that we use to simulate our system is called multilevel monte carlo simulation ( mmc ) @xcite . there are several advantages in using this technique for the simulation of the fls over the usual metropolis monte carlo ( mc ) method . in the discrete model , we work with @xmath35 fls with the @xmath38-axis discretized into @xmath68 planes , thus resulting in @xmath35 beads in each of the @xmath68 planes . in the usual mc method one would displace a few of these beads in a plane by small random displacements inside a two dimensional box and then would accept or reject the move based on a probability given by the boltzmann factor . a big disadvantage of using this technique is that it is difficult to move beads appreciably from their original positions over a number of mc steps . the reason for this is that a bead in a plane belonging to a fl finds itself in a local harmonic potential generated by the kinetic energy term involving this bead and the beads belonging to the same fl on either side of the plane . this harmonic potential becomes stronger and stronger at lower temperatures and magnetic fields . as a result , in the usual mc simulations beads keep moving around inside these local harmonic cages and end up sampling only a small part of the phase space . the other problem with the usual mc method is that there is no natural easy way of implementing fl cutting . if there are two fls twisted around each other and if it is energetically favorable for them to reconnect each other in such a way as to lower their free energy then this step should be permitted in the mc method without regard to the question if this process occurs in reality . this is so because in the mc simulations phase space is sampled according to the probability distribution and all one needs is to generate configurations weighted by the boltzmann factor , and the path followed in configuration space has nothing to do with any real dynamics . these two main drawbacks of the usual mc method are easily overcome in the mmc technique . first , one moves bigger chunks of fls encompassing beads in several planes . this way one can avoid local harmonic traps . ( this is like taking an aerial route to a destination rather than going through the zigzag maze of roads . ) this is much in the spirit of fourier space monte carlo where one first samples modes with smaller wave numbers and then move onto higher modes . the method of creating new fl configurations is based on the concept of the conditional probabilities . it is called the bisection method @xcite because one starts sampling beads by iteratively bisecting the fls . at each stage of the division , the beads belonging to that stage are moved with some conditional probability factor @xmath123 . it is important to make sure that the probabilities are chosen in such a way that detailed balance is satisfied at each stage of the division . one notes that the @xmath123 s may not be the actual boltzmann factors for the beads to be moved at different levels . but what is required is that when all @xmath123 s are multiplied together , they cancel in such a way so as to leave the correct boltzmann factor for the whole move . thus , inherent in this algorithm is the fact that a move would finally be accepted only if it has been accepted at each stage of the bisection method . the power of this method lies in choosing the appropriate @xmath123 s . if these conditional probability factors are chosen judiciously , most of the rejections would take place at the initial stages of the bisection process when not too much computational effort has been spent yet . the cutting and reconnection of fls is implemented naturally in a mmc method : permutation among the 3 or 4 lines chosen to be moved becomes the first among the many hierarchical steps one goes through before a move is finally accepted and the position of the beads updated accordingly . we typically moved a total of 15 to 20 beads distributed over 5 planes . permutations were sampled by a random walk algorithm through the space of permutations @xcite(see appendix b ) . in the case of ybco we worked with a field of @xmath124 g. working in the primitive approximation of the action would require the use of a smaller value of the dimensionless parameter @xmath59 , which would require slicing of the @xmath38-direction into a large number of planes . to avoid this , the matrix - squaring method @xcite has been used to get the effective action for bigger values of @xmath59 . for example nordborg and blatter @xcite use a value of @xmath125 and they work with 100 planes . in this simulation a value of @xmath126 has been used and the @xmath38-axis has been sliced into 75 planes . choosing a bigger value of @xmath127 by utilizing the matrix - squaring method makes it easier to equilibrate the system as compared to working with the primitive approximation . for bscco , we did not use the matrix - squaring method because of the complications involved due to the few extra terms in free energy which contribute depending on whether @xmath51 is smaller or bigger than @xmath128 . here , we used the natural inter - layer spacing @xmath88 to calculate the parameter @xmath127 at different temperatures and then used the mmc technique to efficiently sample the configurational space . as mentioned previously , in the present simulations we included only the josephson coupling . this approximation works well with ybco but it could be less satisfactory for bscco because of its high anisotropy . for very anisotropic materials the electromagnetic coupling becomes important @xcite . as discussed in section 2 , ryu _ et al . _ @xcite estimated that for anisotropy of magnitude @xmath93 the josephson interaction still dominates over the electromagnetic interaction by an order of magnitude , but this will not be the case for @xmath106 which can characterize some samples . @xcite discuss how to include the electromagnetic interaction in a mc simulation , but they only consider the opposite limit where the josephson coupling is totally neglected . to our knowledge there is no satisfactory treatment of both couplings included on equal footings . we carried out preliminary simulations which show that the inclusion of the electromagnetic coupling does not shift the position of the transition line much at a field of @xmath129 g , thus supporting our current conclusions . these results will be reported elsewhere @xcite . simulations were usually carried out for @xmath130 and @xmath131 fls . the decay of the structure factors at the transition temperature becomes sharper when one uses a larger number of fls . however , no appreciable shift in the transition temperature is seen while working with the smaller ( @xmath132 ) or the larger system ( @xmath133 ) . we did not run our simulations for even larger systems as it becomes computationally very time consuming . typical simulation times were @xmath134 mc steps . each mc step involved moving 3 - 4 lines in 5 planes simultaneously . we usually averaged over 10 - 15 realizations of the columnar disorders , though some results have an average over as many as 20 realizations of the disorder . columnar disorder is modeled as an array of straight cylindrical wells of typical radius @xmath135=25 - 35 placed randomly throughout the cross - sectional region of the sample and oriented along the @xmath38-direction @xcite . each columnar defect is of length @xmath136 . the density of the columnar pins can be varied by changing their number for a given cross sectional area , and the strength is controlled by a positive dimensionless parameter @xmath137 . if a bead happens to wander inside a columnar well , we include an extra free energy of @xmath138 . the defect concentration was taken to be a 20 percent ratio of defects to fls which means @xmath139 . the strength of disorder was set at @xmath140 for bscco . for ybco , @xmath141 was found to be too large in the sense that the transition became too broad and useful information could not be extracted . thus , for ybco we kept @xmath142 . other parameters used for ybco are @xmath143 @xmath144 , @xmath145 , @xmath146 . parameters for bscco are as follows : @xmath147 , @xmath148 , @xmath149 , @xmath150 . in this section we describe many different physical quantities that were monitored during the simulation . from the variation of these quantities with the temperature we can extract important conclusions about the different phases of the system . in terms of the reduced temperature , the energy @xmath151 can be simply written as @xcite : @xmath152 where , @xmath153 and @xmath154-\frac{(2r_{g}- ( |\mathbf{r}_{i , m}-\mathbf{r}_{i , m+1}|))^{2}}{2\lambda ^{2}\tau } \big\rangle,\ ] ] for @xmath111 , whereas @xmath155 any discontinuous jump in @xmath156 would indicate a fot . from this discontinuous jump in energy , @xmath113 , we can also calculate the jump in entropy @xmath157 , @xmath158 the translational structure factor @xmath159 is defined as , @xmath160 where @xmath161 stands for the mc average , @xmath162 is a reciprocal lattice vector and is of the form @xmath163 and @xmath164 and @xmath165 are some integers . @xmath166 represent the basis vectors of the reciprocal lattice @xmath167 where @xmath168 , @xmath65 is the nearest neighbor distance and @xmath169 are the unit vectors along the hexagonal unit cell such that @xmath170 in the simulations , @xmath164 and @xmath165 in eq . ( [ struc ] ) are chosen to be 1 and 0 or 0 and 1 . these choices correspond to the first bragg peak . we will normally write @xmath171 as simply @xmath172 . this quantity gives important information about the phase of the system . in an ordered phase where fls sit on a triangular lattice , @xmath172 is of the order of @xmath35 . in the disordered phase , it saturates to almost zero as the system size increases . in the simulations , however , there is a problem with measuring @xmath173 , especially in the presence of disorder . we will return to this point at the end of the section . we use delaunay triangulation to measure the hexatic order parameter , @xmath174 , which is defined as , @xmath175 where @xmath176 denotes the number of the nearest neighbors of a bead at position @xmath177 , @xmath54 and it is 6 for a perfect hexagonal lattice , @xmath178 stands for the bond angle , that is the angle that vector @xmath179 makes with an arbitrary axis . just like the @xmath172 , this quantity too has a large value in an ordered phase and saturates at a finite value for a system of finite size . as we allow permutations of fls , we can define a number @xmath180 as that fraction of the total number of fls which belong to loops that are bigger than the size of a `` simple '' loop . a simple loop is defined as a set of @xmath68 beads connected end to end , @xmath68 being the total number of planes . loops of size @xmath181 , @xmath182 ... start proliferating at and above the transition temperature and in the corresponding 2d boson system this proliferation is related to the onset of the superfluidity . the transverse fl fluctuations are measured by @xmath183 which is independent of @xmath184 . at the transition temperature @xmath185 undergoes a large increase for large @xmath38 . we want to emphasize one important aspect of measuring the translational structure factor . the usual way of measuring this quantity is by choosing @xmath186 corresponding to the first bragg peak i.e. , @xmath187 in eq . ( [ struc ] ) . now , it might happen that the configuration of the fls comes close to making an almost perfect hexagonal lattice but its basis vectors are not aligned along the usual major axes of the rhombically shaped cell encompassing the system . if we use the reciprocal lattice vector @xmath188 to measure the structure factor of such a configuration , we would end up getting a very small value for @xmath173 and might wrongly conclude that the system is in a very disordered state . this happened many times in our simulations ; we got a very low value of the translational structure factor while the hexatic order was indicating a high degree of orderliness in the fl lattice . to remedy this situation the translational structure factor is measured at 60 different angles , 1 degree apart , and choose that number which gives the largest possible value of the @xmath189 corresponding to some angle @xmath190 . after implementing this technique we find that @xmath173 and @xmath174 , which differ so much initially , almost follow each other . \1 . for ybco , simulations were carried out at a magnetic field of @xmath124 g. at this field we have @xmath191 . by the argument given in ref . , we conclude that our results should hold qualitatively for any @xmath49 such that @xmath192 . we checked our simulation results against those by nordborg and blatter @xcite . first we verified that our code was working fine by comparing our results against those given in ref . . working in the limit of @xmath193 , we kept @xmath59 fixed as @xmath80 was increased . we got a sharp transition at @xmath194 . a jump was also seen in the energy as defined in the previous section and was found to be @xmath195 , again the same as in ref . . the effect of introducing columnar disorder on the structure factor at the first bragg peak is shown in fig . [ all1 ] . in fig . [ all2 ] we depict the jump in the energy for a pure as well as for a disordered system . at various disorder strengths ( @xmath33 s ) for @xmath132 . a clear shift in transition point towards higher @xmath80 s is seen for @xmath196 for @xmath132 . a jump in the energy is seen up to @xmath197 . for @xmath198 , the rise in the energy is soothed out . ] we note that at lower strengths of the disorder , the melting line is almost unaffected and the transition takes place at @xmath199 . however , as we increase the strength of the disorder , the melting transition shifts towards higher values of @xmath80 which means that the melting line shifts towards higher temperatures and/or higher magnetic fields . for @xmath137 up to @xmath200 no change is seen in the melting curve or in the jump in the free energy functional . however , at @xmath137 equal to @xmath201 the structure factor comes down at around @xmath202 and , the jump in the energy becomes gradual . this finding is in agreement with several experimental studies where a change in the irreversibility line is seen with the introduction of columnar disorder @xcite . next , we look at the fl entanglement . @xmath203 denotes the number of fls which do not form a simple loop . by looking at the @xmath180 vs. @xmath114 graph in fig . [ all3 ] . . for @xmath204 a sharp jump in the fl entanglement is seen . for @xmath198 , the rise of @xmath205 with @xmath80 becomes gradual as compared with the clean system . ] it is clear that the entanglement is suppressed by as much as almost one order of magnitude just after and in the vicinity of the original transition line at @xmath194 . this result is expected , as it is well known that point disorder helps in line wandering and entanglement while the columnar disorder has just the opposite effect @xcite since a nearby fl is induced to align along the columnar defect , and thus its transverse fluctuations are reduced . unfortunately , there have been few studies on columnar disorder in ybco . @xcite work with vanishingly small magnetic field and infinite pinning strengths , in the vicinity of the lower branch of the melting line . figure [ 64ybco ] shows the effect of columnar disorder on a system of @xmath131 fls in ybco . for @xmath133 . smooth lines are provided as a visual aid . transition for @xmath206 and @xmath207 are seen almost at the same @xmath80 s as with @xmath132 . ] this graph shows a transition at almost the same @xmath114 as in fig . this is a confirmation that finite size effects are not important in our simulations . \2 . for the bscco system the @xmath156 vs. @xmath40 graphs for different @xmath33 values are depicted in fig . [ 36energy ] ( 36 fls ) and fig . [ 64energy ] ( 64 fls ) . at @xmath208 g for @xmath132 . lines were added to help visualize the energy jumps . a discontinuous change in energy is seen at @xmath209 k ( @xmath210 k ) for @xmath211 respectively . for @xmath212 only a change in slope can be seen at @xmath213k ( @xmath214 k ) instead of a discontinuous jump . ] at @xmath215 g for @xmath133 . a jump in energy can be seen at @xmath216 k ( @xmath217 k ) for @xmath211 respectively . again no discontinuity is observed for @xmath212 , but only a change in slope at @xmath218 k ( @xmath219 k ) . ] for the pure system ( @xmath220 ) we see a sharp jump in the energy at exactly the point where the translational as well as hexatic structure factors sharply decline . this jump in energy is a clear signature of a first order transition . from the energy jump we can compute the jump in entropy ( @xmath157 ) . the jump in energy at the transition is around @xmath221 k per vortex per layer , which gives @xmath222 which is small compared to the experimental value of @xmath223 at @xmath224 g. however , these values for @xmath157 , @xmath225 and @xmath49 are in qualitative agreement with the same system studied using a different model @xcite . introduction of the columnar defects of a finite strength shows some interesting effects . we put columnar pins at random positions with a concentration fixed at 20 percent of the fls . we study the bscco system for pins of low strength ( @xmath226 ) as well as for a higher strength ( @xmath141 ) . columnar disorder of strength up to @xmath226 appear to have little effect on the system . this can be seen from the energy vs. temperature graph in figs . [ 36energy ] and [ 64energy ] , where the curve for the pure system and the one in the presence of low disorder strength ( @xmath207 ) fall on top of each other , except in the transition region where the jump in case of the system with columnar pins is more gradual . also , from figs . [ 36stru ] and [ 64stru ] we see that the @xmath172 is not much affected in the presence of the columnar disorder of strength @xmath207 . the same is true for the hexatic order depicted in fig . [ 64hexa ] . this means that for the lower strengths of the disorder , the vortex - vortex interaction is dominant in the range of the temperature where we carried out simulations . as a result the first order vortex - lattice melting transition ( fot ) line is almost not affected . g for @xmath132 . no shift in transition point is seen for @xmath207 . in the presence of columnar pins with @xmath212 , the transition temperature increases from @xmath227 k to @xmath228 k. also , we can see that @xmath172 starts to rise close to @xmath229 k ] the nature of fl melting changes , however , when the strength of the disorder is increased to @xmath212 as seen in figs . [ 36stru ] and [ 64stru ] . g for @xmath133 . for @xmath207 no significant change in the structure factor is observed . in the presence of columnar pins with @xmath212 , transition temperature increasing from @xmath230 k to @xmath231 k. also , we can see that @xmath172 starts to rise close to @xmath232 k ] we see that at the lower temperatures the order parameter is suppressed . as we increase the temperature , the order parameter starts to rise and joins the melting curve of the pure system and then falls along it at even higher temperatures . [ 64hexa ] indicates that the hexagonal structure factor also shows a similar behavior . g for @xmath133 . when compared with the previous figure , it can be seen that @xmath174 almost follows @xmath172 . ] this clearly shows that the fls start to disengage from the pinning centers as they wiggle more . at higher temperatures , columnar pins with the strength chosen here are not effective in pinning the vortex system . this becomes clear from the fact that the melting curve for the pure system and the one with columnar disorder join each other at a temperature below the fot . further interpretation and a possible explanation of this phenomenon will be discussed in more detail in the conclusions section below . this convergence of the curves for the pure and disordered cases is also borne out in the @xmath156 vs. @xmath40 graphs in fig . [ 36energy ] and fig . [ 64energy ] . initially at low temperatures there is a big difference in energies of the systems with no disorder ( @xmath233 ) and high disorder ( @xmath141 ) . however as the temperature is increased , the two curves come closer and finally merge together in the liquid phase . the jump in energy in presence of columnar defects can still be seen in the @xmath156 vs. @xmath40 graph for a disorder strength of @xmath226 . this tends to suggest that the fot in a bscco system is not affected by disorders kept at as many as 20 percent of the total number of lines used in the simulations as long as we keep the strength of the disorder less or equal to @xmath226 . for @xmath141 , the rise of energy with temperature becomes much more gradual and we do not see any discontinuous jump in energy at any temperature . on the other hand an abrupt change in slope of the energy vs temperature graph is observed , suggesting a discontinuity in the specific heat characterizing a second order transition . in fig . [ 64low ] and fig . [ 64high ] , the snapshots of fls , projected on a plane , at temperatures less than the transition temperature and at a temperature bigger than the transition temperature are shown . fls for @xmath234 . fls have been projected onto one plane . ] and @xmath234 . fls have been projected onto a single plane . columnar defects are not drawn to scale . some fls on the boundary do not seem to make loops . that is only because virtual images of fls outside the cell are not shown . ] from fig . [ 64low ] it is easily seen that at low temperatures most columnar defects have captured fls . also , the fls make simple loops and have cleverly set themselves so as to make a hexagonal lattice and yet occupy as many defects as possible . the transverse fluctuations of the trapped fls are greatly suppressed at low temperatures . at higher temperature beyond the transition point , we see that columnar defects are not occupied any more and a lot of fls are entangled . inspection of snapshots like fig . [ 64low ] gives support to the assertion of sen _ et al . _ that the bose glass consists of patches of ordered regions with only short range positional and orientational order . this phase is different from the bragg - glass in systems with point pins which is characterized by quasi - long - range order . figure [ 64wander ] shows the mean squared displacements of the fls in the @xmath38-direction at different temperatures for a pure system . -direction at @xmath215 g for @xmath133 . here @xmath88 is the distance between adjacent planes . a large increase in line wandering occurs at @xmath235 k. ] at lower temperatures @xmath236 saturates for large @xmath38 but in the liquid state it grows linearly @xcite . a large gap in @xmath236 for large values of @xmath38 seen at the transition temperature signals the onset of the entanglement of the fls . in the presence of the columnar disorder of the strength @xmath141 we see from fig . [ 64wander_col ] that @xmath236 at temperatures less than the transition temperature is suppressed compared to the corresponding @xmath236 in the pure system . -direction at @xmath215 g for @xmath133 in presence of disorder of strength @xmath212 . the big jump in @xmath185 has moved to @xmath237 k now . ] this result is in agreement with the findings in ref . . also the big gap in @xmath236 occurring at the melting transition has moved towards a higher temperature signaling a shift in the position of the melting transition in the presence of columnar defects . for lower strengths of disorder ( @xmath238 for ybco and @xmath239 for bscco ) no appreciable shift in the melting line was seen . for these strengths , a sharp drop in the translational and hexatic structure factors takes place at the transition . also , the jump in energy at the transition is not affected much as compared with the case when there is no disorder present . this suggests that for smaller concentrations of the columnar defects the transition still remains first order . this result is in agreement with ref . where even with the introduction of the columnar disorder no shift in transition temperatures is seen as long as the concentration of columnar disorder introduced is small . it is found that for ybco as well as bscco , the melting transition shifts towards higher values of temperature and magnetic field when random disorder is introduced provided its strength exceeds a threshold which is different for ybco and bscco . the size of the shift , for a given concentration of defects , depends on the strength of the disorder . for ybco , a considerable shift in the melting line towards higher temperatures and magnetic fields was seen for @xmath226 and @xmath240 . this shift was bigger for @xmath240 than for @xmath226 . similarly , for bscco , a large shift in the melting line towards a higher temperature was found for @xmath141 . these results are in tune with numerous experimental findings @xcite as well as theoretical prediction @xcite where the irreversibility line is seen to shift towards higher temperature and magnetic fields in the presence of columnar defects . the qualitative reason for this effect is that due to the interaction with the columnar defects the transverse thermal fluctuations of the fls are reduced and thus the melting transition , as determined from the lindemann criterion , takes place at at a higher temperature . at @xmath141 the jump in energy is not discernible any more . instead , a change of slope corresponding to a specific heat discontinuity is observed . this means that the transition is probably not of first order in the presence of columnar disorder of higher strengths , but it is rather a continuous ( second order ) transition . the most dramatic outcome of this study for bscco is that for some values of the applied field and defect strength both the translational and hexatic structure factors start to rise at a certain temperature as the transition is approached from the lower temperature side . this is an unusual result , that to our knowledge , has not been seen in previous simulations . this fact can be explained as follows . at low temperatures , the free energy is dominated by energy effects rather than entropy considerations , and pinning effects are dominant . as a result most columnar defects capture a fl , while the rest of the fls adjust themselves in positions such as to minimize the free energy . however , as the temperature is increased , fls start to decouple from the defect sites , which allow the vortices pinned at interstitial positions to move themselves into a more ordered arrangement , thus increasing the structure factors compared to the situation at lower temperatures . as was mentioned before , due to its high anisotropy , fls in bscco behave more like a collection of two dimensional pancakes rather than rigid rods . these pancakes are comparatively more difficult to get pinned all at once by a columnar defect . on the other hand , fls are much stiffer in ybco and it is easier for a columnar defect to capture a fl all along its length . it can be shown ( eq . ( 9.49 ) in ref . ) that the depinning temperature for fls , @xmath241 , is inversely proportional to the anisotropy parameter @xmath23 . thus , we expect fl depinning to occur at comparatively smaller temperatures in bscco than in ybco . since the melting temperatures are not that different between these materials at the fields we consider , the depinning process for bscco occurs further below the melting temperature than in ybco . this allows the structure factor to increase above the depinning temperature before its ultimate decline at the melting transition . for ybco , the depinning takes place close to the melting temperature , and it is difficult to detect any rise in the structure factors , especially for small systems , because it is masked by the decrease in order due to the stronger thermal fluctuations . in principle , the rise of the order parameters in the presence of low amount of columnar defects as the melting transition temperature is approached could be observed experimentally , if the appropriate parameters are tuned correctly . in small angle neutron scattering ( sans ) @xcite one can measure the integrated intensity over a bragg peak of wave vector @xmath242 which is proportional to the translational structure factor @xmath243 measured in the simulation . another commonly used technique is muon spin rotation @xcite which gives information about the width of the magnetic field distribution in the sample . this is not directly proportional to the structure factor at a given @xmath242 , but is rather given by @xmath244 ^ 2},\ ] ] where @xmath245 is the mean square deviation of vortices from their average position . it is possible to measure this quantity in the simulations but this was not done it in the present work . this quantity might not show the unusual rise describe above since it is dominated by @xmath246 which is very likely monotonically increasing with temperature yilding a monotonically decreasing line width . thus in order to look for the effect observed in this paper we suggest using the sans technique to look at a bscco sample with columnar pins described by a matching field of about 25 g and an applied field of about 125 g. it is not clear to us what is the corresponding @xmath33 parameter describing the pinning strength of the experimental defects . according to blatter _ @xcite @xmath33 lies in the range 0.1 - 1 . this work is supported by the us department of energy ( doe ) , grant no . de - fg02 - 98er45686 . consider a rhombically shaped region with side @xmath247 and angle @xmath248 unit vectors are @xmath249 , @xmath250 , with @xmath251 . in practice we took @xmath252 but we leave the discussion here more general . the green s function @xmath253 which describe the 2d coulomb interaction between one vortex and another including all its images , as is implied by the periodic boundary conditions is given by the solution to london s equation ( see e.g. ref . ) @xmath254 with the parameter @xmath255 setting the scale for the range of the interaction . the solution is given by @xmath256 with @xmath257 where @xmath258 runs over all reciprocal lattice vectors spanned by@xmath259 for @xmath260 . substituting in eq . ( [ expansion ] ) we obtain @xmath261}{l^{2}\sin ^{2 } \theta /(2\pi \lambda ) ^{2}+n_{1}^{2}-2n_{1}n_{2}\cos \theta + n_{2}^{2}}.\ ] ] we are now going to carry out the summation over @xmath262 analytically . this can be done by using the formula @xmath263 we have subtracted the constant @xmath177 from @xmath264 to ensure that @xmath265 on the contour of integration when @xmath38 has a large negative imaginary component . the contour of integration is a square with sides parallel to the real and imaginary axes with the origin in the middle , in the limit that its size goes to infinity . in our case @xmath266 with appropriate values of @xmath72 , @xmath267 , and @xmath95 . this function has two simple poles , and the residues can be easily evaluated . the final answer becomes ( relabeling @xmath268 as @xmath61 ) @xmath269 where @xmath270 this expression is simpler than the one used by nordborg and blatter since it does not have different expressions for even and odd n. it also converges faster in certain regions . essentially the same method is used for permutation sampling as was used in ref . . the only difference is that we use permutation space of only the neighboring lines . this is so because even if we were able to get a permutation step , involving a large number of lines , accepted at the first stage of the algorithm , it would be very likely to get rejected at the following stages . so we work with only 3 - 5 lines . sufficiently long segments ( typically 5 planes ) of a number of lines were cut . care should be taken to make sure that chosen fls are the nearest neighbors in the plane where the reconnection of fls is going to take place . these points can be implemented easily with the concept of linked lists and pointers @xcite . also , care is to be taken that even though a fl may be far away from some other fl in the rhombically shaped unit cell , it can still permute with it through one of the images of the latter . these few simple points are very important to implement the whole procedure correctly . just for a check , we tried with the sampling procedure given in ref . . this gave results in good agreement with the sampling procedure given above . w. e. lawrence and s. doniach , in _ proceedings of lt 12 , kyoto,1970 _ , edited by e. kanda ( keigaku , tokyo , 1971),p . 361 ; s. doniach , in _ high temperature superconductivity , proceedings _ , edited by k. s. bedell et al . ( addison - wesley , redwood city , 1989 ) , p. 406 .

the effect of columnar pins on the flux - lines melting transition in high - temperature superconductors is studied using path integral monte carlo simulations . we highlight the similarities and differences in the effects of columnar disorder on the melting transition in yba@xmath0cu@xmath1o@xmath2 ( ybco ) and the highly anisotropic bi@xmath0sr@xmath0cacu@xmath0o@xmath3 ( bscco ) at magnetic fields such that the mean separation between flux - lines is smaller than the penetration length . for pure systems , a first order transition from a flux - line solid to a liquid phase is seen as the temperature is increased . when adding columnar defects to the system , the transition temperature is not affected in both materials as long as the strength of an individual columnar defect ( expressed as a flux - line defect interaction ) is less than a certain threshold for a given density of randomly distributed columnar pins . this threshold strength is lower for ybco than for bscco . for higher strengths the transition line is shifted for both materials towards higher temperatures , and the sharp jump in energy , characteristic of a first order transition , gives way to a smoother and gradual rise of the energy , characteristic of a second order transition . also , when columnar defects are present , the vortex solid phase is replaced by a pinned bose glass phase and this is manifested by a marked decrease in translational order and orientational order as measured by the appropriate structure factors . for bscco , we report an unusual rise of the translational order and the hexatic order just before the melting transition . no such rise is observed in ybco .

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Proceed to summarize the following text: in a series of earlier communications @xcite we have proposed the augmented space recursion ( asr ) as an efficient computational technique for the calculation of configuration averaged electronic properties of random binary alloys . the method is a combination of the augmented space formalism @xcite and the recursion method of haydock @xcite . when coupled with the local density functional approximation based tight - binding linearized muffin - tin orbital @xcite it provides a first - principles determination of electronic structure and total energy calculations for disordered alloys . effects such as short - ranged ordering and local lattice distortions due to size mismatch of the constituent atoms can very easily be incorporated in the methodology @xcite . recently the method has been applied to study dispersion and line widths for phonons in disordered alloys . since recursion can take into account disorder in beyond nearest - neighbour force constants as well as large environmental effects , this application was a step forward in the theory of phonons in disordered alloys @xcite . it has been pointed out repeatedly in the above publications that the asr with its terminator approximation @xcite goes beyond the standard single - site mean field theories and accurately takes off - diagonal disorder into account . a number of attempts have been made for practical implementation of mean - field theories beyond the single site approximation . one of the most sophisticated developments among the mean - field theories , is the travelling cluster approximation of gray and kaplan @xcite and its latest version : the itinerant coherent potential approximation ( icpa ) @xcite , where a small number of classes of selected excitations in augmented space are preserved . in spite of the simplification so introduced , the resulting equations are still formidable enough to discourage further generalization to larger clusters beyond the nearest neighbour . in the cluster cpa proposed by mookerjee @xcite , in which one retains disorder fluctuations only from a chosen cluster and replaces the rest of the system by an effective medium , one is restricted by the limitation of solving number of coupled self - consistent equations . the number of equations increases with the size of chosen cluster , which again discouraged extension to larger clusters beyond the nearest neighbour . this brings us back to the attempt at the implementation of the asr for large environment fluctuations , implying accurate evaluation of a large sequence of recursion coefficients . we shall discuss this in some detail in the following section . the asr , in its practical implementation to real alloy systems , has several practical problems . the formalism maps a disordered hamiltonian , described in a hilbert space spanned by a countable basis in which preferably its representation is sparse , onto an ordered hamiltonian in an enlarged space . this enlarged or _ augmented _ space is constructed by taking the outer product of the configuration space of the random variables of the disordered hamiltonian together with original hilbert space in which the disordered hamiltonian was described . the main difficulty in the implementation of recursion in this augmented space is its enormous rank . a system of @xmath1 sites with binary distribution has an augmented space of rank @xmath2 . this enormous rank of the augmented space and its description on the computer is what had discouraged earlier workers in implementing the asr , although the augmented space formalism was first proposed about thirty years ago @xcite . the second problem was the recursive propagation of errors . very small errors during recursion ( inevitable in numerical implementations ) tends to add up leading to loss of orthogonality in the recursion basis . this leads to appearance of ghost states . haydock and te @xcite has overcome this difficulty in proposing the _ dynamical recursion_. we refer the reader to their paper for details . in this communication we shall describe the reduction of the effective rank of the augmented space on which the recursion is carried out , using the local point - group symmetries in both the hilbert space and the configuration space . such reduction is well known in various manipulations in reciprocal space for ordered systems . however , such a reduction has not been implemented in real space recursion , although it had been proposed by gallagher @xcite quite some time ago . it had been shown by gallagher that if we start recursion with a state belonging to an irreducible subspace of a hilbert space , subsequent recursion always stays within this subspace . this allows us significant reduction in rank of the required subspace and makes this method practically feasible . further , this method with its terminator approximation , retains the herglotz properties of the configuration averaged green function . the recursion method addresses inversions of infinite matrices @xcite . once a sparse representation of an operator in hilbert space , @xmath3 , is known in a countable basis , the recursion method obtains an alternative basis in which the operator becomes tri - diagonal . this basis and the representations of the operator in it are found recursively starting from @xmath4 : @xmath5 where , @xmath6 for n@xmath7 2 . in this basis the representation of @xmath3 is tridiagonal : @xmath8^{1/2}}\ = \ \beta_n \enskip ; \enskip \langle u_n\vert \tilde{h}\vert u_{m}\rangle \ = \ 0 \quad m\geq n+2\end{aligned}\ ] ] and the averaged green function can be written as a continued fraction : @xmath9 the asymptotic part of the continued fraction is obtained from the initial set of coefficients @xmath10 for @xmath11 , using the idea of _ terminators _ @xcite . for small values of @xmath1 we have large inaccuracies and the more the structure in the spectral density of @xmath3 the larger is the @xmath1 needed to maintain the window of accuracy required by us . in earlier communications we have described how to deal with random binary alloys @xcite within the framework of the tight - binding linearized muffin - tin orbital method ( tb - lmto ) @xcite and using the augmented space formalism ( asf ) . we refer the readers to the seminal papers mentioned earlier for details . here we shall introduce only the salient features which will be required by us subsequently in this communication . we begin by setting up a muffin - tin potential with centres at the atomic sites @xmath12 on a lattice . next , we inflate the muffin - tins into atomic spheres and start from a most - localized tb - lmto representation of the hamiltonian : @xmath13 the _ potential parameters _ , @xmath14 and @xmath15 , are diagonal matrices in the angular momentum indeces , and @xmath16 here @xmath17 is a composite label @xmath18 for the angular momentum quantum numbers . @xmath19 is the random site - occupation variable which takes values 0 or 1 with probability @xmath20 or @xmath21 @xmath22 respectively , depending upon whether the muffin - tin labelled by @xmath12 is occupied by @xmath23 or @xmath24-type of atom . the @xmath25 and @xmath26 are projection and transfer operators in the hilbert space @xmath27 spanned by the tight - binding basis @xmath28 . we can associate the random variable @xmath19 with an operator @xmath29 whose eigenvalues correspond to the observed values of @xmath19 , and whose corresponding eigenvectors , @xmath30 are the _ state vectors _ of the variable . these state vectors of the set of @xmath1 random variables @xmath19 of rank @xmath31 span a space called configuration space @xmath32 with configurations of the kind @xmath33 where @xmath34 if we define the configuration @xmath35 as the _ reference _ configuration , then any other configuration may be uniquely labelled by the _ cardinality sequence _ : @xmath36 , which is the sequence of positions where we have a @xmath37 configuration . the _ cardinality sequence _ of the _ reference _ configuration is the null sequence @xmath38 the augmented space theorem states that a(\{n_r } ) where , @xmath39 ( @xmath40 ) is the spectral density of the self - adjoint operator @xmath29 , which is such that the probability density of @xmath19 is given by : @xmath41 @xmath29 is an operator in the space of configurations @xmath42 of the variable @xmath19 . this is of rank 2 and is spanned by the _ states _ @xmath43 . @xmath44 the expended hamiltonian @xmath45 in the augmented space is constructed by replacing all the random variables @xmath19 by the corresponding operators @xmath29 . it is an operator in the augmented space @xmath46 = @xmath47 . so the asf maps a disordered hamiltonian described in a hilbert space @xmath27 onto an ordered hamiltonian in an enlarged space @xmath46 , where the space @xmath46 is constructed by augmenting the configuration space @xmath48 of the random variables of the disordered hamiltonian together with the hilbert space @xmath27 of the disordered hamiltonian . the configuration space @xmath32 is of rank 2@xmath49 if there are n muffin - tin spheres in the system . we may rewrite the expression for the configuration average of the green operator as : @xmath50 where , @xmath51\enskip { \cal p}_r \otimes { \cal p}_{l } \otimes { \cal i } \\ \bhat & = & \sum_{rl } b\left[\rule{0mm}{4 mm } ( e - c_l)/\delta_l \right ] \enskip { \cal p}_{r } \otimes { \cal p}_{l } \otimes { \cal p}_{\downarrow}^r \\ \chat & = & \sum_{rl } c\left[\rule{0mm}{4 mm } ( e - c_l)/\delta_l \right ] \enskip { \cal p}_{r } \otimes { \cal p}_{l}\otimes \left\ { { \cal t}^r_{\uparrow\downarrow } + { \cal t}^r_{\downarrow\uparrow } \right\ } \\ \shat & = & \sum_{rl } \sum_{r'l ' } s_{rl , r'l ' } \enskip { \cal t}_{rr ' } \otimes { \cal t}_{ll ' } \otimes { \cal i } \\ \phantom{x } & & \\ \dhat & = & \sum_{rl } a\left ( \delta^{1/2}_l \right)\enskip { \cal p}_r \otimes { \cal p}_{l } \otimes { \cal i } + \sum_{rl } b\left ( \delta^{1/2}_l \right ) \enskip { \cal p}_{r}\otimes { \cal p}_{l}\otimes { \cal p}_{\downarrow}^r \ldots \\ & & + \sum_{rl } c\left ( \delta^{1/2}_l \right ) \quad { \cal p}_{r } \otimes { \cal p}_{l}\otimes \left\ { { \cal t}^r_{\uparrow\downarrow } + { \cal t}^r_{\downarrow\uparrow } \right\ } \end{aligned}\ ] ] and @xmath52 since , @xmath53 the ket @xmath54 is not normalized . we first write the above in terms of a normalized ket @xmath55 = @xmath56^{-1/2 } \vert 1 \ } $ ] . we now have : @xmath57 where , @xmath58 / a\left ( 1/\delta_{l } \right)\right\}\enskip { \cal p}_{r } \otimes { \cal p}_l\otimes { \cal p}_{\downarrow}^{r } \nonumber\\ \fl \chat^{\prime } = \sum_{rl } \left\{c\left [ ( c_l - e)/\delta_l\right ] / a\left ( 1/\delta_l \right)\right\}\enskip { \cal p}_{r } \otimes { \cal p}_{l } \otimes \left\ { { \cal t}^{r}_{\uparrow\downarrow } + { \cal t}^{r}_{\downarrow\uparrow } \right\}\nonumber \\ \fl \shat^{\prime } = \sum_{rl } \sum_{r'l'}\left[a\left ( 1/\delta_l \right)\right]^{-1/2 } \ s_{rl , r'l ' } \ \left [ a\left ( 1/\delta_{l ' } \right)\right]^{-1/2 } { \cal t}_{rr'}\otimes{\cal t}_{ll ' } \otimes { \mathaccent " 7e { \cal i}}\nonumber\\ \label{kasr2 } \end{aligned}\ ] ] this equation is now exactly in the form in which the recursion method may now be applied . the computational burden is considerably reduced due to this diagonal formulation , the recursion now becomes energy dependent as is clear from the form of the effective hamiltonian as shown in and discussed in @xcite . the recursion formalism of the ordered hamiltonian was free of this constraint . this energy dependence makes the recursion technique computationally unsuitable because to obtain the green functions we have to carry out recursion per energy point of interest . this problem has been tackled using _ seed recursion technique _ @xcite . the idea is to choose a few seed points across the energy spectrum uniformly , carry out recursion over those points and then interpolate the values of coefficients across the band . in this way one may reduce computation time . for example , if one is interested in an energy spectrum of 200 points , in the bare diagonal formulation recursion has to be carried out at all the 200 points but in the seed recursion technique one needs to perform recursions only at 10 - 15 points . the whole idea stems from the fact that in most of the cases of binary alloys , it is seen that the recursion coefficients @xmath59 and @xmath60 vary quite weakly across the energy spectrum . at this point we note that the above expression for the averaged @xmath61 is _ exact_. we mentioned earlier that recursion on the augmented space is not computationally feasible because of its large rank . for a binary alloy with @xmath1 sites and with only @xmath62-orbitals on them , the rank of the augmented space is @xmath2 . implementing recursion on this huge space for a sufficient number of steps to ensure accuracy is often not feasible on available computers . however , if we exploit the symmetry both of the underlying lattice in real space and of the configuration space ( which arises due to the homogeneity of the disorder and arrangements of atoms on the underlying lattice ) , the rank of the irreducible part of the augmented space in which the recursion is effectively carried out becomes tractable . the conceptual advantages in asf include apart from analyticity , translational and underlying lattice symmetries automatically built into the augmented space hamiltonian . this allows us to involve the idea of utilizing symmetry operations present in both the real and configuration spaces , in the context of recursion method , reducing the rank of hamiltonian drastically and making the implementation of asf feasible . since the augmented space recursion essentially retains all the properties of real space recursion but is described in a much enlarged space , it will be useful to consider symmetry operations in real space recursion first and then to consider those in the full augmented space . during the recursion , the basis member @xmath63 is generated from the starting state @xmath64 by repeated application of the hamiltonian . if the starting state belongs to an irreducible sub - space of the hilbert space then all subsequent states generated from recursion will belong to the same irreducible sub - space . physically , we may understand this as follows : the recursion states @xmath63 carry information of distant environment of the starting state . if the hamiltonian is nearest - neighbour _ only _ , then the state @xmath65 , which arises by the application of the hamiltonian on @xmath66 is a combination of states in the nearest shell with which @xmath64 couple via the hamiltonian . similarly , @xmath63 is a combination of n - th neighbour shell with which @xmath64 is coupled via the hamiltonian . if @xmath67 is a point group symmetry of the hamiltonian , then all @xmath68-th neighbour - shell states which are related to one another through the symmetry operator must have equal coupling to @xmath64 . hence , it is useful to consider among the @xmath68-th neighbour - shell states of which @xmath63 is a linear combination , only those belonging to the irreducible subspace and redefine the hamiltonian operation . = 3.0 in as an example , take a nearest - neighbour @xmath62-state anderson model on a square lattice , with a binary distribution of its diagonal elements . we shall label the tight - binding basis by the position of the lattice points in units of the lattice constant : e.g. @xmath69 where @xmath70 are integers @xmath71 . the starting state @xmath72 belongs to the one dimensional representation of the point group of a square lattice . this state then couples with linear combinations of states on neighbour shells which are symmetric under square rotations : @xmath73 the shows the grouping together of sites with the local symmetry of the square lattice on the first three nearest neighbour shells of the central site . the first and second groups in are coloured gray in and the last group coloured black . the labels on the groups ( shown on the left sides of the equations ) are confined _ only _ to the upper right quadrant of the lattice . if we go up to @xmath1 shells ( for large @xmath1 ) there are about @xmath74 states in the diamond shaped nearest - neighbour cluster . however , there are only @xmath75 states with square symmetry . so within this reduction we can work only in 1/8-th of the lattice , provided we incorporate proper weights to the states to reproduce the correct matrix elements . if @xmath76 and @xmath77 are two states coupled to each other via the hamiltonian , and both belong to the same irreducible subspace , and if @xmath78 are states obtained by operating on @xmath76 by the symmetry group operations of the real space lattice , then @xmath79 is called the _ weight _ associated with the state labelled by @xmath12 . if we wish to retain only the states in the irreducible subspace and throw out the others and yet obtain the same results , we have to redefine the hamiltonian matrix elements as follows : @xmath80 = 5.4 in = 6.5 in = 6.in in the above prescription , the new irreducible basis introduced by us , reflects only the symmetry of the underlying lattice and holds good for a model system which has @xmath62-like orbitals only . but for a real system , the tb - lmto minimal basis contains members with other symmetries as well . for example , in a cubic lattice with a @xmath81 minimal basis , we have basis members with @xmath62 , @xmath82 , @xmath83 and @xmath84 symmetries . the symmetry of the orbitals is reflected in the two - centered slater - koster integrals and this prohibits overlap integrals at certain positions , called _ symmetric positions _ with respect to the overlapping orbitals . a few of these symmetric positions for a simple cubic lattice are shown in figure [ fig2 ] . for the sake of clarity we have shown only the projections on @xmath85 plane . we have indicated the positive and negative parts of the orbital lobes by different shades . from ( b ) it is easy to argue that since the hamiltonian is spherically symmetric and the product @xmath86 is positive in the upper right quadrant and negative in the upper left quadrant : @xmath87 . the same is true for @xmath88 . similarly , @xmath89 . on the other hand , @xmath90 . element & expression & factor & to be zero + @xmath91 & @xmath92 & @xmath93 & + @xmath94 & @xmath95 @xmath96 & @xmath97 & = 0 + @xmath98 & ' '' '' @xmath99 @xmath100 & @xmath101 & =0 or = 0 + @xmath102 & @xmath103(@xmath104-@xmath104 ) @xmath100 & @xmath105 & = + @xmath106 & ' '' '' [ @xmath107 - 1/2(@xmath104+@xmath104 ) ] @xmath100 & @xmath108 & @xmath109 + @xmath110 & @xmath111@xmath104 @xmath112+(1-@xmath104 ) @xmath113 & @xmath114 & + @xmath115 & @xmath111 ( @xmath116 ) & @xmath117 & =0 or = 0 + @xmath118 & @xmath111 [ @xmath119@xmath104 @xmath120 + ( 1 - 2@xmath104 ) @xmath121 & @xmath122 & = 0 + @xmath123 & @xmath124 ( @xmath120 -2 @xmath125 ) & @xmath126 & =0 or = 0 @xmath127 + & & & @xmath127 or = 0 + @xmath128 & ' '' '' [ @xmath129 ( @xmath104-@xmath104 ) @xmath120 + ( 1-@xmath104+@xmath104 ) @xmath121 & @xmath130 & =0 + @xmath131 & ' '' '' ( @xmath104-@xmath104)[@xmath129 @xmath120 - @xmath121 & @xmath132 & = 0 ot = + @xmath133 & ' '' '' @xmath134[@xmath104-(@xmath104+@xmath104)/2 ] @xmath120-@xmath119 @xmath104 @xmath135 & @xmath136 & =0 + @xmath137 & ' '' '' 3@xmath104@xmath104 @xmath138+(@xmath104+@xmath104 - 4@xmath104@xmath104 ) @xmath139 & @xmath140 & + & @xmath127 + ( @xmath104+@xmath104@xmath104 ) @xmath141 & & + @xmath142 & ' '' '' @xmath134 3@xmath104 @xmath138+(1 - 4@xmath104 ) @xmath143 + ( @xmath104 - 1 ) @xmath141 & @xmath144 & =0 or = 0 + @xmath145 & ' '' '' ( @xmath104-@xmath104 ) @xmath134 ( 3/2)@xmath138 - 2 @xmath146 + ( 1/2 ) @xmath147 & @xmath148 & =0 or = 0 @xmath127 + & & & @xmath127 or = + @xmath149 & ' '' '' @xmath134(3/2)(@xmath104-@xmath104 ) @xmath138-[1 + 2(@xmath104-@xmath104 ) ] @xmath146 @xmath127 & @xmath150 & = 0 or = 0 + & @xmath127 + [ 1 + 1/2(@xmath104-@xmath104 ) ] @xmath147 & & + @xmath151 & ' '' '' @xmath119 @xmath134 [ @xmath104 - 1/2(@xmath104+@xmath104 ) ] @xmath138 - 2 @xmath104 @xmath146 @xmath127 & @xmath152 & =0 or = 0 + & @xmath127 + 1/2(1+@xmath104 ) @xmath147 & & + @xmath153 & ' '' '' 3/4(@xmath104-@xmath104)@xmath104 @xmath138 + [ @xmath104+@xmath104-(@xmath104-@xmath104)@xmath104]@xmath146@xmath127 & @xmath155 & + & @xmath127 + [ @xmath104 + 1/4(@xmath104-@xmath104)@xmath104 ] @xmath141 & & + @xmath156 & ' '' '' @xmath119 ( @xmath104-@xmath104)@xmath134 [ @xmath104/2 - 1/4(@xmath104+@xmath104 ) ] @xmath138 -@xmath104 @xmath146@xmath127 & @xmath157 & = + & @xmath127 + 1/4(1+@xmath104 ) @xmath141 & & + @xmath158 & ' '' '' [ @xmath104 - 1/2(@xmath104+@xmath104)]@xmath104 @xmath138 + 3@xmath104(@xmath104+@xmath104)@xmath146@xmath127 & @xmath159 & + & @xmath127 + 3/4(@xmath104+@xmath104)@xmath104 @xmath141 & & + the above is an illustration , detailed arguments for the orbital symmetry has been discussed by harrison @xcite . the provides the conditions for obtaining the orbital based symmetry factor @xmath160 for a lattice with cubic symmetry . we shall describe the direction of @xmath161 by the direction cosines ( @xmath162 . the table gives the details for the calculation of the symmetry factor @xmath163 on a cubic lattice . a look at the figures [ fig2 ] and [ fig3 ] shows the reason for introduction of this factor . for the @xmath62-orbital overlaps the symmetry factor is always 1 ( figure 1(a ) ) . all six neighbours in the cubic lattice in the directions ( 100),(@xmath16400),(010),(0@xmath1640 ) , ( 001 ) and ( 00@xmath164 ) are equivalent with the same overlap integrals @xmath92 ( only four are shown in the @xmath165 plane ) . we may then reduce the lattice and retain only the neighbour in the ( 100 ) direction and scale the overlap @xmath92 by the appropriate weight ( i.e. 2 in this case ) as discussed earlier . for the @xmath166 , however , the overlaps in the ( 010 ) , ( 0@xmath1640 ) , ( 001 ) and ( 00@xmath164 ) are zero . this can be deduced immediately by noticing that the overlap products in these directions are positive for @xmath167 and negative for @xmath168 . on reduction , although they are related to one another by the symmetry operations of the cubic lattice , we can not club the neighbours in the ( 100 ) and ( @xmath16900 ) direction with those in the ( 010),(0@xmath1640),(001 ) and ( 00 @xmath164 ) and will have to put these overlaps to be zero by introducing the symmetry factor @xmath163 . @xmath170 where @xmath163 is 0 if @xmath171 is a symmetric position of @xmath12 with respect to @xmath17 and @xmath172 , otherwise it is 1 . illustrates some of the overlaps involving the @xmath173 states with @xmath83 and @xmath84 symmetries . with this simple reduction procedure the real space part can be reduced to 1/8-th , keeping only the sites in the ( @xmath174 ) octant and suitably renormalizing the hamiltonian matrix elements as described earlier . = 5 in -1 cm = 5 in -1 cm = 4 in we still have not exhausted all the symmetries in the full augmented space . as discussed earlier , this space is a direct product of the real ( lattice - orbital ) space and the configuration space which are disjoint . as a consequence the symmetry operations apply independently to each of them . since the disorder is homogeneous , the cardinality sequence in configuration space itself has the symmetry of the underlying lattice . to see this , let us look at the where we show a part of a cubic lattice where the central site is occupied by a configuration labelled by @xmath175 , while two of the six nearest neighbours are occupied by configurations labelled by @xmath37 , and four of them by @xmath175s . we note that the twelve configurations in the first three rows of the figure , where the two @xmath37s sit at distances @xmath176 times the lattice constant , are related to one another by the symmetry operations of the cubic lattice . for example , the second to the fourth configurations on the top row can be obtained from the first by the rotations @xmath177 , @xmath178 and @xmath179 respectively . the configurations are described by _ cardinality sequences _ ( as described earlier ) . the cardinality sequences for the four configurations on the top row of are : @xmath180 , @xmath181 , @xmath182 and @xmath183 . from the figure it is easy to see that : @xmath184 = 3.5in=2.5 in this equivalence of the configurations on the lattice is quite independent of the symmetries of the hamiltonian in real space discussed earlier . thus , in augmented space , equivalent states are @xmath185 and the set @xmath186 for all different symmetry operators @xmath187 of the underlying lattice . again the symmetry of the orbitals also rules out the operation of the hamiltonian at certain symmetric positions discussed earlier . = 4.0 in = 2.5 in .comparison between system size and cpu time ( in seconds ) taken for recursion on a p4/256 machine for a full fcc lattice and the reduced lattice in real space . [ cols="^,^,^,^,^,^,^ " , ] [ tab3 ] for the case of the ordered fcc lattice , the shows how the size of the map increases as we increase the number of nearest neighbour shells from a starting site , both with and without reduction . table 2 shows the details of the cpu time and storage space reduction for recursion after applying the symmetry reduction . we have carried out calculations both on a simple @xmath62-state system on a fcc lattice , as well as for @xmath188 ( with @xmath81 minimal tb - lmto basis ) also on a fcc lattice . for the calculation of a disordered binary alloy on a fcc lattice , the shows the enormous decrease in the size of the augmented space map after application of symmetry reduction for a seven nearest neighbour map on a fcc lattice . tabulates the reduction in storage and cpu time for a 7 shell , 11 step recursion in augmented space carried over 15 seed points using tb - lmto potential parameters and structure matrix to built the hamiltonian . the power of symmetry reduction of storage space is more evident in this example . such reduction will allow us to stretch our nearest neighbour map up to 9 - 10 shells , stepping up our accuracy . the indicates the reduction in cpu time as we increase the number of recursion steps . further , since the number of sites in the map decrease , the number of individual applications of the @xmath189 hamiltonian also decrease significantly , as do the number of operations involved in taking various inner products . this will reduce the inherent cumulative error of the recursion technique and lessen the probability of the appearance of _ ghost bands _ which often plague recursion calculations . we have shown that recursion calculations can be carried out much faster and for many more recursive steps exactly , if we perform the recursion on a subspace of the original augmented space reduced by using the symmetries of both the underlying lattice and random configurations on the lattice . this will allow us to obtain results for disordered binary alloys with enhanced accuracy required for first - principles , self - consistent , density functional based calculations . in this communication we have described the details of the implementation of this symmetry reduction and the modifications required in the standard recursion method . we propose using the symmetry reduced version of the augmented space recursion in our future work on disordered alloys . 99 saha t. , dasgupta i and mookerjee a. , 1994 , l245 saha t. , dasgupta i. and mookerjee a. , 1996 1979 mookerjee a. , 1973 l205 ; 1973 1340 haydock r. , heine v. and kelly m.j . , 1972 2845 andersen o. k. , 1975 b * 12 * 3060 jepsen o. and andersen o. k. , 1971 1763 mookerjee a. and prasad r. , 1993 17724 saha t. , dasgupta i. and mookerjee a. , 1994 13267 saha t. and mookerjee a.,1995 2915 alam a. and mookerjee a. , 2003 ( to be published ) ; cond - mat/0307697 beer n. and pettifor d. g. ( 1982 ) , in _ the electronic structure of complex systems _ , ed . p. phariseau and w. m. temmerman , nato asi series b , v.113 , p 769 lucini m.u . and nex c.m.m . , 1987 , 3125 kaplan t. and gray l.j . , 1976 l303 , l483 kaplan t. and gray l.j . , 1977 3260 ghosh s. , leath p.l . , and cohen m. h. , 2002 , 214206 kumar v. , mookerjee a. and srivastava v.k . , 1982 , 1939 srivastava v.k . , choudhry v. and mookerjee a. , 1983 4555 haydock r. and te r.l . , 1995 _ computer phys . communication _ * 90 * 81 gallagher j. ( 1978 ) , _ ph . d. thesis , _ university of cambridge , u. k. harrison w.a . , 1999 _ elementary electronic structure _ ( world scientific , singapore ) p 546 . biswas p. sanyal b. , mookerjee a. , huda a. , halder a. and ahmed m. , 1997 _ int . phys . _ * b 11 * 3715 ghosh s. , das n. and mookerjee a. , 1999 _ int . j. mod . _ * b 21 * 723 haydock r. , _ solid state physics _ ed . h. ehrenreich , f. seitz and d. turnbull ( academic press , new york ) ( 1980 )

we present here an efficient method which systematically reduces the rank of the augmented space and thereby helps to implement augmented space recursion for any real calculation . our method is based on the symmetry of the hamiltonian in the augmented space and keeping recursion basis vectors in the irreducible subspace of the hilbert space . @xmath0 # 1 2\ { r } # 1 # 1

You are an expert at summarizing long articles. Proceed to summarize the following text: there is a number of new physics ( np ) models beyond the standard model ( sm ) of particle physics . motivated by the hierachy and/or the fine - tuning problem in the sm , most np models propose new states with tev - scale masses . a few examples are the susy models , models with extra gauge bosons ( @xmath6 models ) , and the models with extra dimensions . when the masses of the np states are heavier than the center of mass ( cm ) energy of the collider , the effects of the np can be measured indirectly in terms of the deviations of the sm observables such as the total cross section and various asymmetries . the deviations from the sm in the scattering processes are determined by the mass , spin , and coupling strength of the new states being exchanged by the initial state particles . the question of how to distinguish new states with different spins and couplings at the low energies arises at the sub - tev @xmath7 collider . while the cern large hadron collider ( lhc ) will probe np models with the tev - scale masses , we certainly need the precision measurements to distinguish signature of one model from the others . the precision measurements of the four - fermion scattering at the @xmath7 collider are expected to efficiently reveal the nature of the intermediate states being exchanged by the fermions . the angular distributions and the asymmetries induced by various new states provide information of the spin and coupling of the interactions . at the international linear collider ( ilc ) with the center of mass energy @xmath8 gev , the tev masses could not be observed directly as the resonances since they are heavier than @xmath9 . low energy taylor expansion is a good approximation for the signals induced from the np models and the corrections will be characterized by the higher dimensional operators . for the 4-fermion scattering , the leading - order np signals from the states with spin-0 and spin-1 such as leptoquarks , sneutrino with @xmath10-parity violating interactions @xcite and @xmath6 will appear as the dimension-6 contact interaction at low energies . as a candidate for the np state with spin-2 , the interaction induced by the ( massive ) gravitons , @xmath11 , can be characterized at the low energies by the effective interaction of the form @xmath12 @xcite , a dimension-8 operator . in the viewpoint of effective field theory , this effective interaction does not need to be originated from the exhanges of massive graviton states and it is not the most generic form of the interaction containing dimension-8 operators . however , it certainly has the gravitational interpretation due to the use of the symmetric energy - momentum tensor @xmath13 . it can be thought of as the low - energy effective interaction induced by exchange of the kk gravitons ( in add @xcite and rs @xcite scenario ) interacting with matter fields in the non - chiral fashion . in the braneworld scenario where the sm particles are identified with the open - string states confined to the stack of d - branes subspace , and gravitons are the closed - string states propagating freely in the bulk spacetime @xcite , table - top experiments @xcite and astrophysical observations allow the quantum gravity scale to be as low as tevs @xcite . since the string scale , @xmath14 , in this scenario is of the same order of magnitude as the quantum gravity scale , it is possible to have the string scale to be as low as a tev . the tev - scale stringy excitations would appear as the string resonances ( sr ) in the @xmath15 processes at the lhc @xcite . the most distinguished signals would be the resonances in the dilepton invariant mass distribution appearing at @xmath16 each resonance would contain various spin states degenerate at the same mass . these srs can be understood as the stringy spin excitations of the zeroth modes which are identified with the gauge bosons in the sm . they naturally inherit the chiral couplings of the gauge bosons . in this article , it will be shown that the leading - order stringy excitations of the exchanging modes identified with the gauge bosons in the four - fermion interactions will contain both spin-1 and 2 . their couplings will be chiral , inherited from the chiral coupling of the zeroth mode ( identified with the gauge boson ) . namely , we construct the tree - level stringy amplitudes with chiral spin-2 interactions ( in addition to a stringy dimension-8 spin-1 contribution contrasting to the dimension-6 contributions from other @xmath17 model at low energies ) which can not be described by the non - chiral effective interaction of the form @xmath12 as stated above . this article is organized as the following . in section ii , we discuss briefly the construction of the stringy amplitudes in the four - fermion scattering as is introduced in the previous work @xcite , the comments on the chiral interaction are stated and emphasized . in section iii , the low - energy stringy corrections are approximated . the angular momentum decomposition reveals the contribution of each spin state induced at the leading order . in section iv , we calculate the angular left - right , forward - backward , and center - edge asymmetries . the extensions to partially polarized beams are demonstrated in section v. in section vi , we make concluding remarks and discussions . the low - energy ( @xmath18 ) expressions for the asymmetries @xmath19 induced by the sm and the np models ( kk gravitons and sr ) up to the order of @xmath20 are given in the appendix . the 4-fermion processes that we will consider are the scattering of the initial electron and positron into the final states with one fermion and one antifermion , @xmath21 . we will ignore the masses of the initial and final - states particles and therefore consider only the processes with @xmath22 where @xmath23 . the physical process will be identified as @xmath24 . the @xmath25 and @xmath1-channel are labeled @xmath26 and @xmath27 respectively . to make a relatively model - independent phenomenological study of the tev - scale stringy kinematics , we adopt the parametrised bottom - up " approach for the construction of the tree - level string amplitudes @xcite . in this approach , the gauge structure and the assignment of chan - paton matrices to the particles are not explicitly specified . the main requirement is that the relevant string amplitudes reproduce the sm amplitudes in the low energy limit ( @xmath28 , field theory limit ) . this implies that we identify the zeroth - mode string states as the gauge bosons of the sm . the expression for the open - string 4-fermion tree - level helicity amplitude for the process @xmath29 with @xmath30 is @xcite @xmath31 where @xmath32 and the regge slope @xmath33 . the interaction factors from the exchange of photon and @xmath34 ( chiral ) are given by @xmath35 here @xmath36 and the @xmath37 coupling @xmath38 . the neutral current couplings are @xmath39 . the chan - paton parameter @xmath40 represents the tree - level stringy interaction which can not be determined in the field theory limit @xmath41 since @xmath42 . the amplitude for other helicity combination @xmath43 is given by @xmath44 and an index exchange in the @xmath45 factor @xmath46 since the veneziano - like function @xmath47 and @xmath48 are appearing with the chiral couplings ( the factor @xmath45 ) in the amplitudes , the piece of the stringy states induced by this function will have the similar chiral couplings to those of the sm . on the other hand , the purely stringy interaction piece , proportional to @xmath40 , are assumed to be non - chiral and the values of @xmath40 are set to the same value for all helicity combinations . for @xmath49 , we can use taylor expansion to approximate the leading order stringy corrections to the amplitudes . they are in the form of the dimension-8 operator @xmath50 the amplitude in eq . ( [ eq:3 ] ) becomes @xmath51 again , for @xmath52 , the amplitude is @xmath44 and @xmath53 of the above . the stringy correction can be decomposed into the contribution from the angular momentum states , @xmath54 . using the wigner functions @xmath55 and @xmath56 , @xmath57 where @xmath58 is the angle between the incoming electron and the outgoing antifermion in the c.m . frame . from eq . ( [ eq:12]-[eq:13 ] ) , we can see that the couplings of the @xmath54 states , proportional to @xmath59 , inherit the chirality from the coupling @xmath45 of the zeroth mode gauge boson exhange in the sm . this is the distinctive feature of the couplings of the string states in this bottom - up " approach . because of the chirality of the coupling , the @xmath60 interaction in these stringy amplitudes can not be described by the effective interaction of the form @xmath12 . as long as we couple the spin-2 state @xmath61 to the energy - momentum tensor @xmath62 which does not contain information of the chirality of the fermions , the interaction will always be non - chiral . therefore , the effective interaction of this kind can never describe the _ chiral _ stringy interaction induced by the worldsheet stringy spin excitations in the string models under consideration ( i.e. the models which address chiral weak interaction ) . as a comparison , we give the sm - kk amplitudes where the kk part is induced by the effective interaction of the form @xmath12 as @xmath63 for @xmath64 and @xmath65 . the kk - gravitons contributions can be represented by the wigner functions as @xmath66 the chiral spin-1 and spin-2 stringy interaction will lead to remarkable and unique phenomenological signatures in the 4-fermion scattering at the electron - positron collider even when compared with the contributions from kk gravitons as we will see in the following . the left - right ( @xmath67 ) and forward - backward ( @xmath68 ) asymmetries quantify the degrees of the _ chirality _ of the interaction under consideration regardless of the spin of the intermediate states . on the other hand , the center - edge asymmetry ( @xmath69 ) does not contain any information on the chirality of the spin-1 interaction ( at least in the massless limit ) @xcite . for spin-2 , @xmath69 shows dependence on the chirality of the couplings of the intermediate states in the scattering as we will present in section v. therefore , we can use @xmath69 to distinguish np models with spin-2 mediator from the models mediated by the spin-1 state . among the class of models with spin-2 interactions , we can use the @xmath70 to distinguish one model from another as demonstrated in fig . [ .1-fig ] and fig . [ 10-fig]-[12-fig ] for the case of sr versus kk - gravitons . the angular left - right asymmetry is defined as a function of @xmath71 as @xmath72 where @xmath73 is the cross section of the scattering of the 100% left(right)-handedly polarized electron beam with the 100% right(left)-handedly polarized positron beam . the angular left - right asymmetries induced by the stringy corrections are plotted as in fig . [ .1-fig ] in comparison to the kk - graviton model . it is interesting to comment that the angular left - right asymmetries induced by the stringy corrections differ significantly from the sm distributions only in the quark ( @xmath1 and @xmath2 ) final states . the difference in @xmath74 is hardly visible . this feature is similar to the asymmetries induced by the kk gravitons @xcite . = 5.5 in the forward - backward asymmetry is defined as @xmath75 where @xmath76 is the number of events in the forward(backward ) direction . the numerical values of the unpolarised forward - backward asymmetries for the sm and the stringy amplitudes with @xmath77 gev , @xmath78 tev , and @xmath79 are @xmath80 the deviations of the values with the stringy corrections from the sm values are linearly dependent on @xmath40 and @xmath81 at the leading order as we can see from the expression in the polarized beams section with @xmath82 . this is true only for the scattering at low energies comparing to the string scale . at higher energies @xmath83 or around the srs at @xmath84 , the forward - backward asymmetries become very small due to the non - chiral choice of the chan - paton parameters @xmath40 which efficiently dilutes the chirality of the interaction @xcite . the center - edge asymmetry is defined as a function of the cut of the central region @xmath85 as @xmath86 . \label{eq:2.10}\end{aligned}\ ] ] the deviations of the center - edge asymmetry from the sm values of the unpolarised beams , @xmath87 , are plotted with respect to @xmath85 in fig . [ 4-fig ] . a few interesting remarks are worthwhile making . there is distinctive feature between @xmath69 of the lepton @xmath74 and the quarks , @xmath1 and @xmath2 . the effect of the non - chiral purely stringy interaction represented by the value of @xmath40 is opposite for @xmath74 and @xmath1 . on the other hand , the features of @xmath74 and @xmath2-type quark are , for the small value of @xmath88 , roughly the same . for large value of @xmath89 , the deviation from the sm of the @xmath2-type becomes negative and appears very distinctive from the corresponding value of the @xmath74 . this strange behaviour is originated from the competing contributions between @xmath90 appearing in the @xmath91 terms and the terms of order @xmath92 . it should be emphasized that the numerical results as in fig . [ 4-fig ] are calculated using full expression upto the order of @xmath91 . they are different from the results obtained from the approximation using the leading order upto @xmath92 as is given in section v. the difference becomes very obvious in the @xmath2-type quark with high value of @xmath89 . let @xmath93 be the degrees of the longitudinal polarization of the @xmath94 beam defined as ( @xmath95 ) @xmath96 where @xmath97 is the number of particle @xmath98 with the left(right)-handed helicity in the beams . the polarized differential cross section can be expressed as @xmath99 \label{eq:3.1}\end{aligned}\ ] ] where @xmath100 upper(lower ) signs are for @xmath101 . @xmath102 represents the scattering involving the left(right)-handed electron . in practice , the observable left - right asymmetry is defined with respect to the partially polarized beams of electron and positron by taking difference of the total number of events when the polarizations of the beams are inverted . it is therefore given by @xmath103 where @xmath104 . for @xmath105 , @xmath106 is @xmath107 , while @xmath106 becomes @xmath108 when @xmath109 . the full expressions of @xmath110 for the kk - gravitons and sr models are given in the appendix . with the partially polarised beams of electron and positron , we can calculate @xmath68 using eq . ( [ eq:3.1 ] ) . the full expressions for the asymmetries induced by kk gravitons and sr are given in the appendix . up to the leading order of @xmath92 in the energy taylor expansion , the polarized forward - backward asymmetry induced by the stringy corrections is @xmath111 @xmath112 \label{eq:3.3 } \end{aligned}\ ] ] where the einstein summation convention is implied for @xmath113 . @xmath114 and @xmath115 . the other combination is obtained by exchanging @xmath116 . the sm value is given by @xmath117 with @xmath118 and @xmath119 . [ 10-fig]-[12-fig ] show @xmath68 induced by kk gravitons and sr in comparison to the sm values with respect to the cm energy . assuming the string scale @xmath78 tev and the effective quantum gravity scale @xmath120 tev , the differences between the asymmetries induced by the two models appear at higher cm energies even in the case when they are indistinguishable at the low cm energies . this is due to the fact that while the _ chirality _ of the stringy interaction keeps the terms of both order @xmath92 and @xmath91 in the expression of @xmath68 , the non - chiral graviton interaction , on the other hand , keeps only the term of order @xmath121 in the numerator @xmath122 , and only term of order @xmath123 in the denominator @xmath124 ( see appendix ) . this leads to distinguishable aspects of the two models . with the partially polarised beams of electron and positron , we can calculate @xmath69 using eq . ( [ eq:3.1 ] ) . the full expressions for the asymmetries induced by kk gravitons and sr are given in the appendix . the deviations from the sm values for various polarizations of the beams for the stringy model are plotted as in fig . [ 7-fig]-[9-fig ] . up to the leading order of @xmath92 in the energy taylor expansion , the polarized center - edge asymmetry induced by the stringy corrections is @xmath125 @xmath126 where the sm value is given by @xmath127 eq . ( [ eq : c3 ] ) is unique for the spin-1 contribution in the 4-fermion scattering . any np particles with spin-1 will not change this @xmath85-dependence . remarkably , the spin-2 contributions , either in the form of kk @xcite or the string states ( eq . ( [ eq:3.5 ] ) ) , will induce the deviations from this sm value proportional to @xmath128 . similar to the @xmath68 case , the _ chirality _ of the stringy interaction keeps terms of both order @xmath92 and @xmath91 in the expression of @xmath129 while in the case of kk gravitons , there is no terms of order @xmath121 in the constant term @xmath130 and the denominator @xmath131 . the differences become obvious when @xmath132 is relatively large ( @xmath133 ) as shown in fig . [ 4-fig ] . = 5.5 in = 4.0 in = 4.0 in = 4.0 in = 4.0 in = 4.0 in = 4.0 in we have constructed and approximated the tree - level stringy amplitudes of the scattering @xmath134 . the low - energy stringy corrections for the 4-fermion processes contain both spin-1 and spin-2 contributions with the chiral couplings inherited from the zeroth mode states identified with the gauge bosons in the sm . the chirality of the couplings is diluted by the non - chiral choice of the purely - stringy piece of the stringy interaction , characterized by @xmath40 . the contributions from both stringy spin-1 and spin-2 are of dimension-8 in nature . the low - energy dimension-8 spin-1 contribution is remarkably distinctive from the dimension-6 contributions in other @xmath17 models . the chirality of stringy spin-2 interaction also leads to unique phenomenological features distinguishable from the non - chiral spin-2 interaction induced by the kk - gravitons ( or massive gravitons ) . then we investigated the signatures of the tev - scale string model at the @xmath7 collider in comparison to the kk - gravitons using angular left - right , forward - backward , and center - edge asymmetries . the deviations of the asymmetries from the sm values are investigated separately for each model . all asymmetries show significant differences between the low - energy corrections of the two models . for the @xmath7 collider with variable center - of - mass energies from 500 to 1000 gev and assuming @xmath135 tev , the forward - backward asymmetries show drastic differences between stringy signals and the kk - graviton ones . the center - edge asymmetries also show significant differences between the two models if the chan - paton parameter @xmath132 ( representing purely stringy piece of the interaction ) in the sr model is sufficiently large ( @xmath133 ) . the origin of the differences between the two models is mainly ( another reason is the fact that sr also has spin-1 contribution in addition to spin-2 ) the _ chirality _ of the interactions . while the string interaction is chiral , the interaction induced by the kk gravitons , couple to the energy - momentum tensor @xmath62 , is non - chiral . as we can see in the appendix from the full espressions of @xmath19 , the chirality of the stringy interaction keeps almost all of the terms of order @xmath92 and @xmath91 while the non - chirality of the interaction of the kk gravitons eliminates certain terms of order @xmath121 and @xmath123 in the asymmetries . specifically , the @xmath136 term in @xmath110 contains only the sm term in the case of kk gravitons , in contrast to the stringy case . in @xmath68 , as discussed above , @xmath137 in the case of kk gravitons does not contain the term of order @xmath138 . for @xmath129 , @xmath139 does not contain terms of order of @xmath121 in the case of kk gravitons . this is in contrast to the processes where the stringy interaction is non - chiral as well as having only the spin-2 contributions such as in the scattering @xmath140 ; in which case the two models give exactly the same low - energy signatures @xcite . in the intermediate energy range ( @xmath141 ) with @xmath142 , since the deviations induced by kk gravitons and sr depend only on @xmath143 , the results in this article therefore are also valid for the clic with center - of - mass energies 3 to 6 tev with @xmath144 tev . i would like to thank tao han for helpful discussions . this work was supported in part by the u.s . department of energy under contract number de - fg02 - 01er41155 . @xmath145 [ [ section ] ] @xmath146 @xmath147 [ [ section-1 ] ] @xmath148 @xmath149 @xmath150 [ [ section-2 ] ] @xmath151 [ [ section-3 ] ] @xmath152 @xmath153 a. antoniadis , n. arkani - hamed , s. dimopoulos , and g. dvali , phys . lett . * b436 * , 257 ( 1998)[hep - ph/9804398 ] ; n. arkani - hamed , s. dimopoulos and g.r . dvali , phys . * b429 * , 263 ( 1998 ) [ hep - ph/9803315 ] . l. randall and r. sundrum , phys . lett . * 83 * , 3370 ( 1999 ) [ hep - ph/9905221 ] . a. antoniadis , n. arkani - hamed , s. dimopoulos , and g. dvali , phys . lett . * b436 * , 257 ( 1998)[hep - ph/9804398 ] . g. shiu and s.h . tye , phys . rev . * d58 * , 106007 ( 1998)[hep - th/9805157 ] ; 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we investigate the tev - scale stringy signals of the four - fermion scattering at the electron - positron collider with the center of mass energy @xmath0 gev . the nature of the stringy couplings leads to distinguishable asymmetries comparing to the other new physics models . specifically , the stringy states in the four - fermion scattering at the leading - order corrections are of spin-1 and 2 with the chiral couplings inherited from the gauge bosons identified as the zeroth - mode string states . the angular left - right , forward - backward , center - edge asymmetries and the corresponding polarized - beam asymmetries are investigated . the low - energy stringy corrections are compared to the ones induced by the kaluza - klein ( kk ) gravitons . the angular left - right asymmetry of the scattering with the final states of @xmath1 and @xmath2-type quarks , namely @xmath3 and @xmath4 , shows significant deviations from the standard model values . the center - edge and forward - backward asymmetries for all final - states fermions also show significant deviations from the corresponding standard model values . the differences between the signatures induced by the stringy corrections and the kk gravitons are appreciable in both angular left - right and forward - backward asymmetries . @xmath5 0.4 cm

You are an expert at summarizing long articles. Proceed to summarize the following text: how can one best bound the complexity of an algebraic set in terms of the complexity of its defining polynomials ? over the complex numbers ( or any algebraically closed field ) , bzout s theorem @xcite bounds the number of roots , for a system of multivariate polynomials , in terms of the degrees of the polynomials . over finite fields , weil s famous mid-20@xmath11 century result @xcite bounds the number of points on a curve in terms of the genus of the curve ( which can also be bounded in terms of degree ) . these bounds are optimal for dense polynomials . for sparse polynomials , over fields that are not algebraically closed , these bounds can be much larger than necessary . for example , descartes rule @xcite tells us that a univariate real polynomial with exactly @xmath2 monomial terms always has less than @xmath12 real roots , even though the terms may have arbitrarily large degree . is there an analogue of descartes rule over finite fields ? despite the wealth of beautiful and deep 20@xmath11-century results on point - counting for curves and higher - dimensional varieties over finite fields , varieties defined by sparse _ univariate _ polynomials were all but ignored until @xcite ( see lemma 7 there , in particular ) . aside from their own intrinsic interest , refined univariate root counts over finite fields are useful in applications such as cryptography ( see , e.g. , @xcite ) , the efficient generation of pseudo - random sequences ( see , e.g. , @xcite ) , and refined estimates for certain exponential sums over finite fields ( * ? ? ? * proof of theorem 4 ) . for instance , estimates on the number of roots of univariate tetranomials over a finite field were a key step in establishing the uniformity of the _ diffie - helman distribution _ * proof of thm . 8 , sec . 4 ) a quantitative statement important in justifying the security of cryptosystems based on the discrete logarithm problem . we are thus interested in the number of roots of sparse univariate polynomials over finite fields . the polynomial @xmath13 having two terms and exactly @xmath0 roots in @xmath14 might suggest that there is no finite field analogue of descartes rule . however , the roots of @xmath13 consist of @xmath15 and the roots of @xmath16 , and the latter roots form the unit group @xmath17 . for an arbitrary binomial @xmath18 $ ] with @xmath19 and @xmath20 and @xmath21 nonzero , the roots consist of @xmath15 ( if @xmath22 ) and the roots of @xmath23 . note that the number of roots of @xmath23 in @xmath14 is either @xmath15 or @xmath24 . in the latter case , the roots form a coset of a subgroup of @xmath5 . for polynomials with three or more terms , the number of roots quickly becomes mysterious and difficult , and , as we shall demonstrate in this paper , may exhibit very different behaviors in the two extreme cases where ( a ) @xmath0 is a large power of a prime , and ( b ) @xmath0 is a large prime . to fix notation , we call a polynomial @xmath25 $ ] with @xmath26 and @xmath27 for all @xmath28 a _ ( univariate ) @xmath2-nomial_. the best current upper bounds on the number of roots of @xmath6 in @xmath14 , as a function of @xmath0 , @xmath2 , and the coset structure of the roots of @xmath6 , can be summarized as follows , using @xmath29 for set cardinality : [ thm : z ] let @xmath1 $ ] be any univariate @xmath2-nomial with degree @xmath3 and exponent set @xmath30 containing @xmath15 . set @xmath31 , @xmath32 , @xmath33 , and let @xmath34 denote the maximum cardinality of any coset ( of any subgroup of @xmath35 contained in @xmath36 . then : + 0 . ( special case of ( * ? ? ? * thm . 1 ) ) @xmath37 . 1.1 ) @xmath36 is a union of no more than @xmath38 cosets , each associated to one of two subgroups @xmath39 of @xmath5 , where @xmath40 , @xmath41 , and @xmath42 can be determined within @xmath43 bit operations . 1.2 ) for @xmath44 we have @xmath45 and , if we have in addition that @xmath0 is a square and @xmath46 , then @xmath47 . ( see ( * ? ? ? 2.2 & 2.3 ) ) for any @xmath48 we have @xmath49 . + [ 1]furthermore , @xmath50 . @xmath51 for any fixed @xmath48 , _ dirichlet s theorem _ ( see , e.g. , ( * ? ? ? * thm . 8.4.1 , pg . 215 ) ) implies that there are infinitely many prime @xmath0 with @xmath52 . for such pairs @xmath53 the bound from assertion ( 0 ) is tight : the roots of @xmath54 are the disjoint union of @xmath55 cosets of size @xmath56 . ( there are no @xmath57-cosets for this @xmath2-nomial . ) however , assertions ( 1 ) and ( 3 ) tell us that we can get even sharper bounds by making use of the structure of the cosets inside @xmath36 . for instance , when @xmath44 and @xmath46 , assertion ( 2 ) yields the upper bound @xmath58 , which is smaller than @xmath59 for @xmath60 . while assertion ( 3 ) might sometimes not improve on the upper bound @xmath61 , it is often the case that @xmath34 is provably small enough for a significant improvement . for instance , when @xmath62 and @xmath63 for all @xmath64 , we have @xmath65 and then @xmath66 . [ thm : opt ] let @xmath67 with @xmath48 and @xmath68 prime . if @xmath69 then the polynomial @xmath70 has @xmath71 and exactly @xmath72 roots in @xmath14 . furthermore , if @xmath73 , then the polynomial @xmath74 has @xmath75 , @xmath76 , and exactly @xmath77 roots in @xmath14 . theorem [ thm : opt ] is proved in section [ sec : trick ] below . the polynomials @xmath78 show that the bounds from assertions ( 1 ) and ( 3 ) of theorem [ thm : z ] _ are within a factor of @xmath79 of being optimal _ , at least for @xmath80 and a positive density of prime powers @xmath0 . note in particular that @xmath81 shows that assertion ( 2 ) of theorem [ thm : z ] is _ optimal _ for square @xmath0 and @xmath46 . ( see ( * ? ? ? 1.3 ) for a different set of extremal trinomials when @xmath0 is an odd square . ) the second family @xmath82 reveals similar behavior for a different family of prime powers . optimally bounding the maximal number of roots ( or cosets of roots ) for the case of _ prime _ @xmath0 is already more subtle in the trinomial case : we are unaware of any earlier lower bound that is a strictly increasing function of @xmath0 . ( note , for instance , that @xmath83 vanishes on just cosets of roots , assuming @xmath84 mod @xmath85 . ) in section [ sec : harder ] , we shall prove the following theorem . [ thm : grh ] for any @xmath86 and prime @xmath87 , consider @xmath88 as an element of @xmath89 $ ] . then @xmath90 , and there exists an infinite family of primes @xmath68 such that the trinomial @xmath91 has exactly @xmath92 distinct roots in @xmath9 where this theorem is proved by using results from algebraic number theory on the splitting of primes in number fields . in particular , we use a classic estimate of lagarias , montgomery , and odlyzko @xcite ( reviewed in section [ sec : harder ] below ) on the least prime ideal possessing a frobenius element exhibiting a particular galois conjugacy class . the latter result is in turn heavily based on the effective chebotarev density theorem of lagarias and odlyzko @xcite . cohen @xcite applied an effective function field analogue of the chebotarev density theorem to trinomials of the form @xmath95 , where @xmath21 varies in @xmath9 and @xmath68 goes to infinity . his results tell us that when @xmath68 is large , there always exist @xmath96 so that @xmath95 splits completely over @xmath9 , and the least such @xmath68 satisfies @xmath94 unconditionally . this gives the existence of @xmath97 ( and infinitely many primes @xmath68 ) with @xmath98 having many roots in @xmath9 , but gives no information on how to easily find such @xmath21 . the novelty of theorem [ thm : grh ] is thus an even simpler family of trinomials with number of roots in @xmath9 increasing as @xmath68 runs over an infinite sequence of primes . assuming grh , the rate of increase matches cohen s lower bound . could the existence of trinomials @xmath6 with @xmath46 and , say , @xmath99 roots in @xmath9 ( as one might conjecture from theorem [ thm : opt ] ) be obstructed by @xmath68 not being a square ? we feel that @xmath68 being prime is such an obstruction and , based on experimental results below , we conjecture the following upper bound : ( see also ( * ? ? ? 1.5 & sec . 4 ) for other refined conjectures and heuristics in this direction . ) it is a simple exercise to show that , to compute the maximal number of roots in @xmath9 of trinomials with @xmath46 , one can restrict to the family of trinomials in the conjecture . for any @xmath107 , let @xmath108 be the least prime for which there exists a univariate trinomial @xmath109 with @xmath110 and exactly @xmath92 distinct roots in @xmath111 . we did a computer search to find the values of @xmath108 for @xmath112 . they are ... in particular , for each @xmath113 , a full search was done so that the trinomial @xmath109 below has the least degree among all trinomials over @xmath111 having exactly @xmath92 roots in @xmath111 . ( it happens to be the case that , for @xmath113 , we can also pick the middle degree monomial of @xmath109 to be @xmath114 . ) by rescaling the variable as necessary , we have forced @xmath115 to be among the roots of each of the trinomials below . it is easily checked via the last part of assertion ( 3 ) of theorem [ thm : z ] that @xmath116 for each @xmath113 . the least prime @xmath117 for which there is a trinomial @xmath118 with @xmath119 and exactly @xmath120 roots in @xmath121 is currently unknown ( as of july 2016 ) . better and faster code should hopefully change this situation soon . jingguo bi ; qi cheng ; and j. maurice rojas , _ `` sub - linear root detection , and new hardness results , for sparse polynomials over finite fields , '' _ proceedings of issac ( international symposium on symbolic and algebraic computation , june 2629 , boston , ma ) , pp . 6168 , acm press , 2013 . ran canetti ; john b. friedlander ; sergei konyagin ; michael larsen ; daniel lieman ; and igor e. shparlinski , _ `` on the statistical properties of diffie - hellman distributions , '' _ israel j. math . 120 ( 2000 ) , pp . 2346 . alexander kelley , _ `` roots of sparse polynomials over a finite field , '' _ in proceedings of twelfth algorithmic number theory symposium ( ants - xii , university of kaiserslautern , august 29 september 2 , 2016 ) , to appear . also available as math arxiv preprint 1602.00208 . jeff lagarias and andrew odlyzko , _ `` effective versions of the chebotarev density theorem , '' _ algebraic number fields : @xmath123-functions and galois properties ( proc . durham , durham , 1975 ) , 409464 , academic press , london , 1977 . david eugene smith and marcia l. latham , _ the geometry of ren descartes , _ translated from the french and latin ( with a facsimile of descartes 1637 french edition ) , dover publications inc . , new york ( 1954 ) .

suppose @xmath0 is a prime power and @xmath1 $ ] is a univariate polynomial with exactly @xmath2 monomial terms and degree @xmath3 . to establish a finite field analogue of descartes rule , bi , cheng , and rojas ( 2013 ) proved an upper bound of @xmath4 on the number of cosets in @xmath5 needed to cover the roots of @xmath6 in @xmath5 . here , we give explicit @xmath6 with root structure approaching this bound : for @xmath0 a @xmath7-st power of a prime we give an explicit @xmath2-nomial vanishing on @xmath8 distinct cosets of @xmath5 . over prime fields @xmath9 , computational data we provide suggests that it is harder to construct explicit sparse polynomials with many roots . nevertheless , assuming the generalized riemann hypothesis , we find explicit trinomials having @xmath10 distinct roots in @xmath9 .

You are an expert at summarizing long articles. Proceed to summarize the following text: supermassive black holes ( smbhs ) appear to be present in the nucleus of most , and perhaps all , nearby galaxies ( see , e.g. , reviews by @xcite and @xcite ) . the correlations between the masses of the smbhs and various global properties of the host galaxies suggest that evolution of smbhs is closely related to the evolution of galaxies . in particular , in hierarchical structure formation models , galaxies are built up by mergers between lower mass progenitors . each merger event is expected to deliver the nuclear smbhs ( e.g. @xcite ) , along with a significant amount of gas @xcite , to the central regions of the new post merger galaxy . there is some evidence for nuclear supermassive black hole binaries ( smbhbs ) , which would be expected to be produced in galaxy mergers . direct x ray imaging of an active nucleus @xcite has revealed a smbh binary at a separation of @xmath10kpc , and @xcite recently identified a candidate smbhb , at @xmath11 times smaller separation , from its optical spectrum . a radio galaxy is also known to have a double core with a projected separation of @xmath12 pc @xcite , and several other observations of radio galaxies , such as the wiggled shape of jets indicating precession ( e.g. * ? ? ? * ) , the x shaped morphologies of radio lobes ( e.g. * ? ? ? * ; * ? ? ? * ) , the interruption and recurrence of activity in double double radio galaxies ( e.g. * ? ? ? * ; * ? ? ? * ) , and the elliptical motion of the unresolved core of 3c66b @xcite below . ] have all been interpreted as indirect evidence for smbh binaries down to sub pc scales . two interesting conclusions may be inferred from the above observations . first , while there is evidence for a handful of nuclear smbhbs , these objects appear to be rare . this suggests that if binaries do form frequently , then they coalesce ( or at least their orbital separation decays to undetectably small values ) in a small fraction of the hubble time . second , smbhbs can apparently produce bright emission , with a luminosity comparable to active galactic nuclei ( agn ) , before they coalesce . in general , the circumbinary gas , delivered to the nucleus in galactic mergers , can both play a catalyst role in driving rapid smbhb coalescence @xcite , and could also accrete onto one or both smbhs , accounting for bright emission during the orbital decay . the dense nuclear gas around the bh binary is expected to cool rapidly , and settle into a rotationally supported , circumbinary disk ( e.g. * ? ? ? * ; * ? ? ? the dynamical evolution of a smbhb embedded in such a thin disk has been studied in various idealized configurations ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? the generic conclusion of these studies is that initially , the orbital decay is relatively slow , and is dominated by viscous angular momentum exchange with the gas disk , whereas at small separations , the decay is much more rapid , and is eventually dominated by gravitational wave ( gw ) emission . whether the decaying smbhbs produce bright electromagnetic ( em ) emission is comparatively much less well understood . if the disk is thin , the torques from the binary create a central cavity , nearly devoid of gas , within a region about twice the orbital separation ( for a nearly equal mass binary , e.g. * ? ? ? * ) , or a narrower gap around the orbit of the lower mass bh in the case of unequal masses @xmath13 ( e.g. * ? ? ? * ) . in the latter case , the lower mass hole `` ushers '' the gas inward as its orbit decays , producing a prompt and luminous signal during coalescence . in the former case , if the central cavity were indeed truly empty , no gas would reach the smbhbs , and bright emission could not be produced . however , numerical simulations suggest residual gas inflow into the cavity @xcite , which may plausibly accrete onto the bhs , producing non negligible em emission . yr after the coalescence , caused by the gas outside the cavity falling in , after a delay set by the disk viscous time @xcite . such an afterglow is interesting , for example for a follow up to smbh merger events detected by the _ laser interferometric space antenna ( lisa ) _ , but not relevant to the idea proposed in the present paper . ] finally , smbhbs recoil at the time of their coalescence due to the emission of gravitational waves @xcite . the gas disk will respond promptly ( on the local orbital timescale ) to such a kick , which may produce shocks , and transient em emission , after coalescence @xcite . the kick , however , can begin building up during the late inspiral phase @xcite , possibly resulting in some emission even before the final coalescence . during the late stages of coalescence , emission may also be produced by viscous heating of the disk by the gws themselves @xcite . the luminosity , spectrum , and time evolution of any em emission produced by coalescing smbhbs , especially during the last , gw driven stages , remains uncertain . _ however , any emission produced during the inspiral stage is likely to be variable . _ for example , recent numerical simulations of an equal mass binary on parsec scales @xcite , and of both equal and unequal mass binaries on sub parsec scales @xcite find that the circumbinary gas disk is perturbed into eccentric orbits by the rotating quadrupole potential of the binary , and that the rate of residual accretion across the edge of the cavity is modulated , tracking the orbital period . the luminosity is likely to be directly tied to the mass accretion rate , and therefore may vary periodically . however , even if the gas accretion rate were steady , one would expect periodic flux variations , due to the orbital motion of the binary ( kocsis & loeb , in preparation ) . in this paper , we address the question : _ given their expected rate of orbital decay , could the population of coalescing smbhbs be identified statistically in an observational survey for periodically variable sources ? _ given that the interpretation of individual smbhb candidates have so far remained ambiguous , with alternative explanations possible for each source , the potential for such a statistical identification should be explored . to answer this question , we first utilize steady state thin disk models to study the orbital decay of a smbhb , embedded in a circumbinary disk . the decay is described by the residence time @xmath14 the sources spend at each orbital radius @xmath15 , or at the corresponding orbital timescale @xmath16 . in the limiting case of a purely gw driven evolution , which becomes valid at small orbital separations ( typically at @xmath17 several@xmath18 schwarzschild radii , but with large variations ; see below ) and remains valid until the final minutes of the merger ( the so called `` plunge '' stage ) , the residence time is given by @xmath19 . at larger separations , the viscous interaction between the binary and the disk drives the binary evolution . the residence time in this regime becomes dependent on assumptions about the properties of the disk and the nature of the binary disk interaction , which we will explore in this paper . in general , @xmath20 , with the generic value of @xmath21 well below @xmath22 significantly flatter than the @xmath23 relation in the gw driven stage . we then hypothesize that ( i ) non negligible emission ( at a fair fraction of the eddington luminosity ) is maintained throughout the orbital decay , and ( ii ) the luminosity varies periodically on the orbital time the first assumption allows us to identify coalescing smbhbs with luminous quasars . the second assumption implies that as the orbit of a binary decays , its variability timescale decreases . among sources at redshift @xmath24 with similar inferred bh masses , the observed incidence rate @xmath25 of periodic variability on the time scale @xmath26 , is then proportional to the residence time @xmath27 . at short periods , the @xmath25 could therefore show a characteristic power law dependence on @xmath28 indicative of a gw driven evolution , whereas at longer periods ( and , as we will discuss , for lower bh masses ) the dependence will be flatter , due to viscosity driven evolution . we quantify the requirements that such periodically variable sources be identifiable , based on their incidence rate , in an optical or x ray survey . luminosity variations at a fraction @xmath29 of the eddington luminosity would correspond to a periodically varying flux component with amplitude @xmath30 for bhbs at @xmath31 , or to @xmath32 $ ] magnitudes in the optical . we find that these periodic sources are either too faint or too rare to have been found in existing variability surveys . however , if the overall luminosity is indeed a non negligible fraction of the binary s eddington luminosity , then a long duration future survey , sensitive to periods of weeks to tens of weeks , could look for periodically variable sources , and identify a population of sources obeying well defined scaling laws . the discovery of a population of such periodically variable sources could have several implications . at long periods and low bh masses , the scaling index @xmath21 between the residence time and the period @xmath33 will probe the physics of the circumbinary accretion disk and viscous orbital decay . at shorter periods and higher masses ( roughly , at @xmath34 few weeks for @xmath35 ) , the identification of a @xmath36 power law would confirm that the orbital decay is driven by gws . this would amount to an indirect , statistical detection of gw driven smbhbs , independent of any direct detection of gws by _ lisa_. this would also confirm that circumbinary gas is present at small orbital radii and is being perturbed by the bhs and would thus serve as a proof of concept for finding _ lisa _ electromagnetic counterparts . the rest of this paper is organized as follows . in [ sec : diskmodels ] , we discuss the evolution of binaries with different masses and mass - ratios , embedded in a circumbinary gas disk . we describe simplified models for the disk and for the binary disk interaction , and emphasize that the binaries probe the distinct physical regimes in the disk , before gws take over their evolution . in [ sec : observations ] , we discuss the possibility of searching for a population of coalescing smbhbs among a catalog of luminous quasars , either based on their variability , or on shifts of their spectral lines . we discuss modest constraints available from existing surveys , and comment on specific recently detected individual smbhb candidates . we then quantify the requirements for a detection in a future survey . in [ sec : summary ] , we briefly summarize our results and offer our conclusions . when necessary in this paper , we adopt the background cosmological parameters @xmath37 , @xmath38 , and @xmath39 @xcite . in this section , we describe the evolution of the orbital separation of a smbh binary . the basic picture we adopt is that the binary is embedded in a thin circumbinary disk , with the plane of the disk aligned with the binary s orbit @xcite . initially , the orbital decay is dominated by viscous angular momentum exchange with the gas disk . however , the time scale for viscous decay decreases relatively slowly as the orbital separation @xmath15 decreases ( @xmath40 ; see below ) whereas the time scale to decay due to gravitational radiation decreases steeply ( @xmath41 ) . therefore , generically , there exists a critical orbital radius @xmath42 , below which the decay is dominated by gravitational radiation . to describe the evolution quantitatively , we make several simplifying assumptions . the circumbinary gas is assumed to form a standard geometrically thin , optically thick , radiatively efficient , steady state accretion disk @xcite . we assume zero eccentricity for both the binary and for the disk ( justified by @xcite , however see @xcite ) , and we assume co planarity between the disk and the binary @xcite . all of these assumptions may fail in the late stages of the merger ( even before gw driven decay begins ) . however , under these assumptions , the disk structure and the orbital decay have simple limiting power law solutions , with the power law indices depending on the choice for the underlying physics . these solutions are useful to describe the possible evolution of the binary , and to illustrate the point that the decay rate is generically a different much flatter function of @xmath16 than the @xmath43 behavior in the gw driven case . we emphasize that our aim here is not to provide accurate , self consistent solutions for the co evolution of the smbh binary and circumbinary disk . rather , we derive only gross scaling laws in various regimes our main point is that these regimes and associated uncertainties , which are large , can in principle be probed observationally . we adopt the following notation throughout this paper . we refer the reader to @xcite and @xcite for general introductions to accretion disks . * _ physical constants : _ @xmath44 is the gravitational constant ; @xmath45 is the speed of light ; @xmath46 is the boltzmann constant ; @xmath47 is the stefan boltzmann constant ; @xmath48 is the thompson cross section ; @xmath49 is the mean mass per electron in units of hydrogen atom mass , @xmath50 , which satisfies @xmath51 for a fully ionized gas of both hydrogen and helium ; @xmath52 is the mass fraction of hydrogen ; @xmath53 is the mean molecular weight ; @xmath54 is the electron scattering opacity ; and @xmath55 ( t/{\rm k})^{-7/2}$ ] is the rosseland mean absorption opacity in the free free regime @xcite . * _ bh parameters : _ @xmath56 and @xmath57 are the individual bh masses ; @xmath58 is the total bh mass ; @xmath59 is the mass ratio ; @xmath60 is the normalized symmetric mass ratio ; @xmath61 is the reduced mass ; @xmath15 is the binary separation ; @xmath62 is the location of the lower mass secondary , measured from the center of mass of the binary ; @xmath63 is the schwarzschild radius corresponding to the total mass ; @xmath64 is the eddington luminosity for a bh of mass @xmath0 ; @xmath65 is the eddington accretion rate with a radiative efficiency @xmath66 ; and @xmath67 is the characteristic time scale associated with eddington accretion . * _ disk parameters : _ @xmath68 is the vertical scale height ( the effective geometrical semi thickness of the disk ) ; @xmath69 is the volumic gas density ; @xmath70 is the surface density ; @xmath71 is the gas pressure ; @xmath72 is the radiation pressure ; @xmath73 is the total pressure ; @xmath74 ; @xmath75 is the ( midplane ) gas temperature ; @xmath76 is the effective temperature defined such that the locally emitted flux through an infinitesimal disk surface element is @xmath77 ; @xmath78 is the keplerian orbital angular velocity ; @xmath79 is the outer radius of the gap in the punctured circumbinary disk , measured from the center of mass of the binary ; @xmath80 is the anomalous dynamical viscosity ; @xmath81 is the anomalous kinematic viscosity ; @xmath21 is the standard viscosity parameter of thin accretion disks ; @xmath82 is a constant , either 0 or 1 , determining whether viscosity scales with the total or just the gas pressure , so that @xmath83 ; @xmath84 is the opacity of the disk material ; @xmath85 is the vertical optical depth ; and @xmath86 is a constant defined such that @xmath87 . quantities with a @xmath88 subscript ( e.g. @xmath89 ) denote parameters in a steady state disk around a single unperturbed accreting bh , computed at the radius @xmath79 . similarly , quantities with a 0 subscript are evaluated at the position of the secondary @xmath90 . with the above definitions , we proceed to define the dimensionless quantities @xmath91 , @xmath92 , @xmath93 , @xmath94 , @xmath95 , @xmath96 , @xmath97 , @xmath98 , @xmath99 , and @xmath100 . note that radii are measured from the center of mass of the binary . we adopt the set of fiducial values @xmath101 , @xmath102 , @xmath103 , @xmath104 , is appropriate for a one zone model where all the energy is dissipated near the midplane and the opacity is constant vertically @xcite . for reference , we note that @xcite and @xcite adopt different values of @xmath105 and @xmath106 , respectively . ] and denote values relative to these fiducial choices with a tilde , e.g. @xmath107 . our fiducial binary+disk model is therefore chosen to be @xmath108 . since all of our expressions can be written as products of power laws in the physical parameters , the resulting expressions become tractable in these units . we next collect the basic expressions from the literature for accretion disk models under different physical conditions . we quote the equations for a range of different steady thin disks , valid for a single accreting bh @xcite . we distinguish several cases : _ ( i ) _ whether the radiation or gas pressure provides the dominant vertical support , _ ( ii ) _ whether the opacity is dominated by electron scattering , @xmath109 , or free free absorption , @xmath110 , and _ ( iii ) _ whether the viscosity @xmath80 is proportional to the total pressure or the gas pressure ( also known as @xmath21 and @xmath111 disk models , respectively ) . based on these choices , the accretion disk can be divided radially into three distinct regions @xcite : 1 . _ inner region : _ radiation pressure and electron - scattering opacity dominate , @xmath112 , @xmath113 , valid inside @xmath114 where @xmath115 is defined in equations ( [ e : gas / radb1 ] ) and ( [ e : gas / radb0 ] ) below . 2 . _ middle region : _ gas pressure and electron - scattering opacity dominate , @xmath116 , @xmath117 , valid between @xmath118 , where @xmath119 is defined in equation ( [ e : es / ff ] ) below . _ outer region : _ gas pressure and free - free opacity dominate , @xmath116 , @xmath120 , valid outside of @xmath121 . in region ( 1 ) , it makes a difference whether the viscosity is proportional to the total pressure or just the gas pressure , labeled below by @xmath122 or 1 , ( i.e. @xmath21 or @xmath111 disk ) respectively . in all cases , we assume that the disk is optically thick , i.e. @xmath123 . we obtain @xmath124 and @xmath125 following @xcite or @xcite , @xmath126 where @xmath122 or 1 , and the radial dependence is implicit in @xmath127 and @xmath111 . here , @xmath128 which satisfies @xmath129 the asymptotic limits of equations ( [ e : sigma_definition ] ) and ( [ e : h_definition ] ) can be obtained in regions ( 13 ) , using equation ( [ e : beta_definition ] ) . the results are _ inner region : _ @xmath130 _ middle region : _ @xmath131 _ outer region : _ @xmath132 the boundaries between the inner / middle and middle / outer regions can be found from equations ( [ e : sigma_definition])-([e : beta_definition ] ) , by requiring @xmath133 and @xmath134 , respectively . note that @xmath135 depends on radius implicitly through the density and the temperature . using the ( mid - plane ) temperature given by @xcite , @xmath136 we find that the transitions are located at the radii @xmath137 note that the middle and outer regions differ only in their opacity laws , and the equations in these two regions are equivalent ( this can be seen by setting @xmath138 ) . since @xmath139 , @xmath68 , @xmath69 , and @xmath75 scale with a low power of @xmath140 , the radial dependence ends up being similar in the middle and outer regions . the distinction between these equations is nevertheless useful , since we can assume that @xmath141 and @xmath142 are constants in the middle and outer regions , respectively . we emphasize that equations ( [ e : sigma_inb1])-([e : h_out ] ) represent only a very non - exhaustive subset of solutions even for radiatively efficient steady thin accretion disks . in particular , at large radii , there are several effects that can invalidate the disk model described by these equations . first , these solutions assume that the self gravity of the disk is negligible . this assumption becomes invalid at radii where the toomre @xmath143parameter equals unity , @xmath144 beyond these radii , the disk is commonly believed to be unstable to fragmentation . second , at large radii , the disk can also become optically thin ( see * ? ? ? * where solutions can be obtained by fixing the toomre parameter in the outermost region at @xmath145 ) . at these binary separations , the disks may not actually be geometrically thin @xcite , and slim or thick solutions might instead be relevant . third , beyond the radii where the disk temperature falls below @xmath146k , the gas becomes neutral . the corresponding change in opacity will modify the disk structure , and the disk may become susceptible to ionization instabilities ( although see @xcite ) . finally , at large radii ( where the orbital velocity @xmath147 km / s ) , the gravitational potential of the galaxy can no longer be ignored . these regimes , however , turn out to correspond to separations larger than we are interested in the present paper , for bh masses above @xmath148 ( as will be shown in figures [ fig : tres_q1 ] and [ fig : tres_q0.01 ] below ) . we have verified our solutions numerically by substituting them back into the fundamental conservation equations of thin accretion disks @xcite . moreover , equations ( [ e : sigma_inb1])-([e : h_middle ] ) agree with those quoted in @xcite . in their eq . 20 should be @xmath149 , and @xmath150 in their eq . 21 should be @xmath151 and @xmath152 in the inner and middle regions , respectively . ] equations ( [ e : sigma_inb0])-([e : h_in ] ) are also consistent with @xcite , for the @xmath153 , @xmath154 , @xmath122 model . it is also reassuring that equations ( [ e : sigma_out])-([e : h_out ] ) are consistent with those in ( * ? ? ? 8.1 , p. 244 ) . , which should instead read as @xmath155 , so that @xmath156 is satisfied for all @xmath157 . ] also note that , owing to the weak dependence on @xmath158 , our numerical factors are very similar to those in @xcite , even though @xmath110 is defined to be two orders of magnitude larger there than the value we adopted here ( to be consistent with most other textbooks ) . here we collect and summarize the most important formulae describing the interaction between a binary and the accretion disk in order to identify the mechanism that drives the orbital decay of the binary during the final stages of the merger , as a function of binary separation ( the choices being gw driven inspiral and tidal viscous torques ) . this will allow us to explicitly compute the residence time @xmath14 that an individual binary spends at each orbital separation @xmath15 , or at the corresponding orbital timescale @xmath16 . the formulae collected in this section will also allow us to quantify the binary separation at which the viscous evolution of the disk is decoupled from the increasingly rapid , gw driven orbital decay of the binary . we provide results for @xmath21 and @xmath111disks , and give analytic results as a function of binary and disk parameters . in general , the evolution of a smbh binary in a thin disk is analogous to planetary migration ( see , e.g. @xcite ) . in the limit of a very low mass companion ( @xmath159 ) , the interaction between the planet and the disk is linear . in addition to co - rotation resonances , the density waves excited in the gas at discrete lindblad resonances with the binary exert a large net torque on the binary , leading to rapid , so called type i migration , which occurs on a time scale much shorter than the local viscous time scale ( e.g. , @xcite ) . if the binary is massive enough for the tidal torque to dominate over the viscous torque in the disk , the interaction becomes non linear , and a gap is opened in the disk , extending to the outer radius @xmath160 . the condition for a gap to open is that the mass ratio exceeds the critical value @xmath161 ( e.g. * ? ? ? * note that @xmath162 is evaluated at the position of the secondary @xmath90 ) . for binaries that are not in the gw driven regime , and for which the disk mass exceeds the mass of the secondary ( see below ) , this typically translates into the very modest requirement @xmath163 . this is satisfied for all smbh binaries that may produce the electromagnetic signatures we discuss below . the exceptions are the so called extreme mass ratio binary inspirals ( emri s ) with @xmath164 ( i.e. a stellar mass object coalescing with a smbh ) . in this paper , we focus on smbh binaries , and therefore in the rest of this paper , we neglect type i migration . if the secondary s mass satisfies the above gap opening threshold , but is still small compared to the local disk mass , then it acts as an angular momentum bridge for the disk , and the secondary s orbital evolution is simply determined by the viscous diffusion time , @xmath165 where we have used @xmath166 which follows directly from angular momentum conservation in steady disks @xcite . the orbital decay of the binary in this limit is analogous to _ disk dominated type - ii planetary migration_. in practice , the assumption that the local disk mass exceeds the secondary s mass often fails . in this case , analogous to _ `` planet dominated '' type - ii migration _ , the angular momentum of the binary can still be absorbed by the gaseous disk outside the gap , and the viscosity of the gas can drive the binary toward merger . however , migration is slower , and the time scale in this regime , @xmath167 , is longer than @xmath168 . an estimate of the slowing factor is @xmath169 , where @xmath170 is a measure of the lack of local disk mass dominance ( * ? ? ? * but note that our @xmath171 is denoted by @xmath172 in their original definition ) , which is less than unity in this case , and @xmath173 is a constant defined as @xmath174 thus , the separation of the binary in this case is driven inward on the timescale @xmath175 note that the viscous time scale @xmath168 in disk dominated limit ( eq . [ e : armitage_natarayan ] ) should be evaluated at the position of the secondary @xmath62 , while the quantities entering the time scale @xmath167 for the secondary dominated type - ii migration of more massive binaries ( @xmath176 , eq . [ e : t_ii ] ) , should be evaluated at the outer edge of the cavity , @xmath160 @xcite . in order to avoid a discontinuous jump in the migration time scale at the @xmath177 transition , below we will omit this distinction , and evaluate both time scales at @xmath178 . using the steady thin disk model outlined above , we can calculate the rate at which the binary is driven inward by the gas . we will also estimate the rate at which the inner edge of the punctured gaseous disk follows the binary due to its viscosity . from the preceding discussion , we see that both the viscous time scale and the orbital decay rate depend on whether the binary is located in the inner / middle / outer region of the disk ; and also on whether the local disk mass is larger / smaller than the mass of the smaller smbh . for completeness , we here obtain and quote the residence time as a function of orbital radius and orbital time , in each of these @xmath179 regimes . we then construct the self consistent evolution of individual binaries , with different masses and mass - ratios , across the relevant regimes . we first consider the timescale @xmath168 , and assume that the secondary perturbs the disk at the radius @xmath180 . this is the relevant regime initially , at large binary separations , when the disk mass enclosed within the secondary s orbit is large . in this regime , we find , @xmath181 the above can be expressed as a function of the orbital time of the binary , @xmath182 which results in @xmath183 the measure of disk dominance can be calculated by substituting the viscous time scale into equation ( [ e : q_b ] ) , @xmath184 in order to decide whether the evolution indeed follows the disk dominated decay ( on the viscous timescale @xmath168 ) or the secondary dominated decay ( on the longer time scale @xmath167 ) , one should examine whether @xmath185 or @xmath176 is satisfied , respectively . from equations ( [ e : q_b-1])-([e : q_b-4 ] ) , we find that the transition occurs at @xmath186 note that with the exception of very unequal masses @xmath187 , the transition takes place well in the outer region of the disk , with @xmath188 . at smaller radii , the binary is driven viscously on the timescale @xmath167 ( rather than @xmath168 ) . the `` secondary dominated '' type - ii decay timescales relevant at these radii can be obtained by substituting equations ( [ e : t_nur-1])-([e : t_nur-4 ] ) into equation ( [ e : t_ii ] ) @xmath189 or , in terms of @xmath16 using equation ( [ e : t_orb ] ) , @xmath190 finally , at a still smaller radius , the orbital decay will be dominated by gravitational wave emission . the gw driven decay timescale in the leading order ( newtonian ) approximation , is @xmath191 this approximation is adequate for our purposes , since post newtonian corrections do not become appreciable until the final @xmath192 day of the merger ( see , e.g. , figure 5 in @xcite ) . note that @xmath193 defined above differs from the total time to merger , ( defined as the binary separation decreasing to zero ) , which is often used in the literature , and which occurs at @xmath194 . what is the radius at which @xmath193 becomes smaller than the time scale for type - ii orbital decay ? let us express this transition in terms of the radius @xmath195 that satisfies @xmath196 , where @xmath197 is a fixed constant of order unity : @xmath198 the corresponding critical radius is around @xmath199 for system parameters near the assumed fiducial values . the critical radius , however , is significantly closer in for very massive , and very unequal mass binaries ( i.e. for @xmath200 and @xmath201 ; see fig . [ fig : tres_r_q0.01 ] below ) . interestingly , the critical radius is quite insensitive to the bh mass and accretion rate ( i.e. to @xmath202 and @xmath203 ) . note that the viscous timescale , @xmath168 , describing gas accretion , is faster than @xmath167 , which indicates that at the time when gw starts driving the evolution , the viscous inward diffusion of gas can initially still follow the binary . however , the comparison of equations ( [ e : t_nur-1])-([e : t_nur-4 ] ) and equation ( [ e : t_gw ] ) shows that as the binary orbit shrinks further , the viscous time - scale always decreases less rapidly than the gw inspiral timescale , so that eventually the evolution of the gaseous disk will decouple from that of the binary . let us find the critical radius , @xmath204 , where gw inspiral outpaces viscous gas accretion . we find that in most cases , this critical radius is not relevant for the orbital decay of the bhs themselves , because the transition to secondary driven orbital decay always takes place before gws start dominating the decay . , equal mass binaries , and only if @xmath205 is assumed in this case , the gw inspiral takes over in a radiation pressure dominated disk , in the disk dominated regime , i.e. before the transition to the secondary dominated regime . ] however , this critical radius is relevant for the behavior of the disk : it provides an estimate for the time when the punctured disk decouples from the gw driven binary , and effectively stops evolving ( and also for the size of the inner gap at this time and onward ) . by requiring @xmath206 , where @xmath207 is a constant coefficient of order unity , we obtain : @xmath208 the appropriate choices for @xmath209 and @xmath207 are poorly known , but @xmath210 may be reasonably taken to be @xmath211 when the binary is first driven by gw emission , rather than by tidal interaction with the gas . the simplest choice for @xmath207 , adopted in many previous studies , is also @xmath212 ( e.g. * ? ? ? * ; * ? ? ? however , the gas inflow rate across the edge of the central gap will be increased due to the steep density and pressure gradient @xcite , which will delay the decoupling . this motivated @xcite to adopt @xmath213 ( the value describing the limiting case of an infinitely sharp edge ) . adopting @xmath214 in equations ( [ e : r_ii / gw1])-([e : r_ii / gw4 ] ) then yields the radius where the binary evolution changes from being viscosity driven to gw driven , and @xmath215 in equations ( [ e : r_nu / gw1])-([e : r_nu / gw4 ] ) gives the separation at which the disk totally decouples from the binary and the radius of the gap `` freezes '' . these expressions generalize the results of @xcite , who restricted their analysis to the @xmath205 case , and focused on the behavior of the disk at decoupling , rather than the orbital evolution of the binary . in particular , @xcite evaluate disk conditions at the single radius at the edge of the gap , at the time of decoupling , and do not discuss the transition from the disk to the secondary dominated decay , or other details of the binary s orbital decay . the binary separation at decoupling is of order @xmath216 for both the gas pressure dominated models and the radiation pressure dominated case with @xmath205 . in these cases , the transition between viscosity and gw driven decay and the disk decoupling take place in relatively quick succession , since @xmath217 depends weakly on @xmath207 . the delay between these two events is much longer for the radiation pressure dominated regime when @xmath122 , since in this case the viscosity , which is proportional to the total , rather than just the gas pressure , is much larger , and the gas can follow the binary nearly all the way to merger ( at least for large @xmath203 ) . in this case , the result is also extremely sensitive to the accretion rate and the binary mass ratio . generically , for a fixed total binary mass , the decoupling occurs at the largest separations for nearly equal masses . interestingly , the decay rate of a given individual binary can decelerate and accelerate during its evolution , according to the variations in the local disk environment at each instantaneous binary separation . the evolutionary tracks of binaries with four different choices for the total mass ( @xmath218 , and @xmath219 ) and two different mass ratios ( @xmath220 and @xmath201 ) are shown in figures [ fig : tres_q1 ] and [ fig : tres_q0.01 ] . in both figures , we assume that the viscosity is proportional to the total pressure ( @xmath122 ) . the motivation for this choice is to illustrate the effect of the additional radiation pressure related viscosity on the orbital decay ( which is not present in the @xmath205 case ) . we note that a phenomenological @xmath122 disk is known to suffer from a formal thermal instability ( e.g. * ? ? ? * ) ; recent magnetohydrodynamical simulations , however , found such disks thermally stable ( while accounting for the correlation between viscosity and radiation pressure * ? ? ? these figures show the residence time as a function of the orbital time . they demonstrate that the evolution of the binary in most cases proceeds through the following distinct stages . _ ( i ) disk dominated viscous evolution . _ initially , at large separations ( shown in blue curves ) , the binary is strongly coupled to the circumbinary disk and evolves on the viscous time scale @xmath221 ( analogous to `` disk - dominated '' planetary migration ) . the radius of the gap follows the binary . during this stage , @xmath222 is proportional to @xmath223@xmath224 ( the range corresponding to the choice @xmath122 _ vs. _ @xmath205 ) for radiation pressure , or @xmath223@xmath225 ( the range corresponding to the choice of dominant opacity being electron scattering or free free absorption ) for gas pressure dominated disks ( see eqs . [ e : t_nur-1][e : t_nur-4 ] ) . these decay rates translate into @xmath226@xmath227 , or @xmath228@xmath229 , in the two cases respectively ( see eqs . [ e : t_nuorb-1][e : t_nuorb-4 ] ) . note , however , that for nearly equal mass binaries , the separations have to be quite large to correspond to this disk dominated regime falling into the outer regions of the disk , which are unstable to fragmentation ( the orbital radii where the disks are marginally toomre stable are marked with large dots ) . therefore , depending on the behavior of the gas disk beyond this radius , this early stage of disk dominated viscous evolution may exist only for unequal mass binaries . as shown in figure [ fig : tres_q0.01 ] , disk dominated viscous evolution may be realized in a stable disk for binaries with @xmath230 and @xmath231 ; in these cases , the binaries are in the free free opacity and gas pressure dominated regions of the disk , so the relevant scaling is @xmath232 . _ ( ii ) secondary dominated viscous evolution . _ as the binary separation shrinks below @xmath233 ( @xmath234 ) for mass ratios @xmath235 ( @xmath231 ) , the binary mass starts to dominate over the local disk mass , and the binary evolves more slowly , according to `` secondary dominated '' decay ( analogous to `` planet - dominated '' type - ii migration ) . during this stage , the gw emission is still negligible , and the decay time scale can be obtained from equations ( [ e : t_iir-1])-([e : t_iir-4 ] ) , and @xmath236@xmath237 for radiation pressure ( with @xmath238 ) , or @xmath239@xmath240 for gas pressure dominated ( with electron scattering vs. free free opacity ) disks , implying that @xmath241@xmath242 , and @xmath243@xmath244 , in the two cases respectively ( see eqs . [ e : t_iiorb-1][e : t_iiorb-4 ] ) . as can be seen from figures [ fig : tres_q1 ] and [ fig : tres_q0.01 ] , on orbital time scales between weeks to years , each of these scalings is relevant for some choice of binary masses . however , the transition to gw domination always takes place either in the `` inner '' or `` middle '' disk region . _ ( iii ) gw dominated evolution . _ still later , within the radius @xmath245 for systems with parameters close to the fiducial values , the binary s orbital evolution starts to be driven primarily by gws , but the outer edge of the gap can still diffuse inward and follow the binary . during this stage , the decay time scale is @xmath246 . _ ( iv ) gas disk decoupled . _ finally , within @xmath247 the binary is entirely driven by gws and the binary falls in much more quickly than the outer edge of the gap is able to move inward . the above ordering of events is valid for a broad range of binary and disk parameters . note that the ultimate fate of the gas inside the binary s orbit is left unspecified in our considerations ( see , e.g. * ? ? ? * for a possible outcome ) . in addition to the above sequence of events describing the evolution of individual binaries , several interesting conclusions can be drawn from figures [ fig : tres_q1 ] and [ fig : tres_q0.01 ] . 1 . _ coalescing binaries have a non - negligible abundance . _ first , binaries with masses in the range @xmath248 may be both bright and common enough to be detectable in a survey , provided they have bright emission . indeed , figures [ fig : tres_q1 ] and [ fig : tres_q0.01 ] show that these binaries spend a non negligible fraction ( @xmath249 ) of their total fiducial lifetime of @xmath2 years at orbital time scales between 1 day @xmath3 1 year ( the total lifetime will be justified below ) . it is feasible , in principle , to look for variability on these time scales , and the residence times shown on the figures suggest that these variables may not be uncommon among bright agn . we will discuss this possibility further in [ sec : periodic ] below . 2 . _ disk and gw driven evolution may both be observationally relevant . _ figures [ fig : tres_q1 ] and [ fig : tres_q0.01 ] also show that the transition from gas to gw driven evolution can occur within this `` observational window '' . for example , at the fixed orbital time scale of @xmath250 weeks , equal mass binaries above @xmath251 are gw driven , and below this mass , they are gas driven . secondary dominated evolution can not be ignored . _ essentially all binaries at the orbital times relevant for actual surveys ( again , between 1 day @xmath3 1 year ) that are gas driven are in the regime of `` secondary dominated '' type - ii orbital decay ( referred to as stage _ ( ii ) _ above ) . likewise , the transition from `` gas driven '' to `` gw driven '' evolution always occurs from the `` secondary dominated '' type - ii decay regime . in previous works whose primary focus was on the behavior of gas at ( and after ) the time of decoupling ( e.g. @xcite ) , this intermediary step , which is important for the orbital decay of the binary , is not discussed . 4 . _ observed binaries could probe all three disk regions . _ interestingly , among the @xmath248 binaries with 1 day @xmath3 1 year , it appears that all three of the disk regions ( inner / middle / outer ) enumerated in [ subsec : thindisks ] can be observationally relevant ( i.e. , gas driven binaries can be found in each of these three disk regions ) . viscous evolution is non negligible even in the lisa regime . _ the comparison of figures [ fig : tres_q1 ] and [ fig : tres_q0.01 ] shows that unequal mass binaries evolve more rapidly when they are gas driven . consequently , they make the transition to the gw driven stage quite late in their evolution . in particular , binaries enter _ lisa _ s detection range at the approximate observed gw frequency of @xmath252 mhz . this corresponds to an observed orbital time ( on earth ) of @xmath253 week . we find that at this orbital time , viscous evolution is not necessarily negligible . figures [ fig : tres_q1 ] and [ fig : tres_q0.01 ] show that equal mass binaries with @xmath254 , and @xmath201 binaries with @xmath255 are just making the transition to the gw - driven regime as they enter the _ lisa _ band . _ total decay time in a stable disk is consistent with quasar lifetime . _ as figures [ fig : tres_q1 ] and [ fig : tres_q0.01 ] show , the residence time at the radius at which @xmath256 is , in all cases , close to ( although somewhat shorter ) than the fiducial quasar lifetime of @xmath2 years . it is plausible that smbhs become luminous , and act as quasars , only once they are embedded in stable circumbinary accretion disk . the fact that it takes @xmath257 years for the binary to evolve from the outer edge of a stable disk to coalescence is therefore consistent with the idea proposed in this paper , that there is a one to one correspondence between coalescing smbhs and quasars ( although , as mentioned above , there are caveats that can invalidate the steady disk models at the relevant large radii ) . the possible implication of conclusion no . 5 above for _ lisa _ merits some further elaboration . as discussed , e.g. , in @xcite , individual binaries can contribute to the _ lisa _ data stream in several ways . sources can be divided into two types , based on whether they evolve significantly on a time scale of @xmath258 years , the duration of the _ lisa _ experiment . binaries caught at an orbital separation with short enough residence times for the frequency evolution to be measurable are sometimes referred to as `` gravitational sirens '' or `` gravitational inspirals '' . figures [ fig : tres_q1 ] and [ fig : tres_q0.01 ] show that during the last several years of the coalescence , the orbital evolution is always strongly gw dominated , even for the lowest mass bhs , and therefore the gw waveform of these rapidly evolving sources ( including those whose actual coalescence is detected by _ lisa _ ) will not be affected by the gas disk . binaries that have a much longer residence time at some fixed frequency in _ s band represent `` stationary '' sources whose frequency remains roughly constant during the _ lisa _ mission lifetime . these sources could , in principle , be individually detectable by _ lisa_. however , in practice , they are likely to accumulate sufficient signal to noise for detection only in the last few hundred years of their coalescence ( see , e.g. , figure 2 in @xcite for the detectability of @xmath259 binaries as a function of their look back time from the merger ) . figures [ fig : tres_q1 ] and [ fig : tres_q0.01 ] show that viscous processes can significantly speed up the evolution of binaries only at somewhat larger look back times ( note that the look back time is 4 times shorter than the residence time in the pure gw driven case ) . the cumulative signal from a collection of faint stationary sources can , however , still add up to an unresolved background that is detectable , depending on the the cosmic evolution of the bh merger rate and the instrumental noise of _ lisa_. the presence of the gas disks could reduce any such background that is present ( compared to a prediction that assumes pure gw driven evolution at _ lisa_frequencies ) . in figures [ fig : tres_q1 ] and [ fig : tres_q0.01 ] , we have showed the evolution of the binary as a function of its orbital period . this will be particularly useful for assessing the detectability of such binaries in a survey for periodically variable sources ( [ sec : periodic ] below ) . in figures [ fig : tres_r_q1 ] and [ fig : tres_r_q0.01 ] , we show , instead , the evolution of the same set of binaries , but as a function of their orbital separation . the @xmath260axis on these figures is shown in units of @xmath261 , with the corresponding orbital velocities shown by the labels on the top axis . this figure directly reveals that relatively more massive binaries ( @xmath262 ) spend a significant time at orbital velocities of several thousand @xmath263 . such orbital speeds may be detectable in the spectra of individual sources , providing an alternative to the detection based on periodic flux variations ( see [ sec : alternatives ] below ) . finally , the conclusions enumerated above also highlight the large uncertainty in the residence times predicted in figures [ fig : tres_q1 ] and [ fig : tres_q0.01 ] , caused by our idealized treatment of `` secondary dominated '' type - ii orbital decay . one immediate additional source of uncertainty is the choice of @xmath122 vs. @xmath205 . before entering the gw driven regime , most of the equal mass binaries ( fig . [ fig : tres_q1 ] ) are in the gas pressure dominated region of the disk , but unequal mass binaries ( fig . [ fig : tres_q0.01 ] ) are in the radiation pressure dominated region . therefore , whether the viscosity is proportional to the total pressure or just the gas pressure makes little difference to the near equal mass binaries . however , it makes a significant difference for unequal mass binaries with @xmath264 . to show this explicitly , in figure [ fig : tres_b_q0.01 ] , the upper vs. lower curves contrast the evolution in the @xmath205 vs @xmath122 case , respectively . as expected , once the binary approaches the radiation pressure dominated regime , the evolution is significantly slower in the @xmath205 case . the difference is most pronounced for the most massive ( @xmath265 ) binary . for this system , the transition to gw domination also occurs at a larger orbital time ( @xmath266 weeks for @xmath205 , vs. @xmath267 weeks for @xmath122 ) . for simplicity , above we calculated the timescales @xmath168 and @xmath167 in steady thin disk models . however , as noted above , this highly idealized model makes several crucial assumptions . in particular , ( * ? ? ? * hereafter ipp ) considered the tidal viscous interaction of an unequal mass binary ( @xmath159 ) with a _ time - dependent _ accretion disk . they assumed that the accretion disk is initially described by the steady state solution for a single bh , and then considered the modifications due to tidal torques from a secondary bh . the torques are turned on suddenly at some moment @xmath268 , when the secondary , whose mass is @xmath269 , is at an orbital radius @xmath270 that encloses a disk mass @xmath271 . the torques are assumed to be concentrated in a narrow ring near the secondary s orbit , which results in a pile up of material near the outer edge of the disk cavity . they found ( see their eq . 58 ) that this results in a decay time scale of @xmath272 where @xmath273 is the initial steady state accretion rate , @xmath274 is the initial viscous time ( at @xmath275 ) , @xmath276 is the disk dominance parameter at @xmath275 ( see eq . [ e : q_b ] ) , @xmath277 is the time - dependent position of the secondary and @xmath278 is its initial position , and @xmath279 ^ 2 $ ] , with the dimensionless time @xmath280 , implying that @xmath281 where @xmath282 , and @xmath283 , @xmath82 , and @xmath173 are defined such that @xmath284 , @xmath285 , @xmath286[(2c+a)/(2c+1)]^{-(2c+a)/(a+1)}$ ] , @xmath287 $ ] , and @xmath168 is the unperturbed viscous timescale given by equation ( [ e : armitage_natarayan ] ) and calculated explicitly below . refers to the standard gas pressure dominated accretion disk , which we generalize to radiation dominated disks below . ] if the opacity is dominated by electron scattering , then @xmath288 , @xmath205 , @xmath289 , @xmath290 , and @xmath291 , while for the free - free process @xmath292 , @xmath293 , @xmath294 , @xmath295 , and @xmath296 . from equations ( [ e : t_ipp ] ) and ( [ e : tau_ipp ] ) we find @xmath297 here , the radial evolution given by equation ( [ e : t_ivanov ] ) is ( at least initially ) not a simple power law . most importantly , as noted by @xcite , the pile up of the disk material causes the binary decay to slow down _ even more _ than estimated for a steady disk based on the `` disk dominance '' parameter ( eq . [ e : t_ii ] above ) . in figures [ fig : tres_ipp_q1 ] and [ fig : tres_ipp_q0.01 ] , we illustrate the impact of allowing the disk to evolve . in the above approach of @xcite , we have to specify when the interaction between the secondary and the disk is turned on . in figure [ fig : tres_ipp_q1 ] , we assume that the interaction begins as soon as the disk dominance parameter reaches @xmath177 . in figure [ fig : tres_ipp_q0.01 ] , we delay the onset of the interaction to @xmath298 . in both figures , the new dotted ( magenta ) curves denote the binary s residence time in the time dependent disk . note that in the latter case , in figure [ fig : tres_ipp_q0.01 ] , the residence time undergoes a discrete jump when the disk binary interaction is turned on : the binary stalls , and does not move initially , until the mass of material that has piled up is of the order of the secondary s mass . as the figures show , these residence times are indeed significantly longer than in the steady disks . at relatively late times after the interaction is assumed to turn on , the residence times asymptote to power law forms . most significantly , for each of the binaries shown in figures [ fig : tres_ipp_q1 ] and [ fig : tres_ipp_q0.01 ] , the transition to the gw driven regime occurs significantly earlier due to the evolution of the disk ; just before this transition to the gw driven regime , the residence times are longer by @xmath299 two orders of magnitude compared to a steady disk . in the rest of this paper , we will discuss identifying coalescing smbhbs with quasars , and interpreting the residence time @xmath300 as the duty cycle for exhibiting periodic variability on the observed time scale @xmath301 . our broad justification for these hypotheses is the generic idea , advanced in numerous other works , that quasars are activated in major galaxy mergers ( e.g. @xcite and references therein ) . since smbhs are believed to be common in galactic nuclei ( at least at low redshifts @xmath302 ; see @xcite and references therein ) there could then arguably be a one to one correspondence between the quasar phenomenon and smbhb coalescences . the total quasar lifetime , defined as the cumulative duration ( possibly over multiple episodes ) for an individual source to produce bright emission near the eddington limit , is generally believed to be @xmath303 yr , based on several lines of observational evidence @xcite . as discussed above , this value is consistent with the time scale it takes for a binary smbh to evolve to coalescence , starting from the outer edge of a gravitationally stable thin @xmath21disk . therefore , we hypothesize that the luminous quasar phase coincides with this last stage in the merger of the two smbhs . of course , it is possible that the quasar phase occurs either _ long before _ or _ after _ coalescence in either case , there would be no bright emission to observe during the last stages , as hypothesized here . we next assume that during the coalescence , the binary produces a steady luminosity @xmath304 ( which evolves only on long time scales @xmath305 ) , with roughly periodic fluctuations of amplitude @xmath306 and period @xmath307 about this steady mean luminosity . as argued in the introduction , periodic variations could be reasonably expected if the luminosity is tied to the mass accretion rate , with the latter modulated on the orbital period . even in the absence of such modulations , the emission could vary owing to the orbital motion and emission geometry of the binary ( kocsis & loeb , in preparation ) . in the absence of a quantitative model for the electromagnetic emission , we will assume the amplitude @xmath306 is unknown , and below we will ask whether a particular assumed @xmath306 may be detectable . since the residence time @xmath300 decreases continuously as the binary separations shrinks , variability with decreasing periods @xmath16 would be exhibited by a diminishing fraction @xmath308 of bright quasars . _ our main point is that an observational survey can attempt to identify such periodically variable sources . _ the total number of such periodic sources will be @xmath309 , where @xmath310 represents the merger rate between bhs within the survey volume ( or more precisely , the activation rate of smbh coalescence events ) . in general , the merger rate depends on redshift and on both bh masses , or @xmath311 , and should include only those sources with a luminosity above the survey detection threshold . to account for the latter condition , the light curve of each smbhb , @xmath312 needs to be known ( here @xmath313 could , for example , refer to the look back time before merger ) . the merger rate @xmath314 can be modeled using the dark matter halo merger rate with a recipe of associating bhs with halos , and the light curve @xmath315 can then be constrained by matching the observed quasar luminosity function ( e.g. * ? ? ? however , a large range of such bh population models can fit the observational data ( e.g. * ? ? ? * ; * ? ? ? * ) . to proceed , we instead make the simple assumption that each bh binary produces a constant mean luminosity of @xmath316 for a total duration @xmath317 during its lifetime , where @xmath318 is a constant of order unity . reasonable fiducial values appropriate to the bright quasar phase are @xmath319 @xcite and , as mentioned above , @xmath320 yr @xcite . note , in particular , that the quasar lifetime @xmath317 is known to be much shorter than the hubble time , and @xmath310 , which is likely determined by the galaxy merger rate , and proceeds on a cosmological time scale , can reasonably be assumed to be constant during @xmath317 . under the above assumptions , the fraction @xmath25 of objects with luminosity @xmath321 that display periodic variability on the time - scale @xmath322 is simply given by the ratio @xmath323 . this ratio can be read off directly from figures [ fig : tres_q1]-[fig : tres_ipp_q0.01 ] . note that this conclusion still holds if the quasar emission is intermittent ; we require only that the quasar is `` on '' for the duration @xmath300 when the binary orbital timescale is @xmath16 . most importantly , under these assumptions , the predicted number @xmath324 is a fixed fraction of the total number @xmath325 of quasars , and is independent of the merger rate , as long as the latter is constant during @xmath317 . we can then associate @xmath325 with the _ observed _ number of bright agn . in particular , in the gw dominated regime , we have the simple prediction @xmath326^{8/3 } m_{7}^{-5/3 } q_s^{-1}.\ ] ] note that in this equation , @xmath327 is the variability time - scale as observed on earth ( assumed to equal the redshifted orbital time ) ; the quasar lifetime @xmath328 is evaluated in the quasar s rest frame . before we proceed , we emphasize that there are many complications over the above , simplified picture . first , luminous quasar activity requires a near eddington mass accretion rate , with the gas reaching within several schwarzschild radii of one or both bhs . it is unclear whether abundant gas will indeed be present this close to the bhs , especially since during the late stages of the merger , the gas is evacuated from the inner disk by the binary s torques , and the exterior gas disk is eventually unable to follow the rapidly decaying bh binary . furthermore , in the final , gw dominated regime , the @xmath329 scaling strictly holds only if any residual circumbinary gas has negligible impact on the orbital decay . this requirement could , in fact , contradict the assumption that the binary is producing bright emission during this stage . second , in order for the emission to be periodically variable , the gas has to respond rapidly to the gravitational perturbations from the binary . scale for this response is of order the _ local _ orbital time ; variability on the orbital time scale of the _ binary _ itself therefore again requires gas close to the binary s orbital radius . if the central cavity were indeed truly empty , no gas would reach the smbhbs , and bright emission could not be produced . on the other hand , an empty cavity is certainly an idealization , and detailed models for the joint disk + binary evolution are required to assess the plausibility of our assumptions . conversely , the observations envisioned here will constrain such models ( which , again , is the main point of the present paper ) . in support of our assumptions , we note , however , that gas could be present near the bhs in the case of unequal masses ( so that the torques are reduced ) , or if the disk remains thick , making it difficult for the binary to open and maintain a nearly empty central cavity . numerical simulations indeed suggest residual gas inflow into the cavity @xcite , which may plausibly accrete onto the bhs ( with both bhs possibly forming their smaller individual accretion disks ; @xcite ) , producing non negligible em emission . simulations have also shown , in the context of proto planetary disks , that when the circumbinary disk is sufficiently thick , the mass flow rate across the gap is increased @xcite . such residual inflow onto a smbh binary has been invoked to explain the @xmath19212yr periodic emission from the quasar oj287 @xcite . more recently , the large velocity offsets seen in the spectrum of the quasar sdss j092712.65 + 294344.0 @xcite have been interpreted with a similar model , including gas inflow onto a luminous smbh binary @xcite ; a similar interpretation was invoked for the binary quasar candidate recently identified by @xcite . there are additional caveats that will hamper the identification of any periodic sources , even if they exist and produce bright enough luminosity to be detectable . the eddington ratio of bright agn is already known to have a significant scatter ( @xmath330 dex ; @xcite ) . the light curve of the merging binary is also likely to evolve , rather than having a simple `` tophat '' shape . it is possible , in particular ( e.g. * ? ? * ) that merging smbhs spend a significantly longer time ( @xmath331 yr ) at lower luminosities , ( @xmath332 ) . this will complicate the interpretation of any observed variability ( i.e. , converting the observed ratio @xmath333 at @xmath322 to @xmath334 will require knowing the probability distribution of eddington ratios ) . this , however , can be alleviated by considering only the _ relative _ abundance of periodically variable objects at different values of @xmath322 , instead of the absolute number of sources that show periodic variability . in this case , the only assumption required is that @xmath318 does not evolve significantly during the observed range of @xmath322 this should be reasonable over a factor of a @xmath192 few range in orbital radius or in @xmath16 . furthermore , even if there is a range of different bh masses , among sources with a similar luminosity , producing variability with the same period , figures [ fig : tres_q1 ] and [ fig : tres_q0.01 ] show that more massive bhbs will move much more quickly through a fixed @xmath335 . given that there are most likely fewer of the more massive bhbs to begin with , the set of all sources with the same @xmath322 will be heavily dominated by the lowest mass bhbs , caught at their relevant orbital radius . this still leaves the caveat , however , that the source is significantly sub eddington during the late stages of coalescence . in this case , the periodic sources will be harder to detect both because they are fainter , and also because they will also be rarer ( among the long lived and therefore more numerous , near eddington quasars with a similar luminosity ) . another caveat is that at fixed @xmath16 and @xmath0 , the distribution of @xmath1 is unknown , and can depend on @xmath0 . however , bright agn activity is thought to be activated only in relatively major mergers . a smaller satellite galaxy , falling onto a larger central galaxy that is more than @xmath267 times more massive , may not experience the torques needed to bring its gaseous nucleus , with the low mass bh , close to the center of the larger galaxy , for the bh - bh merger to take place @xcite ; the dynamical friction time for small galaxies themselves can also be too long ( e.g. * ? ? ? * ) , and/or the small satellites can be tidally stripped before reaching the central regions of the larger galaxy ( e.g. * ? ? ? these arguments , coupled with the well established correlations between the mass of a smbh and its host galaxy ( e.g. @xcite , see also the introduction ) , suggest that the @xmath1distribution among binaries associated with quasars may not extend to values significantly below @xmath336 . finally , for simplicity , in our estimates we have assumed circular orbits , both for the binary and the disk gas . it has been shown that the binary disk interaction could drive both the smbhs and the gas to have significant eccentricities @xcite . such eccentricities should leave characteristic asymmetric signatures in the modulated mass accretion rate ( see figure 8 in @xcite ) . the resulting light curves may exhibit corresponding features , which could be resolved , given sufficient time sampling . in practice , allowing for eccentricities will most likely further complicate the interpretation of any observed period distribution , especially if the time sampling is too coarse to explicitly reveal any asymmetric features . despite the caveats listed in the previous section , it is plausible that the periodic sources envisioned here exist , and we propose that they can be looked for , in a suitably designed survey . most importantly , figures [ fig : tres_q1 ] and [ fig : tres_q0.01 ] show that the expected variability timescale can be in a suitable range for a statistical detection , with a duty cycle of @xmath337 yr over the range from @xmath5day to @xmath192yr . this suggests that such periodic sources may not be too rare . what will be the practical limitations for discovering the population of periodic sources ? clearly , there has to be a sufficient number of sources , observed over a range of variability time scales for a representative statistical sampling , and the brightness variations of these sources must be detectable . in addition , the individual light curves have to be sampled well enough to confirm their periodic nature : this will be necessary to distinguish the coalescing smbh binaries from other types of variable objects . besides discovering the periodic sources , the idea proposed here is to measure the dependence of @xmath338 on @xmath322 possibly to use the @xmath339 scaling to demonstrate that the periodic variability comes from perturbations by the orbital motion during the gw inspiral . for this , the survey also needs to cover at least a factor of several range in @xmath322 . the above issues will place requirements on ( i ) the sensitivity and ( ii ) solid angle , as well as on the ( iii ) total duration and ( iv ) sampling rate for a survey . we can use the simple disk models and the idealized picture discussed above , to roughly delineate these requirements . for simplicity of discussion , let us assume that all sources are at @xmath31 . in reality , quasars ( and therefore major bh mergers ) have a broad distribution with a peak around this redshift ; clearly this will have to be taken into account in designing an actual survey . for simplicity , let us also fix the mass ratio @xmath220 . in reality , there should be a distribution of values , perhaps in the range @xmath340 , for the mergers that activate bright quasar activity @xcite . this would not significantly affect our conclusions , unless @xmath1 frequently extends well below 0.1 . imagine a survey with a sensitivity that corresponds to detecting the periodic variability of bhbs with a mass @xmath341 at @xmath31 , covering a solid angle @xmath342 . ( a real survey , of course , will have a completeness for variability detection that is not a step function ) . let us assume that the variable flux corresponds to a fraction @xmath343 of the steady mean luminosity , @xmath344 . if the survey volume contains a total of @xmath325 smbhbs with the luminosity @xmath304 , then the periodic variable fraction , @xmath345 , can be determined down to the smallest value @xmath346 ( i.e. to find at least one periodic source ) . fixing the values of @xmath347 and @xmath348 ( as well as @xmath341 , @xmath24 and @xmath1 ) , this corresponds to a minimum variability time scale @xmath349 that can be probed . let us define the requirement that this minimum is @xmath350 weeks . assuming that the longest variability time scale of interest is around @xmath351 year ( so that the periodic nature of the variations can be convincingly demonstrated over a multi year survey ) , this will offer a factor of three range in @xmath322 for mapping out the @xmath338 _ vs. _ @xmath352 dependence . for example , with the steepest possible ( pure gw driven ) scaling @xmath339 , a survey volume containing a single source with @xmath353 weeks would then contain @xmath354 sources with a similar luminosity but with a @xmath355 week period . to fix some numbers , let us set @xmath356 , @xmath357 , and @xmath358 yr . for reference , the eddington luminosity of a @xmath359 bh at @xmath31 , assuming a @xmath192 10% bolometric correction , corresponds to an optical magnitude of @xmath29924 mag ( in the @xmath360 band ) . let us also impose the ( somewhat ad hoc ) requirement that the survey volume should contain at least @xmath361 sources with a detectable flux variations at the period of @xmath362 weeks . in the gw driven stage , there will then be at least 5 detectable periodic sources with a period of @xmath363 weeks ; in the gas driven regime , where the scaling @xmath25 _ vs. _ @xmath322 is flatter , there will be a larger number of @xmath364week period sources . in figure [ fig : survey ] , the curves show the sky coverage required to satisfy these criteria , as a function of the @xmath360band variable magnitude corresponding to the detection limit of the survey . the bh masses producing the corresponding steady @xmath360 magnitude ( which , in our fiducial model , is 2.5mag brighter than the variable magnitude ) are shown on the top axis . this figure assumes @xmath220 . we used the fitting formula by @xcite for the bolometric quasar luminosity function ( lf ) @xmath365 to compute the the total number @xmath325 of quasars at @xmath31 , per solid angle @xmath342 , in a redshift range of @xmath366 , i.e. @xmath367 , where @xmath368 is the cosmological volume element , and @xmath369 is the bolometric luminosity corresponding to the steady magnitude threshold @xmath360 . we then used equation ( [ eq : nvar ] ) for @xmath25 to obtain the total number @xmath370 of variable sources at observed period of @xmath353 weeks and at @xmath355 weeks . requiring @xmath338(@xmath355 weeks)@xmath371 then yields the solid angle @xmath342 as a function of @xmath360 . note that the quasar lf is almost a pure power law up to @xmath372 . the break between @xmath360 = 26 - 27 mag in the solid curves corresponds to the transition between gw and gas driven orbital decay . in particular , the figure shows that smbhs with a mass above / below @xmath373 are in the gw / gas driven regime , respectively . figure [ fig : survey ] shows that there is a clear trade off between survey depth and area : the required sky coverage scales with the survey flux limit approximately as @xmath374 , with a steepening for shallow surveys with limiting magnitudes @xmath375 ( due to the decline at the bright end of the quasar lf ) , and a flattening for very deep surveys with limiting magnitudes @xmath376 ( because the smbhbs are in the gas driven regime and their residence times at fixed @xmath16 are shorter than in the pure gw driven regime ) . from figure [ fig : survey ] , we conclude that , for example , a 1 sq . degree survey , detecting smbhbs whose steady luminosity is @xmath377 mag , with a variability at the level of @xmath378 mag , with sufficient sampling and duration to cover periods of 2060 weeks , represents an example for the minimum specification for the survey parameters ( i)-(iv ) . in this example , the mass of the bhs being detected is @xmath379 . figures [ fig : tres_q1 ] and [ fig : tres_q0.01 ] show that at orbital periods of @xmath380 weeks , these bhs are all in the gw driven regime when @xmath235 , but may be in the viscosity driven regime for @xmath159 . surveys that go deeper and cover a smaller area will begin probing the gas driven evolutionary stages . in figure [ fig : survey2 ] , we examine how the required sky coverage changes when the parameters @xmath1 , @xmath343 or @xmath318 are modified . the middle ( black ) curve shows the sky coverage required to find 20 sources at @xmath381 weeks ( intermediate between the red and black curves in figure [ fig : survey ] ) , with our fiducial parameters , @xmath220 , @xmath357 , @xmath382 . the green curve corresponds to changing the mass ratio to @xmath201 ; this increases / decreases the residence time in the gw / gas driven regimes relative to the @xmath220 case ( compare figs . [ fig : tres_q1 ] and [ fig : tres_q0.01 ] ) , and therefore reduces / increases the required solid angle coverage . the top pair of ( blue ) curves in figure [ fig : survey2 ] show variations when either @xmath343 or @xmath318 is decreased by a factor of 10 ( to 0.03 or 0.01 , upper and lower of the pair , respectively ) . note that the survey volume requirement is more sensitive to @xmath343 ( whereas the critical bh mass is equally sensitive to either ) . similarly , the bottom pair of ( red ) curves show variations when either @xmath343 or @xmath318 is increased by a factor of 10 . the small break visible at @xmath383 mag in this case corresponds to the transition from the middle to the outer disk region for @xmath384 smbhbs at @xmath385 weeks ( see fig . [ fig : tres_q1 ] ) . each dashed curve shows the mass of the smbhb corresponding to the effective @xmath360 magnitude limit ( bh masses are labeled on the right @xmath386-axis ) . the required survey volume also shifts linearly with the assumed total quasar lifetime @xmath317 . if we knew where to look ( i.e. , if _ lisa _ delivers a candidate for an on going merger , with sufficiently accurate localization on the sky ) , it would be possible to perform a deep , targeted observation for variability on short time scales ; between several minutes up to @xmath387 hours within the last @xmath192 month of merger ( this possibility is discussed in detail in * ? ? ? * ) . however , each source will spend only a @xmath192month at such short variability time scales , and a random search , in the absence of a preferred direction on the sky , would then have to monitor @xmath388 agn to find a single example of such a late stage periodic source . alternatively , one may monitor @xmath389 agn for @xmath12 years , to look for ( slowly evolving ) periods , on the timescale of @xmath192 a day . the slow decrease in the period , which would be a smoking gun for gw inspiral , will be challenging to observe in real time for individual objects . existing observations from radio to x ray bands have shown that the luminosity of quasars and other active galactic nuclei varies on time scales from hours to several years ( see , e.g. , the articles in @xcite or the recent review by @xcite ) . in fact , variability often aids in the identification of agn ( and may conversely be a major obstacle in identifying the periodic signal proposed here ) . while variability is detected in a large fraction of all agn , there are only a handful of sources whose structure function shows clear _ periodic _ variability on long ( @xmath390 weeks ) time scales ( see , e.g. @xcite for a review focusing on searches for periodic variability ) . examples include a handful of blazars , whose historical light curves show periodic outbursts on timescales of a year to a decade , or even longer ( see , e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * and references in these papers , for individual objects ) . @xcite monitored 25 low redshift , optically selected quasars for variability over several decades in infrared bands . they identified one object , the radio loud quasar pg 1535 + 547 , whose structure function shows a periodic component with a period of @xmath387 yr . this source has a bolometric luminosity of @xmath391 , implying a bh mass of @xmath392 . figure [ fig : tres_q1 ] shows that equal mass smbh binaries with this mass are in the gw driven regime , and @xmath393 may exhibit a 10year period . in comparison , figure [ fig : tres_q0.01 ] shows that unequal mass binaries with this total mass may be in the gas driven regime , and periodic variability may be exceedingly rare . thus , we conclude that the identification of one periodic object is roughly consistent with it being an example of a gw driven , near equal mass binary . however , there are only 4 objects in the sample studied by @xcite with luminosities above @xmath394 , prohibiting robust conclusions . very large area variability surveys , such as in the sloan digital sky survey ( sdss ) , are shallow , and detect variability down to only @xmath395 mag @xcite . with our fiducial @xmath396 , this corresponds to periodic agn whose steady luminosities are @xmath397 mag . figure [ fig : survey ] shows that such a survey would have missed the periodic variations discussed above . figure [ fig : survey2 ] shows that if the variable fraction is much larger , @xmath398 , then @xmath399 variable sources with a yearly period would be detectable but their periodic nature could be demonstrated only with sufficient time sampling , extending over a decade . a recent , much deeper optical survey by the subaru telescope @xcite for variable objects provides interesting constraints on the scenario envisioned here . the completeness function in this survey , defined as the probability to detect flux variations of an object with a variable component @xmath360 , goes from @xmath192unity to @xmath192zero between @xmath400 to @xmath401 ( see fig . 8 in @xcite ) . in the fiducial case with @xmath382 and @xmath357 , the limiting variability magnitude @xmath402 mag corresponds to the mean steady magnitude of @xmath403 mag , and bh mass of @xmath404 . at @xmath31 , the subaru survey has a completeness of 0.5 at this magnitude ( see fig . 11 in @xcite ) and covers an area of 0.9 sq . this combination of sensitivity and area lies very close ( just below ) the curves in figure [ fig : survey ] . using the @xcite quasar lf , and assuming a completeness of 0.5 , we find that the subaru survey should detect @xmath405 agn ; this is in nearly exact agreement with their quoted result ( 489 deg@xmath406 ) . we further find that of these sources , @xmath407 would vary with observed periods of 20 weeks , 60 weeks , and 1000 days ( adopting a probability of 0.5 for detecting variability , from fig . 8 in @xcite ) . figure 12 in @xcite shows that they found several dozen sources that varied , at least once in their life , on all of these timescales . unfortunately , we do not know whether these sources are periodic or not , and therefore the subaru survey results represent only an upper limit on the fraction of periodic sources . nevertheless , this already suggests that the @xmath408 bhs at the limit of the survey can not produce variability at the level significantly exceeding our fiducial @xmath409 . the ultra deep _ hubble space telescope ( hst ) _ variability surveys ( see , e.g. , the recent review by * ? ? ? * and references therein ) discovered galaxies whose nuclei varied by magnitudes down to @xmath410 . the observations were taken a year apart in the hubble deep field north ( hdfn ) and the groth survey strip ( gss ) , whose areas are @xmath411 and @xmath412 sq . degrees , respectively . while the solid angle of the hdfn dataset is too small to yield useful constraints , the fiducial case with @xmath356 and @xmath357 in figure [ fig : survey ] shows that the gss dataset just reaches the sensitivity / area combination of @xmath413 mag and @xmath412 sq . degrees required to find flux variations from @xmath414 smbhbs . approximately @xmath415 of agn were found to vary by magnitudes down to @xmath416 in this dataset ( see * ? ? ? * for more details ) , suggesting that @xmath417 of agn containing smbhbs with this mass can produce variability at the @xmath418 level . agn are also known to vary on long times scales in x - ray bands . systematic and unbiased variability surveys sensitive to times scales of weeks , such as those in soft x rays in the rosat all sky survey @xcite or in hard x rays in _ swift_/bat data @xcite however , have been restricted to the brightest agn , while deeper surveys , such as those by _ rxte _ @xcite , of the _ chandra _ deep field north @xcite and south @xcite , and by _ xmm _ @xcite have only monitored up to a few hundred sources . these observations do suggest that a large fraction of agns vary in x ray bands on time scales of a day to a year , but whether the variations are periodic have not been determined . in the 9month duration observations covered by _ swift_/bat data , @xcite find a strong anti correlation between luminosity and variability ( with no source with luminosity @xmath419 showing significant variability ) ; @xcite report a similar trend from an x - ray variability analysis of 66 agn in the lockman hole . these findings would be consistent with the trend that the most massive smbhbs ( @xmath420 ) spend less time at a fixed orbital timescale of @xmath421 weeks ( see figures [ fig : tres_q1 ] and [ fig : tres_q0.01 ] ) . the results of @xcite suggest that absorbed sources vary more than unabsorbed ones , which may be particularly relevant for finding the periodic smbh binary sources envisioned here , which are undergoing the last stages of their merger , and may be heavily obscured and visible primarily in x ray bands . while the deep existing optical surveys come close to placing useful constraints on the scenario envisioned here , future surveys , designed to uncover source populations with periodic variations on times scales of tens of weeks , should be able to either discover these populations , or place stringent limits on their existence . many large optical / ir surveys are being planned or built , motivated largely by finding type ia supernovae ( sne ) for cosmological studies ( see , e.g. , * ? ? ? * for a recent review ) . the most ambitious of these , such as lsst and pan - starrs-4 will be all sky surveys , and should be able detect variability to @xmath422mag , allowing detections well beyond the most pessimistic case shown in figure [ fig : survey2 ] . the proposed alpaca survey @xcite will cover 1,000 sq . degrees to 23 - 25 mag in 5 optical bands , and would already reach the sensitivity / area combination probing these pessimistic scenarios . in addition to producing periodic variability , there could be several other methods to prove or disprove the presence of a smbh binary . first , the orbital motion of the binary may cause relative shifts in the quasar s emission lines . for example , in a configuration in which the broad lines arise from gas close to one of the two ( moving ) bhs , and the narrow lines arise from material farther away , which is close to rest at the systemic redshift , such a shift could arise between the narrow and broad emission lines @xcite . similarly , if both bhs carry their own accretion disks , extending to a few schwarzschild radii , and produce broad lines @xcite , then there could be two sets of broad emission lines , super imposed with a similar relative velocity shift . the magnitude of these shifts may be of order the orbital velocity ( @xmath423 km / s at @xmath424 schwarzschild radii ) , which could be detectable either in individual objects , or else statistically for the population . @xcite recently reported a candidate smbh binary , with two sets of broad emission lines separated by @xmath425 . the spectrum of this source can also be interpreted with a single bh+disk system @xcite ; indeed , this interpretation is favored by the lack of any change in the velocity offset over the course of @xmath426 year @xcite . nevertheless , it is interesting to note that , with the binary parameters reported for this source ( assuming random orientation , and an expected orbital speed of @xmath427 ) , @xmath428 , @xmath429 , @xmath430 , @xmath431 years , and @xmath432 , we find the evolutionary track of the proposed system to be virtually indistinguishable from the @xmath200 , @xmath201 case shown in figure [ fig : tres_r_q0.01 ] . at its currently observed orbital separation of @xmath432 , the binary would be in the gas driven regime , close to outer radius of the formally gravitationally stable disk ( i.e. , the system is just outside the marked @xmath256 point in this figure ) , with a residence time of @xmath433 years . this could indeed make this observed separation common among quasars with @xmath200 smbhbs . the figure also shows that the residence time at fixed orbital velocity decreases steeply with bh mass , suggesting that fainter binary quasars with similar orbital speeds would be much less common . in the last stages of coalescence , the gws emitted by such a system would induce periodic modulations in the arrival times of pulses from background radio pulsars ; at 200ns timing sensitivity , these modulations would be detectable from smbhbs out to a distance of @xmath434mpc ( * ? ? ? * note that this study already applies the idea to the source 3c66b mentioned in 1 , whose elliptical motion was interpreted as due to a smbhbs , and rules out the smbhb hypothesis ) . in this paper , we followed the evolution of smbh binaries , starting from large separations , to coalescence . we find , in agreement with earlier works , that the orbital decay is initially generically driven by viscous binary disk interactions , whereas gws dominate the last stages . in a refinement of earlier results , we also find that just prior to the transition to gw driven evolution , the viscous orbital decay is generically in the `` secondary dominated '' type ii migration regime ( the mass of the secondary is larger than the enclosed disk mass ) . this is slower than the disk dominated type ii migration that has sometimes been assumed in the past , and , as a result , smbh binaries spend a significant fraction of their time at orbital periods of @xmath192days to @xmath192 a year , where they may not be rare , and may be identifiable . we emphasized the large uncertainties in the residence times in this regime for example , time dependent disk models predict even slower decay . we also find that observations of bhs with a mass range of @xmath435 over this range of periods could find binaries located in all three physically distinct regions of the circumbinary disk . thus , several aspect of disk physics could potentially also be probed in future observations of a population of smbh binaries . we also find that viscous processes may contribute to the orbital decay rate even after the binaries enter _ lisa _ s frequency range , for low and/or unequal mass binaries ( @xmath436 or @xmath437 ) . while viscous processes are strongly sub dominant for rapidly evolving `` inspiral '' sources , detected during the last few years of their coalescence , the presence of the gaseous disk could reduce any background of unresolved stationary sources at frequencies near the low frequency end of the _ lisa _ range ( @xmath438 mhz ) . we considered the possibility that there may be a one to one correspondence between the activation of luminous agn and smbh coalescences , with a fraction of agn exhibiting periodic flux variations . given that the interpretation of individual smbhb candidates have so far remained ambiguous , we proposed that a statistically large sample should aid in the identification of these binary bh sources . our main conclusion is that future surveys in optical and x ray bands , which can be sensitive to periodic variations in the emission from @xmath439 supermassive black hole binaries , on timescales of @xmath5 tens of weeks , at the level of @xmath211% of the eddington luminosity , could look for a population of such sources , with the aim of determining the fraction @xmath25 of sources , at a given redshift and luminosity , as a function of @xmath322 . in our simplified models for the binary disk interaction , this time scale of tens of weeks corresponds to the orbital time when binaries with @xmath440 make their transition from viscous to gw driven evolution . in the latter regime , for sources with @xmath441 , gravitational radiation predicts the scaling @xmath442 . the discovery of a population of periodic sources whose abundance obeys this scaling would confirm that the orbital decay is indeed driven by gws , and also that circumbinary gas is present at small orbital radii and is being perturbed by the bhs . deviations from the @xmath443 power law for lower mass bhs would constrain the structure of the circumbinary gas disk and viscosity driven orbital decay . there is certainly a possibility that the periodic sources envisioned here do not exist ( e.g. , because the smbh binary does not produce bright and variable emission during its gw emitting stage , at orbital separations of @xmath444 schwarzschild radii ) . nevertheless , we argued that existing surveys already approach the required combination of sky coverage and depth , and future surveys , designed to make observations for several years , with a sampling rate of a few days , could yield a positive detection and identify periodic source populations . this would bring rich scientific rewards , possibly including the indirect detection of gravitational waves , driving the orbital decay of these sources . zh thanks george djorgovski and tuck stebbins for stimulating discussions , and mamoru doi and tomoki morokuma for sharing their subaru variability search results prior to publication , which originally inspired this paper . we also thank zsolt frei and david hogg for useful comments , chris stubbs , michael strauss and richard mushotzky for advice on variability surveys , and the anonymous referee for comments that significantly improved this paper . km thanks the aspen center for physics , where a part of the work reported here was performed , for their hospitality . this work was supported by the polnyi program of the hungarian national office for research and technology ( nkth ) and by nasa atfp grant nnx08ah35 g . bk acknowledges support from otka grant 68228 .

supermassive black hole binaries ( smbhbs ) in galactic nuclei are thought to be a common by product of major galaxy mergers . we use simple disk models for the circumbinary gas and for the binary - disk interaction to follow the orbital decay of smbhbs with a range of total masses ( @xmath0 ) and mass ratios ( @xmath1 ) , through physically distinct regions of the disk , until gravitational waves ( gws ) take over their evolution . prior to the gw driven phase , the viscous decay is generically in the stalled `` secondary dominated '' regime . smbhbs spend a non negligible fraction of a fiducial time of @xmath2 years at orbital periods between days @xmath3 year , and we argue that they may be sufficiently common to be detectable , provided they are luminous during these stages . a dedicated optical or x ray survey could identify coalescing smbhbs statistically , as a population of periodically variable quasars , whose abundance obeys the scaling @xmath4 within a range of periods around @xmath5 tens of weeks . smbhbs with @xmath6 , with @xmath7 , would probe the physics of viscous orbital decay , whereas the detection of a population of higher mass binaries , with @xmath8 , would confirm that their decay is driven by gws . the lowest mass smbhbs ( @xmath9 ) enter the gw - driven regime at short orbital periods , when they are already in the frequency band of the _ laser interferometric space antenna _ ( _ lisa _ ) . while viscous processes are negligible in the last few years of coalescence , they could reduce the amplitude of any unresolved background due to near stationary _ lisa _ sources . we discuss modest constraints on the smbhb population already available from existing data , and the sensitivity and sky coverage requirements for a detection in future surveys . smbhbs may also be identified from velocity shifts in their spectra ; we discuss the expected abundance of smbhbs as a function of their orbital velocity .

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Proceed to summarize the following text: the structure of the pulsar magnetosphere has been an active area of research for more that 30 years , starting with the pioneering work by goldreich & julian ( 1969 ) . despite the fact that real astrophysical radio - pulsars are believed to be oblique rotating magnetic dipoles , much of the theoretical effort has been devoted to a significantly simpler case of an aligned rotating magnetic dipole ( i.e. , a `` non - pulsing pulsar '' ) , in which case the problem becomes stationary and axisymmetric . the situation becomes even more tractable if one also assumes the space around the pulsar to be filled with a plasma that , on one hand , is dense enough to shorten out the longitudinal electric fields , thus providing the basis for the ideal - magnetohydrodynamic approximation , and , on the other hand , is at the same time tenuous enough for all the non - electromagnetic forces to be negligible , thus enabling one to regard the plasma as force - free . all these simplifications have lead early on to the derivation , simultaneously by several researchers , of the main equation governing the structure of the magnetosphere , the so - called pulsar equation ( scharlemann & wagoner 1973 ; michel 1973b ; okamoto 1974 ) , which is essentially a special - relativistic generalization of the well - known force - free grad shafranov equation . this equation is a quasi - linear elliptic second - order partial differential equation with a regular singular surface , the so - called light cylinder . despite the fact that this , in general non - linear , equation was first derived thirty years ago , most of the attempts to solve it , both analytical and numerical , have been limited , until very recently , to its linear special cases and to the region inside the light cylinder ( e.g. , scharlemann & wagoner 1973 ; michel 1973a ; beskin et al . 1983 ; beskin & malyshkin 1998 ) . these self - imposed restrictions can of course be attributed to the considerations of mathematical convenience and tractability ; however , they are not very well physically motivated and thus may be too restrictive to be relevant to real astrophysical systems . at the same time , by the late 1990s , the available computing resources and numerical techniques have become sufficiently powerful for the most general nonlinear problem , involving the regions on both sides of the light cylinder , to become tractable . in their pioneering work , contopoulos et al . ( 1999 , hereafter ckf ) have obtained the first ( and apparently unique ) numerical solution of the general problem , albeit with a rather poor spatial resolution . their approach has later been used by ogura & kojima ( 2003 ) who have obtained essentially the same results , but with a higher numerical resolution . both of these groups have focussed on the global structure of the solution ; consequently , they have not paid enough attention , in our opinion , to some key but subtle issues regarding the separatrix between the open and closed field - line regions and , especially , the point of intersection of this separatrix with the equatorial plane . as a result , the separatrix has not , we believe , been treated correctly , and , in particular , the separatrix equilibrium condition ( okamoto 1974 ; lyubarskii 1990 ) has not been satisfied close to the intersection point . we thus suspect that their solution is probably not quite correct . in this paper we try to clarify how the magnetic field near the separatrix and especially around the separatrix equator intersection point should be treated , as this question appears to be extremely important for setting up the _ correct _ boundary conditions for the global problem . in [ sec - basic ] we review the basic equations describing the ideal - mhd force - free magnetosphere of an aligned rotating dipole ; we pay special attention to the so - called light cylinder regularity condition and discuss how this condition could be used to find the correct form of the poloidal - current function and , simultaneously , to obtain the solution that passes smoothly across the light cylinder . then , in [ sec - y - point ] , we investigate some general properties of the magnetic field near the putative separatrix between the open and closed field regions ; we use the separatrix equilibrium condition to show that , if the separatrix intersects the equator at the light cylinder ( as has been assumed in the numerical simulations by ckf and by ogura & kojima 2003 ) , then the poloidal current and hence the toroidal magnetic field have to vanish on the last open field line above the separatrix . we then use these findings in [ sec - analysis ] to construct a unique self - similar asymptotic solution of the pulsar equation in the vicinity of such an intersection point ; in particular , we find all the power - law exponents describing the field near this point and also the angle between the separatrix and the equator . at the end of this section we show that a light surface ( a surface where the electric field becomes equal to the magnetic field and therefore where the particle drift velocity reaches the speed of light ) has to appear right outside the light cylinder ( it starts at the intersection point and extends outward at a finite angle with respect to the equator ) . we thus conclude that the only way to get a force - free solution that would be valid at least some finite distance beyond the light cylinder , is to consider the case of a separatrix equator intersection point lying some finite distance inside the light cylinder . we investigate the magnetic field structure around such a point in [ sec - t - point ] . we find that in this case the poloidal - current function may stay finite , but its derivative with respect to the poloidal flux has to go to zero on the last open field line ; we then find the asymptotic behavior of the magnetic field around the intersection point corresponding to this case . finally , we present our conclusions in [ sec - conclusions ] , where we also discuss the implications of our results for the past and future numerical studies of the axisymmetric pulsar magnetospheres . we consider the magnetosphere of an aligned rotating magnetic dipole under the assumptions of stationarity , axial symmetry with respect to the rotation axis @xmath0 , and reflection symmetry with respect to the equator @xmath1 . we shall include the effects of special relativity but will ignore general - relativistic effects ( i.e. , work in euclidean space ) . we shall assume that the dipole is surrounded by a very tenuous , but highly conducting plasma . the first of these assumptions enables us to neglect all non - electromagnetic forces ( i.e. , gravity , inertial , and pressure forces ) , and thus to conclude that the structure of the magnetosphere is governed by the relativistic force - free equation : @xmath2 here , the electric charge density @xmath3 and the electric current density @xmath4 are related to the electric and magnetic field through the steady - state maxwell equations : @xmath5 the second of our assumptions concerning the magnetospheric plasma , i.e. , the assumption of infinite conductivity , enables us to use ideal magnetohydrodynamics ( mhd ) : @xmath6 these equations ( with the appropriate boundary conditions ) should be sufficient for determining the structure of the magnetosphere . as is well known , an axisymmetric magnetic field @xmath7 can be described in terms of two functions , the poloidal magnetic flux ( per one radian in the azimuthal direction ) @xmath8 and the poloidal current @xmath9 , as @xmath10 where @xmath11 is the azimuthal ( or toroidal ) angle . in cylindrical coordinates the magnetic field components then are : @xmath13 next , by applying faraday s law @xmath14 in a steady state , we see that @xmath15 , and hence @xmath16 ; together with the assumption of axial symmetry , this gives @xmath17 that is , the electric field is purely poloidal . since , as it follows from equation ( [ eq - force - free ] ) or ( [ eq - ideal - mhd ] ) , the electric field must be perpendicular to the magnetic field , we can write it as @xmath18 where @xmath19 is the poloidal magnetic field , and @xmath20 is the unit vector in the @xmath11-direction . by substituting this relationship into @xmath15 , we find that @xmath21 , which has the meaning of the angular velocity of magnetic field lines , is constant along the field lines , @xmath22 . this is the well - known ferraro isorotation law . in the problem we are interested in , all the field lines are tied , at least at one end , to the surface of the pulsar , which is assumed to be rotating as a solid body with some uniform angular velocity @xmath23 . we also assume that our ideal - mhd assumption is valid all the way up to the pulsar surface , i.e. , that there is no appreciable gap between this surface and the ideal - mhd region . therefore , from now on , we shall assume that all the field lines rotate with the same angular velocity was considered , e.g. , by beskin et al . ( 1983 ) and by beskin & malyshkin ( 1998 ) . ] @xmath24 from equation ( [ eq - e ] ) we find @xmath25 where @xmath26 is the cylindrical radius normalized to the radius of the light cylinder ( lc ) @xmath27 . thus , we see that @xmath28 inside the lc and @xmath29 outside the lc . now let us turn to the force - free equation ( [ eq - force - free ] ) . the toroidal component of this equation , together with equation ( [ eq - e_phi=0 ] ) , gives : @xmath30 \times [ \nabla\psi\times\nabla\phi ] = 0 \quad \rightarrow \quad \tilde{i}=\tilde{i}(\psi ) \ , , \label{eq - i = i(psi)}\ ] ] and the poloidal component ( perpendicular to @xmath31 ) of equation ( [ eq - force - free ] ) can be written as @xmath32 - { { 1+x^2}\over x}\ , { \partial\psi\over{\partial x } } = - i i'(\psi ) \ , , \label{eq - pulsar}\ ] ] where we have introduced @xmath33 and where we also incorporated @xmath34 into @xmath35 on the right - hand side ( rhs ) by defining @xmath36 then the magnetic field components can be rewritten in terms of @xmath37 as @xmath38 equation ( [ eq - pulsar ] ) is the famous _ pulsar equation _ ( scharlemann & wagoner 1973 ; michel 1973b ; okamoto 1974 ) , also known in literature as the relativistic force - free grad shafranov equation ( e.g. , beskin 1997 ) . it is an elliptic second - order partial differential equation ( pde ) for the flux function @xmath37 ; the left - hand side ( lhs ) of this equation is linear , whereas the rhs is , in general , nonlinear . one very important feature of equation ( [ eq - pulsar ] ) is that it has a regular singular surface at the light cylinder @xmath39 . both indicial ( or characteristic ) exponents for this equation are equal to zero and so a general solution of equation ( [ eq - pulsar ] ) diverges logarithmically at this surface ( e.g. , bender & orszag 1978 ) . on physical grounds , one imposes an additional condition that both the function @xmath37 and its 1st and 2nd derivatives remain finite near the lc . for any given function @xmath35 , it is indeed possible to find such non - divergent solutions separately on each side of the lc . then , as can be seen from equation ( [ eq - pulsar ] ) itself , the contribution of the first term on the lhs [ the term proportional to @xmath40 vanishes as @xmath41 , and thus such non - divergent solutions on both sides of the lc have to satisfy the well - known lc regularity condition ( e.g. , scharlemann & wagoner 1973 ; okamoto 1974 ; beskin 1997 ) : @xmath42 this condition is very important and merits a few extra words of discussion . our force - free equation with a constant field - line angular velocity @xmath21 is a 2nd - order equation with one , a priori unknown , integral of motion , @xmath35 , and one singular surface , the lc . according to general theory ( beskin 1997 ) , one then needs two boundary conditions to set up the problem properly . we believe that the correct approach here is to set the boundary conditions for the function @xmath8 at both the inner boundary ( inside the lc : on the surface of the star , the symmetry axis , the separatrix between the open and closed field - line regions , and maybe the equator ) and at whatever outer boundary ( outside the lc : the equator and infinity or the light surface ) . at the same time the poloidal current function @xmath35 should not be prescribed explicitly at any boundary of the domain ; instead , it is to be determined from the matching condition at the singular surface , i.e. , the lc ( see below ) . thus , our position differs from that of beskin ( see beskin et al . 1983 , and beskin 1997 ) who proposed prescribing @xmath35 explicitly at the surface of the pulsar . indeed , if one considers the region inside the lc only , then , upon prescribing both @xmath8 and @xmath35 on the inner boundary ( the pulsar surface , etc . ) , one should be able to obtain a solution that is regular at the lc by using the regularity condition ( [ eq - regularity - lc ] ) at the other boundary of this inner region , i.e. , at the lc . thus , the solution in this inner region is then completely determined ; as such , it is totally independent of what happens outside the lc . this does not seem physical ; indeed , one should then be able to take this solution and continue it smoothly across the lc , thus prescribing both @xmath43 and @xmath44 at the outer side of the lc . but then the problem of finding a solution in the region outside the lc becomes over - determined , as we now have two conditions at @xmath39 , in addition to any conditions at the outer boundary . for example , in the very important particular case of the domain under consideration extending all the way to radial infinity , one can not actually prescribe any specific boundary conditions at infinity because equation ( [ eq - pulsar ] ) has a regular singularity there . instead , however , one imposes a regularity condition at infinity . in spherical polar coordinates ) . we hope that this does not cause any confusion . ] ( @xmath45 ) this regularity condition can be written as @xmath46 this regularity condition has a very simple physical meaning : it is the condition of the force balance in the @xmath47-direction between the toroidal magnetic field @xmath48 and the poloidal electric field @xmath49 ( which both become much larger than the poloidal magnetic field @xmath50 as one approaches infinity ) . thus , in this case the region outside the lc does not have any boundary conditions at all ( apart from the condition @xmath51 at the equator ) , but has two regularity conditions , one at the lc , and the other at infinity . these two regularity conditions are already sufficient to uniquely determine the solution @xmath52 for a given @xmath35 . therefore , if , in addition to these two regularity conditions , one also tries to impose the function @xmath43 along the lc , the system becomes over - determined . thus we come to the conclusion that prescribing @xmath35 at the inner boundary is not appropriate . one could , however , try to use the outer boundary condition ( or the regularity condition at infinity ) as the condition that fixes @xmath35 . to do this , one first sets the inner boundary conditions , picks an initial guess for @xmath35 , and uses ( [ eq - regularity - lc ] ) to obtain a regular solution inside the lc ; one then continues this solution smoothly across the lc and solves equation ( [ eq - pulsar ] ) in the outer region as an initial - value problem ( with both @xmath8 and @xmath53 specified at @xmath39 ) ; as a result , one gets a mismatch at the outer boundary between the obtained solution and the desired outer boundary condition [ or , in the case of an infinite domain , one would presumably fail to get a convergent solution satisfying the regularity condition ( [ eq - regularity - infinity ] ) ] . then one iterates with respect to @xmath35 until this outer - boundary mismatch is zero ( or until a solution regular at infinity is achieved ) . in reality , however , such an approach may not be practical as it involves solving an initial - value problem for an elliptic equation , which is not a well - posed problem . instead , we advocate the approach adopted by ckf . in this approach one considers the regions inside the lc ( @xmath54 ) and outside the lc ( @xmath55 ) separately . first , one makes an initial guess for @xmath35 and prescribes the corresponding boundary conditions for @xmath8 at the inner boundary of the the region inside the lc and at the outer boundary of the the region outside the lc [ or , if this outer boundary is at the radial infinity , imposes the regularity condition ( [ eq - regularity - infinity ] ) there ] . then , one uses , again separately in each region , the regularity condition ( [ eq - regularity - lc ] ) in lieu of a boundary condition at the lc [ in practice , one can think of ( [ eq - regularity - lc ] ) as of a mixed - type dirichlet von neumann boundary condition relating the value of @xmath8 at the surface @xmath39 to the value of its derivative normal to this surface ] . thus one obtains two solutions , one inside and the other outside the lc , that correspond to the same function @xmath35 and are both regular at @xmath39 . however , these solutions also depend on their respective boundary or regularity conditions set at the inner / outer boundaries of the domain . hence , in general , although they are both regular at the lc , these solutions are not going to coincide at @xmath39 . the mismatch @xmath56 depends on both the chosen function @xmath35 and on the inner and outer boundary conditions . one then iterates with respect to @xmath35 in order to minimize this mismatch . we believe that for a given set of boundary conditions there should be only one choice of @xmath35 for which the mismatch @xmath57 vanishes and hence the function @xmath8 becomes continuous along the entire lc , i.e. , @xmath58 . once this special function @xmath35 is found , it then follows from ( [ eq - regularity - lc ] ) , which is satisfied separately on both sides of the lc , that @xmath53 is also continuous , and so the entire solution passes smoothly across the lc . thus , one can say that the function @xmath35 is determined by the regularity condition ( [ eq - regularity - lc ] ) applied separately on both sides of the lc , plus the condition of matching of the two solutions . another , equivalent way to put it , is to say that @xmath35 is determined by the ( non - trivial ! ) requirement that the derivative @xmath59 actually _ exists _ at the lc [ whereby @xmath35 is expressed in terms of this derivative via equation ( [ eq - regularity - lc ] ) ] . this approach has been implemented successfully in the pioneering numerical work by ckf and then subsequently by ogura & kojima ( 2003 ) . both these groups have used a relaxation procedure to arrive at @xmath35 that corresponded to a unique solution that was both continuous and smooth at the lc . the main focus of this paper is the behavior of the magnetic field in the vicinity of the point of intersection of the separatrix @xmath60 between the region of closed field lines ( region i ) and the region of open field lines ( region ii ) with the equator @xmath61 ( see fig . [ fig - geometry ] ) . we shall generally call this point the _ separatrix intersection point_. our interest in this non - trivial problem is fueled by the belief that understanding the key features of this behavior is absolutely crucial to devising the proper boundary conditions and providing verification benchmarks for any future global numerical investigations of a force - free pulsar magnetosphere . at the same time , the treatment of the magnetic field around this very special point is expected to require a certain degree of subtlety and delicacy , as noted , for example , by beskin et al . ( 1983 ) and by ogura & kojima ( 2003 ) . among the questions that need to be answered are : what is the radial position @xmath62 of the intersection point ( namely , whether this point lies at the lc , @xmath63 , of inside the lc , @xmath64 ) ? and what is the angle @xmath65 at which the separatrix approaches the equator at this point , the three a priori possibilities being @xmath66 ( in which case we ll call this point the cusp - point ) , @xmath67 ( the y - point ) , and @xmath68 ( the t - point ) ? in our analysis , we shall assume that equations ( [ eq - force - free ] ) and ( [ eq - ideal - mhd ] ) that describe our system are valid almost everywhere in the vicinity of the separatrix intersection point , i.e. , everywhere with perhaps the exception of measure - zero regions . thus , we shall allow for the presence in our system of current sheets of infinitesimal thickness , across which the magnetic field can experience a finite jump . we limit our consideration to the situation where such current sheets can be present only along the separatrix between regions i and ii for @xmath69 and along the equatorial ( @xmath61 ) separatrix between the upper and lower open - field regions for @xmath70 . thus , we have one current sheet that lies on the equator at @xmath70 and at @xmath71 splits into two symmetrical current sheets ( one in each hemisphere ) lying along the separatrix @xmath72 ( see , e.g. , okamoto 1974 ) . at the same time we assume that our equations apply everywhere else at least in some region around the intersection point , including the portion of this region that lies outside the lc ( in the case @xmath63 ) . let us first assume that the separatrix between closed and open field - line regions reaches the lc ( i.e. , @xmath63 ) , and let us consider the separatrix equilibrium condition ( e.g. , okamoto 1974 ; lyubarskii 1990 ) . indeed , whether or not the separatrix contains a current sheet ( which we assume to be infinitesimally thin ) , it must satisfy the condition of force balance , which is obtained by integrating equation ( [ eq - force - free ] ) across the separatrix : ) may not be valid inside a current sheet as other forces may also be important in the pressure balance there . however , gravity and the plasma inertia are small because the separatrix current layer is assumed to be very thin ; and the plasma pressure gradient , while not necessarily small inside the current layer , gives , when integrated across the layer , just the difference between the values of the pressure on both sides outside the separatrix , both of which are small by assumption . ] @xmath73 by using equation ( [ eq-|e| ] ) and also the fact that @xmath74 in region i , we can rewrite this condition as @xmath75\ , ( 1-x^2 ) = ( b_\phi^{ii})^2 = { 1\over{r_{lc}^4}}\ , { i^2(\psi_s)\over{x^2 } } \ , , \label{eq - sepx - equil-2}\ ] ] where @xmath76 marks the distance from the intersection point ( @xmath61 ) along the separatrix . now , as the separatrix approaches the lc ( @xmath41 , @xmath77 ) , we get @xmath78 = { { i_s^2}\over{r_{lc}^4 } } \ , , \label{eq - sepx - equil - endpt}\ ] ] where we have defined @xmath79 as the value of the poloidal current of the last open field line @xmath80 ( above the separatrix current sheet ) . since @xmath81 is constant along this last open field line , and hence is independent of @xmath76 , we immediately see that in order to satisfy equation ( [ eq - sepx - equil - endpt ] ) in the limit @xmath41 , we must have either : + _ ( i ) _ @xmath82 , i.e. , all poloidal current that flows out of the pulsar must return back to the pulsar along open magnetic field lines , with no finite poloidal current flowing along the separatrix ; of the compatibility condition by beskin et al . ( 1983 ) and beskin & malyshkin ( 1998 ) . ] simultaneously , from equation ( [ eq - sepx - equil - endpt ] ) it then follows that @xmath83 , and hence no finite toroidal current flows along the separatrix either ( we make a very natural assumption that @xmath84 is in the same direction as @xmath85 ) ; thus the separatrix in this case is actually not a current sheet ; + or , + _ ( ii ) _ a finite @xmath86 but a divergent poloidal magnetic field in the closed - field region , @xmath87 , as @xmath77 , @xmath88 . note that this situation can not be dismissed automatically ; indeed , close to the lc the magnitude of the electric field becomes very close to that of the poloidal magnetic field , with the infinitely strong electric forces balancing out the infinitely strong magnetic ones to a high degree . then , the separatrix current sheet would have to carry back to the star a finite poloidal current @xmath81 ( thus closing the poloidal current circuit ) , and also a toroidal corotation current with the surface current density @xmath89 divergent near the lc : @xmath90 ; there would also have to be a divergent surface charge density on the separatrix : @xmath91 . note that these conclusions , derived from the separatrix equilibrium condition near the lc , are valid regardless of the angle @xmath65 between the separatrix and the equator at the intersection point . in our analysis in the next section we shall assume that there are no infinitely strong fields anywhere in the system , and thus shall dismiss situation _ ( ii ) _ ( @xmath63 , @xmath92 , finite @xmath86 ) as unphysical . we shall thus concentrate our attention on situation _ ( i ) _ ( @xmath82 ) . we shall try to analyze the structure of the pulsar equation ( [ eq - pulsar ] ) and obtain the asymptotic solution of this equation in the vicinity of the intersection point ( @xmath63 , @xmath93 ) . in this section we perform an asymptotic analysis of the pulsar equation ( [ eq - pulsar ] ) in the vicinity of the separatrix intersection point under the assumption that it is located at the lc , i.e. , @xmath63 , @xmath93 . our approach is actually very similar to the analysis of the magnetic field near the endpoint of a non - relativistic reconnecting current layer , performed previously by uzdensky & kulsrud ( 1997 ) . in addition to assuming @xmath63 , we shall , in this section , make use of the following two assumptions : + 1 ) the intersection point is a finite - angle y - point , i.e. , the separatrix approaches the equator at an angle @xmath65 , @xmath67 , where we measure angles from the radial vector lying on the equator and directed _ toward _ the star , as shown in figure [ fig - geometry ] ; + 2 ) the poloidal current is fully closed in the open - field region , i.e. , @xmath94 ; this assumption is motivated by the arguments presented in the previous section . furthermore , we also assume symmetry with respect to the equator and therefore consider only the upper half - space , @xmath95 . when considering the @xmath96-dependent coefficients in equation ( [ eq - pulsar ] ) in the vicinity of the y - point , we can , to lowest order in @xmath97 , replace @xmath96 by 1 everywhere except where it appears in a combination like @xmath98 ; thus we can rewrite this equation as @xmath99 + 2 \ , { { \partial\psi}\over{\partial\xi } } = - ii'(\psi ) \ , , \label{eq - pulsar - endpt}\ ] ] where we introduced a new coordinate , @xmath100 . instead of using coordinates @xmath101 , it is actually more convenient to use polar coordinates @xmath102 defined by @xmath103 when making the transition to these coordinates , we shall make use of the following expressions : @xmath104 as well as the usual expression for two - dimensional ( 2-d ) laplacian in polar coordinates : @xmath105 in these coordinates the poloidal magnetic field components are written as follows ( we do not need these expressions now but will need them later ) : @xmath106 finally , by using expressions ( [ eq - coord - derivatives - xi])([eq-2d - laplacian ] ) , we can rewrite the pulsar equation ( [ eq - pulsar - endpt ] ) in our polar coordinates ( [ eq - polar_coords-1])([eq - polar_coords-2 ] ) as @xmath107 now we need to solve this equation separately in region i ( region of closed field lines , @xmath108 ) and in region ii ( region of open field lines , @xmath109 ) and match the two solutions together at the separatrix @xmath110 . since in this paper we are dealing only with the local structure of the magnetic field near the y - point , we can , without any loss of generality , adopt the convention of counting the poloidal magnetic flux @xmath8 from the separatrix @xmath110 ( instead of the usual convention of counting @xmath8 from the rotation axis ) . thus , we shall set @xmath111 and , correspondingly , @xmath112 in region i and @xmath113 in region ii . very close to the y - point , i.e. at distances much smaller than the light cylinder radius ( @xmath114 ) , the system lacks any natural length scale . hence , we can expect the radial dependence of @xmath8 to be a power law , which enables us to make the following _ self - similar ansatz _ : @xmath115 [ we put a minus sign in equation ( [ eq - psi_ii ] ) in order to have @xmath116 . ] then , the magnetic field components are given by : + in region i : @xmath117 and in region ii : @xmath118 notice that the condition that magnetic field does not diverge near @xmath119 imposes the restriction @xmath120 whereas for the case where the poloidal magnetic flux function in region i diverges as @xmath121 [ i.e. , case _ ( ii ) _ in [ sec - y - point ] ] , we would have @xmath122 . as for the poloidal current function @xmath35 that appears on the rhs of equation ( [ eq - pulsar - polar ] ) , it takes very different forms in regions i and ii . in region i there should be no toroidal field , so @xmath123 in the open - field region ii , @xmath35 is not zero but does approach zero in the limit @xmath124 . since there is no natural a priori magnetic field scale in the vicinity of the y - point , we shall again employ a self - similar ansatz , i.e. , assume that @xmath125 scales as a power of @xmath126 near @xmath127 : @xmath128 so that the right - hand side of equation ( [ eq - pulsar - polar ] ) becomes @xmath129 where @xmath130 upon substituting relationships ( [ eq - psi_i ] ) and ( [ eq - i - regioni ] ) ( for the closed - field region ) and ( [ eq - psi_ii ] ) and ( [ eq - rhs - regionii ] ) ( for the open - field region ) into the main equation ( [ eq - pulsar - polar ] ) , we obtain two ordinary differential equations ( odes ) for the functions @xmath131 and @xmath132 , along with a relationship between the power - law indices @xmath133 and @xmath134 . in particular , in region i we get a homogeneous linear 2nd - order ode for @xmath131 : @xmath135 which is to be supplemented by two boundary conditions : @xmath136 similarly , in region ii we get a non - linear 2nd - order ode for @xmath132 : @xmath137 together with the relationship @xmath138 that follows from the requirement that the lhs of equation ( [ eq - pulsar - polar ] ) scale as the same power of @xmath139 as the rhs , i.e. , @xmath140 . [ at this point , however , we have to remark that one can not a priori exclude the possibility @xmath141 ; in such a case the poloidal current would go to zero near the separatrix so rapidly that its contribution to the pulsar equation would become completely negligible . however , as we shall discuss later in this section , it is possible to show that no continuous solutions exist in this case . ] the boundary conditions for equation ( [ eq - pulsar - ii ] ) are @xmath142 notice that the constant @xmath143 that appears on the rhs of equation ( [ eq - pulsar - ii ] ) is in fact unimportant ; it just sets the scale of variation of the function @xmath132 and , hence , the overall scale of the magnetic field strength . thus , we can rescale @xmath143 away by incorporating it into the solution ; we do this by defining a new variable @xmath144 then , equation ( [ eq - pulsar - ii ] ) becomes @xmath145 with the same boundary conditions @xmath146 thus , we see that in our problem the magnetic field structure near the y - point is completely characterized by four finite dimensionless parameters : the separatrix angle @xmath65 and the three power - law indices @xmath147 , @xmath133 , and @xmath134 . our goal is to obtain a unique solution of our problem , that is to determine the values of these parameters and , simultaneously , determine the functions @xmath131 and @xmath132 . in fact , equation ( [ eq - beta - alpha2 ] ) already gives us one relationship between the power exponents , but we still need three more relationships to fix all four parameters . therefore , we shall now discuss the conditions that will help us obtain these three additional relationships . the second condition [ the first being equation ( [ eq - beta - alpha2 ] ) ] that links our dimensionless parameters is rather obvious ; it is the condition of force - balance across the separatrix @xmath110 and it will give us a relationship between @xmath147 and @xmath133 . we have already discussed this condition in [ sec - y - point ] [ see eqs . ( [ eq - sepx - equil-1])([eq - sepx - equil - endpt ] ) ] ; with @xmath148 , @xmath149 , and @xmath82 , this condition can be written simply as @xmath150^i = [ b_{\rm pol}^2 ( r,\theta_0)]^{ii } \ , . \label{eq - sepx - equil-3}\ ] ] since in our @xmath102-coordinates the separatrix is a radial line @xmath110 , and since we expect the poloidal magnetic field on the two sides of the separatrix to be in the same direction , condition ( [ eq - sepx - equil-3 ] ) simply means that @xmath151 . then , using equations ( [ eq - br - i])([eq - btheta - ii ] ) , we immediately obtain a second relationship between the power exponents : @xmath152 and also a relationship between the normalizations of the functions @xmath131 and @xmath132 , cast in terms of their derivatives at @xmath110 : @xmath153 the remaining two relationships between the dimensionless parameters come from the properties of equations ( [ eq - pulsar - i ] ) and ( [ eq - pulsar - ii - g ] ) themselves and can not , unfortunately , be written out as explicit algebraic equations containing these parameters . this fact , however , can prevent us neither from describing and discussing the physical and mathematical conditions on which these two additional relationships are based , nor from using these conditions to obtain the actual unique values of the parameters . first , we shall show that there is a unique one - to - one relationship between the separatrix angle @xmath65 and the power exponent @xmath154 . this relationship comes from analyzing equation ( [ eq - pulsar - i ] ) for the closed - field region i. for any given @xmath65 this homogeneous linear equation with boundary conditions ( [ eq - bc - i ] ) is an eigen - value problem for the coefficient @xmath155 . it is in fact very easy to see why this must be the case . indeed , the boundary conditions ( [ eq - bc - i ] ) do not give us any scale for @xmath156 ; if some function @xmath131 is a solution of the problem ( [ eq - pulsar - i])([eq - bc - i ] ) , then @xmath157 with an arbitrary multiplier @xmath158 will also be a valid solution . thus , the normalization of @xmath131 is arbitrary : if , for given @xmath65 and @xmath154 , a non - trivial solution of the problem ( [ eq - pulsar - i ] ) ([eq - bc - i ] ) exists , then there will be infinitely many such solutions ; in particular , there will be a solution with @xmath159 . thus , we can impose an additional boundary condition @xmath159 . then , however , we have three boundary conditions for a 2nd - order differential equation , which means that for arbitrary @xmath65 and @xmath154 the system is over - determined . hence , such a solution will exist not for all values of @xmath65 and @xmath154 ; the condition that it actually does exist gives us a certain relationship between @xmath65 and @xmath154 : for a given @xmath65 we would get an infinite discrete spectrum of the allowed values of @xmath154 . of these , however , we are interested only in the lowest one because we want a solution without direction reversals of the magnetic field , i.e. , with @xmath160 everywhere . from the practical point of view , the easiest way to determine the relationship between @xmath154 and @xmath65 is the following . we scan over the values of @xmath154 ; for each given @xmath154 we use our freedom of normalization of @xmath131 to set @xmath159 . thus we now have an initial - value problem with two conditions at @xmath161 : @xmath159 and @xmath162 . with these two conditions and with the value of @xmath163 given , we can integrate equation ( [ eq - pulsar - i ] ) forward in @xmath47 ( we do it numerically using a 2nd - order runge kutta scheme ) until the point where @xmath131 becomes zero . this point is then declared the correct value of the separatrix angle @xmath65 for the given @xmath154 . the function @xmath164 we have obtained as a result of this procedure is shown in figure [ fig - theta0 ] , whereas figure [ fig - f ] shows the behavior of the function @xmath131 for several selected values of @xmath154 . we see that @xmath164 is a monotonically decreasing function with @xmath165 rad [ instead of @xmath166 that one would get if the y - point were at a finite distance inside the lc ! ] , and @xmath167 . , corresponding to the case of zero poloidal current ( see also beskin et al . 1983 ) ; this is because in his analysis he had in fact assumed that @xmath168 and then derived the corresponding value of @xmath65 . ] the asymptotic behavior of @xmath164 in the limit @xmath169 is @xmath170 , which is in fact very easy to obtain analytically . , we expect @xmath171 , and then we can neglect the @xmath172 term in equation ( [ eq - pulsar - i ] ) . as a result , we get a simple harmonic equation @xmath173 , with @xmath162 , @xmath174 ; the solution of this equation , positive everywhere in the domain @xmath175 , is @xmath176 ; it corresponds to @xmath177 . ] actually , the exact analytical fit for our numerical curve @xmath164 appears to be @xmath178 with @xmath179 . with the dependence @xmath164 thus determined , and with @xmath134 and @xmath133 related to @xmath154 via equations ( [ eq - beta - alpha2 ] ) and ( [ eq - alpha1=alpha2 ] ) , respectively , we now have everything expressed in terms of @xmath154 . the fourth ( and final ! ) condition that will help us determine @xmath154 and hence @xmath65 and the rest of the power exponents is the light cylinder regularity condition . this condition states that at the lc @xmath180 [ which is a regular singular point for equation ( [ eq - pulsar - ii - g ] ) ] the function @xmath181 should be regular , namely , it should be a continuously differentiable function , with finite 1st and 2nd derivatives . this condition can be written as @xmath182 and it is just a particular manifestation of the general lc regularity condition ( [ eq - regularity - lc ] ) . we use this condition to fix the unique value of @xmath154 ( see the general discussion of this issue at the end of [ sec - basic ] ) . here is how we do it in practice . first , we divide up region ii into two sub - regions : region ii@xmath183 ( inside the lc : @xmath184 ) and region ii@xmath185 ( outside the lc : @xmath186 ) . then , we scan over @xmath154 ; for each value of @xmath154 , we first determine the corresponding value of @xmath65 using the procedure outlined above . once @xmath65 for a given @xmath154 is found , we solve equation ( [ eq - pulsar - ii - g ] ) separately in regions ii@xmath183 and ii@xmath185 ( again , using a numerical shooting method in conjunction with the 2nd - order runge - kutta integration scheme ) . in each of these regions we use regularity condition ( [ eq - lc - regularity - g ] ) at @xmath180 and one of the boundary conditions ( [ eq - bc - ii - g ] ) at the other end of the region , i.e. , @xmath187 in region ii@xmath183 and @xmath188 in region ii@xmath185 . note that the use of the regularity condition ( [ eq - lc - regularity - g ] ) guarantees that the solutions in _ each _ of the two regions are regular . however , in general ( i.e. , for an arbitrarily chosen @xmath154 ) , the two solutions obtained in this manner do not match each other at the lc , the mismatch @xmath189 [ and hence the mismatch in @xmath190 related to @xmath191 via eq . ( [ eq - lc - regularity - g ] ) ] being @xmath154-dependent . we find that there is only one , special value of @xmath154 for which @xmath191 vanishes and the solution continues smoothly across the lc as one passes from region ii@xmath183 into region ii@xmath185 . this special value , which we call @xmath192 , is declared the correct value of @xmath154 . numerically , we find @xmath193 and , correspondingly , @xmath194 this is the way in which the unique correct solution of the problem is obtained . the functions @xmath131 and @xmath195 corresponding to this value of @xmath154 are plotted in figures [ fig - f_0 ] and [ fig - g_0 ] , respectively . now let us make a little digression and step back to discuss the possibility of finding a solution in the case @xmath196 . recall that in this case the dominant balance of the lowest - order ( in @xmath139 ) terms in the pulsar equation does not include the contribution @xmath197 due to the poloidal current . at the same time , note that relationships ( [ eq - alpha1=alpha2 ] ) and ( [ eq - sepx - f=-g ] ) , derived from the separatrix force - balance condition , still hold . hence , the solution in region ii would have to satisfy a linear equation that is identical to equation ( [ eq - pulsar - i ] ) for @xmath131 on the other side of the separatrix . this fact , combined with the condition ( [ eq - sepx - f=-g ] ) , implies that the solution just has to pass smoothly across @xmath110 , and thus this point [ which is just an ordinary point for equation ( [ eq - pulsar - i ] ) ] does not have any special significance ; it is just the point where the solution changes sign . thus , the general solution in this case takes the form @xmath198 , where @xmath131 satisfies the homogeneous linear equation ( [ eq - pulsar - i ] ) in the entire domain @xmath199 $ ] with the homogeneous boundary conditions @xmath162 and @xmath200 . in addition , the solution has to pass smoothly across the regular singular point @xmath180 ( the light cylinder ) ; the regularity condition at this point is simply @xmath201 . such a solution would start positive with a zero derivative at @xmath161 , then decrease and change sign at some angle @xmath202 , reach a minimum at the lc [ since @xmath201 ] , and finally increase and go back to zero at the equator @xmath204 . however , the extra regularity condition at @xmath180 makes the system over - determined . indeed , counting two boundary conditions at @xmath205 , an arbitrary normalization of the solution [ e.g. , @xmath159 ] , and the lc regularity condition , we now have four conditions for our second - order equation with one free parameter @xmath154 . thus , unless we are somehow extremely lucky , the solution of this problem does not exist . one can show that this is indeed the case ; it turns out that a solution of equation ( [ eq - pulsar - i ] ) in the region inside the lc ( @xmath207 $ ] ) satisfying both the boundary condition @xmath162 and the lc regularity condition @xmath201 exists only for even values of @xmath208 , whereas a solution in region ( @xmath209 $ ] ) satisfying both the boundary condition @xmath200 and the lc regularity condition exists only for odd values of @xmath210 . this demonstrates that , in the case @xmath196 , it is impossible to obtain a solution that is continuous at the lc . this leaves us with the case @xmath211 ( for which we have just found a unique suitable solution ) as the only possibility . thus we have managed to obtain a _ unique _ asymptotic solution of the ideal - mhd , force - free system ( [ eq - force - free])([eq - ideal - mhd ] ) in the vicinity of the separatrix intersection point ( @xmath63 , @xmath93 ) . this solution , characterized by dimensionless parameters ( [ eq - alpha_0 ] ) and ( [ eq - beta&theta_0 ] ) , satisfies the separatrix equilibrium condition and is continuous and smooth at the lc . let us now discuss the physical relevance of our solution to the pulsar problem . note that the obtained unique solution of our problem has a finite value of the derivative @xmath212 at the equator @xmath213 , namely , @xmath214 , and thus has a non - zero radial magnetic field just above ( and just below ) the equatorial current sheet . this actually spells bad news for the applicability of our solution . indeed , one more condition that needs to be satisfied for the solution to be physically relevant and that we have not discussed so far is the condition @xmath215 with @xmath216 related to @xmath217 via equation ( [ eq-|e| ] ) , this condition can be expressed as @xmath218 we see that this condition is satisfied trivially everywhere inside the lc , but is not automatically satisfied outside the lc . in particular , in our solution , and in fact in any solution characterized by @xmath82 , the toroidal field on the last open field line @xmath72 is zero , whereas the poloidal magnetic ( and hence electric ) field is not zero , as our explicit solution shows . this means that the inequality ( [ eq - b > e-2 ] ) is violated immediately outside the lc as one moves along the equator . more generally , this inequality is violated beyond the so - called light surface ( defined as the surface where @xmath219 ) that emanates from the y - point . it is in fact not difficult to find the location of this light surface in our solution . indeed , according to equations ( [ eq - br - ii])([eq - btheta - ii ] ) , we have @xmath220\ , , \label{eq - bpol^2}\ ] ] and , according to equations ( [ eq - b_phi-2 ] ) , ( [ eq - i - regionii ] ) , and ( [ eq - beta - alpha2 ] ) , @xmath221 using the result ( [ eq - bpol^2 ] ) , we can rewrite this as @xmath222 upon comparing this result with @xmath223 we conclude that the light surface @xmath224 emanates from the y - point at a finite angle @xmath225 which is determined from the algebraic equation [ provided that the solution @xmath181 is known ] : @xmath226\ , \cos\theta_{ls } \ , . \label{eq - theta_ls-1}\ ] ] for our solution described by the parameters ( [ eq - alpha_0])([eq - beta&theta_0 ] ) , we find @xmath227 which corresponds to the angle @xmath228 . beyond the ls ( @xmath229 ) the force - free equation ( [ eq - force - free ] ) is not applicable . the solution we have obtained in this section is valid in the region @xmath230 , but , because the process of obtaining this solution also involved the region @xmath229 [ e.g. , via the boundary condition @xmath231 , our solution may not be the correct solution of the overall problem . in fact , we can make an even more general statement : any relevant to our problem finite - field magnetic configuration with the separatrix intersecting the equator at the lc can not be almost everywhere ( that is , again , everywhere except at a finite number of infinitesimally thin current sheets ) ideal and force - free beyond the lc ! indeed , as we saw previously , if one insists on having the separatrix intersection point at the lc , then one has to contend either with having @xmath232 divergent near @xmath39 or with having @xmath233 . the latter case , however , is characterized by zero toroidal magnetic field everywhere along the last open field line @xmath72 . that the poloidal ( radial ) magnetic field does not vanish identically near the equatorial current sheet , it then follows that @xmath234 , and so the force - free equation ( [ eq - force - free ] ) becomes inapplicable . and , as our particular solution shows , the condition @xmath235 is violated not only on some singular surfaces of measure zero , such as the equatorial current sheet ( that would be acceptable to us at this stage , since we do expect our simple force - free , ideal - mhd assumptions to break down there anyway ) , but in a volume of non - zero extent in both poloidal directions , e.g. , in the region ( @xmath236,@xmath237 ) . thus we have to conclude that if one is to keep the hope of finding a force - free solution that would be valid all the way up to infinity ( in the limit of vanishing plasma density ) , or at least to some finite distance beyond the lc , then one has to make concessions regarding the anticipated location of the separatrix / equator intersection point . in other words , the requirement that @xmath238 outside the lc means that @xmath81 must be non - zero and this , in turn , forces one to conclude that the intersection point must be located at some finite distance inside the lc , i.e. , @xmath64 . this scenario has actually been the subject of several studies in the past 20 years ( e.g. , beskin et al . 1983 ; lyubarskii 1990 ; beskin & malyshkin 1998 ) . according to equation ( [ eq - sepx - equil-2 ] ) , the corresponding finite-@xmath81 configuration has to be characterized by @xmath239 staying finite in the limit @xmath240 , and , therefore , the separatrix should approach the equator at a right angle , @xmath68 , corresponding to a t - point ( see lyubarskii 1990 ) . let us now ask what other characteristic features should such a solution possess ? here we present a few simple facts that can be gleaned immediately by inspecting the force - free equation ( [ eq - pulsar ] ) and its regularity condition ( [ eq - regularity - lc ] ) . first , since beyond @xmath71 the last open field line , @xmath72 , is assumed to lie along the equator @xmath61 , the derivative @xmath241 should be zero everywhere along this line ( for @xmath70 ) . in particular , we should have @xmath242 , but then it follows from equation ( [ eq - regularity - lc ] ) that either @xmath243 or @xmath244 has to be zero . since here we are considering the case @xmath86 , we conclude that @xmath35 has to approach @xmath80 with a zero slope , @xmath245 . thus , we see that @xmath246 can not be a simple linear function , unless it is constant everywhere . next , by substituting the result @xmath247 , together with @xmath248 , into equation ( [ eq - pulsar ] ) , we find that @xmath249 for all @xmath70 . this means that the radial magnetic field also approaches the equator @xmath61 with a zero slope , @xmath250 . now let us see what one can deduce by investigating the regularity condition ( [ eq - regularity - lc ] ) just above the equatorial current sheet . according to what we have just established , above the equator the function @xmath35 can be expanded as @xmath251 where @xmath252 in order to satisfy the condition @xmath253 , as @xmath124 ( here we again set @xmath111 to simplify notation ) . then we find , to lowest order in @xmath126 , @xmath254 now , the radial magnetic field just above the equator is , in general , not zero , so we can expand @xmath255 to lowest order in @xmath256 as @xmath257 where @xmath258 is in fact just the rescaled electric field above the equator , as can be seen from equations ( [ eq - e ] ) and ( [ eq - b_r-2 ] ) : @xmath259 upon substituting equations ( [ eq - tpoint - rhs ] ) and ( [ eq - tpoint - psi - z>0 ] ) into the regularity condition ( [ eq - regularity - lc ] ) , we get @xmath260^{\beta-1 } \ , , \label{eq - tpoint - regularity - lc - z>0}\ ] ] and , therefore , @xmath261 unless @xmath262 [ and @xmath263 if @xmath264 . we can now use this result to analyze the magnetic field in the vicinity of the point @xmath265 . equation ( [ eq - pulsar ] ) in the vicinity of this point can be written as @xmath266 + { { 1+x_0 ^ 2}\over x_0}\ , { \partial\psi\over{\partial\zeta } } = -\ , ii'(\psi ) \ , , \label{eq - tpoint - pulsar}\ ] ] where @xmath267 . now , just as we did in [ sec - analysis ] , we can make a transition to the polar coordinates : @xmath268 , @xmath269 . in these polar coordinates equation ( [ eq - tpoint - pulsar ] ) becomes @xmath270 + { { 1+x_0 ^ 2}\over x_0}\ , \biggl[{\partial\psi\over{\partial r}}\cos\theta- { \partial\psi\over{r\partial\theta}}\sin\theta \biggr ] = -\ , ii'(\psi ) \ , , \label{eq - tpoint - pulsar - polar}\ ] ] again , similarly to what we did in [ sec - analysis ] [ see eq . ( [ eq - psi_ii ] ) ] , let us assume that the magnetic flux function in region ii ( the region of open field lines , @xmath271 ) is a power law of @xmath139 near the point ( @xmath272 ) : @xmath273 then , using equations ( [ eq - tpoint - rhs ] ) and ( [ eq - tpoint - beta=2 ] ) , we can express the rhs of equation ( [ eq - tpoint - pulsar - polar ] ) as @xmath274 whereas the lhs of this equation can be expressed as @xmath275 \nonumber \\ & - & \ , { { 1+x_0 ^ 2}\over x_0}\ , r^{\alpha_2 - 1}\ , \biggl [ \alpha_2 g(\theta)\cos\theta - g'(\theta ) \sin\theta \biggr ] \ , . \label{eq - tpoint - lhs - polar}\end{aligned}\ ] ] in this expression , terms of the lowest order in @xmath139 are those proportional to @xmath276 ; at the same time we see from equation ( [ eq - tpoint - rhs - polar ] ) that the rhs of equation ( [ eq - tpoint - pulsar - polar ] ) scales with @xmath139 as @xmath277 . therefore , we see that in this case of the separatrix intersecting the equator inside the lc , @xmath64 , the contribution of the rhs is completely negligible ( as @xmath278 ) . thus , the dominant balance in equation ( [ eq - tpoint - pulsar - polar ] ) dictates that the terms of order @xmath276 should balance each other and hence we get a very simple homogeneous linear ode for the function @xmath132 : @xmath279 the boundary conditions for this equation are @xmath280 and so the solution with the lowest @xmath133 ( corresponding to the simplest magnetic field topology in this region ) obviously is @xmath281 and , correspondingly , @xmath282 thus , @xmath283 . must be equal to zero according to the lc regularity condition , whereas lyubarskii assumed that @xmath284 and hence had a finite , non - vanishing @xmath285 . ] as for region i ( the region of closed field lines ) , the magnetic field there is , to lowest order in @xmath139 , purely vertical and finite at @xmath240 , so we can write @xmath286 where @xmath287 is related to @xmath81 via the separatrix equilibrium condition ( [ eq - sepx - equil-2 ] ) : @xmath288 we express hope that these simple findings will be helpful in setting up or checking the correctness of future numerical attempts to solve this nontrivial problem . in this paper we have considered the axisymmetric force - free magnetosphere of an aligned rotating magnetic dipole , under the additional assumptions of ideal mhd and of the uniformity of the field - line angular velocity [ @xmath289 . this fundamental model problem is of great importance to any attempts to understand the workings of radio - pulsars . more specifically , we have focussed most of our attention on the structure of the magnetic field in the vicinity of the point of intersection of the separatrix ( between the closed- and open - field regions ) and the equator . we call this point the separatrix intersection point . the unique singular nature of this point makes it play an extremely important role in the overall global problem ; in particular , without a thorough understanding of the subtleties of the magnetic field behavior near this point , it is impossible to prescribe the correct global boundary conditions in any sensible way . we start , however , by discussing ( in [ sec - basic ] ) the basic equations governing the global force - free pulsar magnetosphere . we give special attention to the role played by the light cylinder regularity condition in determining the poloidal electric current . after this general discussion , the rest of the paper is devoted entirely to the analysis of the separatrix intersection point . we first consider the separatrix equilibrium condition in the vicinity of this point ( see [ sec - y - point ] ) and find that if it lies at the light cylinder , then all the poloidal current @xmath35 has to return back to the pulsar in the open - field region above the equator , i.e. , there should be no singular current running in a current sheet along the equator and the separatrix . we then perform ( in [ sec - analysis ] ) an asymptotic analysis of the relativistic grad shafranov , or pulsar , equation in the vicinity of such an intersection point located at the light cylinder . we find a unique self - similar solution that can be described by the power law @xmath290 , where @xmath139 is the distance form the intersection point and @xmath291 , and by the equator separatrix angle @xmath292 rad . however , a further analysis of this solution in the region outside the light cylinder shows that a light surface ( a surface where @xmath293 ) appears just outside the light cylinder ; in particular , we find that this light surface originates right at the intersection point and extends outward at a finite angle with respect to the equator . the appearance of a light surface in this case is , of course , not surprising , taking into account the fact that ( in this case of the separatrix intersection point lying at the light cylinder ) the poloidal electric current @xmath125 and hence the toroidal magnetic field have to become zero on the last open field line . we therefore conclude that the only possibility for an ideal - mhd force - free magnetosphere above the putative equatorial current sheet to extend at least some finite distance beyond the light cylinder , is for the separatrix intersection point to be located inside , as supposed to right at , the light cylinder . ( of course we understand that the exact location of this point can only be found as a part of a global solution of the pulsar equation and that it is impossible to determine it from our local analysis ) . these findings give us the motivation to consider the case when the intersection point lies at some finite distance inside the light cylinder . in [ sec - t - point ] we examine the behavior of the function @xmath35 near the last open field line @xmath72 ( above the equatorial current sheet ) for this case ; we find that the derivative @xmath294 has to go to zero as @xmath295 , whereas the current itself can approach a finite value @xmath86 . we then perform an asymptotic analysis of the magnetic field around the intersection point ; it is essentially a somewhat simplified and trivialized analogue of our analysis in [ sec - analysis ] . we find that the separatrix approaches the equator at a right angle , @xmath68 , and that the field in the region of open field lines behaves simply as @xmath296 , while in the closed - field region the magnetic field is simply vertical and finite near the intersection point ( we call this configuration the t - point ) . finally , we would like to make several remarks regarding the numerical simulations by ckf and by ogura & kojima ( 2003 ) : \1 ) the midplane boundary conditions ( @xmath297 , @xmath298 ; @xmath51 , @xmath299 ) adopted by both groups have automatically assumed that the separatrix intersection point lies at the lc , and thus have precluded them from even considering the possibility @xmath64 . it is interesting to note , however , that from the magnetic contour plots presented by ogura & kojima ( 2003 ) it does seem that the actual intersection point lies a little bit inside the lc . \2 ) neither ckf , nor ogura & kojima ( 2003 ) have discussed or even mentioned the separatrix equilibrium condition ; thus it is not clear whether this condition has been satisfied in their simulations . in the light of our present findings and of the fact that both of these groups have found @xmath86 , we suspect that the equilibrium condition has not , in fact , been satisfied in their studies , at least close to the separatrix intersection point . this suspicion is strengthened by the fact that both groups have reported having experienced some difficulties near the separatrix . \3 ) ogura & kojima ( 2003 ) have reported that they had found @xmath300 at some finite distance outside the lc , whereas ckf have claimed that in their solution @xmath301 everywhere . the origin of this discrepancy is not clear . it may be attributed to the difference in numerical resolution , although neither of the two groups have conducted very extensive convergence studies . i am grateful to leonid malyshkin and anatoly spitkovsky for interesting and insightful discussions . this research was supported by the national science foundation under grant no . phy99 - 07949 .

we investigate the axisymmetric magnetosphere of an aligned rotating magnetic dipole surrounded by an ideal force - free plasma . we concentrate on the magnetic field structure around the point of intersection of the separatrix between the open and closed field - line regions and the equatorial plane . we first study the case where this intersection point is located at the light cylinder . we find that in this case the separatrix equilibrium condition implies that all the poloidal current must return to the pulsar in the open - field region , i.e. , that there should be no finite current carried by the separatrix / equator current sheet . we then perform an asymptotic analysis of the pulsar equation near the intersection point and find a unique self - similar solution ; however , a light surface inevitably emerges right outside the light cylinder . we then perform a similar analysis for the situation where the intersection point lies somewhere inside the light cylinder , in which case a finite current flowing along the separatrix and the equator is allowed . we find a very simple behavior in this case , characterized by a 90-degree angle between the separatrix and the equator and by finite vertical field in the closed - field region . finally , we discuss the implications of our results for global numerical studies of pulsar magnetospheres .

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Proceed to summarize the following text: the primary aim of the efar project ( wegner 1996 ; paper 1 ) is to use the tight correlations between the global properties of early - type galaxies embodied in the fundamental plane ( fp : djorgovski & davis 1987 , dressler 1987 ) to measure relative distances to clusters of galaxies in order to investigate peculiar motions and the mass distribution on large scales . however these global relations also constrain the dynamical properties and evolutionary histories of early - type galaxies . for example , renzini & ciotti ( 1993 ) show that the tilt of the fp implies a range in mass - to - light ratio @xmath2 among ellipticals of less than a factor of three , while the low scatter about the fp implies a scatter in @xmath2 at any location in the plane of less than 12% . similar reasoning has been used to constrain the star formation history of cluster ellipticals using the colour magnitude relation ( bower 1992 , kodama & arimoto 1997 ) . recently the fp , and colour magnitude relations have been followed out to higher redshifts and used to show that the early - type galaxies seen at @xmath9@xmath101 differ from present - day early - type galaxies in a manner consistent with passive evolutionary effects ( van dokkum & franx 1996 , ziegler & bender 1997 , kelson 1997 , ellis 1997 , kodama & arimoto 1997 , stanford 1998 , kodama 1998 , bender 1998 , van dokkum 1998 ) . in this paper we consider the relation between the central velocity dispersion @xmath11 and the strength of the magnesium lines at a rest wavelength of 5174 for the early - type galaxies in the efar sample . this relation connects the dynamical properties of galaxy cores with their stellar populations . the remarkably small scatter about this relation ( burstein 1988 , guzmn 1992 , bender 1993 , jrgensen 1996 , bender 1998 ) , and its distance - independent nature , make it a potentially useful constraint on models of the star formation history of early - type galaxies and a test for environmental variations in the fp ( burstein 1988 , bender 1996 ) . there are , however , some problems with using the relation for probing galaxy formation . two of these problems are apparent from the stellar population models ( worthey 1994 , vazdekis 1996 ) : ( i ) both age and metallicity contribute to the mg linestrengths in comparable degree , so that a spread in linestrengths could be due to either a range of ages or a range of metallicities or some combination ; ( ii ) the mg linestrengths are not particularly sensitive indicators at fixed metallicity a difference in age of a factor of ten only results in a change of 0.05 - 0.1 mag , while at fixed age a difference of 1 dex in metallicity gives a change of 0.10.2 mag . thus mg linestrength measurements must be accurate in order to yield useful constraints on the ages and metallicities of stellar populations , and the relation on its own can only supply constraints on combinations of age and metallicity and not one or the other separately . recently trager ( 1997 ) has suggested that the tightness of the relation may be the result of a ` conspiracy ' , in that there appears to be an anti - correlation between the ages and metallicities of the stellar populations in early - type galaxies at fixed mass which acts to reduce the scatter in the mg linestrengths . trager takes the accurate h@xmath12 , mg and fe linestrengths from gonzlez ( 1993 ) and applies the stellar population models of worthey ( 1994 ) to derive ages and abundances from line indices with different dependences on age and metallicity . he finds that at fixed velocity dispersion the ages and abundances lie in a plane of almost constant mg linestrength , leading him to predict little scatter in the relation even for large differences in age or metallicity a factor of ten in age ( from 1.5 gyr to 15 gyr ) gives a spread in of only 0.010.02 mag . this conclusion depends on the appropriateness of the single stellar population models and requires confirmation from further high - precision linestrength measurements . it can also be tested using the high - redshift samples now becoming available . in a similar vein , a number of authors ( ferreras 1998 , shioya & bekki 1998 , bower 1998 ) have recently re - examined whether the apparent passive evolution of the colour magnitude relation out to @xmath9@xmath101 really implies a high redshift for the bulk of the star - formation in elliptical galaxies . they conclude that in fact such evolution can be consistent with a rather broad range of ages and metallicities if the galaxies assembling more recently are on average more metal - rich than older galaxies of similar luminosity . as well as studies focussing on the evolution of the galaxy population , there have also been investigations of possible variations with local environment . guzmn ( 1992 ) have suggested that there are systematic variations in the relation which affect estimates of relative distances based on the fp . they report a significant offset in the zeropoint of the relation between galaxies in the core of the coma cluster and galaxies in the cluster halo . jrgensen and co - workers ( 1996 , 1997 ) examine a sample of 11 clusters and find a weak correlation between mg linestrength and local density within the cluster which is consistent with this result . similar offsets are claimed between field and cluster ellipticals by de carvalho & djorgovski ( 1992 ) and jrgensen ( 1997 ) , although burstein ( 1990 ) find no evidence of environmental effects . such systematic differences could result from different star - formation histories in different density environments , producing variations in the mass - to - light ratio of the stellar population . fp distance measurements would then be subject to environment - dependent systematic errors leading to spurious peculiar motions . where data for field and cluster ellipticals come from different sources , however , the possibility also exists that any zeropoint differences are due to uncertainties in the relative calibrations rather than intrinsic environmental differences . the relation has thus become an important diagnostic for determinations of both the star formation history and the peculiar motions of elliptical galaxies . here we examine the relation in the efar sample , which includes more than 500 early - type galaxies drawn from 84 clusters spanning a wide range of environments . in 2 we summarise the relevant properties of the sample and the techniques used to determine the and linestrength indices , the central velocity dispersions @xmath11 , and the errors in these quantities . we present the relation in 3 and investigate how it varies from cluster to cluster within our sample , and with cluster velocity dispersion , x - ray luminosity and x - ray temperature . in 4 we compare our results with the predictions of stellar population models in order to derive constraints on the ages , metallicities and mass - to - light ratios of early - type galaxies in clusters . in particular , we consider the constraints on the dispersion in the ages and metallicities from the intrinsic scatter in the relation on its own , and in combination with the intrinsic scatter in the fp . our conclusions are given in 5 . here we give a short description of our sample and dataset , with emphasis on the velocity dispersions and line indices used in this paper . the interested reader can find more detail on the sample selection in paper 1 ( wegner 1996 ) ; on the measurement , calibration and error estimation procedures for the spectroscopic parameters in paper 2 ( wegner 1998 ) ; and on the structural and morphological properties of the galaxies in paper 3 ( saglia 1997 ) . the efar sample of galaxies comprises 736 mostly early - type galaxies in 84 clusters . these clusters span a range of richnesses and lie in two regions toward hercules corona borealis and perseus pisces cetus at distances of between 6000 and 15000 . in addition to this program sample we have also observed 52 well - known galaxies in coma , virgo and the field in order to provide a calibrating link to previous studies . the efar galaxies are listed in table 2 of paper 1 , and comprise an approximately diameter - limited sample of galaxies larger than about 20 arcsec with the visual appearance of ellipticals . photometric imaging ( paper 3 ) shows that 8% are cds , 12% are pure es and 49% are bulge - dominated e / s0s ; thus 69% of the sample are early - type galaxies , with the remaining 31% being spirals or barred galaxies . we have obtained spectroscopy for 666 program galaxies , measuring redshifts , velocity dispersions and linestrength indices ( paper 2 ) . we have used the redshifts we obtained together with literature redshifts for other galaxies in the clusters in order to assign program galaxies to physical clusters . we have used the combined redshift data for these physical clusters to estimate cluster mean redshifts and velocity dispersions . the early - type galaxies in our sample span a wide range in luminosity , size and mass : they have absolute magnitudes from @xmath13=@xmath824 to @xmath818 ( @[email protected] ; h@xmath15=50kms@xmath16mpc@xmath16 ) , effective radii from 1 to 70 ( @xmath17=9.1 ) and central velocity dispersions from less than 100 to over 400 ( @xmath18=220 ) . the sample is thus dominated by early - type galaxies with luminosities , sizes and masses typical of giant ellipticals . we summarise here the procedures used in measuring the redshifts , velocity dispersions and mg linestrengths ; full details are given in paper 2 . = redshifts and velocity dispersions were measured from each observed galaxy spectrum using the iraf task fxcor . linestrength indices on the lick system were determined using the prescription given by gonzlez ( 1993 ) . the and indices were both measured : because it is the index most commonly measured in previous work , and because it could be measured for more objects ( as it requires a narrower spectral range ) and is better - determined ( being less susceptible to variations in the non - linear continuum shape ) . we find it more convenient to express the ` atomic ' index in magnitudes like the ` molecular ' index rather than as an equivalent width in ngstroms , since this puts these two indices on similar footings . the conversion is @xmath19 where @xmath20 is the index bandpass ( 32.5 for ) . error estimates for each quantity were derived from detailed monte carlo simulations , calibrated by comparisons of the estimated errors with the results obtained from repeat measurements ( over 40% of our sample had at least two spectra taken ) . two sorts of corrections were applied to the dispersions and linestrengths : ( i ) an aperture correction , based on that of jrgensen ( 1995 ) , to account for different effective apertures sampling different parts of the galaxy profile , and ( ii ) a run correction to remove systematic errors between different observing setups . after applying these corrections , individual measurements for each galaxy were combined using a weighting scheme based on the estimated errors and the overall quality of the spectrum . = = the median estimated errors in the final combined values are @xmath21 ( @xmath22 dex ) , @xmath23 mag and @xmath24 mag . the distribution of estimated errors for each quantity is shown in the upper panel of figure [ fig : errsum ] . the lower panel of the figure shows how the error estimates were calibrated against the repeat observations : the distribution of the ratio of rms error to estimated error for objects with repeat measurements is compared to the predicted distribution assuming the estimated errors are the true errors . the initial error estimates from the simulations have been re - scaled to give the best match ( under a k - s test ) to the rms errors from the repeat measurements . a re - scaling by factors of 0.85 and 1.15 respectively gives good agreement for the errors in @xmath11 and ; adding 0.005 mag likewise gives good agreement for the errors in . a comparison with the literature ( paper 2 , figure 13 ) shows that our dispersions are consistent with previous measurements by davies ( 1987 ) , guzmn ( 1993 ) , jrgensen ( 1997 ) , lucey ( 1997 ) and whitmore ( 1985 ) . for the subset of galaxies in common , we compared our linestrengths with the definitive lick system measurements of trager ( 1998 ) in order to derive the small zeropoint corrections required to calibrate our measurements to the lick system ( paper 2 , figures 14 & 15 ) ; the overlap of our measurements with those of lucey ( 1997 ) also shows consistency ( figure 16 , paper 2 ) . in this section we investigate the global relation found amongst the entire sample of efar galaxies with early - type morphological classifications ( cd , e or e / s0 ; see definitions in paper 3 ) for which we obtained linestrength measurements . the relation is shown in figure [ fig : mgsig]a and the relation in figure [ fig : mgsig]b . in order to fit linear relations with intrinsic scatter in the presence of significant measurement errors in both variables , arbitrary censoring of the dataset and a broad sample selection function , we have developed a comprehensive maximum likelihood ( ml ) fitting procedure ( saglia , in preparation ) . excluding galaxies with dispersions less than 100or selection probabilities less than 10% , and also outliers with low likelihoods , the ml fits to the relation ( 490 galaxies ) and the relation ( 423 galaxies ) are : @xmath25 these fits are shown in figure [ fig : mgsig ] as solid lines . the ratio of the slopes of these relations is consistent with the -relation we obtained in paper 2 : @[email protected] . monte carlo simulations of the dataset and fitting process , the results of which are displayed in figure [ fig : mgsigsim ] , show that there is no bias in the ml estimates of the slopes and zeropoints , and provide reliable estimates of the uncertainties in the fit . = + = the ml fits can be compared to simple regressions of and on @xmath26 . these regressions are shown in the figure as dashed lines , and are : @xmath27 as expected , the simple regressions yield slopes which are biased low due to the presence in the data of errors in the abscissa as well as the ordinate , and also the intrinsic scatter in the relation . slightly less - biased results are obtained by least squares regression minimising the orthogonal residuals ( jrgensen 1996 ) : @xmath28 these least squares fits and their uncertainties are obtained using the slopes regression program written by e.d.feigelson and described in isobe ( 1990 ) and feigelson & babu ( 1992 ) . the uncertainties are under - estimated because these regressions do not properly account for the measurement errors or the selection functions . we conclude that previous determinations of the slope of the relation are likely to be biased low whenever the dataset being fitted had significant errors in the velocity dispersions ( as is generally the case ) . hereafter we adopt the ml fits to the relation . the distributions of the residuals in and about the ml fits are shown in the insets to figures [ fig : mgsig]a and [ fig : mgsig]b . in order to minimise the effects of outliers , we robustly estimate the scatter about the relations as half the range spanned by the central 68% of the data points . we find an _ observed _ scatter of [email protected] mag about the relation and [email protected] mag about the relation . excluding outliers , the distributions of residuals are very well fitted by gaussians parametrised by the median residual and the robustly estimated scatter . there is no evidence for a tail of negative residuals such as noted by burstein ( 1988 ) and jrgensen ( 1996 ) . as the latter authors point out , the presence of such a tail is sensitive to the adopted slope of the relation . some giant ellipticals do , however , have intrinsically weak mg linestrengths for their velocity dispersions ( schweizer 1990 ) . the estimates of the _ intrinsic _ scatter about the relations that are provided by the ml fit may be exaggerated by outliers or by deviations of the underlying distribution of galaxies in the plane from a bivariate gaussian . we therefore drop the assumption of an intrinsic bivariate gaussian distribution in the plane and use monte carlo simulations based on the observed distribution of dispersions and linestrengths and their estimated errors ( accounting for both measurement errors and run correction errors ) . these simulations assume only that there is a global linear relation about which there is gaussian intrinsic scatter . we vary this intrinsic scatter and compute the robust estimate of the observed scatter about the fit ( the half - width of the central 68% of the residuals ) for the simulated distributions . the results of these simulations are presented in figure [ fig : mgscat ] , which shows the normalised likelihood distributions for the intrinsic scatter in and given the observed scatter . we find that to account for the observed scatter in the relations we require an intrinsic scatter of [email protected] mag for and [email protected] mag for . the ratio of the intrinsic scatter in to the intrinsic scatter in is slightly lower than expected from the observed relation , @[email protected] ( see paper 2 ) . = .comparison of relation fits [ cols="<,^,^,^,^ " , ] = = table [ tab : mgsig ] compares our fits to the relations obtained by other authors , and gives the observed scatter @xmath30mg@xmath31 and the intrinsic scatter @xmath30mg@xmath32 in the relations obtained in each case . for both and the slopes we obtain are about 25% steeper than those obtained by most previous authors . this is not due to a difference in our data , but stems from our use of the ml method rather than regressions . in this situation regressions are biased towards flatter slopes than the true relation because they ignore the intrinsic scatter , the presence of errors in both variables and the selection function of the dataset . the standard or orthogonal regression fits to our data , which our simulations show under - estimate the slope of the relations , give results very similar to those obtained by other authors . if we divide the sample by morphological type , we find that the cds have a zeropoint which is 0.009 mag higher than that of the other early - type galaxies in , and 0.014 mag higher in . these differences in the zeropoints are readily apparent in the distributions of residuals about the global relations ( see the insets to figures [ fig : mgsig]a&b ) , and are significant at the 3@xmath11-level . despite these zeropoint offsets , including or excluding the cds changes the scatter about the ml fit by less than its uncertainty , as they make up only 10% of the whole sample . we find no significant differences , however , if we compare the relations for the two volumes of space , the hercules - corona - borealis and perseus - pisces - cetus regions , from which our sample is drawn . the two regions have relations with slopes and zeropoints which are consistent both with each other and with the overall relations , providing a check that there are no gross systematic environmental differences between these two regions . we do not have enough galaxies per cluster to fit both the slope and the zeropoint of the relations on a cluster - by - cluster basis , even in our best - sampled clusters . we therefore limit ourselves to investigating the variation in the zeropoint . to this end we measured the median offset in and from the global fits given above for the clusters with three or more linestrength measurements ( 75 clusters for and 72 for ) . note that we only used galaxies that are cluster members based on their redshifts ( see paper 2 ) . the results are not changed significantly if we use all clusters , or only clusters with five or more measurements . = = the top panels of figure [ fig : mgclus ] show these zeropoint offsets as a function of cluster i d number ( cid ) , while the middle panels show the distributions of the offset values . the robustly - estimated scatter in the zeropoint offsets is [email protected] mag in and [email protected] mag in , showing that the relations are remarkably uniform among the aggregates of galaxies in the efar sample . the bottom panels in the figure plot the same offsets as a function of redshift , showing that there is no dependence of the relations on relative distance within the sample . this scatter in the zeropoint offsets could purely be a consequence of a galaxy - to - galaxy scatter in a global relation , or it could also require a variation in the zeropoint of the relation from cluster to cluster . these possibilities were examined by extending the simulations described in the previous section , adding a further source of scatter to the relation in the form of an intrinsic variation between clusters in the zeropoint of the relation . for simplicity we assume that this variation also has a gaussian distribution . we find that if we make the extreme assumption that there is cluster - to - cluster scatter but no intrinsic scatter between galaxies within a cluster , then zeropoint variations between clusters with an rms of 0.009 mag in and 0.015 mag in are required to recover the observed cluster - to - cluster scatter . however this model under - predicts the observed scatter about the global relation , giving [email protected] mag for and [email protected] mag for compared to the actual values of [email protected] mag and [email protected] mag . on the other hand , if we assume that there is no zeropoint variation between clusters , then the intrinsic scatter between galaxies required to recover the observed scatter in the global relation ( 0.016 mag in and 0.023 mag in ; see previous section ) predicts a scatter in the cluster zeropoints of [email protected] mag in and [email protected] mag in , which is consistent with the observed values of [email protected] mag and [email protected] mag within the joint errors . we conclude that there is no evidence for significant intrinsic zeropoint variations between clusters , since sampling a galaxy population drawn from a single global relation with intrinsic scatter consistent with the observations can account for the zeropoint differences between our clusters . as there is very little change in the zeropoint of the relation from cluster to cluster , it follows that there can be at most only a weak dependence of the zeropoint on the properties of the clusters . here we investigate the effect of cluster properties on the stellar populations as reflected in the zeropoints , considering cluster velocity dispersions , x - ray luminosities and x - ray temperatures ( all indicators of cluster mass ) . the cluster dispersions come from table 7 of paper 2 , using redshifts both from efar and from the zcat catalogue ( huchra 1992 ; version of 1997 may 29 ) . x - ray luminosities and temperatures are available for 26 of our 84 clusters in the homogeneous and flux - limited catalogue of x - ray properties of abell clusters by ebeling ( 1996 ) based on rosat all - sky survey data . the x - ray luminosities are determined to a typical precision of about 20% . in order to have comparable precision in the cluster velocity dispersions , we only use clusters with dispersions computed from at least 20 galaxy redshifts ; this also leaves 26 clusters , 17 of which are in common with the x - ray subsample . figure [ fig : mgrich ] shows the offsets in the relations as functions of @xmath33 , @xmath34 and @xmath35 . applying the spearman rank correlation statistic , we find that there is no significant correlation between the offsets and any of these quantities , and thus no evidence for a trend in the zeropoint of the relation with cluster mass . weighted regressions give best - fit relations and their uncertainties : @xmath36 if we take a complementary approach , splitting the clusters into two subsamples about the median value of @xmath37 and fitting the relations to the galaxies of the high-@xmath37 and low-@xmath37 clusters separately , we again find no significant differences in the slopes or the zeropoints of the fits , which are compatible with the global fits obtained above . there are at least four main questions which can be addressed using the above results . \(i ) what are the theoretical implications of the lack of correlation between the mass of a cluster and the zeropoint of the relation for cluster galaxies ? \(ii ) what effect do the stellar population differences implied by the observed variations in the relation have on fundamental plane estimates of distances and peculiar velocities ? \(iii ) what constraint does the intrinsic scatter about the relation place on the spread in age , metallicity and mass - to - light ratio amongst early - type galaxies in clusters ? \(iv ) what further constraints on these quantities result from combining the scatter about the relation with the intrinsic scatter about the fundamental plane ? the small scatter in the zeropoint of the relation from cluster to cluster , and in particular the lack of correlation between the zeropoint and the cluster mass , seems to imply that the mass over - density on mpc scales in which an early - type galaxy is found has little connection with its stellar population and star - formation history . the variation of the relation with cluster properties has previously been studied in a sample of 11 nearby clusters by jrgensen ( 1996 ) and jrgensen ( 1997 ) . following guzmn ( 1992 ) , these authors look for a trend in offsets with the ` local density ' _ within _ clusters . the estimator of local density used is @xmath38 , where @xmath39 is the projected distance of the galaxy from the cluster centre . since @xmath39 is only a lower limit on the galaxy s true distance from the cluster centre , this is a rather poor estimator of the true local density . jrgensen find that the residuals in show a weak trend0.009 ( jrgensen , priv.comm . ) ] with local density , @xmath40 . since the residuals do _ not _ correlate with radius within the cluster ( see figure 3 of jrgensen ( 1997 ) ) , but _ do _ show a significant correlation with cluster velocity dispersion , @xmath41 ( least - squares fit to the data in figure 5 of jrgensen ( 1997 ) ) , we would argue that a more straightforward interpretation of their results is a correlation of zeropoint with total cluster mass rather than local density . a correlation of this amplitude is formally consistent at the 2@xmath11 level with the distribution of offsets versus @xmath33 for the efar data ( see equation [ eqn : clusfitmg2 ] ) ; transforming jrgensen s result via the relation gives a correlation which is consistent at the 1.4@xmath11 level with equation [ eqn : clusfitmgb ] . we conclude that any correlation between the relation zeropoint and the cluster mass is sufficiently weak ( of order @xmath42 or less ) that it is not reliably established by the existing data , which are consistent with no correlation at all . semi - analytic models for the formation of elliptical galaxies , which previously neglected metallicity effects ( see kauffmann 1996 , baugh 1996 ) , are only now beginning to incorporate chemical enrichment and successfully reproduce the general form of the observed colour magnitude and relations ( kauffmann & charlot 1998 ) . in consequence , there are as yet no reliable predictions for the variation of the relation zeropoint with cluster mass . the limits given above , together with limits on the difference in zeropoints for field and cluster ellipticals ( burstein 1990 , de carvalho & djorgovski 1992 , jrgensen 1997 ) , should provide valuable additional constraints and encourage further development of chemical enrichment models within a hierarchical framework for galaxy and cluster formation . we now consider the effects on fp distance estimates of systematic differences in the stellar populations of early - type galaxies from cluster to cluster . in [ ssec : clusvars ] we found that the observed cluster - to - cluster variations in the zeropoint were consistent with sampling a single global relation with intrinsic scatter between galaxies , and did not _ require _ intrinsic scatter between clusters . here we turn the question around and ask how much intrinsic cluster - to - cluster scatter is _ allowed _ by the observations . from simulations using the model described in [ ssec : clusvars ] , incorporating intrinsic scatter both between galaxies and between clusters , we find that the maximum cluster - to - cluster scatter allowed within the 1@xmath11 uncertainties in the scatter in the global relation and the cluster zeropoints is approximately 0.005 mag in and 0.010 mag in . for our best - fit ml relations and a fp given by @xmath39@xmath43@xmath44 with @[email protected] , this level of cluster - to - cluster scatter would lead to rms errors in fp distance estimates of up to 10% . these systematic errors , resulting from differences in the mean stellar populations between clusters , would apply even to clusters in which the fp distance errors due to stellar population differences between galaxies had been made negligible by observing many galaxies in the cluster . we emphasise that our results here do not _ require _ any cluster - to - cluster scatter , but are _ consistent _ with cluster - to - cluster scatter corresponding to systematic distance errors between clusters with an rms of up to 10% . we therefore can not determine from the relation _ alone _ whether systematic differences in the mean stellar populations between clusters contribute significantly ( or at all ) to the errors in fp estimates of distances and peculiar velocities . a more effective way of testing for such systematic differences is by directly comparing each cluster s zeropoint offset from the global relation to the ratio of its fp and hubble distance estimates ; this approach will be investigated in a future paper . to answer the questions concerning the typical age , metallicity and mass - to - light ratio of early - type galaxies which were raised at the start of this discussion , we need to employ stellar population models . we use the predictions from the single stellar population models of worthey ( 1994 ) and vazdekis ( 1996 ) , noting the many caveats given by these authors regarding their models . to simplify our analysis , we fit , and @xmath46 as linear functions of logarithmic age ( @xmath47 , with @xmath48 in gyr ) and metallicity ( @xmath49 ) , for galaxies with ages greater than 4 gyr and metallicities in the range @xmath80.5 to @xmath500.5 . for the model of worthey ( 1994 ; salpeter imf ) we obtain @xmath51 figure [ fig : mgbmlrw ] compares this fit to worthey s model in the case of the predicted dependence of and @xmath46 on age and metallicity . the figure shows that for ages of 5 gyr or greater the fit and the model are in satisfactory agreement for all metallicities . = for the model of vazdekis ( 1996 ; bimodal imf , @xmath52=1.35 ) we have @xmath53 in agreement with the fit obtained by jrgensen ( 1997 ) . there is good agreement between the predictions of the two models for the dependence of and on age and metallicity , and moderately good agreement for the dependence of @xmath54 . note that the same change in the mg indices is produced by changes in age , @xmath55 , and metallicity , @xmath56 , if @xmath57 . this is the ` 3/2 rule ' of worthey ( 1994 ) , which applies to many of the lick line indices , leaving them degenerate with respect to variations in age and metallicity . however age and metallicity produce the same change in @xmath46 only if @xmath58 or 1/4 , so that measurements of mass - to - light ratios can in principle be combined with mg linestrengths to break the age / metallicity degeneracy . in the following analysis we infer the dispersion in the ages and metallicities of early - type galaxies by comparing the scatter in the relation with the predictions of the single stellar population models described in the previous section . this analysis uses the stellar population models to predict differential changes in the quantities of interest , and not absolute values . it is also important to remember that by the dispersion in age or metallicity we mean the dispersion in these quantities at fixed @xmath26 or , equivalently , the dispersion after the overall trend with @xmath26 is accounted for . thus the dispersion in age or metallicity we infer is the dispersion at fixed galaxy mass , not the distribution of ages and metallicities as a function of galaxy mass ( which is related to the slope of the relation and the distribution of galaxies along it ) . single stellar populations models specified by ( amongst other parameters ) a unique age and a unique metallicity can only provide an approximation to real galaxies , whose stellar contents must necessarily span a range ( though perhaps a narrow one ) of ages and metallicities . since the global mg indices can be quite sensitive to the detailed metallicity distribution ( greggio 1997 ) , some of the scatter we observe may be due to galaxy - to - galaxy differences in the shape of the metallicity distribution rather than a dispersion in the mean metallicity or age . a further complication is presented by the over - abundance of mg with respect to fe ( compared to the solar ratio ) in the cores of early - type galaxies ( peletier 1989 , gorgas 1990 , worthey 1992 ) . as a comparison of figures [ fig : mgsig ] & [ fig : mgbmlrw ] shows , the models discussed in the previous section are unable to account for the highest observed mg linestrengths . tantalo ( 1998 ) have produced single stellar population models including the effects of [ mg / fe ] variations and find that @xmath59 + 0.089 \delta\log t + 0.166 \delta\log z / z_\odot \label{eqn : tantalo}\ ] ] comparing this equation with those above , we see that the differential dependence on age and metallicity is similar to that predicted by worthey ( 1994 ) and vazdekis ( 1996 ) . however , any intrinsic scatter in the [ mg / fe]@xmath11 relation will contribute additionally to the intrinsic scatter in the relation and reduce the dispersion in age and metallicity required to account for the observations . for these reasons , and also because of other potential sources of intrinsic scatter such as dark matter , rotation , anisotropy , projection effects and broken homology , the estimates of the dispersion in age and metallicity derived here must be considered as upper limits . with these caveats in mind , we proceed to use the model fits given in the previous section to infer the dispersion in age or metallicity based on the observed intrinsic scatter of 0.016 mag in and 0.023 mag in . for ease of interpretation we quote the dispersions in age and metallicity as the fractional dispersions @xmath60 and @xmath61 . in applying the models in what follows , we adopt the mean of the coefficients for the two models and give the dispersions in age and metallicity corresponding to the intrinsic scatter about the relation . using the intrinsic scatter obtained from the relation would give results that are @xmath1030% smaller , since the observed ratio of the intrinsic scatters is @[email protected] , rather than about 2 as would be expected either from the observed relation or from the models . we use the scatter in rather than because our goal is to establish upper limits on the dispersions in age and metallicity . the estimated errors in the intrinsic scatter lead to uncertainties in the dispersions of 510% . if age variations in single stellar populations are the only source of scatter then the dispersion in age is @xmath3=67% , whereas if metallicity variations are the sole source then the dispersion in metallicity is @xmath4=43% . similarly , the observed difference in the relation zeropoint for the cd galaxies implies that these objects are either older or more metal - rich than normal e or e / s0 galaxies . if the zeropoint differences are interpreted as age differences , cds are on average 40% older than typical e or e / s0 galaxies ( as old as the oldest early - type galaxies ) ; if the zeropoint differences are interpreted as metallicity differences , cds have metallicities on average 25% higher than typical e or e / s0 galaxies ( as high as the most metal - rich early - type galaxies ) . we can also use the model fits to estimate the approximate change in @xmath54 corresponding to a change in the mg line indices . if these changes are caused by age variations alone , then we find that @xmath63 and @xmath64 ; if , however , they are due only to variations in metallicity we have @xmath65 and @xmath66 . thus the change in @xmath67 is about 5 times larger if the observed change in the mg indices is due to age differences rather than metallicity differences . the intrinsic scatter in the relation implies a dispersion in mass - to - light ratio of 50% if due to age variations , but only 10% if due to metallicity variations . this predicted scatter in @xmath2 is in fact a scatter in luminosity or surface brightness ( since that is all the models deal with ) . we can therefore readily establish the effect of this scatter on distances estimated using the fundamental plane ( fp ) if the scatter in @xmath2 is uncorrelated with the galaxies sizes and dispersions , as indeed is the case for the efar sample ( at least for galaxies with @xmath11@xmath68100 ) . for a fp given by @xmath39@xmath43@xmath69 , with @xmath39 the effective radius and @xmath70 the mean surface brightness within this radius , if the scatter in @xmath2 is simply a scatter in @xmath70 we have @xmath71 . most determinations of the fp , including our own , yield @xmath72 ( dressler 1987 , jrgensen 1996 , saglia 1998 ) . combining this relation with the dependence of @xmath2 on the mg line indices obtained above , we find that the scatter in the relation corresponds to an intrinsic scatter in relative distances estimated from the fp of 40% if due to age variations , or 8% if due to metallicity variations . as the intrinsic scatter in the fp is found to be in the range 1020% ( djorgovski & davis 1987 , jrgensen 1993 , jrgensen 1996 ) , one can not explain both the scatter in the relation and the scatter in the fp as the result of age variations alone or metallicity variations alone ( unless the single stellar population models are incorrect or there are significant galaxy - to - galaxy differences in the metallicity distributions ) . suitable combinations of age variations and metallicity variations _ can _ , however , account for the measured intrinsic scatter in both the and fp relations . as a simple model , we assume that the scatter in the fp and the relations ( at fixed @xmath26 ) is entirely due to variations in age and metallicity ( at fixed galaxy mass ) . these variations are further assumed to have gaussian distributions in @xmath47 and @xmath49 with dispersions @xmath73 and @xmath74 and correlation coefficient @xmath7 ( @xmath81@xmath75@xmath7@xmath751 ) . while a gaussian distribution of metallicities at fixed galaxy mass is a reasonable initial hypothesis for describing variations in the chemical enrichment process , the single - peaked shape of the assumed lognormal distribution for the mean ages may not realistically represent the star - formation history ( even for galaxies of the same mass ) . the dispersion in age inferred under this model should therefore be considered only as a general indication of the time - span over which early - type galaxies of fixed mass formed the bulk of their stellar population . writing the scatter in mg linestrengths and fp residuals as @xmath76 and @xmath77 and the dispersion in @xmath47 and @xmath78 as @xmath79 and @xmath80 , this simple model relates the scatter in the observed quantities to the dispersion in age and metallicity by : @xmath81 here @xmath82 and @xmath83 are the coefficients of @xmath47 and @xmath78 for mg , and @xmath84 and @xmath85 the coefficients for @xmath86 , derived from the mean of the linear fits to the two stellar population models given in [ ssec : models ] . = = figure [ fig : scatter ] shows the constraints on the variations in age and metallicity ( assumed for now to be uncorrelated ) which are imposed by the measured intrinsic scatter in the relations and the intrinsic dispersion in @xmath46 inferred from the intrinsic scatter in the fp . the intrinsic scatter we find about the and relations is then consistent with dispersions in age and metallicity on an elliptical locus defined by equation [ eqn : modelmg ] ( with @xmath7=0 ) in the @xmath3@xmath4 plane . the different loci for and ( the solid lines in figure [ fig : scatter ] ) result from the difference between the observed ratio of the scatter in to that in and the predicted ratio from the model , and give some indication of uncertainties both in the intrinsic scatter about the relations and in the model predictions . a second constraint is similarly obtained from the intrinsic scatter in distance ( in @xmath87 ) about the fp using equation [ eqn : modelfp ] ( again with @xmath7=0 ) . the dashed lines in figure [ fig : scatter ] correspond to intrinsic scatter about the fp of 10% , 15% and 20% . the important point to note about the figure is that , as mentioned in [ ssec : models ] , the dependences of the mg linestrengths and mass - to - light ratio on age and metallicity are quite different , so that ( if variations in age and metallicity are uncorrelated ) the two sets of constraints are nearly orthogonal . thus the region of the @xmath5@xmath4 plane that is consistent with the scatter in both the relation and the fp is quite limited . if we use the intrinsic scatter in the relation and assume a 20% intrinsic scatter in @xmath87 about the fp ( at the upper end of the quoted range see , , djorgovski & davis ( 1987 ) or jrgensen ( 1996 ) ) , we obtain approximate upper limits on the dispersions in age and metallicity of @xmath3=32% and @xmath4=38% . if , however , we use the intrinsic scatter in the relation and adopt an intrinsic fp scatter of 10% ( as obtained for coma by jrgensen 1993 ) , then we obtain approximate lower limits of @xmath3=15% and @xmath88=27% . similar arguments allow us to evaluate the relative contributions of the dispersions in age and metallicity to the errors in distance estimates derived from the fp . for the fiducial case ( @xmath30fp=20% , @xmath30=0.016 mag and @xmath7=0 ) , where @xmath3=32% and @xmath4=38% , the mean stellar population model implies that the dispersion in age gives an intrinsic fp scatter of 19% while the dispersion in metallicity gives 7% . in fact for most of the plausible range of dispersions in age and metallicity shown in figure [ fig : scatter ] , it is the dispersion in age which dominates the intrinsic scatter about the fp . only for the lowest plausible age dispersion and the highest plausible metallicity dispersion ( @xmath5=11% and @xmath4=43% , corresponding to @xmath30fp=10% and @xmath30=0.016 mag ) does the contribution to the fp scatter from the dispersion in metallicity achieve equality with the contribution from the dispersion in age . = the constraints on the dispersions change if there is a significant correlation ( or anti - correlation ) between the variations in age and metallicity . figure [ fig : scatter_rho ] shows how the constraints corresponding to the upper limits @xmath30fp=20% and @xmath30=0.016 mag ( corresponding to the thick lines in figure [ fig : scatter ] ) are modified as the correlation coefficient @xmath7 varies over its full range from @xmath81 to 1 . note that for @xmath89 we have @xmath90 . the main point to extract from this figure is that if the variations in age and metallicity have a correlation coefficient in the range @xmath80.5@xmath91@xmath7@xmath911 , then the dispersions in age and metallicity vary by only @xmath296% and @xmath2912% respectively about the values inferred in the uncorrelated case . only if the age and metallicity variations are strongly anti - correlated ( @xmath7@xmath6@xmath81 ; younger galaxies are more metal - rich ) do we obtain significantly different solutions , with a broader allowed range in both age and metallicity ( @xmath3 as large as 57% and @xmath4 as large as 80% ) . this conclusion is complementary to that reached by ferreras ( 1998 ) , who find that the apparently passive evolution of the colour magnitude relation observed in high - redshift clusters does not necessarily imply a common epoch of major star - formation if younger galaxies are on average more metal - rich . we can test the degree of correlation between the variations in age and metallicity by examining the joint distribution of residuals about the and fp relations . this distribution is shown for the efar data set in figure [ fig : dmgdfp]a . there is no evidence for a correlation between the residuals in this figure ; the spearman rank correlation coefficient between the residuals is 0.084 , and is not significant at the 2@xmath11 level . in order to investigate the expected distribution of residuals in the presence of the estimated measurement errors , we have performed monte carlo simulations of the efar data using the models for the dispersion in age and metallicity discussed above . figure [ fig : dmgdfp]b shows a simulation with @xmath3=32% and @xmath4=38% ; these are the values derived from the intrinsic scatter in the relation and a fp scatter of 20% when there is no correlation between age and metallicity . the simulated distribution resembles the observed distribution , although there is a weak but significant anti - correlation between the residuals ( due to the dominance of the age variations in the fp residuals ) which is not apparent in the efar data . over 100 such simulations , a two - dimensional k - s test ( press 1992 ) gives a median probability of 0.3% that this distribution and the observed distribution are the same . figures [ fig : dmgdfp]c&d show simulated distributions for the cases where the intrinsic scatter in is due to age alone or metallicity alone . neither case is consistent with the observed distribution , supporting the claim that neither age nor metallicity can be solely responsible for the scatter in both the relation and the fp . figures [ fig : dmgdfp]e h show simulated distributions for four cases where the variations in age and metallicity are correlated ( with @xmath7=+1 , + 0.5 , @xmath80.5 and @xmath81 respectively ) . the perfectly correlated and perfectly anti - correlated cases are not consistent with the observed distribution . however figure [ fig : dmgdfp]g shows that a distribution with no significant correlation between the and fp relation residuals is produced when @[email protected] . a two - dimensional k - s test gives a median probability over 100 such simulations of 1.7% that this distribution and the observed distribution are the same . this relatively low probability may reflect a problem with the model , although it may simply be due to sampling uncertainty ( the probabilities under this test vary between simulations with an rms of a factor of 6 ) or non - gaussian outliers in the efar residuals . the point to be emphasised is that a model with a moderate degree of anti - correlation between age and metallicity appears to give significantly better agreement with the observed distribution than a model in which age and metallicity are uncorrelated . we have examined the relation for early - type galaxies in the efar sample . we fit global and relations ( equations [ eqn : mlmgbp ] and [ eqn : mlmgtwo ] ) that have slopes about 25% steeper than those obtained by most previous authors . this difference results not from the data itself but from an improved fitting procedure : we apply a comprehensive maximum likelihood approach which correctly accounts for the biases introduced by both the sample selection function and the significant errors in both mg and @xmath11 . the _ observed _ scatter about the relations is 0.022 mag in and 0.031 mag in ; the _ intrinsic _ scatter in the global relations , estimated from monte carlo simulations , is 0.016 mag in and 0.023 mag in . with too few galaxies per cluster to reliably determine the full relation for each cluster separately , we fix the slopes of the relations at their global values in order to investigate the variation in the zeropoint from cluster to cluster . we find that the zeropoint has an observed scatter between clusters of 0.012 mag in and 0.019 mag in , and that this observed scatter is consistent with the small number of galaxies sampled in each cluster being drawn from a single global relation with intrinsic scatter between galaxies as given above the observations do not _ require _ any scatter in the zeropoint between clusters . the _ allowed _ range for the intrinsic scatter between clusters corresponds to cluster - to - cluster systematic errors in fundamental plane distances and peculiar velocities with an rms anywhere in the range 010% . we therefore can not determine from the relation _ alone _ whether systematic differences in the mean stellar populations between clusters contribute significantly ( or at all ) to the errors in distances and peculiar velocities obtained using the fundamental plane . we have also examined the variation in the relation with cluster properties . our cluster sample ranges from poor clusters to clusters as rich as coma , having velocity dispersions from 300 to 1000 and x - ray luminosities spanning 0.38@xmath9210@xmath93ergs@xmath16 . we do not detect a significant correlation of zeropoint with cluster velocity dispersion , x - ray luminosity or x - ray temperature , nor is there any significant difference in the relations obtained by fitting the galaxies in the high-@xmath37 clusters and low-@xmath37 clusters separately . the predominant factor in the production of mg in these early - type galaxies ( and presumably other @xmath45-elements and perhaps their metallicity and star - formation history in general ) is thus _ galaxy _ mass and not _ cluster _ these observations place constraints on semi - analytic models for the formation of elliptical galaxies , which are now beginning to incorporate chemical enrichment and should soon be able to make reliable predictions for the variation of the relation with cluster mass . we apply the single stellar population models of worthey ( 1994 ) and vazdekis ( 1996 ) to place upper limits on the global dispersion in the ages , metallicities and @xmath2 ratios of early - type galaxies of given mass using the intrinsic scatter in the global relation . we infer an upper limit on the dispersion in @xmath54 of 50% if the scatter in is due to age differences alone , or 10% if it is due to metallicity differences alone . these correspond to upper limits on the dispersion in relative galaxy distances estimated from the fundamental plane ( fp ) of 40% ( age alone ) or 8% ( metallicity alone ) . since the intrinsic scatter in the fp is found to be 1020% , one can not ( within the context of the single stellar population models ) explain both the scatter in the relation and the scatter in the fp as the result of age variations alone or metallicity variations alone . we therefore determine the joint range of dispersions in age and metallicity which are consistent with the measured intrinsic scatter in both the and fp relations . for a simple model in which the galaxies have independent gaussian distributions in @xmath47 and @xmath78 , we find upper limits of @xmath3=32% and @xmath88=38% at fixed galaxy mass . if the variations in age and metallicity are not independent , but have correlation coefficient @xmath7 , we find that so long as @xmath7 is in the range @xmath80.5 to 1 these limits on the dispersions in age and metallicity change by only @xmath296% and @xmath2912% respectively . only if the age and metallicity variations are strongly anti - correlated ( @xmath7@xmath6@xmath81 ) do we obtain significantly higher upper limits , with @xmath3 as large as 57% and @xmath4 as large as 80% . the distribution of the residuals from the and fp relations is only marginally consistent with a model having no correlation between age and metallicity , and is better - matched by a model in which age and metallicity variations are moderately anti - correlated ( @xmath3@xmath640% , @xmath4@xmath650% and @xmath7@[email protected] ) , with younger galaxies being more metal - rich . stronger bounds on the dispersion in age and metallicity amongst early - type galaxies of given mass will require more precise measurements of the deviations from the relation and the fundamental plane and also improved models for the dependence of the line indices and mass - to - light ratio on age and metallicity . further powerful constraints can also be obtained by measuring the intrinsic scatter in the and fp relations at higher redshifts , since the linestrengths and mass - to - light ratio have different dependences on age . mmc acknowledges the support of a dist collaborative research grant . db was partially supported by nsf grant ast90 - 16930 . rld thanks the lorenz centre and prof . de zeeuw . rkm received partial support from nsf grant ast90 - 20864 . rps acknowledges the financial support by the deutsche forschungsgemeinschaft under sfb 375 . gw is grateful to the serc and wadham college for a year s stay in oxford , to the alexander von humboldt - stiftung for making possible a visit to the ruhr - universitt in bochum and to nsf grants ast90 - 17048 and ast93 - 47714 for partial support . the entire collaboration benefitted from nato collaborative research grant 900159 and from the hospitality and monetary support of dartmouth college , oxford university , the university of durham and arizona state university . support was also received from pparc visitors grants to oxford and durham universities and pparc rolling grant ` extragalactic astronomy and cosmology in durham 1994 - 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we have examined the relation for early - type galaxies in the efar sample and its dependence on cluster properties . a comprehensive maximum likelihood treatment of the sample selection and measurement errors gives fits to the global relation of @xmath0 and @xmath1 . the slope of these relations is 25% steeper than that obtained by most other authors due to the reduced bias of our fitting method . the intrinsic scatter in the global relation is estimated to be 0.016 mag in and 0.023 mag in . the relation for cd galaxies has a higher zeropoint than for e and s0 galaxies , implying that cds are older and/or more metal - rich than other early - type galaxies with the same velocity dispersion . we investigate the variation in the zeropoint of the relation between clusters . we find it is consistent with the number of galaxies observed per cluster and the intrinsic scatter between galaxies in the global relation . we find no significant correlation between the zeropoint and the cluster velocity dispersion , x - ray luminosity or x - ray temperature over a wide range in cluster mass . these results provide constraints for models of the formation of elliptical galaxies . however the relation on its own does not place strong limits on systematic errors in fundamental plane distance estimates due to stellar population differences between clusters . we compare the intrinsic scatter in the and fundamental plane ( fp ) relations with stellar population models in order to constrain the dispersion in ages , metallicities and @xmath2 ratios for early - type galaxies at fixed velocity dispersion . we find that variations in age alone or metallicity alone can not explain the measured intrinsic scatter in both and the fp . we derive the joint constraints on the dispersion in age and metallicity implied by the scatter in the and fp relations for a simple gaussian model . we find upper limits on the dispersions in age and metallicity at fixed velocity dispersion of 32% in @xmath3 and 38% in @xmath4 if the variations in age and metallicity are uncorrelated ; only strongly anti - correlated variations lead to significantly higher upper limits . the joint distribution of residuals from the and fp relations is only marginally consistent with a model having no correlation between age and metallicity , and is better - matched by a model in which age and metallicity variations are moderately anti - correlated ( @xmath5@xmath640% , @xmath4@xmath650% and @xmath7@[email protected] ) , with younger galaxies being more metal - rich . galaxies : distances and redshifts galaxies : elliptical and lenticular , cd galaxies : stellar content galaxies : formation galaxies : evolution

You are an expert at summarizing long articles. Proceed to summarize the following text: the following is a special case of theorem 8.5a of ivanov s survey article @xcite . it is a consequence of ideas laid out in the papers of ivanov on the mapping class groups of higher genus orientable surfaces and the work of korkmaz @xcite on the complex of curves associated to an @xmath69-punctured sphere . for simplicity we shall write @xmath72 for the 2-sphere with @xmath69 points removed . if @xmath73 is an orientable surface ( without boundary ) , we denote by @xmath74 the group of all diffeomorphisms of @xmath73 ( not necessarily respecting orientation ) modulo diffeotopy . [ ivanovkthm ] let @xmath75 . if @xmath76 is an isomorphism between finite index subgroups @xmath77 , then @xmath78 is the restriction to @xmath79 of an inner automorphism of @xmath80 ( conjugation by some element @xmath81 ) . [ defncommg ] let @xmath82 be a group . we define the _ abstract commensurator group of @xmath82 _ to be @xmath83 where @xmath84 if they agree on a finite index subgroup of @xmath82 . we note that the group structure on @xmath85 is given by composition of isomorphisms after appropriate restriction of their domains to finite index subgroups . [ normalizers ] let @xmath86 , @xmath75 , let @xmath87 denote a finite index subgroup and @xmath88 the normalizer of @xmath79 in @xmath89 . then * @xmath90 , and * @xmath91 . \(i ) there is a natural homomorphism @xmath92 which takes @xmath93 to conjugation by @xmath94 restricted to the finite index subgroup @xmath95 . it follows from theorem [ ivanovkthm ] that this map is surjective . to show that it is injective , we observe that @xmath89 has trivial `` virtual centre '' , that is , any element which centralizes a finite index subgroup is necessarily trivial . this is because any mapping class @xmath96 is determined by its action on a finite number of isotopy classes of simple closed curves on the surface @xmath73 , and therefore by its action on the @xmath6-th powers @xmath97 of the corresponding dehn twists for any choice of @xmath6 . given a finite index subgroup @xmath98 centralized by @xmath99 we can always find a sufficiently large @xmath6 such that @xmath100 contains all @xmath97 . then @xmath99 is determined by its action on @xmath100 and so must be the identity in @xmath89 . \(ii ) there is now an obvious homomorphism @xmath101 which factors through the inverse of the isomorphism just described , and realizes each automorphism of @xmath79 as conjugation by an element of @xmath89 . it follows easily that this map is injective with image precisely the normalizer @xmath88 of @xmath79 in @xmath89 . let @xmath102 denote the group of permutations of the set @xmath103 , for @xmath104 . if @xmath105 then we consider @xmath106 as a subgroup of @xmath102 consisting of those permutations which fix the subset @xmath107 . notice that each mapping class of the @xmath69-punctured sphere @xmath72 induces a permutation of the punctures . moreover , the mapping class group acts on the orientation class of the surface . thus we have a surjective group homomorphism @xmath108 with kernel the group of pure orientation preserving mapping classes . note that , strictly speaking , the definition of @xmath109 depends on a labelling of the punctures by the numbers @xmath110 through @xmath69 . * notation . * fix @xmath0 . we shall write @xmath111 and @xmath112 , @xmath113 , @xmath114 respectively , for the finite index subgroup of @xmath89 generated by those orientation preserving diffeomorphisms which fix the last 1 , 2 , 3 punctures of @xmath47 respectively . these three subgroups are , in other words , the preimages under @xmath109 of the subgroups @xmath115 , with @xmath116 respectively . in particular we have @xmath117=2(n+2)$ ] , @xmath118=2(n+1)(n+2)$ ] and @xmath119=2n(n+1)(n+2)$ ] . it is well known that the @xmath120-string braid group , @xmath36 , is isomorphic to the mapping class group of the @xmath120-punctured disk _ relative _ to the boundary of the disk . ( that is , diffeotopies are required in this case to fix the boundary of the disk _ pointwise _ ) . the centre @xmath41 of this mapping class group is generated by the dehn twist about a curve parallel to the boundary , and is precisely the kernel of the natural homomorphism to the mapping class group @xmath111 ( induced by inclusion of the punctured disk in @xmath47 ) . thus we may realise the group @xmath44 as the finite index subgroup @xmath112 of @xmath89 . it is less well - known ( see @xcite ) that the artin groups @xmath66 and @xmath43 , respectively , are isomorphic to the subgroups of the braid group @xmath36 leaving fixed one , respectively two , of the punctures in the disk . we note that the latter of these two groups intersects the centre of the braid group trivially , from which we deduce that * @xmath121 , * @xmath122 , and * @xmath123 . these isomorphisms may be described explicitly as follows . distribute the punctures @xmath124 ( @xmath125 ) evenly along half the equator in the order @xmath126 so that @xmath127 and @xmath128 are antipodal ( see figure [ fig2 ] ) . then the @xmath24th standard generator of @xmath36 maps to the positive braid twist @xmath129 exchanging @xmath124 with @xmath130 . the @xmath24th standard generator of @xmath66 maps to @xmath131 if @xmath132 and @xmath129 otherwise . the @xmath24th standard generator of @xmath43 maps to @xmath133 if @xmath134 and @xmath129 if @xmath135 . ( this maps @xmath43 into the subgroup of @xmath89 fixing @xmath136 , and @xmath137 . renumbering the punctures gives @xmath114 . ) isomorphic to @xmath44 , @xmath45 and @xmath43 , @xmath0 , fix the subsets @xmath138 and @xmath139 respectively.,width=302 ] [ abc ] ( i ) : : @xmath140 ( ii ) : : there exist short exact sequences @xmath141 let @xmath79 denote one of @xmath112 , @xmath113 , or @xmath114 , and @xmath142 the set of 1,2 or 3 punctures fixed by all elements of @xmath79 . it is easily checked that @xmath88 is exactly the group of all mapping classes ( orientation preserving or not ) which leave @xmath142 setwise invariant . it follows that @xmath143 is a direct product of @xmath144 with the group @xmath145 of permutations of the set @xmath142 . statements ( i ) and ( ii ) now follow immediately from the corresponding statements in corollary [ normalizers ] . it is easy to see , with the aid of figure [ fig2 ] , that the short exact sequences in the proposition above are split in the cases @xmath112 and @xmath113 . the group @xmath146 may be generated by a reflection in the equatorial plane and a rotation about an axis in the equatorial plane which exchanges the points @xmath127 and @xmath128 . only the reflection leaves @xmath112 invariant . for @xmath114 , the exact sequence splits providing @xmath147 mod @xmath28 . to see this , think of @xmath47 as a sphere with @xmath148 punctures arranged symmetrically along the equator and , if @xmath149 , the remaining two punctures at the north and south poles . then there is an orientation - preserving action of @xmath150 on @xmath47 generated by an order three rotation @xmath151 about the vertical axis through the poles and an order two rotation @xmath152 about a horizontal axis through one ( @xmath153 odd ) or two ( @xmath153 even ) punctures at the equator . if @xmath154 is a puncture fixed by @xmath152 , then the orbit of @xmath154 under @xmath151 is preserved by @xmath150 , hence for an appropriate numbering of the punctures ( namely , so that @xmath155 ) the action of @xmath150 lies in @xmath156 and is faithful on @xmath142 . this splits the @xmath150 factor . the @xmath63 factor is realized by reflection through the equatorial plane . [ nosplit ] if @xmath157 with @xmath158 mod @xmath28 , then @xmath156 does not contain a subgroup isomorphic to @xmath150 . in this case , the exact sequence for @xmath114 in the previous proposition does not split . suppose @xmath159 contains a subgroup @xmath100 isomorphic to @xmath150 and let @xmath79 be the subgroup of @xmath100 generated by a 3-cycle . then @xmath79 is a subgroup of @xmath46 , hence @xmath79 acts by permutations on the set of punctures @xmath160 . every @xmath79 orbit consists of either @xmath110 or @xmath28 punctures . since @xmath161 , the number of fixed points of @xmath79 must also be congruent to @xmath110 mod @xmath28 . if @xmath79 fixes @xmath162 or more punctures , then it lies in a subgroup of @xmath89 isomorphic to @xmath114 which is impossible since @xmath114 is torsion - free ( since , by @xcite , it is a subgroup of the braid group on @xmath37 strings which is known to be torsion free ) . thus @xmath79 must have a unique fixed puncture . since @xmath79 is normal in @xmath100 , all of @xmath100 must fix this puncture and hence @xmath100 lies in a subgroup of @xmath89 isomorphic to @xmath112 . by a theorem of bestvina , @xcite thm 4.5 , every finite subgroup of @xmath112 is cyclic , so we arrive at a contradiction . it has been observed by several authors @xcite@xcite@xcite that there exists a semidirect product decomposition @xmath163 ( see @xcite for further discussion of this decomposition . ) the coxeter graph of @xmath42 is an @xmath6-cycle and the generator of the cyclic factor in the semi - direct product acts on @xmath42 via an order @xmath6 rotation of this graph . the centre of @xmath66 is the subgroup @xmath164 of the cyclic factor . thus we also have @xmath165 here we interpret @xmath42 as the subgroup of @xmath113 consisting of mapping classes of `` zero angular momentum '' about the axis through the two fixed punctures . more precisely , suppose that the punctures of @xmath47 are arranged so that @xmath127 and @xmath128 are placed at the north and south pole respectively , and the remaining points @xmath166 are equally spaced ( in that cyclic order ) around the equator , see figure [ fig3 ] . define @xmath167 to be the subgroup of @xmath89 generated by the braid twists @xmath129 which exchange the points @xmath124 and @xmath130 , for @xmath60 , indices taken mod @xmath6 . the isomorphism @xmath168 is given by sending the @xmath24th standard generator of @xmath42 to @xmath129 . note that @xmath167 is a finite index subgroup of @xmath89 of index @xmath169=n[\ga:\ga_b]=2n(n+1)(n+2)$ ] . in particular , @xmath170 . of @xmath111 isomorphic to @xmath42.,width=302 ] [ atilde ] let @xmath171 be as above . then @xmath172 . consequently @xmath173 and there is an exact sequence @xmath174 where @xmath58 denotes the dihedral group of order @xmath175 . moreover , a splitting of this sequence is given by realizing the group of outer automorphisms as the direct product of the group of graph automorphisms of @xmath42 with the group of order @xmath176 generated by the inversion @xmath177 which inverts each standard generator . we suppose that the punctures of @xmath47 are arranged as shown in figure [ fig3 ] . the outer automorphism group of @xmath113 may then be generated by the reflection in the equatorial plane and a rotation exchanging the two poles ( this gives a splitting of the exact sequence of proposition [ abc ] slightly different to the one previously mentioned ) . it follows easily that @xmath167 is invariant by these mapping classes , and hence a characteristic subgroup of @xmath113 . in particular , we have @xmath178 . let @xmath179 . to prove that @xmath172 it suffices now to check that every mapping class which normalizes @xmath167 leaves @xmath142 invariant . but if @xmath180 then @xmath181 normalizes @xmath182 , and this is just to say that @xmath99 induces a symmetry of the puncture set which leaves @xmath142 invariant , as required . the remaining statements are now easily deduced from corollary [ normalizers ] and proposition [ abc ] . in summary , propositions [ abc ] and [ atilde ] give us explicitly the automorphism groups of @xmath44 , @xmath45 , @xmath42 , and @xmath43 , for all @xmath0 , as well as the fact that the abstract commensurator group is in each case isomorphic to @xmath46 . this proves theorem [ main ] with the exception of the first isomorphism in each of part ( ii)(a ) and ( b ) . in the next section we shall recover the group @xmath183 from the above result on @xmath184 . in a similar fashion , the result of @xcite on automorphisms of the braid group is easily recovered from the computation of @xmath185 . these steps will complete the proof of theorem [ main ] . let @xmath79 be a group with nontrivial centre @xmath41 . by a _ transvection _ of @xmath79 we mean a homomorphism @xmath186 given by @xmath187 , where @xmath188 denotes a function @xmath189 . the fact that @xmath190 is a homomorphism requires that @xmath188 is also a homomorphism , and it is easily checked that the composite of two transvections is again a transvection . we note , however , that transvections are typically not automorphisms . it is easily shown that a transvection is an automorphism ( resp . injective , resp . surjective ) if and only if its restriction to the centre @xmath41 is an automorphism ( resp . injective , resp . surjective ) . let @xmath191 denote the group of automorphisms of @xmath79 which are transvections . then , almost by definition , we have an exact sequence @xmath192 note that @xmath193 always maps isomorphically onto @xmath194 . in general there is an issue , however , as to whether _ every _ outer automorphism of @xmath195 lifts to an automorphism of @xmath79 , and as to whether the extension splits . in the case of an irreducible finite type artin group @xmath32 , the extension splits at least over a finite index subgroup @xmath196 of @xmath197 defined as follows . let @xmath198 be the length homomorphism which maps each generator of @xmath32 to @xmath110 . recall that the centre @xmath41 of @xmath32 is infinite cyclic . let @xmath199 be a generator of @xmath41 and let @xmath200 . the length homomorphism on @xmath32 descends to a `` length homomorphism '' @xmath201 . define @xmath202 , respectively @xmath196 , to be the group of length preserving or reversing automorphisms , that is , automorphisms @xmath203 such that @xmath204 ( respectively @xmath205 ) . [ autstar ] suppose @xmath32 is an irreducible artin group of finite type . then the natural map @xmath206 is an isomorphism . if , in addition , the abelianization of @xmath32 is infinite cyclic ( i.e. , @xmath32 is not of type @xmath207 , or @xmath208 ) , then @xmath209 is trivial and @xmath210 . note that a non - trivial transvection @xmath211 of @xmath32 can never be length preserving , and can only be length reversing if @xmath212 ( and @xmath213 ) . since @xmath214 for all irreducible finite type artin groups we conclude that @xmath215 is trivial . for any length preserving @xmath216 , we define a lift @xmath217 as follows . ( the lift for length reversing automorphisms is defined analogously ) . by definition , for any @xmath218 , @xmath219 for some @xmath220 with @xmath221 mod @xmath222 . since @xmath41 is generated by an element of length @xmath222 , the coset @xmath223 contains a unique representative @xmath224 satisfying @xmath225 . define @xmath226 . to verify that @xmath227 is a homomorphism , note that if @xmath228 and @xmath229 , then @xmath230 so it follows that @xmath231 . moreover , it is straightforward to verify that if @xmath232 and @xmath233 are two elements of @xmath196 , then @xmath234 lifts to @xmath235 . in particular , the lift of an automorphism is also an automorphism . thus , @xmath236 defines a section for the projection @xmath109 , and @xmath109 is an isomorphism . if the abelianization of @xmath32 is infinite cyclic , then the length homomorphism @xmath237 can be identified with the abelianization map @xmath238 . hence every automorphism of @xmath32 is either length preserving or length reversing . the second statement of the proposition follows . we remark , however , that while @xmath242 when @xmath32 has infinte cyclic abelianization , it is not always the case that @xmath243 ( or equivalently that @xmath244 ) . in particular , for @xmath32 of type @xmath68 with @xmath69 odd , @xmath245 , and @xmath246 is isomorphic to the group of units of @xmath247 , while @xmath248 . [ trans ] if @xmath32 is an irreducible artin group of type @xmath249 , @xmath0 , or @xmath208 , then @xmath250 . if @xmath32 is of type @xmath251 then @xmath252 . moreover , in all these cases @xmath253 contains @xmath254 as a finite index subgroup . we refer the reader to @xcite and @xcite for descriptions of the coxeter graphs , and central elements respectively , for each of the irreducible finite type artin groups . in each case listed above , the abelianization of @xmath32 is @xmath255 , and the generator @xmath199 of @xmath41 maps to an element @xmath256 , where @xmath257 for type @xmath258 , @xmath259 for type @xmath208 , and @xmath260 for type @xmath31 . any homomorphism @xmath261 can be obtained by composing the abelianisation homomorphism with a map @xmath262 given by @xmath263 for a pair of integers @xmath264 . the associated transvection then satisfies @xmath265 where @xmath266 , or @xmath267 , depending on the case ( @xmath268 for type @xmath208 ) . on the other hand @xmath190 is an automorphism if and only if @xmath269 . when @xmath0 this is only possible if @xmath270 and @xmath271 , or @xmath272 , respectively . this allows exactly an infinite cyclic group of transvection automorphisms ( all of which act by the identity on @xmath41 ) . in the remaining case , type @xmath251 , we have the constraint @xmath273 , and so @xmath271 or @xmath274 . we note that @xmath275 is given by the presentation @xmath276 , where @xmath277 . a transvection of the second type ( @xmath278 ) is realized by the automorphism @xmath279 and @xmath280 . since in general we have @xmath281 it is easily checked that @xmath209 is an infinite dihedral group with @xmath282 acting as a direction reversing involution . the subgroup @xmath196 is finite index in @xmath197 ( since @xmath283 takes values in the finite group @xmath284 and @xmath285 , being finitely generated , admits only a finite number of homomorphisms to @xmath284 ) . hence the inverse image @xmath286 is a finite index subgroup of @xmath33 . by the previous proposition , this subgroup splits as a semi - direct product @xmath287 . for @xmath32 of type @xmath68 , we have the following presentation : @xmath288 in this case the outer automorphism groups were computed in @xcite : @xmath289 where @xmath94 denotes the graph involution @xmath290 , and the @xmath63 factor in each case is generated by the inversion automorphism @xmath291 . in the case @xmath69 even , the infinite cyclic factor @xmath65 is generated by the automorphism @xmath292 and @xmath293 , and contains the group of transvections as a subgroup of index @xmath294 , when @xmath295 . ( note that @xmath152 fixes both elements @xmath296 and @xmath297 , and that the centre is generated by @xmath298 ) . it follows that , in the case @xmath299 with @xmath295 even , the subgroup @xmath287 has index @xmath294 in the entire automorphism group . this behaviour seems to be repeated in the case @xmath300 , for @xmath0 , where , as we will see below , @xmath301 is a subgroup of index @xmath176 in @xmath33 . the type @xmath251 artin group is however exceptional in this regard . in this case the group of transvections is generated by @xmath302 and the element @xmath303 , from which it follows that @xmath287 is the whole automorphism group . we do not know what the index is ( or whether the subgroup is proper ) in the case of @xmath304 . we first show that the map @xmath306 is surjective and splits . recall from section [ sect2 ] that @xmath307 . it is clear from the proof of proposition [ atilde ] that elements of @xmath308 act on this semidirect product by leaving the characteristic subgroup @xmath167 invariant and by mapping the generator of the cyclic factor @xmath309 either to itself or its inverse . any such automorphism lifts uniquely to an automorphism of @xmath310 . thus we have a section to the map @xmath311 . we claim that the @xmath63 factor generated by the inversion @xmath177 also commutes with transvections . this follows from the fact that @xmath177 restricts to a length reversing automorphism of the center @xmath41 and hence takes the generator @xmath199 of @xmath41 to @xmath313 . for any transvection @xmath314 and any generator @xmath16 , @xmath315 for some @xmath153 , so @xmath316 by proposition [ trans ] , @xmath250 and the second claim of the proposition now follows . * the @xmath65 factor in the decomposition @xmath317 is generated by the element @xmath318 of @xmath66 and it acts on @xmath167 via rotation of the @xmath39 graph . these rotations thus become inner automorphisms in @xmath319 . the remaining graph automorphisms ( the reflections ) give rise to one of the @xmath63 factors in @xmath319 . ( the other @xmath63 factor is generated by the inversion @xmath177 . ) a generator of this @xmath63 is represented by the automorphism @xmath320 defined by @xmath321 and hence @xmath322 . in particular , @xmath320 does not lie in @xmath323 ( as long as @xmath0 ) , and this latter group is index 2 in the full automorphism group . ( by contrast , when @xmath32 is of type @xmath324 , the involution @xmath320 just described is exactly the exceptional transvection @xmath325 already mentioned in the preceding discussion ) . if @xmath79 is a group whose centre @xmath41 is a finite rank free abelian group , the injective transvections which are not automorphisms of @xmath79 have as image a proper finite index subgroup of @xmath79 and therefore represent infinite order elements of @xmath326 ( _ not _ equivalent to automorphisms ) . in the case of a finite type artin group , this implies that the abstract commensurator group must be large . it suffices to prove the proposition for irreducible @xmath32 . let @xmath198 be the length homomorphism . let @xmath199 be the generator of @xmath328 of positive length @xmath200 . then for every @xmath329 , there is a homomorphism @xmath330 defined by @xmath331 . the associated transvection takes @xmath332 to @xmath333 . in particular , it takes @xmath199 to @xmath334 . we denote this transvection by @xmath335 . it is straightforward to verify that @xmath336 . thus , @xmath335 and @xmath337 are commuting elements of @xmath327 and to find an infinitely generated free abelian subgroup of @xmath327 , it suffices to find an infinite sequence of integers @xmath338 which are relatively prime and satisfy @xmath339 mod @xmath222 . this can be done inductively by setting @xmath340 and @xmath341 . the transvections @xmath342 are then linearly independent .

we observe that , for fixed @xmath0 , each of the artin groups of finite type @xmath1 , @xmath2 , and affine type @xmath3 and @xmath4 is a central extension of a finite index subgroup of the mapping class group of the @xmath5-punctured sphere . ( the centre is trivial in the affine case and infinite cyclic in the finite type cases ) . using results of ivanov and korkmaz on abstract commensurators of surface mapping class groups we are able to determine the automorphism groups of each member of these four infinite families of artin groups . # 1#1 hs a rank @xmath6 _ coxeter matrix _ is a symmetric @xmath7 matrix @xmath8 with integer entries @xmath9 where @xmath10 for @xmath11 , and @xmath12 for all @xmath13 . given any rank @xmath6 coxeter matrix @xmath8 , the _ artin group _ of type @xmath8 is defined by the presentation @xmath14 adding the relations @xmath15 to this presentation yields a presentation of the coxeter group of type @xmath8 generated by standard reflections @xmath16 and such that the rotation @xmath17 has order @xmath18 , for all @xmath19 . a coxeter matrix @xmath8 and its artin group @xmath20 are said to be of _ finite type _ if the associated coxeter group @xmath21 is finite , and of _ affine ( or euclidean ) type _ if @xmath21 acts as a proper , cocompact group of isometries on some euclidean space with the generators @xmath22 acting as affine reflections . the information contained in the coxeter matrix @xmath8 is often displayed in the form of a graph , the _ coxeter graph _ , whose vertices are numbered @xmath23 and which has an edge labelled @xmath18 between the vertices @xmath24 and @xmath25 whenever @xmath26 or @xmath27 . with this particular convention , one usually suppresses the labels which are equal to @xmath28 ( but not the corresponding edges ! ) . note that the absence of an edge between two vertices indicates that the corresponding generators of @xmath20 commute . we say that an artin group is _ irreducible _ if its coxeter graph is connected and observe that every artin group is isomorphic to a direct product of irreducible artin groups corresponding to the connected components of its coxeter graph . in this paper we concern ourselves with the following four infinite families of artin groups @xmath29 ( see figure [ fig1 ] ) : the finite types @xmath30 , @xmath31 and affine types @xmath3 and @xmath4 , of rank @xmath0 in each case . ( we refer to @xcite for the classification of irreducible finite and affine type coxeter systems . ) for each artin group @xmath32 on this list , we determine its automorphism group @xmath33 , its outer automorphism group @xmath34 , and the abstract commensurator group @xmath35 of the group modulo its centre ( see definition [ defncommg ] ) . , @xmath31 and affine type @xmath3 , @xmath4 , with rank @xmath0 . note that , by convention , the subscript in each case indicates the geometric dimension of the artin group.,width=453 ] the artin group @xmath36 is well - known as the braid group on @xmath37 strings , and the automorphism groups of the braid groups were determined by dyer and grossman in @xcite . in this paper we exploit the fact that the artin groups of type @xmath38 , @xmath39 and @xmath40 also have descriptions as braid groups ( see @xcite ) in order to study their automorphism groups . note that the centre of any irreducible artin group of finite type is an infinite cyclic group , which we generally denote by @xmath41 . ( on the other hand , the groups @xmath42 and @xmath43 have trivial centres . ) as explained in section [ sect1 ] , each of the groups @xmath44 , @xmath45 , @xmath42 and @xmath43 is a finite index subgroup of the mapping class group @xmath46 of the @xmath5-punctured 2-sphere @xmath47 . using this fact , and appealing to the work of ivanov and korkmaz on the structure of surface mapping class groups @xcite , we show : [ main ] for each @xmath0 , we have * @xmath48 for each @xmath49 . + * * * _ ( dyer , grossman @xcite ) _ @xmath50 * * @xmath51 and @xmath52 * * @xmath53 * * @xmath54 . in addition , we show that the exact sequence @xmath55 splits in each of the above cases with the exception of the case @xmath56 with @xmath57 ( see proposition [ nosplit ] ) . in the case of @xmath42 , for example , the dihedral factor @xmath58 is realised by the group of `` graph automorphisms '' induced from symmetries of the coxeter graph , which in this case is an @xmath6-cycle . every artin group admits an involution @xmath59 for all @xmath60 which is never inner ( because it reverses the sign of the length function @xmath61 , defined by @xmath62 ) . this accounts for the central @xmath63 factor in each of the outer automorphism groups above . note that all of the outer automorphism groups in the theorem are finite with the exception of @xmath64 . the infinite cyclic factor in @xmath64 is generated by a `` transvection '' which multiplies each generator @xmath16 by some element of the center @xmath41 . in section [ sect3 ] , we study the group of transvections of a finite type artin group . transvection homomorphisms exist in any artin group with non - trivial center , but in general , they are not automorphisms . in particular , transvection automorphisms do not occur when the artin group has abelianisation @xmath65 , as in the case of the braid groups . however , @xmath66 has abelianisation @xmath67 and likewise for artin groups of dihedral type @xmath68 when @xmath69 is even . ( the groups @xmath70 , for @xmath71 , were computed in @xcite and are discussed in section [ sect3 ] ) . by contrast , transvection homomorphisms appear in the abstract commensurator group of _ every _ finite type artin group . in proposition [ bigcomm ] , we show that the abstract commensurator group of any finite type artin group contains an infinitely generated free abelian group generated by transvections . the first author would like to thank the institut de mathematiques de bourgogne for their hospitality during the development of this paper .

You are an expert at summarizing long articles. Proceed to summarize the following text: in paper @xcite , we surveyed period variations of superhumps in su uma - type dwarf novae ( for general information of su uma - type dwarf novae and superhumps , see @xcite ) . @xcite indicated that evolution of superhump period ( @xmath3 ) is generally composed of three distinct stages : early evolutionary stage with a longer superhump period ( stage a ) , middle stage with systematically varying periods ( stage b ) , final stage with a shorter , stable superhump period ( c ) . it was also shown that the period derivatives ( @xmath4 ) during stage b is correlated with @xmath3 , or binary mass - ratios ( @xmath5 ) . although this relation commonly applies to classical su uma - type dwarf novae , wz sge - type dwarf novae , a subtype of su uma - type dwarf novae with very infrequent superoutbursts , tend to deviate from this picture : they rarely show a distinct stage b c transition , and some objects show relatively small period derivatives , and they frequently exhibit unusual multiple post - superoutburst rebrightenings . the origin of these relations is not yet well understood . we have extended the survey in order to test whether this picture is applicable to new su uma - type systems and to the degree of diversity between systems in a larger sample . we include in this paper newly recorded objects and superoutbursts since the publication of @xcite . some of new observations have led to revisions of analysis in @xcite . we also include a few past superoutburst not analyzed in the previous studies . the structure of the paper follows the scheme in @xcite , in which we mostly restricted to superhump timing analysis . we also include some more details if the paper provides the first solid presentation of individual objects . the data were obtained under campaigns led by the vsnet collaboration @xcite . in some objects , we used archival data for published papers , and the public data from the aavso international databasehttp://www.aavso.org / data / download/@xmath6 . ] . the majority of the data were acquired by time - resolved ccd photometry by using 30 cm - class telescopes , whose observational details on individual objects will be presented in future papers dealing with analysis and discussion on individual objects . the list of outbursts and observers is summarized in table [ tab : outobs ] . the data analysis was performed just in the same way described in @xcite . in this paper , we introduce bayesian extension to traditional method ( appendix ) , and used them if they give significantly improved results . the derived @xmath3 , @xmath7 and other parameters are listed in table [ tab : perlist ] as in the same format in @xcite . the definitions of parameters @xmath8 and @xmath7 are the same as in @xcite . as in @xcite , we present comparisons of @xmath0 diagrams between different superoutbursts since this has been one of the motivations of these surveys ( cf . @xcite ) . ccccc subsection & object & year & observers or references & i d + [ obj : kxaql ] & kx aql & 2010 & mhh , aavso , kis , got , dpv , oku & + [ obj : nncam ] & nn cam & 2009 & aavso , dem , mhh , jsh , imi & + [ obj : v591cen ] & v591 cen & 2010 & mlf , gbo & + [ obj : zcha ] & z cha & 2010 & aavso , kis & + [ obj : pucma ] & pu cma & 2009 & ous , mhh , kis & + [ obj : aqcmi ] & aq cmi & 2010 & gbo & + [ obj : gzcnc ] & gz cnc & 2010 & oku , oao , mhh , dem , ous & + [ obj : gocom ] & go com & 2010 & dem , mhh , oku , dpv & + [ obj : tvcrv ] & tv crv & 2009 & kis & + [ obj : v337cyg ] & v337 cyg & 2010 & imi , ku , oku , dpv & + [ obj : v1113cyg ] & v1113 cyg & 2003 & @xcite & + [ obj : v1454cyg ] & v1454 cyg & 2009 & hmb , mhh , imi , aavso & + [ obj : aqeri ] & aq eri & 2010 & nel , mhh , kis , ioh & + [ obj : vxfor ] & vx for & 2009 & mlf , nyr , nel , aavso , sto , mhh , kis & + [ obj : awgem ] & aw gem & 2010 & hyn & + [ obj : irgem ] & ir gem & 2010 & dem & + [ obj : v592her ] & v592 her & 2010 & aavso , ous , mhh , oku , imi , dpv , rit , hmb , ku , boy , oao , mev & + [ obj : v660her ] & v660 her & 2009 & aavso & + [ obj : v844her ] & v844 her & 2009 & dpv , aavso & + & v844 her & 2010 & mhh & + [ obj : cthya ] & ct hya & 2010 & mhh & + [ obj : v699oph ] & v699 oph & 2010 & dem , ous & + [ obj : v1032oph ] & v1032 oph & 2010 & mhh , dem , aavso , kis , rui & + [ obj : v2051oph ] & v2051 oph & 2010 & sto , ous , kis & + [ obj : efpeg ] & ef peg & 2009 & mhh , imi , ous , aavso & + [ obj : v368peg ] & v368 peg & 2009 & ctx , aavso , imi , nyr , hmb & + [ obj : uvper ] & uv per & 2010 & dpv , pxr , dem , aavso & + [ obj : eipsc ] & ei psc & 2009 & got , kis & + [ obj : ektra ] & ek tra & 2009 & nel & + [ obj : suuma ] & su uma & 2010 & dpv , sac & + [ obj : bcuma ] & bc uma & 2009 & nyr , mhh & + [ obj : eluma ] & el uma & 2010 & mhh , ioh , dpv , hmb & + [ obj : iyuma ] & iy uma & 2009 & @xcite , dpv , aavso & + [ obj : ksuma ] & ks uma & 2010 & ous & + [ obj : mruma ] & mr uma & 2010 & ous , oku & + [ obj : tyvul ] & ty vul & 2010 & mev , aavso , imi , bst & + [ obj : j0423 ] & 1rxs j0423 & 2010 & ous , mhh , dem & + [ obj : j0532 ] & 1rxs j0532 & 2009 & dpv , mhh & + [ obj : asas2243 ] & asas j2243 & 2009 & hmb , kra , mhh , jsh , gbo , imi , dem , ioh , aavso & + [ obj : lanning420 ] & lanning 420 & 2010 & ku , aavso , dem , bxs , mhh , hmb , rui & + [ obj : pg0149 ] & pg 0149 & 2009 & imi , hmb , mhh & + [ obj : j1715 ] & rx j1715 & 2009 & @xcite , mhh & + [ obj : j0129 ] & sdss j0129 & 2009 & jsh , mhh , aavso , boy & + [ obj : j0310 ] & sdss j0310 & 2009b & imi & + [ obj : j0732 ] & sdss j0732 & 2010 & aavso , jsh & + [ obj : j0838 ] & sdss j0838 & 2010 & mhh , dem & + [ obj : j0839 ] & sdss j0839 & 2010 & dem , jsh & + [ obj : j0903 ] & sdss j0903 & 2010 & dem , vol , aavso , mas , mhh & + + + + + + + + + + + [ cols="^,^,^,^,^",options="header " , ] kx aql has long been known as a dwarf nova with a low outburst frequency . @xcite and @xcite reported two outbursts in 1967 and 1972 from aavso observations . @xcite examined photographic plate archives and estimated an interval between outbursts to be @xmath9 1000 d. the object underwent a very bright ( @xmath10 ) outburst in 1980 november , which was first recorded by s. fujino on november 9 . the long duration of the outburst inferred from the vsolj and aavso observations is sufficient to imply the su uma - type nature . additional short outbursts were recorded in 1994 decemberhttp://www.kusastro.kyoto - u.ac.jp / vsnet / mail/1994/vsnet - obs / msg00155.html@xmath6 . ] and 2007 may ( baavss - alert 582 ) , which later turned out to be a normal outburst . @xcite first presented a spectrum of this object clearly showing a dwarf nova at a low mass - transfer rate . @xcite listed kx aql as a candidate for a wz sge - type dwarf nova . @xcite obtained a @xmath2 of 0.06035(3 ) d from a radial - velocity study . the long - waited superoutburst was finally recorded in 2010 march by t. gomez ( cvnet - outburst 3626 ) . subsequent observations detected superhumps ( vsnet - alert 11859 , 11862 , 11884 ) . the outburst was followed by a rebrightening ( vsnet - alert 11895 ) . the times of superhump maxima are listed in table [ tab : kxaqloc2010 ] . there was an apparent break in the period evolution around @xmath11 . we identified this break as a stage b c transition since the observed duration of the outburst after the break was only 7 d ( in contrast to @xmath914 d for typical durations of superoutbursts of short-@xmath2 su uma - type dwarf novae ) and the superhump period was almost constant after the break . the relatively low recorded maximum brightness ( @xmath12 ) for a superoutburst also suggests that the true maximum was missed . the period for stage b was not determined due to the shortness of the observed segment . we adopted a mean superhump period of 0.06128(2 ) d with the pdm method ( figure [ fig : kxaqlshpdm ] ) . the recorded @xmath13 of 1.5 % is slightly low for this @xmath2 ( cf . figure 15 in @xcite ) , but not as low as those of wz sge - type dwarf novae with similar@xmath2 . this interpretation is consistent with the presence of a single rebrightening , rather than multiple ones in wz sge - type systems with similar @xmath2 . the scaled @xmath14 is 0.08 based on the refined relation in @xcite . ( 88mm,110mm)fig1.eps ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 55265.6366 & 0.0007 & @xmath170.0110 & 145 + 1 & 55265.6941 & 0.0018 & @xmath170.0149 & 52 + 7 & 55266.0852 & 0.0025 & 0.0077 & 56 + 17 & 55266.6967 & 0.0028 & 0.0050 & 53 + 26 & 55267.2439 & 0.0010 & @xmath170.0005 & 183 + 27 & 55267.3082 & 0.0005 & 0.0023 & 148 + 38 & 55267.9827 & 0.0004 & 0.0012 & 59 + 43 & 55268.2993 & 0.0007 & 0.0108 & 74 + 50 & 55268.7192 & 0.0014 & 0.0008 & 43 + 54 & 55268.9648 & 0.0003 & 0.0008 & 126 + 55 & 55269.0288 & 0.0007 & 0.0033 & 46 + 59 & 55269.2728 & 0.0006 & 0.0016 & 238 + 70 & 55269.9473 & 0.0004 & 0.0006 & 53 + 71 & 55270.0091 & 0.0008 & 0.0010 & 60 + 75 & 55270.2611 & 0.0025 & 0.0073 & 127 + 76 & 55270.3120 & 0.0017 & @xmath170.0032 & 113 + 82 & 55270.6850 & 0.0019 & 0.0013 & 52 + 87 & 55270.9918 & 0.0026 & 0.0010 & 7 + 99 & 55271.7243 & 0.0026 & @xmath170.0035 & 44 + 103 & 55271.9691 & 0.0008 & @xmath170.0044 & 62 + 119 & 55272.9490 & 0.0017 & @xmath170.0072 & 16 + + + + the superoutburst in 2009 november is the first superoutburst of this object whose entire evolution was first observed in detail [ for general information of this object , see @xcite ] . the times of superhump maxima are listed in table [ tab : nncamoc2009 ] . the @xmath0 diagram clearly shows all a c stages . there was a rise in the light curve in accordance with the stage a b transition ( figure [ fig : nncam2009oc ] ) . the mean @xmath3 and @xmath7 for stage b ( @xmath18 ) were 0.07426(1 ) d and @xmath19 , respectively . there was a distinct stage b c transition around @xmath15 = 87 . the mean period for stage a was calculated for @xmath20 although there was significant period shortening even during stage a. @xcite also recently reported on two superoutbursts of this object in 2007 and 2009 . @xcite did not distinguish stages a c during the 2009 superoutburst treated in this paper probably due to fragmentary observational coverage . ( 88mm,90mm)fig2.eps ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 55137.7356 & 0.0010 & @xmath170.0396 & 71 + 1 & 55137.8129 & 0.0019 & @xmath170.0366 & 69 + 2 & 55137.8989 & 0.0009 & @xmath170.0249 & 71 + 12 & 55138.6733 & 0.0006 & 0.0064 & 72 + 13 & 55138.7472 & 0.0004 & 0.0060 & 70 + 14 & 55138.8239 & 0.0003 & 0.0084 & 67 + 15 & 55138.8974 & 0.0003 & 0.0076 & 63 + 19 & 55139.1977 & 0.0002 & 0.0107 & 109 + 20 & 55139.2719 & 0.0002 & 0.0106 & 147 + 21 & 55139.3475 & 0.0003 & 0.0119 & 90 + 39 & 55140.6834 & 0.0004 & 0.0102 & 46 + 40 & 55140.7574 & 0.0003 & 0.0099 & 69 + 41 & 55140.8316 & 0.0003 & 0.0098 & 68 + 42 & 55140.9059 & 0.0003 & 0.0098 & 63 + 61 & 55142.3164 & 0.0003 & 0.0085 & 152 + 76 & 55143.4293 & 0.0008 & 0.0068 & 68 + 77 & 55143.5065 & 0.0011 & 0.0097 & 33 + 78 & 55143.5796 & 0.0016 & 0.0084 & 25 + 79 & 55143.6544 & 0.0009 & 0.0089 & 36 + 87 & 55144.2482 & 0.0005 & 0.0083 & 140 + 103 & 55145.4305 & 0.0009 & 0.0017 & 41 + 104 & 55145.5047 & 0.0007 & 0.0015 & 39 + 105 & 55145.5767 & 0.0012 & @xmath170.0008 & 30 + 106 & 55145.6509 & 0.0007 & @xmath170.0008 & 40 + 107 & 55145.7255 & 0.0013 & @xmath170.0006 & 32 + 116 & 55146.3882 & 0.0010 & @xmath170.0066 & 65 + 117 & 55146.4641 & 0.0006 & @xmath170.0050 & 63 + 118 & 55146.5391 & 0.0005 & @xmath170.0043 & 104 + 119 & 55146.6146 & 0.0007 & @xmath170.0031 & 39 + 129 & 55147.3495 & 0.0012 & @xmath170.0113 & 33 + 130 & 55147.4291 & 0.0028 & @xmath170.0061 & 37 + 134 & 55147.7171 & 0.0013 & @xmath170.0153 & 39 + + + + v591 cen was discovered by @xcite ( see also @xcite ) . although the discovery observation was already suggestive of a superoutburst , this object had long been overlooked due to the apparent small outburst amplitude . the lack of cv - type signature in spectroscopic observation by @xcite even led to a non - cv classification . monitoring for outbursts , however , continued because of its outbursting nature and the likely presence of superoutbursts . in 2006 , b. monard succeeded in obtaining astrometry of the outbursting object , which was 3 arcsec distant from the supposed quiescent counterpart ( vsnet - alert 9158 ) . the true v591 cen was invisible down to 20 mag on dss images . the object again underwent a bright outburst in 2010 april ( vsnet - alert 11915 ) , and the existence of superhumps was finally confirmed ( vsnet - alert 11919 ; figure [ fig : v591censhpdm ] ) . astrometry of the outbursting object also confirmed the identification in 2006 ( vsnet - alert 11922 ) . the times of superhump maxima are listed in table [ tab : v591cenoc2010 ] . the data clearly shows a positive period derivative @xmath7 = @xmath21 , typical for this short @xmath3 . ( 88mm,110mm)fig3.eps ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 55296.1418 & 0.0008 & 0.0022 & 111 + 1 & 55296.2051 & 0.0007 & 0.0052 & 109 + 2 & 55296.2634 & 0.0011 & 0.0033 & 111 + 17 & 55297.1647 & 0.0009 & 0.0001 & 110 + 18 & 55297.2252 & 0.0008 & 0.0003 & 111 + 19 & 55297.2849 & 0.0011 & @xmath170.0003 & 112 + 20 & 55297.3425 & 0.0014 & @xmath170.0030 & 70 + 86 & 55301.3271 & 0.0032 & 0.0019 & 267 + 87 & 55301.3806 & 0.0020 & @xmath170.0050 & 268 + 88 & 55301.4362 & 0.0018 & @xmath170.0097 & 267 + 89 & 55301.4977 & 0.0017 & @xmath170.0084 & 268 + 102 & 55302.2868 & 0.0039 & @xmath170.0033 & 268 + 133 & 55304.1630 & 0.0018 & 0.0036 & 61 + 134 & 55304.2236 & 0.0030 & 0.0040 & 103 + 135 & 55304.2836 & 0.0015 & 0.0036 & 329 + 136 & 55304.3447 & 0.0014 & 0.0044 & 281 + 137 & 55304.4012 & 0.0014 & 0.0006 & 264 + 138 & 55304.4612 & 0.0027 & 0.0004 & 73 + + + + we analyzed aavso and our observations of the 2010 january superoutburst of z cha . the observation was performed during the late stage of the superoutburst . after removing observations within 0.10 @xmath2 of eclipses for the superoutburst plateau and 0.08 @xmath2 for later observations , we determined times of superhump maxima ( table [ tab : zchaoc2010 ] ) . the times for @xmath22 are secondary humps , which are likely traditional late superhumps . the peaks of persistent ordinary superhumps were not determined because they fell on orbital humps and around eclipses . these superhumps ( @xmath23 ) most likely correspond to stage c superhumps . the period [ 0.07698(5 ) d ] determined from the superhump maxima is close to the period [ 0.07681(6 ) d ] of stage c superhumps in 1982 @xcite . although post - superoutburst observation suggests the appearance of traditional late superhumps , this signal was not well traced due to the strong orbital modulation and limited coverage . a better continuous observation is necessary to test whether dominant superhumps in the post - superoutburst stage are stage c superhumps or traditional late superhumps ( see also a discussion on the nature of late superhumps in @xcite ) . ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 55200.0935 & 0.0008 & 0.0032 & 152 + 13 & 55201.0794 & 0.0003 & @xmath170.0086 & 209 + 14 & 55201.1575 & 0.0003 & @xmath170.0072 & 165 + 15 & 55201.2328 & 0.0001 & @xmath170.0087 & 124 + 27 & 55202.1711 & 0.0021 & 0.0086 & 24 + 28 & 55202.2385 & 0.0015 & @xmath170.0007 & 22 + 65 & 55205.0878 & 0.0006 & 0.0088 & 158 + 66 & 55205.1638 & 0.0005 & 0.0081 & 159 + 67 & 55205.2385 & 0.0007 & 0.0060 & 94 + 78 & 55206.0940 & 0.0007 & 0.0173 & 130 + 79 & 55206.1667 & 0.0011 & 0.0132 & 62 + 80 & 55206.2399 & 0.0007 & 0.0097 & 159 + 91 & 55207.0442 & 0.0015 & @xmath170.0303 & 168 + 92 & 55207.1319 & 0.0009 & @xmath170.0194 & 169 + + + + we observed the 2009 superoutburst of this object ( table [ tab : pucmaoc2009 ] ) . the nightly superhump profiles ( figure [ fig : pucma2009prof ] ) suggests that the superhump period first increased until bjd 2455164 , and then decreased . the stages listed in table [ tab : perlist ] reflect this interpretation . a comparison of @xmath0 diagrams between different superoutbursts is shown in figure [ fig : pucmacomp ] ) . it would be worth noting that the @xmath0 diagram for the 2008 superoutburst ( superoutburst preceded by a precursor ) matched others only if @xmath15 was counted from the start of the main superoutburst , rather than from the precursor . this suggests that superhumps were excited around the ignition of the main superoutburst , rather than during the precursor . ( 88mm,110mm)fig4.eps ( 88mm,70mm)fig5.eps ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 55159.1299 & 0.0009 & @xmath170.0110 & 66 + 1 & 55159.1892 & 0.0008 & @xmath170.0096 & 65 + 35 & 55161.1572 & 0.0007 & @xmath170.0083 & 53 + 36 & 55161.2158 & 0.0003 & @xmath170.0076 & 84 + 37 & 55161.2724 & 0.0005 & @xmath170.0088 & 82 + 53 & 55162.2057 & 0.0010 & @xmath170.0011 & 193 + 54 & 55162.2609 & 0.0004 & @xmath170.0038 & 186 + 55 & 55162.3181 & 0.0005 & @xmath170.0043 & 179 + 87 & 55164.1812 & 0.0008 & 0.0076 & 85 + 88 & 55164.2404 & 0.0008 & 0.0090 & 248 + 89 & 55164.2954 & 0.0009 & 0.0061 & 249 + 90 & 55164.3611 & 0.0019 & 0.0139 & 102 + 121 & 55166.1436 & 0.0011 & 0.0032 & 150 + 122 & 55166.2110 & 0.0007 & 0.0128 & 334 + 123 & 55166.2661 & 0.0005 & 0.0100 & 263 + 124 & 55166.3215 & 0.0006 & 0.0075 & 165 + 137 & 55167.0993 & 0.0035 & 0.0334 & 128 + 138 & 55167.1327 & 0.0019 & 0.0089 & 208 + 139 & 55167.1960 & 0.0006 & 0.0144 & 316 + 140 & 55167.2478 & 0.0005 & 0.0083 & 273 + 141 & 55167.3057 & 0.0013 & 0.0083 & 168 + 174 & 55169.1975 & 0.0022 & @xmath170.0087 & 84 + 175 & 55169.2371 & 0.0010 & @xmath170.0270 & 85 + 176 & 55169.2990 & 0.0007 & @xmath170.0229 & 82 + 192 & 55170.2172 & 0.0017 & @xmath170.0304 & 78 + + + + although the su uma - type nature of this object has been well established , http://www.kusastro.kyoto - u.ac.jp / vsnet / dne / aqcmi.html@xmath6 . ] there has unfortunately no solid publication on superhumps in this system . the only known superoutburst since the 1997 observation was in 2008 september , which was too badly placed for time - series photometry . the superoutburst in 2010 april brought the first chance to record superhumps since its recognition as an su uma - type dwarf nova . the times of superhump maxima are listed in table [ tab : aqcmioc2010 ] . although the course of the superhump evolution was not fully recorded , there appears to have been a discontinuous period change around @xmath24 . we attributed this to stage a b transition because the observation recorded the early stage of the superoutburst . the period derivative was not determined because of a gap in observation . the mean @xmath3 during the entire observation was 0.06622(1 ) d ( figure [ fig : aqcmishpdm ] ) . this period needs to be treated with caution because it was likely derived from different stages of superhump evolution . ( 88mm,110mm)fig6.eps ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 55295.9806 & 0.0006 & @xmath170.0022 & 116 + 1 & 55296.0489 & 0.0007 & @xmath170.0002 & 112 + 15 & 55296.9775 & 0.0005 & 0.0017 & 104 + 16 & 55297.0431 & 0.0005 & 0.0011 & 121 + 106 & 55302.9994 & 0.0009 & @xmath170.0004 & 68 + + + + gz cnc is a variable star discovered by takamizawa ( tmzv34 , vsnet - obs 10504 ) , which later turned out to be a dwarf nova @xcite . @xcite observed an outburst in 2000 february . based on the relatively slow rise to an outburst maximum and the lack of periodic modulations , they concluded that the object is likely an ss cyg - type dwarf nova with a long @xmath2 . in 2002 , @xcite noticed an unusually increase in the number of outburst detections , and suggested the similarity to a proposed intermediate polar v426 oph . most surprisingly , radial - velocity studies by @xcite yielded an @xmath2 of 0.08825(28 ) d , placing the object at the lower edge of the period gap . although this @xmath2 was seemingly incompatible with behavior of outbursts in this object , later observations have detected abundant short outbursts which are compatible with a short @xmath2-system . the mystery , however , remained why the object did not develop superhumps during its long , 2000 february outburst . later on , the object again exhibited a long outburst in 2007 december ( vsnet - alert 9783 ) , which was unfortunately not observed for searching superhumps . in 2009 january , this object underwent a long , bright outburst ( vsnet - alert 10984 ) . follow - up observations , however , did not detect superhumps ( vsnet - alert 11002 ) . gz cnc was then considered as a rare object below the period gap without a signature of an su uma - type dwarf nova . in 2010 march , the object again underwent a long , bright outburst ( vsnet - alert 11855 , 11863 ) . subsequent observations eventually detected superhumps ( vsnet - alert 11881 , 11888 , 11890 , 11894 ) . the object is now confirmed to be a rare object exhibiting three classes of outbursts : normal ( narrow ) outbursts , long ss cyg - type ( wide ) outbursts and su uma - type superoutbursts ( figure [ fig : gzcnclc ] ) . the mystery of the 2000 outburst could be understood if this outburst was a `` long '' outburst failing to trigger the tidal instability . the only other known object having the same property is tu men ( cf . @xcite ; @xcite ) , unusual su uma - type dwarf nova above the period gap . if gz cnc is indeed a `` borderline '' su uma - type dwarf nova , the object may be analogous to bz uma , which also showed an usually slow rise to a full outburst ( cf . @xcite ) . the times of superhump maxima during the 2010 superoutburst are listed in table [ tab : gzcncoc2010 ] . the @xmath0 diagram can be reasonably interpreted without a phase jump only if we assume a very large decrease in the period as in ax cap ( figure [ fig : gzcncoc ] ) . the mean @xmath3 for @xmath25 was 0.09277(2 ) d ( pdm method ) , which is equivalent to @xmath13 = 5.1 % . this fractional superhump excess is one of the largest among su uma - type dwarf novae below the period gap ( cf . figure 15 in @xcite ) . the mean period 0.08972(12 ) d after the period decrease corresponds to @xmath13 = 1.7 % . since this period was fairly close to @xmath2 , this period decrease may have reflected the dominance of orbital humps during the late course of the superoutburst . ( 88mm,100mm)fig7.eps ( 88mm,100mm)fig8.eps ( 88mm,110mm)fig9.eps ( 88mm,110mm)fig10.eps ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 55269.0989 & 0.0007 & @xmath170.0405 & 64 + 1 & 55269.1935 & 0.0007 & @xmath170.0371 & 52 + 11 & 55270.1286 & 0.0008 & @xmath170.0146 & 175 + 26 & 55271.5221 & 0.0007 & 0.0099 & 86 + 27 & 55271.6139 & 0.0015 & 0.0104 & 55 + 32 & 55272.0723 & 0.0006 & 0.0124 & 226 + 33 & 55272.1535 & 0.0072 & 0.0024 & 48 + 36 & 55272.4347 & 0.0036 & 0.0099 & 57 + 37 & 55272.5312 & 0.0012 & 0.0150 & 93 + 42 & 55273.0023 & 0.0016 & 0.0298 & 53 + 43 & 55273.0952 & 0.0029 & 0.0315 & 74 + 44 & 55273.1874 & 0.0016 & 0.0325 & 88 + 64 & 55274.9832 & 0.0010 & 0.0029 & 97 + 65 & 55275.0761 & 0.0008 & 0.0046 & 276 + 66 & 55275.1721 & 0.0016 & 0.0093 & 160 + 87 & 55277.0421 & 0.0006 & @xmath170.0373 & 346 + 88 & 55277.1295 & 0.0007 & @xmath170.0412 & 173 + + + + the 2010 superoutburst was particularly well observed and stages b and c were very clearly defined ( table [ tab : gocomoc2010 ] ) . these observations have confirmed the finding in @xcite and @xcite . figure [ fig : gocomcomp2 ] illustrates a comparison of @xmath0 diagrams between different superoutbursts . the 2010 observation better recorded the stage c superhumps with higher accuracy , while the 2003 observation well recorded the earlier part . a combination of these superoutbursts very well demonstrates the `` canonical '' period variation in a short-@xmath3 system . it is not clear whether the 2010 superoutburst was preceded by a precursor outburst as in the 2003 one . ( 88mm,70mm)fig11.eps ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 55286.1498 & 0.0006 & 0.0028 & 93 + 23 & 55287.5958 & 0.0004 & 0.0005 & 61 + 24 & 55287.6574 & 0.0004 & @xmath170.0009 & 37 + 38 & 55288.5362 & 0.0006 & @xmath170.0036 & 41 + 39 & 55288.5989 & 0.0003 & @xmath170.0038 & 61 + 40 & 55288.6630 & 0.0004 & @xmath170.0027 & 55 + 47 & 55289.1040 & 0.0002 & @xmath170.0026 & 127 + 48 & 55289.1663 & 0.0004 & @xmath170.0032 & 127 + 54 & 55289.5443 & 0.0004 & @xmath170.0030 & 50 + 55 & 55289.6075 & 0.0003 & @xmath170.0028 & 66 + 56 & 55289.6694 & 0.0005 & @xmath170.0039 & 66 + 63 & 55290.1132 & 0.0014 & @xmath170.0008 & 40 + 64 & 55290.1758 & 0.0004 & @xmath170.0012 & 40 + 65 & 55290.2384 & 0.0009 & @xmath170.0015 & 40 + 70 & 55290.5554 & 0.0005 & 0.0006 & 64 + 71 & 55290.6181 & 0.0004 & 0.0004 & 61 + 79 & 55291.1213 & 0.0019 & @xmath170.0002 & 36 + 94 & 55292.0701 & 0.0008 & 0.0041 & 135 + 95 & 55292.1369 & 0.0006 & 0.0079 & 159 + 96 & 55292.1997 & 0.0006 & 0.0077 & 96 + 101 & 55292.5144 & 0.0005 & 0.0076 & 60 + 102 & 55292.5763 & 0.0004 & 0.0065 & 64 + 103 & 55292.6419 & 0.0005 & 0.0091 & 66 + 117 & 55293.5184 & 0.0004 & 0.0041 & 60 + 118 & 55293.5817 & 0.0003 & 0.0045 & 39 + 130 & 55294.3331 & 0.0006 & 0.0002 & 66 + 131 & 55294.3949 & 0.0008 & @xmath170.0009 & 62 + 132 & 55294.4595 & 0.0004 & 0.0007 & 77 + 133 & 55294.5220 & 0.0005 & 0.0003 & 114 + 134 & 55294.5837 & 0.0004 & @xmath170.0011 & 132 + 143 & 55295.1460 & 0.0009 & @xmath170.0055 & 178 + 144 & 55295.2120 & 0.0014 & @xmath170.0024 & 38 + 150 & 55295.5881 & 0.0005 & @xmath170.0041 & 63 + 158 & 55296.0832 & 0.0028 & @xmath170.0128 & 28 + + + + we observed an superoutburst in 2009 december . the times of superhump maxima are listed in table [ tab : tvcrvoc2009 ] . a combined @xmath0 diagram ( figure [ fig : tvcrvcomp2 ] ) strengthens the assertion in @xcite that the @xmath0 behavior is common between different superoutbursts . ( 88mm,70mm)fig12.eps ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 55183.2495 & 0.0004 & 0.0028 & 125 + 1 & 55183.3107 & 0.0003 & @xmath170.0010 & 144 + 16 & 55184.2870 & 0.0005 & @xmath170.0005 & 108 + 47 & 55186.3019 & 0.0006 & @xmath170.0024 & 143 + 62 & 55187.2796 & 0.0008 & @xmath170.0005 & 142 + 63 & 55187.3426 & 0.0007 & @xmath170.0025 & 121 + 77 & 55188.2587 & 0.0008 & 0.0028 & 143 + 78 & 55188.3227 & 0.0006 & 0.0018 & 144 + 92 & 55189.2347 & 0.0116 & 0.0030 & 114 + 93 & 55189.2941 & 0.0009 & @xmath170.0027 & 144 + 94 & 55189.3610 & 0.0015 & @xmath170.0008 & 82 + + + + we observed the 2010 superoutburst of this object . the times of superhump maxima are listed in table [ tab : v337cygoc2010 ] . although the period derivative was not well determined , we detected a stage b c transition . the obtained parameters are listed in table [ tab : perlist ] . ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 55421.4715 & 0.0012 & @xmath170.0046 & 118 + 1 & 55421.5390 & 0.0007 & @xmath170.0072 & 153 + 29 & 55423.5087 & 0.0007 & @xmath170.0009 & 75 + 42 & 55424.4236 & 0.0007 & 0.0024 & 73 + 43 & 55424.4944 & 0.0006 & 0.0032 & 72 + 44 & 55424.5626 & 0.0007 & 0.0012 & 77 + 45 & 55424.6340 & 0.0011 & 0.0025 & 54 + 51 & 55425.0473 & 0.0078 & @xmath170.0050 & 36 + 52 & 55425.1288 & 0.0019 & 0.0064 & 63 + 95 & 55428.1525 & 0.0013 & 0.0149 & 30 + 126 & 55430.3163 & 0.0065 & 0.0050 & 119 + 138 & 55431.1449 & 0.0010 & @xmath170.0079 & 128 + 139 & 55431.2128 & 0.0020 & @xmath170.0101 & 147 + + + + @xcite recently reported new observations of 2003 and 2005 superoutbursts , and claimed the presence of a large negative period derivative . upon examination of their observations , it has become evident that they observed the apparently late stage of a superoutburst in 2003 . using the typical duration ( @xmath26 d ) of superoutbursts in this system , their observation probably started @xmath9 94 cycles after the start of the superoutburst , and they most likely caught a stage b c transition . a combined @xmath0 diagram based on this interpretation , supplemented by early observations reported in @xcite , is shown in figure [ fig : v1113cygcomp ] . as judged from this figure , the evolution of the superhump period is not particularly unusual in this system . ( 88mm,70mm)fig13.eps this object was only partly observed during the 2006 superoutburst @xcite . the 2009 superoutburst was detected during its early stage ( i. miller , baavss - alert 2020 , vsnet - alert 11376 ) . a delay of @xmath276.5 d in the full growth of ordinary superhumps was recorded ( vsnet - alert 11393 , 11395 ) . the new observation now clarified the alias selection [ 0.05769(2 ) d with the pdm method ] and safely excluded the 0.0610 d earlier reported ( figure [ fig : v1454cygshpdm ] ) . the @xmath7 for stage b was @xmath28 , fairly common for this short @xmath3 . although there was likely a stage b c transition after @xmath29 , we could not determine the period of stage c superhumps due to the lack of observations . an analysis of the earlier observation ( bjd before 2455058 ) has yielded a weak signal of 0.05777(2 ) d. although the extraction of times of superhump maxima was difficult due to the low amplitudes , this period suggests that this interval involved stage a evolution rather than a manifestation of early superhumps . ( 88mm,110mm)fig14.eps ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 55058.1434 & 0.0004 & 0.0055 & 122 + 1 & 55058.2036 & 0.0013 & 0.0081 & 65 + 39 & 55060.3868 & 0.0010 & 0.0001 & 57 + 40 & 55060.4431 & 0.0024 & @xmath170.0013 & 54 + 41 & 55060.4996 & 0.0010 & @xmath170.0025 & 56 + 42 & 55060.5540 & 0.0035 & @xmath170.0057 & 54 + 50 & 55061.0219 & 0.0009 & 0.0009 & 109 + 51 & 55061.0756 & 0.0019 & @xmath170.0031 & 107 + 52 & 55061.1361 & 0.0024 & @xmath170.0003 & 121 + 53 & 55061.1919 & 0.0020 & @xmath170.0021 & 121 + 54 & 55061.2405 & 0.0033 & @xmath170.0112 & 115 + 56 & 55061.3703 & 0.0016 & 0.0033 & 54 + 57 & 55061.4258 & 0.0009 & 0.0012 & 52 + 58 & 55061.4806 & 0.0009 & @xmath170.0017 & 56 + 59 & 55061.5380 & 0.0011 & @xmath170.0020 & 57 + 60 & 55061.5956 & 0.0042 & @xmath170.0021 & 52 + 75 & 55062.4672 & 0.0019 & 0.0046 & 57 + 76 & 55062.5230 & 0.0027 & 0.0027 & 57 + 86 & 55063.0867 & 0.0032 & @xmath170.0101 & 114 + 87 & 55063.1513 & 0.0064 & @xmath170.0032 & 121 + 88 & 55063.2097 & 0.0155 & @xmath170.0024 & 86 + 92 & 55063.4463 & 0.0024 & 0.0035 & 56 + 93 & 55063.5010 & 0.0033 & 0.0005 & 57 + 94 & 55063.5635 & 0.0023 & 0.0054 & 57 + 95 & 55063.6254 & 0.0029 & 0.0095 & 30 + 160 & 55067.3705 & 0.0019 & 0.0066 & 57 + 161 & 55067.4330 & 0.0020 & 0.0115 & 54 + 163 & 55067.5267 & 0.0037 & @xmath170.0102 & 57 + 164 & 55067.5890 & 0.0019 & @xmath170.0055 & 54 + + + + we observed another superoutburst in 2010 ( table [ tab : aqerioc2010 ] ) . there was a relatively large scatter in the @xmath0 diagram after the rapid fading due to its faintness . we therefore did not attempt to determine the period during stage c. the mean @xmath3 appears to confirm the previous period determination ( cf . figure [ fig : aqericomp2 ] ) . ( 88mm,70mm)fig15.eps ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 55201.0094 & 0.0004 & 0.0067 & 54 + 1 & 55201.0715 & 0.0011 & 0.0064 & 37 + 16 & 55202.0038 & 0.0003 & 0.0025 & 102 + 31 & 55202.9335 & 0.0013 & @xmath170.0039 & 65 + 32 & 55202.9972 & 0.0003 & @xmath170.0027 & 223 + 33 & 55203.0595 & 0.0004 & @xmath170.0027 & 73 + 48 & 55203.9967 & 0.0006 & @xmath170.0016 & 90 + 49 & 55204.0482 & 0.0048 & @xmath170.0126 & 51 + 64 & 55204.9952 & 0.0008 & @xmath170.0017 & 100 + 81 & 55206.0660 & 0.0022 & 0.0081 & 45 + 82 & 55206.1207 & 0.0028 & 0.0004 & 37 + 145 & 55210.0545 & 0.0018 & 0.0023 & 142 + 146 & 55210.1137 & 0.0021 & @xmath170.0008 & 146 + 161 & 55211.0574 & 0.0012 & 0.0067 & 68 + 162 & 55211.1045 & 0.0059 & @xmath170.0086 & 65 + 163 & 55211.1695 & 0.0035 & @xmath170.0060 & 101 + 193 & 55213.0543 & 0.0016 & 0.0065 & 61 + 194 & 55213.1111 & 0.0015 & 0.0009 & 91 + + + + vx for was discovered in 1990 as a probable dwarf nova showing balmer , hei and possibly heii emission lines @xcite . the large outburst amplitude and the outburst lasting for more than 10 d @xcite were already suggestive of an su uma - type superoutburst . the object has been listed as a good candidate for a wz sge - type dwarf nova @xcite . the 2009 outburst , first - ever since the initial discovery , was detected by r. stubbings on 2009 sep . 14 at a visual magnitude of 13.0 ( vsnet - alert 11471 ) . ordinary superhump were soon observed following this outburst detection ( vsnet - alert 11474 ) . although the early appearance of ordinary superhumps was initially considered as unfavorable for the wz sge - type interpretation , a later retrospective detection of earlier positive observation in the asas-3 data ( @xmath30 = 12.61 on september 10 ) indicated that the early stage of the outburst was missed when early superhumps were expected ( vsnet - alert 11492 ) . using the empirical classification of wz sge - type dwarf novae introduced in @xcite , t. kato suggested that the object is expected to undergo multiple rebrightenings as in eg cnc based on relatively small @xmath7 obtained from early observations and a relatively long @xmath3 ( vsnet - alert 11492 ) . the object indeed underwent five rebrightenings ( vsnet - alert 11521 , 11526 , 11536 , 11577 , 11602 ; figure [ fig : vxforlc ] ) , making it the first object predicted for its multiple rebrightening in real - time . the overall behavior is very similar to that of asas [email protected] @xcite . the mean @xmath3 during the plateau phase was 0.061355(7 ) d ( pdm method , figure [ fig : vxforshpdm ] ) . the times of superhump maxima during the plateau phase and fading stage are listed in table [ tab : vxforoc2009 ] . the superhumps persisted during the post - superoutburst and rebrightening phase . we obtained @xmath31 d , slightly short than that of the superoutburst plateau phase ( figure [ fig : vxforlatepdm ] ) . there was some indication of a signal around @xmath32 d ( small excess signal in figure [ fig : vxforlatepdm ] ) and transient appearance of this period ( cf . vsnet - alert 11513 ) . although this signal needs to be confirmed with better data , this might be analogous to long - period late - stage superhumps in wz sge - type dwarf novae ( @xcite ; @xcite ) . ( 88mm,70mm)fig16.eps ( 88mm,110mm)fig17.eps ( 88mm,110mm)fig18.eps ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 55090.0917 & 0.0002 & @xmath170.0100 & 991 + 6 & 55090.4665 & 0.0004 & @xmath170.0026 & 258 + 7 & 55090.5274 & 0.0004 & @xmath170.0029 & 258 + 8 & 55090.5873 & 0.0003 & @xmath170.0042 & 230 + 19 & 55091.2630 & 0.0007 & @xmath170.0019 & 104 + 20 & 55091.3216 & 0.0006 & @xmath170.0046 & 104 + 21 & 55091.3893 & 0.0004 & 0.0019 & 252 + 22 & 55091.4483 & 0.0003 & @xmath170.0003 & 229 + 23 & 55091.5073 & 0.0015 & @xmath170.0025 & 51 + 24 & 55091.5683 & 0.0015 & @xmath170.0028 & 116 + 32 & 55092.0586 & 0.0010 & @xmath170.0022 & 42 + 33 & 55092.1223 & 0.0005 & 0.0003 & 68 + 34 & 55092.1822 & 0.0009 & @xmath170.0011 & 66 + 35 & 55092.2450 & 0.0008 & 0.0005 & 84 + 36 & 55092.3078 & 0.0007 & 0.0021 & 30 + 48 & 55093.0420 & 0.0003 & 0.0016 & 349 + 49 & 55093.0999 & 0.0003 & @xmath170.0018 & 127 + 50 & 55093.1617 & 0.0003 & @xmath170.0012 & 159 + 51 & 55093.2244 & 0.0006 & 0.0003 & 95 + 55 & 55093.4687 & 0.0003 & @xmath170.0003 & 223 + 56 & 55093.5293 & 0.0003 & @xmath170.0009 & 261 + 57 & 55093.5909 & 0.0003 & @xmath170.0006 & 261 + 66 & 55094.1453 & 0.0015 & 0.0028 & 51 + 67 & 55094.2044 & 0.0011 & 0.0007 & 41 + 68 & 55094.2659 & 0.0008 & 0.0010 & 162 + 69 & 55094.3264 & 0.0018 & 0.0002 & 68 + 85 & 55095.3094 & 0.0012 & 0.0037 & 126 + 87 & 55095.4308 & 0.0008 & 0.0026 & 258 + 88 & 55095.4932 & 0.0006 & 0.0038 & 214 + 89 & 55095.5545 & 0.0005 & 0.0039 & 213 + 99 & 55096.1696 & 0.0013 & 0.0068 & 52 + 100 & 55096.2289 & 0.0011 & 0.0049 & 46 + 101 & 55096.2960 & 0.0020 & 0.0108 & 33 + 115 & 55097.1400 & 0.0049 & @xmath170.0024 & 24 + 132 & 55098.1979 & 0.0017 & 0.0147 & 22 + 133 & 55098.2561 & 0.0012 & 0.0117 & 32 + 137 & 55098.4994 & 0.0014 & 0.0100 & 258 + 149 & 55099.2215 & 0.0059 & @xmath170.0025 & 23 + 185 & 55101.4220 & 0.0094 & @xmath170.0061 & 134 + 186 & 55101.4843 & 0.0031 & @xmath170.0050 & 134 + 187 & 55101.5307 & 0.0027 & @xmath170.0198 & 134 + 201 & 55102.3908 & 0.0103 & @xmath170.0169 & 125 + 202 & 55102.4749 & 0.0167 & 0.0061 & 135 + 203 & 55102.5323 & 0.0018 & 0.0022 & 135 + 204 & 55102.5907 & 0.0018 & @xmath170.0006 & 133 + + + + the times of superhump maxima during the 2010 superoutburst is listed in table [ tab : awgemoc2010 ] . a stage b c transition was well recorded . although only parts of superoutbursts were observed in individual years , the combined @xmath0 diagram ( figure [ fig : awgemcomp ] ) clearly demonstrates the universal pattern of period evolution . ( 88mm,70mm)fig19.eps ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 55257.3860 & 0.0002 & @xmath170.0033 & 72 + 1 & 55257.4640 & 0.0003 & @xmath170.0040 & 40 + 2 & 55257.5425 & 0.0004 & @xmath170.0042 & 69 + 24 & 55259.2830 & 0.0004 & 0.0040 & 81 + 25 & 55259.3619 & 0.0004 & 0.0041 & 80 + 26 & 55259.4398 & 0.0005 & 0.0033 & 80 + 51 & 55261.4091 & 0.0004 & 0.0041 & 69 + 52 & 55261.4873 & 0.0005 & 0.0035 & 61 + 53 & 55261.5602 & 0.0012 & @xmath170.0024 & 37 + 101 & 55265.3439 & 0.0029 & 0.0018 & 43 + 102 & 55265.4189 & 0.0008 & @xmath170.0020 & 78 + 103 & 55265.4946 & 0.0006 & @xmath170.0050 & 79 + + + + the times of superhump maxima during the 2010 superoutburst is listed in table [ tab : irgemoc2010 ] . although the object was observed on only two nights , the period is in good agreement with earlier observations . the observations was likely performed during stage b. ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 55265.4201 & 0.0004 & 0.0003 & 65 + 1 & 55265.4903 & 0.0004 & @xmath170.0003 & 67 + 56 & 55269.3867 & 0.0003 & 0.0002 & 71 + 57 & 55269.4573 & 0.0004 & @xmath170.0000 & 73 + 58 & 55269.5280 & 0.0007 & @xmath170.0002 & 50 + + + + v592 her was discovered as a possible fast nova in 1968 on sonneberg plates @xcite . @xcite further discovered a second outburst in 1986 on historical plates . in 1998 , another outburst was recorded which led to the identification of this object as being a wz sge - type dwarf nova ( @xcite ; @xcite ) . @xcite first identified the true superhump period of 0.05648(2 ) d , although their observation was not very sufficient to determine its period variation . the 2010 outburst of this object was detected by m. reszelski at a relatively faint magnitude of 14.16 ( unfiltered ccd magnitude , vsnet - obs 67929 ) . subsequent observations detected growing superhumps ( vsnet - alert 12092 , 12094 , 12095 ) . this new outburst has confirmed the selection of the superhump period in ( @xcite ; figure [ fig : v592hershpdm ] ) . the times of superhump maxima are listed in table [ tab : v592heroc2010 ] . all a c stages for superhump evolution were clearly detected ( figure [ fig : v592heroc2010 ] ) . we obtained a clearly positive @xmath7 of @xmath33 during stage b , which is likely a more reliable value than the 1998 estimate thanks to the greatly improved statistics . as judged from this relatively large period derivative and the presence of stage c evolution , this object is less likely an extreme wz sge - type dwarf nova with a very small period variation what was supposed from the 1998 data @xcite . the relatively short ( @xmath9 12 yr ) recurrence time would qualify the system as a wz sge - type dwarf nova similar to hv vir @xcite . the lack of repetitive rebrightenings would also support this classification . there was a weaker signal slightly shorter than the superhump period ( figure [ fig : v592hershpdm ] ) , which might be attributed to the orbital period . although the detection was not statistically significant , a similar periodicity may have been present before the appearance of ordinary superhumps ( figure [ fig : v592hereshpdm ] ) . since this period agrees with the candidate period from radial - velocity study ( @xcite ; see @xcite for the alias selection ) , we adopted it as a candidate for the orbital period , which corresponds to a fractional superhump excess of 0.9 % . ( 88mm,110mm)fig20.eps ( 88mm,90mm)fig21.eps ( 88mm,110mm)fig22.eps ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 55413.7374 & 0.0014 & @xmath170.0009 & 53 + 1 & 55413.7957 & 0.0016 & 0.0008 & 53 + 7 & 55414.1260 & 0.0019 & @xmath170.0086 & 82 + 12 & 55414.4196 & 0.0008 & 0.0018 & 53 + 13 & 55414.4805 & 0.0012 & 0.0061 & 54 + 17 & 55414.7105 & 0.0005 & 0.0096 & 48 + 18 & 55414.7679 & 0.0003 & 0.0105 & 53 + 35 & 55415.7294 & 0.0004 & 0.0094 & 52 + 36 & 55415.7856 & 0.0006 & 0.0089 & 38 + 40 & 55416.0066 & 0.0015 & 0.0035 & 58 + 41 & 55416.0661 & 0.0023 & 0.0063 & 79 + 42 & 55416.1268 & 0.0061 & 0.0104 & 127 + 46 & 55416.3490 & 0.0003 & 0.0062 & 33 + 47 & 55416.4049 & 0.0002 & 0.0054 & 104 + 48 & 55416.4612 & 0.0003 & 0.0051 & 71 + 49 & 55416.5176 & 0.0006 & 0.0049 & 42 + 51 & 55416.6290 & 0.0003 & 0.0031 & 62 + 52 & 55416.6836 & 0.0005 & 0.0010 & 69 + 53 & 55416.7427 & 0.0004 & 0.0035 & 52 + 57 & 55416.9637 & 0.0049 & @xmath170.0020 & 71 + 58 & 55417.0236 & 0.0006 & 0.0013 & 175 + 59 & 55417.0835 & 0.0017 & 0.0046 & 89 + 60 & 55417.1378 & 0.0013 & 0.0022 & 82 + 61 & 55417.1935 & 0.0064 & 0.0013 & 50 + 64 & 55417.3606 & 0.0005 & @xmath170.0015 & 89 + 65 & 55417.4180 & 0.0005 & @xmath170.0007 & 143 + 66 & 55417.4725 & 0.0009 & @xmath170.0027 & 55 + 70 & 55417.7001 & 0.0008 & @xmath170.0017 & 35 + 71 & 55417.7584 & 0.0005 & 0.0000 & 51 + 88 & 55418.7157 & 0.0010 & @xmath170.0053 & 52 + 89 & 55418.7719 & 0.0005 & @xmath170.0057 & 51 + 94 & 55419.0581 & 0.0008 & @xmath170.0026 & 27 + 95 & 55419.1071 & 0.0019 & @xmath170.0102 & 68 + 96 & 55419.1677 & 0.0021 & @xmath170.0062 & 66 + 99 & 55419.3370 & 0.0008 & @xmath170.0068 & 61 + 100 & 55419.3968 & 0.0009 & @xmath170.0036 & 71 + 101 & 55419.4498 & 0.0009 & @xmath170.0072 & 75 + 102 & 55419.5057 & 0.0004 & @xmath170.0080 & 62 + 103 & 55419.5641 & 0.0006 & @xmath170.0061 & 38 + 104 & 55419.6192 & 0.0007 & @xmath170.0077 & 62 + 105 & 55419.6739 & 0.0006 & @xmath170.0096 & 72 + 106 & 55419.7355 & 0.0009 & @xmath170.0046 & 53 + 117 & 55420.3571 & 0.0011 & @xmath170.0058 & 62 + 118 & 55420.4101 & 0.0011 & @xmath170.0095 & 69 + 119 & 55420.4693 & 0.0011 & @xmath170.0069 & 20 + 120 & 55420.5254 & 0.0006 & @xmath170.0074 & 52 + 130 & 55421.0909 & 0.0033 & @xmath170.0081 & 62 + 135 & 55421.3703 & 0.0018 & @xmath170.0119 & 60 + 136 & 55421.4374 & 0.0021 & @xmath170.0013 & 68 + 141 & 55421.7181 & 0.0012 & @xmath170.0037 & 52 + 142 & 55421.7706 & 0.0016 & @xmath170.0078 & 41 + 159 & 55422.7367 & 0.0016 & @xmath170.0043 & 52 + 176 & 55423.6891 & 0.0089 & @xmath170.0144 & 42 + 181 & 55423.9909 & 0.0046 & 0.0042 & 149 + 182 & 55424.0429 & 0.0032 & @xmath170.0004 & 113 + 183 & 55424.1055 & 0.0044 & 0.0056 & 60 + 194 & 55424.7301 & 0.0029 & 0.0073 & 52 + + + + ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 195 & 55424.7736 & 0.0030 & @xmath170.0058 & 42 + 209 & 55425.5776 & 0.0017 & 0.0056 & 47 + 210 & 55425.6348 & 0.0026 & 0.0061 & 59 + 212 & 55425.7491 & 0.0019 & 0.0072 & 53 + 216 & 55425.9826 & 0.0042 & 0.0142 & 162 + 229 & 55426.7172 & 0.0027 & 0.0127 & 53 + 230 & 55426.7657 & 0.0012 & 0.0046 & 45 + 234 & 55426.9957 & 0.0026 & 0.0081 & 82 + 235 & 55427.0454 & 0.0012 & 0.0012 & 113 + 247 & 55427.7308 & 0.0014 & 0.0071 & 53 + 252 & 55428.0234 & 0.0046 & 0.0166 & 62 + 264 & 55428.6724 & 0.0063 & @xmath170.0138 & 36 + 265 & 55428.7468 & 0.0027 & 0.0039 & 52 + 282 & 55429.7099 & 0.0048 & 0.0045 & 52 + 283 & 55429.7621 & 0.0097 & 0.0000 & 52 + 300 & 55430.7126 & 0.0048 & @xmath170.0120 & 50 + + + + the times of superhump maxima during the 2009 superoutburst are listed in table [ tab : v660heroc2009 ] . since the late stage of the superoutburst was only observed , the recorded superhumps are likely stage c superhumps . ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 55056.4789 & 0.0008 & @xmath170.0032 & 30 + 12 & 55057.4501 & 0.0007 & @xmath170.0004 & 36 + 25 & 55058.5061 & 0.0019 & 0.0067 & 9 + 49 & 55060.4337 & 0.0049 & @xmath170.0023 & 17 + 61 & 55061.4035 & 0.0098 & @xmath170.0008 & 23 + + + + two further superoutbursts in 2009 february march ( table [ tab : v844heroc2009 ] ) and in 2010 april may ( table [ tab : v844heroc2010 ] ; see also vsnet - alert 11959 for the outburst detection ) were observed . both superoutbursts were relatively faint and short ones . during the former superoutburst we obtained a clearly positive @xmath7 of @xmath34 ( likely for stage b ) . the object underwent yet another superoutburst between them : 2009 october november ( vsnet - alert 11622 ) . there was also a normal outburst in 2009 june ( vsnet - alert 11289 , 11290 ) . the object was thus unusually active for this star ( cf . @xcite ; @xcite ) . as suggested in @xcite and @xcite , these superoutburst would provide an excellent opportunity to study the dependence of period derivatives on the extent of the superoutburst . as already seen , the @xmath7 of the 2009 superoutburst is not significantly different from those of longer superoutbursts . although the superhumps had already developed at the epoch of initial observation ( @xmath9 1.5 d after the initial outburst detection ) , we could nt severely constrain the delay time in evolution of superhumps since there was a three - day gap of observation before this detection . the 2010 observation started @xmath9 1 d after the rising phase of the outburst , and superhumps were already present . although these superhumps were possibly stage a superhumps ( see figure [ fig : v844hercomp2 ] ) , this observations seems to support the hypothesis in @xcite that the delay time in development of superhumps is shorter in smaller superoutbursts . ( 88mm,70mm)fig23.eps ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 54887.4721 & 0.0003 & 0.0024 & 87 + 1 & 54887.5268 & 0.0002 & 0.0012 & 104 + 2 & 54887.5827 & 0.0002 & 0.0012 & 109 + 3 & 54887.6379 & 0.0002 & 0.0005 & 125 + 72 & 54891.4923 & 0.0003 & @xmath170.0038 & 101 + 73 & 54891.5472 & 0.0003 & @xmath170.0049 & 108 + 74 & 54891.6031 & 0.0004 & @xmath170.0049 & 108 + 90 & 54892.4986 & 0.0008 & @xmath170.0042 & 105 + 91 & 54892.5626 & 0.0008 & 0.0039 & 108 + 92 & 54892.6132 & 0.0006 & @xmath170.0014 & 107 + 93 & 54892.6681 & 0.0009 & @xmath170.0024 & 93 + 108 & 54893.5116 & 0.0006 & 0.0023 & 106 + 109 & 54893.5667 & 0.0005 & 0.0014 & 103 + 110 & 54893.6264 & 0.0023 & 0.0052 & 60 + 111 & 54893.6807 & 0.0015 & 0.0035 & 51 + + + + ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 55316.1758 & 0.0002 & @xmath170.0000 & 101 + 53 & 55319.1315 & 0.0002 & 0.0001 & 134 + 54 & 55319.1870 & 0.0002 & @xmath170.0001 & 177 + + + + the times of superhump maxima during the 2010 superoutburst is listed in table [ tab : cthyaoc2010 ] . this observation first time recorded a stage a b transition and growth of superhumps . ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 55269.0068 & 0.0049 & @xmath170.0021 & 91 + 1 & 55269.0698 & 0.0024 & @xmath170.0057 & 143 + 14 & 55269.9458 & 0.0013 & 0.0047 & 86 + 15 & 55270.0122 & 0.0005 & 0.0045 & 129 + 90 & 55275.0002 & 0.0009 & @xmath170.0014 & 142 + + + + our new observation of the 2010 superoutburst ( table [ tab : v699ophoc2010 ] ) clearly caught a stage b c transition . a comparison of @xmath0 diagrams is shown in figure [ fig : v699ophcomp ] . ( 88mm,70mm)fig24.eps ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 55362.5280 & 0.0023 & @xmath170.0044 & 12 + 14 & 55363.5125 & 0.0007 & @xmath170.0008 & 77 + 22 & 55364.0674 & 0.0041 & @xmath170.0064 & 47 + 27 & 55364.4254 & 0.0016 & 0.0014 & 41 + 28 & 55364.4981 & 0.0007 & 0.0039 & 68 + 42 & 55365.4813 & 0.0007 & 0.0063 & 65 + 56 & 55366.4592 & 0.0008 & 0.0033 & 69 + 57 & 55366.5288 & 0.0008 & 0.0029 & 60 + 70 & 55367.4392 & 0.0013 & 0.0025 & 62 + 71 & 55367.5086 & 0.0008 & 0.0018 & 62 + 84 & 55368.4083 & 0.0132 & @xmath170.0092 & 39 + 85 & 55368.4882 & 0.0020 & 0.0005 & 67 + 99 & 55369.4666 & 0.0047 & @xmath170.0019 & 65 + + + + v1032 oph was originally discovered as a candidate rr lyr - type variable star @xcite . based on its bright uv emission , @xcite conducted a systematic observation which led to a conclusion that the object is a likely su uma - type dwarf nova . the 2010 outburst was detected by e. muyllaert ( vsnet - alert 11898 ) . e. de miguel and h. maehara confirmed that this object is an eclipsing su uma - type dwarf nova ( vsnet - alert 11904 , 11905 ) . the times of recorded eclipses , determined with the kwee and van woerden ( kw ) method @xcite , are summarized in table [ tab : v1032ophecl ] . we obtained an ephemeris of @xmath35 in the following analysis , we removed observations within 0.07 @xmath2 of eclipses . the times of superhump maxima are listed in table [ tab : v1032ophoc2010 ] . due to the faintness of the object and overlapping eclipsing feature , the scatter is relatively large . the last maximum ( @xmath36 ) may have been a traditional late superhump with an @xmath90.5 phase offset , when the observation was performed after a brightness drop from the superoutburst . a linear fit to the observed epochs for @xmath37 yielded a mean period of 0.08529(12 ) d. this period is in agreement with 0.08534(5 ) d determined with the pdm method ( figure [ fig : v1032ophshpdm ] ) . we adopted the latter because of a smaller error . although the period derivative could not be unambiguously determined , restricting to well - defined maxima ( @xmath38 ) and allowing a large period variation ( as in mn dra ) , we can derive a global @xmath7 of @xmath39 . this value needs to be confirmed by future observations . the fractional superhump excess amounts to 5.3 % , which is one of the largest among su uma - type dwarf novae below the period gap ( cf . subsection [ obj : gzcnc ] ) . ( 88mm,110mm)fig25.eps cccc @xmath15 & minimum & error & @xmath0 + 0 & 55286.6829 & 0.0008 & 0.0002 + 32 & 55289.2765 & 0.0006 & 0.0001 + 49 & 55290.6544 & 0.0007 & -0.0000 + 69 & 55292.2763 & 0.0010 & 0.0008 + 74 & 55292.6809 & 0.0006 & 0.0001 + 89 & 55293.8967 & 0.0005 & 0.0000 + 90 & 55293.9775 & 0.0004 & -0.0002 + 99 & 55294.7075 & 0.0006 & 0.0002 + 102 & 55294.9505 & 0.0004 & 0.0001 + 103 & 55295.0310 & 0.0006 & -0.0004 + 111 & 55295.6801 & 0.0006 & 0.0001 + 114 & 55295.9231 & 0.0004 & 0.0000 + 115 & 55296.0041 & 0.0005 & -0.0001 + 122 & 55296.5710 & 0.0005 & -0.0005 + 123 & 55296.6517 & 0.0004 & -0.0009 + 212 & 55303.8664 & 0.0011 & -0.0002 + 213 & 55303.9478 & 0.0009 & 0.0001 + 238 & 55305.9747 & 0.0006 & 0.0006 + 296 & 55310.6754 & 0.0005 & 0.0001 + + + ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 55286.6842 & 0.0023 & 0.0062 & 74 + 24 & 55288.7046 & 0.0018 & @xmath170.0174 & 24 + 30 & 55289.2294 & 0.0011 & @xmath170.0036 & 167 + 41 & 55290.1909 & 0.0040 & 0.0210 & 126 + 43 & 55290.3086 & 0.0017 & @xmath170.0316 & 73 + 47 & 55290.6862 & 0.0006 & 0.0053 & 73 + 65 & 55292.2223 & 0.0023 & 0.0083 & 147 + 66 & 55292.3013 & 0.0072 & 0.0022 & 141 + 85 & 55293.9201 & 0.0012 & 0.0028 & 34 + 86 & 55294.0063 & 0.0015 & 0.0039 & 32 + 94 & 55294.6949 & 0.0015 & 0.0110 & 65 + 97 & 55294.9549 & 0.0018 & 0.0156 & 37 + 98 & 55295.0222 & 0.0083 & @xmath170.0023 & 26 + 100 & 55295.2186 & 0.0035 & 0.0238 & 37 + 106 & 55295.6961 & 0.0060 & @xmath170.0097 & 60 + 117 & 55296.6072 & 0.0006 & @xmath170.0355 & 69 + + + + the times of superhump maxima during the 2010 superoutburst are listed in table [ tab : v2051ophoc2010 ] . although the outburst was detected during its relatively early stage ( vsnet - alert 12049 ) , the subsequent observational coverage was rather insufficient . the observed superhumps were likely stage b superhumps . we could not meaningfully determine @xcite . ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 55382.9240 & 0.0003 & 0.0014 & 107 + 1 & 55382.9876 & 0.0003 & 0.0008 & 104 + 2 & 55383.0492 & 0.0002 & @xmath170.0019 & 78 + 3 & 55383.1165 & 0.0003 & 0.0012 & 94 + 14 & 55383.8220 & 0.0014 & 0.0000 & 34 + 17 & 55384.0142 & 0.0006 & @xmath170.0004 & 92 + 18 & 55384.0865 & 0.0005 & 0.0076 & 89 + 34 & 55385.1019 & 0.0009 & @xmath170.0048 & 71 + 35 & 55385.1613 & 0.0027 & @xmath170.0096 & 40 + 65 & 55387.1037 & 0.0013 & 0.0057 & 37 + + + + ef peg is one of representatives of long-@xmath3 su uma - type dwarf novae with infrequent outbursts . @xcite reported mildly negative @xmath7 for the 1991 and 1997 superoutbursts . the object underwent an outburst in 2009 december ( vsnet - alert 11738 , baavss - alert 2177 , vsnet - alert 11740 ) , first time since its 2001 superoutburst . the outburst was caught during its early stage and the development of superhumps was recorded . the times of superhump maxima are listed in table [ tab : efpegoc2009 ] . due to the short visibility in the evening sky , the number of maxima was relatively small . the @xmath0 diagram , however , clearly shows the presence of stage a during the evolutionary stage of superhumps ( @xmath40 ) , and subsequent phase of a slow period decrease . we attributed the latter phase to a transition from stage b to c and identified the periods in table [ tab : perlist ] . a comparison of @xmath0 variations between different superoutbursts is presented in figure [ fig : efpegcomp2 ] . among long-@xmath3 systems , the @xmath0 variation looks similar to those of ax cap and sdss j1627 ( cf . @xcite ) with a discontinuous period variation between stages b and c. it would be noteworthy that all systems are known to show only rare superoutbursts . ( 88mm,70mm)fig26.eps ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 55187.3273 & 0.0007 & @xmath170.0296 & 229 + 6 & 55187.8804 & 0.0008 & 0.0006 & 353 + 7 & 55187.9609 & 0.0007 & @xmath170.0061 & 521 + 10 & 55188.2402 & 0.0005 & 0.0117 & 287 + 11 & 55188.3234 & 0.0004 & 0.0078 & 277 + 30 & 55189.9852 & 0.0014 & 0.0135 & 296 + 68 & 55193.2913 & 0.0005 & 0.0073 & 268 + 75 & 55193.8931 & 0.0008 & @xmath170.0010 & 464 + 79 & 55194.2507 & 0.0005 & 0.0080 & 80 + 87 & 55194.9431 & 0.0008 & 0.0031 & 440 + 125 & 55198.2372 & 0.0006 & @xmath170.0151 & 77 + + + + we also observed the 2009 superoutburst . this outburst was one of the brightest in recent years ( cf . vsnet - alert 11507 ) . the times of superhump maxima are listed in table [ tab : v368pegoc2009 ] , which clearly shows a stage b c transition . the superhumps were not very apparent on the first night ( bjd 2455102 ) , and it was likely that the development of superhumps took more than 1 d. the relatively large @xmath7 for stage b strongly depends on @xmath41 , and may not be real . a comparison of @xmath0 diagrams between different superoutbursts is shown in figure [ fig : v368pegcomp2 ] . despite its brightness , the behavior of the 2009 superoutburst was not strikingly different from that of other superoutbursts . ( 88mm,70mm)fig27.eps ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 55102.8556 & 0.0009 & @xmath170.0046 & 109 + 35 & 55105.3047 & 0.0030 & @xmath170.0110 & 31 + 36 & 55105.3797 & 0.0003 & @xmath170.0062 & 73 + 37 & 55105.4512 & 0.0004 & @xmath170.0048 & 69 + 38 & 55105.5209 & 0.0005 & @xmath170.0053 & 43 + 40 & 55105.6602 & 0.0008 & @xmath170.0063 & 24 + 55 & 55106.7133 & 0.0003 & @xmath170.0056 & 110 + 97 & 55109.6732 & 0.0006 & 0.0078 & 153 + 98 & 55109.7456 & 0.0005 & 0.0100 & 156 + 99 & 55109.8159 & 0.0004 & 0.0102 & 148 + 113 & 55110.7965 & 0.0006 & 0.0087 & 36 + 114 & 55110.8657 & 0.0006 & 0.0076 & 33 + 115 & 55110.9358 & 0.0008 & 0.0076 & 26 + 125 & 55111.6361 & 0.0004 & 0.0064 & 74 + 126 & 55111.7060 & 0.0004 & 0.0061 & 102 + 127 & 55111.7756 & 0.0006 & 0.0056 & 56 + 128 & 55111.8544 & 0.0065 & 0.0142 & 13 + 137 & 55112.4750 & 0.0006 & 0.0033 & 57 + 138 & 55112.5466 & 0.0007 & 0.0049 & 73 + 177 & 55115.2731 & 0.0007 & @xmath170.0048 & 49 + 178 & 55115.3412 & 0.0009 & @xmath170.0068 & 46 + 179 & 55115.4090 & 0.0016 & @xmath170.0092 & 50 + 180 & 55115.4822 & 0.0018 & @xmath170.0061 & 50 + 181 & 55115.5469 & 0.0018 & @xmath170.0115 & 49 + 220 & 55118.2843 & 0.0009 & @xmath170.0103 & 30 + + + + we observed the 2010 superoutburst during this middle and final stages ( table [ tab : uvperoc2010 ] ) . although a stage b c transition was recorded , the long gap in observation hindered precise determination of periods . we only list representative values for stage b. late - stage superhumps superimposed on the rapid fading from the superoutburst plateau were clearly recorded as in the 1992 superoutburst . a comparison of @xmath0 diagrams between different superoutbursts is shown in figure [ fig : uvpercomp2 ] . this figure is an improvement of the corresponding one presented in @xcite . ( 88mm,70mm)fig28.eps ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 55203.4188 & 0.0003 & @xmath170.0041 & 74 + 1 & 55203.4864 & 0.0004 & @xmath170.0029 & 74 + 2 & 55203.5521 & 0.0003 & @xmath170.0037 & 73 + 3 & 55203.6189 & 0.0004 & @xmath170.0033 & 62 + 13 & 55204.2845 & 0.0004 & @xmath170.0016 & 71 + 15 & 55204.4177 & 0.0005 & @xmath170.0012 & 40 + 16 & 55204.4847 & 0.0004 & @xmath170.0006 & 46 + 17 & 55204.5518 & 0.0006 & 0.0001 & 34 + 29 & 55205.3527 & 0.0005 & 0.0042 & 41 + 30 & 55205.4212 & 0.0005 & 0.0063 & 35 + 31 & 55205.4878 & 0.0006 & 0.0065 & 37 + 32 & 55205.5524 & 0.0009 & 0.0047 & 28 + 133 & 55212.2538 & 0.0004 & @xmath170.0001 & 63 + 134 & 55212.3179 & 0.0008 & @xmath170.0024 & 67 + 135 & 55212.3847 & 0.0009 & @xmath170.0019 & 62 + 136 & 55212.4535 & 0.0018 & 0.0004 & 48 + 137 & 55212.5190 & 0.0011 & @xmath170.0004 & 62 + + + + we observed the 2009 superoutburst first detected by asas-3 on june 8 at @xmath42 . due to the poor seasonal condition , we could only determine the mean superhump period of 0.04635(5 ) d ( with the pdm method ) , in agreement with previous measurements . the times of superhump maxima are listed in table [ tab : eipscoc2009 ] . the overall light curve of the outburst suggests that the main superoutburst plateau lasted less than 10 d and experienced a rebrightening on june 18 . the course of the outburst could have been similar to the 2005 one ( @xcite ; @xcite ) . the shortness of the superoutburst plateau in such short-@xmath2 systems with evolved secondaries would require a special explanation . ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 54994.1841 & 0.0011 & 0.0006 & 44 + 1 & 54994.2292 & 0.0005 & @xmath170.0006 & 67 + 10 & 54994.6467 & 0.0004 & 0.0001 & 29 + + + + we observed the 2009 superoutburst . a clear stae a b transition was recorded ( table [ tab : ektraoc2009 ] ) . although the period variation was alost absent during the supposed stage b ( @xmath43 ) , this may have been a result of fragmentary observations of stages b and c. ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 55027.0310 & 0.0029 & @xmath170.0009 & 74 + 1 & 55027.0787 & 0.0026 & @xmath170.0181 & 118 + 29 & 55028.9255 & 0.0002 & 0.0101 & 159 + 30 & 55028.9901 & 0.0003 & 0.0098 & 100 + 76 & 55031.9722 & 0.0003 & 0.0043 & 180 + 77 & 55032.0358 & 0.0003 & 0.0030 & 184 + 137 & 55035.9273 & 0.0011 & @xmath170.0023 & 108 + 138 & 55035.9924 & 0.0006 & @xmath170.0022 & 137 + 139 & 55036.0559 & 0.0005 & @xmath170.0036 & 139 + + + + the 2010 january superoutburst was observed for its early and late stages ( table [ tab : suumaoc2010 ] ) . although we could only measure the mean period of stage b , the value is in agreement with those obtained during previous superoutbursts . the humps observed for @xmath44 may not be genuine superhumps . ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 55219.5098 & 0.0002 & @xmath170.0003 & 132 + 1 & 55219.5883 & 0.0002 & @xmath170.0009 & 149 + 2 & 55219.6689 & 0.0002 & 0.0005 & 150 + 7 & 55220.0702 & 0.0014 & 0.0061 & 89 + 11 & 55220.3809 & 0.0002 & 0.0002 & 145 + 12 & 55220.4613 & 0.0002 & 0.0014 & 156 + 13 & 55220.5392 & 0.0002 & 0.0002 & 153 + 14 & 55220.6182 & 0.0004 & @xmath170.0001 & 157 + 15 & 55220.6987 & 0.0003 & 0.0013 & 122 + 25 & 55221.4871 & 0.0004 & @xmath170.0018 & 155 + 26 & 55221.5668 & 0.0004 & @xmath170.0012 & 151 + 27 & 55221.6454 & 0.0004 & @xmath170.0018 & 154 + 50 & 55223.4634 & 0.0004 & @xmath170.0044 & 96 + 51 & 55223.5457 & 0.0004 & @xmath170.0012 & 118 + 163 & 55232.4126 & 0.0012 & 0.0006 & 75 + 164 & 55232.4926 & 0.0012 & 0.0014 & 83 + + + + bc uma underwent a superoutburst in 2009 september october after a period of 6.7 yr ( cf . vsnet - alert 11514 ) . the start of the outburst was not well constrained dur to the poor visibility in the morning sky . superhumps were observed despite unfavorable seasonal conditions ( cf . vsnet - alert 11540 , 11550 , 11559 ) . the times of superhump maxima are listed in table [ tab : bcumaoc2009 ] . there was a clear stage b c transition and the @xmath7 for stage b ( @xmath45 ) was @xmath46 . we attributed the interval @xmath47 to likely stage a superhumps based on comparison with other superoutbursts ( figure [ fig : bcumacomp2 ] ) . if this identification is correct , the superhumps during this superoutburst appears to have taken a longer time to fully develop . ( 88mm,70mm)fig29.eps ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 55105.2681 & 0.0022 & @xmath170.0189 & 70 + 2 & 55105.3853 & 0.0013 & @xmath170.0310 & 43 + 56 & 55108.9240 & 0.0016 & 0.0181 & 182 + 92 & 55111.2424 & 0.0004 & 0.0101 & 49 + 125 & 55113.3725 & 0.0007 & 0.0076 & 45 + 126 & 55113.4388 & 0.0004 & 0.0093 & 84 + 138 & 55114.2105 & 0.0021 & 0.0056 & 40 + 139 & 55114.2800 & 0.0004 & 0.0104 & 86 + 140 & 55114.3451 & 0.0005 & 0.0109 & 72 + 141 & 55114.4106 & 0.0005 & 0.0117 & 85 + 142 & 55114.4740 & 0.0005 & 0.0105 & 86 + 143 & 55114.5395 & 0.0005 & 0.0114 & 87 + 144 & 55114.6037 & 0.0004 & 0.0109 & 87 + 170 & 55116.2768 & 0.0005 & 0.0039 & 113 + 203 & 55118.3985 & 0.0013 & @xmath170.0070 & 67 + 204 & 55118.4593 & 0.0007 & @xmath170.0108 & 72 + 205 & 55118.5248 & 0.0006 & @xmath170.0099 & 73 + 206 & 55118.5888 & 0.0005 & @xmath170.0105 & 73 + 207 & 55118.6512 & 0.0014 & @xmath170.0128 & 45 + 219 & 55119.4198 & 0.0007 & @xmath170.0196 & 46 + + + + although el uma was discovered as an eruptive object relatively early in the history @xcite , only little had been known until recent years . @xcite listed this object among candidate wz sge - type dwarf novae . based on the similarity of its quiescent sdss color to those of known wz sge - type dwarf novae , we started monitoring since 2008 . another outburst at @xmath48 = 13.7 in 2003 april was found on an archival image @xcite . on 2009 january 13 , h. maehara finally detected this object in outburst at an unfiltered ccd magnitude of 17.4 ( vsnet - alert 11771 ) the object further brightened to a magnitude of 14.9 on january 16 ( vsnet - alert 11772 ) . the object has been confirmed to be a hydrogen - rich dwarf niva in outburst by spectroscopy ( takahashi and kinugasa , private communication ) . although the nature of this brightening was unclear at the time , the detection of additional sequence of outbursts ( vsnet - alert 11789 , 11808 ) , led to an interpretation that they are post - superoutburst rebrightenings of a wz sge - type superoutburst , whose main superoutburst was missed ( vsnet - alert 11795 ) . the detection of modulations attributable to superhumps seems to strengthen this interpretation ( vsnet - alert 11799 ) . we include this object based on this interpretation . a period analysis of the rebrightening phase , after subtracting the trends of outbursts ( cf . @xcite ) , has yielded a period of 0.06045(6 ) d ( figure [ fig : elumashpdm ] ) . although this period needs to be confirmed by future observations , its relatively long @xmath3 appears to be similar to that of vx for ( subsection [ obj : vxfor ] ) that underwent multiple rebrightenings ( figure [ fig : elumalc ] ) . there is a common tendency that the quiescent interval preceding the last rebrightening is longer than the other intervals between rebrightenings . this object is a good candidate for a cv passed the period minimum in evolution ( see a discussion in @xcite ) . further radial - velocity study in quiescence in encouraged in order to determine the orbital period and the nature of the period during the rebrightening phase . ( 88mm,110mm)fig30.eps ( 88mm,110mm)fig31.eps although the 2009 superoutburst of this object was reported in @xcite , we provide greatly improved results by combining newly available observations . the times of superhump maxima ( table [ tab : iyumaoc2009 ] ) now clearly illustrate the presence of all a c stages , and a definitely positive @xmath7 = @xmath49 for stage b superhumps , confirming the suggestion in @xcite based on the combined @xmath0 diagram ( cf . figure [ fig : iyumacomp2 ] ) . although there was a signature of distinct stages during stage a , we listed a mean period in table [ tab : perlist ] . there was a very clear signature of a stage b c transition thanks to the high quality of data . although the phases of superhump at late epochs ( @xmath50 ) deviate from extrapolations of stage c superhumps , they can still be interpreted as a continuation of stage c superhump , rather than traditional late superhumps . this object showed a strong `` textbook '' beat phenomenon between superhumps and orbital modulations ( figure [ fig : iyumabeatoc ] ) . the period of the beat phenomenon was shorter ( @xmath9 2.5 d ) during stage b , while it became longer ( @xmath9 3.0 d ) during stage c. these periods are in very good agreement with the expected beat periods for stage b and c superhumps , 2.45 d and 3.08 d , respectively . this can be understood if the variation of the beat period reflects the variation of the angular velocity of the apsidal motion of the elliptical accretion disk . the close correlation between the beat period and the superhump period suggests that the change in the angular velocity of the global apsidal motion is more responsible for the stage b c transition rather than the appearance of a more localized new component . ( 88mm,70mm)fig32.eps ( 88mm,110mm)fig33.eps ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 54934.6865 & 0.0078 & @xmath170.0294 & 52 + 9 & 54935.3637 & 0.0011 & @xmath170.0372 & 77 + 10 & 54935.4455 & 0.0021 & @xmath170.0315 & 74 + 11 & 54935.5222 & 0.0017 & @xmath170.0309 & 64 + 12 & 54935.5987 & 0.0025 & @xmath170.0305 & 33 + 22 & 54936.3877 & 0.0013 & @xmath170.0026 & 105 + 23 & 54936.4637 & 0.0010 & @xmath170.0027 & 121 + 24 & 54936.5418 & 0.0010 & @xmath170.0007 & 82 + 30 & 54937.0005 & 0.0003 & 0.0014 & 144 + 31 & 54937.0753 & 0.0002 & 0.0001 & 201 + 34 & 54937.3054 & 0.0002 & 0.0019 & 59 + 35 & 54937.3817 & 0.0003 & 0.0021 & 68 + 36 & 54937.4578 & 0.0003 & 0.0020 & 63 + 37 & 54937.5331 & 0.0003 & 0.0012 & 66 + 43 & 54937.9895 & 0.0007 & 0.0010 & 119 + 47 & 54938.3025 & 0.0010 & 0.0096 & 66 + 48 & 54938.3787 & 0.0009 & 0.0097 & 68 + 49 & 54938.4553 & 0.0008 & 0.0102 & 67 + 50 & 54938.5309 & 0.0007 & 0.0097 & 58 + 56 & 54938.9851 & 0.0003 & 0.0073 & 123 + 57 & 54939.0607 & 0.0002 & 0.0067 & 125 + 58 & 54939.1363 & 0.0003 & 0.0062 & 82 + 64 & 54939.5901 & 0.0007 & 0.0034 & 86 + 65 & 54939.6675 & 0.0002 & 0.0047 & 122 + 69 & 54939.9721 & 0.0003 & 0.0049 & 83 + 70 & 54940.0471 & 0.0004 & 0.0038 & 123 + 71 & 54940.1227 & 0.0007 & 0.0033 & 108 + 74 & 54940.3503 & 0.0009 & 0.0026 & 61 + 83 & 54941.0404 & 0.0017 & 0.0077 & 104 + 84 & 54941.1189 & 0.0009 & 0.0101 & 97 + 87 & 54941.3452 & 0.0003 & 0.0080 & 136 + 88 & 54941.4247 & 0.0004 & 0.0115 & 133 + 89 & 54941.4983 & 0.0004 & 0.0090 & 105 + 90 & 54941.5746 & 0.0002 & 0.0091 & 130 + 100 & 54942.3361 & 0.0002 & 0.0096 & 151 + 101 & 54942.4111 & 0.0002 & 0.0085 & 254 + 102 & 54942.4877 & 0.0003 & 0.0090 & 203 + 103 & 54942.5638 & 0.0004 & 0.0090 & 134 + 108 & 54942.9498 & 0.0008 & 0.0144 & 53 + 109 & 54943.0254 & 0.0006 & 0.0139 & 88 + 110 & 54943.1010 & 0.0004 & 0.0134 & 62 + 113 & 54943.3315 & 0.0012 & 0.0156 & 68 + 114 & 54943.4078 & 0.0008 & 0.0158 & 67 + 115 & 54943.4832 & 0.0007 & 0.0151 & 62 + 116 & 54943.5539 & 0.0008 & 0.0097 & 27 + 122 & 54944.0106 & 0.0005 & 0.0098 & 86 + 123 & 54944.0867 & 0.0005 & 0.0097 & 88 + 128 & 54944.4631 & 0.0004 & 0.0056 & 51 + 130 & 54944.6139 & 0.0008 & 0.0042 & 65 + 135 & 54944.9922 & 0.0004 & 0.0020 & 177 + 136 & 54945.0690 & 0.0004 & 0.0027 & 213 + + + + ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 137 & 54945.1445 & 0.0006 & 0.0021 & 79 + 138 & 54945.2196 & 0.0008 & 0.0011 & 79 + 141 & 54945.4462 & 0.0010 & @xmath170.0006 & 37 + 142 & 54945.5220 & 0.0010 & @xmath170.0010 & 55 + 153 & 54946.3601 & 0.0009 & @xmath170.0000 & 67 + 154 & 54946.4323 & 0.0008 & @xmath170.0040 & 66 + 155 & 54946.5090 & 0.0006 & @xmath170.0033 & 65 + 156 & 54946.5854 & 0.0010 & @xmath170.0030 & 30 + 161 & 54946.9622 & 0.0016 & @xmath170.0067 & 44 + 162 & 54947.0420 & 0.0016 & @xmath170.0030 & 61 + 166 & 54947.3439 & 0.0007 & @xmath170.0055 & 72 + 167 & 54947.4212 & 0.0007 & @xmath170.0044 & 156 + 168 & 54947.4962 & 0.0005 & @xmath170.0055 & 134 + 169 & 54947.5722 & 0.0006 & @xmath170.0056 & 103 + 175 & 54948.0264 & 0.0008 & @xmath170.0081 & 139 + 179 & 54948.3261 & 0.0006 & @xmath170.0128 & 67 + 180 & 54948.4012 & 0.0005 & @xmath170.0138 & 67 + 181 & 54948.4776 & 0.0008 & @xmath170.0135 & 64 + 182 & 54948.5506 & 0.0015 & @xmath170.0166 & 64 + 189 & 54949.0981 & 0.0026 & @xmath170.0018 & 191 + 194 & 54949.4680 & 0.0016 & @xmath170.0124 & 66 + 195 & 54949.5410 & 0.0007 & @xmath170.0156 & 67 + 201 & 54950.0120 & 0.0041 & @xmath170.0012 & 169 + 202 & 54950.0831 & 0.0091 & @xmath170.0062 & 149 + 215 & 54951.0703 & 0.0056 & @xmath170.0084 & 108 + + + + the times of superhump maxima during the 2010 superoutburst are listed in table [ tab : ksumaoc2010 ] . since there was a large gap in observations , we did not attempt to determine @xmath7 from these data . the observation was probably recorded during stages b and c. ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 55304.0436 & 0.0004 & @xmath170.0009 & 92 + 1 & 55304.1148 & 0.0003 & 0.0002 & 100 + 2 & 55304.1837 & 0.0003 & @xmath170.0011 & 99 + 3 & 55304.2563 & 0.0006 & 0.0013 & 99 + 99 & 55310.9915 & 0.0013 & 0.0034 & 99 + 100 & 55311.0610 & 0.0011 & 0.0028 & 100 + 101 & 55311.1262 & 0.0011 & @xmath170.0021 & 96 + 102 & 55311.1953 & 0.0011 & @xmath170.0031 & 100 + 103 & 55311.2691 & 0.0012 & 0.0005 & 93 + 113 & 55311.9785 & 0.0015 & 0.0086 & 87 + 114 & 55312.0382 & 0.0023 & @xmath170.0019 & 95 + 115 & 55312.1023 & 0.0018 & @xmath170.0079 & 75 + + + + the times of superhump maxima during the 2010 superoutburst are listed in table [ tab : mrumaoc2010 ] . since there were relatively large gaps in observations , we did not attempt to determine @xmath7 from these data . the period of the presumable stage c superhumps is listed in table [ tab : perlist ] . an updated comparison of @xmath0 diagrams between different superoutbursts is given in figure [ fig : mrumacomp2 ] . ( 88mm,70mm)fig34.eps ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 55303.9991 & 0.0002 & 0.0004 & 155 + 1 & 55304.0645 & 0.0005 & 0.0007 & 107 + 2 & 55304.1286 & 0.0005 & @xmath170.0001 & 97 + 61 & 55307.9730 & 0.0012 & 0.0074 & 42 + 108 & 55311.0210 & 0.0009 & @xmath170.0011 & 150 + 109 & 55311.0877 & 0.0009 & 0.0007 & 157 + 110 & 55311.1519 & 0.0006 & @xmath170.0002 & 158 + 111 & 55311.2214 & 0.0011 & 0.0043 & 113 + 112 & 55311.2787 & 0.0012 & @xmath170.0035 & 89 + 123 & 55311.9932 & 0.0012 & @xmath170.0043 & 138 + 124 & 55312.0565 & 0.0016 & @xmath170.0061 & 155 + 125 & 55312.1196 & 0.0047 & @xmath170.0079 & 71 + 184 & 55315.9675 & 0.0006 & 0.0031 & 37 + 185 & 55316.0361 & 0.0019 & 0.0067 & 44 + + + + the 2010 superoutburst of this su uma - type dwarf nova was relatively well - observed during its later stage . the times of superhump maxima are listed in table [ tab : tyvuloc2010 ] . the period evolution now clearly demonstrates the presence of stage b c transition , whose existence was suggested from the 2003 observation @xcite . the sudden change in the superhump period at this transition favors the suggestion in @xcite that this object is analogous to ax cap and sdss j1627 . ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 55368.7467 & 0.0008 & @xmath170.0048 & 41 + 1 & 55368.8262 & 0.0004 & @xmath170.0055 & 39 + 9 & 55369.4721 & 0.0007 & @xmath170.0017 & 202 + 10 & 55369.5530 & 0.0003 & @xmath170.0010 & 188 + 13 & 55369.7944 & 0.0009 & @xmath170.0003 & 40 + 14 & 55369.8748 & 0.0008 & @xmath170.0002 & 39 + 21 & 55370.4365 & 0.0007 & @xmath170.0003 & 135 + 22 & 55370.5183 & 0.0008 & 0.0012 & 160 + 23 & 55370.5991 & 0.0008 & 0.0018 & 89 + 25 & 55370.7518 & 0.0030 & @xmath170.0060 & 20 + 26 & 55370.8390 & 0.0007 & 0.0009 & 40 + 27 & 55370.9180 & 0.0008 & @xmath170.0004 & 32 + 33 & 55371.3982 & 0.0090 & @xmath170.0017 & 64 + 34 & 55371.4837 & 0.0011 & 0.0036 & 158 + 35 & 55371.5635 & 0.0008 & 0.0031 & 154 + 37 & 55371.7267 & 0.0017 & 0.0059 & 37 + 38 & 55371.8037 & 0.0033 & 0.0026 & 39 + 39 & 55371.8884 & 0.0033 & 0.0070 & 22 + 46 & 55372.4457 & 0.0004 & 0.0026 & 105 + 47 & 55372.5236 & 0.0004 & 0.0002 & 163 + 48 & 55372.6074 & 0.0005 & 0.0037 & 74 + 58 & 55373.4079 & 0.0011 & 0.0017 & 51 + 59 & 55373.4872 & 0.0005 & 0.0007 & 156 + 60 & 55373.5673 & 0.0006 & 0.0005 & 146 + 75 & 55374.7689 & 0.0014 & @xmath170.0016 & 27 + 76 & 55374.8473 & 0.0019 & @xmath170.0035 & 27 + 88 & 55375.8077 & 0.0021 & @xmath170.0061 & 58 + 89 & 55375.8919 & 0.0033 & @xmath170.0022 & 109 + + + + in @xcite , we reported observations of the 2008 superoutburst of this object ( = hs 0417@xmath17445 , hereafter 1rxs j0423 ) . the 2010 superoutburst was again fortunately detected during its rising stage and early stage evolution of superhumps was recorded . the times of superhump maxima are listed in table [ tab : j0423oc2010 ] . a stage a b and transition was clearly recorded . the shorter superhump period after @xmath51 probably corresponds to stage c superhumps . although the later half of the stage b and the stage b c transition itself were not observed , there was a possible indication of a positive @xmath7 during the stage b. the parameters are listed in table [ tab : perlist ] . figure [ fig : j0423occomp ] illustrates the comparison of @xmath0 diagrams between the 2008 and 2010 superoutbursts . although the later parts of the @xmath0 diagrams were similar , there was a distinction during the early stage this may have been a result of the presence of a precursor outburst and early appearance of superhumps during the 2008 superoutburst ( cf . the @xmath0 evolution in the 2010 superoutburst resembled those of ordinary su uma - type dwarf novae than in the 2008 superoutburst . ( 88mm,70mm)fig35.eps ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 55442.1696 & 0.0006 & @xmath170.0155 & 124 + 1 & 55442.2491 & 0.0005 & @xmath170.0141 & 94 + 12 & 55443.1233 & 0.0003 & @xmath170.0007 & 114 + 13 & 55443.1995 & 0.0004 & @xmath170.0028 & 309 + 14 & 55443.2795 & 0.0008 & @xmath170.0010 & 108 + 25 & 55444.1410 & 0.0004 & @xmath170.0003 & 268 + 26 & 55444.2188 & 0.0002 & @xmath170.0007 & 470 + 27 & 55444.2979 & 0.0003 & 0.0001 & 406 + 38 & 55445.1623 & 0.0007 & 0.0037 & 67 + 39 & 55445.2402 & 0.0018 & 0.0034 & 42 + 40 & 55445.3196 & 0.0007 & 0.0045 & 73 + 79 & 55448.3718 & 0.0009 & 0.0049 & 47 + 80 & 55448.4501 & 0.0004 & 0.0050 & 79 + 81 & 55448.5259 & 0.0004 & 0.0025 & 76 + 82 & 55448.6063 & 0.0005 & 0.0047 & 79 + 83 & 55448.6825 & 0.0010 & 0.0026 & 62 + 88 & 55449.0717 & 0.0020 & 0.0005 & 157 + 89 & 55449.1554 & 0.0008 & 0.0061 & 240 + 90 & 55449.2320 & 0.0009 & 0.0043 & 475 + 91 & 55449.3099 & 0.0005 & 0.0041 & 205 + 92 & 55449.3840 & 0.0024 & @xmath170.0001 & 49 + 93 & 55449.4716 & 0.0016 & 0.0093 & 80 + 94 & 55449.5447 & 0.0015 & 0.0041 & 78 + 95 & 55449.6240 & 0.0009 & 0.0052 & 79 + 96 & 55449.7020 & 0.0016 & 0.0048 & 42 + 102 & 55450.1716 & 0.0009 & 0.0050 & 218 + 103 & 55450.2496 & 0.0007 & 0.0047 & 197 + 114 & 55451.1109 & 0.0021 & 0.0053 & 90 + 115 & 55451.1783 & 0.0080 & @xmath170.0056 & 114 + 116 & 55451.2728 & 0.0026 & 0.0106 & 108 + 140 & 55453.1084 & 0.0032 & @xmath170.0318 & 77 + 141 & 55453.2028 & 0.0042 & @xmath170.0157 & 87 + 142 & 55453.2896 & 0.0033 & @xmath170.0071 & 85 + + + + the period evolution of this su uma - type dwarf nova during the 2005 and 2008 superoutbursts has been described in @xcite and @xcite . we also observed the 2009 superoutburst which was accompanied by a precursor outburst , as in the 2005 one ( figure [ fig : j05322009oc ] ) . the times of superhump maxima are listed in table [ tab : j0532oc2009 ] . superhumps were already present during the fading stage of the rebrightening , and the period was smoothly decreasing as in the 2005 one @xcite . this indicates that the stage b , rather than stage a , already started during the precursor outburst . this behavior very well reproduced the features observed during the 2005 superoutburst . the observed @xmath7 for stage b ( @xmath52 ) was @xmath53 , similar to the one observed in 2008 , but is larger than in 2005 . the 2009 outburst was also well - observed during the post - superoutburst stage . although times of individual superhumps were not sufficiently measured due to strong flickering , the period analysis has yielded a periodicity of 0.05690(2 ) d ( bjd 2455081.22455090.6 , figure [ fig : j0532lateshpdm ] ) , which is in agreement with the period of stage c superhumps . the signal from the orbital period , if present , was still much smaller than the superhump signal . this analysis suggests the long endurance of superhumps even after the termination of the superoutburst . ( 88mm,90mm)fig36.eps ( 88mm,110mm)fig37.eps ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 55068.3087 & 0.0016 & 0.0047 & 106 + 4 & 55068.5335 & 0.0008 & 0.0011 & 114 + 5 & 55068.5967 & 0.0016 & 0.0072 & 72 + 19 & 55069.3875 & 0.0004 & @xmath170.0013 & 106 + 20 & 55069.4457 & 0.0005 & @xmath170.0003 & 114 + 21 & 55069.5014 & 0.0004 & @xmath170.0016 & 111 + 22 & 55069.5574 & 0.0003 & @xmath170.0027 & 113 + 71 & 55072.3504 & 0.0003 & @xmath170.0075 & 112 + 72 & 55072.4075 & 0.0004 & @xmath170.0075 & 114 + 73 & 55072.4669 & 0.0009 & @xmath170.0052 & 36 + 74 & 55072.5226 & 0.0004 & @xmath170.0066 & 72 + 75 & 55072.5791 & 0.0003 & @xmath170.0071 & 107 + 105 & 55074.2935 & 0.0008 & @xmath170.0056 & 108 + 123 & 55075.3249 & 0.0019 & @xmath170.0019 & 115 + 124 & 55075.3828 & 0.0014 & @xmath170.0012 & 102 + 125 & 55075.4411 & 0.0009 & 0.0000 & 97 + 126 & 55075.4958 & 0.0011 & @xmath170.0023 & 108 + 127 & 55075.5598 & 0.0011 & 0.0045 & 114 + 128 & 55075.6117 & 0.0015 & @xmath170.0006 & 77 + 140 & 55076.3047 & 0.0014 & 0.0072 & 112 + 141 & 55076.3688 & 0.0013 & 0.0142 & 114 + 142 & 55076.4220 & 0.0010 & 0.0103 & 113 + 143 & 55076.4764 & 0.0009 & 0.0076 & 112 + 144 & 55076.5335 & 0.0006 & 0.0076 & 107 + 145 & 55076.5907 & 0.0010 & 0.0077 & 110 + 209 & 55080.2372 & 0.0005 & 0.0001 & 145 + 210 & 55080.2932 & 0.0004 & @xmath170.0011 & 172 + 213 & 55080.4643 & 0.0011 & @xmath170.0013 & 63 + 214 & 55080.5134 & 0.0017 & @xmath170.0093 & 54 + 215 & 55080.5705 & 0.0006 & @xmath170.0092 & 44 + + + + asas [email protected] ( hereafter asas j2243 ) was selected as a dwarf nova by p. wils ( cvnet - discussion 1320 ; @xcite ) . the 2009 outburst and the superhumps were detected by i. miller ( cvnet - outburst 3358 ) . the object showed well - developed superhumps and their time - evolution was intensively studied . the times of superhump maxima are listed in table [ tab : asas2243oc2009 ] . as also reported in @xcite , a textbook stage b c transition was observed . the @xmath7 during stage b was @xmath54 ( @xmath55 ) . of @xmath56 . our analysis of their timing data yielded @xmath7 of @xmath57 ( @xmath58 ) by our definition . ] the object also showed a post - superoutburst rebrightening ( figure [ fig : asas22432009oc ] ) . ( 88mm,90mm)fig38.eps ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 55113.0454 & 0.0002 & 0.0005 & 132 + 1 & 55113.1144 & 0.0003 & @xmath170.0003 & 131 + 2 & 55113.1842 & 0.0003 & @xmath170.0001 & 132 + 5 & 55113.3918 & 0.0003 & @xmath170.0017 & 64 + 6 & 55113.4596 & 0.0003 & @xmath170.0036 & 75 + 7 & 55113.5328 & 0.0003 & @xmath170.0001 & 67 + 10 & 55113.7392 & 0.0002 & @xmath170.0028 & 182 + 11 & 55113.8079 & 0.0002 & @xmath170.0039 & 182 + 15 & 55114.0866 & 0.0003 & @xmath170.0040 & 134 + 23 & 55114.6442 & 0.0002 & @xmath170.0041 & 184 + 24 & 55114.7131 & 0.0003 & @xmath170.0049 & 180 + 25 & 55114.7841 & 0.0003 & @xmath170.0036 & 179 + 26 & 55114.8539 & 0.0005 & @xmath170.0035 & 104 + 29 & 55115.0633 & 0.0004 & @xmath170.0033 & 109 + 30 & 55115.1365 & 0.0007 & 0.0003 & 64 + 36 & 55115.5479 & 0.0010 & @xmath170.0067 & 16 + 38 & 55115.6896 & 0.0004 & @xmath170.0044 & 55 + 47 & 55116.3174 & 0.0004 & @xmath170.0039 & 57 + 48 & 55116.3881 & 0.0003 & @xmath170.0030 & 83 + 49 & 55116.4581 & 0.0005 & @xmath170.0027 & 69 + 50 & 55116.5269 & 0.0007 & @xmath170.0036 & 31 + 51 & 55116.6073 & 0.0017 & 0.0071 & 39 + 52 & 55116.6714 & 0.0013 & 0.0014 & 49 + 53 & 55116.7374 & 0.0005 & @xmath170.0023 & 104 + 54 & 55116.8085 & 0.0022 & @xmath170.0009 & 69 + 56 & 55116.9463 & 0.0019 & @xmath170.0025 & 30 + 62 & 55117.3650 & 0.0009 & @xmath170.0020 & 80 + 63 & 55117.4344 & 0.0005 & @xmath170.0023 & 87 + 64 & 55117.5049 & 0.0005 & @xmath170.0015 & 76 + 65 & 55117.5769 & 0.0018 & 0.0008 & 22 + 71 & 55117.9965 & 0.0006 & 0.0020 & 196 + 72 & 55118.0654 & 0.0005 & 0.0012 & 176 + 75 & 55118.2775 & 0.0011 & 0.0042 & 26 + 76 & 55118.3461 & 0.0009 & 0.0030 & 35 + 77 & 55118.4148 & 0.0010 & 0.0021 & 58 + 78 & 55118.4790 & 0.0013 & @xmath170.0034 & 56 + 79 & 55118.5559 & 0.0013 & 0.0038 & 57 + 89 & 55119.2535 & 0.0025 & 0.0043 & 24 + 91 & 55119.3946 & 0.0012 & 0.0059 & 56 + 92 & 55119.4650 & 0.0007 & 0.0066 & 65 + 93 & 55119.5342 & 0.0009 & 0.0060 & 32 + 94 & 55119.6046 & 0.0007 & 0.0068 & 90 + 95 & 55119.6740 & 0.0005 & 0.0065 & 81 + 96 & 55119.7461 & 0.0005 & 0.0088 & 83 + 97 & 55119.8138 & 0.0005 & 0.0068 & 79 + 100 & 55120.0233 & 0.0004 & 0.0072 & 143 + 101 & 55120.0937 & 0.0004 & 0.0079 & 149 + 106 & 55120.4399 & 0.0005 & 0.0055 & 63 + 108 & 55120.5797 & 0.0007 & 0.0060 & 73 + 109 & 55120.6490 & 0.0004 & 0.0055 & 91 + 110 & 55120.7186 & 0.0004 & 0.0054 & 88 + 111 & 55120.7870 & 0.0006 & 0.0041 & 88 + 118 & 55121.2762 & 0.0015 & 0.0053 & 113 + 119 & 55121.3420 & 0.0004 & 0.0014 & 156 + 120 & 55121.4116 & 0.0007 & 0.0013 & 182 + 121 & 55121.4811 & 0.0008 & 0.0011 & 149 + 122 & 55121.5515 & 0.0009 & 0.0017 & 92 + 123 & 55121.6190 & 0.0003 & @xmath170.0005 & 182 + + + + ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 124 & 55121.6885 & 0.0003 & @xmath170.0006 & 182 + 125 & 55121.7587 & 0.0005 & @xmath170.0002 & 184 + 126 & 55121.8274 & 0.0004 & @xmath170.0012 & 130 + 131 & 55122.1763 & 0.0009 & @xmath170.0008 & 131 + 133 & 55122.3165 & 0.0015 & @xmath170.0001 & 24 + 134 & 55122.3975 & 0.0012 & 0.0112 & 140 + 135 & 55122.4514 & 0.0007 & @xmath170.0046 & 220 + 136 & 55122.5235 & 0.0057 & @xmath170.0022 & 85 + 137 & 55122.6033 & 0.0005 & 0.0079 & 178 + 138 & 55122.6636 & 0.0007 & @xmath170.0016 & 181 + 139 & 55122.7466 & 0.0010 & 0.0117 & 178 + 140 & 55122.7979 & 0.0007 & @xmath170.0067 & 180 + 151 & 55123.5594 & 0.0026 & @xmath170.0120 & 94 + 152 & 55123.6415 & 0.0013 & 0.0004 & 178 + 157 & 55123.9831 & 0.0009 & @xmath170.0066 & 147 + 167 & 55124.6724 & 0.0008 & @xmath170.0144 & 92 + 173 & 55125.0856 & 0.0013 & @xmath170.0195 & 149 + 174 & 55125.1588 & 0.0046 & @xmath170.0160 & 73 + + + + lanning 420 was selected as a uv - bright transient object @xcite , which was considered to be a possible nova because of the absence on the digitized sky survey image . @xcite listed lanning cvs and further investigated these objects . s. brady indeed detected an outburst in 2007 and another one in 2010 , which turned out to be a superoutburst ( baavss alert 2374 ) . follow - up observation confirmed the presence of superhumps ( vsnet - alert 12131 , 12132 , 12138 ; figure [ fig : lanning420shpdm ] ) . the times of superhump maxima are listed in table [ tab : lanning4202010 ] . there was a distinct shortening of the superhump period around @xmath59 , which we interpreted as a stage b c transition . the last two epochs were measured during the early post - superoutburst stage . we included these epochs because short-@xmath3 tend to show persistent superhumps during this stage and the times of maxima were in good agreement with extrapolated stage c superhumps . the measured period derivative during the stage b had a relatively large error because only the late stage of the stage b was observed . observations of the earlier stage are crucial to better characterize the period variation in this system . ( 88mm,110mm)fig39.eps ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 55438.5841 & 0.0005 & @xmath170.0084 & 20 + 1 & 55438.6469 & 0.0005 & @xmath170.0070 & 26 + 2 & 55438.7074 & 0.0005 & @xmath170.0078 & 24 + 3 & 55438.7691 & 0.0005 & @xmath170.0075 & 25 + 4 & 55438.8277 & 0.0018 & @xmath170.0102 & 24 + 14 & 55439.4478 & 0.0011 & @xmath170.0033 & 51 + 15 & 55439.5051 & 0.0018 & @xmath170.0074 & 46 + 16 & 55439.5672 & 0.0011 & @xmath170.0066 & 89 + 17 & 55439.6314 & 0.0012 & @xmath170.0037 & 86 + 18 & 55439.6895 & 0.0010 & @xmath170.0069 & 88 + 19 & 55439.7535 & 0.0008 & @xmath170.0042 & 76 + 20 & 55439.8132 & 0.0012 & @xmath170.0059 & 46 + 21 & 55439.8767 & 0.0009 & @xmath170.0037 & 60 + 22 & 55439.9369 & 0.0008 & @xmath170.0048 & 56 + 23 & 55439.9982 & 0.0012 & @xmath170.0049 & 45 + 24 & 55440.0589 & 0.0015 & @xmath170.0055 & 125 + 25 & 55440.1225 & 0.0030 & @xmath170.0032 & 115 + 31 & 55440.4990 & 0.0049 & 0.0054 & 20 + 32 & 55440.5511 & 0.0022 & @xmath170.0039 & 21 + 33 & 55440.6082 & 0.0073 & @xmath170.0081 & 19 + 34 & 55440.6785 & 0.0020 & 0.0008 & 49 + 35 & 55440.7379 & 0.0006 & @xmath170.0010 & 56 + 36 & 55440.7997 & 0.0009 & @xmath170.0006 & 31 + 37 & 55440.8611 & 0.0006 & @xmath170.0005 & 45 + 38 & 55440.9243 & 0.0007 & 0.0014 & 56 + 39 & 55440.9852 & 0.0023 & 0.0009 & 163 + 40 & 55441.0392 & 0.0054 & @xmath170.0064 & 128 + 41 & 55441.1128 & 0.0018 & 0.0059 & 253 + 42 & 55441.1713 & 0.0019 & 0.0031 & 115 + 46 & 55441.4149 & 0.0030 & 0.0014 & 15 + 47 & 55441.4792 & 0.0028 & 0.0044 & 21 + 48 & 55441.5345 & 0.0020 & @xmath170.0017 & 25 + 49 & 55441.5992 & 0.0038 & 0.0017 & 39 + 50 & 55441.6608 & 0.0016 & 0.0019 & 48 + 51 & 55441.7225 & 0.0008 & 0.0023 & 73 + 52 & 55441.7874 & 0.0017 & 0.0059 & 60 + 53 & 55441.8443 & 0.0020 & 0.0015 & 53 + 54 & 55441.9112 & 0.0015 & 0.0071 & 56 + 55 & 55441.9710 & 0.0019 & 0.0056 & 56 + 62 & 55442.4036 & 0.0011 & 0.0088 & 21 + 63 & 55442.4636 & 0.0010 & 0.0075 & 71 + 64 & 55442.5285 & 0.0026 & 0.0111 & 21 + 65 & 55442.5846 & 0.0016 & 0.0059 & 21 + 66 & 55442.6469 & 0.0029 & 0.0068 & 21 + 76 & 55443.2586 & 0.0018 & 0.0053 & 110 + 78 & 55443.3918 & 0.0051 & 0.0159 & 35 + 79 & 55443.4458 & 0.0014 & 0.0085 & 48 + 80 & 55443.5039 & 0.0015 & 0.0053 & 46 + 81 & 55443.5651 & 0.0017 & 0.0051 & 48 + 82 & 55443.6274 & 0.0015 & 0.0061 & 45 + 83 & 55443.6835 & 0.0083 & 0.0009 & 17 + 92 & 55444.2430 & 0.0024 & 0.0085 & 122 + 93 & 55444.3012 & 0.0021 & 0.0053 & 63 + 94 & 55444.3635 & 0.0024 & 0.0064 & 15 + 95 & 55444.4301 & 0.0130 & 0.0116 & 21 + 96 & 55444.4863 & 0.0023 & 0.0065 & 20 + + + + ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 97 & 55444.5442 & 0.0020 & 0.0031 & 15 + 98 & 55444.6082 & 0.0026 & 0.0058 & 19 + 99 & 55444.6637 & 0.0014 & @xmath170.0001 & 19 + 109 & 55445.2851 & 0.0155 & 0.0081 & 65 + 116 & 55445.7083 & 0.0024 & 0.0020 & 17 + 117 & 55445.7670 & 0.0032 & @xmath170.0006 & 16 + 118 & 55445.8309 & 0.0017 & 0.0019 & 16 + 124 & 55446.1980 & 0.0028 & 0.0011 & 111 + 125 & 55446.2583 & 0.0046 & 0.0001 & 80 + 126 & 55446.3132 & 0.0059 & @xmath170.0063 & 60 + 174 & 55449.2331 & 0.0071 & @xmath170.0301 & 117 + 175 & 55449.2880 & 0.0028 & @xmath170.0365 & 109 + + + + this object ( hereafter pg 0149 ) was originally discovered as an eruptive object with strong uv excess , and was suspected to be a supernova @xcite . @xcite selected this object during the course of the sdss survey and classified it as a cv ( dwarf nova ) . the object as been monitored as a dwarf nova since then . @xcite photometrically identified a @xmath2 of 0.08242(3 ) d. the 2009 september outburst of this object was detected by the the catalina real - time transient survey ( crts , @xcite)http://nesssi.cacr.caltech.edu / catalina/@xmath6 . for the information of the individual catalina cvs , see @xmath60http://nesssi.cacr.caltech.edu / catalina / allcv.html@xmath6 . ] (= css090911:015152@xmath1140047 ) . superhumps were detected immediately following the announcement ( vsnet - alert 11465 ) . the mean @xmath3 with the pdm method was 0.08495(2 ) d ( figure [ fig : pg0149shpdm ] ) . the times of superhump maxima are summarized in table [ tab : pg0149oc2009 ] . there was a clear stage b c transition with a positive @xmath7 = @xmath61 ( @xmath62 ) . such a positive @xmath7 is rare for su uma - type dwarf novae with this @xmath3 . the behavior in the superhump period resemble that of long-@xmath3 systems like qw ser ( cf . @xcite ; @xcite ) . ( 88mm,110mm)fig40.eps ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 55086.5129 & 0.0008 & @xmath170.0038 & 44 + 1 & 55086.5987 & 0.0003 & @xmath170.0028 & 89 + 11 & 55087.4448 & 0.0007 & @xmath170.0055 & 57 + 12 & 55087.5312 & 0.0005 & @xmath170.0040 & 114 + 13 & 55087.6173 & 0.0005 & @xmath170.0029 & 89 + 23 & 55088.4654 & 0.0006 & @xmath170.0036 & 40 + 24 & 55088.5484 & 0.0006 & @xmath170.0054 & 83 + 55 & 55091.1917 & 0.0006 & 0.0065 & 180 + 59 & 55091.5320 & 0.0007 & 0.0073 & 77 + 60 & 55091.6176 & 0.0008 & 0.0080 & 89 + 70 & 55092.4661 & 0.0006 & 0.0076 & 91 + 71 & 55092.5517 & 0.0007 & 0.0083 & 130 + 72 & 55092.6371 & 0.0006 & 0.0089 & 111 + 93 & 55094.4113 & 0.0014 & 0.0005 & 40 + 94 & 55094.4977 & 0.0009 & 0.0020 & 45 + 95 & 55094.5776 & 0.0009 & @xmath170.0029 & 44 + 103 & 55095.2556 & 0.0008 & @xmath170.0039 & 146 + 154 & 55099.5741 & 0.0013 & @xmath170.0145 & 78 + + + + this object ( hereafter rx j1715 ) is a cv identified from the rosat north ecliptic pole survey @xcite . @xcite reported the detection of a 1.64(2 ) hr period from radial - velocity study . @xcite identified the dwarf - nova type behavior and detected superhumps during a superoutburst in 2009 august . we analyzed the combined data from @xcite ( from the aavso database ) and our own observations . as reported in @xcite , these observations covered the final stage of the superoutburst . the times of maxima are listed in table [ tab : j1715oc2009 ] . since the @xmath0 s for maxima for @xmath63 largely deviate from the earlier trend , these humps are less likely a continuation of superhumps observed during the plateau phase ( @xmath64 ) . restricting to @xmath64 , we obtained a mean @xmath3 of 0.07074(4 ) d with the pdm method ( figure [ fig : j1715shpdm ] ) . and is in agreement with the corresponding period of 0.07086(78 ) d by @xcite within respective errors . the @xmath13 against @xmath2 was 3.5 % . since the observation covered the final part of the superoutburst , these superhumps are likely stage c superhumps . although @xcite interpreted observed humps for @xmath63 ( rapid fading stage and post - superoutburst stage ) as orbital humps , the reported period [ 0.06944(90 ) d ] is somewhat different from the one reported from radial - velocity study . our analysis also confirmed a significant phase offset during the rapid fading stage ( figure [ fig : j1715prof ] ) . we could not , however , determine whether the newly arising signals were from orbital humps or were from traditional late superhumps because the amplitude on jd 2455069 was very small and it was difficult to derive a meaningful period based on the last two nights of observations . future observations are needed to confirm the nature of these large - amplitude humps . since stage c superhumps frequently endure during the post - superoutburst stage in many well - observed systems ( cf . @xcite ) , the early disappearance of superhumps in rx j1715 appears to be rather unique . the case may be similar to dt oct ( cf . @xcite ) , which showed an early switch to traditional late superhumps . although both relatively low outburst amplitudes and the relatively strong x - ray emission are common to these objects , the frequency of outbursts in rx j1715 appears to be smaller @xcite . a further detailed comparison between these objects might be fruitful . ( 88mm,110mm)fig41.eps ( 88mm,110mm)fig42.eps ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 55065.3776 & 0.0007 & 0.0037 & 61 + 1 & 55065.4528 & 0.0010 & 0.0078 & 87 + 2 & 55065.5181 & 0.0008 & 0.0021 & 30 + 23 & 55067.0079 & 0.0080 & @xmath170.0000 & 86 + 39 & 55068.1405 & 0.0021 & @xmath170.0041 & 134 + 42 & 55068.3635 & 0.0014 & 0.0058 & 62 + 43 & 55068.4183 & 0.0014 & @xmath170.0105 & 90 + 44 & 55068.4953 & 0.0015 & @xmath170.0045 & 101 + 47 & 55068.7020 & 0.0021 & @xmath170.0110 & 37 + 48 & 55068.7722 & 0.0024 & @xmath170.0118 & 27 + 63 & 55069.8143 & 0.0045 & @xmath170.0353 & 37 + 63 & 55069.8756 & 0.0019 & 0.0259 & 38 + 64 & 55069.9378 & 0.0010 & 0.0171 & 32 + 75 & 55070.7114 & 0.0019 & 0.0092 & 27 + 76 & 55070.7752 & 0.0016 & 0.0020 & 25 + 77 & 55070.8431 & 0.0024 & @xmath170.0012 & 26 + 78 & 55070.9205 & 0.0012 & 0.0051 & 24 + + + + this object ( hereafter sdss j0129 ) is an am cvn - type cv selected during the course of the sdss @xcite , who reported broad hei emission lines . the orbital period has not been reported yet . the object was reported in outburst on 2009 november 29 at a unfiltered ccd magnitude of 14.5 ( cvnet - outburst 3479 , vsnet - alert 11702 ) . two days after this detection , the object started to fade rapidly . immediately following this transient fading , the object experienced rebrightenings on december 4 and 18 ( vsnet - alert 11707 , 11737 ) . the overall behavior of the outburst ( figure [ fig : j0129lc ] ) was extremely similar to the 2003 august superoutburst of v803 cen ( @xcite ; vsnet - alert 11709 ) . the object developed hump signals during its terminal stage of the plateau phase ( figures [ fig : j0129shpdm ] , [ fig : j0129shprof ] ) . although these humps may reflect orbital modulations , we identified them as superhumps based on extreme analogy with v803 cen @xcite . the mean period determined with the pdm method was 0.01805(10 ) d. the period is also very similar to that of v803 cen ( @xmath65 = 0.018686(4 ) d = 1614.5(4 ) s , @xcite ) . the times of maxima during the fading branch from the superoutburst are listed in table [ tab : j0129oc2009 ] . ( 88mm,110mm)fig43.eps ( 88mm,110mm)fig44.eps ( 88mm,110mm)fig45.eps ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 55166.3297 & 0.0030 & @xmath170.0020 & 40 + 1 & 55166.3491 & 0.0037 & @xmath170.0007 & 40 + 2 & 55166.3686 & 0.0012 & 0.0007 & 41 + 4 & 55166.4080 & 0.0013 & 0.0039 & 41 + 5 & 55166.4231 & 0.0006 & 0.0009 & 52 + 6 & 55166.4433 & 0.0013 & 0.0031 & 62 + 7 & 55166.4567 & 0.0012 & @xmath170.0016 & 61 + 8 & 55166.4739 & 0.0015 & @xmath170.0026 & 22 + 18 & 55166.6542 & 0.0011 & @xmath170.0032 & 17 + 19 & 55166.6730 & 0.0011 & @xmath170.0024 & 17 + 20 & 55166.6947 & 0.0017 & 0.0012 & 17 + 22 & 55166.7329 & 0.0019 & 0.0031 & 17 + 23 & 55166.7458 & 0.0017 & @xmath170.0020 & 17 + 25 & 55166.7840 & 0.0010 & 0.0000 & 17 + 26 & 55166.8034 & 0.0012 & 0.0013 & 17 + 28 & 55166.8384 & 0.0010 & 0.0001 & 17 + 29 & 55166.8547 & 0.0017 & @xmath170.0016 & 16 + 30 & 55166.8764 & 0.0008 & 0.0020 & 16 + + + + in addition to the 2004 superoutburst of this object ( hereafter sdss j0310 ) , two superoutbursts were recorded in 2009 . table [ tab : j0310oc2009b ] gives the times of superhump maxima during its second superoutburst in 2009 ( 2009 november , designated as 2009b in table in order to avoid confusion with the 2009 superoutburst described in @xcite ) . the mean @xmath3 during this superoutburst was 0.06786(3 ) d ( pdm method , alias selected based on the 2004 observation ) , which is shorter than the period obtained during the 2004 superoutburst . although this difference may have resulted from different stages observed in different outbursts , the difference appears to be larger than those associated with typical stage b c transitions @xcite . this needs to be clarified by future observations . the shortest interval between superoutbursts was 284 d , which is likely the supercycle of this object . ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 55144.5155 & 0.0003 & 0.0028 & 67 + 1 & 55144.5777 & 0.0003 & @xmath170.0029 & 73 + 31 & 55146.6161 & 0.0009 & 0.0001 & 61 + + + + this object ( hereafter sdss j0732 ) was selected using sdss and crts data as a candidate dwarf nova by @xcite . the object was detected in outburst by j. shears on 2009 december 31 at an unfiltered ccd magnitude of 16.2 ( cvnet - outburst 3528 ; @xcite ) . subsequent observations confirmed the presence of superhumps ( cvnet - outburst 3535 , figure [ fig : j0732shpdm ] ) . the times of superhump maxima determined from aavso observations are listed in table [ tab : j0732oc2010 ] . the object showed a clear stage b c transition around @xmath66 . the @xmath7 for stage b was @xmath67 . of @xmath68 . they used a data set including our present data set . we do not attempt to choose a better @xmath7 between ours and theirs , since @xmath7 is dependent on the segment used , and the observed baseline for the stage b was too short . the @xmath7 determined from the timing data in @xcite was @xmath69 ( @xmath70 ) while it was @xmath71 for @xmath72 ( both values are by our definition of @xmath7 ) . ] relatively few objects with similar @xmath3 , including rz leo and qy per , are known to show positive @xmath7 @xcite . although the relatively large ( 5.2 mag ) outburst amplitude also might suggest a low mass - transfer rate as in rz leo and qy per , the outburst frequency estimated in @xcite is much higher than in these objects . an exact determination of @xmath7 during the next superoutburst is desired . ( 88mm,110mm)fig46.eps ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 55199.6270 & 0.0020 & @xmath170.0080 & 42 + 1 & 55199.7096 & 0.0007 & @xmath170.0050 & 83 + 2 & 55199.7887 & 0.0006 & @xmath170.0054 & 76 + 3 & 55199.8666 & 0.0006 & @xmath170.0071 & 81 + 4 & 55199.9462 & 0.0008 & @xmath170.0071 & 82 + 10 & 55200.4238 & 0.0018 & @xmath170.0071 & 60 + 14 & 55200.7446 & 0.0006 & @xmath170.0047 & 82 + 15 & 55200.8244 & 0.0007 & @xmath170.0044 & 68 + 16 & 55200.9040 & 0.0006 & @xmath170.0044 & 82 + 17 & 55200.9833 & 0.0011 & @xmath170.0047 & 73 + 26 & 55201.7046 & 0.0006 & 0.0003 & 82 + 27 & 55201.7859 & 0.0007 & 0.0020 & 82 + 28 & 55201.8656 & 0.0008 & 0.0020 & 74 + 29 & 55201.9427 & 0.0010 & @xmath170.0004 & 82 + 30 & 55202.0233 & 0.0012 & 0.0005 & 83 + 38 & 55202.6663 & 0.0010 & 0.0068 & 71 + 39 & 55202.7456 & 0.0007 & 0.0065 & 82 + 40 & 55202.8253 & 0.0006 & 0.0066 & 76 + 41 & 55202.9044 & 0.0009 & 0.0061 & 82 + 42 & 55202.9841 & 0.0011 & 0.0062 & 75 + 59 & 55204.3438 & 0.0017 & 0.0129 & 88 + 60 & 55204.4230 & 0.0011 & 0.0125 & 90 + 75 & 55205.6122 & 0.0006 & 0.0077 & 73 + 76 & 55205.6924 & 0.0007 & 0.0083 & 83 + 77 & 55205.7727 & 0.0008 & 0.0090 & 82 + 78 & 55205.8501 & 0.0007 & 0.0069 & 79 + 79 & 55205.9295 & 0.0007 & 0.0067 & 82 + 90 & 55206.8014 & 0.0006 & 0.0031 & 76 + 91 & 55206.8798 & 0.0008 & 0.0019 & 82 + 92 & 55206.9602 & 0.0010 & 0.0027 & 67 + 114 & 55208.7037 & 0.0013 & @xmath170.0049 & 82 + 115 & 55208.7840 & 0.0011 & @xmath170.0042 & 75 + 116 & 55208.8664 & 0.0010 & @xmath170.0014 & 83 + 117 & 55208.9413 & 0.0011 & @xmath170.0061 & 75 + 118 & 55209.0165 & 0.0023 & @xmath170.0104 & 82 + 126 & 55209.6628 & 0.0019 & @xmath170.0009 & 83 + 127 & 55209.7402 & 0.0013 & @xmath170.0031 & 76 + 128 & 55209.8098 & 0.0014 & @xmath170.0130 & 76 + 129 & 55209.8958 & 0.0014 & @xmath170.0067 & 82 + + + + this object ( hereafter sdss j0838 ) underwent another superoutburst in 2010 in addition to the 2007 and 2009 ones discussed in @xcite . the 2010 superoutburst was notable in that it was preceded by a prominent precursor outburst ( vsnet - alert 11910 ) . although only two superhump maxima were measured , we listed them in table [ tab : j0838oc2010 ] . the interval between maxima is in agreement with the period of stage c superhumps in 2009 . since the 2010 observation was undertaken during the early stage of the superoutburst , the period is expected to be close to the @xmath3 at the start of stage b. as discussed in @xcite , this period is expected to be close to @xmath3 for stage c superhumps . the present finding is consistent with this interpretation . there was a slight indication of modulation at a period around 0.0709 d and an amplitude of 0.07 mag . although the presence of this signal is not conclusive , the periodicity might suggest a rare evolution of @xmath3 starting from @xmath2 , as recorded in qz vir @xcite . this needs to be confirmed by future observations . cccc @xmath15 & max & error & @xmath16 + 0 & 55294.9551 & 0.0008 & 99 + 1 & 55295.0289 & 0.0004 & 153 + + + this object ( hereafter sdss j0839 ) was selected as a cv during the course of sdss @xcite . although the spectrum was suggestive of that of a dwarf nova , no outburst had been recorded . on 2010 april 8 , k. itagaki detected an outbursting object which can be identified with this cv ( vsnet - alert 11911 ) . subsequent observations have confirmed the presence of superhumps ( vsnet - alert 11916 , 11921 , 11926 , 11930 ) . the times of superhump maxima are listed in table [ tab : j0839oc2010 ] . there was an apparent break in the @xmath0 diagram around @xmath24 , which is likely a stage b c transition . the periods given in table [ tab : perlist ] are based on this interpretation . the mean @xmath3 determined with the pdm method is 0.078423(7 ) d ( figure [ fig : j0839shpdm ] ) . ( 88mm,110mm)fig47.eps ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 55295.3531 & 0.0002 & @xmath170.0003 & 190 + 1 & 55295.4309 & 0.0002 & @xmath170.0009 & 220 + 2 & 55295.5099 & 0.0004 & @xmath170.0003 & 83 + 13 & 55296.3732 & 0.0004 & 0.0006 & 79 + 14 & 55296.4518 & 0.0003 & 0.0007 & 78 + 15 & 55296.5311 & 0.0007 & 0.0016 & 41 + 26 & 55297.3919 & 0.0008 & @xmath170.0000 & 78 + 27 & 55297.4694 & 0.0005 & @xmath170.0009 & 78 + 38 & 55298.3319 & 0.0006 & @xmath170.0008 & 128 + 39 & 55298.4115 & 0.0003 & 0.0003 & 246 + + + + this object ( hereafter sdss j0903 ) is a cv selected during the course of the sdss @xcite , who suspected its eclipsing nature . @xcite confirmed that this object is a deeply eclipsing cv with a short orbital period of 0.059073543(9 ) d. the 2010 outburst , the first - ever outburst reported in real - time , was detected by crts (= css100522:090351@xmath1330036 ; cf . vsnet - alert 11994 ) . subsequent observations have confirmed the eclipsing su uma - type nature of this object ( vsnet - alert 12006 ; figure [ fig : j0903shpdm ] ) . the times of mid - eclipses were determined with the kw method ( table [ tab : j0903ecl ] ) . we obtained an updated orbital ephemeris ( equation [ equ : j0903ecl ] ) using our data and times of eclipses in @xcite . ccccc @xmath15 & minimum & error & @xmath0 & source + 0 & 53800.394700 & 0.000006 & -0.00001 & 1 + 2 & 53800.512854 & 0.000006 & -0.00000 & 1 + 34 & 53802.403198 & 0.000006 & -0.00001 & 1 + 35 & 53802.462279 & 0.000006 & -0.00000 & 1 + 36 & 53802.521361 & 0.000006 & 0.00001 & 1 + 37 & 53802.580442 & 0.000006 & 0.00001 & 1 + 38 & 53802.639501 & 0.000006 & -0.00000 & 1 + 50 & 53803.348386 & 0.000006 & 0.00000 & 1 + 51 & 53803.407462 & 0.000006 & 0.00000 & 1 + 52 & 53803.466529 & 0.000006 & -0.00000 & 1 + 53 & 53803.525601 & 0.000006 & -0.00000 & 1 + 26086 & 55341.38690 & 0.00030 & 0.00022 & 2 + 26087 & 55341.44648 & 0.00036 & 0.00073 & 2 + 26102 & 55342.33163 & 0.00038 & -0.00022 & 2 + 26104 & 55342.44915 & 0.00038 & -0.00085 & 2 + 26119 & 55343.33624 & 0.00039 & 0.00013 & 2 + 26136 & 55344.34126 & 0.00034 & 0.00091 & 2 + 26137 & 55344.39964 & 0.00039 & 0.00020 & 2 + 26152 & 55345.28500 & 0.00030 & -0.00053 & 2 + 26154 & 55345.40339 & 0.00046 & -0.00029 & 2 + 26169 & 55346.29034 & 0.00029 & 0.00056 & 2 + 26170 & 55346.34966 & 0.00032 & 0.00080 & 2 + 26171 & 55346.40821 & 0.00045 & 0.00028 & 2 + 26188 & 55347.41127 & 0.00056 & -0.00091 & 2 + 26193 & 55347.70712 & 0.00027 & -0.00043 & 2 + 26204 & 55348.35745 & 0.00037 & 0.00009 & 2 + 26205 & 55348.41652 & 0.00035 & 0.00009 & 2 + 26209 & 55348.65271 & 0.00030 & -0.00001 & 2 + 26210 & 55348.71183 & 0.00036 & 0.00003 & 2 + 26221 & 55349.36176 & 0.00037 & 0.00015 & 2 + 26222 & 55349.42067 & 0.00040 & -0.00001 & 2 + 26243 & 55350.66096 & 0.00034 & -0.00026 & 2 + 26244 & 55350.71996 & 0.00044 & -0.00034 & 2 + 26255 & 55351.36974 & 0.00028 & -0.00037 & 2 + 26256 & 55351.42824 & 0.00041 & -0.00095 & 2 + 26272 & 55352.37433 & 0.00072 & -0.00003 & 2 + + + + @xmath73 in the following analysis , we removed observations within 0.08 @xmath2 of eclipses . the times of superhump maxima are listed in table [ tab : j0903oc2010 ] . there was a clear stage b c transition around @xmath74 . the @xmath7 for stage b was @xmath75 . this clearly positive @xmath7 in deeply eclipsing short - period su uma - type dwarf novae confirmed the finding in xz eri ( @xcite ; @xcite ) . the fractional superhump excesses for stages b and c were 2.1 % and 1.7 % , respectively . other parameters are listed in table [ tab : perlist ] . ( 88mm,110mm)fig48.eps ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 55340.4222 & 0.0009 & 0.0072 & 51 + 31 & 55342.2882 & 0.0009 & 0.0033 & 26 + 32 & 55342.3424 & 0.0007 & @xmath170.0028 & 57 + 33 & 55342.4003 & 0.0019 & @xmath170.0052 & 31 + 48 & 55343.3142 & 0.0013 & 0.0039 & 19 + 49 & 55343.3624 & 0.0016 & @xmath170.0083 & 55 + 50 & 55343.4267 & 0.0017 & @xmath170.0043 & 54 + 65 & 55344.3326 & 0.0008 & @xmath170.0032 & 57 + 81 & 55345.2974 & 0.0008 & @xmath170.0035 & 57 + 82 & 55345.3554 & 0.0047 & @xmath170.0058 & 48 + 83 & 55345.4122 & 0.0040 & @xmath170.0093 & 51 + 98 & 55346.3301 & 0.0017 & 0.0038 & 83 + 99 & 55346.3923 & 0.0017 & 0.0056 & 47 + 114 & 55347.3027 & 0.0090 & 0.0112 & 74 + 116 & 55347.4195 & 0.0024 & 0.0075 & 51 + 121 & 55347.7186 & 0.0012 & 0.0050 & 52 + 132 & 55348.3800 & 0.0009 & 0.0028 & 54 + 133 & 55348.4419 & 0.0017 & 0.0044 & 29 + 137 & 55348.6809 & 0.0007 & 0.0021 & 62 + 149 & 55349.4016 & 0.0010 & @xmath170.0010 & 50 + 170 & 55350.6692 & 0.0019 & @xmath170.0002 & 49 + 182 & 55351.3798 & 0.0022 & @xmath170.0133 & 29 + + + + this object ( hereafter sdss j1152 ) is a cv selected during the course of the sdss @xcite , who suspected its eclipsing nature . @xcite established that this cv is deeply eclipsing and determined its period to be 0.06770(28 ) d. the 2009 june outburst was detected by e. muyllaert at a ccd magnitude of 16.4 on june 9 ( cvnet - outburst 3158 ) . superhumps and eclipses were soon detected ( vsnet - alert 11288 ) , establishing the su uma - type nature of this object . the times of superhump maxima determined observations outside the eclipses are listed in table [ tab : j1152oc2009 ] . the intervening clouds made the epoch of @xmath15 = 39 rather uncertain . a period analysis with the pdm method has yielded a @xmath3 of 0.0689(1 ) d ( figure [ fig : j1152shpdm ] ) . since the signal is broad due to the poor phase coverage , we used a bayesian modeling of the light curve using the template superhump light curve ( see appendix ) . the best period by this method is 0.06887(4 ) d , corresponding to @xmath13 of 1.7 % . we adopted this period in the figure and table . although the basic nature of the object is well - established , both orbital and superhump periods need to be refined by further observations . ( 88mm,110mm)fig49.eps ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 54994.0621 & 0.0005 & 0.0056 & 125 + 39 & 54996.7376 & 0.0068 & @xmath170.0135 & 9 + 67 & 54998.6919 & 0.0021 & 0.0062 & 18 + 68 & 54998.7564 & 0.0021 & 0.0016 & 28 + + + + this object ( hereafter j1250 ) is a cv selected during the course of the sdss @xcite , who suggested a high inclination cv . @xcite confirmed that this is indeed a deeply eclipsing cv with a very short orbital period of 0.058735687(4 ) d. shortly before @xcite becomes available , the object was found to be in outburst in 2008 january ( s. brady , cvnet - discussion 1104 ) , who successfully detected eclipses and reported a period of 0.059 d. based on these and 2009 observations , we have determined eclipse times and updated the ephemeris . both the 2008 and 2009 outbursts were superoutbursts , and the times of superhump maxima are listed in tables [ tab : j1250oc2008 ] and [ tab : j1250oc2009 ] determined from observations outside the eclipses . a pdm analysis of the 2008 observations has yielded a mean @xmath3 of 0.06032(5 ) d ( figure [ fig : j1250shpdm ] ) . the @xmath7 for @xmath76 , apparently stage b , was @xmath77 . the period apparently decreased after this , suggesting a transition to stage c. the @xmath13 of 2.7 % for the mean @xmath3 is relatively large for this @xmath2 . using the @xmath13@xmath14 relation in @xcite , this @xmath13 corresponds to @xmath14 = 0.14 . assuming a moderate white - dwarf mass , this @xmath14 would place the object around the upper ( massive secondary ) boundary defined by sdss j0903 and sdss j1507 in figure 2 of @xcite rather than around the period bounce . this identification appears to be favored by the relatively large @xmath7 and rather frequent outbursts ( the interval of two superoutburst was @xmath9 650 d ) . further determination of system parameters of this object will improve our knowledge of cv evolution near the period minimum . ( 88mm,110mm)fig50.eps ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 54491.9263 & 0.0010 & 0.0039 & 6 + 44 & 54494.5762 & 0.0018 & @xmath170.0002 & 18 + 45 & 54494.6331 & 0.0012 & @xmath170.0037 & 41 + 46 & 54494.6926 & 0.0014 & @xmath170.0044 & 39 + 47 & 54494.7553 & 0.0008 & @xmath170.0020 & 42 + 48 & 54494.8165 & 0.0007 & @xmath170.0011 & 41 + 77 & 54496.5690 & 0.0022 & 0.0021 & 17 + 78 & 54496.6287 & 0.0011 & 0.0015 & 24 + 79 & 54496.6906 & 0.0015 & 0.0031 & 24 + 80 & 54496.7482 & 0.0011 & 0.0003 & 22 + 81 & 54496.8082 & 0.0009 & @xmath170.0000 & 24 + 82 & 54496.8696 & 0.0013 & 0.0011 & 24 + 83 & 54496.9293 & 0.0010 & 0.0005 & 20 + 146 & 54500.7289 & 0.0033 & 0.0000 & 22 + 147 & 54500.7882 & 0.0029 & @xmath170.0010 & 19 + + + + ccccc @xmath15 & max & error & @xmath16 + 0 & 55144.2715 & 0.0015 & 77 + 1 & 55144.3338 & 0.0004 & 54 + + + this object ( hereafter sdss j1502 ) is an eclipsing cv selected during the course of the sdss @xcite . @xcite performed high - speed photometry in quiescence and obtained orbital parameters [ @xmath7 = 0.05890961(5 ) d and estimated @xmath14 = 0.109(3 ) ] . @xcite attributed this cv to a cv before the `` period bounce '' . the 2009 july superoutburst of this object was detected by j. shears ( baavss - alert 1997 ) . subsequent observations detected superhumps ( cf . vsnet - alert 11332 ; figure [ fig : j1502shpdm ] ) . @xcite also reported an analysis of the same superoutburst using the slightly different data set from ours . the times of mid - eclipses were determined with the kw method ( table [ tab : j1502ecl ] ) . we obtained an updated orbital ephemeris ( equation [ equ : j1502ecl ] ) using our data and times of eclipses in @xcite and @xcite . we only used times of eclipses in @xcite which were not covered by our observations . we also disregarded @xmath78 eclipse due to its large @xmath0 . ccccc @xmath15 & minimum & error & @xmath0 & source + 0 & 53799.640618 & 0.000004 & 0.00013 & 1 + 2 & 53799.758414 & 0.000007 & 0.00011 & 1 + 17 & 53800.642070 & 0.000006 & 0.00012 & 1 + 18 & 53800.700966 & 0.000006 & 0.00011 & 1 + 19 & 53800.759901 & 0.000006 & 0.00014 & 1 + 52 & 53802.703911 & 0.000002 & 0.00013 & 1 + 68 & 53803.646461 & 0.000003 & 0.00013 & 1 + 69 & 53803.705371 & 0.000006 & 0.00013 & 1 + 70 & 53803.764277 & 0.000003 & 0.00013 & 1 + 854 & 53849.94908 & 0.00035 & -0.00009 & 2 + 866 & 53850.65615 & 0.00012 & 0.00006 & 2 + 867 & 53850.71498 & 0.00011 & -0.00002 & 2 + 868 & 53850.77384 & 0.00021 & -0.00007 & 2 + 869 & 53850.83284 & 0.00012 & 0.00002 & 2 + 870 & 53850.89165 & 0.00014 & -0.00008 & 2 + 871 & 53850.95066 & 0.00013 & 0.00003 & 2 + 883 & 53851.65760 & 0.00014 & 0.00005 & 2 + 884 & 53851.71630 & 0.00014 & -0.00016 & 2 + 886 & 53851.83418 & 0.00016 & -0.00010 & 2 + 887 & 53851.89334 & 0.00014 & 0.00015 & 2 + 888 & 53851.95250 & 0.00014 & 0.00040 & 2 + 900 & 53852.65899 & 0.00018 & -0.00002 & 2 + 901 & 53852.71800 & 0.00015 & 0.00008 & 2 + 902 & 53852.77661 & 0.00011 & -0.00022 & 2 + 903 & 53852.83568 & 0.00014 & -0.00006 & 2 + 1308 & 53876.69411 & 0.00004 & 0.00004 & 2 + 6454 & 54179.84201 & 0.00009 & -0.00021 & 2 + 6455 & 54179.90009 & 0.00009 & -0.00104 & 2 + 6457 & 54180.01899 & 0.00011 & 0.00004 & 2 + 13787 & 54611.82514 & 0.00012 & -0.00025 & 2 + 13788 & 54611.88411 & 0.00012 & -0.00019 & 2 + 13789 & 54611.94325 & 0.00017 & 0.00004 & 2 + 13802 & 54612.70876 & 0.00019 & -0.00027 & 2 + 13803 & 54612.76795 & 0.00012 & 0.00001 & 2 + 13804 & 54612.82684 & 0.00018 & -0.00001 & 2 + 13805 & 54612.88550 & 0.00012 & -0.00026 & 2 + 13806 & 54612.94456 & 0.00027 & -0.00011 & 2 + 20758 & 55022.48413 & 0.00005 & 0.00081 & 2 + 20760 & 55022.60121 & 0.00032 & 0.00007 & 3 + 20761 & 55022.66052 & 0.00036 & 0.00047 & 3 + 20762 & 55022.71932 & 0.00026 & 0.00036 & 3 + 20763 & 55022.77857 & 0.00031 & 0.00070 & 2 + 20777 & 55023.60263 & 0.00029 & 0.00003 & 3 + 20778 & 55023.66133 & 0.00028 & -0.00018 & 2 + 20779 & 55023.72066 & 0.00021 & 0.00024 & 3 + 20780 & 55023.77951 & 0.00018 & 0.00018 & 3 + 20781 & 55023.83814 & 0.00024 & -0.00010 & 3 + 20782 & 55023.89720 & 0.00040 & 0.00005 & 3 + 20794 & 55024.60306 & 0.00050 & -0.00101 & 2 + 20796 & 55024.72187 & 0.00026 & -0.00001 & 3 + 20797 & 55024.78077 & 0.00022 & -0.00002 & 3 + 20798 & 55024.83956 & 0.00028 & -0.00014 & 3 + 20807 & 55025.37023 & 0.00027 & 0.00034 & 3 + + + + + ccccc @xmath15 & minimum & error & @xmath0 & source + 20808 & 55025.42910 & 0.00030 & 0.00030 & 3 + 20811 & 55025.60573 & 0.00042 & 0.00020 & 3 + 20812 & 55025.66472 & 0.00014 & 0.00028 & 3 + 20813 & 55025.72362 & 0.00016 & 0.00027 & 3 + 20814 & 55025.78256 & 0.00017 & 0.00030 & 3 + 20815 & 55025.84150 & 0.00023 & 0.00034 & 3 + 20824 & 55026.37130 & 0.00026 & -0.00005 & 3 + 20825 & 55026.43013 & 0.00033 & -0.00013 & 3 + 20828 & 55026.60726 & 0.00011 & 0.00027 & 2 + 20829 & 55026.66588 & 0.00029 & -0.00002 & 3 + 20830 & 55026.72440 & 0.00022 & -0.00041 & 3 + 20831 & 55026.78290 & 0.00033 & -0.00082 & 3 + 20832 & 55026.84210 & 0.00100 & -0.00053 & 3 + 20835 & 55027.01943 & 0.00032 & 0.00008 & 3 + 20836 & 55027.07836 & 0.00028 & 0.00010 & 3 + 20837 & 55027.13703 & 0.00053 & -0.00014 & 3 + 20841 & 55027.37299 & 0.00026 & 0.00018 & 3 + 20845 & 55027.60852 & 0.00038 & 0.00007 & 3 + 20846 & 55027.66772 & 0.00040 & 0.00036 & 3 + 20847 & 55027.72659 & 0.00021 & 0.00032 & 3 + 20848 & 55027.78561 & 0.00022 & 0.00043 & 2 + 20849 & 55027.84470 & 0.00051 & 0.00061 & 3 + 20858 & 55028.37421 & 0.00027 & -0.00006 & 3 + 20859 & 55028.43318 & 0.00019 & -0.00000 & 3 + 20860 & 55028.49206 & 0.00023 & -0.00003 & 3 + 20861 & 55028.55083 & 0.00026 & -0.00017 & 3 + 20864 & 55028.72779 & 0.00037 & 0.00006 & 3 + 20881 & 55029.72935 & 0.00035 & 0.00016 & 3 + 20882 & 55029.78814 & 0.00036 & 0.00004 & 3 + 20883 & 55029.84720 & 0.00033 & 0.00019 & 3 + 20894 & 55030.49541 & 0.00105 & 0.00040 & 2 + 20895 & 55030.55404 & 0.00024 & 0.00012 & 2 + 20898 & 55030.73046 & 0.00014 & -0.00019 & 2 + 20898 & 55030.73050 & 0.00033 & -0.00015 & 3 + 20899 & 55030.78951 & 0.00034 & -0.00005 & 3 + 20900 & 55030.84853 & 0.00035 & 0.00006 & 3 + 20915 & 55031.73159 & 0.00076 & -0.00052 & 3 + 20916 & 55031.79037 & 0.00043 & -0.00065 & 2 + 20917 & 55031.84916 & 0.00138 & -0.00077 & 3 + 20928 & 55032.49776 & 0.00068 & -0.00017 & 3 + 20929 & 55032.55681 & 0.00079 & -0.00003 & 3 + 20932 & 55032.73351 & 0.00064 & -0.00006 & 3 + 20934 & 55032.85137 & 0.00065 & -0.00002 & 3 + 20949 & 55033.73470 & 0.00075 & -0.00033 & 3 + 20950 & 55033.79383 & 0.00077 & -0.00011 & 3 + 20966 & 55034.73608 & 0.00118 & -0.00041 & 2 + 20968 & 55034.85449 & 0.00133 & 0.00018 & 3 + 21031 & 55038.56538 & 0.00065 & -0.00023 & 3 + 21758 & 55081.39289 & 0.00168 & 0.00009 & 2 + + + + + @xmath79 in the following analysis , we removed observations within 0.08 @xmath2 of eclipses . the times of superhump maxima are listed in table [ tab : j1502oc2009 ] . there was a likely stage b c transition around @xmath80 . the values given in table [ tab : perlist ] were estimated following this interpretation . the @xmath7 for stage b was @xmath81 , which may have been underestimated because the earliest part of the outburst was not sufficiently observed . although @xcite detected a pattern of @xmath0 variation basically to ours , we used our values based on larger set of data . was reported to be @xmath82 , our analysis of their timing data for @xmath83 yielded a @xmath7 of @xmath84 by our definition . ] the maxima for @xmath85 may not be superhumps , probably strongly affected by orbital modulations , and are excluded from this period analysis . ( 88mm,110mm)fig51.eps ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 55022.6300 & 0.0004 & 0.0039 & 92 + 1 & 55022.6895 & 0.0003 & 0.0029 & 78 + 16 & 55023.5921 & 0.0004 & @xmath170.0009 & 69 + 17 & 55023.6541 & 0.0008 & 0.0007 & 9 + 18 & 55023.7122 & 0.0004 & @xmath170.0016 & 145 + 19 & 55023.7736 & 0.0003 & @xmath170.0007 & 179 + 20 & 55023.8343 & 0.0004 & @xmath170.0004 & 147 + 21 & 55023.8946 & 0.0012 & @xmath170.0005 & 74 + 34 & 55024.6790 & 0.0005 & @xmath170.0017 & 73 + 35 & 55024.7401 & 0.0002 & @xmath170.0011 & 95 + 36 & 55024.8015 & 0.0003 & @xmath170.0002 & 95 + 46 & 55025.4060 & 0.0003 & 0.0000 & 91 + 47 & 55025.4652 & 0.0003 & @xmath170.0012 & 91 + 49 & 55025.5904 & 0.0004 & 0.0031 & 29 + 50 & 55025.6449 & 0.0002 & @xmath170.0028 & 154 + 51 & 55025.7061 & 0.0001 & @xmath170.0020 & 279 + 52 & 55025.7648 & 0.0001 & @xmath170.0038 & 230 + 53 & 55025.8260 & 0.0003 & @xmath170.0029 & 187 + 61 & 55026.3178 & 0.0011 & 0.0054 & 70 + 62 & 55026.3694 & 0.0005 & @xmath170.0035 & 110 + 63 & 55026.4387 & 0.0005 & 0.0054 & 108 + 66 & 55026.6156 & 0.0014 & 0.0010 & 24 + 67 & 55026.6726 & 0.0004 & @xmath170.0024 & 127 + 68 & 55026.7367 & 0.0005 & 0.0012 & 115 + 69 & 55026.7966 & 0.0006 & 0.0007 & 111 + 73 & 55027.0410 & 0.0006 & 0.0033 & 91 + 74 & 55027.1001 & 0.0003 & 0.0020 & 109 + 79 & 55027.4019 & 0.0003 & 0.0017 & 94 + 82 & 55027.5840 & 0.0005 & 0.0025 & 52 + 83 & 55027.6446 & 0.0005 & 0.0027 & 71 + 84 & 55027.7070 & 0.0005 & 0.0046 & 75 + 86 & 55027.8266 & 0.0003 & 0.0033 & 75 + 89 & 55028.0090 & 0.0006 & 0.0045 & 54 + 90 & 55028.0678 & 0.0003 & 0.0028 & 131 + 91 & 55028.1284 & 0.0007 & 0.0030 & 85 + 95 & 55028.3691 & 0.0003 & 0.0020 & 105 + 96 & 55028.4293 & 0.0003 & 0.0017 & 135 + 97 & 55028.4904 & 0.0004 & 0.0024 & 157 + 98 & 55028.5513 & 0.0003 & 0.0029 & 101 + 101 & 55028.7293 & 0.0006 & @xmath170.0005 & 59 + 118 & 55029.7516 & 0.0007 & @xmath170.0055 & 88 + 119 & 55029.8127 & 0.0006 & @xmath170.0048 & 92 + 120 & 55029.8733 & 0.0006 & @xmath170.0046 & 81 + 134 & 55030.7147 & 0.0005 & @xmath170.0093 & 76 + 135 & 55030.7763 & 0.0005 & @xmath170.0081 & 87 + 136 & 55030.8392 & 0.0007 & @xmath170.0057 & 88 + 152 & 55031.7872 & 0.0023 & @xmath170.0246 & 34 + 153 & 55031.8347 & 0.0023 & @xmath170.0375 & 34 + 163 & 55032.4860 & 0.0016 & 0.0095 & 47 + 164 & 55032.5491 & 0.0017 & 0.0121 & 46 + 167 & 55032.7254 & 0.0010 & 0.0072 & 48 + 168 & 55032.7880 & 0.0008 & 0.0093 & 46 + 169 & 55032.8425 & 0.0011 & 0.0034 & 49 + 183 & 55033.6874 & 0.0037 & 0.0023 & 31 + 184 & 55033.7398 & 0.0070 & @xmath170.0058 & 48 + 185 & 55033.7996 & 0.0020 & @xmath170.0065 & 49 + 186 & 55033.8721 & 0.0038 & 0.0056 & 27 + 200 & 55034.7195 & 0.0013 & 0.0070 & 47 + 201 & 55034.7818 & 0.0016 & 0.0088 & 49 + 202 & 55034.8427 & 0.0026 & 0.0093 & 49 + + + + this object ( hereafter sdss j1610 ) was initially selected as a dwarf nova by @xcite . the 2009 july outburst was detected by the crts (= css090727:161028 + 090739 ) . double - peaked strong emission lines , characteristic to a relatively high inclination dwarf nova , in sdss spectrum was announced ( vsnet - alert 11350 ) . a remarkable growth of superhumps was observed four days after the outburst detection ( vsnet - alert 11366 ) , suggesting a substantial delay in the growth of superhumps . further observations clarified the potential wz sge - type characters ( vsnet - alert 11367 , 11368 , 11381 ; figure [ fig : j1610shpdm ] . the times of ordinary superhumps are listed in table [ tab : j1610oc2009 ] . there was a clear stage a b transition around @xmath86 . the @xmath7 for the well - observed segment of stage b was @xmath87 ( @xmath88 ) . although an extrapolation to @xmath89 has yielded a @xmath7 of @xmath90 , the identification of superhumps was slightly ambiguous due to the faintness . we thus adopted the former value for the @xmath7 of this object . an analysis of the light curve before the growth of ordinary superhumps detected early superhumps ( figure [ fig : j1610eshpdm ] ) with a period of 0.05687(1 ) d ( pdm and bayesian methods ) , confirming both the wz sge - type nature and a relatively high inclination . the resultant @xmath13 of 1.6 % is relatively large among wz sge - type dwarf novae ( cf . @xcite ) , consistent with a relatively large @xmath7 . ( 88mm,110mm)fig52.eps ( 88mm,110mm)fig53.eps ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 55042.4471 & 0.0006 & @xmath170.0220 & 120 + 10 & 55043.0435 & 0.0016 & @xmath170.0053 & 99 + 11 & 55043.1014 & 0.0011 & @xmath170.0053 & 184 + 16 & 55043.3959 & 0.0012 & @xmath170.0007 & 29 + 17 & 55043.4515 & 0.0016 & @xmath170.0031 & 28 + 26 & 55043.9811 & 0.0003 & 0.0048 & 61 + 27 & 55044.0393 & 0.0002 & 0.0049 & 60 + 28 & 55044.0986 & 0.0004 & 0.0063 & 60 + 33 & 55044.3907 & 0.0003 & 0.0085 & 23 + 34 & 55044.4468 & 0.0006 & 0.0066 & 26 + 35 & 55044.5061 & 0.0013 & 0.0080 & 17 + 43 & 55044.9678 & 0.0004 & 0.0059 & 58 + 44 & 55045.0245 & 0.0006 & 0.0045 & 56 + 45 & 55045.0839 & 0.0004 & 0.0060 & 46 + 51 & 55045.4319 & 0.0005 & 0.0061 & 114 + 52 & 55045.4864 & 0.0005 & 0.0026 & 76 + 61 & 55046.0067 & 0.0006 & 0.0012 & 49 + 62 & 55046.0641 & 0.0011 & 0.0006 & 23 + 68 & 55046.4086 & 0.0009 & @xmath170.0028 & 29 + 68 & 55046.4086 & 0.0010 & @xmath170.0027 & 30 + 78 & 55046.9894 & 0.0012 & @xmath170.0017 & 181 + 79 & 55047.0467 & 0.0012 & @xmath170.0023 & 167 + 80 & 55047.1042 & 0.0008 & @xmath170.0029 & 60 + 85 & 55047.3919 & 0.0008 & @xmath170.0050 & 31 + 86 & 55047.4519 & 0.0009 & @xmath170.0029 & 27 + 102 & 55048.3792 & 0.0009 & @xmath170.0033 & 22 + 103 & 55048.4370 & 0.0010 & @xmath170.0034 & 28 + 137 & 55050.4060 & 0.0028 & @xmath170.0055 & 28 + 268 & 55058.0072 & 0.0025 & 0.0009 & 121 + 269 & 55058.0659 & 0.0035 & 0.0017 & 123 + + + + this object was selected as a cv candidate by @xcite . although the object was not recognized as a cv during the course of the sdss , we employed the sdss designation in this paper ( hereafter sdss j1625 ) . the 2010 outburst of this object was detected by the crts (= css100705:162520@xmath1120309 ) , a discussion on the sdss spectrum and the earlier discovery of the object by @xcite can be found in vsnet - alert 12052 , 12053 . short - term variations were detected soon after this outburst detection ( vsnet - alert 12054 , 12059 ) , which later turned out to be developing superhumps ( vsnet - alert 12061 , 12062 ) . fully developed superhumps and the course of period evolution were subsequently observed ( vsnet - alert 12064 , 12065 , 12066 , 12068 , 12071 ; figure [ fig : j1625shpdm ] ) . the object entered the rapid decline phase 45 d after the development of superhumps , which was unexpectedly early ( vsnet - alert 12079 ) . the object was also unusual in its rebrightening phenomenon soon after the rapid decline ( vsnet - alert 12087 ) . the times of superhump maxima are listed in table [ tab : j1625oc2010 ] . the epochs for @xmath91 correspond to the growing stage of superhumps ( stage a ) . the epochs for @xmath92 were maxima after the rebrightening and these humps may not be true superhumps . we also excluded @xmath93 ( just prior to the rebrightening ) in determining period variation due to its low signal . the @xmath7 for @xmath94 was @xmath95 , which is unusually large for this long @xmath3 = 0.09605(5 ) d ( mean period based on timing analysis ) . there was also little indication of a stage b c transition during the superoutburst plateau . these unusual evolution may be related to the unexpectedly early fading during the superoutburst plateau ( see figure [ fig : j1625oc2010 ] ) . this object appears to add another variety of superhump evolution in long-@xmath3 systems ( cf . @xcite , subsection 4.10 ) . ( 88mm,110mm)fig54.eps ( 88mm,90mm)fig55.eps ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 55384.4489 & 0.0003 & @xmath170.0363 & 363 + 2 & 55384.6553 & 0.0002 & @xmath170.0224 & 176 + 6 & 55385.0497 & 0.0003 & @xmath170.0128 & 161 + 10 & 55385.4459 & 0.0002 & @xmath170.0015 & 587 + 11 & 55385.5446 & 0.0002 & 0.0010 & 365 + 12 & 55385.6431 & 0.0001 & 0.0032 & 179 + 20 & 55386.4255 & 0.0001 & 0.0159 & 732 + 21 & 55386.5198 & 0.0001 & 0.0140 & 662 + 30 & 55387.3886 & 0.0003 & 0.0169 & 265 + 31 & 55387.4833 & 0.0003 & 0.0153 & 289 + 41 & 55388.4373 & 0.0002 & 0.0071 & 328 + 42 & 55388.5327 & 0.0002 & 0.0064 & 291 + 43 & 55388.6296 & 0.0002 & 0.0070 & 194 + 44 & 55388.7250 & 0.0002 & 0.0062 & 158 + 51 & 55389.3941 & 0.0003 & 0.0018 & 382 + 52 & 55389.4895 & 0.0003 & 0.0010 & 430 + 53 & 55389.5855 & 0.0002 & 0.0007 & 189 + 54 & 55389.6808 & 0.0003 & @xmath170.0001 & 200 + 61 & 55390.3514 & 0.0027 & @xmath170.0031 & 135 + 62 & 55390.4523 & 0.0004 & 0.0016 & 338 + 63 & 55390.5440 & 0.0006 & @xmath170.0029 & 144 + 72 & 55391.4104 & 0.0005 & @xmath170.0025 & 193 + 73 & 55391.5071 & 0.0007 & @xmath170.0021 & 170 + 82 & 55392.3723 & 0.0010 & @xmath170.0027 & 102 + 83 & 55392.4739 & 0.0008 & 0.0026 & 130 + 84 & 55392.5750 & 0.0009 & 0.0075 & 144 + 85 & 55392.6648 & 0.0008 & 0.0010 & 190 + 93 & 55393.4316 & 0.0034 & @xmath170.0019 & 153 + 94 & 55393.5346 & 0.0025 & 0.0049 & 52 + 104 & 55394.4882 & 0.0013 & @xmath170.0037 & 43 + 114 & 55395.4413 & 0.0026 & @xmath170.0127 & 49 + 130 & 55397.0091 & 0.0028 & 0.0156 & 200 + 134 & 55397.3893 & 0.0035 & 0.0108 & 94 + 135 & 55397.4389 & 0.0042 & @xmath170.0358 & 149 + + + + this object ( hereafter sdss j1637 ) is a cv selected during the course of the sdss @xcite . @xcite reported the presence of high and low states and suggested the dwarf nova - type classification . the 2004 outburst of this object was detected by r. stubbings at a visual magnitude of 15.0 on 2004 march 27 ( vsnet - outburst 6212 ) . the presence of superhumps was soon confirmed ( vsnet - alert 8084 , 8086 , 8088 ) . the mean superhump period determined from the observation was 0.06910(4 ) d ( pdm method , figure [ fig : j1637shpdm ] ) , and the times of superhump maxima are listed in table [ tab : j1637oc2004 ] . since the outburst entered its rapid decline stage @xmath9 4 d after the initial detection , these superhumps were very likely stage c superhumps . using the orbital period of 0.06739(1 ) d determined by @xcite , we obtained an @xmath13 of 2.5 % . ( 88mm,110mm)fig56.eps ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 53093.2412 & 0.0016 & @xmath170.0023 & 145 + 1 & 53093.3132 & 0.0012 & 0.0004 & 76 + 2 & 53093.3832 & 0.0012 & 0.0014 & 37 + 4 & 53093.5206 & 0.0015 & 0.0004 & 18 + 5 & 53093.5896 & 0.0008 & 0.0002 & 155 + 29 & 53095.2464 & 0.0027 & @xmath170.0030 & 36 + 30 & 53095.3206 & 0.0025 & 0.0020 & 38 + 31 & 53095.3890 & 0.0022 & 0.0013 & 30 + 43 & 53096.2189 & 0.0034 & 0.0012 & 87 + 44 & 53096.2853 & 0.0028 & @xmath170.0016 & 147 + + + + this object ( hereafter sdss j1653 ) is a cv selected during the course of the sdss @xcite , who detected superhumps with a period of 1.58 hr during one of its superoutburst . the 2010 superoutburst was detected by the crts ( cf . vsnet - alert 11936 ) . the times of superhump maxima are listed in table [ tab : j1653oc2010 ] . it is evident from these data that these observations recorded a stage b c transition . the derived periods for each stage are listed in table [ tab : perlist ] . although the observation only covered the late part of the superoutburst , the evolution of superhump period appears to be typical . as judged from these periods , @xcite appears to have recorded stage b superhumps . ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 55304.1037 & 0.0009 & @xmath170.0007 & 67 + 1 & 55304.1673 & 0.0007 & @xmath170.0022 & 59 + 2 & 55304.2313 & 0.0004 & @xmath170.0032 & 66 + 3 & 55304.2970 & 0.0005 & @xmath170.0025 & 66 + 5 & 55304.4272 & 0.0003 & @xmath170.0024 & 73 + 6 & 55304.4913 & 0.0004 & @xmath170.0033 & 67 + 7 & 55304.5579 & 0.0004 & @xmath170.0018 & 72 + 50 & 55307.3670 & 0.0017 & 0.0110 & 55 + 51 & 55307.4263 & 0.0014 & 0.0052 & 65 + 52 & 55307.4906 & 0.0010 & 0.0046 & 69 + 93 & 55310.1499 & 0.0009 & @xmath170.0025 & 45 + 94 & 55310.2197 & 0.0015 & 0.0023 & 40 + 215 & 55318.0817 & 0.0041 & @xmath170.0045 & 22 + + + + this object ( hereafter sdss j2048 ) is a cv selected during the course of the sdss @xcite , who reported the detection of a white dwarf in the spectrum , suggesting a dwarf nova with a low mass - transfer rate . the outburst in 2009 october was detected by e. muyllaert ( cvnet - outburst 3367 ) . subsequent observations by i. miller detected superhumps ( cvnet - outburst 3383 ) . @xcite observed the object in quiescence and obtained an orbital period of 0.060597(2 ) d. we identified the superhump period of 0.06166(2 ) d based om this @xmath2 ( figure [ fig : j2048shpdm ] . the times of superhump maxima are listed in table [ tab : j2048oc2009 ] . the @xmath13 was 1.8 % . ( 88mm,110mm)fig57.eps ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 55119.4241 & 0.0008 & @xmath170.0013 & 45 + 16 & 55120.4152 & 0.0012 & 0.0029 & 32 + 31 & 55121.3347 & 0.0007 & @xmath170.0029 & 63 + 32 & 55121.4006 & 0.0011 & 0.0014 & 56 + + + + this object ( hereafter ot j0406 ) was discovered by k. itagaki in 2008 @xcite . the 2010 outburst , detected by the crts , is the second known superoutburst of this object . the outburst was apparently detected in its late stage . the times of superhump maxima are listed in table [ tab : j0406oc2010 ] . we attribute these superhumps to stage c superhumps . the mean period with the pdm method was 0.07996(3 ) d , close to that ( 0.07992 d ) recorded during the 2008 superoutburst @xcite . we adopted this period in table [ tab : perlist ] . ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 55246.9899 & 0.0009 & @xmath170.0011 & 126 + 12 & 55247.9497 & 0.0010 & @xmath170.0000 & 132 + 25 & 55248.9899 & 0.0011 & 0.0015 & 205 + 49 & 55250.9069 & 0.0030 & 0.0010 & 156 + 62 & 55251.9432 & 0.0027 & @xmath170.0014 & 207 + + + + this object ( = [email protected] 1257@xmath170089884 , hereafter ot j0506 ) is a dwarf nova discovered by @xcite ( see also vsnet - alert 11686 for the initial announcement ) . the object was rising at the time of the discovery and superhumps were subsequently detected ( vsnet - alert 11688 ; @xcite ) . since the discovery announcement was made sufficiently early , the early stages of the outburst was well observed . the times of superhump maxima are listed in table [ tab : j0506oc2009 ] . the mean superhump period excluding the initial night was 0.06928(2 ) d ( pdm method , figure [ fig : j0506shpdm ] ) . ( 88mm,110mm)fig58.eps ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 55160.5113 & 0.0034 & @xmath170.0010 & 24 + 1 & 55160.5777 & 0.0008 & @xmath170.0039 & 48 + 2 & 55160.6485 & 0.0007 & @xmath170.0024 & 48 + 23 & 55162.1113 & 0.0006 & 0.0045 & 130 + 24 & 55162.1810 & 0.0007 & 0.0049 & 149 + 52 & 55164.1160 & 0.0010 & @xmath170.0012 & 102 + 53 & 55164.1854 & 0.0011 & @xmath170.0011 & 140 + 56 & 55164.3920 & 0.0009 & @xmath170.0025 & 12 + 81 & 55166.1293 & 0.0023 & 0.0016 & 146 + 82 & 55166.2028 & 0.0099 & 0.0058 & 92 + 85 & 55166.4085 & 0.0029 & 0.0035 & 7 + 88 & 55166.6129 & 0.0018 & @xmath170.0001 & 13 + 96 & 55167.1688 & 0.0020 & 0.0013 & 157 + 97 & 55167.2387 & 0.0020 & 0.0018 & 14 + 98 & 55167.3056 & 0.0017 & @xmath170.0006 & 12 + 99 & 55167.3755 & 0.0011 & @xmath170.0000 & 14 + 100 & 55167.4428 & 0.0015 & @xmath170.0021 & 12 + 101 & 55167.5116 & 0.0016 & @xmath170.0026 & 13 + 102 & 55167.5774 & 0.0022 & @xmath170.0061 & 12 + + + + this object ( hereafter ot j1026 ) is a dwarf nova discovered by k. itagaki @xcite . although @xcite reported on late - stage observations of the 2009 superoutburst , the observational coverage was short . we observed the 2010 superoutburst , detected by i. miller ( baavss - alert 2245 ) . we first time succeeded in recording both stages b and c ( table [ tab : j1026oc2010 ] ) . the resultant period is in disagreement with the 2009 result . since the present coverage is much better than the 2009 observation , the present values are more reliable . a reanalysis of the 2009 data could not yield a continuous @xmath0 diagram , suggesting a discontinuous phase jump ( or appearance of stronger secondary superhump peaks ) in the final stage of the 2009 superoutburst . ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 55270.5082 & 0.0002 & @xmath170.0012 & 67 + 1 & 55270.5783 & 0.0003 & 0.0002 & 75 + 2 & 55270.6456 & 0.0003 & @xmath170.0011 & 75 + 26 & 55272.2894 & 0.0009 & @xmath170.0048 & 30 + 27 & 55272.3616 & 0.0005 & @xmath170.0013 & 76 + 28 & 55272.4296 & 0.0005 & @xmath170.0020 & 58 + 29 & 55272.5006 & 0.0006 & 0.0004 & 68 + 43 & 55273.4636 & 0.0004 & 0.0024 & 60 + 44 & 55273.5311 & 0.0008 & 0.0012 & 36 + 57 & 55274.4268 & 0.0039 & 0.0045 & 47 + 58 & 55274.4972 & 0.0021 & 0.0063 & 67 + 59 & 55274.5603 & 0.0012 & 0.0007 & 72 + 113 & 55278.2660 & 0.0013 & @xmath170.0004 & 53 + 114 & 55278.3322 & 0.0012 & @xmath170.0029 & 76 + 115 & 55278.4048 & 0.0011 & 0.0010 & 69 + 116 & 55278.4718 & 0.0008 & @xmath170.0006 & 76 + 117 & 55278.5393 & 0.0008 & @xmath170.0018 & 61 + 118 & 55278.6091 & 0.0007 & @xmath170.0006 & 72 + + + + this object (= css100217:104411@xmath1211307 , hereafter ot j1044 ) was discovered by the crts on 2010 february 17 . the large outburst amplitude was already suggestive of a wz sge - type like outburst ( vsnet - alert 11817 ) . subsequent observation soon established the presence of early superhumps ( vsnet - alert 11820 , 11828 , 11829 ; figure [ fig : j1044shpdm ] ) . the object later developed ordinary superhumps ( vsnet - alert 11836 , 11837 ) . there was a single post - superoutburst rebrightening ( vsnet - alert 11885 ) . the times of ordinary superhumps are listed in table [ tab : j1044oc2010 ] . it was most likely that we only sufficiently observed stage a ( @xmath70 ) and stage c ( @xmath96 ) . although the @xmath0 analysis suggests a mean period of @xmath90.0605 d during stage b , we did not adopt this value in table [ tab : perlist ] . the superhump period during stage c was , however , reliably determined ( figure [ fig : j1044shpdm ] ) . the period [ 0.06024(4 ) d ] suggests an @xmath13 of 1.9 % against the period [ 0.05909 ( 1 ) d ] of early superhumps . the waveform of superhumps became double - humped during the rapid fading stage ( @xmath97 ) , we only listed maxima having the same phases as in @xmath98 . the last two points ( @xmath99 , @xmath100 ) were obtained during the fading stage of a rebrightening . although these humps may have been different from earlier superhumps , we included them because wz sge - type dwarf novae are known to exhibit long - enduring superhumps even during the rebrightening phase . all the observed features , including the outburst amplitude of @xmath96 mag , the existence of early superhumps , relatively large @xmath13 , the likely existence of stage c and the single post - superoutburst rebrightening resemble those of large - amplitude borderline wz sge - type dwarf novae such as bc uma and rz leo ( objects showing `` type - c '' wz sge - type outbursts in @xcite ) . ( 88mm,110mm)fig59.eps ( 88mm,110mm)fig60.eps ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 55250.5223 & 0.0010 & @xmath170.0081 & 65 + 1 & 55250.5847 & 0.0008 & @xmath170.0062 & 49 + 2 & 55250.6440 & 0.0013 & @xmath170.0074 & 25 + 8 & 55251.0082 & 0.0030 & @xmath170.0065 & 60 + 9 & 55251.0696 & 0.0004 & @xmath170.0056 & 95 + 10 & 55251.1309 & 0.0003 & @xmath170.0049 & 228 + 11 & 55251.1920 & 0.0002 & @xmath170.0044 & 229 + 24 & 55251.9822 & 0.0004 & @xmath170.0011 & 101 + 25 & 55252.0459 & 0.0009 & 0.0019 & 100 + 26 & 55252.1050 & 0.0003 & 0.0005 & 312 + 27 & 55252.1643 & 0.0002 & @xmath170.0008 & 302 + 28 & 55252.2251 & 0.0004 & @xmath170.0004 & 218 + 41 & 55253.0074 & 0.0019 & @xmath170.0052 & 61 + 92 & 55256.1063 & 0.0017 & 0.0060 & 105 + 113 & 55257.3823 & 0.0041 & 0.0106 & 32 + 114 & 55257.4322 & 0.0019 & @xmath170.0000 & 61 + 115 & 55257.4891 & 0.0011 & @xmath170.0037 & 67 + 116 & 55257.5505 & 0.0013 & @xmath170.0028 & 65 + 195 & 55262.3421 & 0.0014 & 0.0059 & 21 + 196 & 55262.4117 & 0.0011 & 0.0149 & 87 + 197 & 55262.4689 & 0.0009 & 0.0116 & 85 + 198 & 55262.5285 & 0.0008 & 0.0106 & 89 + 199 & 55262.5863 & 0.0010 & 0.0079 & 96 + 200 & 55262.6474 & 0.0011 & 0.0084 & 76 + 212 & 55263.3763 & 0.0028 & 0.0109 & 30 + 213 & 55263.4288 & 0.0010 & 0.0028 & 92 + 214 & 55263.4911 & 0.0010 & 0.0046 & 93 + 215 & 55263.5524 & 0.0017 & 0.0053 & 94 + 216 & 55263.6158 & 0.0012 & 0.0082 & 97 + 217 & 55263.6732 & 0.0017 & 0.0051 & 49 + 227 & 55264.2828 & 0.0014 & 0.0093 & 62 + 228 & 55264.3384 & 0.0012 & 0.0043 & 76 + 229 & 55264.3995 & 0.0016 & 0.0048 & 86 + 230 & 55264.4557 & 0.0015 & 0.0005 & 96 + 231 & 55264.5205 & 0.0016 & 0.0047 & 82 + 232 & 55264.5774 & 0.0011 & 0.0011 & 93 + 233 & 55264.6271 & 0.0026 & @xmath170.0097 & 86 + 244 & 55265.2995 & 0.0022 & @xmath170.0033 & 78 + 245 & 55265.3571 & 0.0012 & @xmath170.0062 & 91 + 246 & 55265.4192 & 0.0027 & @xmath170.0047 & 97 + 247 & 55265.4772 & 0.0013 & @xmath170.0072 & 92 + 248 & 55265.5405 & 0.0026 & @xmath170.0045 & 100 + 249 & 55265.6008 & 0.0043 & @xmath170.0047 & 85 + 376 & 55273.2741 & 0.0018 & @xmath170.0204 & 63 + 377 & 55273.3328 & 0.0023 & @xmath170.0222 & 63 + + + + this object (= css100603:112253@xmath17111037 , hereafter ot j1122 ) is a transient discovered by the crts . the object was soon suspected to be a large - amplitude dwarf nova ( vsnet - alert 12020 ) . h. maehara detected short - period superhumps ( vsnet - alert 12025 ; figure [ fig : j1122shpdm ] ) indicating that the object is an unusual short-@xmath2 cv with an evolved secondary . j. greaves pointed out that this object is included in the public sdss dr7 archive . the spectrum indicates that the object is hydrogen - rich , rather than an am cvn - type helium cv ( vsnet - alert 12025 ) , although the strength of helium emission lines suggests an helium enrichment ( vsnet - alert 12026 ) . the times of superhump maxima are listed in table [ tab : j1122oc2010 ] . there is an indication of a systematic period decrease after @xmath101 . we attributed this to a stage b c transition and give the derived parameters based in this interpretation in table [ tab : perlist ] . more detailed analysis and discussion will be presented in maehara et al . , in preparation . ( 88mm,110mm)fig61.eps ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 55350.9981 & 0.0034 & @xmath170.0001 & 95 + 1 & 55351.0405 & 0.0033 & @xmath170.0032 & 89 + 36 & 55352.6341 & 0.0007 & 0.0019 & 54 + 37 & 55352.6784 & 0.0007 & 0.0007 & 83 + 44 & 55352.9942 & 0.0015 & @xmath170.0012 & 96 + 45 & 55353.0346 & 0.0033 & @xmath170.0061 & 95 + 49 & 55353.2263 & 0.0004 & 0.0040 & 148 + 52 & 55353.3564 & 0.0022 & @xmath170.0021 & 45 + 72 & 55354.2685 & 0.0006 & 0.0022 & 189 + 73 & 55354.3118 & 0.0012 & 0.0001 & 189 + 93 & 55355.2223 & 0.0007 & 0.0029 & 188 + 94 & 55355.2656 & 0.0007 & 0.0007 & 219 + 95 & 55355.3096 & 0.0012 & @xmath170.0006 & 211 + 96 & 55355.3646 & 0.0048 & 0.0089 & 48 + 132 & 55356.9831 & 0.0030 & @xmath170.0066 & 95 + 137 & 55357.2187 & 0.0012 & 0.0020 & 83 + 138 & 55357.2635 & 0.0012 & 0.0014 & 94 + 139 & 55357.3070 & 0.0011 & @xmath170.0005 & 101 + 154 & 55357.9839 & 0.0046 & @xmath170.0044 & 42 + + + + we reported on the detection of superhumps and period variation in this object (= css090530:144011@xmath1494734 , hereafter ot j1440 ) in @xcite . we present here a re - analysis after incorporating the data in @xcite . the times of superhump maxima are listed in table [ tab : j1440oc2009 ] . it has now become evident that @xmath102 corresponds to stage a , when the superhumps were indeed still in development and the object once started to fade temporarily ( figure [ fig : j14402009oc ] ) . the period break reported in @xcite and @xcite after @xmath103 this is most likely a stage b c transition . it is unusual for such a short-@xmath3 system to show a stage b c transition at such an early stage . the apparent lack of distinct positive @xmath7 during the stage b is also unusual . the development of the superhump period the presence of a likely precursor ( or a stagnation ) in the light curve are similar to those of bz uma , which has been proposed to critically exhibit the su uma - type phenomenon @xcite . since bz uma is known to show many normal outbursts in addition to a very rare superoutburst ( @xcite ; @xcite ) , a further systematic observation of ot j1440 is encouraged to explore the similarity between these objects and potential cause of the unusual period behavior . ( 88mm,90mm)fig62.eps ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 54983.0245 & 0.0032 & @xmath170.0095 & 69 + 1 & 54983.0946 & 0.0101 & @xmath170.0039 & 64 + 7 & 54983.4733 & 0.0017 & @xmath170.0123 & 68 + 8 & 54983.5397 & 0.0015 & @xmath170.0104 & 68 + 15 & 54984.0046 & 0.0014 & 0.0029 & 191 + 16 & 54984.0697 & 0.0011 & 0.0034 & 223 + 22 & 54984.4581 & 0.0004 & 0.0048 & 164 + 23 & 54984.5231 & 0.0005 & 0.0052 & 168 + 24 & 54984.5869 & 0.0006 & 0.0046 & 74 + 38 & 54985.4914 & 0.0007 & 0.0059 & 106 + 39 & 54985.5564 & 0.0010 & 0.0063 & 105 + 53 & 54986.4604 & 0.0006 & 0.0072 & 71 + 54 & 54986.5254 & 0.0005 & 0.0077 & 64 + 69 & 54987.4837 & 0.0010 & @xmath170.0017 & 110 + 70 & 54987.5528 & 0.0014 & 0.0029 & 125 + 119 & 54990.7082 & 0.0008 & @xmath170.0028 & 47 + 120 & 54990.7752 & 0.0014 & @xmath170.0003 & 102 + 121 & 54990.8374 & 0.0014 & @xmath170.0026 & 169 + 122 & 54990.8997 & 0.0015 & @xmath170.0049 & 143 + 135 & 54991.7401 & 0.0078 & @xmath170.0031 & 98 + 136 & 54991.8083 & 0.0044 & 0.0006 & 119 + + + + this object (= css080505:163121@xmath1103134 , hereafter ot j1631 ) underwent another superoutburst in 2010 ( i. miller , baavss - alert 2268 ) . the times of superhump maxima are listed in table [ tab : j1631oc2010 ] . although there seems to have a change in the period between @xmath104 and @xmath105 , the change was apparently too large to be attributed to a stage a b or b c transition . it is likely that observation at @xmath104 was insufficient to derive a reliable maximum . disregarding observations before bjd 2455307 , we obtained a period of 0.06395(2 ) d with the pdm method . ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 55306.6220 & 0.0008 & @xmath170.0103 & 42 + 14 & 55307.5341 & 0.0009 & 0.0047 & 68 + 15 & 55307.6010 & 0.0010 & 0.0074 & 69 + 55 & 55310.1595 & 0.0018 & 0.0026 & 133 + 71 & 55311.1836 & 0.0020 & 0.0014 & 138 + 72 & 55311.2404 & 0.0012 & @xmath170.0059 & 108 + + + + this object (= css090622:170344@xmath1090835 , hereafter ot j1703 ) was discovered by the crts on 2009 june 22 . the detection of superhumps ( vsnet - alert 11297 , 11298 ) led to a classification as an su uma - type dwarf nova . although there still remain possibilities of aliases , we have adopted @xmath3 = 0.06085(2 ) d ( figure [ fig : j1703shpdm ] ) based on the best period determined from the single - night observation ( vsnet - alert 11298 ) . the times of superhump maxima are listed in table [ tab : j1703oc2009 ] . the outburst appears to have been caught when the amplitude of superhumps decayed ( the initial observation was performed @xmath9 5 d after the crts detection ) . according to crts observations , this object were further detected in outburst on 2009 july 22 and 29 . since there were no previous crts detections before 2009 june , there may have been an enhanced outburst activity ( either normal outbursts or post - superoutburst rebrightenings ) in 2009 july . ( 88mm,110mm)fig63.eps ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 55009.4145 & 0.0021 & 0.0011 & 94 + 1 & 55009.4716 & 0.0012 & @xmath170.0026 & 100 + 2 & 55009.5366 & 0.0014 & 0.0016 & 103 + 51 & 55012.5157 & 0.0004 & @xmath170.0000 & 58 + + + + this object ( hereafter ot j1821 ) was discovered by k. itagaki ( vsnet - alert 11952 ) . the sdss color suggested that the object is a dwarf nova in outburst . superhumps were subsequently detected ( vsnet - alert 11956 ; figure [ fig : j1821shpdm ] ) . the times of superhump maxima are listed in table [ tab : j1821oc2010 ] . these observations covered the final part of the superoutburst , and the recorded superhumps were likely stage c superhumps . the lack of variation in the period is compatible with this interpretation . ( 88mm,110mm)fig64.eps ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 55312.1638 & 0.0008 & @xmath170.0034 & 80 + 1 & 55312.2544 & 0.0039 & 0.0052 & 50 + 18 & 55313.6433 & 0.0018 & @xmath170.0016 & 53 + 48 & 55316.1072 & 0.0011 & @xmath170.0005 & 46 + 49 & 55316.1883 & 0.0021 & @xmath170.0014 & 59 + 74 & 55318.2444 & 0.0055 & 0.0023 & 118 + 85 & 55319.1458 & 0.0119 & 0.0007 & 19 + 86 & 55319.2259 & 0.0284 & @xmath170.0013 & 31 + + + + this transient ( hereafter ot j2138 ) was independently discovered by d .- a . yi @xcite and s. kaneko @xcite . @xcite spectroscopically confirmed that this is an outbursting dwarf nova ( see also vsnet - alert 11971 ) . the spectrum by @xcite indicated the presence of highly excited emission lines , suggesting that this is a wz sge - type outburst [ cf . vsnet - alert 11974 , see also vsnet - alert 11987 ( k. kinugasa , gunma astronomical observatory ) and follow - up spectroscopy by @xcite ] . the earliest indication of short - period modulations ( likely early superhumps ) was detected by g. masi ( vsnet - alert 11977 , see also 11978 , 11985 ; figure [ fig : j2138eshpdm ] ) . ordinary superhumps subsequently appeared ( vsnet - alert 11990 ; figure [ fig : j2138shpdm ] ) , strengthening the identification as a wz sge - type dwarf nova . @xcite reported a detection of another outburst in 1942 , which is the only known outburst other than the present one . the times of maxima for ordinary superhumps are listed in table [ tab : j2138oc2010 ] . although there was an indication of double maxima during the late plateau phase , we did not attempt to distinguish the different peaks , and listed most prominent ones . the overall @xmath0 diagram ( figure [ fig : j2138humpall ] ) bears a high degree of similarity to that of asas j0025 in 2004 @xcite . and the low amplitudes of post - superoutburst superhumps . ] the main difference is that ot j2138 did not fade from the plateau before entering stage c. combined with the apparent presence of multiple hump peaks , this stage in ot j2138 appears to phenomenologically correspond to the phase between the rapid fading from the plateau and rebrightening in asas j0025 . the apparent absence of a rebrightening in ot j2138 may be explained if this object somehow succeeded in maintaining the plateau phase ( or avoiding the quenching of the outbursting state ) when asas j0025 once returned to quiescence before the rebrightening . the predominance of type - d superoutbursts ( superoutbursts without a rebrightening , see @xcite ) in very short-@xmath2 systems may be understood as this kind of variety . in table [ tab : perlist ] , we listed intervals exhibiting best - defined maxima and omitted a phase with multiple peaks . the post - superoutburst stage with @xmath106 showed a slightly shorter superhump period and the signal was persistent even during the very late post - superoutburst stage ( @xmath65 = 0.05487(1 ) d , figure [ fig : j2138latepdm ] ) . there was no indication of a longer late - stage superhumps as observed in gw lib , v455 and , eg cnc , wz sge and several newly discovered wz sge - type dwarf novae ( @xcite ; @xcite ) . more detailed analysis and discussion will be presented in maehara et al . , in preparation . ( 88mm,110mm)fig65.eps ( 88mm,110mm)fig66.eps ( 160mm,160mm)fig67.eps ( 88mm,110mm)fig68.eps ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 55330.8319 & 0.0006 & 0.0019 & 89 + 6 & 55331.1881 & 0.0013 & 0.0274 & 309 + 36 & 55332.8555 & 0.0002 & 0.0412 & 76 + 37 & 55332.9109 & 0.0002 & 0.0414 & 83 + 38 & 55332.9650 & 0.0002 & 0.0404 & 116 + 41 & 55333.1300 & 0.0005 & 0.0401 & 125 + 42 & 55333.1838 & 0.0002 & 0.0388 & 662 + 43 & 55333.2391 & 0.0002 & 0.0390 & 832 + 44 & 55333.2940 & 0.0002 & 0.0387 & 503 + 54 & 55333.8428 & 0.0002 & 0.0363 & 128 + 55 & 55333.8963 & 0.0002 & 0.0347 & 149 + 56 & 55333.9519 & 0.0002 & 0.0351 & 106 + 60 & 55334.1724 & 0.0004 & 0.0352 & 145 + 61 & 55334.2265 & 0.0003 & 0.0342 & 477 + 62 & 55334.2810 & 0.0004 & 0.0335 & 427 + 66 & 55334.5003 & 0.0009 & 0.0324 & 35 + 67 & 55334.5554 & 0.0004 & 0.0324 & 53 + 68 & 55334.6094 & 0.0003 & 0.0313 & 53 + 73 & 55334.8845 & 0.0002 & 0.0307 & 152 + 80 & 55335.2689 & 0.0002 & 0.0293 & 110 + 86 & 55335.5978 & 0.0005 & 0.0275 & 151 + 87 & 55335.6535 & 0.0005 & 0.0281 & 117 + 90 & 55335.8173 & 0.0003 & 0.0265 & 102 + 91 & 55335.8749 & 0.0004 & 0.0290 & 130 + 103 & 55336.5340 & 0.0014 & 0.0267 & 52 + 104 & 55336.5871 & 0.0008 & 0.0246 & 134 + 105 & 55336.6417 & 0.0009 & 0.0241 & 171 + 108 & 55336.8067 & 0.0003 & 0.0238 & 153 + 109 & 55336.8625 & 0.0002 & 0.0244 & 244 + 110 & 55336.9171 & 0.0002 & 0.0239 & 145 + 114 & 55337.1408 & 0.0024 & 0.0271 & 56 + 115 & 55337.1938 & 0.0005 & 0.0250 & 97 + 116 & 55337.2511 & 0.0002 & 0.0272 & 387 + 122 & 55337.5806 & 0.0007 & 0.0260 & 179 + 123 & 55337.6346 & 0.0007 & 0.0249 & 182 + 127 & 55337.8545 & 0.0002 & 0.0242 & 104 + 128 & 55337.9101 & 0.0002 & 0.0247 & 106 + 129 & 55337.9647 & 0.0004 & 0.0242 & 146 + 132 & 55338.1396 & 0.0018 & 0.0338 & 131 + 133 & 55338.1893 & 0.0005 & 0.0284 & 500 + 134 & 55338.2408 & 0.0004 & 0.0248 & 578 + 137 & 55338.4063 & 0.0006 & 0.0248 & 92 + 138 & 55338.4633 & 0.0003 & 0.0267 & 113 + 139 & 55338.5180 & 0.0005 & 0.0263 & 163 + 140 & 55338.5746 & 0.0008 & 0.0278 & 141 + 141 & 55338.6259 & 0.0008 & 0.0239 & 159 + 145 & 55338.8472 & 0.0004 & 0.0248 & 89 + 146 & 55338.9023 & 0.0003 & 0.0247 & 74 + 156 & 55339.4574 & 0.0005 & 0.0287 & 175 + 157 & 55339.5065 & 0.0005 & 0.0227 & 208 + 158 & 55339.5705 & 0.0017 & 0.0315 & 85 + 159 & 55339.6238 & 0.0017 & 0.0297 & 76 + 162 & 55339.7897 & 0.0029 & 0.0302 & 51 + 163 & 55339.8426 & 0.0006 & 0.0280 & 82 + 164 & 55339.8972 & 0.0007 & 0.0275 & 82 + 187 & 55341.1909 & 0.0016 & 0.0535 & 82 + + + + ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 188 & 55341.2308 & 0.0011 & 0.0383 & 264 + 189 & 55341.2788 & 0.0010 & 0.0311 & 292 + 192 & 55341.4479 & 0.0015 & 0.0348 & 78 + 193 & 55341.5139 & 0.0028 & 0.0457 & 75 + 199 & 55341.8388 & 0.0019 & 0.0400 & 118 + 200 & 55341.8928 & 0.0009 & 0.0388 & 119 + 201 & 55341.9568 & 0.0011 & 0.0477 & 158 + 204 & 55342.1406 & 0.0011 & 0.0662 & 44 + 209 & 55342.3890 & 0.0021 & 0.0390 & 46 + 210 & 55342.4457 & 0.0012 & 0.0405 & 113 + 212 & 55342.5794 & 0.0020 & 0.0640 & 101 + 213 & 55342.6214 & 0.0016 & 0.0508 & 84 + 217 & 55342.8330 & 0.0023 & 0.0419 & 169 + 218 & 55342.8921 & 0.0011 & 0.0460 & 174 + 228 & 55343.4363 & 0.0005 & 0.0389 & 116 + 229 & 55343.4904 & 0.0008 & 0.0379 & 115 + 230 & 55343.5673 & 0.0012 & 0.0597 & 89 + 231 & 55343.6096 & 0.0092 & 0.0469 & 99 + 233 & 55343.7196 & 0.0036 & 0.0466 & 51 + 234 & 55343.7661 & 0.0036 & 0.0381 & 62 + 235 & 55343.8313 & 0.0023 & 0.0481 & 95 + 236 & 55343.8757 & 0.0011 & 0.0374 & 116 + 241 & 55344.1558 & 0.0013 & 0.0419 & 90 + 242 & 55344.2173 & 0.0009 & 0.0483 & 103 + 246 & 55344.4345 & 0.0012 & 0.0449 & 107 + 247 & 55344.4832 & 0.0011 & 0.0386 & 107 + 248 & 55344.5569 & 0.0009 & 0.0571 & 88 + 249 & 55344.5973 & 0.0021 & 0.0424 & 57 + 267 & 55345.5896 & 0.0014 & 0.0425 & 81 + 268 & 55345.6420 & 0.0013 & 0.0399 & 83 + 270 & 55345.7540 & 0.0050 & 0.0416 & 60 + 271 & 55345.8132 & 0.0038 & 0.0457 & 59 + 272 & 55345.8721 & 0.0171 & 0.0495 & 44 + 274 & 55345.9678 & 0.0007 & 0.0350 & 83 + 278 & 55346.1947 & 0.0004 & 0.0413 & 398 + 279 & 55346.2512 & 0.0004 & 0.0427 & 508 + 284 & 55346.5280 & 0.0011 & 0.0439 & 46 + 285 & 55346.5840 & 0.0022 & 0.0448 & 130 + 286 & 55346.6330 & 0.0022 & 0.0387 & 116 + 290 & 55346.8566 & 0.0019 & 0.0418 & 104 + 291 & 55346.9111 & 0.0011 & 0.0411 & 104 + 292 & 55346.9569 & 0.0019 & 0.0319 & 165 + 295 & 55347.1368 & 0.0010 & 0.0464 & 177 + 296 & 55347.1814 & 0.0025 & 0.0359 & 198 + 297 & 55347.2432 & 0.0016 & 0.0425 & 80 + 302 & 55347.5203 & 0.0019 & 0.0440 & 47 + 303 & 55347.5682 & 0.0049 & 0.0368 & 84 + 304 & 55347.6304 & 0.0009 & 0.0440 & 113 + 308 & 55347.8539 & 0.0005 & 0.0469 & 103 + 309 & 55347.9080 & 0.0006 & 0.0459 & 104 + 310 & 55347.9612 & 0.0008 & 0.0440 & 67 + 313 & 55348.1229 & 0.0014 & 0.0403 & 98 + 314 & 55348.1771 & 0.0002 & 0.0394 & 2499 + 315 & 55348.2376 & 0.0018 & 0.0448 & 76 + 321 & 55348.5653 & 0.0005 & 0.0418 & 76 + + + + ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 322 & 55348.6266 & 0.0011 & 0.0480 & 101 + 323 & 55348.6800 & 0.0008 & 0.0462 & 45 + 331 & 55349.1111 & 0.0008 & 0.0364 & 119 + 332 & 55349.1650 & 0.0007 & 0.0352 & 264 + 333 & 55349.2184 & 0.0008 & 0.0334 & 358 + 334 & 55349.2800 & 0.0004 & 0.0399 & 396 + 339 & 55349.5518 & 0.0004 & 0.0362 & 72 + 340 & 55349.6091 & 0.0017 & 0.0383 & 82 + 341 & 55349.6629 & 0.0007 & 0.0370 & 50 + 346 & 55349.9285 & 0.0013 & 0.0270 & 49 + 347 & 55349.9937 & 0.0006 & 0.0371 & 53 + 349 & 55350.1040 & 0.0011 & 0.0371 & 142 + 350 & 55350.1519 & 0.0008 & 0.0299 & 262 + 351 & 55350.2145 & 0.0012 & 0.0374 & 363 + 352 & 55350.2652 & 0.0005 & 0.0329 & 325 + 362 & 55350.8202 & 0.0011 & 0.0367 & 53 + 363 & 55350.8695 & 0.0005 & 0.0309 & 73 + 364 & 55350.9235 & 0.0007 & 0.0299 & 73 + 368 & 55351.1500 & 0.0012 & 0.0358 & 291 + 369 & 55351.2004 & 0.0004 & 0.0311 & 374 + 370 & 55351.2553 & 0.0004 & 0.0309 & 383 + 376 & 55351.5870 & 0.0013 & 0.0319 & 57 + 377 & 55351.6347 & 0.0004 & 0.0244 & 57 + 387 & 55352.1884 & 0.0006 & 0.0270 & 47 + 388 & 55352.2459 & 0.0008 & 0.0293 & 236 + 394 & 55352.5725 & 0.0009 & 0.0252 & 58 + 395 & 55352.6220 & 0.0009 & 0.0196 & 59 + 399 & 55352.8451 & 0.0006 & 0.0223 & 61 + 400 & 55352.9027 & 0.0005 & 0.0247 & 73 + 401 & 55352.9523 & 0.0006 & 0.0191 & 151 + 406 & 55353.2247 & 0.0008 & 0.0160 & 240 + 407 & 55353.2892 & 0.0011 & 0.0254 & 175 + 410 & 55353.4490 & 0.0002 & 0.0198 & 184 + 411 & 55353.5024 & 0.0006 & 0.0181 & 172 + 412 & 55353.5589 & 0.0006 & 0.0195 & 104 + 413 & 55353.6161 & 0.0015 & 0.0216 & 58 + 414 & 55353.6714 & 0.0007 & 0.0218 & 41 + 417 & 55353.8271 & 0.0016 & 0.0120 & 59 + 418 & 55353.8895 & 0.0013 & 0.0194 & 73 + 419 & 55353.9565 & 0.0010 & 0.0312 & 121 + 420 & 55353.9933 & 0.0008 & 0.0129 & 56 + 428 & 55354.4397 & 0.0005 & 0.0183 & 120 + 429 & 55354.4925 & 0.0004 & 0.0160 & 123 + 430 & 55354.5502 & 0.0008 & 0.0186 & 80 + 431 & 55354.5990 & 0.0003 & 0.0123 & 58 + 432 & 55354.6537 & 0.0018 & 0.0119 & 58 + 446 & 55355.4283 & 0.0004 & 0.0148 & 123 + 447 & 55355.4806 & 0.0006 & 0.0119 & 122 + 448 & 55355.5368 & 0.0010 & 0.0130 & 72 + 449 & 55355.5925 & 0.0008 & 0.0136 & 56 + 450 & 55355.6435 & 0.0004 & 0.0095 & 56 + 456 & 55355.9745 & 0.0008 & 0.0098 & 52 + 465 & 55356.4668 & 0.0003 & 0.0060 & 155 + 466 & 55356.5203 & 0.0004 & 0.0044 & 169 + 467 & 55356.5748 & 0.0010 & 0.0037 & 25 + 477 & 55357.1070 & 0.0034 & @xmath170.0152 & 54 + + + + ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 478 & 55357.1882 & 0.0026 & 0.0108 & 256 + 479 & 55357.2414 & 0.0009 & 0.0089 & 272 + 480 & 55357.2894 & 0.0009 & 0.0018 & 119 + 482 & 55357.4003 & 0.0016 & 0.0024 & 61 + 483 & 55357.4555 & 0.0010 & 0.0025 & 62 + 484 & 55357.5187 & 0.0014 & 0.0107 & 63 + 485 & 55357.5628 & 0.0011 & @xmath170.0004 & 52 + 486 & 55357.6172 & 0.0004 & @xmath170.0011 & 54 + 487 & 55357.6766 & 0.0027 & 0.0032 & 29 + 496 & 55358.1624 & 0.0008 & @xmath170.0071 & 259 + 497 & 55358.2249 & 0.0011 & 0.0003 & 301 + 498 & 55358.2725 & 0.0052 & @xmath170.0072 & 269 + 501 & 55358.4491 & 0.0007 & 0.0040 & 59 + 502 & 55358.4989 & 0.0006 & @xmath170.0013 & 61 + 503 & 55358.5534 & 0.0006 & @xmath170.0020 & 46 + 504 & 55358.6081 & 0.0017 & @xmath170.0024 & 54 + 505 & 55358.6608 & 0.0010 & @xmath170.0048 & 45 + 508 & 55358.8288 & 0.0004 & @xmath170.0022 & 59 + 509 & 55358.8805 & 0.0006 & @xmath170.0055 & 73 + 510 & 55358.9432 & 0.0006 & 0.0020 & 73 + 514 & 55359.1547 & 0.0013 & @xmath170.0070 & 177 + 515 & 55359.2161 & 0.0009 & @xmath170.0007 & 222 + 516 & 55359.2561 & 0.0006 & @xmath170.0159 & 224 + 519 & 55359.4297 & 0.0011 & @xmath170.0075 & 60 + 520 & 55359.4892 & 0.0004 & @xmath170.0033 & 61 + 521 & 55359.5382 & 0.0007 & @xmath170.0093 & 35 + 527 & 55359.8668 & 0.0004 & @xmath170.0115 & 73 + 528 & 55359.9224 & 0.0019 & @xmath170.0110 & 73 + 537 & 55360.4107 & 0.0006 & @xmath170.0187 & 61 + 538 & 55360.4622 & 0.0019 & @xmath170.0223 & 61 + 545 & 55360.8531 & 0.0004 & @xmath170.0173 & 73 + 546 & 55360.9034 & 0.0005 & @xmath170.0221 & 73 + 547 & 55360.9613 & 0.0011 & @xmath170.0193 & 49 + 555 & 55361.4070 & 0.0009 & @xmath170.0146 & 53 + 556 & 55361.4620 & 0.0008 & @xmath170.0147 & 62 + 557 & 55361.5089 & 0.0022 & @xmath170.0229 & 54 + 558 & 55361.5579 & 0.0083 & @xmath170.0291 & 60 + 559 & 55361.6198 & 0.0009 & @xmath170.0223 & 56 + 577 & 55362.5991 & 0.0005 & @xmath170.0352 & 56 + 578 & 55362.6583 & 0.0006 & @xmath170.0311 & 42 + 594 & 55363.5412 & 0.0005 & @xmath170.0301 & 39 + 595 & 55363.5908 & 0.0020 & @xmath170.0356 & 56 + 596 & 55363.6413 & 0.0010 & @xmath170.0402 & 54 + 613 & 55364.5834 & 0.0011 & @xmath170.0351 & 55 + 614 & 55364.6486 & 0.0036 & @xmath170.0251 & 52 + 631 & 55365.5662 & 0.0010 & @xmath170.0446 & 56 + 632 & 55365.6194 & 0.0026 & @xmath170.0464 & 54 + 638 & 55365.9491 & 0.0005 & @xmath170.0475 & 85 + 649 & 55366.5640 & 0.0007 & @xmath170.0389 & 42 + 650 & 55366.6105 & 0.0009 & @xmath170.0475 & 55 + 651 & 55366.6678 & 0.0011 & @xmath170.0453 & 29 + 668 & 55367.6003 & 0.0006 & @xmath170.0498 & 52 + 669 & 55367.6548 & 0.0011 & @xmath170.0505 & 46 + 685 & 55368.5408 & 0.0009 & @xmath170.0464 & 36 + 686 & 55368.6132 & 0.0028 & @xmath170.0291 & 53 + 687 & 55368.6551 & 0.0008 & @xmath170.0423 & 47 + + + + ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 704 & 55369.5719 & 0.0015 & @xmath170.0626 & 55 + 705 & 55369.6239 & 0.0012 & @xmath170.0657 & 52 + 722 & 55370.5701 & 0.0011 & @xmath170.0566 & 50 + 777 & 55373.5876 & 0.0012 & @xmath170.0706 & 47 + 778 & 55373.6503 & 0.0012 & @xmath170.0630 & 50 + 813 & 55375.5696 & 0.0014 & @xmath170.0729 & 55 + 814 & 55375.6237 & 0.0045 & @xmath170.0740 & 56 + 815 & 55375.6692 & 0.0025 & @xmath170.0836 & 35 + 831 & 55376.5545 & 0.0034 & @xmath170.0802 & 49 + 833 & 55376.6535 & 0.0149 & @xmath170.0914 & 53 + 849 & 55377.5468 & 0.0006 & @xmath170.0800 & 37 + 850 & 55377.5936 & 0.0013 & @xmath170.0884 & 56 + 851 & 55377.6485 & 0.0013 & @xmath170.0886 & 54 + 868 & 55378.5821 & 0.0014 & @xmath170.0921 & 57 + 869 & 55378.6184 & 0.0171 & @xmath170.1109 & 56 + 886 & 55379.5563 & 0.0020 & @xmath170.1100 & 45 + 887 & 55379.6170 & 0.0012 & @xmath170.1044 & 46 + 888 & 55379.6683 & 0.0015 & @xmath170.1082 & 39 + 923 & 55381.5977 & 0.0025 & @xmath170.1081 & 55 + 924 & 55381.6300 & 0.0050 & @xmath170.1309 & 54 + 1104 & 55391.5100 & 0.0004 & @xmath170.1725 & 49 + 1104 & 55391.5492 & 0.0018 & @xmath170.1333 & 58 + 1140 & 55393.5096 & 0.0004 & @xmath170.1572 & 92 + 1155 & 55394.3357 & 0.0019 & @xmath170.1579 & 53 + 1156 & 55394.3778 & 0.0007 & @xmath170.1710 & 71 + 1174 & 55395.3714 & 0.0011 & @xmath170.1695 & 82 + 1176 & 55395.4992 & 0.0019 & @xmath170.1519 & 54 + 1233 & 55398.5953 & 0.0007 & @xmath170.1977 & 54 + 1234 & 55398.6551 & 0.0006 & @xmath170.1930 & 43 + 1251 & 55399.6012 & 0.0011 & @xmath170.1839 & 53 + 1342 & 55404.5667 & 0.0038 & @xmath170.2344 & 42 + 1343 & 55404.6583 & 0.0030 & @xmath170.1978 & 42 + + + + this object (= css100615:215815@xmath1094709 , hereafter ot j2158 ) was discovered by the crts in 2010 june . astrokolkhoz team immediately detected superhumps ( vsnet - outburst 11306 ) . the times of superhump maxima are listed in table [ tab : j2158oc2010 ] . the mean period determined with the pdm method was 0.07755(9 ) d. the large amplitudes ( 0.35 mag , figure [ fig : j2158shpdm ] ) and waveform suggest that these superhumps are stage b superhumps at their early evolution . ( 88mm,110mm)fig69.eps ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 55366.5092 & 0.0008 & 0.0001 & 25 + 1 & 55366.5868 & 0.0003 & 0.0000 & 50 + 2 & 55366.6644 & 0.0005 & @xmath170.0000 & 56 + 4 & 55366.8190 & 0.0003 & @xmath170.0007 & 108 + 5 & 55366.8979 & 0.0005 & 0.0005 & 74 + + + + this object (= css090727:223003@xmath17145835 , hereafter ot j2230 ) was discovered by the crts in 2009 july . the large outburst amplitude exceeding 6 mag was immediately noted ( vsnet - alert 11351 ) . superhump - like modulations were soon reported ( vsnet - alert 11353 ) . it became evident by later observations that these modulations were early superhumps of a wz sge - type dwarf nova rather than ordinary superhumps ( vsnet - alert 11365 , 11388 ) . we obtained a mean period of 0.05841(1 ) d with the pdm method ( figure [ fig : j2230eshpdm ] ) . doubly humped modulations characteristic to early superhumps ( cf . @xcite ) are clearly seen . due to the faintness of the object , the appearance of ordinary superhumps was not recorded . the outburst lasted for at least 18 d. ( 88mm,110mm)fig70.eps this object (= mls100904:234441@xmath17001206 , hereafter ot j2344 ) was discovered by the crts in 2010 september during the course of the mount lemmon survey . p. wils notified this event ( cvnet - outburst 3828 ) . immediately following this announcement , superhumps were clearly detected ( vsnet - alert 12143 , 12144 ; figure [ fig : j2344shpdm ] ) . the times of superhump maxima are listed in table [ tab : j2344oc2010 ] . these superhumps were likely stage c superhumps since there was little evidence of period variation and the object started a rapid fading 4 d after the start of the observation . ( 88mm,110mm)fig71.eps ccccc @xmath15 & max & error & @xmath0 & @xmath16 + 0 & 55444.4661 & 0.0009 & 0.0024 & 38 + 1 & 55444.5425 & 0.0005 & 0.0020 & 77 + 2 & 55444.6163 & 0.0005 & @xmath170.0008 & 104 + 9 & 55445.1507 & 0.0066 & @xmath170.0035 & 76 + 12 & 55445.3861 & 0.0010 & 0.0019 & 38 + 13 & 55445.4615 & 0.0012 & 0.0005 & 39 + 14 & 55445.5387 & 0.0005 & 0.0010 & 39 + 15 & 55445.6111 & 0.0011 & @xmath170.0033 & 39 + 26 & 55446.4576 & 0.0005 & @xmath170.0006 & 170 + 27 & 55446.5316 & 0.0007 & @xmath170.0034 & 195 + 28 & 55446.6107 & 0.0011 & @xmath170.0009 & 149 + 38 & 55447.3778 & 0.0009 & @xmath170.0009 & 172 + 39 & 55447.4547 & 0.0009 & @xmath170.0008 & 173 + 40 & 55447.5373 & 0.0020 & 0.0052 & 112 + 48 & 55448.1513 & 0.0081 & 0.0054 & 111 + 51 & 55448.3756 & 0.0011 & @xmath170.0005 & 172 + 52 & 55448.4519 & 0.0014 & @xmath170.0009 & 172 + 53 & 55448.5283 & 0.0018 & @xmath170.0011 & 173 + 60 & 55449.0629 & 0.0026 & @xmath170.0035 & 40 + 61 & 55449.1448 & 0.0042 & 0.0017 & 21 + + + + the new data for su uma - type dwarf novae have improved the statistics presented in @xcite . although the presentations in @xcite covered a wide range of relations , we restrict only to key figures after combining the samples with @xcite in this paper . figure [ fig : pdotpsh2 ] represents the relation between @xmath3 and @xmath7 during stage b. the enlarged figure ( corresponding to figure 10 in @xcite ) is only shown here . the estimated density map is overlaid for a better visualization . the new data generally confirmed the predominance of positive period derivatives during stage b in systems with superhump periods shorter than 0.07 d , in agreement with the tendency reported in @xcite . ( 120mm,80mm)fig72.eps in figure [ fig : pdotpsh2 ] , there appears to be a systematic difference in @xmath7 for systems with @xmath3 longer than 0.075 d. we have selected samples for @xmath107 from @xcite and this paper . after using the same criterion for samples described in @xcite and rejecting poorly determined ( nominal error of @xmath7 exceeding @xmath108 ) superoutbursts , we applied student s @xmath109-test . the test indicated that these samples are different at a significance level of 0.09 . this test suggests that the properties of samples are different between these two data sets . it is already apparent all well - determined @xmath7 s in this paper for this @xmath3 region has positive @xmath7 s , while many systems had negative @xmath7 s in @xcite . this might have been caused by a small number of long-@xmath3 samples in this paper , since most of newly discovered su uma - type dwarf novae have short @xmath3 . there would also be a possibility , however , that most of long-@xmath3 with frequent superoutbursts were already discovered at the time of @xcite and that long-@xmath3 systems in this papers were heavily biased toward systems with infrequent superoutbursts . this possibility would be supported considering that all systems with supercycles longer than 1000 d ( v1251 cyg , rz leo , qy per ) have large positive @xmath7 ( see also a discussion in @xcite subsection 4.10 ) while systems with @xmath110 are restricted to systems with supercycles shorter than 400 d ( dh aql , v1316 cyg , cu vel ) . although a direct correlation coefficient ( 0.20 ) between @xmath111 supercycle and @xmath7 is not significant , these results seem to strengthen the idea in @xcite that the degree of period variation is more diverse in long-@xmath3 systems . there is , however , a case of iy uma , with frequent superoutbursts , which showed a positive @xmath7 in this new sample . it is not yet clear whether some of long-@xmath3 have different @xmath7 s between different superoutbursts or whether this was a mere consequence of poor sampling in the past . future studies in these viewpoints would be fruitful . the apparent lack of systematic difference between different superoutbursts of the same system is one of the main conclusions in @xcite . we further examined this issue using new materials . the new data strengthen this conclusions , including well - observed superoutbursts preceded by precursor outbursts ( e.g. pu cma and 1rxs j0532 . it would be noteworthy 1rxs j0532 almost completely reproduced the behavior in 2005 , which has been reported to be rather unusual @xcite . this suggests that peculiarities found in certain systems are specific to these systems , rather than a chance occurrence . this might suggest the existence of parameters specific to systems in addition to general parameters such as @xmath14 or @xmath2 . in v844 her , our observations give additional support to the hypothesis that the delay time in development of superhumps is shorter in significantly smaller superoutbursts . in the 2009 superoutburst of iy uma , we recorded a strong beat phenomenon ( in the total luminosity ) between superhumps and orbital variation . the beat period largely varied between stage b and c , and were in very good agreement with the beat periods expected from superhump periods during the corresponding stages . the close correlation between the beat period and the superhump period suggests that the change in the angular velocity of the global apsidal motion is more responsible for the stage b c transition rather than the appearance of a more localized new component with a different period ( or angular velocity ) , such as a bright spot on the disk . future research focusing on this relation between the beat period and superhump period will be an important key in clarifying the nature of period variations of superhumps . the new sample includes three new wz sge - type objects with established early superhumps , sdss j1610 , ot j1044 and ot j2230 , and one wz sge - type object with likely early superhumps ( ot j2138 ) . all objects have @xmath3 shorter than 0.061 d. we also suggest that two systems , vx for and el uma , are wz sge - type dwarf novae with multiple rebrightenings . although the stage with early superhumps was not observed in vx for , the observed @xmath7 for the first time predicted for its multiple rebrightening in real - time . this success strengthens the relationship between @xmath7 and type of rebrightenings in wz sge - type dwarf novae proposed in @xcite . although el uma was only observed for its supposed rebrightening phase , all the known properties are strongly suggestive of a wz sge - type system with multiple rebrightenings . the @xmath0 analysis of ot j2138 and its comparison to asas j0025 suggest an interpretation that the frequent absence of rebrightenings in very short-@xmath2 objects can be a result of sustained superoutburst plateau when usual su uma - type dwarf novae return to quiescence preceding a rebrightening . although this phenomenon may be somehow related to a small binary separation resulting stronger tidal torque and thereby stronger tidal dissipation to maintain the outbursting state , the exact mechanism should await further clarification . the authors are grateful to observers of vsnet collaboration and vsolj observers who supplied vital data . we acknowledge with thanks the variable star observations from the aavso international database contributed by observers worldwide and used in this research . this work is deeply indebted to outburst detections and announcement by a number of variable star observers worldwide , including participants of cvnet , baa vss alert and avson networks . we are grateful to d. boyd and his collaborators for making the data for ot j1440 available . the ccd operation of the bronberg observatory is partly sponsored by the center for backyard astrophysics . the ccd operation by peter nelson is on loan from the aavso , funded by the curry foundation . we are grateful to the catalina real - time transient survey team for making their real - time detection of transient objects available to the public . we are also grateful to h. takahashi and k. kinugasa for making unpublished spectra of el uma available to us . we are grateful to the anonymous referee for pointing out the apparent difference of distribution of @xmath7 between this paper and @xcite . we explored three applications of bayesian statistics in period analysis employed in this paper . bayesian statistics ( see e.g. @xcite ) provide a framework in estimating a posterior probability density function ( pdf ) of model parameters from a combination of the observed data , a likelihood function defined by the model , and a prior pdf of the model parameters . according to the bayes theorem , the posterior pdf of the model parameters @xmath112 is we usually employ markov - chain monte carlo ( mcmc ) method in order to obtain the pdf . the metropolis - hastings algorithm ( @xcite ; @xcite ) , one of the best known of mcmc implementation , is as follows : starting with @xmath119 , and at the @xmath120-th step of mcmc , we obtain @xmath121 . we then move to a new point @xmath122 following a proposal density distribution @xmath123 . the new point is accepted with a probability multivariate gaussian movements are frequently used to cancel out @xmath125 and @xmath123 . if the proposal is rejected , @xmath126 . after discarding initial ` burn - in ' steps , we sample the remaining steps of the markov chain to obtain the pdf . typical numbers of steps are @xmath127@xmath128 for the entire chain and @xmath129@xmath130 for the ` burn - in ' steps . as reviewed in @xcite , the @xmath0 diagrams of superhump maxima in su uma - type dwarf novae generally follow discrete stages : most frequently a segment with a positive @xmath7 and a later segment with a discontinuously shorter constant period ( stages b and c in @xcite ) . determining the parameters for periods and @xmath7 and the time of transition is not a trivial task for the classical statistics . @xcite dealt with this problem using the bayesian statistics and the mcmc method . we formulate this problem for a wider usage . using this likelihood , or its combination with priors , we can obtain the pdf with the mcmc method . figure [ fig : ocsamp1 ] is a sample application to superhump timings in qz vir in 2004 @xcite . the initial @xmath136 was chosen as 85 and other initial parameters were determined by linear least - squares fitting . the @xmath140 s of the multivariate gaussian distribution for the proposal density were 0.04 times 1-@xmath140 errors of the initial fitting . the total length of the chain was @xmath130 and the first 5000 samples were discarded for obtaining the pdf . the mean parameters and standard errors were @xmath141 , @xmath142 , @xmath143 , @xmath144 and @xmath145 . the posterior means of two parameters @xmath146 and @xmath136 are shown in figure [ fig : ocsamp2 ] . when expected values of some parameters are empirically known . this application of priors would be useful when @xmath7 of a certain object is known from observation of other superoutbursts while a particular observation has a significant gap to determine parameters by itself . figure [ fig : ocsampprior ] demonstrates the effect of priors for observations with a gaps . the data are times of superhumps of qz vir during the 2007 superoutburst . an unconstrained model gives a negative @xmath7 of @xmath148 and a break at @xmath149 ( dashed curve ) . incorporating prior knowledge that this object has a positive @xmath7 , i.e. setting a prior with @xmath150 and @xmath151 , we get a reasonable fit ( solid curve ) with a break at @xmath152 . since this example is for a demonstration purpose of bayesian approach , we did not use these values in the main text . although a proper way of using priors in such a problem need to be further investigated , this formulation would provide a way of analyzing a badly sampled observations when we have firm knowledge in choosing the prior . we consider a bayesian extension of parameter fitting of observed light curves with a cyclic function . in this case , @xmath153 is the observations at @xmath154 , and the parameter space is @xmath155 defined by the model although a directed application of the original mcmc to solve the aliasing problem is limited due to the slow mixing between widely separate peaks , it is well suited for sampling the detailed structure of individual peaks . figure [ fig : v1454cygmcmc ] illustrates the dependence on the model functions . although a spline - interpolated model light curve usually gives a smaller standard error for parameter estimates , the difference is not usually very big . in the present application to v1454 cyg , spline and sine fits give @xmath164 and @xmath165 , respectively . these analysis has confirmed the results , both period and error estimate , of the pdm analysis . this method would be advantageous when the model description becomes more complex than what the original pdm method describes . the pdm method evaluates dispersions of the observed data against phase - binned averages . this prescription sometimes produces spurious scatters in the resultant theta diagram when the number of data ( and data in each bin ) is small . instead of getting phase - averaged light curves ( at given trial periods ) using a small number of discrete bins , we introduce a bayesian approach for obtaining continuous phase - averaged light curves . we model light curves by a set of parameters @xmath166 ( at a trial period ) for phases @xmath167 ) , where @xmath16 is the number of phase bins , which are large ( @xmath168 ) enough to describe continuous functions . following a standard technique in bayesian analysis , we then express the condition of smoothness of @xmath112 by introducing a prior function assuming that second order differences of @xmath175 follow a normal distribution @xmath176 @xmath177 } , \ ] ] where @xmath178 and @xmath179 reflecting the cyclic condition . although the smoothing parameter @xmath183 can be estimated from individual fits by different trial periods , we adopt a constant @xmath183 for all periods . the value is estimated from the best fit , i.e. from the period the giving smallest dispersion . using this smooth phase - average light curve , we can then estimate overall variance as prescribed as in @xcite .

as an extension of the project in @xcite , we collected times of superhump maxima for 61 su uma - type dwarf novae mainly observed during the 20092010 season . the newly obtained data confirmed the basic findings reported in @xcite : the presence of stages a c , as well as the predominance of positive period derivatives during stage b in systems with superhump periods shorter than 0.07 d. there was a systematic difference in period derivatives for systems with superhump periods longer than 0.075 d between this study and @xcite . we suggest that this difference is possibly caused by the relative lack of frequently outbursting su uma - type dwarf novae in this period regime in the present study . we recorded a strong beat phenomenon during the 2009 superoutburst of iy uma . the close correlation between the beat period and superhump period suggests that the changing angular velocity of the apsidal motion of the elliptical disk is responsible for the variation of superhump periods . we also described three new wz sge - type objects with established early superhumps and one with likely early superhumps . we also suggest that two systems , vx for and el uma , are wz sge - type dwarf novae with multiple rebrightenings . the @xmath0 variation in ot j213806.6@xmath1261957 suggests that the frequent absence of rebrightenings in very short-@xmath2 objects can be a result of sustained superoutburst plateau at the epoch when usual su uma - type dwarf novae return to quiescence preceding a rebrightening . we also present a formulation for a variety of bayesian extension to traditional period analyses .

You are an expert at summarizing long articles. Proceed to summarize the following text: [ pddefn ] a group @xmath3 of homeomorphisms of a topological space @xmath11 acts _ properly discontinuously _ on @xmath11 if , for every compact subset @xmath12 of @xmath11 , @xmath13 classically , a discrete group @xmath3 of isometries of a riemannian manifold @xmath11 is a crystallographic group if @xmath3 acts properly discontinuously on @xmath11 , and the quotient @xmath14 is compact . the @xmath3-translates of any fundamental domain for @xmath14 form a tessellation of @xmath11 . these notions generalize to any homogeneous space , even without an invariant metric . [ tessdefn ] let * @xmath1 be a lie group and * @xmath0 be a closed subgroup of @xmath1 . a discrete subgroup @xmath3 of @xmath1 is a _ crystallographic group _ for @xmath2 if 1 . @xmath3 acts properly discontinuously on @xmath2 ; and 2 . @xmath15 is compact . we say that @xmath2 has a _ tessellation _ if there exists a crystallographic group @xmath3 for @xmath2 . crystallographic groups and the corresponding tessellations have been studied for many groups @xmath1 . ( a brief recent introduction to the subject is given in @xcite . ) the classical bieberbach theorems ( * ? ? ? * chap . 1 ) deal with the case where @xmath1 is the group of isometries of euclidean space @xmath16 . as another example , the auslander conjecture @xcite asserts that if @xmath1 is the group of all affine transformations of @xmath17 , then every crystallographic group has a solvable subgroup of finite index . in addition , the case where @xmath1 is solvable has been discussed in @xcite . in this paper , we focus on the case where @xmath1 is a simple lie group , such as @xmath18 , @xmath19 , or @xmath20 . [ standing ] throughout this paper : 1 . [ standing - g ] @xmath1 is a linear , semisimple lie group with only finitely many connected components ; and 2 . [ standing - h ] @xmath0 is a closed subgroup of @xmath1 with only finitely many connected components . [ hdisconnected ] because @xmath21 is finite ( hence compact ) , it is easy to see that @xmath2 has a tessellation if and only if @xmath22 has a tessellation . also , if @xmath2 has a tessellation , then @xmath23 has a tessellation . furthermore , the converse holds in many situations . ( see [ disconnected ] for a discussion of this issue . ) thus , there is usually no harm in assuming that both @xmath1 and @xmath0 are connected ; we will feel free to do so whenever it is convenient . on the other hand , because @xmath19 is usually not connected ( it usually has two components ( * ? ? ? * lem . 10.2.4 , p. 451 ) ) , it would be somewhat awkward to make this a blanket assumption . [ classical ] there are two classical cases in which @xmath2 is well known to have a tessellation . 1 . if @xmath2 is compact , then we may let @xmath24 ( or any finite subgroup of @xmath1 ) . 2 . [ classical - borel ] if @xmath0 is compact , then we may let @xmath3 be any cocompact lattice in @xmath1 . ( a. borel @xcite proved that every connected , simple lie group has a cocompact lattice . ) thus , the existence of a tessellation is an interesting question only when neither @xmath0 nor @xmath2 is compact . ( in this case , any crystallographic group @xmath3 must be infinite , and can not be a lattice in @xmath1 . ) given @xmath1 ( satisfying [ standing]([standing - g ] ) ) , we would like to find all the subgroups @xmath0 ( satisfying [ standing]([standing - h ] ) ) , such that @xmath2 has a tessellation . this seems to be a difficult problem in general . ( see the surveys @xcite and @xcite for a discussion of the many partial results that have been obtained , mainly under the additional assumption that @xmath0 is reductive . ) however , it can be solved in certain cases of low real rank . in particular , as we will now briefly explain , the problem is very easy if @xmath25 or @xmath26 . most of this paper is devoted to solving the problem for certain cases where @xmath27 . if @xmath25 ( that is , if @xmath1 is compact ) , then @xmath2 must be compact ( and @xmath0 must also be compact ) , so @xmath2 has a tessellation , but this is not interesting . if @xmath28 , then there are some interesting homogeneous spaces , but it turns out that none of them have tessellations . [ calabimarkuscircle ] @xmath29 is transitive on @xmath30 , so @xmath30 is a homogeneous space for @xmath1 . it does not have a tessellation , for reasons that we now explain . let @xmath12 be the unit circle , so @xmath12 is a compact subset of @xmath31 . we claim that @xmath32 , for every @xmath33 ( cf . figure [ circle+ellipse ] ) . to see this , note that , because @xmath34 , the ellipse bounded by @xmath35 has the same area as the disk bounded by @xmath12 , so @xmath35 can not be contained in the interior of the disk bounded by @xmath12 , and can not contain @xmath12 in its interior . thus , @xmath35 must be partly inside @xmath12 and partly outside , so @xmath35 must cross @xmath12 , as claimed . let @xmath3 be any discrete subgroup of @xmath1 . the preceding paragraph implies that @xmath36 , for every @xmath37 . if @xmath3 acts properly discontinuously on @xmath30 , then , because @xmath12 is compact , this implies that @xmath3 is finite . so the quotient @xmath38 is not compact . therefore @xmath3 is not a crystallographic group . we have shown that no subgroup of @xmath1 is a crystallographic group , so we conclude that @xmath30 does not have a tessellation . ): @xmath39 , for every @xmath40 , so no infinite subgroup of @xmath41 acts properly discontinuously . ] this example illustrates the _ calabi - markus phenomenon _ : if there is a compact subset @xmath12 of @xmath2 , such that @xmath32 , for every @xmath33 , then no infinite subgroup of @xmath1 acts properly discontinuously on @xmath2 ( see [ calabimarkus ] ) . thus , @xmath2 does not have a tessellation , unless @xmath2 is compact ( see [ cds->notess ] ) . we will see in section [ cartansect ] that the following proposition can be proved quite easily from basic properties of the cartan projection . rrank1-cds [ rrank1-cm ] if @xmath28 , and @xmath0 is not compact , then there is a compact subset @xmath12 of @xmath2 , such that @xmath32 , for every @xmath33 . [ rank1->notess ] if @xmath28 , and neither @xmath0 nor @xmath2 is compact , then @xmath2 does not have a tessellation . we now consider groups of real rank two . the obvious example is @xmath42 , but , in this case , once again , none of the interesting homogeneous spaces have tessellations . moreover , the same is true when real numbers are replaced by complex numbers or quaternions . the case where @xmath43 relies on beautiful methods of y. benoist and f. labourie @xcite or g. a. margulis @xcite , which we describe in section [ 1dsect ] . [ sl3->notess ] if @xmath44 and neither @xmath0 nor @xmath2 is compact , then @xmath2 does not have a tessellation . it is important to note that some interesting homogeneous spaces do have tessellations . [ g = lxh ] suppose @xmath45 , and let @xmath3 be a cocompact lattice in @xmath46 . then @xmath3 acts properly discontinuously on @xmath47 , and @xmath48 is compact . so @xmath2 has a tessellation . the following easy lemma generalizes this example to the situation where @xmath1 is a more general product of @xmath46 and @xmath0 , not necessarily a direct product . construct - tess [ g = lh ] let @xmath0 and @xmath46 be closed subgroups of @xmath1 , such that * @xmath49 , * @xmath50 is compact ; and * @xmath46 has a cocompact lattice @xmath3 . then @xmath2 has a tessellation . ( namely , @xmath3 is a crystallographic group for @xmath2 . ) for @xmath51 or @xmath52 , this lemma leads to some interesting examples found by r. kulkarni ( * ? ? ? 6.1 ) and t. kobayashi ( * ? ? ? [ kulkarnieg ] there are natural embeddings @xmath53 furthermore , identifying @xmath54 with @xmath55 yields an embedding @xmath56 similarly , identifying @xmath57 with @xmath58 yields an embedding @xmath59 thus , we may think of @xmath60 and @xmath61 as subgroups of @xmath62 ; and we may think of @xmath63 and @xmath64 as subgroups of @xmath65 . with the above understanding , we see that @xmath60 is the stabilizer of a vector of norm @xmath66 . since @xmath61 is transitive on the set of such vectors , we have @xmath67 similarly , @xmath68 then lemma [ g = lh ] implies that each of the following four homogeneous spaces has a tessellation : * @xmath69 , * @xmath70 , * @xmath71 , and * @xmath72 . when discussing @xmath73 or @xmath52 , we always assume @xmath74 . this causes no harm , because @xmath75 is locally isomorphic to @xmath76 ( * ? ? ? * ( x ) , p. 520 ) , and @xmath77 is locally isomorphic to @xmath78 ( * ? ? ? * ( vi ) , p. 519 ) . when @xmath79 is even , h. oh and d. witte @xcite provided a complete description of all the ( closed , connected ) subgroups @xmath0 , such that @xmath80 has a tessellation , but their classification is not quite complete when @xmath79 is odd . in this paper , we extend the work of oh and witte to obtain analogous results for homogeneous spaces of @xmath81 . we also give a much shorter proof of the main results of @xcite the same techniques should yield significant results for homogeneous spaces of the other simple groups of real rank two , although the calculations seem to be difficult . on the other hand , the groups of higher real rank require different ideas . once one knows that a tessellation of @xmath2 exists , it would be interesting to find _ all _ of the crystallographic groups for @xmath2 and , for each crystallographic group , describe the possible tessellations . these are much more delicate questions , which we do not address at all . ( w. goldman @xcite , f. salein @xcite , t. kobayashi @xcite , and a. zeghib @xcite have interesting results in some special cases . ) in the remainder of this introduction , we state the specific results for homogeneous spaces of @xmath73 and @xmath52 . [ d(h)-defn ] for any connected lie group @xmath0 , let @xmath82 where @xmath83 is any maximal compact subgroup of @xmath0 . this is well defined , because all the maximal compact subgroups of @xmath0 are conjugate ( * ? ? ? 15.3.1(iii ) , pp . 180181 ) . [ d(g ) ] if @xmath0 is semisimple , we have the iwasawa decomposition @xmath84 ( * ? ? * thm . 6.5.1 , pp . 270271 ) , from which it is obvious that @xmath85 . this yields the following calculations ( see [ d(su2 ) ] and [ d(sp ) ] ) ) : * @xmath86 . . * @xmath88 . * @xmath89 . * @xmath90 . [ d(h)=dimh ] if @xmath91 ( for some iwasawa decomposition @xmath92 of @xmath1 ) , then @xmath93 ( see [ ansc ] and [ solvable]([solvable - nocpct ] ) ) . [ simdefn ] for subgroups @xmath94 and @xmath95 of @xmath1 , we write @xmath96 if there is a compact subset @xmath12 of @xmath1 , such that @xmath97 and @xmath98 . note that @xmath99 is not invariant under the equivalence relation @xmath100 . for example , the cartan decomposition @xmath101 implies that @xmath102 , but we have @xmath103 . the following two theorems state a version of the main results for even @xmath79 . suf - known [ ow - known ] assume @xmath104 , and let @xmath0 be a closed , connected , subgroup of @xmath1 , such that neither @xmath0 nor @xmath2 is compact . the homogeneous space @xmath2 has a tessellation if and only if 1 . @xmath105 ; and 2 . either @xmath106 or @xmath107 . suf - known [ iw - known ] assume @xmath108 , and let @xmath0 be a closed , connected , subgroup of @xmath1 , such that neither @xmath0 nor @xmath2 is compact . the homogeneous space @xmath2 has a tessellation if and only if 1 . @xmath109 ; and 2 . either @xmath110 or @xmath111 . the subgroups @xmath0 that arise in theorems [ ow - known ] and [ iw - known ] can also be described more explicitly ( cf . [ soeventess ] and [ sueventess ] below ) . t. kobayashi @xcite conjectured that if @xmath0 is reductive and it is impossible to construct a tessellation of @xmath2 by using a generalization of lemma [ g = lh ] ( see [ construct - tess ] ) , then @xmath2 does not have a tessellation . the following lists three special cases of this general conjecture . [ notesssu / sp ] the homogeneous spaces 1 . [ notesssu / sp - so2/su1 ] @xmath112 , 2 . [ notesssu / sp - su2/sp1 ] @xmath113 , and 3 . [ notesssu / sp - su2/su1 ] @xmath114 do not have tessellations . if this conjecture is true , then , for odd @xmath79 , there is no interesting example of a homogeneous space of @xmath73 or @xmath52 that has a tessellation . suf->complete [ su1m->complete ] assume @xmath115 and let @xmath0 be any closed , connected subgroup of @xmath1 , such that neither @xmath0 nor @xmath2 is compact . if conjecture [ notesssu / sp ] is true , then @xmath2 does not have a tessellation . the proof of theorem [ su1m->complete ] assumes the following special case proved by r. kulkarni ( * ? ? ? 2.10 ) . in short , kulkarni noted that the euler characteristic of @xmath15 must both vanish ( because the euler characteristic of @xmath2 vanishes ) and not vanish ( by the gauss - bonnet theorem ) . ( other results in the same spirit , obtaining a contradiction from the study of characteristic classes of @xmath116 , appear in @xcite . ) [ so2n / so1odd - notess ] if @xmath79 is odd , then @xmath117 does not have a tessellation . let us give a more explicit description of the closed , connected subgroups @xmath0 of @xmath62 or @xmath65 , such that @xmath2 has a tessellation . this shows that if @xmath79 is even , then the kulkarni - kobayashi examples ( [ kulkarnieg ] ) and certain deformations are essentially the only interesting homogeneous spaces of @xmath73 or @xmath52 that have tessellations . [ kandefn ] fix an iwasawa decomposition @xmath92 . thus , * @xmath118 is a maximal compact subgroup , * @xmath119 is the identity component of a maximal split torus , and * @xmath120 is a maximal unipotent subgroup . the following two results are stated only for subgroups of @xmath121 , because the general case reduces to this ( see [ hcanbean ] ) . the reason is basically that @xmath0 contains a connected , cocompact subgroup that is conjugate to a subgroup of @xmath121 . ( clearly , if @xmath122 is any cocompact subgroup of @xmath0 , then @xmath2 has a tessellation if and only if @xmath123 has a tessellation . ) this is not quite true in general , but the following lemma provides a satisfactory substitute , by showing that it becomes true after enlarging @xmath0 by a compact amount . hcanbean after replacing @xmath0 by a conjugate subgroup , there is a closed , connected subgroup @xmath124 of @xmath1 , such that @xmath125 and @xmath126 are compact , where @xmath127 denotes the identity component of @xmath128 . sufeventess [ soeventess ] assume @xmath104 , and let @xmath0 be a closed , connected , nontrivial , proper subgroup of @xmath121 . the homogeneous space @xmath2 has a tessellation if and only if @xmath0 is conjugate to a subgroup @xmath122 , such that either 1 . [ soeventess - so ] @xmath129 ; or 2 . [ soeventess - sp ] @xmath122 belongs to a certain family @xmath130 of deformations of @xmath131 , described explicitly in theorem [ hbthm ] ( with @xmath132 ) . sufeventess [ sueventess ] assume @xmath108 , and let @xmath0 be a closed , connected , nontrivial , proper subgroup of @xmath121 . the homogeneous space @xmath2 has a tessellation if and only if @xmath0 is conjugate to a subgroup @xmath122 , such that either 1 . [ sueventess - su ] @xmath122 belongs to a certain family @xmath133}}}\}$ ] of deformations of @xmath134 , described explicitly in theorem [ suegs ] ; or 2 . [ sueventess - sp ] @xmath122 belongs to a certain family @xmath130 of deformations of @xmath135 , described explicitly in theorem [ hbthm ] ( with @xmath136 ) . the proof of theorem [ soeventess ] ( or [ ow - known ] ) in @xcite requires a list @xcite of all the homogeneous spaces of @xmath73 that admit a proper action of a noncompact subgroup of @xmath52 . ( the list was obtained by very tedious case - by - case analysis . it was extended to homogeneous spaces of @xmath52 in @xcite . ) the following proposition ( [ tessan->dim>1,2 ] ) provides an _ a priori _ lower bound on @xmath137 , and it turns out that the classification of the interesting subgroups of large dimension can be achieved fairly easily ( see [ suflargesect ] ) . this is the main reason that we are able to give reasonably short complete proofs of theorems [ ow - known ] , [ iw - known ] , [ su1m->complete ] , [ soeventess ] , and [ sueventess ] . [ tessan->dim>1,2 ] suppose @xmath51 or @xmath52 , and let @xmath0 be a closed , connected , nontrivial subgroup of @xmath121 . if @xmath2 has a tessellation , then @xmath138 this research was partially supported by a grant from the national science foundation ( dms-9801136 ) . much of the work was carried out during productive visits to the isaac newton institute for mathematical sciences ( cambridge , u.k . ) ; we would like to thank the newton institute for the financial support that made the visits possible . would like to thank the mathematics department of oklahoma state university for their warm and generous hospitality and marc burger both for pointing out a mistake in the original statement and proof of theorem [ sufeventess ] and for many enlightening conversations . the main problem in this paper is to determine whether or not a homogeneous space @xmath2 has a tessellation . this requires some method to determine whether or not a given discrete subgroup @xmath3 of @xmath1 acts properly discontinuously on @xmath2 . y. benoist and t. kobayashi ( independently ) demonstrated that the cartan projection @xmath139 is an effective tool to study this question . it is the foundation of almost all of our work in later sections . in this section , we introduce the cartan projection , and describe some of its basic properties . first , however , we recall the notion of a proper action ( a generalization of properly discontinuous actions ) and of a cartan - decomposition subgroup . at the end of the section , we use the cartan projection to briefly discuss the question of when there is a loss of generality in assuming that @xmath1 is connected . a topological group @xmath46 of homeomorphisms of a topological space @xmath11 acts _ properly _ on @xmath11 if , for every compact subset @xmath12 of @xmath11 , @xmath140 it is important to note that a _ discrete _ group of homeomorphisms of @xmath11 acts properly on @xmath11 if and only if it acts properly discontinuously on @xmath11 . for the special case where @xmath141 is a homogeneous space , the following lemma restates the definition of a proper action in more group - theoretic terms [ proper<>chc ] a closed subgroup @xmath46 of @xmath1 acts properly on @xmath2 if and only if , for every compact subset @xmath12 of @xmath1 , the intersection @xmath142 is compact . if @xmath12 is any compact subset of @xmath1 , then @xmath143 is a compact subset of @xmath2 ; furthermore , any compact subset of @xmath2 is contained in one of the form @xmath144 . we have @xmath145 this has the following well - known , easy consequence . [ chcproper ] suppose @xmath0 , @xmath94 , @xmath46 , and @xmath146 are closed subgroups of @xmath1 . if * @xmath46 acts properly on @xmath2 , and * there is a compact subset @xmath12 of @xmath1 , such that @xmath147 and @xmath148 , then @xmath146 acts properly on @xmath149 . the following definition describes the subgroups to which the calabi - markus phenomenon applies ( cf . example [ calabimarkuscircle ] ) . we say that @xmath0 is a _ cartan - decomposition subgroup _ of @xmath1 if @xmath150 ( see notation [ simdefn ] ) . [ aiscds ] from the cartan decomposition @xmath101 , we know that @xmath119 is a cartan - decomposition subgroup . [ cdsconj ] any conjugate of a cartan - decomposition subgroup is a cartan - decomposition subgroup . [ calabimarkus ] if @xmath0 is a cartan - decomposition subgroup of @xmath1 , and @xmath3 is a discrete subgroup of @xmath1 that acts properly discontinuously on @xmath2 , then @xmath3 is finite . because @xmath0 is a cartan - decomposition subgroup , there is a compact subset @xmath12 of @xmath1 , such that @xmath151 . however , from lemma [ proper<>chc ] , we know that @xmath152 is finite . therefore @xmath153 is finite . the following well - known , easy fact is a direct consequence of the calabi - markus phenomenon . it is an important first step toward determining which homogeneous spaces have tessellations . [ cds->notess ] if @xmath0 is a cartan - decomposition subgroup of @xmath1 , such that @xmath2 is not compact , then @xmath2 does not have a tessellation . [ a+defn ] * if @xmath1 is connected , let @xmath154 be the ( closed ) positive weyl chamber of @xmath119 in which the roots occurring in the lie algebra of @xmath120 are positive ( cf . [ kandefn ] ) . thus , @xmath154 is a fundamental domain for the action of the ( real ) weyl group of @xmath1 on @xmath119 . * in the general case , let @xmath154 be a closed , convex fundamental domain for the action of the ( real ) weyl group of @xmath1 on @xmath119 , such that @xmath154 is contained in the ( closed ) positive weyl chamber of @xmath119 in which the roots occurring in the lie algebra of @xmath120 are positive . for each element @xmath155 of @xmath1 , the cartan decomposition @xmath9 implies that there is an element @xmath156 of @xmath154 with @xmath157 . in fact , the element @xmath156 is unique , so there is a well - defined function @xmath158 given by @xmath159 . we remark that the function @xmath139 is continuous and proper ( that is , the inverse image of any compact set is compact ) . the following crucial result of y. benoist provides a uniform estimate on the variation of @xmath139 over disks of bounded radius . ( a related result was proved , independently and simultaneously , by t. kobayashi ( * ? ? ? the proof is both elementary and elegant . however , it requires a bit of notation , so we postpone it to [ calcsect - benoist ] ( and , for concreteness , we will assume that @xmath1 is either @xmath73 or @xmath52 in the proof ) . [ bddchange ] for any compact subset @xmath12 of @xmath1 , there is a compact subset @xmath160 of @xmath119 , such that @xmath161 , for all @xmath33 . for subsets @xmath162 and @xmath163 of @xmath154 , we write @xmath164 if there is a compact subset @xmath12 of @xmath119 , such that @xmath165 and @xmath166 . [ simvsmu ] for any subgroups @xmath94 and @xmath95 of @xmath1 , we have @xmath96 if and only if @xmath167 . ( @xmath168 ) let @xmath12 be a compact subset of @xmath1 , such that @xmath169 and @xmath170 . choose a corresponding compact subset @xmath160 of @xmath119 , as in proposition [ bddchange ] . then @xmath171 and , similarly , @xmath172 . ( @xmath173 ) let @xmath12 be a compact subset of @xmath119 , such that @xmath174 and @xmath175 . then @xmath176 and , similarly , @xmath177 . the special case where @xmath178 ( and @xmath94 is closed and almost connected ) can be restated as follows . [ cdsvsmu ] @xmath0 is a cartan - decomposition subgroup of @xmath1 if and only if @xmath179 . [ rrank1-cds ] assume that @xmath28 . the subgroup @xmath0 is a cartan - decomposition subgroup of @xmath1 if and only if @xmath0 is noncompact . ( @xmath173 ) we have @xmath180 , and , because @xmath139 is a proper map , we have @xmath181 as @xmath182 in @xmath0 . because @xmath183 , we know that @xmath154 is homeomorphic to the half - line @xmath184 ( with the point @xmath185 in @xmath154 corresponding to the endpoint @xmath186 of the half - line ) , so , by continuity , it must be the case that @xmath187 . then corollary [ cdsvsmu ] implies that @xmath0 is a cartan - decomposition subgroup , but we provide the following direct proof that avoids any appeal to proposition [ bddchange ] . from the definition of @xmath139 , we have @xmath188 . therefore @xmath189 so @xmath0 is a cartan - decomposition subgroup ( by taking @xmath190 in definition [ simdefn ] ) . by using lemma [ proper<>chc ] , the proof of corollary [ simvsmu ] also establishes the following . [ proper<>mu(l ) ] suppose @xmath0 and @xmath46 are closed subgroups of @xmath1 . the subgroup @xmath46 acts properly on @xmath2 if and only if @xmath191 is compact , for every compact subset @xmath12 of @xmath119 . as was mentioned in remark [ hdisconnected ] , we may assume , without loss of generality , that @xmath0 is connected . however , it may not be possible to assume that @xmath1 is connected , because , although there are no known examples , it is possible that the following question has an affirmative answer . [ assumegconn ? ] does there exist a homogeneous space @xmath2 ( satisfying assumption [ standing ] ) , such that @xmath192 has a tessellation , but @xmath2 does not have a tessellation ? if @xmath3 is a crystallographic group for @xmath192 , then it is easy to see that @xmath15 is compact . however , the following example shows that @xmath3 may not act properly discontinuously on @xmath2 . let * @xmath193 , * @xmath194 be the automorphism of @xmath195 that interchanges the two factors ( that is , @xmath196 ) , * @xmath197 ( semidirect product ) , and * @xmath3 be a cocompact lattice in @xmath46 ( cf . [ classical]([classical - borel ] ) ) . then @xmath198 , and @xmath3 is a crystallographic group for @xmath199 ( see example [ g = lxh ] ) . however , @xmath200 , so @xmath3 does not act properly on @xmath2 ( see [ proper<>chc ] with @xmath201 ) . even so , @xmath2 does have a tessellation , because the diagonal embedding @xmath202 is a crystallographic group for @xmath2 . thus , this example does not provide an answer to question [ assumegconn ? ] . in this example , @xmath194 represents an element of the weyl group of @xmath1 that does not belong to the weyl group of @xmath203 . the following proposition shows that this is a crucial ingredient in the construction . let @xmath3 be a crystallographic group for @xmath192 . if the ( real ) weyl group of @xmath1 is same as the ( real ) weyl group of @xmath203 , then @xmath3 is a crystallographic group for @xmath2 . by assumption , we may choose the same fundamental domain @xmath154 for the weyl groups of @xmath1 and @xmath203 . let @xmath204 and @xmath205 be the cartan projections ; then @xmath206 is the restriction of @xmath139 to @xmath203 . for simplicity , assume , without loss of generality , that @xmath207 ( for example , assume @xmath0 is connected ) . then , for any compact subset @xmath12 of @xmath119 , we have @xmath208 is finite ( see [ proper<>mu(l ) ] ) . thus , @xmath3 acts properly discontinuously on @xmath2 ( see [ proper<>mu(l ) ] ) , as desired . a. borel and j. tits ( * ? ? ? 14.6 , p. 147 ) proved that if @xmath1 is zariski connected , then every element of the weyl group of @xmath1 has a representative in @xmath203 . also , any element of the weyl group must act as an automorphism of the root system . thus , we have the following corollary . let @xmath3 be a crystallographic group for @xmath192 . if either * @xmath1 is zariski connected , or * every automorphism of the real root system of @xmath203 belongs to the weyl group of the root system , then @xmath3 is a crystallographic group for @xmath2 . 1 . if @xmath51 , then @xmath1 is zariski connected ( because @xmath209 is connected ( * ? ? ? * thm . 2.1.9 , p. 60 ) ) , so @xmath2 has a tessellation if and only if @xmath192 has a tessellation . 2 . more generally , if @xmath210 or @xmath52 ( with @xmath211 ) , then every automorphism of the real root system of @xmath203 belongs to the weyl group of the root system ( cf . figure [ rootspict ] ) , so @xmath2 has a tessellation if and only if @xmath192 has a tessellation . if @xmath212 , where @xmath194 is the cartan involution of @xmath42 , then @xmath194 represents an element of the weyl group of @xmath1 that does not belong to the weyl group of @xmath203 , so the proposition does not apply to @xmath1 . however , this does not matter : if neither @xmath0 nor @xmath2 is compact , then theorem [ sl3->notess ] implies that @xmath192 has no tessellations , so @xmath2 has no tessellations either . this section recalls a technical result that often allows us to assume that @xmath0 is a subgroup of @xmath121 . it also recalls some basic topological properties of such subgroups , and also recalls a simple observation relating these subgroups to the root spaces of the lie algebra @xmath213 . an element @xmath155 of @xmath1 is : * _ hyperbolic _ if @xmath155 is conjugate to an element of @xmath119 ; * _ unipotent _ if @xmath155 is conjugate to an element of @xmath120 ; * _ elliptic _ if @xmath155 is conjugate to an element of @xmath118 . [ jordandecomp ] each @xmath33 has a unique decomposition in the form @xmath214 , such that * @xmath156 is hyperbolic , @xmath215 is unipotent , and @xmath216 is elliptic ; and * @xmath156 , @xmath215 , and @xmath216 all commute with each other . [ jordancommute ] if @xmath214 is the real jordan decomposition of some element @xmath155 of @xmath1 , then @xmath156 , @xmath215 , and @xmath216 commute , not only with each other , but also with any element of @xmath1 that commutes with @xmath155 . this is because the real jordan decomposition of @xmath217 is @xmath218 if @xmath219 , then the uniqueness of the real jordan decomposition of @xmath155 implies @xmath220 , @xmath221 , and @xmath222 . the following observation is a generalization of the fact that a collection of commuting triangularizable matrices can be simultaneously triangularized . [ simultaneous ] if @xmath0 is abelian ( or , more generally , solvable ) , and is generated by hyperbolic and/or unipotent elements , then @xmath0 is conjugate to a subgroup of @xmath121 . because of the following result , we usually assume @xmath91 ( by replacing @xmath0 with a conjugate of @xmath122 ) . [ hcanbean ] if @xmath0 is connected , then there is a closed , connected subgroup @xmath122 of @xmath1 and a compact , connected subgroup @xmath12 of @xmath1 , such that 1 . [ hcanbean - inan ] @xmath122 is conjugate to a subgroup of @xmath121 ; 2 . [ hcanbean - ch = ch ] @xmath223 is a subgroup of @xmath1 ; and 3 . [ hcanbean - d(h ) ] @xmath224 ( see notation [ d(h)-defn ] ) . moreover , it is easy to see from ( [ hcanbean - ch = ch ] ) that the homogeneous space @xmath2 has a tessellation if and only if @xmath123 has a tessellation . first , let us note that every connected subgroup of @xmath121 is closed ( see [ solvable]([solvable - h = rn ] ) and [ ansc ] ) , so we do not need to show that @xmath122 is closed . second , let us note that ( [ hcanbean - d(h ) ] ) is a consequence of ( [ hcanbean - inan ] ) and ( [ hcanbean - ch = ch ] ) . to see this , let @xmath225 be a maximal compact subgroup of @xmath226 that contains @xmath12 . then a standard argument shows that @xmath227 is a maximal compact subgroup of @xmath0 . ( because all maximal compact subgroups of @xmath226 are conjugate , there is some @xmath228 , such that @xmath229 is a maximal compact subgroup of @xmath0 that contains @xmath227 . since @xmath230 , we know that @xmath12 normalizes @xmath225 , so we may assume @xmath231 ; thus , @xmath155 normalizes @xmath0 . then @xmath232 contains @xmath227 . because @xmath227 is compact , this implies that @xmath155 normalizes @xmath227 . so @xmath233 is a maximal compact subgroup of @xmath0 . ) therefore @xmath234 similarly , @xmath235 . since @xmath236 , we conclude that @xmath224 , as desired . [ hcanbean - ss ] assume @xmath0 is semisimple . we have an iwasawa decomposition @xmath84 ; let @xmath237 and @xmath238 . assume @xmath239 is a one - parameter subgroup . let * @xmath240 be the real jordan decomposition of @xmath241 ( see [ jordandecomp ] ) ; * @xmath242 ; and * @xmath243 be the closure of @xmath244 . ( lemma [ simultaneous ] implies that @xmath122 is conjugate to a subgroup of @xmath121 . ) [ hcanbean - abel ] assume @xmath0 is abelian . we may write @xmath0 as a product of one - parameter subgroups : @xmath245 let @xmath246 be the real jordan decomposition of @xmath247 ( see [ jordandecomp ] ) . note that @xmath248 , @xmath249 , and @xmath250 commute , not only with each other , but also with every @xmath251 , @xmath252 , and @xmath253 ( see [ jordancommute ] ) . let @xmath254 and let @xmath255 . ( lemma [ simultaneous ] implies that @xmath122 is conjugate to a subgroup of @xmath121 . ) the general case . from the levi decomposition @xcite , we know that there is a connected , semisimple subgroup @xmath46 of @xmath0 and a connected , solvable , normal subgroup @xmath256 of @xmath0 , such that @xmath257 ( and @xmath258 is finite ) . let @xmath259 $ ] , so @xmath162 is a connected , normal subgroup of @xmath0 , and @xmath162 is conjugate to a subgroup of @xmath120 ( cf . * cor . 2.7.1 , p. 51 ) ) . by modding out @xmath162 , we ( essentially ) reduce to the direct product of cases [ hcanbean - ss ] and [ hcanbean - abel ] . for @xmath0 and @xmath122 as in lemma [ hcanbean ] , proposition [ an / h = rd ] ( and [ ansc ] ) implies that if @xmath260 , then @xmath261 is not compact ; also , proposition [ solvable]([solvable - nocpct ] ) ( and [ ansc ] ) implies that if @xmath262 , then @xmath122 is not compact . therefore : * @xmath263 if and only if @xmath2 is compact ; and * @xmath264 if and only if @xmath0 is compact . thus , if neither @xmath0 nor @xmath2 is compact , then @xmath122 is a nontrivial , proper subgroup of @xmath121 . everything is this subsection is well known , though somewhat scattered in the literature . the main results are propositions [ solvable ] and [ an / h = rd ] , which , together with corollary [ ansc ] , show that connected subgroups of @xmath121 and their homogeneous spaces are very well behaved topologically . corollary [ fiberbundle ] , on the homology of very simple quotient spaces , is also used in later sections . we begin with the easy case of abelian groups . this lemma generalizes almost verbatim to solvable groups ( see [ solvable ] ) , but the proof in that generality is not as trivial . [ abelian ] let @xmath256 be a @xmath26-connected , abelian lie group . 1 . [ abelian - h = rn ] if @xmath0 is a connected subgroup of @xmath256 , then @xmath0 is closed , simply connected , and isomorphic to @xmath265 , for some @xmath266 . [ abelian - hcapl ] if @xmath0 and @xmath46 are connected subgroups of @xmath256 , then @xmath267 is connected . [ abelian - nocpct ] if @xmath12 is a compact subgroup of @xmath256 , then @xmath12 is trivial . because @xmath256 is abelian and 1-connected , the exponential map is a lie group isomorphism from the additive group of the lie algebra @xmath268 onto @xmath256 . ( [ abelian - h = rn ] ) let @xmath269 . because the exponential map is a lie group isomorphism ( hence a diffeomorphism ) , and because @xmath270 is a closed @xmath266-submanifold of @xmath268 , we know that @xmath271 is a closed @xmath266-submanifold of @xmath256 . of course , @xmath271 is contained in @xmath0 , which is also a @xmath266-submanifold of @xmath256 . because the dimensions are the same , we know that @xmath271 is open in @xmath0 . also , because @xmath271 is closed in @xmath256 , we know that @xmath271 is closed in @xmath0 . therefore @xmath272 ( because @xmath0 is connected ) . finally , we know that @xmath273 is a diffeomorphism from its domain @xmath274 onto its image @xmath0 . ( [ abelian - hcapl ] ) from ( [ abelianpf - h = rn - exp(h ) ] ) , we have @xmath275 and , similarly , @xmath276 . also , because @xmath277 is bijective , we have @xmath278 . therefore @xmath279 is connected . ( [ abelian - nocpct ] ) because @xmath265 is not compact ( for @xmath280 ) , we know , from [ abelian]([abelian - h = rn ] ) , that @xmath281 is trivial ; so @xmath12 is finite . since @xmath282 has no elements of finite order , we conclude that @xmath12 is trivial . as is usual in the theory of solvable groups , the main results of this section are proved by induction , based on modding out some normal subgroup @xmath46 . to be effective , this method requires an understanding of the quotient space @xmath283 . the information we need ( even if @xmath46 is not normal ) comes from the following elementary observation , because @xmath256 is a principal @xmath46-bundle over @xmath283 . [ trivialbundle ] let @xmath284 be a principal @xmath0-bundle over a manifold @xmath11 . 1 . [ trivialbundle - h = rn ] if @xmath0 is diffeomorphic to @xmath17 , then 1 . @xmath284 is @xmath0-equivariantly diffeomorphic to @xmath285 , so 2 . @xmath284 is homotopy equivalent to @xmath11 . [ trivialbundle - m = rn ] if @xmath11 is diffeomorphic to @xmath17 , then 1 . @xmath284 is @xmath0-equivariantly diffeomorphic to @xmath285 , so 2 . @xmath284 is homotopy equivalent to @xmath0 . any principal bundle with a section is trivial ( * ? ? ? * cor . 4.8.3 , p. 48 ) . if either the fiber or the base is contractible , then there is no obstruction to constructing a section ( * ? ? ? 2.7.1(h1 ) , p. 21 ) , so @xmath284 is trivial : @xmath286 . ( the diffeomorphism can be taken to be @xmath0-equivariant , with respect to the natural @xmath0-action on @xmath285 , given by @xmath287 . ) then the conclusions on homotopy equivalence follow from the fact that @xmath17 is contractible ( that is , homotopically trivial ) . we recall the long exact sequence of the fibration @xmath288 : [ htpyexactfibration ] let @xmath0 be a closed subgroup of a lie group @xmath256 . there is a ( natural ) long exact sequence of homotopy groups : @xmath289 [ r / hsc ] let @xmath0 be a closed subgroup of a @xmath26-connected lie group @xmath256 . the homogeneous space @xmath290 is simply connected if and only if @xmath0 is connected . because @xmath256 is 1-connected , we have @xmath291 , so , from ( [ htpyexactfibration ] ) , we know that the sequence @xmath292 is exact . thus , @xmath293 , so the desired conclusion is immediate . as a step toward proposition [ solvable ] , we prove two special cases that describe the topology of normal subgroups . [ r = rd ] if @xmath256 is a @xmath26-connected , solvable lie group , then @xmath256 is diffeomorphic to @xmath294 , for some @xmath99 . we may assume the group @xmath256 is nonabelian ( otherwise , the desired conclusion is given by lemma [ abelian]([abelian - h = rn ] ) ) . then , because @xmath256 is solvable , there is a nontrivial , connected , proper , closed , normal subgroup @xmath46 of @xmath256 . since @xmath283 is simply connected ( see [ r / hsc ] ) , and @xmath295 , we may assume , by induction on @xmath296 , that @xmath283 is diffeomorphic to some @xmath297 . therefore 1 . [ r = rdpf - r = prod ] @xmath256 is diffeomorphic to @xmath298 and 2 . rdpf - l = r ] @xmath46 is homotopy equivalent to @xmath256 ( see [ trivialbundle]([trivialbundle - m = rn ] ) ) . because @xmath256 is @xmath26-connected , ( [ r = rdpf - l = r ] ) implies that @xmath46 is @xmath26-connected ; hence , @xmath46 is a 1-connected , solvable lie group , so we may assume , by induction on @xmath296 , that @xmath46 is diffeomorphic to some @xmath299 . thus , ( [ r = rdpf - r = prod ] ) implies that @xmath256 is diffeomorphic to @xmath300 , as desired . [ rnormal ] if @xmath256 is a @xmath26-connected , solvable lie group , then every connected , closed , normal subgroup of @xmath256 is @xmath26-connected . the following proposition is a nearly complete generalization of lemma [ abelian ] to the class of solvable groups . there are two exceptions : 1 . of course , subgroups of a solvable group may not be abelian , so the conclusion in [ abelian]([abelian - h = rn ] ) that @xmath0 is isomorphic to some @xmath265 must be weakened to the conclusion that @xmath0 is diffeomorphic to some @xmath265 . the intersection of connected subgroups is not always connected ( see [ hcapldisconn ] ) , so we add the restriction that @xmath46 is normal to [ abelian]([abelian - hcapl ] ) . ( we remark that no such restriction is necessary if @xmath301 , because the exponential map is a diffeomorphism from @xmath268 onto @xmath256 in this case @xcite . ) [ hcapldisconn ] let @xmath302 @xmath303 then @xmath256 , being diffeomorphic to @xmath304 , is 1-connected ; and @xmath305 and @xmath306 are connected subgroups . but @xmath307 is not connected . [ solvable ] let @xmath256 be a @xmath26-connected , solvable lie group . 1 . [ solvable - h = rn ] if @xmath0 is a connected subgroup of @xmath256 , then @xmath0 is closed , simply connected , and diffeomorphic to some @xmath294 . [ solvable - hcapl ] if @xmath0 and @xmath46 are connected subgroups of @xmath256 , and @xmath46 is normal , then @xmath267 is connected . [ solvable - nocpct ] if @xmath12 is a compact subgroup of @xmath256 , then @xmath12 is trivial . ( [ solvable - hcapl ] ) we may assume @xmath46 is nontrivial , so @xmath308 . thus , by induction on @xmath296 , using ( [ solvable - h = rn ] ) , we may assume that @xmath309 is a closed , simply connected subgroup of @xmath283 . then , since @xmath310 is homeomorphic to @xmath309 , we see that @xmath311 so lemma [ r / hsc ] implies that @xmath267 is connected . ( [ solvable - h = rn ] ) because @xmath256 is solvable , there is a connected , closed , proper , normal subgroup @xmath46 of @xmath256 , such that @xmath283 is abelian . we know that @xmath46 is 1-connected ( see [ rnormal ] ) , so , by induction on @xmath296 , we may assume that every connected subgroup of @xmath46 is closed and simply connected . from ( [ solvable - hcapl ] ) , we know that @xmath267 is connected , so we conclude that @xmath267 is closed , and @xmath312 from ( [ htpyexactfibration ] ) ( with @xmath0 in the place of @xmath256 , and @xmath46 in the place of @xmath0 ) , together with ( [ pi1(h / hcapl ) ] ) and ( [ pi1(hcapl ) ] ) , we conclude that @xmath313 ; that is , @xmath0 is simply connected . so ( [ r = rd ] ) implies @xmath0 is diffeomorphic to some @xmath294 . because both @xmath309 and @xmath267 are closed , it is not difficult to see that @xmath0 is closed . ( [ solvable - nocpct ] ) because @xmath256 is solvable , there is a connected , closed , proper , normal subgroup @xmath46 of @xmath256 , such that @xmath283 is abelian . we know that @xmath283 is 1-connected ( see [ r / hsc ] ) , so @xmath283 has no nontrivial , compact subgroups ( see [ abelian]([abelian - nocpct ] ) ) ; thus , we must have @xmath314 . therefore , @xmath12 is a compact subgroup of @xmath46 . then , since @xmath46 is 1-connected ( see [ rnormal ] ) , we may conclude , by induction on @xmath296 , that @xmath12 is trivial . [ ansc ] @xmath121 is a @xmath26-connected , solvable lie group . because @xmath1 is linear , it is a subgroup of some @xmath315 . replacing @xmath1 by a conjugate , we may assume that @xmath121 is contained in the group @xmath316 of upper triangular matrices with positive diagonal entries ( cf . [ simultaneous ] ) . the matrix entries provide an obvious diffeomorphism from @xmath316 onto @xmath317 , so @xmath316 is 1-connected . thus , proposition [ solvable]([solvable - h = rn ] ) implies that @xmath121 is simply connected . the following observation will be used in sections [ dimhsect ] and [ existencesect ] . [ fiberbundle ] let @xmath318 be a connected subgroup of @xmath121 , and suppose we have a proper , @xmath319 action of @xmath318 on a manifold @xmath11 . then @xmath11 and @xmath320 have the same homology . because the action is proper , we know that the stabilizer of each point of @xmath11 is compact . however , @xmath318 has no nontrivial compact subgroups ( see [ solvable]([solvable - nocpct ] ) ) . thus , the action is free . because the action is free , proper , and @xmath319 , it is easy to see that the manifold @xmath11 is a principal fiber bundle over the quotient @xmath320 ( * ? ? ? furthermore , the fiber @xmath318 of the bundle is contractible ( see [ solvable]([solvable - h = rn ] ) ) , so lemma [ trivialbundle]([trivialbundle - h = rn ] ) implies that @xmath11 homotopy equivalent to @xmath320 . therefore , the spaces @xmath11 and @xmath320 have the same homology . for the special case where @xmath320 is a homogeneous space of a solvable group , the following more detailed result describes the topology of @xmath320 , not just its homology . [ an / h = rd ] if @xmath0 is any connected subgroup of a @xmath26-connected , solvable lie group @xmath256 , then @xmath290 is diffeomorphic to the euclidean space @xmath294 , for some @xmath99 . because @xmath256 is solvable , it has a nontrivial , connected , closed , abelian , normal subgroup @xmath46 . since @xmath46 is abelian and @xmath267 is connected ( see [ solvable]([solvable - hcapl ] ) ) , we know that @xmath321 is a 1-connected abelian group ( see [ r / hsc ] ) , so it is isomorphic to some @xmath297 ( see [ abelian]([abelian - h = rn ] ) ) . we know @xmath0 is closed ( see [ solvable]([solvable - h = rn ] ) ) . also , since @xmath46 is nontrivial , we have @xmath308 , so we may assume , by induction on @xmath296 , that @xmath322 is diffeomorphic to some @xmath299 . now @xmath256 is a principal @xmath323-bundle over @xmath324 . because @xmath325 , this bundle is trivial ( see [ trivialbundle]([trivialbundle - m = rn ] ) ) : @xmath256 is @xmath323-equivariantly diffeomorphic to @xmath326 . then @xmath327 as desired . the following well - known observation puts an important restriction on the subspaces of @xmath329 that are normalized by a torus . it is an ingredient in our case - by - case analysis of all possible subgroups of @xmath121 in sections [ suflargesect ] and [ proofsect ] . [ rootdecomp ] let * @xmath330 be the set of weights of @xmath119 on @xmath331 ( in other words , the set of all positive real roots of @xmath1 ) ; * @xmath328 be a subgroup of @xmath119 ; * @xmath332 ; * @xmath333 , where the sum is over all @xmath334 , such that the restriction of @xmath194 to @xmath328 is the same as the restriction of @xmath335 to @xmath328 ; * @xmath336 , where the sum is over all @xmath334 , such that the restriction of @xmath194 to @xmath328 is not the same as the restriction of @xmath335 to @xmath328 . if @xmath337 is any @xmath338-subspace of @xmath329 normalized by @xmath328 , then @xmath339 . since @xmath340 , we know that the elements of @xmath341 are simultaneously diagonalizable ( over @xmath338 ) , so their restrictions to the invariant subspace @xmath337 are also simultaneously diagonalizable ( cf . * thms . 26 and 27 in 3.12 , pp . 167168 ) ) . thus , @xmath337 is a direct sum of weight spaces : @xmath342 for each weight @xmath343 of @xmath328 on @xmath337 , we have @xmath344 so @xmath345 and @xmath346 the conclusion follows . in this section , we prove corollary [ tess->dim>1,2 ] , an _ a priori _ lower bound on @xmath137 . on the way , we recall a result of t. kobayashi that will also be used several times in later sections , and we establish that crystallographic groups have only one end . the following theorem is essentially due to t. kobayashi . ( kobayashi assumed that @xmath0 is reductive , but h. oh and d. witte ( * ? ? ? * thm . 3.4 ) pointed out that , by using lemma [ hcanbean ] , this restriction can be eliminated . ) the proof here is based on kobayashi s original argument and the modifications of oh - witte , but uses less sophisticated topology . namely , instead of group cohomology and the spectral sequence of a covering space , we use only some basic properties of homology groups of manifolds ( including lemma [ fiberbundle ] ) . these comments also apply to theorem [ construct - tess ] . [ noncpctdim ] let @xmath0 and @xmath94 be closed , connected subgroups of @xmath1 , and assume there is a crystallographis group @xmath3 for @xmath2 , such that @xmath3 acts properly discontinuously on @xmath149 . then : 1 . [ noncpctdim - notess ] we have @xmath347 . [ noncpctdim - tess ] if @xmath348 , then @xmath349 is compact , so @xmath149 has a tessellation . by lemma [ hcanbean ] , we may assume @xmath350 . ( so @xmath351 and @xmath352 ( see [ d(h)=dimh ] ) . ) from lemma [ fiberbundle ] , we know that @xmath353 and @xmath354 have the same homology . therefore @xmath355 with equality if and only if @xmath356 is compact ( * ? ? ? 8.3.4 , p. 260 ) . similarly , we have @xmath357 combining these two statements , we conclude ( [ noncpctdim - notess ] ) that @xmath358 and , furthermore , ( [ noncpctdim - tess ] ) that equality holds if and only if @xmath349 is compact . [ noncpct - dim - notess ] let @xmath0 and @xmath94 be closed , connected subgroups of @xmath1 , such that @xmath359 . if there is a compact subset @xmath12 of @xmath119 , such that @xmath360 , then @xmath2 does not have a tessellation . suppose @xmath3 is a crystallographic group for @xmath2 . ( this will lead to a contradiction . ) because @xmath3 acts properly discontinuously on @xmath2 , the assumption on @xmath361 implies that @xmath3 also acts properly discontinuously on @xmath149 ( cf . [ proper<>mu(l ) ] ) . so theorem [ noncpctdim]([noncpctdim - notess ] ) yields a contradiction . it is easy to see that crystallographic groups are finitely generated ; we now show that they have only one end ( see [ tess->1end ] ) . [ 1enddefn ] let @xmath318 be a finite generating set for an ( infinite ) group @xmath3 . we say that @xmath3 has _ only one end _ if , for every partition @xmath362 of @xmath3 into three disjoint sets @xmath363 , @xmath364 , and @xmath12 , such that @xmath363 and @xmath364 are infinite , but @xmath12 is finite , there exists @xmath365 and @xmath366 , such that @xmath367 . ( this does not depend on the choice of the generating set @xmath318 . ) the following observation is a straightforward reformulation of definition [ 1enddefn ] ( obtained by letting @xmath368 and @xmath369 ) . [ 1endcomplement ] let @xmath318 be a finite generating set for an infinite group @xmath3 . if @xmath363 and @xmath160 are subsets of @xmath3 , such that * @xmath363 is infinite , * @xmath160 is finite , and * @xmath370 , for every @xmath366 , then the complement @xmath371 is finite . definition [ 1enddefn ] is often stated in the language of cayley graphs : the _ cayley graph _ of @xmath3 , with respect to the generating set @xmath318 , is the graph @xmath372 whose vertex set @xmath163 and edge set @xmath373 are given by : @xmath374 the group @xmath3 has only one end if and only if , for every finite subset @xmath12 of @xmath3 , the graph @xmath375 has only one infinite component . the following lemma is not difficult , but , unfortunately , we do not have a proof that is both short and elementary . [ hn = an ] if @xmath376 , then , for some @xmath377 , the conjugate @xmath378 is normalized by @xmath119 . [ codim>1 ] if @xmath379 , and @xmath2 is not compact , then @xmath2 does not have a tessellation . it suffices to show that @xmath0 is a cartan - decomposition subgroup of @xmath1 ( see [ cds->notess ] ) . we may assume , without loss of generality , that @xmath91 ( see [ hcanbean ] ) ; then @xmath380 ( see [ d(h)=dimh ] and [ d(g ) ] ) . a theorem of b. kostant ( * ? ? ? 5.1 ) implies that @xmath120 is a cartan - decomposition subgroup , so we may assume @xmath381 ; then @xmath382 . therefore @xmath383 hence @xmath384 , so , from lemma [ hn = an ] , we see that , after replacing @xmath0 by a conjugate subgroup , we may assume that @xmath0 is normalized by @xmath119 . then , letting @xmath385 and @xmath386 in ( [ rootdecomp ] ) , we see that @xmath387 . since @xmath384 , we have @xmath388 , so this implies that @xmath389 ; therefore @xmath0 contains @xmath119 . since @xmath119 is a cartan - decomposition subgroup ( see [ aiscds ] ) , this implies @xmath0 is a cartan - decomposition subgroup , as desired . a topological space @xmath11 is _ connected at @xmath390 _ if every compact subset @xmath391 is contained in a compact subset @xmath392 , such that the complement @xmath393 is connected . [ tess->1end ] if @xmath3 is a crystallographic group for @xmath2 , then @xmath3 is finitely generated and has only one end . assume , without loss of generality , that @xmath91 ( see [ hcanbean ] ) . then @xmath0 is torsion free , so @xmath3 must act freely on @xmath2 ; therefore @xmath15 is a compact manifold ( rather than an orbifold ) . because @xmath3 is essentially the fundamental group of @xmath116 ( specifically , @xmath394 ) , and the fundamental group of any compact manifold is finitely generated ( * ? ? ? * thm . 6.16 , p. 95 ) , we know that @xmath3 is finitely generated . from the iwasawa decomposition @xmath92 , we see that @xmath2 is homeomorphic to @xmath395 , and proposition [ an / h = rd ] asserts that @xmath396 is homeomorphic to @xmath294 , for some @xmath99 . obviously , we must have @xmath397 , and we may assume @xmath2 is not compact ( otherwise , @xmath3 is finite , so the desired conclusion is obvious ) , so corollary [ codim>1 ] implies that @xmath398 . thus , we conclude that @xmath2 is connected at @xmath390 . to complete the proof , we use a standard argument ( cf . * @xmath399 , p. 5 ) ) to show that , because @xmath2 is connected at @xmath390 and @xmath15 is compact , the group @xmath3 has only one end . to begin , note that there is a compact subset @xmath391 of @xmath2 , such that @xmath400 . let @xmath401 ( cf . * ( ii ) , p. 195 ) ) . because @xmath3 acts properly discontinuously on @xmath2 , we know that @xmath402 is finite ; let @xmath318 be a finite generating set for @xmath3 , such that @xmath403 suppose @xmath404 , with @xmath405 and @xmath406 . ( we wish to show there exist @xmath365 and @xmath407 , such that @xmath408 ; this establishes that @xmath3 has only one end . ) because @xmath2 is connected at @xmath390 , there is a compact subset @xmath392 of @xmath2 , containing @xmath409 , such that @xmath410 is connected . because @xmath411 , we have @xmath412 because @xmath3 acts properly discontinuously on @xmath2 , we know @xmath413 and @xmath414 are closed ( and neither is contained in @xmath392 ) , so connectivity implies that @xmath415 : there exist @xmath365 and @xmath416 , such that @xmath417 . let @xmath418 ; then @xmath365 , @xmath419 , and @xmath420 so @xmath421 , as desired . [ tess->misswall ] assume @xmath423 . let * @xmath146 and @xmath424 be the two walls of @xmath154 , and * @xmath3 be a crystallographic group for @xmath2 . if @xmath0 is not compact , then there exists @xmath425 , such that , for every compact subset @xmath12 of @xmath119 , the intersection @xmath426 is finite . : ( a ) @xmath427 can not be on both sides of @xmath428 , because @xmath3 has only one end . ( b ) therefore , @xmath427 stays away from @xmath429 . ] suppose there is a compact subset @xmath12 of @xmath119 , such that each of @xmath430 and @xmath431 is infinite . ( this will lead to a contradiction . ) let @xmath318 be a ( symmetric ) finite generating set for @xmath3 ( see [ tess->1end ] ) . we may assume @xmath12 is so large that @xmath432 for every @xmath37 ( see [ bddchange ] ) . we may also assume that @xmath12 is convex and symmetric . because @xmath3 acts properly on @xmath2 , there is a compact subset @xmath391 of @xmath119 , such that @xmath433 ( see [ proper<>mu(l ) ] ) . furthermore , we may assume that @xmath434 . let * @xmath435 , * @xmath436 be the union of all the connected components of @xmath11 that contain a point of @xmath146 , and * @xmath437 . then @xmath363 is infinite ( because @xmath438 is infinite ) . also , for any @xmath439 and @xmath440 , we have @xmath441 , so @xmath442 . since @xmath3 has only one end ( see [ tess->1end ] ) , this implies @xmath443 is finite ( see [ 1endcomplement ] ) . because @xmath444 is infinite , we conclude that @xmath445 . because @xmath428 separates @xmath146 from @xmath424 , and every connected component of @xmath436 contains a point of @xmath146 , we conclude that @xmath446 . this contradicts the fact that @xmath447 . [ tess->misshk ] assume @xmath423 . let 1 . @xmath146 and @xmath424 be the two walls of @xmath154 ; and 2 . @xmath94 and @xmath95 be closed , connected , nontrivial subgroups of @xmath1 , such that @xmath448 for @xmath449 . if @xmath0 is not compact , then any crystallographic group for @xmath2 acts properly discontinuously on either @xmath149 or @xmath450 . suppose @xmath3 acts properly discontinuously on * neither * @xmath149 * nor * @xmath450 . ( this will lead to a contradiction . ) from proposition [ proper<>mu(l ) ] , we know there is a compact subset @xmath12 of @xmath119 , such that each of @xmath451 and @xmath452 is infinite . then , since @xmath453 , we may assume ( by enlarging @xmath12 ) that each of @xmath438 and @xmath431 is infinite . this contradicts the conclusion of proposition [ tess->misswall ] . [ tess->dim>1,2 ] assume @xmath423 . let 1 . @xmath146 and @xmath424 be the two walls of @xmath154 ; and 2 . @xmath94 and @xmath95 be closed , connected , nontrivial subgroups of @xmath1 ; such that @xmath448 for @xmath449 . if @xmath2 has a tessellation , and @xmath0 is not compact , then @xmath454 the desired conclusion is obtained by combining corollary [ tess->misshk ] with theorem [ noncpctdim]([noncpctdim - notess ] ) . for @xmath455 , there does not exist a connected subgroup @xmath456 , such that @xmath453 ( cf . [ sl3-b+ ] ) . thus , corollary [ tess->dim>1,2 ] does not provide a lower bound on @xmath422 in this case . although the following conjecture does not seem to have been stated previously in the literature , it is perhaps implicit in @xcite . [ d(h)=1->notess ? ] if @xmath457 , then @xmath2 does not have a tessellation . in this section , we establish that the conjecture is valid in two cases : if either @xmath458 ( see [ 1drank2 ] ) or @xmath1 is almost simple ( see [ 1dsimple ] ) . each of these illustrates a general theorem : for groups of real rank two , the conjecture follows from a theorem of y. benoist and f. labourie that is based on differential geometry ; h. oh and d. witte observed that , for simple groups , the conjecture follows from a theorem of g. a. margulis that is based on unitary representation theory . the following example is the only case of conjecture [ d(h)=1->notess ? ] that is needed in later sections . ( it is used in the proof of theorem [ sl3->notess ] . ) because @xmath459 and @xmath460 is almost simple , this example is covered both by the theorem of benoist - labourie and by the theorem of margulis , but it would be interesting to have an easy proof . [ wallinsl3 ] assume @xmath461 , for @xmath132 , @xmath462 , or @xmath463 , and let @xmath464 then @xmath149 does not have a tessellation . let us begin with an easy observation . [ 1drank1 ] if @xmath457 and @xmath28 , then @xmath2 does not have a tessellation . we may assume @xmath91 ( see [ hcanbean ] ) . from ( [ rrank1-cds ] ) , we know that @xmath0 is a cartan - decomposition subgroup , so lemma [ cds->notess ] implies that @xmath2 must be compact ; thus , the trivial group @xmath185 is a crystallographic group for @xmath2 . however , since @xmath465 ( see [ d(g ) ] ) , and @xmath185 acts properly discontinuously on @xmath466 , this contradicts theorem [ noncpctdim]([noncpctdim - notess ] ) . the proof of lemma [ 1drank1 ] shows that the dimension of every connected , cocompact subgroup of @xmath1 is at least @xmath467 . this is a result of m. goto and h.c . wang ( * ? ? ? * ( 1.2 ) , p. 263 ) . [ center->notess ] if @xmath0 is reductive and contains an element of @xmath119 in its center , then @xmath2 does not have a tessellation . to illustrate the idea behind theorem [ center->notess ] , we give a direct proof of the following special case ( under an additional technical assumption ( see [ 1da->notesspf - h = z ] ) ) , which is sufficient for our needs . ( note that condition ( [ 1da->notesspf - h = z ] ) is satisfied for the subgroup @xmath94 of proposition [ wallinsl3 ] . ) benoist and labourie prove the general case by using a slightly different 1-form in place of the form @xmath335 that we define in step [ 1da->notesspf - omega ] . [ 1da->notess ] if @xmath0 is a one - dimensional subgroup of @xmath119 , then @xmath2 does not have a tessellation . suppose @xmath3 is a crystallographic group for @xmath2 . ( this will lead to a contradiction . ) by passing to finite - index subgroups , we may assume that @xmath1 and @xmath0 are connected . we will construct a volume form @xmath468 on @xmath15 that is exact : @xmath469 . the integral of @xmath468 over @xmath15 is the volume of @xmath15 , which is obviously not @xmath186 , but stokes theorem implies that the integral of any exact form over a closed manifold is @xmath186 . this is a contradiction . [ 1da->notesspf - omega ] construction of a certain @xmath470-form @xmath471 on @xmath116 . let @xmath472 be the orthogonal complement to @xmath270 , under the killing form . then @xmath473 is an @xmath474-invariant subspace of @xmath213 , such that @xmath475 and @xmath476 . let @xmath477 be the projection with kernel @xmath473 , and let @xmath335 be the corresponding left - invariant @xmath270-valued 1-form on @xmath1 . the space @xmath1 is a homogeneous principal @xmath0-bundle over @xmath2 . it is well known ( * ? ? ? ii.11.1 , p. 103 ) that @xmath335 is the connection form of a @xmath1-invariant connection on this bundle , and that the curvature form @xmath478 of this connection is given by @xmath479 \bigr)$ \qquad for $ x , y \in { \mathfrak{\lowercase{m}}}$.}\ ] ] also , because @xmath0 is abelian , the structure equation ( * ? ? ? ii.5.2 , p. 77 ) implies @xmath480 by identifying the @xmath26-dimensional lie algebra @xmath270 with @xmath338 , we may think of @xmath335 and @xmath478 as ordinary ( that is , @xmath338-valued ) differential forms . because @xmath335 and ( hence ) @xmath478 are left - invariant , they determine well - defined forms @xmath481 and @xmath482 on @xmath483 . ( note that @xmath481 is a connection form on the principal @xmath0-bundle @xmath353 over @xmath15 , and the curvature form of this connection is @xmath482 . ) because @xmath0 is abelian , we have @xmath484 for all @xmath485 ( cf . ii.5.1(c ) , p. 76 ) ) , so the horizontal form @xmath482 determines a well - defined form @xmath471 on the base space @xmath15 . construction , for a certain @xmath486 , of a certain @xmath486-form @xmath487 on @xmath15 . identifying @xmath270 with @xmath338 provides an ordering on the weights of @xmath270 . let * @xmath488 be the @xmath186 weight space of @xmath489 on @xmath473 ; * @xmath490 be the sum of the positive weight spaces of @xmath491 on @xmath473 ; * @xmath492 be the sum of the negative weight spaces of @xmath491 on @xmath473 ; * @xmath493 ; and * @xmath139 be a nontrivial left - invariant @xmath486-form on @xmath1 , such that @xmath494 because @xmath139 is left - invariant , and @xmath495 , the form @xmath139 determines a well - defined differential form @xmath487 on @xmath15 . ( we remark that , because @xmath496 , condition ( [ mu(h+m ) ] ) implies that the form @xmath139 is unique up to a scalar multiple . ) [ 1da->notesspf - volume ] for a certain @xmath497 , the wedge product @xmath498 is a volume form on @xmath15 . let @xmath499 . it suffices to show that the restriction of @xmath478 to @xmath500 is a symplectic form . it is obviously skew , so we need only show that it is nondegenerate . thus , letting @xmath501 \bigr ) = 0 \,\ } & \mbox{{{\upshape(}\ref{omega(m , m)}{\upshape ) } } } \\ & = \{\ , x \in { \mathfrak{\lowercase{m}}}_+ \oplus { \mathfrak{\lowercase{m}}}_- \mid [ x , { \mathfrak{\lowercase{m}}}_+ \oplus { \mathfrak{\lowercase{m}}}_- ] \subset { \mathfrak{\lowercase{m } } } \,\ } & \mbox{(definition of~$\omega$ ) } \\ & = \{\ , x \in { \mathfrak{\lowercase{m}}}_+ \oplus { \mathfrak{\lowercase{m}}}_- \mid \langle { \mathfrak{\lowercase{h } } } \mid [ x , { \mathfrak{\lowercase{m}}}_+ \oplus { \mathfrak{\lowercase{m}}}_- ] \rangle_{\text{killing } } = 0 \,\ } & \mbox{(${\mathfrak{\lowercase{m } } } = { \mathfrak{\lowercase{h}}}^\perp$ ) } , \end{aligned}\ ] ] we wish to show @xmath502 . there is no harm in passing to the complexification @xmath503 of @xmath213 . let @xmath504 be a cartan subalgebra of @xmath503 that contains @xmath270 . because @xmath504 preserves the killing form , centralizes @xmath270 , and normalizes @xmath500 , we know that @xmath505 is @xmath506-invariant ; thus , @xmath505 is a sum of root spaces . suppose there exists a nonzero element @xmath507 of @xmath505 , such that @xmath507 belongs to some root space @xmath508 . ( this will lead to a contradiction . ) there exists @xmath509 , such that @xmath510 \rangle_{\text{killing } } = \alpha(t)\ ] ] for all @xmath511 ( * ? ? ? 8.3(c ) , p. 37 ) . because @xmath512 , we have @xmath513 . then @xmath514 , so @xmath515 . we now know that @xmath516 and @xmath517 , so @xmath518 \rangle_{\text{killing } } = 0 $ ] . we therefore conclude , from the definition of @xmath519 , that @xmath520 . this is a contradiction . the form @xmath471 is exact : we may write @xmath521 . let @xmath522 be the connection form of a flat connection on @xmath483 over @xmath15 . ( since the principal bundle is trivial ( see [ trivialbundle ] ) , it is obvious that there is a flat connection . ) for any vector field @xmath507 on @xmath15 , let @xmath523 be the lift of @xmath507 to a vector field on @xmath353 that is horizontal with respect to the flat connection @xmath522 . since @xmath0 is abelian , there is a well - defined 1-form @xmath524 on @xmath525 given by @xmath526 then @xmath527 \bigr ) \right ) & \mbox{(definition of~$d$ ) } \\ & = \frac{1}{2 } \left ( \widetilde x \bigl ( \overline{\omega}(\widetilde y ) \bigr ) - \widetilde y \bigl ( \overline{\omega}(\widetilde x ) \bigr ) - \overline{\omega } \bigl ( \widetilde{[x , y ] } \bigr ) \right ) & \mbox{(definition of~$\breve{\phi}$ ) } \\ & = \frac{1}{2 } \left ( \widetilde x \bigl ( \overline{\omega}(\widetilde y ) \bigr ) - \widetilde y \bigl ( \overline{\omega}(\widetilde x ) \bigr ) - \overline{\omega } \bigl ( [ \widetilde x,\widetilde y ] \bigr ) \right ) & \mbox{($\breve{\omega}_0 $ is flat ) } \\ & = d \overline{\omega}(\widetilde x , \widetilde y ) & \mbox{(definition of~$d$ ) } \\ & = \overline{\omega}(\widetilde x , \widetilde y ) & \mbox{{{\upshape(}\ref{structeq}{\upshape ) } } } \\ & = \breve{\omega}(x , y ) & \mbox{(definition of~$\breve{\omega}$ ) } . \end{aligned}\ ] ] for simplicity , assume that @xmath528 [ 1da->notesspf - closed ] we have @xmath529 . let * @xmath530 be a basis of @xmath490 , and * @xmath531 be the dual basis of @xmath492 , with respect to the symplectic form @xmath478 on @xmath500 . thus , @xmath532 . let @xmath533 be a basis of @xmath534 , write @xmath535 = \sum_\ell a_{j , k}^\ell z_\ell \pmod { { \mathfrak{\lowercase{h } } } + { \mathfrak{\lowercase{m}}}_+ + { \mathfrak{\lowercase{m}}}_- } , \ ] ] and define @xmath536 = \sum_{j,\ell } a_{j , j}^\ell z_\ell .\ ] ] [ 1da->notesspf - closed - windep ] @xmath537 is independent of the choice of the basis @xmath538 of @xmath490 ( with the understanding that @xmath539 must be the dual basis of @xmath492 ) . let @xmath540 for some @xmath541 with @xmath542 . then @xmath543 so @xmath544 + [ x'_2,y'_2 ] + \sum_{j \ge 3 } [ x'_j , y'_j ] \\ & = \bigl ( [ x_1,y_1 ] + ( b / a)[x_2,y_1 ] \bigr ) + \bigl ( [ x_2,y_2 ] - ( b / a ) [ x_2,y_1 ] \bigr ) + \sum_{j \ge 3 } [ x_j , y_j ] \\ & = \sum_j [ x_j , y_j ] \\ & = w . \end{aligned}\ ] ] since @xmath530 can be transformed into any other basis of @xmath490 by a sequence of elementary operations as in ( [ elemop ] ) , we conclude that @xmath537 is independent of the choice of basis , as desired . we have @xmath545 . substep [ 1da->notesspf - closed - windep ] implies that @xmath537 is centralized by @xmath546 , so @xmath537 is in the center of @xmath547 . let @xmath194 be a cartan involution of @xmath1 with @xmath548 for @xmath549 . substep [ 1da->notesspf - closed - windep ] implies @xmath550 . ( from substep [ 1da->notesspf - closed - windep ] , we see that @xmath537 depends only on @xmath270 and the chosen identification of @xmath270 with @xmath338 ; @xmath194 reverses the choice of identification . ) thus , @xmath537 is a hyperbolic element of the center of @xmath547 . by assumption [ 1da->notesspf - h = z ] , this implies @xmath545 , as desired . completion of step [ 1da->notesspf - closed ] . let @xmath551 be the basis of @xmath552 dual to @xmath553 we may assume @xmath554 . then , because @xmath555 \subset { \mathfrak{\lowercase{m}}}_-$ ] and @xmath556 \subset { \mathfrak{\lowercase{m}}}_+$ ] , we have @xmath557 so @xmath558 therefore @xmath559 from the choice of @xmath560 , we have @xmath561 , so @xmath562 and @xmath563 hence @xmath564 since @xmath545 , we have @xmath565 , so the desired conclusion follows . [ 1da->notesspf - volexact ] @xmath566 is exact . we have @xmath567 a contradiction . from step [ 1da->notesspf - volume ] , we know that @xmath568 on the other hand , step [ 1da->notesspf - volexact ] implies that this integral is zero . this is a contradiction . [ oppinv ] let @xmath569 be the opposition involution in @xmath154 ; that is , for @xmath570 , @xmath571 is the unique element of @xmath154 that is conjugate ( under an element of the weyl group ) to @xmath572 . thus , for all @xmath573 , we have @xmath574 see ( [ sl3-oppinv ] ) for an explicit description of the opposition involution in @xmath455 . for some groups , such as @xmath51 , we have @xmath575 for all @xmath573 ( see [ mu(h-1 ) ] ) ; in such a case , the opposition involution is simply the identity map on @xmath154 . [ 1drank2 ] if @xmath457 and @xmath458 , then @xmath2 does not have a tessellation . suppose @xmath3 is a crystallographic group for @xmath2 . ( this will lead to a contradiction . ) from ( [ 1drank1 ] ) , we know @xmath423 . let @xmath146 and @xmath424 be the two walls of @xmath154 and , for @xmath425 , let @xmath576 . because @xmath429 is a ray ( that is , a one - parameter semigroup ) , it is clear that @xmath456 is a subgroup of @xmath119 . from proposition [ tess->misswall ] , we know that there is some @xmath577 , such that @xmath578 for every compact subset @xmath12 of @xmath119 . since @xmath579 , we have @xmath580 , so this implies that @xmath581 for every compact subset @xmath12 of @xmath119 . also , because @xmath582 , we have @xmath583 , so @xmath584 therefore @xmath585 for every compact subset @xmath12 of @xmath119 . hence , corollary [ proper<>mu(l ) ] implies that @xmath3 acts properly discontinuously on @xmath586 . then , because @xmath587 ( see [ d(h)=dimh ] ) , theorem [ noncpctdim]([noncpctdim - tess ] ) implies that @xmath586 has a tessellation . this contradicts corollary [ 1da->notess ] . [ tempered - def ] the subgroup @xmath0 is _ tempered _ in @xmath1 if there exists a ( positive ) function @xmath588 ( with respect to a left - invariant haar measure on @xmath0 ) , such that , for every unitary representation @xmath589 of @xmath1 , either * @xmath590 for all @xmath591 and all @xmath118-fixed vectors @xmath592 and @xmath343 ; or * some nonzero vector is fixed by every element of @xmath593 . for many examples of tempered subgroups of simple lie groups , see @xcite . [ margulistempered ] if @xmath0 is noncompact and tempered , then @xmath2 does not have a tessellation . suppose @xmath3 is a crystallographic group for @xmath2 . ( this will lead to a contradiction . ) to simplify the notation somewhat ( and because this is the only case we need ) , let us assume @xmath239 is a one - parameter subgroup of @xmath1 . because @xmath1 and @xmath3 are unimodular ( recall that @xmath1 is semisimple ( see [ standing ] ) and @xmath3 is discrete ) , there is a @xmath1-invariant measure ( in fact , a @xmath1-invariant volume form ) on the homogeneous space @xmath483 ( * ? ? ? * lem 1.4 , p. 18 ) . thus , the natural representation @xmath589 of @xmath1 on @xmath594 , defined by @xmath595 is unitary . because @xmath0 is noncompact , and acts properly on @xmath353 , we know that any compact subset of @xmath353 has infinitely many pairwise - disjoint translates ( all of the same measure ) , so we see that @xmath596 therefore , @xmath589 has no nonzero @xmath1-invariant vectors , so , because @xmath0 is tempered , we know that there is some @xmath597 , such that @xmath598 for all @xmath599 and all @xmath118-invariant @xmath600 . because @xmath15 is compact , there is a compact subset @xmath12 of @xmath1 , such that @xmath601 ; let @xmath602 be the image of @xmath12 in @xmath353 . from the choice of @xmath12 , we know , for each @xmath603 , that there is some @xmath604 , such that @xmath605 because @xmath606 is a compact subset of @xmath353 , there is a positive , continuous function @xmath592 on @xmath353 with compact support , such that @xmath607$,}\ ] ] and , by averaging over @xmath118 , we may assume that @xmath592 is @xmath118-invariant . fix some large @xmath608 . because @xmath609 is compact , and @xmath353 has infinite volume ( see [ margulistemperedpf - notcpct ] ) , there is some @xmath118-invariant continuous function @xmath610 on @xmath353 , such that @xmath611 , @xmath612 and @xmath613 we have @xmath614 however , because @xmath597 , we know that @xmath615 . this is a contradiction . we state the following well - known result of representation theory without proof . as is explained in ( * , p. 140 ) , it can be obtained by combining work of r. howe ( * ? ? ? * cor . 7.2 and 7 ) and m. cowling ( * ? ? ? ( the assumption that @xmath616 can be relaxed : it suffices to assume that @xmath1 is not locally isomorphic to @xmath617 or @xmath618 . ) fix any matrix norm @xmath619 on @xmath1 ; for example , we may let @xmath620 . [ expdecay ] if @xmath1 is almost simple , and @xmath616 , then there are constants @xmath621 and @xmath622 such that , for every unitary representation @xmath589 of @xmath1 , either 1 . @xmath623 for all @xmath33 and all @xmath624-fixed vectors @xmath592 and @xmath343 ; or 2 . some nonzero vector is fixed by every element of @xmath593 . although we can not prove theorem [ expdecay ] here , we present an elementary proof of the following related result , which , unfortunately , is qualitative , rather than quantitative . on the other hand , this simple result applies to all vectors , not only the @xmath118-fixed vectors , and it applies to all semisimple groups , including @xmath617 and @xmath618 . it was first proved by r. howe and c. moore ( * ? ? ? 5.1 ) and ( independently ) r. zimmer ( * ? ? ? [ matcoeffs->0 ] if * @xmath1 is connected and almost simple ; * @xmath589 is a unitary representation of @xmath1 on a hilbert space @xmath625 , such that no nonzero vector is fixed by @xmath593 ; and * @xmath626 is a sequence of elements of @xmath1 , such that @xmath627 , then @xmath628 , for every @xmath629 . [ matcoeffs->0-a ] assume @xmath630 . by passing to a subsequence , we may assume @xmath631 converges weakly , to some operator @xmath373 ; that is , @xmath632 let @xmath633 for @xmath634 , we have @xmath635 so @xmath636 . therefore @xmath637 . we have @xmath638 so the same argument , with @xmath639 in the place of @xmath373 and @xmath640 in the place of @xmath641 , shows that @xmath642 . because @xmath589 is unitary , we know that @xmath631 is normal ( that is , commutes with its adjoint ) for every @xmath643 ; thus , the limit @xmath373 is also normal : we have @xmath644 . therefore @xmath645 so @xmath646 . thus , @xmath647 by passing to a subsequence of @xmath626 , we may assume @xmath648 ( see [ horosubseq ] ) . then @xmath649 , so @xmath650 . hence , for all @xmath651 , we have @xmath652 as desired . the general case . from the cartan decomposition @xmath101 , we may write @xmath653 , with @xmath654 and @xmath655 . because @xmath118 is compact , we may assume , by passing to a subsequence , that @xmath656 and @xmath657 converge : say , @xmath658 and @xmath659 . then @xmath660 by case [ matcoeffs->0-a ] . the following example illustrates lemma [ horosubseq ] . let @xmath455 , define @xmath94 as in proposition [ wallinsl3 ] , and suppose @xmath626 is some sequence of elements of @xmath94 , such that @xmath661 . we may write @xmath662 where @xmath663 . by passing to a subsequence , we may assume that either @xmath664 or @xmath665 . if @xmath664 , then , in the notation of ( [ horodefn ] ) , we have @xmath666 if @xmath665 , then @xmath162 and @xmath667 are interchanged . thus , in either case , @xmath337 is the sum of two root spaces of @xmath213 , and @xmath668 is the sum of the two opposite root spaces . it is not difficult to see that @xmath669 $ ] is the sum of @xmath670 and the remaining two root spaces . therefore , we have @xmath671 , so @xmath672 . [ horosubseq ] if @xmath1 and @xmath626 are as in theorem [ matcoeffs->0 ] , and @xmath673 , then , after replacing @xmath626 by a subsequence , we have @xmath674 , where @xmath162 and @xmath667 are defined in ( [ horodefn ] ) . by passing to a subsequence , we may assume @xmath626 is contained in a single weyl chamber , which we may take to be @xmath154 . then , by passing to a subsequence yet again , we may assume , for every positive real root @xmath675 , that either @xmath676 or @xmath677 is bounded . let * @xmath330 be the set of positive real roots ; * @xmath678 be the set of positive simple real roots ; * @xmath679 ; and * @xmath680 . there is a compact subset @xmath12 of @xmath119 , such that @xmath681 , so , because @xmath661 , we know that @xmath328 is not trivial . for each real root @xmath675 , let @xmath682 be the corresponding root subspace of @xmath213 . then @xmath683 now , for @xmath684 , we have @xmath685 if and only if @xmath675 is in the linear span of @xmath686 . thus , we see that @xmath337 is precisely the unipotent radical of the standard parabolic subalgebra @xmath687 corresponding to the set @xmath686 of simple roots ( * ? ? ? * , pp . 8586 ) . similarly , @xmath668 is the unipotent radical of the opposite parabolic algebra @xmath688 . because @xmath1 is simple , the unipotent radicals of opposite parabolics generate @xmath213 ( * ? ? ? * prop . 4.11 , p. 89 ) , so @xmath689 , as desired . [ 1d->tempered ] assume @xmath1 is simple , and @xmath616 . if @xmath0 is a one - parameter subgroup of @xmath121 , then either 1 . @xmath0 is tempered ; or 2 . @xmath690 . write @xmath239 . from the real jordan decomposition ( [ jordandecomp ] ) , we may assume , after replacing @xmath0 by a conjugate subgroup , that @xmath691 , where @xmath692 is a hyperbolic one - parameter subgroup , and @xmath693 is a unipotent one - parameter subgroup , such that @xmath694 and @xmath695 commute with each other . we may assume @xmath696 , so @xmath694 is nontrivial . since the growth of the hyperbolic one - parameter subgroup @xmath694 is exponential , while that of the unipotent one - parameter subgroup @xmath695 is polynomial , there is some @xmath697 , such that @xmath698 for large @xmath599 . since the function @xmath699 is in @xmath700 , it follows from theorem [ expdecay ] that @xmath0 is tempered , as desired . [ 1dinn->notess ] if @xmath457 and @xmath690 , then @xmath2 does not have a tessellation . we have @xmath701 ( see [ d(h)=dimh ] ) , so @xmath0 is a connected , one - dimensional , unipotent subgroup . hence , the jacobson - morosov lemma ( * ? ? ? 9.7.4 , p. 432 ) implies that there exists a connected , closed subgroup @xmath94 of @xmath1 , such that @xmath94 contains @xmath0 , and @xmath94 is locally isomorphic to @xmath41 . then @xmath0 is a cartan - decomposition subgroup of @xmath94 ( see [ rrank1-cds ] ) , so there is a compact subset @xmath12 of @xmath119 , such that @xmath702 ( see [ bddchange ] ) . also , we have @xmath703 . therefore , theorem [ noncpct - dim - notess ] applies . [ 1dsimple ] if @xmath457 and @xmath1 is simple , then @xmath2 does not have a tessellation . we may assume @xmath91 ( see [ hcanbean ] ) , so @xmath701 ( see [ d(h)=dimh ] ) . * if @xmath704 , then lemma [ 1drank1 ] applies . * if @xmath690 , then lemma [ 1dinn->notess ] applies . * if @xmath616 and @xmath696 , then corollary [ 1d->tempered ] implies that @xmath0 is tempered , so theorem [ margulistempered ] applies . y. benoist ( * ? ? ? 1 ) and g.a . margulis ( unpublished ) proved ( independently ) that @xmath707 does not have a tessellation . using benoist s method , h. oh and d. witte ( * ? ? ? * prop . 1.10 ) generalized this result by replacing @xmath41 with any closed , connected subgroup @xmath0 , such that neither @xmath0 nor @xmath708 is compact . the same argument applies even if @xmath338 replaced with either @xmath462 or @xmath463 . however , the proof of benoist ( which applies in a more general context ) relies on a somewhat lengthy argument to establish one particular lemma . here , we adapt benoist s method to obtain a short proof of theorem [ sl3->notess ] that avoids any appeal to the lemma . [ sl3-oppinv ] assume @xmath461 , for @xmath132 , @xmath462 , or @xmath463 . * let @xmath569 be the opposition involution in @xmath154 ( see [ oppinv ] ) ; * let @xmath709 . more concretely , we have @xmath710 [ 1dinsl3 ] if @xmath461 and @xmath457 , then @xmath2 does not have a tessellation . since @xmath423 , the desired conclusion follows from corollary [ 1drank2 ] ; since @xmath1 is simple , it also follows from corollary [ 1dsimple ] . however , we give a proof that requires only the special case described in proposition [ wallinsl3 ] , rather than the full strength of ( [ 1drank2 ] ) or ( [ 1dsimple ] ) . suppose @xmath3 is a crystallographic group for @xmath2 . ( this will lead to a contradiction . ) let @xmath146 and @xmath424 be the two walls of @xmath154 . from proposition [ tess->misswall ] , we know that there exists @xmath425 , such that @xmath711 is finite , for every compact subset @xmath12 of @xmath119 . because @xmath712 , we have @xmath713 . on the other hand , @xmath569 interchanges @xmath146 and @xmath424 . thus , the preceding paragraph implies that @xmath714 is finite , for every compact subset @xmath12 of @xmath119 . for @xmath94 as in ( [ sl3wallsubgrp ] ) , we have @xmath715 , so the conclusion of the preceding paragraph implies that @xmath3 acts properly discontinuously on @xmath149 ( see [ proper<>mu(l ) ] ) . now theorem [ noncpctdim]([noncpctdim - tess ] ) implies @xmath349 is compact ; thus , @xmath149 has a tessellation . this contradicts proposition [ wallinsl3 ] . for completeness , we include the proof of the following simple proposition . [ sl3-b+ ] assume @xmath461 . if @xmath0 is a closed , connected subgroup of @xmath121 with @xmath716 , then @xmath717 . since @xmath91 and @xmath718 , it is easy to construct a continuous , proper map @xmath719 \times { \mathord{\mathbb{r}}}^+ \to h$ ] such that @xmath720 , for all @xmath721 ( cf.figure [ constructphifig ] ) . for example , choose two linearly independent elements @xmath215 and @xmath722 of @xmath270 , and define @xmath723 . ] if we identify @xmath119 with its lie algebra @xmath724 , then @xmath154 is a convex cone in @xmath724 and the opposition involution @xmath569 is the reflection in @xmath154 across the ray @xmath725 . thus , for any @xmath570 , the points @xmath156 and @xmath726 are on opposite sides of @xmath725 , so any continuous curve in @xmath154 from @xmath156 to @xmath726 must intersect @xmath725 . in particular , for each @xmath721 , the curve @xmath727 from @xmath728 to @xmath729 must intersect @xmath725 . thus , we see , from an elementary continuity argument , that @xmath730 \times { \mathord{\mathbb{r}}}^+ \bigr ) \bigr]$ ] contains @xmath725 . therefore , @xmath725 is contained in @xmath428 . : @xmath428 is on both sides of @xmath725 , so it must contain @xmath725 . ] : @xmath427 stays away from @xmath725 , because @xmath717 . also , half of @xmath427 is on each side of @xmath725 , because @xmath580 . this contradicts the fact that @xmath3 has only one end . ] suppose @xmath3 is a crystallographic group for @xmath2 . ( this will lead to a contradiction . ) we may assume @xmath91 ( see [ hcanbean ] ) . let @xmath318 be a ( symmetric ) finite generating set for @xmath3 , and choose a compact , convex , symmetric subset @xmath12 of @xmath119 so large that @xmath731 for every @xmath37 ( see [ bddchange ] ) . from lemma [ 1dinsl3 ] , we know that @xmath718 , so proposition [ sl3-b+ ] implies that @xmath717 . then , because @xmath3 acts properly on @xmath2 , we conclude that @xmath732 is finite ( see [ proper<>mu(l ) ] ) . since @xmath139 is a proper map , this implies that @xmath733 is finite . let @xmath363 and @xmath364 be the two components of @xmath734 . because @xmath712 , we know that @xmath713 . then , because @xmath569 interchanges @xmath363 and @xmath364 , we conclude that @xmath735 . therefore , @xmath736 and @xmath737 have the same cardinality , so they must both be infinite . so @xmath738 because @xmath3 has only one end ( see [ tess->1end ] ) , this implies there exist @xmath739 such that @xmath740 for some @xmath440 . then @xmath741 , @xmath742 , and @xmath743 using the fact that @xmath12 is symmetric and the fact that @xmath12 contains the identity element @xmath185 , we conclude that @xmath744 therefore @xmath745 intersects both @xmath363 and @xmath364 . since @xmath725 separates @xmath363 from @xmath364 , and @xmath12 is connected , this implies that @xmath745 intersects @xmath725 ; hence @xmath746 . this contradicts the fact that @xmath747 ( see [ sl3->notesspf - gammaf ] ) . from this point on , we focus almost entirely on @xmath73 and @xmath52 . ( the only exception is that some of the examples constructed in section [ existencesect ] are for other groups . ) in this section , we define the group @xmath750 , which allows us to provide a fairly unified treatment of @xmath73 and @xmath52 in later sections . * we use @xmath751 to denote either @xmath338 or @xmath462 . * let @xmath752 , so @xmath753 . * we use @xmath754 to denote the purely imaginary elements of @xmath751 , so @xmath755 * for @xmath756 , there exist unique @xmath757 and @xmath758 , such that @xmath759 . ( * warning : * in our notation , the imaginary part of @xmath760 is @xmath761 , _ not _ @xmath762 . ) * for @xmath756 , we use @xmath763 to denote the conjugate @xmath764 of @xmath592 . ( if @xmath132 , then @xmath765 . ) * for a row vector @xmath766 , or , more generally , for any matrix @xmath767 with entries in @xmath751 , we use @xmath768 to denote the conjugate - transpose of @xmath767 . [ sufdefn ] for @xmath769 we define @xmath770 and @xmath771 then : * @xmath772 is a realization of @xmath73 , * @xmath773 is a realization of @xmath52 , and * @xmath774 is the lie algebra of @xmath750 . we choose * @xmath119 to consist of the diagonal matrices in @xmath750 that have nonnegative real entries , * @xmath120 to consist of the upper - triangular matrices in @xmath750 with only @xmath26 s on the diagonal , and * @xmath775 . a straightforward matrix calculation shows that the lie algebra of @xmath121 is @xmath776 [ tworows ] from , we see that the first two rows of any element of @xmath777 are sufficient to determine the entire matrix . in fact , it is also not necessary to specify the last entry of the second row of the matrix . [ d(su2 ) ] from ( [ d(g ) ] ) and ( [ suf - an ] ) , we see that @xmath778 . because @xmath120 is simply connected and nilpotent , the exponential map is a diffeomorphism from @xmath331 to @xmath120 ( indeed , its inverse , the logarithm map , is a polynomial ( * ? ? ? * thm . 8.1.1 , p. 107 ) , so each element of @xmath120 has a unique representation in the form @xmath779 with @xmath780 . thus , each element @xmath781 of @xmath120 determines corresponding values of @xmath592 , @xmath767 , @xmath782 , @xmath783 , @xmath784 and @xmath785 ( with @xmath786 ) . we write @xmath787 for these values . [ simpleroots ] we let @xmath675 and @xmath788 be the simple real roots of @xmath750 , defined by @xmath789 for a ( diagonal ) element @xmath156 of @xmath119 . thus , the positive real roots ( see figure [ rootspict ] ) are @xmath790 concretely : * the root space @xmath682 is the @xmath592-subspace in @xmath331 , * the root space @xmath791 is the @xmath782-subspace in @xmath331 , * the root space @xmath792 is the @xmath767-subspace in @xmath331 , * the root space @xmath793 is the @xmath783-subspace in @xmath331 , * the root space @xmath794 is the @xmath785-subspace in @xmath331 ( this is 0 if @xmath132 ) , and * the root space @xmath795 is the @xmath784-subspace in @xmath331 ( this is 0 if @xmath132 ) . and ( b ) @xmath796 . ] let @xmath797 and , for a given lie algebra @xmath798 , @xmath799 note that if @xmath800 for every @xmath801 , then @xmath802 \subset { { { \mathfrak{\lowercase{d}}}}_{{\mathfrak{\lowercase{h}}}}}$ ] and @xmath803 is contained in the center of @xmath270 ( cf . [ [ u , v ] ] ) . by definition ( see [ a+defn ] ) , we have @xmath804 therefore , from the definition of @xmath675 and @xmath788 ( see [ simpleroots ] ) , we see that @xmath805 for @xmath806 , the division algebra of real quaternions , the group @xmath807 is a realization of @xmath808 . most of the work in this paper carries over , but the upper bound on @xmath137 given in theorem [ maxnolinear ] is not sharp in this case ( and it does not seem to be easy to improve this result to obtain a sharp bound ) . thus , we have not obtained any interesting conclusions about the nonexistence of tessellations of homogeneous spaces of @xmath808 . we now describe how the four important families of homogeneous spaces of example [ kulkarnieg ] are realized in terms of @xmath750 . [ su1ndefn ] let * @xmath811 ; * @xmath812 ; * @xmath813 ; and * @xmath814 . then , for an appropriate choice of the embeddings in example [ kulkarnieg ] , we have @xmath815 and ( if @xmath816 ) we have @xmath817 [ d(sp ) ] from ( [ d(g ) ] ) , ( [ su1nan ] ) and ( [ sp1man ] ) , we see that * @xmath818 and * @xmath819 . the arguments in later sections often require the calculation of @xmath820 , for some @xmath821 , or of @xmath822 $ ] , for some @xmath823 . we now provide these calculations for the reader s convenience . for @xmath824 we have @xmath825 when @xmath826 , this simplifies to : @xmath827 similarly , when @xmath828 , we have @xmath829 for @xmath830 we have @xmath831 } [ u , \tilde u ] = \begin{pmatrix } 0 & 0 & \phi \tilde y - \tilde \phi y & - x \tilde y^{\dagger } + \tilde x y^{\dagger } + \phi \tilde { { \mathord{\mathsf{y}}}}- \tilde \phi { { \mathord{\mathsf{y } } } } & -2 { \operatorname{im}}(x \tilde x^{\dagger } + \phi { \overline{\tilde \eta } } - \tilde \phi { \overline{\eta } } ) \\ & 0 & 0 & -2 { \operatorname{im}}(y \tilde y^{\dagger } ) & \tilde y x^{\dagger } - y \tilde x^{\dagger } + \tilde { { \mathord{\mathsf{y}}}}{\overline{\phi } } - { { \mathord{\mathsf{y}}}}{\overline{\tilde \phi } } \\ & & & \cdots \\ \end{pmatrix } .\ ] ] [ conjugation ] for @xmath832 , we have @xmath833 + \frac{1}{2 } \bigl [ [ u , v ] , v \bigr ] + \frac{1}{3 ! } \bigl [ \bigl [ [ u , v ] , v \bigr ] , v \bigr ] + \cdots .\ ] ] combining this with allows us to calculate the effect of conjugating by an element of @xmath120 . for example , suppose @xmath821 , with @xmath800 and @xmath834 , and suppose @xmath835 . we see , from ( [ [ u , v ] ] ) , that @xmath836 } = 0 $ ] and that @xmath837 } = y_{[u , v ] } = 0 $ ] , so @xmath838 , v \bigr ] = 0 $ ] ( see [ [ u , v ] ] ) . therefore @xmath839 .\ ] ] y. benoist ( * ? ? ? * lem . 2.4 ) showed that calculating values of the cartan projection @xmath139 is no more difficult than calculating the norm of a matrix ( see [ mucalc ] ) . in this section , we describe this elegant method and some of its consequences , in the special case @xmath840 . throughout this section , we assume @xmath840 . we employ the usual big oh and little oh notation : for functions @xmath841 on a subset @xmath507 of @xmath1 , we say @xmath842 if there is a constant @xmath12 , such that , for all @xmath843 with @xmath844 large , we have @xmath845 . ( the values of each @xmath846 are assumed to belong to some finite - dimensional normed vector space , typically either @xmath462 or a space of complex matrices . which particular norm is used does not matter , because all norms are equivalent up to a bounded factor . ) we say @xmath847 if @xmath848 as @xmath182 . ( we use @xmath182 to mean @xmath849 . ) also , we write @xmath850 if @xmath851 and @xmath852 . we use the following norm on @xmath750 , because it is easy to calculate . the reader is free to make a different choice , at the expense of changing @xmath853 to @xmath854 in a few of the calculations . for @xmath855 , we define @xmath856 to be the maximum absolute value among the matrix entries of @xmath781 . that is , @xmath857 [ rhodefn ] define @xmath858 by @xmath859 , so @xmath860 is the second exterior power of the standard representation of @xmath750 . thus , we may define @xmath861 to be the maximum absolute value among the determinants of all the @xmath862 submatrices of the matrix @xmath781 . that is , @xmath863 from ( [ exp ] ) , ( [ exp(phi=0 ) ] ) , and ( [ exp(y=0 ) ] ) , it is clear that the @xmath864 minor in the top right corner is often larger than the other @xmath862 minors , so we give it a special name . [ deltadefn ] for @xmath865 , define @xmath866 [ calcrho ] for @xmath867 , we have @xmath868 and @xmath869 . from ( [ suf - an ] ) , we see that @xmath870 thus , from ( [ a+ ] ) , we see that @xmath871 for @xmath872 ( and , since @xmath156 is diagonal , we have @xmath873 for @xmath874 ) . therefore , the desired conclusions follow from the definitions of @xmath875 and @xmath876 . [ mu = rho ] we have @xmath877 for @xmath855 . choose @xmath878 , such that @xmath879 . because @xmath880 for @xmath881 , and @xmath882 ( since @xmath118 is compact ) , we have @xmath883 and @xmath884 so ( [ h = mu(h ) ] ) holds . similarly , we have @xmath885 so ( [ rho(h)=rho(mu ) ] ) holds . for @xmath867 , we know , from ( [ calcrho ] ) , that @xmath886 and @xmath887 . thus , letting @xmath888 , and using ( [ h = mu(h ) ] ) and ( [ rho(h)=rho(mu ) ] ) , we see that @xmath889 and @xmath890 as desired . proposition [ mu = rho ] generalizes to any reductive group @xmath1 ( * ? ? ? however , one may need to use a different representation in the place of @xmath860 . in fact , if @xmath891 , then @xmath497 representations of @xmath1 are needed ; for @xmath840 , we have @xmath423 , and the two representations we use are @xmath860 and the identity representation @xmath892 . [ gn = hn ] let @xmath893 and @xmath894 be two sequences of elements of @xmath750 . we have @xmath895 if and only if @xmath896 ( @xmath168 ) let @xmath897 . from ( [ mucalc ] ) , we see that @xmath898 for @xmath899 , so , using ( [ ajj ] ) , we have @xmath900 therefore @xmath901 , as desired . ( @xmath173 ) because @xmath12 is compact , we have @xmath902 ( cf . proof of ( [ h = mu(h ) ] ) and ( [ rho(h)=rho(mu ) ] ) ) . then the desired conclusions follow from ( [ h = mu(h ) ] ) and ( [ rho(h)=rho(mu ) ] ) . because @xmath12 is compact , we have @xmath903 and @xmath904 for any @xmath905 ( cf . proof of ( [ h = mu(h ) ] ) and ( [ rho(h)=rho(mu ) ] ) ) . thus , the desired conclusion follows from corollary [ gn = hn ] . because of proposition [ mu = rho ] , we will often need to calculate @xmath844 and @xmath906 . the following observation and its corollary sometimes simplifies the work , by allowing us to replace @xmath781 with @xmath907 . [ mu(h-1 ) ] we have @xmath575 for @xmath855 . define @xmath908 as in ( [ sufdefn ] ) , and choose @xmath878 , such that @xmath879 . for any @xmath867 , we see , using ( [ suf - an ] ) or ( [ ajj ] ) , that @xmath909 , so @xmath910 note that @xmath911 . also , we have @xmath912 and @xmath913 , so it is obvious that @xmath914 and @xmath915 . therefore @xmath916 thus , from the definition of @xmath139 , we conclude that @xmath575 , as desired . the following corollary is obtained by combining lemma [ mu(h-1 ) ] with corollary [ gn = hn ] . [ rho(h-1 ) ] we have @xmath917 and @xmath918 for @xmath855 . for @xmath425 , set @xmath919 from ( [ a+ ] ) , we see that @xmath146 and @xmath424 are the two walls of @xmath154 . from ( [ calcrho ] ) , we have @xmath920 we reproduce the proof of the following result , because it is both short and instructive . ( although we have no need for it here , let us point out that the converse of this proposition also holds , and that there is no need to assume @xmath91 . ) because of this proposition ( and corollary [ cds->notess ] ) , section [ suflargesect ] will study the existence of curves @xmath241 , such that @xmath921 , for @xmath922 . [ cds<>h_m ] let @xmath0 be a closed , connected subgroup of @xmath121 in @xmath750 . if , for each @xmath425 , there is a continuous curve @xmath241 in @xmath0 , such that @xmath923 as @xmath924 , then @xmath0 is a cartan - decomposition subgroup . : if @xmath428 contains a curve near each wall of @xmath154 , then it also contains the interior . ] by hypothesis , there is a continuous , proper map @xmath925 , such that @xmath926 . because @xmath91 , we know that @xmath0 is homeomorphic to some euclidean space @xmath927 ( see [ solvable]([solvable - h = rn ] ) ) suppose , for the moment , that @xmath43 . ( this will lead to a contradiction . ) we know that @xmath928 for @xmath929 . because @xmath917 and @xmath918 ( see [ rho(h-1 ) ] ) , we must also have @xmath928 for @xmath930 . there is no harm in assuming @xmath931 ; then @xmath932 ( because @xmath43 ) , so we conclude that @xmath928 for all @xmath591 . this contradicts the fact that @xmath933 for @xmath934 . we may now assume @xmath718 . then , because @xmath0 is homeomorphic to @xmath927 , it is easy to extend @xmath935 to a continuous and proper map @xmath936 \times { \mathord{\mathbb{r}}}^+ \to h$ ] . from and ( [ gn = hn ] ) , we know that the curve @xmath937 stays within a bounded distance from the wall @xmath429 ; say @xmath938 < c$ ] for all @xmath939 . we may assume @xmath12 is large enough that @xmath940 for all @xmath941 $ ] . then an elementary homotopy argument shows that @xmath942 \times { \mathord{\mathbb{r}}}^+ \bigr ) \bigr]$ ] contains @xmath943 so @xmath944 \times { \mathord{\mathbb{r}}}^+ \bigr ) \bigr ] \approx a^+$ ] . because @xmath945 \times { \mathord{\mathbb{r}}}^+ \bigr ) \bigr]$ ] , we conclude from theorem [ cdsvsmu ] that @xmath0 is a cartan - decomposition subgroup when @xmath28 , the weyl chamber @xmath154 has only one point at infinity . thus , if @xmath0 is any noncompact subgroup , then the closure of @xmath428 must contain this point at infinity . this is why it is easy to prove that any noncompact subgroup of @xmath1 is a cartan - decomposition subgroup ( see [ rrank1-cds ] ) . the idea of proposition [ cds<>h_m ] is that if @xmath423 , then the points at @xmath390 of the weyl chamber @xmath154 form a closed interval . if the closure of @xmath428 contains the two endpoints of this interval , then , by continuity , it must also contain all the points in between . unfortunately , we have no good substitute for this proposition when @xmath946 . the points at @xmath390 of @xmath154 form a closed disk ( topologically speaking ) . it is easy to define a map @xmath947 from one disk to another , such that the image of @xmath947 contains the entire boundary sphere , but does not contain the interior of the disk . thus , it does not suffice to show only that the closure of @xmath428 contains the boundary of the disk at @xmath390 ; rather , one needs additional homotopical information to guarantee that no interior points are missed . [ mu(suorsp ) ] let @xmath948 , and fix some @xmath949 . then @xmath950 and @xmath951 are the two walls of @xmath154 . we have 1 . [ mu(suorsp)-su ] @xmath928 for @xmath952 ; and 2 . [ mu(suorsp)-sp ] @xmath933 for @xmath953 . let @xmath954 or @xmath810 . then @xmath955 is a maximal compact subgroup of @xmath0 . from the cartan decomposition @xmath956 and the definition of @xmath139 , we conclude that @xmath957 . in the notation of , we see ( from definition [ su1ndefn ] ) that @xmath958 , where @xmath959 then , since @xmath960 ( see [ mu(h-1 ) ] ) and @xmath961 for @xmath962 , we conclude that @xmath963 is a wall of @xmath154 . furthermore , we have @xmath964 for @xmath965 ( see [ rho(l ) ] ) , so @xmath966 for @xmath591 ( see [ mu = rho ] ) . [ su1inh ] if there is a continuous curve @xmath967 in @xmath0 , such that @xmath968 , then there is a compact subset @xmath12 of @xmath1 , such that @xmath969 . for any ( large ) @xmath970 , we see from continuity ( more precisely , from the intermediate value theorem ) that there exists @xmath721 , such that @xmath971 then , by assumption and from [ mu(suorsp)]([mu(suorsp)-su ] ) , we have @xmath972 so there is a compact subset @xmath160 of @xmath119 , such that @xmath973 ( see [ gn = hn ] ) . therefore @xmath974 as desired . the following corollary can be proved by a similar argument . ( recall that the equivalence relation @xmath100 is defined in ( [ simdefn ] ) . ) [ hsimsuorsp ] assume @xmath0 is not compact . 1 . [ hsimsuorsp - su ] we have @xmath975 if and only if @xmath928 for @xmath591 . [ hsimsuorsp - sp ] we have @xmath976 if and only if @xmath977 for @xmath591 . because of proposition [ cds<>h_m ] , we will often want to show that a curve @xmath241 satisfies @xmath978 , for some @xmath577 . the following lemma does half of the work . [ owalls ] let @xmath507 be a subset of @xmath750 . 1 . [ owalls - linear ] if @xmath979 for @xmath843 , then @xmath928 for @xmath843 . [ owalls - square ] if @xmath980 for @xmath843 , then @xmath981 for @xmath843 . from ( [ calcrho ] ) and ( [ a+ ] ) , we have @xmath982 and @xmath983 for @xmath867 . thus , letting @xmath888 , and using ( [ h = mu(h ) ] ) and ( [ rho(h)=rho(mu ) ] ) , we have : @xmath984 and @xmath985 so @xmath986 and @xmath987 . the desired conclusions follow . for convenience , we record the following simple observation . ( for the proof , cf . the proof of ( [ h = mu(h ) ] ) and ( [ rho(h)=rho(mu ) ] ) . ) [ conjcurve ] let * @xmath425 , * @xmath33 , and * @xmath967 be a continuous curve in @xmath0 . if @xmath988 , then @xmath989 . the following well - known , elementary observation is used frequently in the later sections . [ o(linear ) ] let @xmath990 be a subspace of a finite - dimensional real vector space @xmath163 , and let @xmath991 and @xmath992 be linear transformations . 1 . [ o(linear)-o ] if @xmath993 ( or , more generally , if @xmath994 ) , then there is a linear transformation @xmath995 , such that @xmath996 for all @xmath997 . therefore @xmath852 on @xmath990 . [ o(linear)-= ] if @xmath998 , then @xmath999 on @xmath990 . ( [ o(linear)-o ] ) by passing to a subspace , we may assume @xmath1000 . then , by modding out @xmath1001 , we may assume @xmath1002 is an isomorphism onto its image . define @xmath1003 by @xmath1004 , and let @xmath995 be any extension of @xmath1005 . for @xmath997 , we have @xmath1006 so @xmath852 . ( [ o(linear)-= ] ) from ( [ o(linear)-o ] ) , we have @xmath852 and @xmath1007 , so @xmath999 . let @xmath270 be a real lie subalgebra of @xmath329 , and assume there does not exist a nonzero element @xmath215 of @xmath270 , such that @xmath1008 and @xmath834 . then there exist @xmath338-linear transformations @xmath1009 , such that @xmath1010 for all @xmath801 . ( similarly , @xmath1011 , @xmath1012 , and @xmath1013 are also functions of @xmath1014 . ) furthermore , we have @xmath1015 . the following well - known result is a generalization of the fact that all norms on a finite - dimensional vector space are equivalent up to a bounded factor . [ quadrforms ] if @xmath163 is any finite - dimensional real vector space , and @xmath1016 are two continuous , homogeneous functions of the same degree , such that @xmath1017 , then @xmath999 . by continuity , the function @xmath1018 attains a non - zero minimum and a finite maximum on the unit sphere . because @xmath1018 is homogeneous of degree zero , these values bound @xmath1018 on all of @xmath1019 . in this section , we show how to construct several families of homogeneous spaces that have tessellations . all of these examples are based on a method of t. kobayashi ( see [ construct - tess ] ) that generalizes example [ kulkarnieg ] . as explained in the comments before theorem [ noncpctdim ] , the following theorem is essentially due to t. kobayashi . [ construct - tess ] if * @xmath0 and @xmath46 are closed subgroups of @xmath1 , with only finitely many connected components ; * @xmath46 acts properly on @xmath2 ; * @xmath1020 ; and * there is a cocompact lattice @xmath3 in @xmath46 , then @xmath2 has a tessellation . ( namely , @xmath3 is a crystallographic group for @xmath2 . ) because @xmath3 is a closed subgroup of @xmath46 , we know that it acts properly on @xmath2 ( see [ chcproper ] ) . thus , it suffices to show that @xmath116 is compact . from lemma [ hcanbean ] , we see that there is no harm in assuming @xmath1021 , and that there is a closed , connected subgroup @xmath1022 of @xmath1 , such that * @xmath1022 is conjugate to a subgroup of @xmath121 , * @xmath1023 , and * @xmath1024 , for some compact subset @xmath12 of @xmath1 . ( unfortunately , we can not assume @xmath1025 : we may not be able to replace @xmath46 with @xmath1022 , because there may not be a cocompact lattice in @xmath1022 . for example , there is not lattice in @xmath121 , because any group with a lattice must be unimodular ( * ? ? ? * rem . 1.9 , p. 21 ) . ) it suffices to show that @xmath1026 is compact . ( because @xmath1027 is compact , and @xmath1028 is compact , this implies that @xmath15 is compact , as desired . ) we know that @xmath1022 acts properly on @xmath2 ( see [ chcproper ] ) , so @xmath1029 acts properly on @xmath1 , with quotient @xmath1026 . therefore , lemma [ fiberbundle ] implies that @xmath1026 has the same homology as @xmath1 ; in particular , @xmath1030 from the iwasawa decomposition @xmath92 , and because @xmath121 is homeomorphic to @xmath1031 ( see [ ansc ] and [ r = rd ] ) , we know that @xmath1 is homeomorphic to @xmath1032 . since @xmath1031 is contractible , this implies that @xmath1 is homotopy equivalent to @xmath118 , so @xmath1 and @xmath118 have the same homology ; in particular , @xmath1033 since @xmath1034 this implies that the top - dimensional homology of the manifold @xmath1035 is nontrivial . therefore @xmath1026 is compact ( * ? ? ? 8.3.4 ) , as desired . our results for @xmath1036 are based on the following special case of the theorem . the converse of this corollary is proved in section [ proofsect ] ( see [ suf - known ] ) . recall the equivalence relation @xmath100 , introduced in notation [ simdefn ] . [ sueventessexists ] let @xmath0 be a closed , connected subgroup of @xmath1036 . if * @xmath1037 ; and * either @xmath1038 or @xmath976 , then @xmath2 has a tessellation . let @xmath1039 and @xmath1040 . by assumption , we have @xmath1041 , for some @xmath1042 ; let @xmath1043 . because @xmath1044 and @xmath1045 are the two walls of @xmath154 ( see [ mu(suorsp ) ] ) , we know that @xmath1046 acts properly on @xmath1047 ( see [ proper<>mu(l ) ] ) ; since @xmath1048 , this implies that @xmath46 acts properly on @xmath2 ( see [ chcproper ] ) . also , we have @xmath1049 ( see [ d(sp ) ] and [ d(g ) ] ) , and there is a cocompact lattice in @xmath46 ( cf . [ classical]([classical - borel ] ) ) . thus , the desired conclusion follows from theorem [ construct - tess ] . the homogeneous spaces described here were found by h. oh and d. witte ( * ? ? ? * thms . 4.1 and 4.6 ) , ( * ? ? ? * thm . 1.5 ) . [ hb - defn ] for any @xmath338-linear @xmath1050 , we define @xmath1051 we write @xmath1052 , rather than @xmath1053 , because @xmath767 is a row vector . it is easy to see , using , for instance , the formula for the bracket in , that if @xmath1054 then @xmath1055 is a real lie subalgebra of @xmath1056 ; we let @xmath1057 denote the corresponding connected lie subgroup of @xmath121 . from ( [ d(h)=dimh ] ) , we have @xmath1058 [ hb = sp1 m ] assume @xmath1059 . by comparing ( [ sp1man ] ) with ( [ hb - defn ] ) , we see that there is a @xmath338-linear map @xmath1060 , such that @xmath1061 ( and @xmath1062 satisfies ( [ vbwb ] ) ) . thus , in general , @xmath1057 is a deformation of @xmath1063 . [ hbthm ] let @xmath1064 be @xmath338-linear . if * condition ( [ vbwb ] ) holds , and * [ hbthm - xbnotinfx ] @xmath1065 , for every nonzero @xmath1066 , then 1 . [ hbthm - mu ] @xmath933 for @xmath1067 ; and 2 . [ hbthm - tess ] @xmath1068 has a tessellation . ( [ hbthm - mu ] ) given @xmath1067 , write @xmath1069 , with @xmath1070 and @xmath1071 . we may assume that @xmath1072 ( by replacing @xmath781 with @xmath907 if necessary ( see [ rho(h-1 ) ] ) ) . it suffices to show @xmath1073 ( for then @xmath1074 , so lemma [ owalls]([owalls - square ] ) applies ) . [ hb - hinn ] assume @xmath156 is trivial . from and ( [ hb - defn ] ) , we see that @xmath1075 so @xmath1076 from ( [ exp(phi=0 ) ] ) and ( [ deltadefn ] ) , we have @xmath1077 from ( [ xbnotinfx ] ) , we see that @xmath1078 for every nonzero @xmath1066 , so lemma [ quadrforms ] implies @xmath1079 also , because @xmath1080 ( and @xmath1081 ) , we have @xmath1082 thus , @xmath1083 so @xmath1084 , as desired . the general case . from case [ hb - hinn ] , we know @xmath1085 . then , because @xmath1086 , we have @xmath1087 then , since @xmath1088 and @xmath1089 , we conclude that @xmath1090 , as desired . ( [ hbthm - tess ] ) from ( [ hbthm - mu ] ) and [ hsimsuorsp]([hsimsuorsp - sp ] ) , we see that @xmath1091 . then , because @xmath1092 ( see [ d(hb ) ] ) , theorem [ sueventessexists ] implies that @xmath1093 has a tessellation . [ bsymplectic ] let @xmath1050 be @xmath338-linear . condition ( [ vbwb ] ) holds if and only if either 1 . @xmath132 ; or 2 . @xmath136 and @xmath1094 , where @xmath1095 and we use the natural identification of @xmath1096 with @xmath1097 . assume @xmath132 . because @xmath1098 for every @xmath1099 , it is obvious that ( [ vbwb ] ) holds . assume @xmath136 . if ( [ vbwb ] ) holds , then @xmath1100 so @xmath1101 is symplectic . the argument is reversible . [ xbnotinfx ] * for @xmath132 , the assumption that @xmath1065 simply requires that @xmath316 have no real eigenvalues . * for @xmath136 , we do not know a good description of the linear transformations @xmath316 that satisfy @xmath1065 , although it is easy to see that this is an open set ( and not dense ) . a family of examples was constructed by h. oh and d. witte ( see [ hbeg ] below ) . * if @xmath79 is odd , then there does not exist @xmath1102 satisfying the assumption that @xmath1065 . for @xmath132 , this is simply the elementary fact that a linear transformation on an odd - dimensional real vector space must have a real eigenvalue . for @xmath1103 , see step [ dim(u)<2npf - u / z ] of the proof of proposition [ dim(u)<2n ] . * if @xmath79 is even , then , by varying @xmath316 , one can obtain uncountably many pairwise non - conjugate subgroups @xmath1057 , such that @xmath1104 has a tessellation . for @xmath132 , this is proved in ( * ? ? ? * thm . 1.5 ) ) . for @xmath136 , a similar argument can be applied to the examples constructed in ( [ hbeg ] ) below . [ hbeg ] assume @xmath79 is even , let @xmath1105 , such that @xmath1101 has no real eigenvalue , and define an @xmath338-linear map @xmath1106 by @xmath1107 . let us verify that @xmath316 satisfies the conditions of theorem [ hbthm ] ( for @xmath136 ) . let @xmath1108 . from the definition of @xmath316 , and because @xmath1105 , we have @xmath1109 suppose @xmath1110 , for some @xmath1111 . because @xmath1112 , we must have @xmath1113 . then @xmath1114 because @xmath1101 has no real eigenvalues , we know that @xmath26 is not an eigenvalue of @xmath1101 , so we conclude that @xmath1115 . similarly , because @xmath1116 is not an eigenvalue of @xmath1101 , we see that @xmath1117 . therefore @xmath1118 these examples are new for @xmath136 , but provide nothing interesting for @xmath132 ( see [ hc]([hc - real ] ) ) . [ suegsdefn ] for @xmath1119 $ ] , we define @xmath1120}}}= { \left\{\ , \begin{pmatrix } t & \phi & x & { \operatorname{re}}\phi + c { \operatorname{im}}\phi & { { \mathord{\mathsf{x}}}}\\ & 0 & 0 & 0 & * \\ & & \dots \\ \end{pmatrix } \mathrel{\left| \vphantom { \left\ { \begin{pmatrix } t & \phi & x & { \operatorname{re}}\phi + c { \operatorname{im}}\phi & { { \mathord{\mathsf{x}}}}\\ & 0 & 0 & 0 & * \\ & & \dots \\ \end{pmatrix } \mid \begin{matrix } t \in { \mathord{\mathbb{r } } } , \\ \phi \in { \mathbb{f } } , \\ x \in { \mathbb{f}}^{n-2 } , \\ { { \mathord{\mathsf{x}}}}\in{{\mathbb{f}}_{\text{imag}}}\end{matrix } \right\ } } \right . } \begin{matrix } t \in { \mathord{\mathbb{r } } } , \\ \phi \in { \mathbb{f } } , \\ x \in { \mathbb{f}}^{n-2 } , \\ { { \mathord{\mathsf{x}}}}\in{{\mathbb{f}}_{\text{imag}}}\end{matrix } \,\right\ } } .\ ] ] it is easy to see , using , for instance , the formula for the bracket in , that @xmath1121}}}$ ] is a real lie subalgebra of @xmath1056 ( even without the assumption that @xmath1122 ) ; we let @xmath1123}}}$ ] be the corresponding connected lie subgroup of @xmath121 . from ( [ d(h)=dimh ] ) , we have @xmath1124 } } } ) & = \dim { { { { \mathfrak{\lowercase{h}}}}_{[c]}}}\\ & = \dim { \mathord{\mathbb{r}}}+ \dim { \mathbb{f}}+ \dim { \mathbb{f}}^{n-2 } + \dim { { \mathbb{f}}_{\text{imag}}}\\ & = 1 + { q}+ { q}(n-2 ) + ( { q}-1 ) \\ & = { q}n . \end{split}\ ] ] [ hc ] let @xmath1125 be embedded into @xmath774 as in [ su1nan ] . 1 . [ hc - real ] if @xmath1126 , then @xmath216 is irrelevant in the definition of @xmath1121}}}$ ] ( because @xmath1127 ) ; therefore @xmath1121}}}={\operatorname{{\mathfrak{\lowercase{su}}}}}(1,n;{\mathord{\mathbb{r}}})\cap ( { \mathfrak{\lowercase{a } } } + { \mathfrak{\lowercase{n}}})$ ] . 2 . if @xmath1128 , then @xmath1129 } } = { \operatorname{{\mathfrak{\lowercase{su}}}}}(1,n;{\mathord{\mathbb{c}}})\cap ( { \mathfrak{\lowercase{a } } } + { \mathfrak{\lowercase{n}}})$ ] . thus , in general , @xmath1121}}}$ ] is either @xmath1130 or a deformation of it . [ suegs ] assume @xmath136 , and @xmath1059 is even . if @xmath1119 $ ] , then 1 . [ suegs - linear ] @xmath928 for @xmath1131}}}$ ] ; and 2 . [ suegs - tess ] @xmath1132}}}$ ] has a tessellation . ( [ suegs - linear ] ) given @xmath1131}}}$ ] , it suffices to show that @xmath979 ( see [ owalls]([owalls - linear ] ) ) . write @xmath1069 , with @xmath1070 and @xmath1133 . we may assume that @xmath1072 ( by replacing @xmath781 with @xmath907 if necessary ( see [ rho(h-1 ) ] ) ) . let @xmath1134 be the real quadratic form @xmath1135 and let @xmath163 be the @xmath338-subspace of @xmath1136 defined by @xmath1137 [ suegspf - q = x2+phi2 ] for @xmath1138 , we have @xmath1139 . for @xmath1140 , we have @xmath1141 ( because @xmath1142 and @xmath1143 is purely imaginary ) . thus , the restriction of @xmath1144 to @xmath163 is positive definite , so the desired conclusion follows from lemma [ quadrforms ] . [ suegspf - u = x2+phi2+xx ] we have @xmath1145 . from ( [ exp(y=0 ) ] ) ( with @xmath1146 ) , we have @xmath1147 and @xmath1148 then , from step [ suegspf - q = x2+phi2 ] , we see that @xmath1149 as desired . completion of the proof . from step [ suegspf - u = x2+phi2+xx ] , we have @xmath1150 also , from ( [ exp(y=0 ) ] ) , we have @xmath1151 thus , it is easy to see that @xmath1152 so the desired conclusion follows from lemma [ owalls]([owalls - linear ] ) . ( [ suegs - tess ] ) from ( [ suegs - linear ] ) and [ hsimsuorsp]([hsimsuorsp - su ] ) , we see that @xmath1123}}}\sim { \operatorname{su}}(1,n)$ ] . then , because @xmath1153 } } } ) = 2n$ ] ( see [ d(hc ) ] ) , theorem [ sueventessexists ] implies that @xmath1132}}}$ ] has a tessellation . proposition [ hcuncountable ] shows that if @xmath136 , then @xmath1123}}}$ ] is not conjugate to @xmath1154}}$ ] unless @xmath1155 ( for @xmath1156 $ ] ) . thus , theorem [ suegs]([suegs - tess ] ) implies that , by varying @xmath216 , one obtains uncountably many nonconjugate subgroups @xmath1123}}}$ ] , such that @xmath1157}}}$ ] has a tessellation . [ g1xg2-tess ] let @xmath1158 be the direct product of two connected , linear , almost simple lie groups @xmath1159 and @xmath1160 of real rank one , with finite center , and let @xmath0 be a nontrivial , closed , connected , proper subgroup of @xmath121 . the homogeneous space @xmath2 has a tessellation if and only if , perhaps after interchanging @xmath1159 and @xmath1160 , there is a continuous homomorphism @xmath1161 , such that @xmath1162 ( @xmath168 ) we may assume @xmath1163 ( by interchanging @xmath1159 and @xmath1160 if necessary ) . [ hcapboth ] assume @xmath1164 and @xmath1165 . for @xmath1166 , we know that @xmath1167 is not compact ( see [ solvable]([solvable - nocpct ] ) ) , so corollary [ rrank1-cds ] implies that there is a compact subset @xmath1168 of @xmath1169 , such that @xmath1170 . then , letting @xmath1171 , we have @xmath151 , so proposition [ cds->notess ] implies that @xmath2 does not have a tessellation . this is a contradiction . assume @xmath1164 and @xmath1172 . from corollaries [ rrank1-cds ] and [ cdsvsmu ] , we know that there is a compact subset @xmath12 of @xmath1173 , such that @xmath1174 . therefore , corollary [ noncpct - dim - notess ] ( with @xmath1159 in the place of @xmath94 ) implies @xmath1175 . then , because @xmath1172 ( and @xmath91 ) , we conclude that @xmath0 is the graph of a homomorphism from @xmath1176 to @xmath1177 , as desired . assume @xmath1178 . from corollary [ tess->dim>1,2 ] , we know that @xmath1179 . then , since @xmath1178 , we conclude that @xmath0 is the graph of a homomorphism from @xmath1177 to @xmath1176 . interchanging @xmath1159 and @xmath1160 yields the desired conclusion . ( @xmath173 ) we verify the hypotheses of theorem [ construct - tess ] , with @xmath1160 in the role of @xmath46 . let @xmath1180 be the image of @xmath0 under the natural homomorphism @xmath1181 . because @xmath91 , we know that @xmath1180 is closed ( see [ solvable]([solvable - h = rn ] ) ) . it is well known ( and follows easily from ( [ proper<>chc ] ) ) that any closed subgroup acts properly on the ambient group , so this implies that @xmath1180 acts properly on @xmath1182 . from the definition of @xmath0 , we have @xmath1172 , so we conclude that @xmath1183 acts properly on @xmath1182 ; equivalently , @xmath1160 acts properly on @xmath2 ( cf . [ proper<>chc ] ) . because @xmath1184 , we have @xmath1185 . also , we have @xmath93 ( see [ d(h)=dimh ] ) and , from the definition of @xmath0 , we have @xmath1186 . therefore @xmath1187 there is a cocompact lattice in @xmath1160 ( cf . [ classical]([classical - borel ] ) ) . so theorem [ construct - tess ] implies that @xmath2 has a tessellation . t. kobayashi observed that , besides the examples with @xmath1188 or @xmath6 ( see [ kulkarnieg ] ) , theorem [ construct - tess ] can also be used to construct tessellations of some homogeneous spaces @xmath2 in which @xmath1 and @xmath0 are simple lie groups with @xmath1189 . he found one pair of infinite families , and several isolated examples . [ kobayashibigeg ] each of the following homogeneous spaces has a tessellation : 1 . [ kobayashibigeg - so4/sp1 ] @xmath1190 ; 2 . @xmath1191 ; 3 . @xmath1192 ; 4 . @xmath1193 ; 5 . @xmath1194 ; 6 . @xmath1195 ; 7 . @xmath1196 ; 8 . . it would be very interesting to find other examples of simple lie groups @xmath1 with reductive subgroups @xmath0 and @xmath46 that satisfy the hypotheses of theorem [ construct - tess ] . let @xmath1198 and @xmath1199 . from [ kobayashibigeg]([kobayashibigeg - so4/sp1 ] ) , we know that @xmath2 has a tessellation . h. oh and d. witte ( * ? ? ? 4.6(2 ) ) pointed out that the deformations @xmath1200 ( where @xmath1057 is as in theorem [ hbthm ] , with @xmath136 ) also have tessellations , but it is not known whether there are other deformations of @xmath123 that also have tessellations . it does not seem to be known whether the other examples in theorem [ kobayashibigeg ] lead to nontrivial deformations , after intersecting @xmath0 with @xmath121 . this section presents a short proof of the results we need from @xcite and @xcite . those papers provide an approximate calculation of @xmath428 , for every closed , connected subgroup @xmath0 of @xmath73 or @xmath52 , respectively , but here we consider only subgroups of large dimension . also , we do not need a complete description of the entire set @xmath428 ; we are only interested in whether or not there is a curve @xmath241 , such that @xmath988 , for some @xmath577 . the main results of this section are theorem [ maxnolinear ] ( for @xmath1201 ) and theorem [ bestnosquare ] ( for @xmath1202 ) . they give a sharp upper bound on @xmath422 , for subgroups @xmath0 that fail to contain such a curve , and , if @xmath79 is even , also provide a fairly explicit description of all the subgroups of @xmath121 that attain the bound . because of the limited scope of this section , the proof here is shorter than the previous work , and we are able to give a fairly unified treatment of the two groups @xmath73 and @xmath52 . the arguments are elementary , but they involve case - by - case analysis and a lot of details , so they are not pleasant to read . [ standingsu2f ] throughout this section : 1 . we use the notation of [ coordssect ] . ( in particular , @xmath132 or @xmath462 , and @xmath752 . ) 3 . @xmath1203 . 4 . @xmath0 is a closed , connected subgroup of @xmath121 that is compatible with @xmath119 ( see [ compatibledefn ] ) , so @xmath1204 ( see [ d(h)=dimh ] ) . ( note that @xmath162 is connected ( see [ solvable]([solvable - hcapl ] ) ) . ) we use the notation of [ calcsect ] . ( in particular , @xmath906 is defined in ( [ rhodefn ] ) and @xmath1207 is defined in ( [ deltadefn ] ) . ) except in subsection [ compatiblesubsect ] , @xmath0 is compatible with @xmath119 ( see [ compatibledefn ] ) . recall that the real jordan decomposition of an element of @xmath1 is defined in ( [ jordandecomp ] ) ; any element @xmath155 of @xmath121 has a real jordan decomposition @xmath1208 ( with @xmath216 trivial ) . if @xmath156 is an element of @xmath119 , rather than only conjugate to an element of @xmath119 , we could say that @xmath155 is compatible with @xmath119 . " we now define a similar , useful notion for subgroups of @xmath121 . lemma [ conjtocompatible ] shows there is usually no loss of generality in assuming that @xmath0 is compatible with @xmath119 , and lemma [ not - semi ] shows that the compatible subgroups can be described fairly explicitly . [ compatibledefn ] let us say that @xmath0 is _ compatible _ with @xmath119 if @xmath1209 , where @xmath1210 , @xmath1205 , and @xmath1211 denotes the centralizer of @xmath328 in @xmath120 . in preparation for the proofs of the main results , let us state a lemma that records a few of the nice properties of jordan components . [ aucgood ] let @xmath214 be the real jordan decomposition of an element @xmath155 of @xmath1 . then 1 . the real jordan decomposition of @xmath1212 is @xmath1213 ; and 2 . @xmath156 , @xmath215 , and @xmath216 all belong to the zariski closure of @xmath1214 . therefore : 1 . [ aucgood - norm ] @xmath156 , @xmath215 , and @xmath216 each normalize any connected subgroup of @xmath1 that is normalized by @xmath155 ; and 2 . [ aucgood - id ] if @xmath1215 , for some @xmath1216 and some @xmath1217-invariant subspace @xmath537 of @xmath213 , then @xmath1218 [ conjtocompatible ] @xmath0 is conjugate , via an element of @xmath120 , to a subgroup that is compatible with @xmath119 . [ conjtocompatiblepf - algebraic ] assume , for the real jordan decomposition @xmath1069 of each element @xmath781 of @xmath0 , that @xmath156 and @xmath215 belong to @xmath0 . let @xmath328 be a maximal split torus of @xmath0 . ( recall that a split torus is a subgroup consisting entirely of hyperbolic elements . ) then @xmath328 is contained in some maximal split torus of @xmath1 , that is , in some subgroup of @xmath1 conjugate to @xmath119 ; replacing @xmath0 by a conjugate , we may assume @xmath1219 . in other words , we now know that @xmath1220 is a maximal split torus of @xmath0 . given @xmath591 , we have the real jordan decomposition @xmath1069 . by assumption , @xmath1221 ; thus , @xmath156 belongs to some maximal split torus @xmath1222 of @xmath0 . a fundamental result of the theory of solvable algebraic groups implies that all maximal split tori of @xmath0 are conjugate via an element of @xmath1223 ( * ? ? ? * thm . 4.21 ) , so there is some @xmath1224 , such that @xmath1225 . then @xmath1226 , being a subgroup of @xmath119 , is a split torus . thus , the maximality of @xmath328 implies that @xmath1227 ; let @xmath1228 . then @xmath1229 since @xmath591 is arbitrary , we conclude that @xmath1230 so @xmath0 is compatible with @xmath119 . the general case . let @xmath1231 be the subgroup of @xmath121 generated by the jordan components of the elements of @xmath0 . ( of course , since every element of @xmath0 has a jordan decomposition , we have @xmath1232 . ) then case [ conjtocompatiblepf - algebraic ] applies to @xmath1180 , so , replacing @xmath0 by a conjugate , we may assume @xmath1233 , where @xmath1234 and @xmath1235 ( see [ conjtocompatiblepf - h = tu ] ) . because @xmath1180 normalizes @xmath0 ( see [ aucgood]([aucgood - norm ] ) ) , we know that @xmath1236 acts as the identity on @xmath1237 , for all @xmath1238 . hence , lemma [ aucgood]([aucgood - id ] ) implies that @xmath1239 acts as the identity on @xmath1237 , for all @xmath1240 ; therefore @xmath1241 \subset h$ ] . also , we have @xmath1242 \subset [ an , an ] \subset n$ ] . thus , letting @xmath1205 , we have @xmath1243 \subset h \cap n = u .\ ] ] because @xmath1244 and @xmath1245 is @xmath1246-invariant , the adjoint action of @xmath1247 on @xmath1245 is completely reducible , so ( [ conjtocompatiblepf - comm ] ) implies that there is a subspace @xmath1248 of @xmath1245 , such that @xmath1249 = 0 $ ] and @xmath1250 . therefore , @xmath1251 , so @xmath1252 let @xmath1253 be the projection with kernel @xmath120 , and let @xmath1254 . then @xmath1255 so @xmath1256 . for any @xmath591 , we know , from ( [ conjtocompatiblepf - hbarcompat ] ) , that there exist @xmath1257 , @xmath1258 and @xmath1259 , such that @xmath1260 . because @xmath1261 , we must have @xmath1262 and , because @xmath1263 , we have @xmath1264 . therefore , we conclude that @xmath1266 , so @xmath0 is compatible with @xmath119 . the preceding proposition shows that @xmath0 is conjugate to a subgroup @xmath122 that is compatible with @xmath119 . the subgroup @xmath122 is usually not unique , however . the following lemma provides one way to change @xmath122 , often to an even better subgroup . [ conjuomega ] assume that @xmath0 is compatible with @xmath119 , and let @xmath1210 . if @xmath1267 , then @xmath1268 is compatible with @xmath119 . let @xmath1269 , @xmath1270 , and @xmath1271 . because @xmath215 centralizes @xmath328 , we have @xmath1272 also , because @xmath1133 , and @xmath120 is normal , we have @xmath1273 , so @xmath1274 since @xmath1133 , we have @xmath1275 , so @xmath1276 and @xmath1277 thus , @xmath1278 as desired . [ not - semi ] if @xmath0 is compatible with @xmath119 , then either 1 . [ not - semi - tu ] @xmath1279 ; or 2 . [ not - semi - not ] there is a positive root @xmath335 , a nontrivial group homomorphism @xmath1280 , and a closed , connected subgroup @xmath162 of @xmath120 , such that 1 . [ not - semi - codim1 ] @xmath1281 ; 2 . [ not - semi - normal ] @xmath162 is normalized by both @xmath1282 and @xmath1283 ; and 3 . [ not - semi - disjoint ] @xmath1284 . because @xmath0 is compatible with @xmath119 , we have @xmath1266 , where @xmath1285 and @xmath1205 . we may assume that @xmath1286 , for otherwise ( [ not - semi - tu ] ) holds . therefore @xmath1287 . because @xmath1288 is a sum of root spaces , this implies that there is a positive root @xmath335 , such that @xmath1289 . because @xmath423 , we have @xmath1290 , so we must have @xmath1291 ( otherwise we would have @xmath1292 , so @xmath1293 ; hence ( [ not - semi - tu ] ) holds ) . therefore , @xmath1294 . because @xmath1295 , we have @xmath1296 $ ] , so there is a nontrivial one - parameter subgroup @xmath1297 in @xmath1298 that is not contained in @xmath162 . because @xmath328 centralizes @xmath1211 , we may write @xmath1299 where @xmath1300 is a one - parameter subgroup of @xmath328 and @xmath1301 is a one - parameter subgroup of @xmath1211 . furthermore , this decomposition is unique , because @xmath1302 . ( in fact , @xmath1299 is the real jordan decomposition of @xmath1303 . ) define @xmath1304 by @xmath1305 for all @xmath599 . ( [ not - semi - codim1 ] ) for all @xmath599 , we have @xmath1306 , which establishes one inclusion of ( [ not - semi - codim1 ] ) . the other will follow if we show that @xmath1307 , so suppose @xmath1308 . then lemma [ dimt]([dimt - a ] ) implies that @xmath1309 , so it follows from lemma [ rootdecomp ] ( with @xmath386 and @xmath385 ) that @xmath1310 , contradicting our assumption that @xmath1311 . ( [ not - semi - normal ] ) because @xmath1312 , we know that each of @xmath694 and @xmath695 normalizes @xmath0 ( see [ aucgood]([aucgood - norm ] ) ) . being in @xmath121 , they also normalize @xmath120 . therefore , they normalize @xmath1313 . ( [ not - semi - disjoint ] ) suppose @xmath1314 . because the intersection @xmath1315 is connected ( see [ not - semi - normal ] and [ solvable]([solvable - hcapl ] ) ) , and @xmath1316 , we must have @xmath1317 . therefore @xmath1318 , so @xmath1319 . this contradicts our assumption that @xmath1286 . [ tnormsu ] if @xmath0 is compatible with @xmath119 , then @xmath1320 normalizes @xmath1223 . [ dimt ] if @xmath1321 , then 1 . [ dimt - a ] @xmath0 contains a conjugate of @xmath119 ; and 2 . [ dimt - cds ] @xmath0 is a cartan - decomposition subgroup . ( [ dimt - a ] ) let @xmath1253 be the projection with kernel @xmath120 , and let @xmath1180 be the zariski closure of @xmath0 . from the structure theory of solvable algebraic groups ( * ? ? ? 10.6(4 ) , pp . 137138 ) , we know that @xmath1322 is the semidirect product of a torus @xmath328 and and unipotent subgroup @xmath1323 . replacing @xmath0 by a conjugate under @xmath120 , we may assume that @xmath340 . since @xmath1324 we must have @xmath1325 , so @xmath1326 normalizes @xmath0 ( see [ zar - norm ] ) . then , since @xmath1325 , we conclude that @xmath1327 ( see [ rootdecomp ] ) . ( [ dimt - cds ] ) from ( [ dimt - a ] ) , we see that , by replacing @xmath0 with a conjugate subgroup , we may assume @xmath1309 . because @xmath119 is a cartan - decomposition subgroup ( see [ aiscds ] ) , this implies @xmath0 is a cartan - decomposition subgroup . the following basic result was used twice in the above arguments . [ zar - norm ] if @xmath0 is a closed , connected subgroup of @xmath1 , then the zariski closure of @xmath0 normalizes @xmath0 . our goal is to prove theorem [ maxnolinear ] ; we begin with some preliminary results . first , an observation that simplifies the calculations in some cases , by allowing us to assume that @xmath1008 . [ x=0 ] let @xmath1328 . if @xmath1329 and @xmath1330 , then there is some @xmath1331 , such that 1 . @xmath1332 , 2 . @xmath1333 , and 3 . @xmath1334 . because @xmath1329 and @xmath1330 , there is some @xmath1335 , such that @xmath1336 . let * @xmath722 be the element of @xmath682 with @xmath1337 , * @xmath1338 , and * @xmath1339 . from ( [ conjugation ] ) , we see that * @xmath1340 , * @xmath1341 , and * @xmath1342 , as desired . [ hinn - linear ] if there does not exist a continuous curve @xmath967 in @xmath162 , such that @xmath1343 , then 1 . [ hinn - linear - phi=0&eta2= ] for every nonzero element @xmath1344 of @xmath803 , we have @xmath1345 ; and 2 . [ hinn - linear - phi=0&=0 ] for every element @xmath215 of @xmath1346 , such that @xmath1347 , we have @xmath1348 ( [ hinn - linear - phi=0&eta2= ] ) suppose there is a nonzero element @xmath1344 of @xmath803 with @xmath1349 . let @xmath1350 ( see [ exp(phi=0 ) ] ) . we have @xmath1351 then , because @xmath1352 , it is easy to see that @xmath1353 . also , we have @xmath1354 , so @xmath1355 , as desired . ( [ hinn - linear - phi=0&=0 ] ) suppose there is an element @xmath215 of @xmath1346 , such that @xmath1356 , and @xmath1357 let @xmath1358 . [ hinnlinearpf - x=0 ] assume @xmath1008 . because @xmath1356 , we must have @xmath1330 . then , from , we know that @xmath1359 . so , from , we see that * @xmath1360 , * @xmath1361 whenever @xmath1362 , and * @xmath1363 whenever @xmath1364 and @xmath1365 . this implies that @xmath1366 . [ hinnlinearpf - y=0 ] assume @xmath834 . this is similar to case [ hinnlinearpf - x=0 ] . ( in fact , this can be obtained as a corollary of case [ hinnlinearpf - x=0 ] by replacing @xmath0 with its conjugate under the weyl reflection corresponding to the root @xmath675 . ) assume @xmath1330 . because @xmath1356 , lemma [ x=0 ] implies there is some @xmath1331 , such that , letting @xmath1339 , we have @xmath1367 , @xmath1368 , and @xmath1369 . we show below that is satisfied with @xmath1370 in the place of @xmath215 , so , from case [ hinnlinearpf - x=0 ] , we conclude that @xmath1371 . thus , the desired conclusion follows from lemma [ conjcurve ] ( with @xmath1201 ) . to complete the proof , we now show that is satisfied with @xmath1370 in the place of @xmath215 . ( this can be verified by direct calculation , but we give a more conceptual proof . ) because @xmath1372 , multiplication by @xmath1373 on the left performs a row operation on the first two rows of @xmath781 ; likewise , multiplication by @xmath155 on the right performs a column operation on the last two columns of @xmath781 . these operations do not change the determinant @xmath1207 : thus @xmath1374 from and the definition of @xmath678 , we see that @xmath1375 because @xmath1347 , we have @xmath1376 , so this simplifies to @xmath1377 thus , is equivalent to the condition that @xmath1378 . then , since @xmath1379 we conclude that is also valid for @xmath1370 . [ dim(u+z)<3 ] if there does not exist a continuous curve @xmath967 in @xmath162 , such that @xmath1343 , then @xmath1380 furthermore , if equality holds , and @xmath136 , then @xmath1381 and @xmath1382 . assume @xmath136 . because @xmath1383 is a quadratic form of signature @xmath1384 on @xmath1385 , we know , from [ hinn - linear]([hinn - linear - phi=0&eta2= ] ) , that @xmath1386 . thus , we may assume @xmath1387 , so there is some @xmath1388 , such that @xmath1389 . [ dim(u+z)<3pf - yy=0 ] assume there exists @xmath1390 , such that @xmath1391 and @xmath1392 . from ( [ [ u , v ] ] ) , we see that @xmath1393 \in { { { \mathfrak{\lowercase{d}}}}_{{\mathfrak{\lowercase{h}}}}}$ ] , with @xmath1394 } = -{\operatorname{im}}(\phi_u { \overline{\eta_z } } ) \neq 0 $ ] and @xmath1395 } = \eta_{[u , z ] } = 0 $ ] . this contradicts [ hinn - linear]([hinn - linear - phi=0&eta2= ] ) . assume there exists @xmath1390 , such that @xmath1396 . from ( [ [ u , v ] ] ) , we see that @xmath1393 $ ] is an element of @xmath803 , such that @xmath1395 } = 0 $ ] , and @xmath1397 } = \phi_u { { \mathord{\mathsf{y}}}}_z$ ] is a purely imaginary multiple of @xmath1011 . so subcase [ dim(u+z)<3pf - yy=0 ] applies ( with @xmath1393 $ ] in the place of @xmath1344 ) . assume @xmath1391 and @xmath1398 , for all @xmath1390 . from [ hinn - linear]([hinn - linear - phi=0&eta2= ] ) , we see that @xmath1399 , so the assumption of this subcase implies @xmath1400 . thus , @xmath1401 so the desired inequality holds . if equality holds , then @xmath1402 and @xmath1403 . thus , we may choose @xmath1390 , such that @xmath1404 , and @xmath1405 , such that @xmath1406 . from the assumption of this subcase , we know that @xmath1398 ; thus , @xmath1407 . therefore , subcase [ dim(u+z)<3pf - yy=0 ] applies , with @xmath1408 in the place of @xmath215 . assume @xmath132 . because @xmath1409 and @xmath1410 , the desired inequality holds unless @xmath1411 and @xmath1387 . thus , we may assume there is some nonzero @xmath1390 and some @xmath1328 , such that @xmath1389 . assume @xmath834 . we may assume @xmath1412 ( by replacing @xmath215 with @xmath1413 , if necessary ) . let @xmath1414 . from , we see that @xmath1415 , but @xmath1416 therefore @xmath1417 , so lemma [ owalls]([owalls - linear ] ) implies that @xmath1343 . this is a contradiction . assume @xmath1330 . let @xmath722 be the element of @xmath791 with @xmath1418 , and let @xmath1419 . then @xmath1368 ( see [ conjugation ] and [ [ u , v ] ] ) . thus , by replacing @xmath0 with the conjugate @xmath1420 ( see [ conjcurve ] ) , we may assume @xmath1008 . for any large real number @xmath939 , let @xmath1421 be the element of @xmath1422 that satisfies @xmath1423 . then , from , we see that @xmath1424 clearly , we have @xmath1425 . a calculation shows that @xmath1426 , and certain other @xmath864 minors also have cancellation . with this in mind , it is not difficult to verify that @xmath1427 ( see ( * ? ? ? * case 3 of pf . of 5.12(@xmath1428 ) ) for details ) . this is a contradiction . [ dim(u)<2n ] if there does not exist a continuous curve @xmath967 in @xmath1429 , such that @xmath1343 , then @xmath1430 furthermore , 1 . [ dim(u)<2n - eqeven ] if equality holds , and @xmath79 is even , then @xmath1431 , for every @xmath1432 ; 2 . [ dim(u)<2n - eq=3 ] if equality holds , and @xmath1433 , then @xmath1434 . by passing to a subgroup , we may assume @xmath1381 . let @xmath163 be the projection of @xmath337 to @xmath1435 ; then @xmath1436 . [ dim(u)<2n-<xy > pf ] assume @xmath1437 for every @xmath1438 . assume @xmath79 is even . from theorem [ hinn - linear]([hinn - linear - phi=0&eta2= ] ) , we know that @xmath163 does not intersect @xmath791 ( or @xmath792 , either , for that matter ) , so @xmath1439 therefore @xmath1440 as desired . ( if equality holds , then we have conclusion ( [ dim(u)<2n - eqeven ] ) . ) [ dim(u)<2npf-<xy>-n=2 ] assume @xmath79 is odd . [ dim(u)<2npf - u / z ] we have @xmath1441 . suppose not : then @xmath1442 ( this will lead to a contradiction . ) let @xmath1443 , so @xmath507 is a @xmath338-subspace of @xmath1444 . for each @xmath1445 , there is some @xmath1446 , such that @xmath1447 ; define @xmath1448 . by the assumption of this case , we know @xmath1449 so @xmath1450 is uniquely determined by @xmath767 ; thus , @xmath1451 is a well - defined @xmath338-linear map . also , again from the assumption of this case , we know that @xmath1452 because @xmath1453 , we have @xmath1454 if @xmath132 ( that is , if @xmath1455 ) , this implies @xmath1456 , so @xmath947 is defined on all of @xmath1457 . because @xmath79 is odd , this implies that @xmath947 has a real eigenvalue , which contradicts . we may now assume @xmath136 . let * @xmath1458 , where @xmath1459 , * @xmath1460 be the projective space of the real vector space @xmath507 , and * @xmath1461 \in { \mathord{\mathbb{p}x}}$ ] , for @xmath1462 , so @xmath1463 is a vector bundle over @xmath1460 . define @xmath1464 by @xmath1465 . any @xmath338-linear transformation @xmath1466 is a continuous function , such that @xmath1467 for all @xmath1445 ; that is , a section of @xmath1463 . thus , @xmath1468 , @xmath947 , and @xmath155 each define a section of @xmath1463 . furthermore , these three sections are pointwise linearly independent over @xmath338 , because implies that @xmath767 , @xmath1469 , and @xmath1470 are linearly independent over @xmath338 , for every nonzero @xmath1445 . on the other hand , the theory of characteristic classes ( * ? ? ? . 4 , p. 39 ) implies that @xmath1463 does not have three pointwise @xmath338-linearly independent sections ( see ( * ? ? ? 8.2 ) for details ) . this is a contradiction . completion of the proof of subcase [ dim(u)<2npf-<xy>-n=2 ] . from step [ dim(u)<2npf - u / z ] , we see that the desired inequality holds . we may now assume @xmath1433 and @xmath1471 . since @xmath1472 , we must have @xmath1455 , so @xmath132 . therefore @xmath1473 , so @xmath1474 as desired . assume there is some @xmath1475 , such that @xmath1476 . [ dimupf - x=0 ] assume @xmath1477 . since @xmath1478 , we must have @xmath1479 . then @xmath1480 for every @xmath1390 ( otherwise [ hinn - linear]([hinn - linear - phi=0&=0 ] ) yields a contradiction ) ; this implies @xmath1481 and @xmath1482 because @xmath1483 , we know that @xmath1484 ; so @xmath1485 since @xmath1486 for every @xmath1390 , but @xmath1487 ( see [ hinn - linear]([hinn - linear - phi=0&eta2= ] ) ) , we must have @xmath1488 for every @xmath1390 . therefore @xmath1489 let @xmath1490 be the natural projection . note that @xmath1491 ( if @xmath1492 , with @xmath1493 , then there is some @xmath599 , such that @xmath1494 . we also have @xmath1495 , so , from [ hinn - linear]([hinn - linear - phi=0&=0 ] ) , we see that @xmath1496 . thus @xmath1497 . so @xmath1498 is 1-dimensional . ) because @xmath1486 for every @xmath1390 , and @xmath337 is a lie algebra , we see , from ( [ [ u , v ] ] ) , that @xmath1499 must be a totally isotropic subspace for the symplectic form @xmath1500 , so @xmath1501 therefore @xmath1502 this completes the proof if @xmath1503 : * if @xmath79 is even , then , because @xmath1504 , we have @xmath1505 . * if @xmath1506 is odd , then @xmath1507 , so @xmath1508 . now let @xmath1433 , and suppose @xmath1509 . ( this will lead to a contradiction . ) because equality is attained in the proof above , we must have @xmath1510 , so there exists @xmath1511 with @xmath1512 . for @xmath599 , let @xmath1513 . then @xmath1514 thus , this expression changes sign , so it must vanish for some @xmath939 . on the other hand , since @xmath1433 , we have @xmath1515 for every @xmath1516 . thus [ hinn - linear]([hinn - linear - phi=0&=0 ] ) yields a contradiction . [ dimupf - y=0 ] assume @xmath1517 . this is similar to subcase [ dimupf - x=0 ] . ( in fact , this can be obtained as a corollary of subcase [ dimupf - x=0 ] by replacing @xmath0 with its conjugate under the weyl reflection corresponding to the root @xmath675 . ) assume @xmath1479 . because @xmath1518 , lemma [ x=0 ] implies there is some @xmath1331 , such that , letting @xmath1519 , we have @xmath1367 , @xmath1368 , and @xmath1520 . there is no harm in replacing @xmath0 with @xmath1521 ( see [ conjcurve ] ) . then subcase [ dimupf - x=0 ] applies ( with @xmath1370 in the place of @xmath722 ) . [ maxnolinear ] recall that assumptions [ standingsu2f ] are in effect . if there does not exist a continuous curve @xmath967 in @xmath0 , such that @xmath1343 , then @xmath1522 furthermore , if equality holds , and @xmath79 is even , then 1 . [ maxnolinear - tu ] @xmath1523 ; 2 . [ maxnolinear - phi0 ] @xmath800 for every @xmath1328 ; 3 . [ maxnolinear-<xy > ] @xmath1524 , for every @xmath1438 ; 4 . [ maxnolinear - z ] @xmath1525 for every nonzero @xmath1390 ; 5 . [ maxnolinear - dimu / z ] @xmath1526 ; and 6 . [ maxnolinear - dimz ] @xmath1527 . let @xmath1528 from lemmas [ dimt]([dimt - a ] ) and [ dim(u+z)<3 ] , and proposition [ dim(u)<2n ] , we have @xmath1529 this implies the desired inequality , unless @xmath1433 , @xmath136 , and we have equality in both lemma [ dim(u+z)<3 ] and proposition [ dim(u)<2n ] . this is impossible , because equality in lemma [ dim(u+z)<3 ] requires @xmath1530 , but proposition [ dim(u)<2n]([dim(u)<2n - eq=3 ] ) implies @xmath1531 . in the remainder of the proof , we assume that equality holds in , and that @xmath79 is even . proposition [ hinn - linear]([hinn - linear - phi=0&eta2= ] ) implies ( [ maxnolinear - z ] ) . [ maxnolinearpf - f = c ] assume @xmath136 . because equality holds , lemma [ dim(u+z)<3 ] implies ( [ maxnolinear - phi0 ] ) and ( [ maxnolinear - dimz ] ) . then proposition [ dim(u)<2n]([dim(u)<2n - eqeven ] ) implies ( [ maxnolinear-<xy > ] ) ( because @xmath1381 ) . since @xmath1381 ( see [ maxnolinear - phi0 ] ) and equality holds in , we have @xmath1532 and @xmath1533 so ( [ maxnolinear - dimu / z ] ) and ( [ maxnolinear - dimz ] ) hold . let @xmath1210 . corollary [ tnormsu ] implies that @xmath328 normalizes @xmath337 , so , from ( [ maxnolinear-<xy > ] ) and lemma [ rootdecomp ] , we see that @xmath1534 . on the other hand , @xmath1535 , so , from equality in , we conclude that @xmath1536 . therefore @xmath1537 . suppose @xmath1538 is any continuous group homomorphism , such that @xmath1539 normalizes @xmath162 . from ( [ maxnolinear-<xy > ] ) and ( [ [ u , v ] ] ) , we see that @xmath1540 , so @xmath343 must be trivial . this implies that [ not - semi]([not - semi - not ] ) can not apply here , so [ not - semi]([not - semi - tu ] ) yields ( [ maxnolinear - tu ] ) . assume @xmath132 . proposition [ dim(u)<2n]([dim(u)<2n - eqeven ] ) implies that @xmath1541 for every @xmath1432 . suppose ( [ maxnolinear - phi0 ] ) is false . then there is some @xmath1328 , such that @xmath1389 . also , because @xmath1542 , we may fix some @xmath1543 then , letting @xmath1544 $ ] , we see , from ( [ [ u , v ] ] ) , that @xmath1545 and @xmath1512 , so @xmath1546 . this contradicts the conclusion of the preceding paragraph . conclusion ( [ maxnolinear-<xy > ] ) follows from ( [ maxnolinear - phi0 ] ) and [ dim(u)<2n]([dim(u)<2n - eqeven ] ) . conclusion ( [ maxnolinear - tu ] ) can be established by arguing as in the last two paragraphs of case [ maxnolinearpf - f = c ] . equations ( [ maxnolinearpf - u / z ] ) and ( [ maxnolinearpf - dimz ] ) establish ( [ maxnolinear - dimu / z ] ) and ( [ maxnolinear - dimz ] ) . our goal is to prove theorem [ bestnosquare ] ; we start with two preliminary results . [ hinn - square ] if there does not exist a continuous curve @xmath967 in @xmath162 , such that @xmath1547 , then 1 . [ hinn - square - indep ] for every element @xmath215 of @xmath1346 , we have @xmath1548 ; 2 . [ hinn - square - eta2neq ] for every element @xmath1344 of @xmath803 , we have @xmath1549 ; and 3 . [ hinn - square - y=0+yy=0 ] for every element @xmath215 of @xmath337 , such that @xmath1389 , @xmath834 , and @xmath1550 , we have @xmath1551 . ( [ hinn - square - indep ] ) suppose there is an element @xmath215 of @xmath1346 , such that @xmath1552 . let @xmath1553 . then , from , we see that @xmath1554 . furthermore , @xmath1555 because @xmath1552 , we have @xmath1556 , so @xmath1557 ; therefore @xmath1558 , so @xmath1559 so lemma [ owalls]([owalls - square ] ) implies that @xmath1547 , as desired . ( [ hinn - square - eta2neq ] ) suppose there is an element @xmath1344 of @xmath803 , such that @xmath1560 ; in other words , we have @xmath1561 . let @xmath1562 ( see [ exp(phi=0 ) ] ) . then @xmath1563 and @xmath1564 so @xmath1565 so lemma [ owalls]([owalls - square ] ) implies that @xmath1547 , as desired . ( [ hinn - square - y=0+yy=0 ] ) suppose there is an element @xmath215 of @xmath337 , such that @xmath1566 , @xmath834 , @xmath1550 , and @xmath1567 . let @xmath1414 . from ( [ exp(y=0 ) ] ) , we see that @xmath1568 ( note that , because @xmath1567 , we have @xmath1569 ) . then @xmath1570 and @xmath1571 so @xmath1572 . thus , lemma [ owalls]([owalls - square ] ) implies that @xmath1547 , as desired . the following lemma obtains a dimension bound from condition [ hinn - square]([hinn - square - indep ] ) . [ dimv < d(n-2 ) ] if @xmath163 is a @xmath338-subspace of @xmath1573 , such that @xmath1574 for every @xmath1575 , then either 1 . [ dimv < d(n-2)-n>3 ] @xmath1576 ; or 2 . [ dimv < d(n-2)-n=3 ] @xmath1433 and @xmath1577 . because @xmath1578 , we may assume that there exist nonzero @xmath1579 , such that @xmath1580 and @xmath1581 ( otherwise , the projection to one of the factors of @xmath1573 is injective when restricted to @xmath163 , so ( [ dimv < d(n-2)-n>3 ] ) holds ) . then @xmath1582 , so , by assumption , we have @xmath1583 . because @xmath1584 and @xmath1585 are nonzero , this implies @xmath1586 . [ dimv < d(n-2)pf - yinfx ] for all @xmath1575 , we have @xmath1587 . we may assume @xmath1588 ( otherwise the desired conclusion is obvious ) . then , since @xmath1589 , we conclude that @xmath1590 . similarly , because @xmath1591 we must have @xmath1592 . therefore @xmath1593 since @xmath1594 , this implies @xmath1595 , so @xmath1587 , as desired . [ dimv < d(n-2)pf - v = fxf ] we have @xmath1596 . given @xmath1575 , step [ dimv < d(n-2)pf - yinfx ] asserts that @xmath1587 . by symmetry ( interchanging the two factors of @xmath1597 ) , we must also have @xmath1598 . so @xmath1599 , as desired . completion of the proof . from step [ dimv < d(n-2)pf - v = fxf ] , we have @xmath1600 if @xmath1504 , then ( [ dimv < d(n-2)-n>3 ] ) holds ; otherwise , ( [ dimv < d(n-2)-n=3 ] ) holds . [ bestnosquare ] recall that assumptions [ standingsu2f ] are in effect . if there does not exist a continuous curve @xmath967 in @xmath0 , such that @xmath1547 , then @xmath1601 . furthermore , if equality holds , then @xmath0 is of the form @xmath1602 , where 1 . [ bestnosquare - t ] @xmath1603 , 2 . [ bestnosquare - u ] @xmath1604 , and 3 . [ bestnosquare - x ] @xmath1605 for every @xmath1606 . note that @xmath1607 ( see [ dimt]([dimt - a ] ) ) and @xmath1608 we have @xmath1609 . suppose not . let @xmath163 be the projection of @xmath1346 to @xmath1610 . we have @xmath1611 and , for every @xmath1612 with @xmath1613 , we have @xmath1614 ( see [ hinn - square]([hinn - square - indep ] ) ) , so lemma [ dimv < d(n-2 ) ] implies that @xmath1433 . therefore @xmath1615 . then , because @xmath1616 , we know that @xmath1617 and @xmath1618 ; thus , there exist @xmath1619 , such that * @xmath1008 , @xmath1330 ; and * @xmath1620 , @xmath1517 . therefore @xmath822 $ ] is a nonzero element of @xmath793 ( see [ [ u , v ] ] ) , so @xmath1621 \bigr ) \neq 0 $ ] . this contradicts lemma [ hinn - square]([hinn - square - eta2neq ] ) . we have @xmath1622 . suppose not : then , because @xmath1623 , there is some @xmath1624 , and , because @xmath1625 , there is some nonzero @xmath1626 , such that @xmath1627 . we must have @xmath1628 ( otherwise [ hinn - square]([hinn - square - eta2neq ] ) yields a contradiction ) ; thus @xmath1629 . we must have @xmath1630 ( otherwise [ hinn - square]([hinn - square - eta2neq ] ) yields a contradiction ) . thus , we see that @xmath1631 is nonconstant as a function of @xmath599 , so [ hinn - square]([hinn - square - eta2neq ] ) yields a contradiction . the desired inequality . we have @xmath1632 as desired . in the remainder of the proof , we assume that @xmath1633 . we must have equality throughout the preceding paragraphs . [ vinalpha+beta ] we have @xmath1634 . suppose not : then there is some @xmath1635 , such that @xmath1636 . let @xmath1637 and @xmath1544 $ ] . then , from , we see that @xmath1545 and @xmath1512 , and that @xmath1638 \in { { \mathfrak{\lowercase{n}}}}_{\alpha+2\beta } + { { \mathfrak{\lowercase{n}}}}_{2\alpha+2\beta}$ ] . from [ hinn - square]([hinn - square - indep ] ) , we have @xmath1639 and @xmath1640 , so @xmath1641 therefore @xmath1642 , so @xmath1643 } \neq 0 $ ] ( see [ [ u , v ] ] ) , so [ hinn - square]([hinn - square - eta2neq ] ) yields a contradiction . we have @xmath1644 . from step [ vinalpha+beta ] , together with the fact that @xmath1645 we conclude that @xmath1646 . therefore , @xmath1647 so @xmath1648 \\ & = [ { \mathfrak{\lowercase{u}}}_{\phi=0 } + { { \mathfrak{\lowercase{d } } } } , { \mathfrak{\lowercase{u}}}_{\phi=0 } + { { \mathfrak{\lowercase{d } } } } ] \\ & = [ { { \mathfrak{\lowercase{n}}}}_{\alpha+\beta } + { { \mathfrak{\lowercase{d } } } } , { { \mathfrak{\lowercase{n}}}}_{\alpha+\beta } + { { \mathfrak{\lowercase{d } } } } ] \\ & = [ { { \mathfrak{\lowercase{n}}}}_{\alpha+\beta},{{\mathfrak{\lowercase{n}}}}_{\alpha+\beta } ] \\ & = { { \mathfrak{\lowercase{n}}}}_{2\alpha+2\beta } . \end{aligned}\ ] ] because @xmath1649 , we must have @xmath1644 . we have @xmath1644 . let @xmath1650 be the projection of @xmath0 to @xmath119 . then there exists @xmath1651 , such that @xmath1652 , and , in the notation of lemma [ rootdecomp ] , we have @xmath1653 because @xmath328 normalizes @xmath337 ( see [ tnormsu ] ) , we know , from lemma [ rootdecomp ] , that @xmath1654 . since @xmath1655 , we know that @xmath1656 projects nontrivially ( in fact , surjectively ) to @xmath682 . on the other hand , we know that @xmath1657 ( otherwise [ hinn - square]([hinn - square - y=0+yy=0 ] ) yields a contradiction ) . therefore @xmath1658 , so there must be a positive root @xmath1659 , such that @xmath1660 . then @xmath1661 ; since @xmath1662 , we must have @xmath1663 . because @xmath1664 , we have @xmath1665 then , since @xmath1666 , we must have @xmath1667 by inspection , we see that this implies @xmath1668 , so we conclude that @xmath1651 , as desired . [ bestnosquarepf - sigma ] we have @xmath1669 , @xmath1603 , and @xmath1670 . since @xmath1671 , it suffices to show @xmath1672 . thus , let us suppose @xmath1673 . ( this will lead to a contradiction . ) we have @xmath1674 ( and recall that @xmath1675 ) , so there is some @xmath1328 , such that @xmath1389 and @xmath1330 . because @xmath1646 , we have @xmath1676 so there is some @xmath1635 , such that @xmath1677 and @xmath1627 . then @xmath822 \in { { \mathfrak{\lowercase{n}}}}_{\alpha+2\beta } + { { \mathfrak{\lowercase{n}}}}_{2\alpha + 2\beta}$ ] , with @xmath1678 } \neq 0 $ ] ( see [ [ u , v ] ] ) , so [ hinn - square]([hinn - square - eta2neq ] ) yields a contradiction . [ bestnosquarepf - tu ] we have @xmath1679 . suppose not : because @xmath1603 , we conclude that there is some nonzero @xmath1680 , such that @xmath1370 normalizes @xmath337 ( see [ not - semi ] ) . if @xmath1681 , then , because @xmath1646 , there is some @xmath1682 , such that @xmath1683 and @xmath1627 . then @xmath1684 \in { { \mathfrak{\lowercase{n}}}}_{\alpha+2\beta } + { { \mathfrak{\lowercase{n}}}}_{2\alpha + 2\beta}$ ] , with @xmath1685 } \neq 0 $ ] ( see [ [ u , v ] ] ) , so [ hinn - square]([hinn - square - eta2neq ] ) yields a contradiction . if @xmath1545 , then , since @xmath1686 , we must have @xmath1687 . there is some @xmath1688 with @xmath1689 . then @xmath1684 \in { { \mathfrak{\lowercase{n}}}}_{\alpha+2\beta } + { { \mathfrak{\lowercase{n}}}}_{2\alpha + 2\beta}$ ] , with @xmath1685 } \neq 0 $ ] ( see [ [ u , v ] ] ) , so [ hinn - square]([hinn - square - eta2neq ] ) yields a contradiction . completion of the proof . ( [ bestnosquare - t ] ) from step [ bestnosquarepf - tu ] , we know that @xmath1690 , and , from step [ bestnosquarepf - sigma ] , that @xmath1603 . ( [ bestnosquare - u ] ) since @xmath1691 , it suffices to show @xmath1692 : given @xmath1693 , we wish to show @xmath1629 . because @xmath1646 , we know that @xmath1694 . thus , all that remains is to show that @xmath1627 . if not , then choosing @xmath1328 with @xmath1389 , we see that @xmath1678 } \neq 0 $ ] ( see [ [ u , v ] ] ) . so [ hinn - square]([hinn - square - eta2neq ] ) yields a contradiction . ( [ bestnosquare - x ] ) from lemma [ hinn - square]([hinn - square - y=0+yy=0 ] ) , we know that @xmath1551 for every @xmath1606 . this section proves two main results . both assume that @xmath1 is either @xmath73 or @xmath52 . 1 . theorem [ suf->complete ] shows that if @xmath79 is odd , and one or two specific homogeneous spaces of @xmath1 do not have tessellations , then no interesting homogeneous space of @xmath1 has a tessellation . 2 . theorem [ sufeventess ] shows that if @xmath79 is even , then certain deformations of the examples found by r. kulkarni and t. kobayashi ( see [ kulkarnieg ] ) are essentially the only interesting homogeneous spaces of @xmath1 that have tessellations . assume conjecture [ notesssu / sp ] is true , and suppose @xmath3 is a crystallographic group for @xmath2 . ( this will lead to a contradiction . ) let @xmath1697 ( see [ su1ndefn ] ) . from ( [ d(sp ) ] ) , we have @xmath1698 and @xmath1699 , where @xmath752 . we may assume that @xmath91 ( see [ hcanbean ] ) , and that @xmath0 is compatible with @xmath119 ( see [ conjtocompatible ] ) . because @xmath0 is not a cartan - decomposition subgroup ( see [ cds->notess ] ) , the contrapositive of proposition [ cds<>h_m ] implies , for some @xmath577 , that there does * not * exist a continuous curve @xmath1700 in @xmath0 , such that @xmath988 . therefore , either theorem [ maxnolinear ] ( if @xmath1201 ) or theorem [ bestnosquare ] ( if @xmath1202 ) implies that @xmath1701 . theorem [ noncpctdim]([noncpctdim - tess ] ) ( combined with the fact that @xmath1702 ) implies that @xmath149 has a tessellation . this contradicts either theorem [ so2n / so1odd - notess ] ( if @xmath132 ) or conjecture [ notesssu / sp][notesssu / sp - su2/su1 ] ( if @xmath136 ) . from lemma [ mu(suorsp ) ] , we know that @xmath361 and @xmath1703 are the two walls of @xmath154 , so corollary [ tess->misshk ] ( combined with the assumption of this case ) implies that @xmath3 acts properly discontinuously on @xmath450 . therefore , since conjecture [ notesssu / sp][notesssu / sp - so2/su1][notesssu / sp - su2/sp1 ] asserts that @xmath450 does not have a tessellation , the contrapositive of theorem [ noncpctdim]([noncpctdim - tess ] ) ( with @xmath95 in the role of @xmath94 ) implies that @xmath1704 . hence , the contrapositive of theorem [ maxnolinear ] implies there is a continuous curve @xmath1700 in @xmath0 , such that @xmath1343 . thus , there is a compact subset @xmath12 of @xmath1 , such that @xmath147 ( see [ su1inh ] ) . since @xmath3 acts properly discontinuously on @xmath2 , this implies that @xmath3 acts properly discontinuously on @xmath149 ( see [ chcproper ] ) . this contradicts the assumption of this case . 1 . [ sufeventess - sp ] there is an @xmath338-linear map @xmath1050 , such that 1 . [ sufeventess - sp - xb ] @xmath1706 for every @xmath1707 ( see [ bsymplectic ] ) , and 2 . [ sufeventess - sp-<x , y > ] @xmath1065 , for every nonzero @xmath766 ( see [ xbnotinfx ] ) , and 3 . [ sufeventess - sp - hb ] @xmath0 is conjugate to @xmath1057 ( see [ hb - defn ] and [ hb = sp1 m ] ) ; or 2 . [ sufeventess - sur ] @xmath132 and @xmath0 is conjugate to @xmath1708 ( see [ su1ndefn ] ) ; or 3 . [ sufeventess - suc ] @xmath136 and there exists @xmath1119 $ ] , such that @xmath0 is conjugate to @xmath1123}}}$ ] ( see [ suegsdefn ] ) . also , we may assume @xmath0 is compatible with @xmath119 ( see [ conjtocompatible ] ) . because @xmath0 is not a cartan - decomposition subgroup ( see [ cds->notess ] ) , proposition [ cds<>h_m ] implies that one of the following two cases applies . 1 . [ sufeventesspf - tu ] @xmath1537 ; 2 . [ sufeventesspf - phi0 ] @xmath800 for every @xmath1328 ; 3 . [ sufeventesspf-<xy > ] @xmath1524 , for every @xmath1438 ; 4 . [ sufeventesspf - z ] @xmath1525 for every nonzero @xmath1390 ; 5 . [ sufeventesspf - dimu / z ] @xmath1526 ; and 6 . [ sufeventesspf - dimz ] @xmath1527 . for any @xmath1390 with @xmath1716 , we know , from ( [ sufeventesspf - z ] ) , that @xmath1099 ; therefore , lemma [ o(linear)]([o(linear)-o ] ) implies there exist @xmath338-linear maps @xmath1717 and @xmath1718 , such that @xmath1719 for all @xmath1390 . more concretely , we may say that there exist @xmath1111 and @xmath1720 , such that @xmath1721 for all @xmath1390 . let @xmath722 be the element of @xmath682 with @xmath1722 , and let @xmath1723 be the conjugate of @xmath0 by @xmath1724 . then @xmath124 satisfies the conditions imposed on @xmath0 ( note that @xmath124 , like @xmath0 , is compatible with @xmath119 ( see [ conjuomega ] ) ) , so there exist @xmath1725 and @xmath1726 , such that @xmath1727 for all @xmath1728 . given @xmath1728 with @xmath1729 , let @xmath1730 . because @xmath1731 , we have @xmath1732 , -v \bigr ] = 0 $ ] , so , from remark [ conjugation ] and , we see that @xmath1733 @xmath1734 and @xmath1735 therefore @xmath1736 since @xmath1737 is arbitrary , this implies @xmath1738 . thus , by replacing @xmath0 with @xmath124 , we may assume that @xmath1739 . this means that @xmath1740 for all @xmath1390 . from ( [ sufeventesspf - dimz ] ) ( and because @xmath136 , so @xmath1741 ) , we know that @xmath1742 , so there is some nonzero @xmath1743 , such that @xmath1744 . ( so @xmath1745 . ) then @xmath1746 , so we see , from ( [ sufeventesspf - z ] ) , that @xmath1747 for every nonzero @xmath1390 . now , since @xmath1748 there is some nonzero @xmath1390 , such that @xmath1749 . we have @xmath1750 because @xmath1751 is pure imaginary , we know that @xmath1752 , so this implies that @xmath1753 . thus , replacing @xmath0 by a conjugate under a diagonal matrix , we may assume @xmath1754 , as desired . thus , in the notation of lemma [ rootdecomp ] , we have @xmath1760 and @xmath1761 , so @xmath1762 , as desired . ( note that this is a direct sum of vector spaces , not of lie algebras : we have @xmath1763 \subset { { { \mathfrak{\lowercase{d}}}}_{{\mathfrak{\lowercase{h}}}}}$ ] . ) for any @xmath1764 with @xmath1008 , we have @xmath1765 so @xmath1766 ( see [ sufeventesspf-<xy > ] ) ; therefore , lemma [ o(linear)]([o(linear)-o ] ) implies there is a @xmath338-linear map @xmath1767 , such that @xmath1768 for all @xmath1764 . then , because @xmath1769 ( see [ sufeventesspf - dimu / z ] ) , we must have @xmath1770 combining this with ( [ sufeventesspf - tu ] ) and the conclusions of steps [ sufeventesssppf - xx = yy ] and [ sufeventesspf - u+z ] , we see that @xmath1771 . therefore @xmath1772 , so conclusion ( [ sufeventess - sp - hb ] ) holds . letting @xmath1773 $ ] , for any @xmath1774 , we see , from , that @xmath1775 and @xmath1776 from step [ sufeventesssppf - xx = yy ] , we know that @xmath1777 , so this implies that conclusion ( [ sufeventess - sp - xb ] ) holds . let @xmath1782 ( so @xmath1783 ) . let @xmath1144 be the sesquilinear form ( or bilinear form , if @xmath132 ) on @xmath1784 defined by @xmath1785 let @xmath1786 from ( [ pdqf ] ) , we see that the restriction of @xmath1787 to @xmath1788 is a ( positive - definite ) inner product . let @xmath1789 be the @xmath1790-orthogonal complement to @xmath1788 . as a form over @xmath751 , @xmath1144 has signature @xmath1791 . thus , as a form over @xmath338 , @xmath1787 has signature @xmath1792 . since @xmath1793 we conclude that @xmath1789 is a @xmath1794-dimensional @xmath338-subspace on which @xmath1787 is negative - definite . choose some nonzero @xmath1795 . multiplying by a real scalar to normalize , we may assume @xmath1796 . because @xmath1797 is transitive on the vectors of norm @xmath1116 , there is some @xmath1798 , such that @xmath1799 . thus , letting @xmath1800 and @xmath1801 , we have @xmath1802 , so , by replacing @xmath270 with the conjugate @xmath1803 , we may assume @xmath1804 . choose some nonzero @xmath1807 , such that @xmath722 is @xmath1790-orthogonal to @xmath215 . multiplying by a real scalar to normalize , we may assume @xmath1808 . by replacing @xmath722 with @xmath1809 if necessary , we may assume @xmath1810 . because @xmath722 is @xmath1790-orthogonal to @xmath1804 , we have @xmath1811 ( see [ uperp ] ) . let @xmath1812 and @xmath1813 . then @xmath1814 so @xmath1815 also , @xmath1816 so @xmath1817 . thus , @xmath486 and @xmath939 are of opposite signs so , because @xmath1818 therefore , we may choose @xmath1819 $ ] , such that @xmath1820 let @xmath1821 then @xmath1822 and @xmath1823 hence , there is some @xmath1824 , such that @xmath1825 and @xmath1826 . thus , replacing @xmath270 with the conjugate @xmath1827 ( cf . [ ghat ] ) , we may assume @xmath1828 . therefore @xmath1829 by combining this with and comparing with ( [ suegsdefn ] ) ( with @xmath136 ) , we conclude that @xmath1830 } } $ ] . by comparing dimensions , we see that equality must hold ; this establishes conclusion ( [ sufeventess - suc ] ) ( because @xmath1831 ) . ( [ suf - known - sim ] ) in each case , there is some @xmath425 , such that @xmath966 for @xmath591 ( see [ hbthm]([hbthm - mu ] ) , [ mu(suorsp)]([mu(suorsp)-su ] ) , and [ suegs]([suegs - linear ] ) ) . then corollary [ hsimsuorsp ] implies either that @xmath1038 ( if @xmath1835 ) or that @xmath976 ( if @xmath1202 ) . suppose @xmath1837}}}g = { h_{[c']}}$ ] , for some @xmath1838 . because all maximal split tori in @xmath1154}}$ ] are conjugate , we may assume that @xmath155 normalizes @xmath1839 . since all roots of @xmath1839 on both @xmath1121}}}$ ] and @xmath1840}}$ ] are positive , @xmath155 can not invert @xmath1841 , so we conclude that @xmath155 centralizes @xmath1841 ; that is , @xmath1842 . in the notation of case [ sufeventesspf - nosquare ] of the proof of theorem [ sufeventess ] , define @xmath1843 ( cf . [ ghat ] ) . then @xmath1844 , so we may assume @xmath1845 ( because @xmath1841 , being a subgroup of @xmath1123}}}$ ] , obviously normalizes @xmath1123}}}$ ] ) . write @xmath1846 . then , because @xmath1837}}}g = { h_{[c']}}$ ] , we must have @xmath1847 } } } } ) = v_{{{{\mathfrak{\lowercase{h}}}}_{[c']}}}$ ] ; hence @xmath1847}}}}^\perp ) = v_{{{{\mathfrak{\lowercase{h}}}}_{[c']}}}^\perp$ ] . for any basis @xmath1848 of @xmath1849}}}}^\perp$ ] with @xmath1850 and @xmath1851 , we have @xmath1852 similarly for any @xmath1790-orthonormal basis @xmath1853 of @xmath1854}}}^\perp$ ] . because @xmath1824 , this implies @xmath1855 . because @xmath1156 $ ] , we conclude that @xmath1836 , as desired . m. cowling , sur les coefficients des reprsentations unitaires des groupes de lie simple , in : p. eymard , j. faraut , g. schiffmann , and r. takahashi , eds . : _ analyse harmonique sur les groupes de lie ii _ ( sminaire nancy - strasbourg 197678 ) , lecture notes in math . # 739 , springer , new york , 1979 , pp . 132178 . r. ellis and m. nerurkar , enveloping semigroup in ergodic theory and a proof of moore s ergodicity theorem , in : j. c. alexander . , ed . , _ dynamical systems ( college park , md , 198687 ) _ , lecture notes in math . # 1342 , springer , new york , 1988 , pp . 172179 . m. gromov , asymptotic invariants of infinite groups , in : g. a. niblo and m. a. roller , eds . , _ geometric group theory , vol . 2 _ ( sussex , 1991 ) , london math . soc . lecture notes # 182 , cambridge univ . press , cambridge , 1993 , pp . 1295 . r. howe , a notion of rank for unitary representations of the classical groups , in : a. fig talamanca , ed . : _ harmonic analysis and group representations _ , ( cime 1980 ) , liguori , naples , 1982 , pp . 223331 . t. kobayashi , discontinuous groups and clifford - klein forms of pseudo - riemannian homogeneous manifolds , in : b. rsted and h. schlichtkrull , eds . , _ algebraic and analytic methods in representation theory , _ academic press , new york , 1997 , pp . 99165 . f. labourie , quelques rsultats rcents sur les espaces localement homognes compacts , in : p. de bartolomeis , f. tricerri and e. vesentini , eds . , _ manifolds and geometry _ , symposia mathematica , v. xxxvi , cambridge u. press , 1996 .

let @xmath0 be a closed , connected subgroup of a connected , simple lie group @xmath1 with finite center . the homogeneous space @xmath2 has a _ tessellation _ if there is a discrete subgroup @xmath3 of @xmath1 , such that @xmath3 acts properly discontinuously on @xmath2 , and the double - coset space @xmath4 is compact . note that if either @xmath0 or @xmath2 is compact , then @xmath2 has a tessellation ; these are the obvious examples . it is not difficult to see that if @xmath1 has real rank one , then only the obvious homogeneous spaces have tessellations . thus , the first interesting case is when @xmath1 has real rank two . in particular , r. kulkarni and t. kobayashi constructed examples that are not obvious when @xmath5 or @xmath6 . h. oh and d. witte constructed additional examples in both of these cases , and obtained a complete classification when @xmath7 . we simplify the work of oh - witte , and extend it to obtain a complete classification when @xmath8 . this includes the construction of another family of examples . the main results are obtained from methods of y. benoist and t. kobayashi : we fix a cartan decomposition @xmath9 , and study the intersection @xmath10 . our exposition generally assumes only the standard theory of connected lie groups , although basic properties of real algebraic groups are sometimes also employed ; the specialized techniques that we use are developed from a fairly elementary level . # 1 latex@info#1 font@info#1

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Proceed to summarize the following text: the hubble space telescope and advances in ground based observing have greatly increased our knowledge of the galaxy population in the distant universe . however , the nature of these galaxies and their evolutionary connections to local galaxies remain poorly understood . luminous , compact , star forming galaxies appear to represent a prominent phase in the early history of galaxy formation @xcite . in particular : * the number density of luminous , compact star forming galaxies rises significantly out to z @xmath1 1 @xcite . * the lyman break galaxies at z @xmath12 2 seen in the hubble deep field are characterized by very compact cores and a high surface brightness @xcite . * sub - millimeter imaging has revealed distant galaxies ( z @xmath1 2@xmath134 ) , half of them compact objects , which may be responsible for as much as half of the total star formation rate in the early universe @xcite . however , little is definitively known of their physical properties , or how they are related to subsets of the local galaxy population . a classification for known examples of intermediate redshift ( 0.4 @xmath0 z @xmath0 0.7 ) luminous , blue , compact galaxies , such as blue nucleated galaxies , compact narrow emission line galaxies , and small blue galaxies , has been developed by @xcite in order to be able to choose samples over a wide redshift range . they have found that the bulk of these galaxies , collectively termed luminous compact blue galaxies ( lcbgs ) , can be distinguished quantitatively from local normal galaxies by their blue color , small size , high luminosity , and high surface brightness . ( see 2.1 for more detail . ) from studies at intermediate redshifts , it has been found that lcbgs are a heterogeneous class of vigorously starbursting , high metallicity galaxies with an underlying older stellar population @xcite . while common at intermediate redshifts , they are rare locally @xcite and little is known about the class as a whole , nor their evolutionary connections to other galaxies . lcbgs undergo dramatic evolution : at z @xmath1 1 , they are numerous and have a total star formation rate density equal to that of grand - design spirals at that time . however , by z @xmath1 0 , the number density and star formation rate density of lcbgs has decreased by at least a factor of ten @xcite . since the lcbg population is morphologically and spectroscopically diverse , these galaxies are unlikely to evolve into one homogeneous galaxy class . @xcite and @xcite suggest that a subset of lcbgs at intermediate redshifts may be the progenitors of local low - mass dwarf elliptical galaxies such as ngc 205 . alternatively , @xcite and @xcite suggest that others may be disk galaxies in the process of building a bulge to become local l@xmath2 spiral galaxies . clearly , to determine the most likely evolutionary scenarios for intermediate redshift lcbgs , it is necessary to know their masses and the timescale of their starburst activity . are they comparable to today s massive or low - mass galaxies ? are they small starbursting galaxies which will soon exhaust their gas and eventually fade ? or are they larger galaxies with only moderate amounts of star formation ? only kinematic line widths that truly reflect the masses of these galaxies , as well as measures of their gas content and star formation rates , can answer these questions . using ionized gas emission line widths , @xcite , @xcite , and @xcite , have found that lcbgs have mass - to - light ratios approximately ten times smaller than typical local l@xmath2 galaxies . however , since ionized gas emission lines may originate primarily from the central regions of galaxies , their line widths may underestimate the gravitational potential @xcite . h emission lines provide a better estimate of the total galaxy mass as they measure the gravitational potential out to larger galactic radii . observations of both h and co ( the best tracer of cold h@xmath14 ) , combined with star formation rates , are necessary to estimate the starburst timescales . with current radio instrumentation , h and co can only easily be measured in very nearby lcbgs , at distances @xmath0 150 mpc for h , and @xmath0 70 mpc for co. therefore , to understand the nature and evolutionary possibilities of higher redshift lcbgs , we have undertaken a survey in h 21 cm emission and multiple rotational transitions of co of a sample of 20 local lcbgs , drawn from the sloan digital sky survey @xcite . this work , paper i , reports the optical photometric properties of our sample and the results of the h 21 cm portion of the survey , including dynamical masses and comparisons with local galaxy types . paper ii @xcite will report the results of a survey of the molecular gas conditions . knowledge of the dynamical masses , combined with gas masses and star formation rates , constrains the evolutionary possibilities of these galaxies . nearby blue compact galaxies ( bcgs ) have been studied extensively at radio and optical wavelengths since @xcite originated the term `` compact galaxy '' and @xcite distinguished between `` red '' and `` blue '' compact galaxies . the term bcg typically refers to galaxies with a compact nature , a high mean surface brightness , and emission lines superposed on a blue continuum . however , many different selection criteria have been used , leading to various definitions of bcgs and samples with a range of properties . for example , the term `` dwarf '' has been used to mean bcgs fainter than @xmath1317 ( e.g. thuan & martin 1981 ; kong & cheng 2002 ) or @xmath1318 blue magnitudes ( e.g. taylor et al . 1994 ) , or an optical diameter less than 10 kpc @xcite . the term `` blue '' has been used to mean blue on the palomar sky survey plate ( e.g. gordon & gottesman 1981 ) , or to have emission lines superposed on a blue background ( e.g. thuan & martin 1981 ) . the term `` compact '' has been used to mean smaller than 1 kpc in optical diameter ( e.g. thuan & martin 1981 ) , or qualitatively compact ( e.g. doublier et al . some , for example @xcite , began using the term blue compact dwarf to refer to the common low luminosity , low metallicity bcgs . as high luminosity ( @xmath1l@xmath2 ) , nearby bcgs are rare , many began to use the term blue compact dwarf for all nearby bcgs , regardless of luminosity . there are only a few of the rare local lcbgs ( by the @xcite definition ) in previous bcg surveys . note that @xcite have recently been studying local luminous bcgs . however , their selection criteria are not as stringent as that of @xcite , and they have been focusing on very low metallicity ( less than 15% solar ) galaxies . intermediate redshift lcbgs with rest - frame properties matching the class definition of @xcite have metallicities at least 40% solar @xcite . four of our local lcbgs have metallicities available in the literature , which range from 40 @xmath13 70 % solar @xcite . by using the selection criteria of @xcite , we ensure that our study is of those local lcbgs defined to be analogs to the widely studied higher redshift lcbgs . the observations , including sample selection and data reduction , are described in @xmath152 . the optical photometric properties , h 21 cm spectra , measurements , and derived properties are presented in @xmath153 , and analyzed in @xmath154 . we compare the derived physical properties to local normal galaxies and higher redshift lcbgs in @xmath155 , and conclude in @xmath156 . we assume h@xmath16 = 70 km s@xmath17 mpc@xmath17 throughout . when we compare our results to those of other authors , we scale their results to this value . when @xcite compared intermediate redshift lcbgs with local normal galaxies , they found that lcbgs can be isolated quantitatively on the basis of color , surface brightness , image concentration , and asymmetry . color and surface brightness were found to give the best leverage for separating lcbgs from normal galaxies . specifically , lcbgs can be defined by a region limited to b@xmath13v @xmath18 0.6 , and a b - band surface brightness within the half - light radius , sbe , brighter than 21 b - mag arc sec@xmath19 . this simple definition differs only slightly from the formal definition in @xcite which uses a color - dependent sbe . @xcite also applied a luminosity cut - off of 25% l@xmath20m@xmath21 @xmath18 @xmath1318.5 ) , to distinguish lcbgs from blue compact galaxies . lcbgs are not so extreme that they are completely separated from the continuum of normal galaxies . the sharp borders used to classify them are artificial , but serve the purpose of defining similar objects over a range of redshifts . lcbgs at intermediate redshifts ( z @xmath1 0.6 ) can be studied in deep spectroscopic surveys ( i @xmath1 24 ) , while the brightest lcbgs at high redshifts ( z @xmath1 3 ) require very deep spectroscopic surveys ( i @xmath1 26 ) . therefore , lcbgs selected in this manner are observable over a wide range of redshifts . using these color , surface brightness , and luminosity criteria , we selected our sample of local lcbgs from the sloan digital sky survey ( sdss ) . begun in 2000 , this survey will ultimately image one quarter of the sky using a large format ccd camera on a 2.5 m telescope at apache point observatory in new mexico . images are taken in five broad bands ( u , g , r , i , z ) which range from 3540 ( u ) to 9130 ( z ) . the survey has a limiting magnitude of 22.2 in g ( 4770 ) and r ( 6230 ) , our two bands of interest . after searching through approximately one million galaxies ( @xmath11500 degree@xmath22 on the sky ) , we identified only 16 nearby ( d @xmath0 70 mpc ) lcbgs . this distance cut - off was chosen to ensure our galaxies could be detected quickly in both h and co. we added four markarian galaxies from the literature which fulfilled our selection criteria and were not yet surveyed by sdss , for a total of 20 local lcbgs . the color and surface brightness characteristics of our local sample , as well as higher redshift ( 0.4 @xmath0 z @xmath0 1 ) samples of lcbgs , are compared to other nearby galaxies in figure 1 . we note that while our sample is not complete , it is representative of local lcbgs ( see castander et al . 2004 ) . as at higher redshifts @xcite , our local lcbg sample is a morphologically heterogeneous mixture of galaxies , including sb to sc spirals , s0 galaxies , polar ring galaxies , peculiar galaxies , and h liners and starbursts , as classified by hyperleda and the nasa@xmath23ipac extragalactic database ( ned ) . approximately one quarter of these galaxies are not classified in either database . roughly one third are members of multiple , sometimes interacting or merging , systems . sdss and digitized sky survey images of our sample of 20 galaxies are shown in figures 2 and 3 . the properties of our sample of local lcbgs are described below , and listed in table 1 . _ full sdss galaxy designation of the form sdss [email protected] , in the j2000 system . in the remainder of the paper , individual galaxies are referred to by an abbreviated sdss name of the form sdssjhhmm@xmath24ddmm . the four non - sdss galaxies are referred to by their markarian ( mrk ) names . _ alternate name . _ mrk and/or ngc designation , if any . _ d@xmath25 . _ hubble distance , in mpc , calculated from the optical redshift ( from sdss , except for the non - sdss galaxies for which we use ned redshifts . ) _ m@xmath21 . _ absolute blue magnitude calculated from the apparent blue magnitude , m@xmath21 , using d@xmath25 . for the sdss galaxies , m@xmath21 = g + 0.30(g@xmath13r ) + 0.18 , b@xmath13v , and r@xmath26(b ) are based on synthetic spectra which fit the observed spectral energy distribution of lcbgs . ] . the r and g magnitudes were calculated from the sdss `` petrosian '' and `` model '' magnitudes : r = r(petro ) and g = r(petro ) @xmath13 [ r(model ) @xmath13 g(model ) ] @xcite . the sdss galactic reddening corrections were applied , but no k corrections were applied as our galaxies are nearby , and many are of unknown spectral type . the k correction is at most 0.015 magnitudes in r and 0.038 magnitudes in g @xcite . no correction was applied for extinction due to inclination , also because of the uncertainty in spectral types . for the non - sdss galaxies , the `` total apparent blue magnitudes '' were used from hyperleda , and corrected for galactic reddening using extinction values from ned . _ b@xmath13v . _ the color transformation is b@xmath13v @xmath27 0.900(g@xmath13r ) + 0.145@xmath28 , where the magnitudes were calculated as above , for the sdss galaxies . the `` total b@xmath13v colors '' from hyperleda were used for the non - sdss galaxies . they were corrected for reddening using extinction values from ned . _ average surface brightness in the b band within the half light or effective radius in b - mag arc sec@xmath19 . for the sdss galaxies , sbe(b ) = m@xmath21 + 2.5 @xmath29 [ 2@xmath30r@xmath31(b ) ] , where r@xmath26(b ) , the effective or half - light radius in the b band , was calculated from the `` petrosian '' effective radii in the g and r bands [ r@xmath26(b ) = 1.30 @xmath32(g ) @xmath13 0.300 @xmath32(r)]@xmath28 . we corrected sbe for cosmological dimming by subtracting 7.5 @xmath29(1 + z ) , where the redshift , z , was calculated using the h velocity ( @xmath153.1 ) . for the non - sdss galaxies , we used sbe from hyperleda ( termed the `` mean effective surface brightness '' ) , and then corrected it for cosmological dimming in the same way . _ ned type . _ morphological type as given by ned , which does not use a homogeneous system . _ hyperleda type . _ morphological type as given by hyperleda , which uses the @xcite system . _ other sources in beam ? _ indicates whether or not each galaxy has other sources at similar velocities within the [email protected] beam of the green bank telescope ( used for the h observations ) , as given by ned . we compared hyperleda and sdss magnitudes for our sdss selected galaxies to ensure that there are no systematic offsets between the two . fourteen galaxies had magnitudes measured in both systems . the median difference between sdss and hyperleda magnitudes ( as calculated in @xmath152.1.2 ) is 0.1 magnitudes . since the standard deviation of these differences is @xmath34 0.8 magnitudes , we estimate the uncertainty in the median magnitude difference to be 0.2 magnitudes . therefore , there is no significant difference between the hyperleda and sdss magnitudes . unfortunately , while fourteen galaxies have magnitudes measured in both systems , only two sdss selected galaxies have b@xmath13v and sbe entries in hyperleda ; for these two galaxies there are no significant differences between the hyperleda and sdss values . h 21 cm observations of 19 of the 20 nearby lcbgs in our sample were made with the 100 meter green bank telescope ( gbt ) at the national radio astronomy observatory in green bank , west virginia between 2002 november 30 and december 6 . the h spectrum of sdssj0934@xmath240014 was acquired by @xcite during gbt commissioning on 2001 november 29 . the main beam half power width is [email protected] at 21 cm @xcite . both linear and circular polarizations were observed using the l - band receiver ( 1.15 @xmath13 1.73 ghz ) . position - switching was used with an offset of @xmath1318@xmath33 in right ascension . the spectral processor was used with a bandwidth of 20 mhz ( @xmath14000 km s@xmath17 ) for most galaxies , although a few were observed with a bandwidth of 5 mhz ( @xmath11000 km s@xmath17 ) . the sample time was 60 s , with each galaxy observed for between 6 and 30 minutes , for a peak signal - to - noise of at least five . the only exceptions were sdssj0218@xmath130757 and sdssj0222@xmath130830 , marginal detections , which were observed for 52 minutes each . the initial data calibration and reduction was performed using the aips++ single dish analysis environment `` dish . '' the position - switched data were calibrated in the standard way taking the difference of the on and off scans divided by the off scans . the dish package automatically calibrates the data into temperature units using tabulated values for the position of the telescope . the individual scans were then combined and a first order baseline was fit to the line - free regions and removed . our wide bandwidths ensured a sufficient line - free region . the individual polarizations were then combined , except when one polarization was significantly noisier than the other , or was affected by radio interference . finally , each spectrum was smoothed to @xmath112 km s@xmath17 channels using boxcar smoothing . the rest of the reduction and analysis was done using our own procedures written in interactive data language ( idl ) . to convert our data from temperature to flux density units , we observed the radio galaxy 3c295 . it makes an ideal calibration source as it has not varied by more than @xmath11@xmath35 since 1976 for 2.8 cm @xmath36 @xmath37 @xmath36 21 cm @xcite . comparing our observations of 3c295 with those of @xcite , we found the gain of the gbt to be 1.9 k jy@xmath17 and applied this calibration to our data . finally , obvious noise spikes were clipped out of the data . data of sdssj0934@xmath240014 , the galaxy observed during telescope commissioning , were reduced and calibrated by @xcite . the spectrum was then smoothed to @xmath112 km s@xmath17 channels . the central 800 km s@xmath17 of the final h 21 cm spectra of the 20 local lcbgs are shown in figure 4 . each galaxy s velocity calculated from optical redshifts is indicated with a triangle . dashed lines indicate the 20% crossings used to measure the line widths ( @xmath153.1 ) . all 20 galaxies were detected in the 21 cm line of h , although two , sdssj0218@xmath130757 and sdssj0222@xmath130830 , were only detected at the 3 and 4 @xmath38 ( respectively ) level . h measurements and derived optical and h quantities are listed in tables 2 , 3 and 4 and described in the following sections . since the gbt beam is [email protected] at 21 cm , the emission from the target galaxies and any other galaxies within the beam and at similar velocities are blended into one spectrum . ( see table 1 for those galaxies with other sources at similar velocities within the gbt beam . ) we find fairly broad h profiles ( 126 km s@xmath17 @xmath18 w@xmath39 @xmath18 362 km s@xmath17 ) and a variety of profile shapes , including double - horned , flat - topped and gaussian . over a third of our profiles appear asymmetric and only half of these are from galaxies known to have other sources within the gbt beam . these asymmetries may indicate asymmetries in the gas density distribution and/or disk kinematics , or unidentified galaxies within the beam @xcite . estimates of the dynamical masses of our local sample of lcbgs constrain their evolutionary possibilities . we estimate the dynamical mass , m@xmath40 , within a radius , r , as @xmath41 the constant c@xmath14 is a geometry - dependent factor which depends on the galactic light profile . for example , the king models of @xcite give 0.9 @xmath0 c@xmath14 @xmath0 1.5 , depending on the ratio of tidal to core radius . as we do not have information on the light profiles of our galaxy sample , we simply choose c@xmath14 = 1 . the line width at 20@xmath35 , corrected for the effects of inclination , w@xmath42 , is used as a measure of twice the maximum rotational velocity , v@xmath43 ( e.g. * ? ? ? w@xmath39 was measured at the points equal to 20% of the peak flux . the first crossing at 20@xmath35 on each side of the emission line was used , once the spectrum became distinguishable from the noise . these crossings are indicated by dashed lines in figure 4 . we find w@xmath39 ranging from 126 @xmath13 362 km s@xmath17 . when @xcite studied lcbgs at intermediate redshifts , they found a similar range of line widths , using optical emission lines . the recessional velocities in the barycentric system , v@xmath44 , were calculated as the midpoint between the 20@xmath35 crossings . the uncertainties listed in table 2 for w@xmath39 , v@xmath44 , and the galaxy distance ( d ) are from uncertainties in measuring the exact w@xmath39 crossings . two galaxies have multiple crossings at the 20% flux level , once the spectrum is distinguishable from the noise . one of these , sdssj0834@xmath240139 , has a wing on the h spectrum which extends over 200 km s@xmath17 . neither sdssj0834@xmath240139 nor its nearby companion have optical velocities coincident with the peak of the h spectrum ; instead , both have velocities in the wing of the spectrum ( see figure 4 ) . for this reason , we used w@xmath39 measured from the last crossing , which is at the edge of the wing . the other galaxy with multiple crossings at 20% is sdssj0904@xmath245136 , which has a low level extension to one side of the spectrum . we used the first crossing at 20% . the uncertainties in these two w@xmath39 measurements are reflected in the large errors associated with w@xmath39 and derived quantities . we have initiated an observing program at the very large array ( vla ) to map these local lcbgs in h to disentangle target galaxy emission from any other galaxies in the field , improving the mass estimates . to correct the line width for the effects of inclination ( _ i _ ) , we divided w@xmath39 by @xmath45(_i _ ) , to give w@xmath42 . each sdss galaxy s inclination was approximated from the sdss data by : @xmath46 the sdss isophotal major and minor axes in the g - band ( 4770 ) were used , except for sdssj0943@xmath130215 where no g band data were available . in that case , r - band ( 6230 ) data were used . the sdss isophotal axes are derived from the ellipticity of the 25 magnitude arc sec@xmath19 isophote in each band . inclinations derived from sdss isophotal axes in r , i ( 7630 ) , and z ( 9130 ) bands agree with inclinations derived from g - band data to within 8@xmath47 . the inclinations from hyperleda were used for the markarian galaxies . thirteen of our sdss galaxies also have inclinations available in hyperleda , which are calculated from the apparent flattening and morphological type of the galaxies . we estimate the dispersion of the differences between sdss and hyperleda inclinations is 12@xmath47 . a further correction can be made to w@xmath42 to account for random motions , giving w@xmath48 . as outlined in @xcite , @xmath49 - 2w_t^2\exp-(w_{20}/w_c)^2\ ] ] and @xmath50 w@xmath51 is the random motion component of the line width ; w@xmath51 = 38 km s@xmath17 . w@xmath52 characterizes the transition region between linear and quadrature summation of rotational and dispersive terms ; w@xmath52 = 120 km s@xmath17 . the formula degenerates to linear summation for giant galaxies , and quadrature summation for dwarf galaxies @xcite . this correction for random motions decreases the line width of the local lcbg sample by 26 to 38 km s@xmath17 , depending on the rotational velocity . it is crucial , when comparing dynamical masses from different studies , that the rotational velocities be calculated in the same way . in general , we calculate the rotational velocities from w@xmath53 . however , when we wish to compare our sample with others ( e.g. * ? ? ? * ) who have not applied this correction for random motions to their line widths , we also do not apply this correction . we indicate dynamical masses calculated without correcting w@xmath42 for random motions as m@xmath54 . unless indicated as such , all dynamical masses are calculated using the line width corrected for random motions , w@xmath53 . note that in all cases , the line widths have been corrected for inclination . it is also crucial , when comparing dynamical masses from different studies , that they be measured within the same radius . typically , the h radius ( r@xmath3 ) is measured at 1 m@xmath7 pc@xmath19 and is used to estimate the total enclosed mass of a galaxy . however , it is only possible to measure r@xmath3 with interferometers in nearby galaxies . by practical necessity then , some other radius must be adopted for our sample and the galaxy s mass is assumed to be spherically distributed in that radius . two different radii are commonly used : r@xmath26 , the effective or half - light radius , and r@xmath10 , the isophotal radius at the limiting surface brightness of 25 b - magnitudes arc sec@xmath19 . we have measurements of both r@xmath26 ( in r and b band from sdss ) and r@xmath10 ( in b - band from hyperleda ) for most galaxies . @xcite found that r@xmath10 = 2.5 r@xmath26 for nearby emission line galaxies measured in r - band . when we compare our values of r@xmath26 to r@xmath10 , we find medians of @xmath55 and @xmath56 for the galaxies with no r@xmath10 ( sdssj0218@xmath130757 , sdssj0222@xmath130830 and sdssj1118@xmath246316 ) or r@xmath26 ( non - sdss galaxies ) measurements available , we estimate these quantities from the above relations . we find our local sample of lcbgs have dynamical masses , measured within r@xmath10 , ranging from 3@xmath410@xmath6 to 1@xmath410@xmath11 m@xmath7 , with a median of 3@xmath410@xmath57 m@xmath7 . the dynamical masses measured within the effective radius ( measured in both the r and b - band ) range from 8@xmath410@xmath5 to 3@xmath410@xmath57 m@xmath7 , with a median of 8@xmath410@xmath58 m@xmath7 . figure 5 compares the dynamical masses measured for local lcbgs to nearby galaxies of hubble type s0a through i m , where `` m '' indicates magellanic , or low luminosity @xcite . note that @xcite did not correct line widths for random motions as we did . therefore , in figure 5 we compare m@xmath54(r @xmath59r@xmath10 ) which have not been corrected for random motions . these dynamical masses are higher than those we list in table 3 , which have been corrected for random motions . as seen in figure 5 , some local lcbgs have dynamical masses as large as l@xmath2 galaxies ( @xmath110@xmath11 m@xmath7 , roberts & haynes 1994 ) . however , at least 75@xmath35 have dynamical masses approximately an order of magnitude smaller than typical local l@xmath2 galaxies , consistent with observations of lcbgs at higher redshifts ( as discussed in @xmath15 1.1 ) . however , our sample includes galaxies down to 0.25 l@xmath2 . galaxies of this luminosity typically have dynamical masses ( within r@xmath10 ) @xmath12@xmath410@xmath57 m@xmath7 @xcite . therefore , it is more appropriate to compare mass - to - light ratios ; see @xmath154.1 . we estimate the random errors associated with the dynamical masses to be approximately 50% . this estimate is made from uncertainties in the inclination ( @xmath34 12@xmath47 , from the earlier comparison of hyperleda and sdss inclinations ) ; uncertainties in radii measurements ( as given by hyperleda and sdss ) ; and uncertainties in measuring w@xmath39 ( as discussed above ) . however , the random errors are overwhelmed by the systematic uncertainties when estimating dynamical masses . we have chosen a structural constant of c@xmath14 = 1 , but this could be as high as 1.5 , as discussed earlier . even more significant is that 45% of our galaxy sample have other galaxies at similar velocities in the beam of the gbt . as the h emission from all sources in the beam and within the bandwidth is blended into one spectrum , we may be overestimating the masses of these galaxies . finally , a fifth of our galaxies have low inclinations ( @xmath18 40@xmath47 ) which lead to large corrections and therefore increasing uncertainties in the rotational velocities and overestimations of the dynamical masses . the dynamical masses we report may be viewed as upper limits we are certainly overestimating the masses in many cases , but not underestimating them . we are pursuing a follow - up program with both the arecibo radio telescope and the vla to address these issues . at 21 cm the arecibo radio telescope has a beam size of [email protected] @xmath4 [email protected] , a third the size of the gbt beam . this decreases the number of target galaxies observed with other galaxies in the beam , decreasing the number of galaxies with overestimates of dynamical masses . we have already observed over 40 lcbgs at arecibo , but results are not yet available . the vla in `` b '' configuration provides the ability to map our galaxies in 21 cm emission to a resolution of 5@xmath33@xmath33 . we have begun a survey of those lcbgs with companions . vla maps allow us to disentangle the emission from each galaxy and more accurately estimate the dynamical mass of the target galaxy . along with dynamical mass , a knowledge of the amount of fuel , i.e. atomic and molecular gas , available for the starburst activity , is critical for narrowing down the evolutionary possibilities for lcbgs . when combined with star formation rates , this gives an estimate of the maximum length of the starburst at the current rate of star formation . we defer a full discussion of this until paper ii @xcite where we present our measurements of the co content of these galaxies , but present the h results here . the h masses are given by : @xmath60 @xcite , where the distance ( measured from h ) is d = v@xmath44 h@xmath61 . the total h flux , @xmath62 , was calculated by numerically integrating under the spectrum where it is distinguishable from the noise . for the two marginal detections , sdssj0218@xmath130757 and sdssj0222@xmath130830 , h masses were calculated using the optical ( sdss ) recessional velocities . we find local lcbgs have h masses ranging from 5@xmath410@xmath5 to 8@xmath410@xmath6 m@xmath7 , with a median of 5@xmath410@xmath6 m@xmath7 . these span the range of h masses in nearby galaxies across the hubble sequence ; the median is that of nearby late - type spiral galaxies @xcite . the uncertainties listed for m@xmath3 ( table 3 ) include uncertainties in the distance and integrated flux density , as listed in table 2 . however , as in the discussion of dynamical mass uncertainties in @xmath153.1 , the h masses for nearly half of our galaxies are most likely overestimates due to the presence of other galaxies within the gbt beam . dynamical mass - to - light ratios indicate whether our local sample of lcbgs are under - massive for their luminosities . we calculated the total blue luminosities , l@xmath21 , of our galaxies from m@xmath21 , assuming m@xmath63 = 5.48 @xcite . we find the median mass - to - light ratio , m@xmath40(r @xmath18 r@xmath10 ) l@xmath8 = 3 m@xmath7 l@xmath64 , with a minimum of 0.6 and a maximum of 9 m@xmath7 l@xmath64 . @xcite find that m@xmath40(r @xmath18 r@xmath10 ) l@xmath8 ranges from 3.6 @xmath13 10 m@xmath7 l@xmath64 in local galaxies across the hubble sequence . the median m@xmath40(r @xmath18 r@xmath10 ) l@xmath8 is fairly constant ranging from 5 to 7 m@xmath7 l@xmath64 . however , as discussed in @xmath153.1 , to compare our mass - to - light ratios to those of local , normal hubble - types in @xcite , we must use dynamical masses which have not been corrected for random motions . figure 6 compares such mass - to - light ratios for our local sample of lcbgs and nearby hubble types . while some local lcbgs have mass - to - light ratios equal to or greater than the median for local hubble types , approximately half the local lcbgs have smaller mass - to - light ratios . many of the galaxies with mass - to - light ratios typical of local hubble types have other galaxies within the beam , possibly leading to an overestimation of the dynamical mass - to - light ratio of the target galaxy . therefore , in general , lcbgs tend to have mass - to - light ratios smaller than local normal galaxies . that is , most are small galaxies undergoing large amounts of star formation and not simply large galaxies with moderate amounts of star formation . we calculated the gas mass fraction , m@xmath3 m@xmath65 ( r @xmath18 r@xmath3 ) for the local sample of lcbgs as a measure of their atomic gas richness . we do not have interferometric observations of these galaxies , so we estimated the hydrogen radius , r@xmath3 , from optical radii . @xcite found for nearby spiral galaxies that r@xmath3 = 2 r@xmath10 , where r@xmath3 is measured at the 1 m@xmath7 pc@xmath19 level . however , @xcite found r@xmath3 ranging from 3 @xmath13 5 r@xmath10 for a small sample of local h galaxies which are similar to lcbgs , but much less luminous ( m@xmath21 @xmath66 @xmath1316 ) . @xcite studied a large sample of nearby galaxies and found that galaxies with smaller optical radii have larger h extensions . @xcite have measured r@xmath3 in one of our lcbgs , mrk 314 , using the vla . they find r@xmath3 = 4 r@xmath10 . therefore , although we estimated r@xmath3 as 2 @xmath4 r@xmath10 following @xcite , we note it may be an underestimate , which would cause us to overestimate the gas mass fraction . we are undertaking interferometric observations of some of our local sample of lcbgs to directly measure r@xmath3 . this will be addressed in a future paper . in local lcbgs , we find the gas mass fraction , m@xmath3 m@xmath65 , ranges from 0.03 to 0.2 , with a median of 0.08 . that is , it ranges from normal hubble type spirals ( @xmath18 0.1 @xcite ) to very gas rich galaxies such as nearby h and irregular galaxies studied by @xcite with gas fractions ranging from 0.2 @xmath13 0.4 . this is similar to the range of 0.01 @xmath13 0.5 found by @xcite for local blue compact galaxies , most of them less luminous than lcbgs . the fraction of hydrogen mass to blue luminosity provides a distance independent alternative measure of the gas richness of galaxies . we find our local lcbgs have m@xmath3 l@xmath8 ranging from 0.09 to 2 m@xmath7 l@xmath67 , with a median of 0.4 m@xmath7 l@xmath67 . this spans the range of nearby galaxies from s0a through i m , the median corresponding to late - type spirals @xcite . @xcite studied a range of blue compact galaxies , ranging from faint blue compact dwarfs to lcbgs , and found the same median value , 0.4 m@xmath7 l@xmath67 . the tully - fisher relationship , a relation between line width and luminosity , is a fundamental scaling relation for non - interacting spiral galaxies . an absolute blue magnitude versus h line width version of the tully - fisher relationship is shown in figure 7 . we have compared our local lcbgs with the tully - fisher relation found for @xmath14500 normal galaxies within @xmath140 mpc @xcite . as seen in figure 7 , the location of local lcbgs is consistent with the @xcite relation , although with a large scatter . @xcite have a 1 @xmath38 scatter of 0.3 magnitudes in m@xmath21 , while the local lcbgs have a 1 @xmath38 scatter of 0.9 magnitudes in m@xmath21 . approximately half the galaxies ( without other sources in the beam ) lie to the left of the tully - fisher relationship , indicating they have lower masses than expected from their luminosities , although this result is not statistically significant . this is consistent with the distribution of lcbg and hubble type galaxy mass - to - light ratios in figure 6 . seven of the galaxies lying to the right of the tully - fisher relationship have other galaxies within the gbt beam , possibly leading to overestimations of the line widths . note that in our comparison of magnitudes , masses , and mass - to - light ratios , as in all other comparisons in this paper , we have adjusted all values to a hubble constant of h@xmath16 = 70 km s@xmath17 mpc@xmath17 we have observed 20 local lcbgs chosen with the same selection criteria as those lcbgs common at higher redshifts . lcbgs can not remain lcbgs for a long period of time : the number density of lcbgs decreases by at least a factor of ten from z @xmath12 0.5 to today . that is , while lcbgs were common in the past , there are very few today . out of approximately a million nearby galaxies observed by the sdss , only about a hundred are lcbgs . ( see castander et al . 2004 for a discussion of the local space density of lcbgs . ) therefore , lcbgs must evolve into some other galaxy type . from studies of intermediate redshift lcbgs , @xcite and @xcite proposed that some may be progenitors of local dwarf elliptical galaxies . alternatively , @xcite and @xcite proposed that some may be disk galaxies in the process of forming a bulge to become present - day l@xmath2 spiral galaxies . two pieces of information are crucial to determining if these possibilities are likely . it is necessary to know the dynamical masses and the duration of the starbursts . we have measured the dynamical masses of our local sample of lcbgs using h . the length of the starburst will not be estimated until we discuss our molecular gas survey in paper ii @xcite . however , we can use our measurements of line width and radius to identify local galaxies comparable to lcbgs , independent of the evolutionary stage of their stellar populations . in figure 8 we plot the effective or half - light radius , r@xmath26 , versus the velocity dispersion , @xmath38 , for our sample of local lcbgs , intermediate redshift ( 0.4 @xmath0 z @xmath0 1 ) lcbgs @xcite , and local samples of elliptical , spiral , magellanic spiral , irregular , and dwarf elliptical galaxies ( guzmn et al . 1996 ; hyperleda ) . we measured the @xmath38 of our sample of galaxies from the h spectra by measuring the moments of the spectra . the measurements were made in the same way as we described in @xmath153.2 for measuring @xmath62 , the zeroth moment . we find similar results if we simply scale w@xmath39 to @xmath38 by assuming the spectra are gaussian , that is , dividing w@xmath39 by 3.6 . the values of @xmath38 for the other galaxy samples are from optical line width measurements , except the measurements for magellanic spirals are from 21 cm h line widths . optical line widths may be smaller than h line widths . for example , @xcite studied a sample of nearby blue compact galaxies and found the ionized gas emission line widths to be systematically smaller than the neutral hydrogen emission line widths . on average , the ratio of w@xmath39 measured from h to w@xmath39 measured from h was 0.66 for their sample of 11 blue compact galaxies . however , the galaxies in @xcite s sample tend to be smaller both in effective radius and h line width than our sample of local lcbgs , suggesting that the difference between the optical and h line widths may be smaller as well @xcite . the r@xmath26 versus @xmath38 plot allows us to compare the dynamical mass properties of our sample of lcbgs with the other galaxy types . these properties are expected to remain constant despite luminosity evolution . we find that while some local lcbgs have @xmath38 consistent with local spiral galaxies , they tend to be too small in r@xmath26 . however , local lcbgs are consistent with the smaller spiral galaxies , magellanic spirals . they are inconsistent with elliptical galaxies , but are consistent with the most massive dwarf elliptical and irregular galaxies . there is much confusion by what various authors mean when discussing dwarf elliptical galaxies ( sometimes called spheroidal galaxies ) . we are referring to galaxies such as ngc 205 and _ not _ the less massive and less luminous galaxies like draco and carina . finally , when we compare our local sample of lcbgs with those lcbgs observed at intermediate redshifts , we find that they occupy the same region of r@xmath26 @xmath13 @xmath38 space , suggesting we are indeed examining a similar mass range in both samples of galaxies . we also compared local lcbgs to other galaxy types in w@xmath39 sin@xmath17(_i _ ) versus r@xmath10 space , where w@xmath39 was measured from h emission for all samples . this allowed us to investigate if the wavelength used to measure the line width , or correcting the line width for inclination , was having an effect on our interpretation of which galaxy types most resemble lcbgs . we could only compare local lcbgs , spirals , magellanic spirals , and irregular galaxies , as h observations of intermediate redshift lcbgs and dwarf ellipticals are rare or non - existent . our findings were entirely consistent with the interpretation from figure 8 . local lcbgs are therefore consistent with the dynamical mass properties of the most massive dwarf ellipticals and irregulars , and lower mass or magellanic spirals . these classes vary in color and magnitude . knowledge of the amount of molecular gas , time scale of starburst and fading , along with the ability of the galaxy to retain its interstellar medium , will allow us to discriminate between these remaining possibilities . we will begin this work in paper ii @xcite . given the diverse nature of lcbgs , it is likely that multiple scenarios apply , each to a different subset of lcbgs . we have performed a single dish 21 cm h survey of 20 local lcbgs chosen to be local analogs to the numerous lcbgs studied at intermediate redshifts ( 0.4 @xmath0 z @xmath0 0.7 ) . our findings have verified results from intermediate redshift lcbg studies . we have found that local lcbgs are a morphologically heterogeneous mixture of galaxies . they are typically gas - rich , with median values of m@xmath3 = 5@xmath410@xmath6 m@xmath7 and m@xmath3 l@xmath8 = 0.4 m@xmath7 l@xmath9 . approximately half have mass - to - light ratios approximately ten times smaller than local galaxies of all hubble types at similar luminosities , confirming that these are indeed small galaxies undergoing vigorous bursts of star formation . this proportion is likely an underestimate , as nearly half our sample of galaxies may have dynamical mass overestimates . by comparing line widths and radii with local galaxy populations , we find that local lcbgs are consistent with magellanic spirals , and the more massive irregulars and dwarf ellipticals . measurements of the length of starburst , amount of fading , and ability of these galaxies to retain their interstellar media will help to constrain the evolutionary possibilities of this galaxy class . we begin to address these issues in paper ii @xcite , where we present the results of a molecular gas survey of these same local lcbgs . _ acknowledgments _ we thank the referee for helpful comments which improved the quality of this paper . we thank rick fisher for providing the h spectrum of sdssj0934 + 0014 . we also thank the operators and staff at the gbt for their help with the observing and reduction , and their hospitality . support for this work was provided by the nsf through award gssp02 - 0001 from the nrao . d. j. p. acknowledges generous support from an nsf mps distinguished international postdoctoral research fellowship , nsf grant ast0104439 . r. g. acknowledges funding from nasa grant ltsa nag5 - 11635 . funding for the creation and distribution of the sdss archive has been provided by the alfred p. sloan foundation , the participating institutions , the national aeronautics and space administration , the national science foundation , the u.s . department of energy , the japanese monbukagakusho , and the max planck society . the sdss web site is http://www.sdss.org/. we have made extensive use of hyperleda ( http://www-obs.univ-lyon1.fr/hypercat/ ) and 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 ( http://nedwww.ipac.caltech.edu/ ) . the digitized sky surveys were produced at the space telescope science institute under u.s . government grant nag w-2166 . the images of these surveys are based on photographic data obtained using the oschin schmidt telescope on palomar mountain and the uk schmidt telescope . llcccclcc mrk 297 & ngc 6052 , ngc 6064 & 67 & @xmath1321.0 & 0.4 & 20.6 & & sc & y + mrk 314 & ngc 7468 & 30 & @xmath1318.5 & 0.4 & 20.2 & e3 , pec ( polar ring ? ) & e & n + mrk 325 & ngc 7673 , mrk 325 & 49 & @xmath1320.0 & 0.4 & 20.0 & sac ? , pec , h ii starburst & sc & y + mrk 538 & ngc 7714 & 40 & @xmath1320.1 & 0.4 & 20.2 & sb(s)b , pec , h ii liner & sbb & y + sdss j011932.95 + 145219.0 & ngc 469 & 59 & @xmath1318.9 & 0.4 & 20.3 & & & y + sdss [email protected] & & 69 & @xmath1318.8 & 0.5 & 20.2 & & & n + sdss [email protected] & & 67 & @xmath1318.6 & 0.4 & 20.1 & & & n + sdss j072849.75 + 353255.2 & & 56 & @xmath1318.9 & 0.4 & 20.3 & s ? & sbc & n + sdss j083431.70 + 013957.9 & & 59 & @xmath1319.1 & 0.6 & 20.6 & sb(s)b & sbb & y + sdss j090433.53 + 513651.1 & mrk 101 & 68 & @xmath1319.7 & 0.6 & 20.3 & s & sc & n + sdss j091139.74 + 463823.0 & mrk 102 & 61 & @xmath1319.3 & 0.5 & 19.1 & s ? & & n + sdss j093410.52 + 001430.2 & mrk 1233 & 70 & @xmath1319.8 & 0.3 & 19.9 & sb & sbc & y + sdss j093635.36 + 010659.8 & & 71 & @xmath1319.1 & 0.6 & 21.0 & & & y + sdss [email protected] & & 68 & @xmath1319.3 & 0.5 & 20.4 & s0 & s0 & n + sdss j111836.35 + 631650.4 & mrk 165 & 46 & @xmath1318.6 & 0.4 & 19.4 & compact starburst & & n + sdss j123440.89 + 031925.1 & ngc 4538 & 67 & @xmath1319.2 & 0.6 & 21.0 & s pec & sbc & n + sdss j131949.93 + 520341.1 & & 67 & @xmath1318.7 & 0.2 & 20.1 & & & y + sdss j140203.52 + 095545.6 & ngc 5414 , mrk 800 & 61 & @xmath1319.7 & 0.5 & 19.7 & pec & & y + sdss j150748.33 + 551108.6 & & 48 & @xmath1318.9 & 0.4 & 20.8 & s & sbc & n + sdss j231736.39 + 140004.3 & ngc 7580 , mrk 318 & 63 & @xmath1319.3 & 0.6 & 20.4 & s ? & sbc & n + lllll mrk 297 & 4739 @xmath34 17 & 68 @xmath34 0.2 & 362 @xmath34 17 & 6.5 @xmath34 0.08 + mrk 314 & 2081 @xmath34 17 & 30 @xmath34 0.2 & 188 @xmath34 17 & 12 @xmath34 0.1 + mrk 325 & 3427 @xmath34 17 & 49 @xmath34 0.2 & 202 @xmath34 17 & 11 @xmath34 0.2 + mrk 538 & 2798 @xmath34 17 & 40 @xmath34 0.2 & 240 @xmath34 17 & 20 @xmath34 0.2 + sdssj0119 + 1452 & 4098 @xmath34 17 & 59 @xmath34 0.2 & 266 @xmath34 17 & 2.4 @xmath34 0.1 + sdssj0218@xmath130757 & & & & 0.41 @xmath34 0.07 + sdssj0222@xmath130830 & & & & 0.61 @xmath34 0.09 + sdssj0728 + 3532 & 3953 @xmath34 17 & 56 @xmath34 0.2 & 216 @xmath34 17 & 7.9 @xmath34 0.1 + sdssj0834 + 0139 & 4215 @xmath34 160 & 60 @xmath34 2 & 329 @xmath34 160 & 6.9 @xmath34 0.2 + sdssj0904 + 5136 & 4782 @xmath34 103 & 68 @xmath34 2 & 204 @xmath34 103 & 4.5 @xmath34 0.2 + sdssj0911 + 4636 & 4281 @xmath34 17 & 61 @xmath34 0.2 & 151 @xmath34 17 & 2.0 @xmath34 0.1 + sdssj0934 + 0014 & 4860 @xmath34 17 & 69 @xmath34 0.2 & 332 @xmath34 17 & 4.8 @xmath34 0.2 + sdssj0936 + 0106 & 4920 @xmath34 17 & 70 @xmath34 0.2 & 255 @xmath34 17 & 3.3 @xmath34 0.09 + sdssj0943@xmath130215 & 4823 @xmath34 17 & 69 @xmath34 0.2 & 230 @xmath34 17 & 2.6 @xmath34 0.06 + sdssj1118 + 6316 & 3218 @xmath34 17 & 46 @xmath34 0.2 & 152 @xmath34 17 & 2.2 @xmath34 0.07 + sdssj1234 + 0319 & 4685 @xmath34 17 & 67 @xmath34 0.2 & 269 @xmath34 17 & 4.4 @xmath34 0.1 + sdssj1319 + 5203 & 4619 @xmath34 17 & 66 @xmath34 0.2 & 202 @xmath34 17 & 7.7 @xmath34 0.09 + sdssj1402 + 0955 & 4267 @xmath34 17 & 61 @xmath34 0.2 & 343 @xmath34 17 & 6.8 @xmath34 0.2 + sdssj1507 + 5511 & 3373 @xmath34 17 & 48 @xmath34 0.2 & 126 @xmath34 17 & 3.7 @xmath34 0.1 + sdssj2317 + 1400 & 4413 @xmath34 17 & 63 @xmath34 0.2 & 254 @xmath34 17 & 6.2 @xmath34 0.1 + llccccccc mrk 297 & 7.0 @xmath34 0.1 & 42 & 486 & 243 & 8.0 & 11 & 2.0 & 2.7 + mrk 314 & 2.5 @xmath34 0.05 & 65 & 170 & 85 & 3.8 & 0.63 & 0.94 & 0.16 + mrk 325 & 6.3 @xmath34 0.1 & 43 & 242 & 121 & 9.6 & 3.3 & 2.4 & 0.83 + mrk 538 & 7.6 @xmath34 0.1 & 50 & 264 & 132 & 11 & 4.5 & 2.8 & 1.1 + sdssj0119 + 1452 & 2.0 @xmath34 0.1 & 84 & 230 & 115 & 5.6 & 1.7 & 1.6 & 0.48 + sdssj0218@xmath130757 & 0.47 @xmath34 0.08 & 52 & & & 4.9 & & 1.3 & + sdssj0222@xmath130830 & 0.65 @xmath34 0.1 & 48 & & & 4.2 & & 1.2 & + sdssj0728 + 3532 & 6.0 @xmath34 0.1 & 34 & 321 & 160 & 5.4 & 3.3 & 1.2 & 0.74 + sdssj0834 + 0139 & 5.9 @xmath34 0.5 & 83 & 293 & 147 & 6.5 & 3.3 & 1.6 & 0.82 + sdssj0904 + 5136 & 4.9 @xmath34 0.3 & 34 & 301 & 150 & 7.4 & 3.9 & 2.0 & 1.0 + sdssj0911 + 4636 & 1.7 @xmath34 0.1 & 33 & 221 & 111 & 5.6 & 1.6 & 1.2 & 0.35 + sdssj0934 + 0014 & 5.4 @xmath34 0.2 & 51 & 378 & 189 & 6.8 & 5.7 & 1.9 & 1.5 + sdssj0936 + 0106 & 3.8 @xmath34 0.1 & 48 & 293 & 146 & 7.1 & 3.5 & 2.0 & 1.0 + sdssj0943@xmath130215 & 2.9 @xmath34 0.07 & 84 & 194 & 97 & 4.9 & 1.1 & 1.8 & 0.39 + sdssj1118 + 6316 & 1.1 @xmath34 0.04 & 82 & 123 & 61 & 3.1 & 0.27 & 0.91 & 0.08 + sdssj1234 + 0319 & 4.7 @xmath34 0.1 & 52 & 293 & 147 & 6.7 & 3.4 & 2.1 & 1.1 + sdssj1319 + 5203 & 7.9 @xmath34 0.1 & 44 & 239 & 120 & 5.3 & 1.8 & 1.2 & 0.41 + sdssj1402 + 0955 & 6.0 @xmath34 0.2 & 55 & 372 & 186 & 8.7 & 7.0 & 1.6 & 1.3 + sdssj1507 + 5511 & 2.0 @xmath34 0.07 & 44 & 144 & 72 & 7.9 & 0.95 & 1.8 & 0.21 + sdssj2317 + 1400 & 5.8 @xmath34 0.1 & 31 & 420 & 210 & 7.3 & 7.5 & 1.7 & 1.8 + lccccc mrk 297 & 0.03 & 39 & 0.2 & 3 & 0.7 + mrk 314 & 0.2 & 3.9 & 0.7 & 2 & 0.4 + mrk 325 & 0.1 & 16 & 0.4 & 2 & 0.5 + mrk 538 & 0.08 & 17 & 0.5 & 3 & 0.7 + sdssj0119 + 1452 & 0.06 & 5.6 & 0.4 & 3 & 0.9 + sdssj0218 - 0757 & & 5.2 & 0.09 & & + sdssj0222 - 0830 & & 4.3 & 0.2 & & + sdssj0728 + 3532 & 0.09 & 5.6 & 1 & 6 & 1 + sdssj0834 + 0139 & 0.1 & 6.8 & 0.9 & 5 & 1 + sdssj0904 + 5136 & 0.07 & 12 & 0.4 & 3 & 0.9 + sdssj0911 + 4636 & 0.05 & 8.2 & 0.2 & 2 & 0.4 + sdssj0934 + 0014 & 0.05 & 13 & 0.4 & 4 & 1 + sdssj0936 + 0106 & 0.06 & 6.8 & 0.6 & 5 & 2 + sdssj0943 - 0215 & 0.1 & 8.2 & 0.4 & 1 & 0.5 + sdssj1118 + 6316 & 0.2 & 4.3 & 0.3 & 0.6 & 0.2 + sdssj1234 + 0319 & 0.07 & 7.4 & 0.6 & 5 & 1 + sdssj1319 + 5203 & 0.2 & 4.7 & 2 & 4 & 0.9 + sdssj1402 + 0955 & 0.04 & 12 & 0.5 & 6 & 1 + sdssj1507 + 5511 & 0.1 & 5.6 & 0.4 & 2 & 0.4 + sdssj2317 + 1400 & 0.04 & 8.2 & 0.7 & 9 & 2 + . the spectra have been smoothed to a resolution of @xmath112 km s@xmath17 and only the central 800 km s@xmath17 are shown . the dashed lines indicate the 20% crossings used to measure the line width at 20% . the triangles indicate the recessional velocities calculated from sdss redshifts for the sdss galaxies ; velocities from ned redshifts are shown for the non - sdss galaxies . those galaxies with other sources at similar velocities within the gbt beam are indicated by a star in the upper right corner . [ fig4 ] ] ( r @xmath59r@xmath10 ) , for local lcbgs , as measured from h observations , is shown as a gray histogram . for comparison , the range of dynamical masses for local hubble type galaxies are indicated with black boxes . the `` m '' in `` sm '' and `` i m '' indicates magellanic or low - luminosity spirals and irregulars . note that in order to compare lcbg dynamical masses with @xcite s results for local hubble types , the dynamical masses plotted here do not include a line width correction for random motions . [ fig5 ] ] ) to l@xmath21 ratios for local lcbgs is shown as a gray histogram . for comparison , the range of mass - to - light ratios for local hubble type galaxies ( s0a to i m ) @xcite are shown . as in figure 5 , for accurate comparisons , these mass - to - light ratios do not include a line width correction for random motions . many lcbgs tend to have mass - to - light ratios smaller than local normal galaxies , consistent with findings at intermediate redshifts . note that we may have overestimated the dynamical masses for most of the lcbgs with higher mass - to - light ratios . [ fig6 ] ] ) versus absolute blue magnitude ( m@xmath21 ) . local lcbgs are indicated by filled circles . those galaxies with other sources at similar velocities within the gbt beam are circled ; their line widths may be overestimated . the solid line indicates the tully - fisher relationship from @xcite ; the dotted lines indicate their 1 @xmath38 scatter of 0.3 m@xmath21 . the location of our local lcbgs is consistent with the tully - fisher relationship , but with a higher 1 @xmath38 scatter of 0.9 m@xmath21 . [ fig7 ] ] ) versus the the velocity dispersion ( @xmath38 ) is plotted for our local lcbgs ( filled circles ) and intermediate redshift ( 0.4 @xmath0 z @xmath0 1 ) lcbgs ( open circles ) @xcite . the regions occupied by the bulk of other local galaxy types@xmath13ellipticals ( e ) , spirals ( s ) , magellanic spirals ( sm ) , dwarf ellipticals ( de ) , and irregulars ( irr)@xmath13are indicated ( guzmn et al . 1996 , hyperleda ) . we have also indicated the approximate locations for some representative galaxies : draco , ngc 205 and m 31 ( hyperleda ) . local lcbgs are consistent with higher mass irregulars and dwarf ellipticals , and lower mass or magellanic spirals , as well as intermediate redshift lcbgs .

we present single - dish h spectra obtained with the green bank telescope , along with optical photometric properties from the sloan digital sky survey , of 20 nearby ( d @xmath0 70 mpc ) luminous compact blue galaxies ( lcbgs ) . these @xmath1l@xmath2 , blue , high surface brightness , starbursting galaxies were selected with the same criteria used to define lcbgs at higher redshifts . we find these galaxies are gas - rich , with m@xmath3 ranging from 5@xmath410@xmath5 to 8@xmath410@xmath6 m@xmath7 , and m@xmath3 l@xmath8 ranging from 0.2 to 2 m@xmath7 l@xmath9 , consistent with a variety of morphological types of galaxies . we find the dynamical masses ( measured within r@xmath10 ) span a wide range , from 3@xmath410@xmath6 to 1@xmath410@xmath11 m@xmath7 . however , at least half have dynamical mass - to - light ratios smaller than nearby galaxies of all hubble types , as found for lcbgs at intermediate redshifts . by comparing line widths and effective radii with local galaxy populations , we find that lcbgs are consistent with the dynamical mass properties of magellanic ( low luminosity ) spirals , and the more massive irregulars and dwarf ellipticals , such as ngc 205 .

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Proceed to summarize the following text: starting with the question of whether the excited - state lifetime of an atom can be modified , physicists have continuously explored ways to engineer the radiative properties of quantum emitters . the first experimental studies were performed in the near field of flat interfaces @xcite , but the change of spontaneous emission is commonly associated with purcell s prediction that a cavity of quality factor @xmath2 and mode volume @xmath3 can accelerate the radiation of a dipolar transition by @xmath4 folds , where @xmath5 is the wavelength in the corresponding medium @xcite . following this recipe , many clever resonator schemes such as high - q open fabry - prot cavities ( fpc ) , monolithic fpcs in form of pillars , whispering gallery mode resonators and photonic crystal structures have been investigated for realizing large purcell factors @xcite . nevertheless , sizable modifications of the spontaneous emission process remain nontrivial because no cavity design has succeeded in providing all the decisive ingredients of large @xmath2 , small @xmath3 ( ideally down to its fundamental value of the order of @xmath6 ) , a facile way of tuning the cavity resonance , and compatibility with emitters of different materials . of the various cavity geometries , open fpcs remain particularly attractive because they nicely lend themselves to the latter two criteria . efforts using open fpcs have usually pursued large @xmath7 via high @xmath2s instead of low @xmath3 , but this choice is accompanied by several disadvantages . first , the resulting narrow cavity linewidths make the cavity extremely sensitive to mechanically , thermally or stress - induced length changes . second , broad transitions in the condensed phase can not couple to very narrow cavity resonances in an efficient manner . this is especially a restriction for room - temperature studies , where homogeneous linewidths of quantum dots , molecules or ions can reach several nanometers . in addition , scattering from finite - sized particles , e.g. from diamond nanocrystals , could quickly spoil the high q of the cavity . furthermore , narrow cavity resonances do not allow simultaneous coupling to different transitions , for instance , for studies of @xmath8 transitions or of nonlinear interactions involving several light beams @xcite . in this article , we explore a new cavity design using a micromirror with a radius of curvature as small as 2.6 @xmath9 fabricated on a cantilever . by realizing a microcavity with a sub-@xmath0 volume and low @xmath2 , we obtain a sizable @xmath7 that would lead to the modification of the spontaneous emission rate by more than one order of magnitude while keeping a spectral bandwidth as broad as 1 thz . the second novel feature of our arrangement is the compatibility of its high numerical aperture ( @xmath10 ) with the recent advances in efficient atom - photon coupling via tight focusing @xcite . third , the open character , scanning capability and low q of our fpc provide the possibility of using it as a scanning microscope @xcite , which can be used for local field - enhanced spectroscopy @xcite and sensing @xcite of very different materials such as semiconductor quantum dots , carbon nanotubes , organic molecules , rare earth ions , and biomolecules . in particular , the low q of the cavity ensures that even broad resonances of these materials couple efficiently to the cavity modes at room temperature . fourth , the cantilever - based feature of our experimental arrangement makes it highly interesting for an unexplored regime of optomechanical investigations @xcite as we discuss in the concluding paragraph of this article . the inset in fig . [ setup](a ) shows the schematics of our cavity made of a flat distributed bragg reflector ( dbr ) and a micromirror fabricated by focused ion beam milling . here , we started with an n - doped silicon cantilever that contained a pedestal ( diameter 8 @xmath9 ) . the area of the micromirror to be milled was divided into pixels of diameter 5 nm . by controlling the ion dose for each pixel , we obtained a spherical surface profile , which was then polished in a final step . figure [ setup](b ) displays an electron microscope image of a cantilever pedestal with four mirrors of different radii of curvature , and fig . [ setup](c ) presents an exemplary topography cross section recorded with an afm . in this case , the mirror has a radius of curvature of @xmath11 accompanied by a root - mean - square surface roughness below 5 nm . an opening aperture diameter of 2.4 @xmath9 provides a concave mirror with numerical aperture @xmath12 following the standard defintion of @xmath13 , where @xmath14 is the angle subtended by half the mirror aperture and @xmath15 is the mirror focal length . we note that fabrication of micromirrors has been previously reported for wet etching @xcite , laser ablation @xcite and focused ion beam milling @xcite , but the majority of the existing works report radii of curvature above ten microns , limiting both @xmath10 and @xmath3 . to our knowledge , we provide the highest numerical aperture for a tunable microcavity reported to date . the micromirrors were coated with 150 nm of gold followed by 50 nm of silicon dioxide as a protective layer , yielding a nominal reflectivity of 96% at @xmath16 nm . taking into account the slight roughness of this mirror ( see figure [ setup](c ) ) , we expect the reflectivity to be reduced to 95% due to residual scattering @xcite . the dbr consisted of 11 layers of @xmath17 stacks finished by a 22 nm layer of @xmath18 to place the field maximum at the mirror surface . the resulting structure had a total thickness of 2.14 @xmath9 with its design band center at 710 nm and a band edge at 841 nm , leading to reflectivities of 99.9% at @xmath16 nm and 99.99% at 745 nm . as displayed in fig . [ setup](a ) , the microcavity was assembled with a piezoelectric transducer ( pzt ) stack for the axial displacement of the cantilever and a pzt scanner for the lateral positioning of the dbr . we point out that the asymmetry of the mirror reflectivities causes a suboptimal impedance matching , which can be easily alleviated by using lower reflectivity dbr . in this work , however , we have not been concerned with coupling efficiencies . intuitively speaking , we want to operate the cavity close to the condition , where the rays from a strongly focused incident beam are retroreflected by the curved micromirror . it follows that a useful figure of merit for mode matching becomes the opening arc half - angle @xmath19 subtended by the mirror aperture ( see the inset of fig . [ setup](a ) ) . we , thus , define @xmath20 as an effective numerical aperture , which amounts to 0.46 in our experiment . we note in passing that according to common textbook knowledge , the stability of a plane - concave fpc becomes compromised as the cavity length becomes comparable to and larger than the mirror curvature @xcite . to perform spectroscopy on the microcavity , we focused laser beams at @xmath5=785 nm and @xmath5=745 nm through a microscope objective with @xmath21 , whereby the waist of the incident laser beam was adjusted to match @xmath22 . we then scanned the axial position of the cantilever and monitored the reflected light on a photodiode . for various @xmath23 modes . the indices @xmath24 signify the best fit obtained for the longitudinal mode number . , width=8 ] the blue spectra in figs . [ cavity - spectra](a ) and ( b ) plot the measured reflection from the microcavity as a function of the change in @xmath25 at 745 nm and 785 nm , respectively . here , we slightly defocused and misaligned the incident beam to obtain some coupling to the higher transverse modes . the red lines mark the resonances associated with the hermite - gaussian modes predicted by the equation @xcite @xmath26~ , \label{cavityeqn1}\ ] ] for a plane - concave cavity with the same nominal geometrical parameters as in the experiment but with infinitely thin mirrors . here , the the term @xmath27 signifies the gouy phase within the paraxial approximation . the symbols in fig . [ cavity - spectra](c ) and ( d ) summarize the dependence of @xmath25 on @xmath28 indices of the transverse @xmath23 modes for each longitudinal parameter @xmath24 . if we now fit these data with the outcome of eq . ( [ cavityeqn1 ] ) , we obtain values for @xmath24 posted in the legends of fig . [ cavity - spectra](c , d ) . the good cumulative comparison between theory and experiment suggests that we have reached the longitudinal mode @xmath29 . the resulting non - integer values of @xmath24 stem from the fact that eq . ( [ cavityeqn1 ] ) neglects the penetration depth of the field in the dbr mirror @xcite as illustrated in fig . [ volume - cuts ] for @xmath5= 745 nm . the field penetration in the mirror and a considerable gouy phase of @xmath30 in our high - na cavity make the analysis of the spectra presented in fig . [ cavity - spectra ] nontrivial . to account for these , we now examine the resonance condition and spectral properties of our cavity starting from basic derivations . at the wavelength of 745 nm . the cavity is formed by a gold - coated micromirror and a flat distributed bragg reflector marked by the dashed lines . the length of the cavity @xmath25 measured between the air - dbr interface and the apex of the curved mirror is shown in each case together with a cut throught the intensity distribution along the red vertical dashed line . the red horizontal dotted lines mark the dbr interface.,width=321 ] the transmission spectrum of a fabry - prot cavity in the absence of absorption is given by the airy function @xcite @xmath31 where @xmath32 is the total round trip phase experienced by the light given by @xmath33 here , @xmath34 is the wavevector , @xmath35 and @xmath36 are the phase shifts introduced by the two cavity mirrors , @xmath25 is the physical separation between the mirrors , and @xmath37 is the gouy phase shift experienced by the mode after propagating the length @xmath25 @xcite . the cavity resonance occurs when @xmath32 is an integer @xmath24 multiple of @xmath38 such that sin(@xmath39)=0 . at this point , one can define the finesse @xmath40 as the ratio @xmath41 , where @xmath42 is the phase separation of two adjacent resonances , and @xmath43 is the full width at half - maximum ( fwhm ) of the phase through a given resonance . if this definition is applied to eq . ( [ eq : airy ] ) , one obtains @xmath44 where @xmath45 is the geometric mean of the two mirror reflectivities . equation ( [ eq : airy ] ) provides an expression for the spectrum of the cavity . however , in the laboratory one does not have an easy access to the phase of the light field . instead , one either scans the frequency @xmath46 of the incident beam at a fixed @xmath25 or changes the latter at a fixed frequency . in most textbook treatments of fabry - prots , thin planar mirrors are considered , and the dependence of @xmath37 , @xmath47 and @xmath36 on @xmath25 and @xmath46 is neglected . in that case , since only the propagation phase @xmath48 changes , the spectra can be plotted as a function of @xmath25 or @xmath46 in a fully equivalent fashion , leading to @xmath49 . here , @xmath50 and @xmath51 denote the separation of two adjacent resonance lengths and frequencies ( commonly known as the free spectral range ) , respectively , whereas @xmath52 and @xmath53 signify the fwhm of the corresponding resonance as the cavity length or frequency are varied . this relation then allows a direct conversion of data obtained from cavity length spectroscopy to the properties of the frequency spectrum . we now allow @xmath37 , @xmath47 and @xmath36 to depend on @xmath25 and @xmath46 . it turns out that the dependence of @xmath47 and @xmath36 on @xmath25 is fairly negligible ( as can be inferred from fig . [ volume - cuts ] ) and @xmath54 varies linearly with @xmath25 in the regime of our cavity parameters , as shown in fig . [ phases](a ) . in this case , eq . ( [ eq : roundtripphase ] ) reads @xmath55 , where @xmath56 and @xmath57 denote the zeroeth and first order terms of the taylor expansion of @xmath54 , and @xmath58 signifies an effective frequency offset . in this case , the finesse can still be calculated as @xmath59 , which is conveniently accessible in the experiment . next , we have to relate @xmath60 and @xmath52 in order to extract the quality factor @xmath61 from our experimental measurements . we , thus , differentiate eq . [ eq : roundtripphase ] with respect to @xmath5 at the cavity resonance to obtain @xmath62~. \label{eq : lengthdispersion}\ ] ] the first term in the main paranthesis is the axial mode number @xmath24 of an ideal cavity , while the inner parenthesis contains corrections to it . in general , these corrections need not be small and can have implicit dependence on @xmath5 . considering @xmath63 , we arrive at @xmath64 \mathcal{f}=q_{\rm eff}\cdot\mathcal{f}~ \label{eq : q - f}\ ] ] with @xcite @xmath65 to obtain the quantities in eq . ( [ eq : qeffexplicit ] ) , we measured the reflectivity spectrum of the dbr and compared the outcome with the calculations obtained from a transfer matrix method ( see fig . [ phases](b ) ) . the corresponding amplitude and phase of the reflection coefficient are shown in fig . [ phases](c ) . the phase shift upon reflection from the gold mirror is also calculated using the dielectric function from ref . @xcite and is plotted in fig . [ phases](d ) . the phase properties of the mirrors at the working wavelengths are summarized in table [ table : mirrorphases ] . the fit requires @xmath66 = 708.5 nm . this slight discrepancy is likely due to finite tolerances in the dbr fabrication process . ( c ) amplitude and phase of the complex reflectivity of the dbr . ( d ) phase shift upon reflection from the gold mimrror.,width=88 ] @xmath67 & -0.5661@xmath68 & -0.1960@xmath68 + @xmath69 & -1.1405@xmath68 & -1.1310@xmath68 + [ table : mirrorphases ] we are now prepared to analyze the cavity length spectra . in fig . [ linewidths](a ) we plot a spectrum recorded at @xmath16 nm , where the incident beam was aligned to minimize the coupling to higher transverse modes . the finesse and @xmath70 extracted from this spectrum are shown by the symbols in fig . [ linewidths](b ) . these data reveal that contrary to the common case of large cavities , @xmath40 and @xmath2 vary strongly with the mode number . to understand this behavior , we set up a simple model based on the propagation of a gaussian beam between the two mirrors . we calculated the position and size of the beam waist @xmath71 after each round trip , whereby we took into account the loss at the finite aperture of the curved mirror ( about 0.1% of the power per round trip for the second longitudinal mode ; i.e. @xmath24=2 ) simply as a scalar factor . the red curve in fig . [ linewidths](b ) shows that the @xmath2 calculated from the cavity decay time is in good agreement with the experimental values . we do notice deviations for the highest mode orders , where @xmath25 is large enough that our model is no longer valid , and diffraction at mirror edges and losses due to beam clipping become important . the nonmonotonous behavior of @xmath2 in fig . [ linewidths](b ) is the result of the competition between finite - aperture losses on the one hand and gain in the photon lifetime for larger @xmath25 on the other . the blue curve in fig . [ linewidths](b ) displays @xmath72 , which again compares well with the experimentally measured values determined from @xmath73 . ( red ) and finesse @xmath72 ( blue ) . symbols : experiment ; curves : model . , width=8 ] the mode volume @xmath3 plays a central role in the performance of a cavity as suggested by the purcell formula in the context of cavity quantum electrodynamics @xcite , the threshold formula in laser theory @xcite and its role in sensing @xcite . this quantity is usually calculated as the integral of the electromagnetic energy density in the cavity mode normalized to the maximum of the field intensity . for fabry - prot resonators with perfect conductor boundaries , one can write @xmath74 , where @xmath75 is the gaussian mode waist . in the presence of a dbr mirror , the extra energy leakage in the dielectric layers makes the calculation of the mode volume nontrivial , especially when @xmath25 becomes comparable with @xmath5 and the dbr penetration depth @xcite . in order to consider this effect and the curvature of the metallic mirror , we performed full - numerical eigenmode simulations ( comsol multiphysics ) to extract their modal properties . for computational simplicity , we neglected the thin protective layer on the micromirror , chose a simulation box size of about @xmath76 and imposed perfectly - matched layers and scattering boundary conditions . the typical mesh sizes were close to @xmath77 near the micromirror , @xmath78 in the dbr and @xmath79 in the furthest regions . figure [ volume - cuts ] displays cross sections of the intensity distribution for three different cavity lengths at @xmath5=745 nm . to evaluate the mode volume from eigenmodes in the presence of radiative losses , we followed the definition provided by ref . @xcite , which allows for the treatment of the purcell factor in complex environments . figure [ volumepurcell ] shows the calculated mode volume as a function of the mode number @xmath24 for @xmath80 nm . we find that the volume of the first accessible mode with @xmath29 is as low as @xmath81 , while it grows almost linearly with @xmath24 . ( magenta ) and purcell factor @xmath82 ( green ) at @xmath5=745nm.,width=283 ] the purcell factor @xmath83 is commonly used as a measure for the ability of an optical resonator to enhance the spontaneous emission rate into a well - defined mode . an alternative formulation of the same physics is sometimes expressed as the cooperativity @xmath84 , where @xmath85 denotes the vacuum rabi frequency of an atom with transition dipole moment @xmath86 . here , @xmath87 denotes the atomic free - space radiative decay rate , and @xmath88 signifies the cavity loss rate . after a simple manipulation , one sees that the central figure of merit remains the ratio @xmath89 , which can reach about 360 , corresponding to @xmath90 as shown in fig . [ volumepurcell ] . we remark that these calculations assumed a reflectivity of 94% for the gold mirror to reach @xmath2 values close to the experimental measurements . while in this work we emphasize the advantages of the low-@xmath2 and low-@xmath3 cavity regime , higher @xmath2s can be easily achieved by using enhanced metal or dielectric layers on the micromirror instead of a gold coating . of course , one would have to keep in mind that the number of bragg layers and their dielectric contrasts would have to be chosen properly to be compatible with the high curvature of the micromirror . we also point out that a particularly attractive application of our cavity design might be for studies at @xmath91 m , where the higher reflectivity of the gold mirror coating and smaller possible values of @xmath92 would yield higher @xmath93 . having considered the basic features of our ultrasmall , broadband and tunable fpc , we now discuss experiments on coupling it to a point - like dipolar scatterer . it is known that the introduction of a foreign object in the cavity adds to the overall optical path of the photons , leading to a red shift of the cavity resonance @xcite . in the case of a subwavelength nanoparticle , one arrives at the shift of the cavity resonance @xmath94 given by @xmath95}~ , \label{lineshifteqn}\ ] ] where @xmath96 is the electric field at position @xmath97 in the cavity @xcite . here , @xmath19 is the complex electric polarizability of the particle , which is closely linked to its absorption and scattering cross sections @xcite . in the quasi - static approximation @xmath98 where @xmath99 is the particle diameter , and @xmath100 and @xmath101 are the dielectric functions of the particle and its surrounding medium , respectively . one might have the intuitive expectation that the introduction of a foreign object into the cavity would cause losses , thus lowering its @xmath2 . although this is true in general , it has been shown that a nano - object can shift the cavity resonance without incurring a notable broadening @xcite . indeed , recently there has been a great deal of activity to exploit the frequency shift of a high - q cavity for sensing nanoparticles such as viruses @xcite . to investigate the effect of a nanoparticle on our cavity , we spin coated gold nanoparticles ( gnp ) of diameter 80 nm on the dbr with an inter - particle spacing larger than several micrometers . to ensure that we can address individual gnps , we recorded afm images of the dbr mirror after spin coating and identified particles that were far enough from their neighbors . by matching the afm image of the particle distribution and its optical equivalent on our setup , we could target individual gnps . finally , we point out in passing that the polarizability of this gnp at the wavelength of interrogation is equivalent to a virus particle with diameter of 200 nm immersed in water . assuming that the particle is placed at the field maximum of a cavity , eq . ( [ lineshifteqn ] ) can be written as @xmath102 which expresses the ratio of a shift @xmath94 to the linewidth @xmath60 of the cavity resonance profile . it is well - known that a high @xmath2 facilitates the detection of a small frequency shift experienced by a narrow resonance line . in this work , we enhance the shift by using very small @xmath3s so that it can be detected even for broad cavity resonances . = 2 cavity resonance in thz as a function of the lateral position of a gold nanoparticle . ( b ) a cross section from ( a ) , displaying fwhm @xmath103 nm . ( c ) cavity resonance shift measured for different longitudinal modes.,width=8 ] figure [ shift](a ) shows the cavity shift as a function of the lateral position of a gnp , and fig . [ shift](b ) displays a cross section from it . we observe a shift as large as 400 ghz , equivalent to 40% of the cavity linewidth , over a gaussian lateral profile with @xmath104 nm . to obtain these data , the dbr mirror holding the gnp was scanned laterally , while the cavity length was scanned at each pixel . the red symbols in fig . [ shift](c ) plot the maxima of cavity resonance shifts for different longitudinal modes . as expected from eq . ( [ lineshifteqn ] ) , the effect of the particle rapidly diminishes for higher @xmath24 modes and larger @xmath3s . to verify the measured data quantitatively , we have fitted them with eq . ( [ lineshifteqn ] ) , leaving @xmath19 as a free parameter . the blue curve shows the best fit obtained for @xmath105 , which is 1.17 times larger than its expected value for a nominal gnp with a diameter of @xmath106 nm ( as specified by the manufacturer ) , @xmath107 of gold obtained from ref . @xcite , and @xmath108 . however , it should be born in mind that the exact knowledge of @xmath19 for a given gnp is highly nontrivial . first , near - field coupling to the dbr surface modifies the gnp plasmon resonance and polarizability @xcite . by using an analytical expression @xcite , we have estimated the polarizability of the gnp to increase by 1.1 times in the presence of the dbr upper layer alone . furthermore , variations in shape , the finite size of the particle and the resulting radiation damping effect @xcite enter beyond the simple expression of eq . ( [ alpha ] ) . considering these effects , our experimental findings are in excellent agreement with theoretical expectations . next , we turn to the effect of the nanoparticle on the cavity finesse for various mode orders . since the nanoparticle scatters some of the light out of the cavity mode and has a finite absorption cross section , one can expect a degradation of @xmath72 . figure [ q](a ) displays a map of @xmath72 as a function of the gnp lateral position in the @xmath29 mode , and the top curve in fig . [ q](b ) plots a cross section from it . we find that @xmath72 decreases by about 7% from 60 to 56 in the presence of the particle while this behavior is substantially changed for higher modes ( see the other plots in fig . [ q](b ) ) . in fact , we find that the particle can even improve @xmath72 . for example , it is increased by about 10% from 6.5 to 7 for @xmath109 . as a function of the particle position for @xmath29 . ( b ) cross sections of @xmath72 vs gnp position for different cavity modes . the mode number @xmath24 is indicated in each measurement . ( c ) a two - dimensional map of the cavity coupling , obtained by normalizing the reflected power to the incident power , as a function of the particle position . ( d ) cross sections of the cavity coupling efficiency for different @xmath24s.,width=321 ] for the shortest cavities , the mirrors are so close that the light is efficiently captured after each round trip . in this case , absorption by the particle determines the quality factor and finesse . as @xmath25 approaches @xmath110 , the unperturbed resonator becomes less stable and @xmath72 is lowered ( see fig . [ linewidths](b ) ) . interestingly , in this regime the addition of a gnp ameliorates the situation . to investigate the modification of the finesse by the nanoparticle further , we have performed numerical calculations that include the nanoparticle . here , we consider the contribution of losses caused by absorption and scattering ( signified by @xmath111 and @xmath112 ) to the cavity quality factor according to @xmath113 . we , thus , first estimate the absorption and scattering power from simulations by calculating the overall resistive losses per optical cycle and the power flow in the directions perpendicular to the cavity axis . figure [ scatteringcoefficients_finesse](a ) shows these quantities for the different modes normalized to the cavity energy per optical cycle . we find that without the nanoparticle , absorption losses decrease while the scattering losses grow with with @xmath24 . a nanoparticle on the cavity axis and in contact with the dbr slightly increases the absorption ( compare red and black lines ) , but it reduces the power scattered outside the cavity ( green line ) in comparison with the case without it ( blue ) . these results confirm that the gnp acts as a mode matching antenna to improve the coupling into the cavity mode . to support this hypothesis further , we measured the amount of light that circulates inside the cavity by monitoring the power reflected from the dbr . figure [ q](c ) plots the two - dimensional map of the coupling efficiency for @xmath29 , and fig . [ q](d ) displays the cross sections for various modes . the similarity of the patterns of the two data columns in fig . [ q](b , d ) shows that the gnp strongly influences the coupling of the incoming laser beam into the cavity mode . at this point , we note that one can also perform interferometric scattering measurements ( iscat ) to detect individual gnps without the micromirror @xcite . using this method , we determined a fwhm of 690 nm for the focus spot of the incident beam in the absence of the cantilever . this value agrees to within 10% with the fwhm that we found from fig . [ shift](b ) , indicating that the cavity mode waist is indeed well matched to the incoming spot size . the black symbols in fig . [ scatteringcoefficients_finesse](b ) display the simulated cavity finesse , whereby we adjusted the reflectivity of the gold mirror to @xmath114 to reduce the finesse by 2.4 times from its expected value for the ideal structure in order to emulate the experimental values . the trend obtained from the numerical simulations agrees very well with that presented in fig . [ linewidths](b ) . in particular , we observe that the cross - over between the absorption and scattering losses at @xmath24=3 ( see fig . [ scatteringcoefficients_finesse](a ) ) results in larger finesse in the presence of the nanoparticle ( green ) for higher modes , verifying the experimental observations in fig . in fact , even the magnitude of the change in @xmath40 agrees well with the experimental findings . nm nanoparticle ( green symbols ) and without it ( black).,width=321 ] the cavity regime studied in this work has brought forth different interesting phenomena , many of which had not been encountered in previous works on microcavities . these include a strong dependence of the cold cavity @xmath2 and @xmath72 on the longitudinal mode order @xmath24 , the largest numerical aperture reported to date and a wavelength - sized mode waist , increase of cavity @xmath72 by a nanoparticle , the combination of a sub-@xmath0 mode volume and frequency tunability , and the fact that one mirror sits on a cantilever . these features are very promising for a number of future studies ranging from biophysics to quantum optics . one of our future goals is to modify the branching ratio of solid - state emitters such as organic molecules or color centers . the energy level scheme of such systems usually involves a narrow transition on the so - called zero - phonon line ( zpl ) accompanied by a broad phonon wing and transitions among other vibrational levels . in order to have a strong transition for quantum optics experiments , it is desirable to exploit the purcell effect to enhance the zpl and thus improve its branching ratio with respect to the other transitions . in our current measurements , @xmath115 reaches a value of @xmath116 , corresponding to @xmath117 , which would be sufficient for turning a typical branching ratio of about 30% for the zpl of aromatic molecules to about 92% . such a nearly perfect two - level system can then act as an efficient source of single narrow - band photons @xcite . another potential use of our cavity concept is for the achievement of few - photon nonlinear effects on single quantum emitters @xcite . examples of such an effect are three - photon mixing , stimulated rayleigh scattering and hyper - raman scattering @xcite . while there have been a few reports of direct cavity - free optical nonlinear studies on single molecules and quantum dots @xcite , enhancements of the optical field in the cavity and of the radiation of a quantum emitter into its mode can boost these effects further . here , a large cavity bandwidth is of crucial importance since simultaneous coupling to several wavelengths would be otherwise not possible . the broad resonance of a microcavity is also a great asset for studying cooperative effect among many quantum emitters with slightly different resonance frequencies or with large homogeneous lines as it occurs in solid - state systems . an example of such phenomena concerns rabi splittings and coupling to polaritonic states at room temperature @xcite . although the coupling of each individual molecule or exciton is negligible in these systems , strong rabi splitting are observed when a large number of emitters join efforts . the combination of a large cooperativity , small @xmath3 and low @xmath2 would allow one to examine polaritonic physics with far fewer emitters than previously accessible both at room temperature and at cryogenic conditions . a further practical advantage of a considerable cooperativity factor combined with a low @xmath2 is that the experimental setup becomes much less sensitive to mechanical instabilities . this is particularly helpful for experiments in high vacuum or in cryostats , where vibrations are omnipresent . indeed , the arrangement of the cavity presented here could be highly interesting for use in atom @xcite and nanoparticle trapping @xcite , especially for on - chip schemes @xcite . for example , the antinode in fig . [ volume - cuts](a ) is ideally suited for this purpose , keeping the particle sufficiently far from mirror surfaces to avoid van der waals forces @xcite . the cantilever - based nature of our experimental arrangement also holds great promise in the context of optomechanics . let us consider a silicon cantilever of thickness 0.7 @xmath9 , length 10 @xmath9 and width 5 @xmath9 , yielding a mechanical oscillation frequency of about 5 mhz and mass of @xmath118 kg . a micromirror on such a cantilever forming a cavity length @xmath119 would result in a nearly maximal field per photon @xmath120 and an optomechanical coupling strength between the photon and a single phonon of @xmath121 hz , which is comparable with the best reported values @xcite . for these experiments , a higher q would further improve the cavity performance . as mentioned earlier , these can be readily increased by using dielectric coatings instead of gold . to respect the small radius of curvature of the micromirror , one would use materials with strong dielectric contrasts to reduce the number of the necessary bragg layers . other applications of the work presented here are in optical sensing and analytics . over the past decades , a large body of literature has been devoted to the use of optical micro- and nanostructures such as waveguides , cavities and plasmonic nanoantennas for detecting small traces of biological matter @xcite . the existing works on microcavity sensing use fairly large mode volumes and require , therefore , high @xmath2s to reach sufficient sensitivity . recently , we demonstrated that single small proteins can be detected via their direct scattering in a label - free fashion if only diffraction - limited imaging is implemented @xcite . in a nutshell , we have shown that the scattering cross section of a protein , as small as it is , is large enough to extinguish a measurable amount of power from a laser beam . if one now maintains a tight focus and circulates the laser beam in a cavity to allow for repeated interactions with the analyte species , one directly wins in the detection sensitivity by @xmath122 folds . based on a similar argument , our tunable and scannable microcavity can also find applications in raman microscopy , where small cross sections would be compensated by larger excitation intensities . again , a large cavity bandwidth is essential to allow a simultaneous coupling of the excitation beam and the raman signal . the physics of microlasers is another area that would benefit from our cavity design . since its invention more than half a century ago , there has been a continuous stream of reports on various fundamental aspects of laser physics , for example of lasing with minimal gain or with very small thresholds . it turns out that again an important figure of merit is given by @xmath93 @xcite . here , a higher @xmath2 increases the degree of coherence although it also minimizes the overlap between the homogeneous spectrum of the gain medium and the cavity mode , resulting in higher thresholds . a lower @xmath3 ( and thus a higher finesse ) pushes away the modes to reduce mode competition . in addition , a large purcell factor ensures a high @xmath123 , which determines the fraction of the overall spontaneous emission funneled in the mode of interest according to @xmath124 . therefore , the regime of our microcavity , simultaneously keeping a low @xmath2 , low @xmath3 and high @xmath7 , has promise in the development of new microlasers , especially in combination with fluidics @xcite . aside from their academic interest , low - threshold microlasers are also highly in demand for applications , where miniaturization and scaling play a role . we acknowledge financial support from the european union ( erc advanced grant singleion and project univsem of the seventh framework program under grant agreement no . 280566 ) , alexander von humboldt foundation ( humboldt professur ) and the max planck society . we thank harald haakh , ioannis chremmos and florian marquardt for fruitful discussions and maksim medvedev , lothar meier and marek piliarik for technical support at various stages of this project . we also thank sara mouradian for her contribution to the very early stage of this work . h.k . and d.w . contributed equally to this work .

we report on the realization of an open plane - concave fabry - prot resonator with a mode volume below @xmath0 at optical frequencies . we discuss some of the less common features of this new microcavity regime and show that the ultrasmall mode volume allows us to detect cavity resonance shifts induced by single nanoparticles even at quality factors as low as @xmath1 . being based on low - reflectivity micromirrors fabricated on a silicon cantilever , our experimental arrangement provides broadband operation , tunability of the cavity resonance , lateral scanning and promise for optomechanical studies .

You are an expert at summarizing long articles. Proceed to summarize the following text: experiments to date exploit wide feshbach resonances and are thus well described by the simplest single - channel hamiltonian , where the two fermion species interact via an attractive contact potential @xmath2 here , @xmath3 ( we set @xmath4 and @xmath5 ) , @xmath6 is the volume , and we define the chemical potential @xmath7 and ` zeeman ' field @xmath8 such that @xmath9 and @xmath10 . at present , only pairing between different hyperfine species of the _ same _ atom has been explored experimentally , so we restrict ourselves to a single mass @xmath11 . the interaction strength @xmath12 is expressed in terms of the s - wave scattering length @xmath13 using the prescription : @xmath14 we also derive the fermi momentum using the average density @xmath15 , so that @xmath16 . throughout our calculations , we will keep @xmath17 fixed . the full phase diagram is parameterised by just a few observables : the temperature @xmath18 , the interaction strength @xmath19 , and the density difference or ` magnetisation ' @xmath20 . to determine the position of the phase boundaries , we must minimise the mean - field free energy density @xmath21,\end{gathered}\ ] ] with respect to the bcs order parameter @xmath22 , where @xmath23 and @xmath24 . such a mean - field analysis provides a reasonable description of the zero temperature phase diagram , but at finite temperature , it neglects the contribution of non - condensed pairs to both the density @xmath25 and magnetisation @xmath26 . this contribution is necessary to approach the transition temperature of an ideal bose gas in the molecular limit , and can be included in the non - condensed phase ( @xmath27 ) through the nozires - schmitt - rink ( nsr ) fluctuation correction to the energy @xcite @xmath28 with @xmath29 + \tanh[\frac{\beta}{2}(\xi_{{{\mathbf p}}+{{\mathbf q } } } - h)]}}{i\omega + \xi_{{{\mathbf p } } } + \xi_{{{\mathbf p}}+{{\mathbf q } } } } \ ; .\end{gathered}\ ] ] this gives an estimate of the effect of pair fluctuations on the second order phase boundary ( but not the first order boundary , where @xmath30 ) . and magnetisation @xmath31 ( inset ) versus interaction @xmath19 . there are four different phases : the normal ( n ) state , the phase - separated ( ps ) state , the ordinary superfluid ( sf ) and the magnetised superfluid ( sf@xmath32 ) . above the line @xmath33 , the normal state is completely polarised ( @xmath34 ) . the red and black lines enclosing the ps state are both first - order phase boundaries , while the sf@xmath32-n transition is second - order , and the sf - sf@xmath32 transition ( green line ) is at least third - order . the tricritical point is represented by orange circles at @xmath35 with @xmath36 or @xmath34 . [ fig : zerot],scaledwidth=45.0% ] considerable insight can be gained by first examining the zero temperature mean - field phase diagram , as shown in fig . [ fig : zerot ] . the general structure parallels that of the two - channel case found in ref . @xcite . since there is a gap in the quasiparticle excitation spectrum @xmath37 of the unpolarised superfluid , the superfluid ground state will remain unchanged for @xmath38 . we see that the @xmath39 superfluid line in the inset of fig . [ fig : zerot ] corresponds to an _ area _ in the @xmath40 versus @xmath19 diagram , which expands as @xmath19 increases . a key feature of the strong coupling side is that for @xmath41 the superfluid state is able to sustain a finite population of majority quasiparticles . this `` gapless '' @xcite superfluid phase is only stable for @xmath42 and it thus possesses only one fermi surface . in the extreme bec limit , this state is straightforwardly understood as an almost ideal mixture of bosonic pairs and fermionic quasiparticles . however , as we move towards unitarity , the bosons and fermions begin to interact more strongly , leading eventually to a first - order phase transition to the normal state . here , a system with fixed @xmath43 will undergo phase separation into normal and superfluid regions if @xmath44 , where @xmath45 denotes the magnetisation in the normal and superfluid phases at @xmath46 , the critical field for the first - order transition . in the bcs limit ( @xmath47 ) , @xmath48 which is less than the quasiparticle gap , so the superfluid state is unmagnetised @xmath49 , and phase separation occurs for arbitrarily low magnetisation , consistent with ref . @xcite . for the moment we neglect the fflo state , but will return to this point later . a crucial observation is that the line @xmath34 to the right of the region of phase separation can be thought of as a continuous zero temperature transition at which the condensate is totally depleted . it is thus natural to identify the point on @xmath34 where phase separation starts as a tricritical point . indeed a landau expansion of the free energy both confirms this and identifies the tricritical point at @xmath35 . and interaction @xmath19 . the plane at temperature @xmath50 is the phase diagram in fig . [ fig : zerot ] . the yellow line represents the locus of tricritical points calculated in the mean - field approximation , while the orange tricritical line corresponds to mean - field theory plus pair fluctuations . the fluctuation correction breaks down in the unitarity regime @xmath51 , and is thus shown as a dotted line . the slice at @xmath52 is based on a mean - field calculation and it shows the region of phase separation terminating in a tricritical point ( yellow circle ) at finite temperature , followed by a second - order phase transition from the superfluid to normal state . note that the boundary between the fflo and normal states ( blue line ) defines a small region of fflo phase confined to the bcs side of the crossover , as explained in the text.,scaledwidth=45.0% ] . the effective interaction is parameterised by the detuning @xmath53 . the colour scheme for tricritical lines is the same as in fig . [ fig : tricritical].,scaledwidth=45.0% ] with this background , we now turn to the analysis of the fate of the tricritical point when temperature is finite , beginning with the mean - field description . it is well known that there exists a finite temperature tricritical point in the bcs limit @xmath54 , which is a natural consequence of having a first - order transition from the superfluid to normal state at @xmath50 and a second - order transition at @xmath39 . first studied by sarma in the context of superconductivity in the presence of a magnetic field @xcite , the bcs tricritical point is located at @xmath55 @xcite , where @xmath56 $ ] ( i.e. at weak coupling all energies scale with @xmath22 ) . this corresponds to a magnetisation @xmath57 , where @xmath58 is the fermi surface density of states . to investigate how the bcs tricritical point is related to the one at zero temperature , we must develop a perturbative expansion of eq . ( [ eq : energy ] ) for small @xmath22 and general @xmath19 . doing so , one finds ( fig . [ fig : tricritical ] ) that the tricritical point at @xmath34 is connected to that in the bcs limit by a line of tricritical points that passes through a maximum somewhere in the ` unitarity ' regime @xmath59 . moreover , for any given value of @xmath60 , the @xmath61 phase diagram is highly reminiscent of the @xmath0he-@xmath1he system , with @xmath31 playing the role of the fraction of @xmath0he . this is not surprising , as the finite @xmath43 system corresponds in general to a mixture of bosonic pairs and fermionic quasiparticles . note that even the gapped superfluid can be magnetised at finite temperature due to thermal excitation of quasiparticles . of course , at @xmath39 the transition into the superfluid state is second order at any point in the bcs - bec crossover . it is interesting to examine how the fflo phase fits in with the basic topology of the phase diagram . in the bcs limit , we already know that the point where the fflo - normal phase boundary meets the normal - superfluid boundary asymptotes to the tricritical point @xcite . assuming that the transition from the fflo state to the normal state is second - order ( although ref . @xcite found it to be weakly first order , this will make a relatively small difference ) , and performing a mean - field analysis , we find that the fflo point of intersection leaves the finite temperature tricritical point with increasing interaction ( see fig . [ fig : tricritical ] ) , leading eventually to the extinction of the fflo phase at @xmath62 . note that although this treatment is somewhat approximate , as we have taken the sf - fflo boundary to be the same as the sf - n boundary in the absence of fflo , the point of intersection will coincide with that derived from a complete mean - field analysis . moreover , despite all our assumptions , we expect the detachment of the point of intersection from the tricritical point and the eventual disappearance of fflo to be robust features , since in the bec regime we essentially have a mixture of bosons and fermions . the inclusion of the fluctuation contribution eq . ( [ fluct ] ) is crucial for recovering the extreme bec limit , where it is clear that the ( second - order ) transition temperature asymptotes to @xmath63 ( with @xmath64 ) , the ideal bec temperature of a gas of bosons of density @xmath65 and mass @xmath66 . more importantly , we find that fluctuations shift the mean - field tricritical line to lower temperatures and magnetisations on the bec side , while leaving the tricritical points on the bcs side largely unchanged , as expected . however , in a broad region around unitarity , we find that the approximation underlying eq . ( [ fluct ] ) generally leads to non - monotonic behavior of @xmath67 , with @xmath68 for small @xmath8 . we interpret this behaviour as a breakdown of the nsr treatment , yielding an unphysical compressibility matrix @xmath69 that is not positive semi - definite . to address this problem , we note that the nsr scheme is a controlled approximation when we introduce resonant scattering with a finite width , with the width being a small fraction of the fermi energy @xcite . the simplest such description is provided by the two - channel model @xcite . the two - channel description of scattering depends upon two parameters : a detuning @xmath53 describing the distance from the resonance , and a width @xmath70 of the resonance measured in units of the fermi energy . the one - channel description is recovered in the @xmath71 limit , while the treatment of gaussian fluctuations is essentially perturbative in @xmath70 , with @xmath72 in eq . ( [ fluct ] ) being replaced with @xmath73 , so in this case the nsr treatment is expected to be accurate . the resulting phase diagram is shown in figure [ fig : tricritical_2chan ] . the zero temperature phase diagram coincides with the result of ref . @xcite . with fluctuations accounted for , and for sufficiently small @xmath70 , we now find a well - behaved line of tricritical points spanning the crossover region . we expect that the true phase boundary at @xmath71 is qualitatively similar . in the @xmath74-@xmath75 plane . the red and black lines are first- and second - order phase boundaries , respectively . the arrows at constant @xmath75 represent the trajectories followed when going from the centre to the edges of a trapped gas . the two trajectories correspond to two different magnetisations of the gas : one greater and one less than the tricritical point @xmath76.,scaledwidth=42.0% ] we now discuss the consequences of our results for trapped gases studied in experiment . modeling the trapped gas by the local density approximation ( lda ) , the spatial dependence of the density induced by the trapping potential @xmath77 is accounted for by a spatially - varying chemical potential @xmath78 , with @xmath8 kept constant . in the @xmath74-@xmath75 plane , we thus move on a horizontal line ( see fig . [ fig : tri_trap ] ) . at sufficiently low temperatures , a trapped gas will consist of a superfluid core surrounded by the normal state . the transition between normal and superfluid states in the trap can be either second or first order , depending on whether @xmath75 is above or below the tricritical point . moreover , as long as the temperature is non - zero , we can always find a sufficiently small @xmath8 so that @xmath75 lies above the tricritical point . this leads us to a key point : if a trapped gas at a given temperature and magnetisation has a first - order transition between its normal and superfluid phases , then we will _ always _ cross the tricritical point by decreasing the magnetisation at fixed temperature . we emphasise that there are qualitative differences between first and second order transitions in a trap : the former yields a discontinuity in the density and magnetization at the phase interface , resulting in a form of phase separation as seen in recent experiments @xcite , while the latter possesses a density that varies smoothly in space . therefore , the magnetisation and temperature at which a tricritical point is crossed should be detectable experimentally . in fact , a critical magnetisation for the onset of phase separation in a trap has been observed experimentally @xcite , and a calculation by chevy supports the idea that this coincides with crossing a tricritical point @xcite . in addition , the order of the transition will have an impact on experiments that use phase separation as a signature of superfluidity @xcite . the presence of a first - order transition in the trap can be even more pronounced if the density discontinuities result in a breakdown of lda . experiments on highly elongated traps already provide evidence for such a breakdown @xcite , and one requires the addition of surface energy terms at the phase interface to successfully model the trapped density profiles @xcite . an outstanding issue is the experimental detection of the gapless sf@xmath32 phase . while optically probing the momentum distribution of the minority species is one promising method for detecting sf@xmath32 @xcite , another possibility is to study density correlations using , for example , shot noise experiments as suggested in ref . a simple mean - field calculation gives ( for the uniform system ) : @xmath79 ^ 2\end{aligned}\ ] ] where @xmath80 is the fermi - dirac distribution . at @xmath50 , the result is a ` hole ' in the correlation function for momenta less than the fermi wavevector of the majority quasiparticles . such a measurement would therefore constitute both a confirmation of the sf@xmath32 phase and a vivid demonstration of the blocking effect of quasiparticles on @xmath81 pairing . in conclusion , we have determined the structure of the finite temperature phase diagram of the two component fermi gas , as a function of both interaction strength and population imbalance , finding a region of phase separation terminating in a tricritical point for general coupling in the bcs - bec crossover . a secondary result of our work is the demonstration that the nsr scheme yields unphysical results in a broad region around unitarity . this is significant , as it is widely viewed as offering a smooth , albeit uncontrolled approximation throughout the crossover . we emphasize that there is no _ a priori _ reason to believe in the accuracy of the nsr scheme without introducing an additional parameter , as we have done here . the ginzburg criterion governing the smallness of fluctuation corrections is satisfied in both the bcs limit where it takes the form @xmath82 , and in the bec limit where @xmath83 is the relevant criterion . but at unitarity the shift in the transition temperature relative to the mean field value will be of order @xmath84 . at the same time the upper critical dimension at the tricritical point is three , so we may expect that our results there will be little changed . finally , we have argued that these tricritical points play an important role in experiments on trapped fermi gases ( see , also , the subsequent related work on trapped gases at unitarity by gubbels et al . indeed , a recent comprehensive study of the temperature dependence of the phase - separated state at unitarity has yielded experimental results consistent with the phase diagram outlined here @xcite .

the two - component fermi gas is the simplest fermion system displaying superfluidity , and as such is relevant to topics ranging from superconductivity to qcd . ultracold atomic gases provide an exceptionally clean realisation of this system , where interatomic interactions and atom spin populations are both independently tuneable . here we show that the finite temperature phase diagram contains a region of phase separation between the superfluid and normal states that touches the boundary of second - order superfluid transitions at a tricritical point , reminiscent of the phase diagram of @xmath0he-@xmath1he mixtures . a variation of interaction strength then results in a line of tricritical points that terminates at zero temperature on the molecular bose - einstein condensate ( bec ) side . on this basis , we argue that tricritical points are fundamental to understanding experiments on polarised atomic fermi gases . over the past decade , experimental progress in the field of cold atomic gases has resulted in unprecedented control over pairing phenomena in two - component fermi gases . the ability to vary the effective interaction between atoms using magnetically tuned feshbach resonances has already permitted the experimental investigation of the crossover from a bec of diatomic molecules to the bardeen - cooper - schrieffer ( bcs ) limit of weakly - bound cooper pairs of fermionic atoms @xcite . a natural extension of these studies is an exploration of the fermi gas with imbalanced spin populations , especially since this system has a far richer phase diagram than the equal spin case . as well as exhibiting a quantum phase transition between the superfluid and normal states , the polarized fermi gas has been predicted to possess exotic superfluid phases such as the inhomogeneous fulde - ferrell - larkin - ovchinnikov ( fflo ) state @xcite , where the pairing of fermions occurs at finite centre - of - mass momentum , and the deformed fermi surface state @xcite . the exact nature of the superfluid states for the polarised fermi gas is still the subject of considerable debate . however , atomic gases provide an ideal testing ground for this system , since the particle numbers can be varied independently from all other experimental parameters , and pioneering experiments have recently been performed @xcite . contrast atomic gases with the case of superconductors , where the magnetic field used to generate a spin imbalance ( via the zeeman effect ) also couples to orbital degrees of freedom . in this work , we elucidate the finite temperature phase diagram of a polarised fermi gas . while much insight has been gained from previous theoretical studies @xcite , so far a key ingredient of the phase diagram has been overlooked : the tricritical point , at which the phase transition between superfluid and normal states switches from first to second order . by determining the behaviour of the tricritical point as a function of interaction strength , we can completely characterise the topology of the phase diagram without recourse to an extensive numerical treatment . specifically , we shall focus on the uniform , infinite system , and concern ourselves almost exclusively with the phase boundary between the normal and homogeneous superfluid states . we will , however , discuss the ramifications of the inferred phase diagram for the trapped system .

You are an expert at summarizing long articles. Proceed to summarize the following text: quantum - mechanical fluctuations during an early epoch of inflation provide a plausible mechanism to generate the energy - density perturbations responsible for observed cosmological structure . while it has been known for quite some time that inflation is consistent with open spatial hypersurfaces ( gott 1982 ; guth & weinberg 1983 ) , attention was initially focussed on models in which there are a very large number of @xmath17-foldings during inflation , resulting in almost exactly flat spatial hypersurfaces for the observable part of the present universe ( guth 1981 ; also see kazanas 1980 ; sato 1981a , b ) . this was , perhaps , inevitable because of strong theoretical prejudice towards flat spatial hypersurfaces and their resulting simplicity . however , to get a very large number of @xmath17-foldings during inflation it seems necessary that the inflation model have a small dimensionless parameter ( j. r. gott , private communication 1994 ; banks et al . 1995 ) , which would require an explanation . attempts to reconcile these favoured " flat spatial hypersurfaces with observational measures of a low value for the clustered - mass density parameter @xmath1 have concentrated on models in which one postulates the presence of a cosmological constant @xmath18 ( peebles 1984 ) . in the simplest flat-@xmath18 model one assumes a scale - invariant ( harrison 1970 ; peebles & yu 1970 ; zeldovich 1972 ) primordial power spectrum for gaussian adiabatic energy - density perturbations . such a spectrum is generated by quantum - mechanical fluctuations during an early epoch of inflation in a spatially - flat model , provided that the inflaton potential is reasonably flat ( fischler , ratra , & susskind 1985 , and references therein ) . it has been demonstrated that these models are indeed consistent with current observational constraints ( e.g. , stompor , grski , & banday 1995 ; ostriker & steinhardt 1995 ; ratra & sugiyama 1995 ; liddle et al . 1996b ; ganga , ratra , & sugiyama 1996b , hereafter grs ) . an alternative , more popular of late , is to accept that the spatial hypersurfaces are not flat . in this case , the radius of curvature for the open spatial sections introduces a new length scale ( in addition to the hubble length ) , which requires a generalization of the usual flat - space scale - invariant spectrum ( ratra & peebles 1994 , hereafter rp94 ) . such a spectrum is generated by quantum - mechanical fluctuations during an epoch of inflation in an open - bubble model ( rp94 ; ratra & peebles 1995 , hereafter rp95 ; bucher et al . 1995 , hereafter bgt ; lyth & woszczyna 1995 ; yamamoto et al . 1995 , hereafter yst ) , provided that the inflaton potential inside the bubble is reasonably flat . such gaussian adiabatic open - bubble inflation models have also been shown to be consistent with current observational constraints ( rp94 ; kamionkowski et al . 1994 ; grski et al . 1995 , hereafter grsb ; liddle et al . 1996a , hereafter llrv ; ratra et al . 1995 ; grs ) . inflation theory by itself is unable to predict the normalization amplitude for the energy - density perturbations . currently , the least controversial and most robust method for the normalization of a cosmological model is to fix the amplitude of the model - predicted large - scale cmb spatial anisotropy by comparing it to the observed cmb anisotropy discovered by the @xmath0-dmr experiment ( smoot et al . 1992 ) . previously , specific open cold dark matter ( cdm ) models have been examined in light of the @xmath0-dmr two - year results ( bennett et al . grsb investigated the cmb anisotropy angular spectra predicted by the open - bubble inflation model ( rp94 ) , and compared large - scale structure predictions of this dmr - normalized model to observational data . cayn et al . ( 1996 ) performed a related analysis for the open model with a flat - space scale - invariant spectrum ( wilson 1983 , hereafter w83 ) , and yamamoto & bunn ( 1996 , hereafter yb ) examined the effect of additional sources of quantum fluctuations ( bgt ; yst ) in the open - bubble inflation model . in this paper , we study the observational predictions for a number of open cdm models . in particular , we employ the power spectrum estimation technique devised by grski ( 1994 ) for incomplete sky coverage to normalize the open models using the @xmath0-dmr four - year data ( bennett 1996 ) . in @xmath19 we provide an overview of open - bubble inflation cosmogonies . in @xmath20 we detail the various dmr data sets used in the analyses here , discuss the various open models we consider , and present the dmr estimate of the cmb rms quadrupole anisotropy amplitude @xmath21 as a function of @xmath1 for these open models . in @xmath22 we detail the computation of several cosmographic and large - scale structure statistics for the dmr - normalized open models . these statistics are confronted by various current observational constraints in @xmath23 . our results are summarized in @xmath24 . the simplest open inflation model is that in which a single open - inflation bubble nucleates in a ( possibly ) spatially - flat , inflating spacetime ( gott 1982 ; guth & weinberg 1983 ) . in this model , the first epoch of inflation smooths away any preexisting spatial inhomogeneities , while simultaneously generating quantum - mechanical zero - point fluctuations . then , in a tunnelling event , an open - inflation bubble nucleates , and for a small enough nucleation probability the observable universe lies inside a single open - inflation bubble . fluctuations of relevance to the late - time universe can be generated via three different quantum mechanical mechanisms : ( 1 ) they can be generated in the first epoch of inflation ; ( 2 ) they can be generated during the tunnelling event ( thus resulting in a slightly inhomogeneous initial hypersurface inside the bubble , or a slightly non - spherical bubble ) ; and ( 3 ) they can be generated inside the bubble . the tunneling amplitude is largest for the most symmetrical solution ( and deviations from symmetry lead to an exponential suppression ) , so it has usually been assumed that the nucleation process ( mechanism [ 2 ] ) does not lead to the generation of significant inhomogeneities . quantum - mechanical fluctuations generated during evolution inside the bubble ( rp95 ) are significant . assuming that the energy - density difference between the two epochs of inflation is negligible ( and so the bubble wall is not significant ) , one may estimate the contribution to the perturbation spectrum after bubble nucleation from quantum - mechanical fluctuations during the first epoch of inflation ( bgt ; yst ) . as discussed by bucher & turok ( 1995 , hereafter bt ) ( also see yst ; yb ) , the observable predictions of these simple open - bubble inflation models are almost completely insensitive to the details of the first epoch of inflation , for the observationally - viable range of @xmath1 . this is because the fluctuations generated during this epoch affect only the smallest wavenumber part of the energy - density perturbation power spectrum , which can not contribute significantly to observable quantities because of the spatial curvature length cutoff " in an open universe ( e.g. , w83 ; kamionkowski & spergel 1994 ; rp95 ) . inclusion of such fluctuations in the calculations alter the predictions for the present value of the rms linear mass fluctuations averaged over an @xmath25 mpc sphere , @xmath26 $ ] , by @xmath27 ( which is comparable to our computational accuracy ) . besides the open - bubble inflation model spectra , a variety of alternatives have also been considered . predictions for the usual flat - space scale - invariant spectrum in an open model have been examined ( w83 ; abbott & schaefer 1986 ; gouda , sugiyama , & sasaki 1991 ; sugiyama & gouda 1992 ; kamionkowski & spergel 1994 ; sugiyama & silk 1994 ; cayn et al . the possibility that the standard formulation of quantum mechanics is incorrect in an open universe , and that allowance must be made for non - square - integrable basis functions has been investigated ( lyth & woszczyna 1995 ) , and other spectra have also been considered ( e.g. , w83 ; abbott & schaefer 1986 ; kamionkowski & spergel 1994 ) . these spectra , being inconsistent with either standard quantum mechanics or the length scale set by spatial curvature , are of historical interest . more recently , the open - bubble inflation scenario has been further elaborated on . yst have considered a very specific model for the nucleation of the open bubble in a spatially - flat de sitter spacetime , and demonstrated a possible additional contribution from a non - square - integrable basis function which depends on the form of the potential , and on the assumed form of the quantum state prior to bubble nucleation . however , since the non - square - integrable basis function contributes only on the very largest scales , the spatial curvature cutoff " in an open universe makes almost all of the model predictions insensitive to this basis function , for the observationally - viable range of @xmath1 ( yst ; yb ) . for example , at @xmath28 its effect is to change @xmath26 $ ] by @xmath29 . an additional possible effect determined for the specific model of an open - inflation bubble nucleating in a spatially - flat de sitter spacetime is that fluctuations of the bubble wall behave like a non - square - integrable basis function ( hamazaki et al . 1996 ; garriga 1996 ; garca - bellido 1996 ; yamamoto , sasaki , & tanaka 1996 ) . while there are models in which these bubble - wall fluctuations are completely insignificant ( garriga 1996 ; yamamoto et al . 1996 ) , there is as yet no computation that accounts for both the bubble - wall fluctuations as well as those generated during the evolution inside the bubble ( which are always present ) , so it is not yet known if bubble - wall fluctuations can give rise to an observationally significant effect . finally , again in this very specific model , the effects of a finite bubble size at nucleation seem to alter the zero bubble size predictions only by a very small amount ( yamamoto et al . 1996 ; cohn 1996 ) . while there is no guarantee that there is a spatially - flat de sitter spacetime prior to bubble nucleation , these computations do illustrate the important point that the spatial curvature length cutoff " in an open universe ( e.g. , rp95 ) does seem to ensure that what happens prior to bubble nucleation does not significantly affect the observable predictions for observationally - viable single - field open - bubble inflation models . it is indeed reassuring that accounting only for the quantum mechanical fluctuations generated during the evolution inside the bubble ( rp94 ) seems to be essentially all that is required to make observational predictions for the single - field open - bubble inflation models . that is , the observational predictions of the open - bubble inflation scenario seem to be as robust as those for the spatially - flat inflation scenario . in this paper , we utilize the dmr four - year 53 and 90 ghz sky maps in both galactic and ecliptic coordinates . we thus quantify explicitly the expected small shifts in the inferred normalization amplitudes due to the small differences between the galactic- and ecliptic - coordinate maps . the maps are coadded using inverse - noise - variance weights derived in each coordinate system . the least sensitive 31 ghz maps have been omitted from the analysis , since their contribution is minimal under such a weighting scheme . the dominant source of emission in the dmr maps is due to the galactic plane . we are unable to model this contribution to the sky temperature to sufficient accuracy to enable its subtraction , thus we excise all pixels where the galactic - plane signal dominates the cmb . the geometry of the cut has been determined by using the dirbe 140 @xmath30 m map as a tracer of the strongest emission , as described completely in banday ( 1996a ) . all pixels with galactic latitude @xmath31 20@xmath32@xmath33 are removed , together with regions towards scorpius - ophiucus and taurus - orion . there are 3881 surviving pixels in galactic coordinates and 3890 in ecliptic . this extended ( four - year data ) galactic plane cut has provided the biggest impact on the analysis of the dmr data ( see grski et al . 1996 , hereafter g96 ) . the extent to which residual high - latitude galactic emission can modify our results has been quantified in two ways . since the spatial morphology of galactic synchrotron , free - free and dust emission seems to be well described by a steeply falling power spectrum ( @xmath34 kogut 1996a , g96 ) , the cosmological signal is predominantly compromised on the largest angular scales . as a simple test of galactic contamination , we perform all computations both including and excluding the observed sky quadrupole . a more detailed approach ( g96 ) notes that a large fraction of the galactic signal can be accounted for by using the dirbe 140 @xmath30 m sky map ( reach 1995 ) as a template for free - free and dust emission , and the 408 mhz all - sky radio survey ( haslam 1981 ) to describe synchrotron emission . a correlation analysis yields coupling coefficients for the two templates at each of the dmr frequencies . we have repeated our model analysis after correcting the coadded sky maps by the galactic templates scaled by the coefficients derived in g96 . in particular , we adopt those values derived under the assumption that the cmb anisotropy is well - described by an @xmath35 = 1 power law model with normalization amplitude @xmath21 @xmath36 18 @xmath30k and coupling coefficient amplitudes . in fact , we have investigated this for a sub - sample of the models considered here in which we varied @xmath1 but fixed @xmath2 and @xmath10 . no statistically significant changes were found in the derived values of either @xmath21 or the coupling coefficients . ] . one might make criticisms of either technique : excluding information from an analysis , in this case the quadrupole components , can obviously weaken any conclusions simply because statistical uncertainties will grow ; at the same time , it is not clear whether the galactic corrections applied are completely adequate . we believe that , given these uncertainties , our analysis is the most complete and conservative one that is possible . the power spectrum analysis technique developed by grski ( 1994 ) is implemented . orthogonal basis functions for the fourier decomposition of the sky maps are constructed which specifically include both pixelization effects and the galactic cut . ( these are linear combinations of the usual spherical harmonics with multipole @xmath37 . ) the functions are coordinate system dependent . a likelihood analysis is then performed as described in grski ( 1994 ) . we consider four open model energy - density perturbation power spectra : ( 1 ) the open - bubble inflation model spectrum , accounting only for fluctuations that are generated during the evolution inside the bubble ( rp94 ) ; ( 2 ) the open - bubble inflation model spectrum , now also accounting for the fluctuations generated in the first epoch of inflation ( bgt ; yst ) ; ( 3 ) the open - bubble inflation model spectrum , now also accounting for both the usual fluctuations generated in the first epoch of inflation and a contribution from a non - square - integrable basis function ( yst ) ; and , ( 4 ) an open model with a flat - space scale - invariant spectrum ( w83 ) . in all cases we have ignored the possibility of tilt or primordial gravity waves , since it is unlikely that they can have a significant effect in viable open models . with the eigenvalue of the spatial scalar laplacian being @xmath38 , where @xmath39 is the radial coordinate spatial wavenumber , the gauge - invariant fractional energy - density perturbation power spectrum of type ( 1 ) above is @xmath40 where @xmath41 is the transfer function and @xmath42 is the normalization amplitude ) generalize the primordial part of the spectrum of eq . ( 1 ) by multiplying it with @xmath43 . as yet , only the specific @xmath44 generalized spectrum ( i.e. , eq . [ 1 ] ) is known to be a prediction of an open - bubble inflation model and therefore consistent with the presence of spatial curvature . it is premature to draw conclusions about open cosmogony on the basis of the @xmath45 version of the spectrum considered by bw . ] . in the simplest example , perturbations generated in the first epoch of inflation introduce an additional multiplicative factor , @xmath46 , on the right hand side of eq . ( 1 ) . for a discussion of the effects of the non - square - integrable basis function see yst and yb . the energy - density power spectrum of type ( 4 ) above is @xmath47 and in this case one can also consider , e.g. , @xmath48 ( w83 ) , but because of the spatial curvature cutoff " in an open model the predictions are essentially indistinguishable . at small @xmath49 the asymptotic expressions are @xmath50 ( type 1 ) , @xmath51 ( type 2 ) , and @xmath52 ( type 4 ) . conventionally , the cmb fractional temperature perturbation , @xmath53 , is expressed as a function of angular position , @xmath54 , on the sky via the spherical harmonic decomposition , @xmath55 the cmb spatial anisotropy in a gaussian model can then be characterized by the angular perturbation spectrum @xmath56 , defined in terms of the ensemble average , @xmath57 the @xmath56 s used here were computed using two independent boltzmann transfer codes developed by ns ( e.g. , sugiyama 1995 ) and rs ( e.g. , stompor 1994 ) . some illustrative comparisons are shown in fig . we emphasize that the excellent agreement between the @xmath56 s computed using the two codes is mostly a reflection of the currently achievable numerical accuracy . currently , the major likely additional , unaccounted for , source of uncertainty is that due to the uncertainty in the modelling of various physical effects . the computations here assume a standard recombination thermal history , and ignore the possibility of early reionization . the simplest open models ( with the least possible number of free parameters ) have yet to be ruled out by observational data ( grsb ; ratra et al . 1995 ; grs ; this paper ) , so there is insufficient motivation to expand the model - parameter space by including the effect of early reionization , tilt or gravity waves values determined from the dmr data here ( assuming no early reionization ) are unlikely to be very significantly affected by early reionization . however , since structure forms earlier in an open model , other effects of early reionization might be more significant in an open model . while it is possible to heuristically account for such effects , an accurate quantitative estimate must await a better understanding of structure formation . ] . for the @xmath58 of types ( 1 ) , ( 2 ) , and ( 4 ) above , we have evaluated the cmb anisotropy angular spectra for a range of @xmath1 spanning the interval between 0.1 and 1.0 , for a variety of values of @xmath2 ( the hubble parameter @xmath59 ) and the baryonic - mass density parameter @xmath10 . the values of @xmath2 were selected to cover the lower part of the range of ages consistent with current requirements ( @xmath60 10.5 gyr , 12 gyr , or 13.5 gyr , with @xmath2 as a function of @xmath1 computed accordingly ; see , for example , jimenez et al . 1996 ; chaboyer et al . the values of @xmath10 were chosen to be consistent with current standard nucleosynthesis requirements ( @xmath61 0.0055 , 0.0125 , or 0.0205 ; e.g. , copi , schramm , & turner 1995 ; sarkar 1996 ) . to render the problem tractable , @xmath56 s were determined for the central values of @xmath62 and @xmath63 , and for the two combinations of these parameters which most perturb the @xmath56 s from those computed at the central values ( i.e. , for the smallest @xmath62 we used the smallest @xmath63 , and for the largest @xmath62 we used the largest @xmath63 ) . specific parameter values are given in columns ( 1 ) and ( 2 ) of tables 16 , and representative anisotropy spectra can be seen in figs . 2 and 3 . we therefore improve on our earlier analysis of the dmr two - year data ( grsb ) by considering a suitably broader range in the ( @xmath10 , @xmath2 ) parameter space . the cmb anisotropy spectra for @xmath58 of type ( 3 ) above were computed for a range of @xmath1 spanning the interval between 0.1 and 0.9 , for @xmath64 and @xmath65 . specific parameter values are given in columns ( 1 ) and ( 2 ) of table 7 , and these spectra are shown in fig . 4 . in fig . 5 we compare the various spectra considered here . the differences in the low-@xmath66 shapes of the @xmath56 s in the various models ( figs . 25 ) are a consequence of three effects : ( 1 ) the shape of the energy - density perturbation power spectrum at low wavenumber ; ( 2 ) the exponential suppression at the spatial curvature scale in an open model ; and ( 3 ) the interplay between the usual " ( fiducial cdm ) sachs - wolfe term and the integrated " sachs - wolfe ( hereafter sw ) term in the expression for the cmb spatial anisotropy . the relative importance of these effects is determined by the value of @xmath1 , and leads to the non - monotonic behaviour of the large - scale @xmath56 s as a function of @xmath1 seen in figs . more precisely , the contributions to the cmb anisotropy angular spectrum from the usual " and integrated " sw terms have a different @xmath66-dependence as well as a relative amplitude that is both @xmath1 and @xmath58 dependent . on very large angular scales ( small @xmath66 s ) , the dominant contribution to the usual " sw term comes from a higher redshift ( when the length scales are smaller ) than does the dominant contribution to the integrated " sw term ( hu & sugiyama 1994 , 1995 ) . as a result , in an open model on very large angular scales , the usual " sw term is cut off more sharply by the spatial curvature length scale than is the integrated " sw term ( hu & sugiyama 1994 ) , i.e. , on very large angular scales in an open model the usual " sw term has a larger ( positive ) effective index @xmath35 than the integrated " sw term . on slightly smaller angular scales the integrated " sw term is damped ( i.e. , it has a negative effective index @xmath35 ) while the usual " sw term plateaus ( hu & sugiyama 1994 ) . as a consequence , going from the largest to slightly smaller angular scales , the usual " term rises steeply and then flattens , while the integrated " term rises less steeply and then drops ( i.e. , it has a peak ) . the change in shape , as a function of @xmath66 , of these two terms is both @xmath1 and @xmath58 dependent . these are the two dominant effects at @xmath67 ; at higher @xmath66 other effects come into play . more specifically , for @xmath68 the curvature length scale cutoff and the precise large - scale form of the @xmath58 considered here are relatively unimportant the cmb anisotropy angular spectrum is quite similar to that for @xmath69 , and the dominant contribution is the usual " sw term . for a @xmath58 that does not diverge at low wavenumber , as with the flat - space scale - invariant spectrum in an open model , for @xmath70 the exponential cutoff " at the spatial curvature length dominates , and the lowest-@xmath66 @xmath56 s are suppressed ( figs . 3 and 5 ) . for this @xmath58 , as @xmath1 is reduced , the usual " term continues to be important on the largest angular scales down to @xmath28 . as @xmath1 is reduced below @xmath71 the integrated " term starts to dominate on the largest angular scales , and as @xmath1 is further reduced the integrated " term also starts to dominate on smaller angular scales . from fig . 3(a ) one will notice that the integrated " sw term peak " first makes an appearance at @xmath72 the central line in the plot at @xmath73 and that as @xmath1 is further reduced ( in descending order along the curves shown ) the integrated " term peak " moves to smaller angular scales . the @xmath74 case is where the integrated " term peaks at @xmath75 , and the damping of this term on smaller angular scales ( @xmath76 ) is compensated for by the steep rise of the usual " sw term the two terms are of roughly equal magnitude at @xmath77 and these effects result in the almost exactly scale - invariant spectrum at @xmath9 ( this case is more scale - invariant than fiducial cdm ) . a discussion of some of these features of the cmb anisotropy angular spectrum in the flat - space scale - invariant spectrum open model is given in cayn et al . ( 1996 ) . open - bubble inflation models have a @xmath58 that diverges at low wavenumber ( rp95 ; note that no physical quantity diverges ) , and this increases the low-@xmath66 @xmath56 s ( figs . 2 and 5 ) relative to those of the flat - space scale - invariant spectrum open model ( figs . 3 and 5 ) . the @xmath56 s for low @xmath1 models increase more than the higher @xmath1 ones , since , for a fixed wavenumber - dependence of @xmath58 , the divergence is more prominent at lower @xmath1 ( rp94 ) . the non - square - integrable basis function ( yst ) contributes even more power on large angular scales , and so , at low-@xmath66 , the @xmath56 s of fig . 4 are slightly larger than those of fig . 2 ( also see fig . 5 ) . again , spectra at lower values of @xmath1 are more significantly influenced . as is clear from figs . 2 and 5 , in an open - bubble inflation model , quantum - mechanical zero - point fluctuations generated in the first epoch of inflation scarcely affect the @xmath56 s , although at the very lowest values of @xmath1 the very lowest order @xmath56 coefficients are slightly modified . the effect is concentrated in this region of the parameter space since the fluctuations in the first inflation epoch only contribute to , and increase , the lowest wavenumber part of @xmath58 . in simple open - bubble inflation models , the precise value of this small effect is dependent on the model assumed for the first epoch of inflation ( bt ) . since the dmr data is most sensitive to multipole moments with @xmath78 810 , one expects the effect at @xmath78 23 to be almost completely negligible ( bt ; also see yst ; yb ) . figs . 35 show that both the flat - space scale - invariant spectrum open model , and the contribution from the non - square - integrable mode , do lead to significantly different @xmath56 s ( compared to those of fig . the results of the dmr likelihood analyses are summarized in figs . 621 and tables 17 and 13 . two representative sets of likelihood functions @xmath79 are shown in figs . 6 and 7 . figure 6 shows those derived from the ecliptic - frame sky maps , ignoring the correction for faint high - latitude foreground galactic emission , and excluding the quadrupole moment from the analysis . figure 7 shows the likelihood functions derived from the galactic - frame sky maps , accounting for the faint high - latitude foreground galactic emission correction , and including the quadrupole moment in the analysis . together , these two data sets span the maximum range of normalizations inferred from our analysis ( the former providing the highest , and the latter the lowest @xmath21 ) . tables 17 give the @xmath21 central values and 1-@xmath80 and 2-@xmath80 ranges for spectra of type ( 1 ) , ( 3 ) , and ( 4 ) above , computed from the appropriate posterior probability density distribution function assuming a uniform prior . each line in tables 17 lists these values at a given @xmath1 for the 8 possible combinations of : ( 1 ) galactic- or ecliptic - coordinate map ; ( 2 ) faint high - latitude galactic foreground emission correction accounted for or ignored ; and , ( 3 ) quadrupole included ( @xmath81 ) or excluded ( @xmath82 ) value of varying cosmological parameters like @xmath10 . since they do not quote derived @xmath21 values for this model we are not able to compare to their results . ] . the corresponding ridge lines of maximum likelihood @xmath21 value as a function of @xmath1 are shown in figs . 810 for some of the cosmological - parameter values considered here . although we have computed these values for spectra of type ( 2 ) above ( i.e. , those accounting for perturbations generated in the first epoch of inflation ) we record only a subset of them in column ( 4 ) of table 13 . these should be compared to columns ( 2 ) and ( 6 ) of table 13 , which show the maximal 2-@xmath80 @xmath21 range for spectra of types ( 1 ) and ( 3 ) . while the differences in @xmath21 between spectra ( 1 ) and ( 2 ) [ cols . ( 2 ) and ( 4 ) of table 13 ] are not totally insignificant , more importantly the differences between the @xmath26 $ ] values for the three spectra [ cols . ( 3 ) , ( 5 ) , and ( 7 ) of table 13 ] are observationally insignificant . the entries in tables 16 illustrate the shift in the inferred normalization amplitudes due to changes in @xmath2 and @xmath10 . these shifts are larger for models with a larger @xmath1 , since these models have cmb anisotropy spectra that rise somewhat more rapidly towards large @xmath66 , so in these cases the dmr data is sensitive to somewhat smaller angular scales where the effects of varying @xmath2 and @xmath63 are more prominent . figure 11 shows the effects that varying @xmath62 and @xmath63 have on some of the ridge lines of maximum likelihood @xmath21 as a function of @xmath1 , and fig . 13 illustrates the effects on some of the conditional ( fixed @xmath1 slice ) likelihood densities for @xmath83 on the whole , for the cmb anisotropy spectra considered here , shifts in @xmath2 and @xmath84 have only a small effect on the inferred normalization amplitude . the normalization amplitude is somewhat more sensitive to the differences between the galactic- and ecliptic - coordinate sky maps , to the foreground high - latitude galactic emission treatment , and to the inclusion or exclusion of the @xmath85 moment . for the purpose of normalizing models , we choose for our 2-@xmath80 c.l . bounds values from the likelihood fits that span the maximal range in the @xmath21 normalizations . specifically , for the lower 2-@xmath80 bound we adopt the value determined from the analysis of the galactic - coordinate maps accounting for the high - latitude galactic emission correction and including the @xmath85 moment in the analysis , and for the upper 2-@xmath80 value that determined from the analysis of the ecliptic - coordinate maps ignoring the galactic emission correction and excluding the @xmath85 moment from the analysis . these values are recorded in columns ( 5 ) and ( 8) of tables 912 , and columns ( 2 ) , ( 4 ) , and ( 6 ) of table 13 ) were used in the likelihood analyses of the various model spectra , and different interpolation methods were used in the determination of the @xmath21 values , there are small ( but insignificant ) differences in the quoted @xmath21 values for some identical models in these tables . ] . figure 12 compares the ridge lines of maximum likelihood @xmath21 value , as a function of @xmath1 , for the four different cmb anisotropy angular spectra considered here , and fig . 14 compares some of the conditional ( fixed @xmath1 slice ) likelihood densities for @xmath21 for these four cmb anisotropy angular spectra . approximate fitting formulae may be derived to describe the above two extreme 2-@xmath80 limits . for the open - bubble inflation model ( rp94 ; bgt ; yst ) , not including a contribution from a non - square - integrable basis function , we have @xmath86 , \eqno(5)\ ] ] which is good to better than @xmath87 for all values of @xmath1 ( and to better than @xmath88 over the observationally - viable range of @xmath89 ) . for those models including a contribution from the non - square - integrable basis function ( yst ) , we have @xmath90 , \eqno(6)\ ] ] mostly good to better than @xmath88 . the flat - space scale - invariant spectrum open model fitting formula is @xmath91 , \eqno(7)\ ] ] generally good to better than @xmath92 , except near @xmath93 and @xmath94 where the deviations are larger . further details about these fitting formulae may be found in stompor ( 1996 ) . the approximate fitting formulae ( 5)(7 ) provide a convenient , portable normalization of the open models . it is important , however , to note that they have been derived using the @xmath21 values determined for a given @xmath2 and @xmath10 , and hence do not account for the additional uncertainty ( which could be as large as @xmath88 ) due to allowed variations in these parameters . we emphasize that in our analysis here we make use of the actual @xmath21 values derived from the likelihood analyses , not these fitting formulae . figures 15 and 16 show projected likelihood densities for @xmath1 , for some of the models and dmr data sets considered here . note that the general features of the projected likelihood densities for the open - bubble inflation model only accounting for the fluctuations generated during the evolution inside the bubble ( spectrum [ 1 ] above ) , are consistent with those derived from the dmr two - year data ( grsb , fig . 3 ) . however , since we only compute down to @xmath95 here , only the rise to the prominent peak at very low @xmath1 ( grsb ) is seen . bw show in the middle left - hand panel of their fig . 11 ( presumably ) the projected likelihood density for @xmath1 for the same open - bubble inflation model , the general features of which are consistent with those derived here . figures 1721 show marginal likelihood densities for @xmath1 , for some of the models and dmr data sets considered here . for the open - bubble inflation model accounting only for the fluctuations generated during the evolution inside the bubble ( rp94 ) , the dmr two - year data galactic - frame ( quadrupole moment excluded and included ) marginal likelihoods are shown in fig . 3 of grsb , and are in general concord with those shown in fig . 17 here ( although , again , only the rise to the prominent low-@xmath1 peak is seen here ) . note that now , especially for the quadrupole excluded case , the peaks and troughs are more prominent ( although still not greatly statistically significant ) . furthermore , comparing the solid line of fig . 17(b ) here to the heavy dotted line of fig . 3 of grsb , one notices that the intermediate @xmath1 peak is now at @xmath96 , instead of at @xmath97 for the dmr two - year data . ( since bw chose not to compute for the case when the quadrupole moment is excluded from the analysis , they presumably did not notice the peak at @xmath98 in the marginalized likelihood density for the open - bubble inflation model see fig . 17 . ) for the open - bubble inflation model now also accounting for both the fluctuations generated in the first spatially - flat epoch of inflation ( bgt ; yst ) , and those from the non - square - integrable basis function ( yst ) , the dmr two - year data ecliptic - frame quadrupole - included marginal likelihood ( shown as the solid line in fig . 3 of yb ) is in general agreement with the dot - dashed line of fig . however , yb did not compute for the case where the quadrupole moment was excluded from the analysis and so did not find the peak at @xmath99 in fig . 19 . given the shapes of the marginal likelihoods in figs . 1721 , it is not at all clear if it is meaningful to derive limits on @xmath1 without making use of other ( prior ) information . as an example , it is not at all clear what to use for the integration range in @xmath1 . focussing on fig . 21(a ) ( which is similar to the other quadrupole excluded cases ) , the only conclusion seems to be that @xmath9 is the value most consistent with the dmr data ( at least amongst those models with @xmath14 some of the models have another peak at @xmath100 , grsb ) . however , when the quadrupole moment is included in the analysis ( as in fig . 21b ) , the open - bubble inflation model peaks are at @xmath12 ( at least in the range @xmath14 , grsb ) , while the flat - space scale - invariant spectrum open model peak is at @xmath11 . at the 95% c.l . no value of @xmath1 over the range considered , 0.11 , is excluded . ( the yb and bw claims of a lower limit on @xmath1 from the dmr data alone are , at the very least , premature . ) the @xmath58 ( e.g. , eqs . [ 1 ] and [ 2 ] ) were determined from a numerical integration of the linear perturbation theory equations of motion . as before , the computations were performed with two independent numerical codes . for some of the model - parameter values considered here the results of the two computations were compared and found to be in excellent agreement . illustrative examples of the comparisons are shown in fig . again , we emphasize that the excellent agreement is mostly a reflection of the currently available numerical accuracy , and the most likely additional , unaccounted for , source of uncertainty is that due to the uncertainty in the modelling of various physical effects . table 8 list the @xmath58 normalization amplitudes @xmath42 ( e.g. , eqs . [ 1 ] and [ 2 ] ) when @xmath101k . examples of the power spectra normalized to @xmath21 derived from the mean of the dmr four - year data analysis extreme upper and lower 2-@xmath80 limits discussed above are shown in figs . one will notice , from fig . 23(e ) , the good agreement between the open - bubble inflation spectra . when normalized to the two extreme 2-@xmath80 @xmath21 limits ( e.g. , cols . [ 5 ] and [ 8 ] of table 10 ) , the @xmath58 normalization factor ( eq . [ 1 ] and table 8) for the open - bubble inflation model ( rp94 ; bgt ; yst ) , may be summarized by , for the lower 2-@xmath80 limit , @xmath102 , \eqno(8)\ ] ] and for the upper 2-@xmath80 limit , @xmath103 . \eqno(9)\ ] ] these fits are good to @xmath104 for @xmath14 . note however that they are derived using the @xmath21 values determined for given @xmath62 and @xmath63 and hence do not account for the additional uncertainty introduced by allowed variations in these parameters ( which could affect the power spectrum normalization amplitude by as much as @xmath105 ) . from fig . 23(e ) , and given the uncertainties , we see that the fitting formulae of eqs . ( 8) and ( 9 ) provide an adequate summary for all the open - bubble inflation model spectra . the extreme @xmath106-@xmath80 @xmath58 normalization factor ( eq . [ 2 ] and table 8) for the flat - space scale - invariant spectrum open model ( w83 ) may be summarized by , for the lower 2-@xmath80 limit , @xmath107 , \eqno(10)\ ] ] and for the upper 2-@xmath80 limit , @xmath108 . \eqno(11)\ ] ] these fits are good to better than @xmath88 for @xmath109 ; again , they are derived from @xmath21 values determined at given @xmath62 and @xmath63 . given the uncertainties involved in the normalization procedure ( born of both statistical and other arguments ) it is not yet possible to quote a unique dmr normalization amplitude ( g96 ) . as a central " value for the @xmath58 normalization factor , we currently advocate the mean of eqs . ( 8) and ( 9 ) or eqs . ( 10 ) and ( 11 ) as required . we emphasize , however , that it is incorrect to draw conclusions about model viability based solely on this central " value . in conjunction with numerically determined transfer functions , the fits of eqs . ( 8)(11 ) allow for a determination of @xmath26 $ ] , accurate to a few percent . here the mean square linear mass fluctuation averaged over a sphere of coordinate radius @xmath110 is @xmath111 ^ 2\right\rangle & = & { 2 \over \pi^2 \left[{\rm sinh } ( \bar\chi)\ , { \rm cosh}(\bar\chi ) - \bar\chi\right]^2 } \nonumber \\ { \ } & { \ } & \times \int^\infty_0 { dk \over ( 1 + k^2)^2 } \left[{\rm cosh}(\bar\chi)\ , { \rm sin}(k\bar\chi ) - k\ , { \rm sinh}(\bar\chi ) \ , { \rm cos}(k\bar\chi ) \right]^2 p(k ) , \end{aligned}\ ] ] which , on small scales , reduces to the usual flat - space expression @xmath112 \int^\infty_0 dk\ , k^2 p(k ) \left[{\rm sin}(k\bar\chi ) - k\bar\chi \ , { \rm cos}(k\bar\chi ) \right]^2/(k\bar\chi)^6 $ ] . if instead use is made of the bardeen et al . ( 1986 , hereafter bbks ) analytic fit to the transfer function using the parameterization of eq . ( 13 ) below ( sugiyama 1995 ) and numerically determined values for @xmath42 , the resultant @xmath113 $ ] values are accurate to better than @xmath87 ( except for large baryon - fraction , @xmath114 , models where the error could be as large as @xmath115 ) . use of the analytic fits of eqs . ( 8)(11 ) for @xmath42 ( instead of the numerically determined values ) slightly increases the error , while use of the bbks transfer function fit parameterized by an earlier version of eq . ( 13 ) below , @xmath116 $ ] , results in @xmath26 $ ] values that could be off by as much as @xmath117 . nevertheless , as has been demonstrated by llrv , the approximate analytic fit to the transfer function greatly simplifies the computation and allows for rapid demarcation of the favoured part of cosmological - parameter space . numerical values for some cosmographic and large - scale structure statistics for the models considered here are recorded in tables 915 . we emphasize that when comparing to observational data we make use of numerically - determined large - scale structure predictions , not those derived using an approximate analytic fitting formula . tables 912 give the predictions for the open - bubble inflation model accounting only for the perturbations generated during the evolution inside the bubble ( rp94 ) , and for the flat - space scale - invariant spectrum open model ( w83 ) . each of these tables corresponds to a different pair of @xmath118 values . the first two columns in these tables record @xmath1 and @xmath2 , and the third column is the cosmological baryonic - matter fraction @xmath119 . the fourth column gives the value of the matter power spectrum scaling parameter ( sugiyama 1995 ) , @xmath120 which is used to parameterize approximate analytic fits to the power spectra derived from numerical integration of the perturbation equations . the quantities listed in columns ( 1)(4 ) of these tables are sensitive only to the global parameters of the cosmological model . columns ( 5 ) and ( 8) of tables 912 give the dmr data 2-@xmath80 range of @xmath21 that is used to normalize the perturbations in the models considered here . the numerical values in table 12 are for @xmath121 gyr , @xmath122 . we did not analyze the dmr data using @xmath56 s for these models , and in this case the perturbations are normalized to the @xmath21 values from the @xmath123 gyr , @xmath124 analyses . ( as discussed above , shifts in @xmath2 and @xmath63 do not greatly alter the inferred normalization amplitude . ) columns ( 6 ) and ( 9 ) of tables 912 give the 2-@xmath80 range of @xmath125 $ ] . these were determined using the @xmath58 derived from numerical integration of the perturbation equations . for about two dozen cases , these rms mass fluctuations determined using the two independent numerical integration codes were compared and found to be in excellent agreement . ( at fixed @xmath21 , they differ by @xmath126 depending on model - parameter values , with the typical difference being @xmath127 . we again emphasize that this is mostly a reflection of currently achievable numerical accuracy . ) . to usually better than @xmath128 accuracy , for @xmath129 , the 2-@xmath80 @xmath130 $ ] entries of columns ( 6 ) and ( 9 ) of tables 912 may be summarized by the fitting formulae listed in table 14 . these fitting formulae are more accurate than expressions for @xmath26 $ ] derived at the same cosmological - parameter values using an analytic approximation to the transfer function and the normalization of eqs . ( 8)(11 ) . for open models , as discussed below , it proves most convenient to characterize the peculiar velocity perturbation by the parameter @xmath131 where @xmath132 is the linear bias factor for @xmath133 galaxies ( e.g. , peacock & dodds 1994 ) . the 2-@xmath80 range of @xmath134 are listed in columns ( 7 ) and ( 10 ) of tables 912 . table 13 compares the @xmath113 $ ] values for spectra of types ( 1)(3 ) above . clearly , there is no significant observational difference between the predictions for the different spectra . in what follows , for the open - bubble inflation model we concentrate on the type ( 1 ) spectrum above . again , the ranges in tables 914 are those determined from the maximal 2-@xmath80 @xmath21 range . table 15 lists central dmr - normalized " values for @xmath130 $ ] , defined as the mean of the maximal @xmath1352-@xmath80 entries of tables 912 . ( the mean of the @xmath1352-@xmath80 fitting formulae of table 14 may be used to interpolate between the entries of table 15 . ) we again emphasize that it is incorrect to draw conclusions about model viability based solely on these central " values for the purpose of constraining model - parameter values by , e.g. , comparing numerical simulation results to observational data one must make use of computations at a few different values of the normalization selected to span the @xmath1352-@xmath80 ranges of tables 912 . the dmr likelihoods do not meaningfully exclude any part of the ( @xmath1 , @xmath2 , @xmath63 ) parameter space for the models considered here . in this section we combine current observational constraints on global cosmological parameters with the dmr - normalized model predictions to place constraints on the range of allowed model - parameter values . it is important to bear in mind that some measures of observational cosmology remain uncertain thus our analysis here must be viewed as tentative and subject to revision as the observational situation approaches equilibrium . to constrain our model - parameter values we have employed the most robust of the current observational constraints . tables 912 list some observational predictions for the models considered here , and the boldface entries are those that are inconsistent with current observational data at the 2-@xmath80 significance level . for each cosmographic or large - scale parameter , we have generally chosen to use constraints from a single set of observations or from a single analysis . we generally use the most recent analyses since we assume that they incorporate a better understanding of the uncertainties , especially those due to systematics . the specific constraints we use are summarized below , where we compare them to those derived from other analyses . the model predictions depend on the age of the universe @xmath62 . to reconcile the models with the high measured values of the hubble parameter @xmath2 , we have chosen to focus on @xmath60 10.5 , 12 , and 13.5 gyr , which are near the lower end of the ages now under discussion . for instance , jimenez et al . ( 1996 ) find that the oldest globular clusters have ages @xmath136 gyr ( also see salaris , deglinnocenti , & weiss 1996 ; renzini et al . 1996 ) , and that it is very unlikely that the oldest clusters are younger than 9.7 gyr . the value of @xmath1 is another input parameter for our computations . as summarized by peebles ( 1993 , @xmath137 ) , on scales @xmath138 mpc a variety of different observational measurements indicate that @xmath1 is low . for instance , virial analyses of x - ray cluster data indicates @xmath139 , with a 2-@xmath80 range : @xmath140 ( carlberg et al . 1996 we have added their 1-@xmath80 statistical and systematic uncertainties in quadrature and doubled to get the 2-@xmath80 uncertainty ) . in a cdm model in which structure forms at a relatively high redshift ( as is observed ) , these local estimates of @xmath1 do constrain the global value of @xmath1 ( since , in this case , it is inconceivable that the pressureless cdm is much more homogeneously distributed than is the observed baryonic mass ) . we hence adopt a 2-@xmath80 upper limit of @xmath141 to constrain the cdm models we consider here . ( this large upper limit allows for the possibility that the models might be moderately biased . ) the boldface entries in column ( 1 ) of tables 912 indicates those @xmath1 values inconsistent with this constraint . column ( 2 ) of tables 912 gives the value of the hubble parameter @xmath2 that corresponds to the chosen values of @xmath1 and @xmath62 . current observational data favours a larger @xmath2 ( e.g. , kennicutt , freedman , & mould 1995 ; baum et al . 1995 ; van den bergh 1995 ; sandage et al . 1996 ; ruiz - lapuente 1996 ; riess , press , & kirshner 1996 ; but also see schaefer 1996 ; branch et al . for the purpose of our analysis here we adopt the @xmath142 value @xmath143 ( 1-@xmath80 uncertainty , tanvir et al . 1995 ) ; doubling the uncertainty , the 2-@xmath80 range is @xmath144 . the bold face entries in column ( 2 ) of tables 912 indicates those model - parameter values which predict an @xmath2 inconsistent with this range . comparison of the standard nucleosynthesis theoretical predictions for the primordial light element abundances to what is determined by extrapolation of the observed abundances to primordial values leads to constraints on @xmath63 . it has usually been argued that @xmath145he and @xmath146li allow for the most straightforward extrapolation from the locally observed abundances to the primordial values ( e.g. , dar 1995 ; fields & olive 1996 ; fields et al . 1996 , hereafter fkot ) . the observed @xmath145he and @xmath146li abundances then suggest @xmath147 , and a conservative assessment of the uncertainties indicate a 2-@xmath80 range : @xmath148 ( fkot ; also see copi et al . 1995 ; sarkar 1996 ) . observational constraints on the primordial deuterium ( d ) abundance should , in principle , allow for a tightening of the allowed @xmath63 range . there are now a number of different estimates of the primordial d abundance , and since the field is still in its infancy it is , perhaps , not surprising that the different estimates are somewhat discrepant . songaila et al . ( 1994 ) , carswell et al . ( 1994 ) , and rugers & hogan ( 1996a , b ) use observations of three high - redshift absorption clouds to argue for a high primordial d abundance and so a low @xmath63 . tytler , fan , & burles ( 1996 ) and burles & tytler ( 1996 ) study two absorption clouds and argue for a low primordial d abundance and so a high @xmath63 . carswell et al . ( 1996 ) and wampler et al . ( 1996 ) examine other absorption clouds , but are not able to strongly constrain @xmath63 . while the error bars on @xmath63 determined from these d abundance observations are somewhat asymmetric , to use these results to qualitatively pick the @xmath63 values we wish to examine we assume that the errors are gaussian ( and where needed add all uncertainties in quadrature to get the 2-@xmath80 uncertainties ) . the large d abundance observations suggest @xmath149 with a 2-@xmath80 range : @xmath150 ( rugers & hogan 1996a ) . when these large d abundances are combined with the observed @xmath145he and @xmath146li abundances , they indicate @xmath151 , with a 2-@xmath80 range : @xmath152 ( fkot ) . the large d abundances are consistent with the standard interpretation of the @xmath145he and @xmath146li abundances , and with the standard model of particle physics ( with three massless neutrino species ) ; they do , however , seem to require a modification in galactic chemical evolution models to be consistent with local determinations of the d and @xmath153he abundances ( e.g. , fkot ; cardall & fuller 1996 ) . the low d abundance observations favour @xmath154 with a 2-@xmath80 range : @xmath155 ( burles & tytler 1996 ) . the low d abundance observations seem to be more easily accommodated in modifications of the standard model of particle physics , i.e. , they are difficult to reconcile with exactly three massless neutrino species ; alternatively they might indicate a gross , as yet unaccounted for , uncertainty in the observed @xmath145he abundance ( burles & tytler 1996 ; cardall & fuller 1996 ) . the low d abundance is approximately consistent with locally - observed d abundances , but probably requires some modification in the usual galactic chemical evolution model for @xmath146li ( burles & tytler 1996 ; cardall & fuller 1996 ) . to accommodate the range of @xmath63 now under discussion , we compute model predictions for @xmath124 ( table 9 ) , 0.007 ( table 12 ) , 0.0125 ( table 10 ) , and 0.0205 ( table 11 ) . we shall find that this uncertainty in @xmath63 precludes determination of robust constraints on model - parameter values . fortunately , recent improvements in observational capabilities should eventually lead to a tightening of the constraints on @xmath63 , and so allow for tighter constraints on the other cosmological parameters . column ( 3 ) of tables 912 give the cosmological baryonic - mass fraction for the models we consider here . the cluster baryonic - mass fraction is the sum of the cluster galactic - mass and gas - mass fractions . assuming that the white et al . ( 1993 ) 1-@xmath80 uncertainties on the cluster total , galactic , and gas masses are gaussian and adding them in quadrature , we find for the 2-@xmath80 range of the cluster baryonic - mass fraction : @xmath156 elbaz , arnaud , & bhringer ( 1995 ) , white & fabian ( 1995 ) , david , jones , & forman ( 1995 ) , markevitch et al . ( 1996 ) , and buote & canizares ( 1996 ) find similar ( or larger ) gas - mass fractions . note that elbaz et al . ( 1995 ) and white & fabian ( 1995 ) find that the gas - mass error bars are somewhat asymmetric ; this non - gaussianity is ignored here . assuming that the cluster baryonic - mass fraction is an unbiased estimate of the cosmological baryonic - mass fraction , we may use eq . ( 15 ) to constrain the cosmological parameters . the boldface entries in column ( 3 ) of tables 9 - 12 indicates those model - parameter values which predict a cosmological baryonic - mass fraction inconsistent with the range of eq . ( 15 ) . viana & liddle ( 1996 , hereafter vl ) have reanalyzed the combined galaxy @xmath58 data of peacock & dodds ( 1994 ) , ignoring some of the smaller scale data where nonlinear effects might be somewhat larger than previously suspected . using an analytic approximation to the @xmath58 , they estimate that the scaling parameter ( eq . [ 13 ] ) in the exponent of eq . ( 13 ) , so the numerical values of their constraint on @xmath157 should be reduced slightly . we ignore this small effect here . ] @xmath158 , with a 2-@xmath80 range , @xmath159 this estimate is consistent with earlier ones than eq . ( 16 ) this is one reason why llrv favour a higher @xmath1 for the open - bubble inflation model than do grsb . ] . it might be of interest to determine whether the wiggles in @xmath58 due to the pressure in the photon - baryon fluid , see figs . 23 , can significantly affect the determination of @xmath157 , especially in large @xmath119 models . ( these wiggles are not well described by the analytic approximation to @xmath58 . ) the boldface entries in column ( 4 ) of tables 912 indicates those model - parameter values which predict a scaling parameter value inconsistent with the range of eq . ( 16 ) . to determine the value of the linear bias parameter @xmath160 , @xmath161 where @xmath162 is the rms fractional perturbation in galaxy number , we adopt the apm value ( maddox , efstathiou , & sutherland 1996 ) of @xmath163 = 0.96 $ ] , with 2-@xmath80 range : @xmath164 where we have added the uncertainty due to the assumed cosmological model and due to the assumed evolution in quadrature with the statistical 1-@xmath80 uncertainty ( maddox et al . 1996 , eq . [ 43 ] ) , and doubled to get the 2-@xmath80 uncertainty . the range of eq . ( 18 ) is consistent with that determined from eqs . ( 7.33 ) and ( 7.73 ) of peebles ( 1993 ) . the local abundance of rich clusters , as a function of their x - ray temperature , provides a tight constraint on @xmath113 $ ] . eke , cole , & frenk ( 1996 , hereafter ecf ) ( and s. cole , private communication 1996 ) find for the open model at 2-@xmath80 : @xmath165 where we have assumed that the ecf uncertainties are gaussian , and that in general it depends weakly on the value of @xmath157 ( and so on the value of @xmath2 and @xmath10 ) see fig . 13 of ecf . in our preliminary analysis here we ignore this mild dependence on @xmath2 and @xmath10 . also note that the constraint of eq . ( 19 ) is approximately that required for consistency with the observed cluster correlation function . ] . the constraints of eq . ( 19 ) are consistent with , but more restrictive than , those derived by vl = 0.60 $ ] for fiducial cdm , which is at the @xmath1662-@xmath80 limit of eq . ( as discussed in ecf , this is because vl normalize to the cluster temperature function at 7 kev , where there is a rise in the temperature function . ) this is one reason why llrv favour a higher value of @xmath1 for the open - bubble inflation model than did grsb . ] . this is because ecf use observational data over a larger range in x - ray temperature to constrain @xmath167 , and also use n - body computations at @xmath168 0.3 and 1 to calibrate the press - schechter model ( which is used in their determination of the constraints ) . furthermore , ecf also make use of hydrodynamical simulations of a handful of individual clusters in the fiducial cdm model ( @xmath69 ) to calibrate the relation between the gas temperature and the cluster mass , and then use this calibrated relation for the computations at all values of @xmath1 . the initial conditions for all the simulations were set using the analytical approximation to @xmath58 , so again it might be of interest to see whether the wiggles in the numerically integrated @xmath58 could significantly affect the determination of the constraints of eq . kitayama & suto ( 1996 ) use x - ray cluster data , and a method that allows for the fact that clusters need not have formed at the redshift at which they are observed , to directly constrain the value of @xmath1 for cdm cosmogonies normalized by the dmr two - year data . their conclusions are in resonable accord with what would be found by using eq . ( 19 ) ( derived assuming that observed clusters are at their redshifts of formation ) . however , kitayama & suto ( 1996 ) note that evolution from the redshift of formation to the redshift of observation can affect the conclusions , so a more careful comparison of these two results is warranted . the boldface entries in columns ( 6 ) and ( 9 ) of tables 912 indicate those model - parameter values whose predictions are inconsistent with the constraints of eq . ( 19 ) ( 1-@xmath80 ) uncertainty of eq . ( 19 ) , approximate analyses based on using the analytic bbks approximation to the transfer function should make use of the more accurate parameterization of eq . ( 13 ) ( rather than that with @xmath169 in the exponent ) , as this gives @xmath26 $ ] to better than @xmath87 in the observationally viable part of parameter space ( provided use is made of the numerically determined values of @xmath42 ) . ] . from large - scale peculiar velocity observational data zaroubi et al . ( 1996 ) estimate @xmath26 = ( 0.85 \pm 0.2)\omega_0{}^{-0.6}$ ] ( 2-@xmath80 ) . it might be significant that the large - scale peculiar velocity observational data constraint is somewhat discordant with ( higher than ) the cluster temperature function constraint . since @xmath170 is less sensitive to smaller length scales ( compared to @xmath26 $ ] ) , observational constraints on @xmath170 are more reliably contrasted with the linear theory predictions . however , since @xmath170 is sensitive to larger length scales , the observational constraints on @xmath170 are significantly less restrictive than the @xmath171 ( 1-@xmath80 ) constraints of eq . ( 19 ) , and so we do not record the predicted values of @xmath170 here . observational constraints on the mass power spectrum determined from large - scale peculiar velocity observations provide another constraint on the mass fluctuations . kolatt & dekel ( 1995 ) find at the 1-@xmath80 level @xmath172 where the 1-@xmath80 uncertainty also accounts for sample variance ( t. kolatt , private communication 1996 ) . since the uncertainties associated with the constraint of eq . ( 19 ) are more restrictive than those associated with the constraint of eq . ( 20 ) , we do not tabulate predictions for this quantity here . however , comparison may be made to the predicted linear theory mass power spectra of figs . 23 , bearing in mind the @xmath173 ( 2-@xmath80 ) uncertainty of eq . ( 20 ) ( the uncertainty is approximately gaussian , t. kolatt , private communication 1996),-@xmath80 , significance level , eq . ( 20 ) provides a strong upper limit on @xmath174 , especially at larger @xmath1 because of the @xmath1 dependence . ] and the uncertainty in the dmr normalization ( not shown in figs . 23 ) . columns ( 7 ) and ( 10 ) of tables 912 give the dmr - normalized model predictions for @xmath134 ( eq . [ 14 ] ) . cole , fisher , & weinberg ( 1995 ) measure the anisotropy of the redshift space power spectrum of the @xmath133 1.2 jy survey and conclude @xmath175 with a 2-@xmath80 c.l . range : @xmath176 where we have doubled the error bars of eq . ( 5.1 ) of cole et al . ( 1995 ) to get the 2-@xmath80 range . cole et al . ( 1995 , table 1 ) compare the estimate of eq . ( 21 ) to other estimates of @xmath134 , and at 2-@xmath80 all estimates of @xmath134 are consistent . it should be noted that the model predictions of @xmath134 ( eq . [ 14 ] ) in tables 912 assume that for @xmath133 galaxies @xmath163 = 1/1.3 $ ] holds exactly , i.e. , they ignore the uncertainty in the rms fractional perturbation in @xmath133 galaxy number , which is presumably of the order of that in eq . ( 18 ) . as the constraints from the deduced @xmath134 values , eq . ( 21 ) , are not yet as restrictive as those from other large - scale structure measures , we do not pursue this issue in our analysis here . the boldface entries in columns ( 7 ) and ( 10 ) of tables 912 indicate those model - parameter values whose predictions are inconsistent with the constraints of eq . ( 21 ) . the boldface entries in tables 912 summarize the current constraints imposed by the observational data discussed in the previous section on the model - parameter values for the open - bubble inflation model ( spectra of type [ 1 ] above ) , and for the flat - space scale - invariant spectrum open model ( type [ 4 ] above ) . the current observational constraints on the models are not dissimilar , but this is mostly a reflection of the uncertainty on the constraints themselves since the model predictions are fairly different . in the following discussion of the preferred part of model - parameter space we focus on the open - bubble inflation model ( rp94 ) . note from table 13 that the large - scale structure predictions of the open - bubble inflation model do not depend on perturbations generated in the first epoch of inflation ( bgt ; yst ) , and also do not depend significantly on the contribution from the non - square - integrable basis function ( yst ) . table 9 corresponds to the part of parameter space with maximized " small - scale power in matter fluctuations . this is accomplished by picking a low @xmath123 gyr ( and so large @xmath2 ) , and by picking a low @xmath124 ( this is the lower 2-@xmath80 limit from standard nucleosynthesis and the observed @xmath145he , @xmath146li , and high d abundances , fkot ) . the tightest constraints on the model - parameter values come from the matter power spectrum observational data constraints on the shape parameter @xmath157 ( table 9 , col . [ 4 ] ) , and from the cluster x - ray temperature function observational data constraints on @xmath26 $ ] ( col . note that for @xmath177 the predicted upper 2-@xmath80 value of @xmath113 = 0.69 $ ] , while ecf conclude that at 2-@xmath80 the observational data requires that this be at least 0.74 , so an @xmath177 case fails this test . the constraints on @xmath134 ( col . [ 7 ] ) are not as restrictive as those on @xmath113 $ ] . for these values of @xmath62 and @xmath178 the cosmological baryonic - mass fraction at @xmath177 is predicted to be 0.033 ( col . [ 3 ] ) , while at 2-@xmath80 white et al . ( 1993 ) require that this be at least 0.039 ( at @xmath179 ) , so again this @xmath177 model just fails this test . given the observational uncertainties , it might be possible to make minor adjustments to model - parameter values so that an @xmath180 model with @xmath181 gyr and @xmath182 is just consistent with the observational data . however , it is clear that current observational data do not favour an open model with @xmath183 the observed cluster @xmath184 $ ] favours a larger @xmath1 while the observed cluster baryonic - mass fraction favours a smaller @xmath1 , and so are in conflict . table 10 gives the predictions for the @xmath121 gyr , @xmath185 models . this value of @xmath63 is consistent with the 2-@xmath80 range determined from standard nucleosynthesis and the observed @xmath145he and @xmath146li abundances : @xmath148 ( fkot , also see copi et al . 1995 ; sarkar 1996 ) . it is , however , somewhat difficult to reconcile @xmath8 with the 2-@xmath80 range derived from the observed @xmath145he , @xmath146li , and current high d abundances @xmath152 ( fkot ) , or with that from the current observed low d abundances @xmath186 ( burles & tytler 1996 ) . in any case , the observed d abundances are still under discussion , and must be viewed as preliminary . in this case , open - bubble inflation models with @xmath187 are consistent with the observational constraints . the current central observational data values for @xmath157 and @xmath134 favour @xmath74 , while that for the cluster baryonic - mass fraction prefers @xmath188 , and that for @xmath130 $ ] favours @xmath189 , so in this case the agreement between predictions and observational data is fairly impressive ( although the tanvir et al . 1995 central @xmath2 value favours @xmath190 ) . note that in this case models with @xmath191 are quite inconsistent with the data . table 11 gives the predictions for @xmath192 gyr , @xmath193 models . this baryonic - mass density value is consistent with that determined from the current observed low d abundances , but is difficult to reconcile with the current standard nucleosynthesis interpretation of the observed @xmath145he and @xmath146li abundances ( cardall & fuller 1996 ) . the larger value of @xmath63 ( and smaller value of @xmath2 ) has now lowered small - scale power in mass fluctuations somewhat significantly , opening up the allowed @xmath1 range to larger values . models with @xmath194 are consistent with the observational data , although the higher @xmath1 part of the range is starting to conflict with what is determined from the small - scale dynamical estimates , and the models do require a somewhat low @xmath2 ( but not yet inconsistently so at the 2-@xmath80 significance level while the tanvir et al . 1995 central @xmath2 value requires @xmath100 , at 2-@xmath80 the @xmath2 constraint only requires @xmath195 ) . the central observational values for @xmath157 , the cluster baryonic - mass fraction , @xmath26 $ ] , and @xmath134 favour @xmath97 , so the agreement with observational data is fairly impressive , and could even be improved by reducing @xmath62 a little to raise @xmath2 . table 12 gives the predictions for another part of model - parameter space . here we show @xmath122 models ( at @xmath121 gyr ) , consistent with the central value of @xmath63 determined from standard nucleosynthesis using the observed @xmath145he , @xmath146li , and high d abundances ( fkot ) . the larger value of @xmath63 ( compared to table 9 ) eases the cluster baryonic - mass fraction constraint , which now requires only @xmath196 . the increase in @xmath63 also decreases the mass fluctuation amplitude , making it more difficult to argue for @xmath177 ; however , models with @xmath197 seem to be consistent with the observational constraints when @xmath4 and @xmath198 gyr . it is interesting that in this case the central observational data values we consider for @xmath157 , for @xmath26 $ ] , and for @xmath134 prefer @xmath9 ; however , that for the cluster baryonic - mass fraction ( as well as that for @xmath2 ) favours @xmath190 ( although at 2-@xmath80 the cluster baryonic - mass fraction constraint only requires @xmath196 ) . hence , while @xmath199 open - bubble inflation models with @xmath200 and @xmath198 gyr are quite consistent with the observational constraints , in this case the agreement between predictions and observations is not spectacular . note that in this case models with @xmath201 are quite inconsistent with the observational data . in summary , open - bubble inflation models based on the cdm picture ( rp94 ; bgt ; yst ) are reasonably consistent with current observational data provided @xmath202 . the flat - space scale - invariant spectrum open model ( w83 ) is also reasonably compatible with current observational constraints for a similar range of @xmath1 . the uncertainty in current estimates of @xmath63 is one of the major reasons why such a large range in @xmath1 is consistent with current observational constraints . our previous analysis of the dmr two - year data led us to conclude that only those open - bubble inflation models near the lower end of the above range ( @xmath203 ) were consistent with the majority of observations ( grsb ) . the increase in the allowed range to higher @xmath1 values @xmath204 can be ascribed to a number of small effects . specifically , these are : ( 1 ) the slight downward shift in the central value of the dmr four - year normalization relative to the two - year one ( g96 ) ; ( 2 ) use of the full 2-@xmath80 range of normalizations allowed by the dmr data analysis ( instead of the 1-@xmath80 range allowed by the galactic - frame quadrupole - excluded dmr two - year data set used previously ) ; ( 3 ) use of the 2-@xmath80 range of the small - scale dynamical estimates of @xmath1 instead of the 1-@xmath80 range used in our earlier analysis ; ( 4 ) we consider a range of @xmath63 values here ( in grsb we focussed on @xmath8 ) ; and ( 5 ) we consider a range of @xmath62 values here ( in grsb we concentrated on @xmath121 gyr ) . we emphasize , however , that the part of parameter space with @xmath205 is only favoured if @xmath63 is large ( @xmath206 ) , @xmath2 is low @xmath207 ) , and the small - scale dynamical estimates of @xmath1 turn out to be biased somewhat low . the observational results we have used to constrain model - parameter values in the previous sections are the most robust currently available . in addition , there are several other observational results which we do not consider to be as robust , and any conclusions drawn from these should be treated with due caution . in this section we summarize several of the more tentative constraints from more recent observations . in our analysis of the dmr two - year data normalized models , we compared model predictions for the rms value of the smoothed peculiar velocity field to results from the analysis of observational data ( bertschinger et al . we do not do so again here since , given the uncertainties , the conclusions drawn in grsb are not significantly modified . in particular , comparison of the appropriate quantities implies that we can treat the old 1-@xmath80 upper limits essentially as 2-@xmath80 upper limits for the four - year analysis . in grsb we used @xmath134 determined by nusser & davis ( 1994 ) , @xmath208 ( 2-@xmath80 ) , to constrain the allowed range of models to @xmath209 . here we use the cole et al . ( 1995 ) estimate , @xmath210 ( 2-@xmath80 ) , which , for the models of table 10 , requires @xmath211 . this value is just slightly below the lower limit ( @xmath212 ) derived from the bertschinger et al . ( 1990 ) results in grsb . we hence conclude that the large - scale flow results of bertschinger et al . ( 1990 ) indicates a lower 2-@xmath80 limit on @xmath1 that is about @xmath213 higher than that suggested by the redshift - space distortion analysis of cole et al . ( 1995).$ ] . ] we however strongly emphasize that the central value of the large - scale flow results of bertschinger et al . ( 1990 ) does favour a significantly larger value of @xmath1 than the rest of the data we have considered here . furthermore , as discussed in detail in grsb , there is some uncertainty in how to properly interpret large - scale velocity data in the open models , particularly given the large sample variance associated with the measurement of a single bulk velocity ( bond 1996 , also see llrv ) . a more careful analysis , as well as more observational data , is undoubtedly needed before it will be possible to robustly conclude that the large - scale velocity data does indeed force one to consider significantly larger values of @xmath1 than is favoured by the rest of the observational constraints ( and hence rules out the models considered here ) . it might be significant that on comparing the mass power spectrum deduced from a refined set of peculiar velocity observations to the galaxy power spectrum determined from the apm survey , kolatt & dekel ( 1995 ) estimate that for the optically - selected apm galaxies @xmath214 with a 2-@xmath80 range , @xmath215 ( note that it has been argued that systematic uncertainties preclude a believable determination of @xmath134 from a comparison of the observed large - scale peculiar velocity field to the @xmath133 1.2 jy galaxy distribution , davis , nusser , & willick 1996 . ) this range is consistent with other estimates now under discussion . the stromlo - apm comparison of loveday et al . ( 1996 ) indicates @xmath216 , with a 2-@xmath80 upper limit of 0.75 , while baugh ( 1996 ) concludes that @xmath217 ( 2-@xmath80 ) , and ratcliffe et al . ( 1996 ) argue for @xmath218 . using the apm range for @xmath163 $ ] , ( 18 ) , the kolatt & dekel ( 1995 ) estimate of @xmath219 , eq . ( 22 ) , may be converted to an estimate of @xmath167 , and at 2-@xmath80 , @xmath220 it is interesting that at @xmath69 the lower part of this range is consistent with that determined from the cluster x - ray temperature function data , eq . ( 19 ) , although at lower @xmath1 eq . ( 23 ) indicates a larger value then does eq . ( 19 ) because of the steeper rise to low @xmath1 . zaroubi et al . ( 1996 ) have constrained model - parameter values by comparing large - scale flow observations to that predicted in the dmr two - year data normalized open - bubble inflation model . they conclude that the open - bubble inflation model provides a good description of the large - scale flow observations if , at 2-@xmath80 , @xmath221 from table 12 we see that an open - bubble inflation model with @xmath222 and @xmath223 provides a good fit to all the observational data considered in @xmath224 . for @xmath223 zaroubi et al . ( 1996 ) conclude that at 2-@xmath80 @xmath225 ( eq . [ 24 ] ) , just above our value of @xmath222 . since the zaroubi et al . ( 1996 ) analysis does not account for the uncertainty in the dmr normalization ( t. kolatt , private communication 1996 ) , it is still unclear if the constraints from the large - scale flow observations are in conflict with those determined from the other data considered here ( and so rule out the open - bubble inflation model ) . it might also be significant that on somewhat smaller length scales there is support for a smaller value of @xmath1 from large - scale velocity field data ( shaya , peebles , & tully 1995 ) . the cluster peculiar velocity function provides an alternate mechanism for probing the peculiar velocity field ( e.g. , croft & efstathiou 1994 ; moscardini et al . 1995 ; bahcall & oh 1996 ) . bahcall & oh ( 1996 ) conclude that current observational data is well - described by an @xmath177 flat-@xmath18 model with @xmath226 and @xmath227 = 0.67 $ ] . this normalization is somewhat smaller than that indicated by the dmr data ( e.g. , ratra & sugiyama 1995 ) . while bahcall & oh ( 1996 ) did not compare the cluster peculiar velocity function data to the predictions of the open - bubble inflation model , approximate estimates indicate that this data is consistent with the open - bubble inflation model predictions for the range of @xmath1 favoured by the other data we consider in @xmath228 see the @xmath26 $ ] values for the allowed models in tables 912 . bahcall & oh ( 1996 ) also note that it is difficult , if not impossible , to reconcile the cluster peculiar velocity observations with what is predicted in high density models like fiducial cdm and mdm . at fixed @xmath113 $ ] , low - density cosmogonies form structure earlier than high density ones . thus observations of structure at high redshift may be used to constrain the matter density . as benchmarks , we note that scaling from the results of the numerical simulations of cen & ostriker ( 1993 ) , in a open model with @xmath229 = 0.8 $ ] galaxy formation peaks at a redshift @xmath230 when @xmath231 and at @xmath232 when @xmath72 . thus the open - bubble inflation model is not in conflict with observational indications that the giant elliptical luminosity function at @xmath233 is similar to that at the present ( e.g. , lilly et al . 1995 ; glazebrook et al . 1995 ; i m et al . 1996 ) , nor is it in conflict with observational evidence for massive galactic disks at @xmath233 ( vogt et al . these models can also accommodate observational evidence of massive star - forming galaxies at @xmath234 ( cowie , hu , & songaila 1995 ) , as well as the significant peak at @xmath235 in the number of galaxies as a function of ( photometric ) redshift found in the hubble deep field ( gwyn & hartwick 1996 ) , and it is not inconceivable that objects like the @xmath236 protogalaxy " candidate ( yee et al . 1996 ; ellingson et al . 1996 ) can be produced in these models . it is , however , at present unclear whether the open - bubble inflation model can accommodate a substantial population of massive star - forming galaxies at @xmath237 ( steidel et al . 1996 ; giavalisco , steidel , & macchetto 1996 ) , and if there are many more examples of massive damped lyman@xmath238 systems like the one at @xmath239 ( e.g. , lu et al . 1996 ; wampler et al . 1996 ; fontana et al . 1996 ) , then , depending on the masses , these might be a serious problem for the open - bubble inflation model . on the other hand , the recent discovery of galaxy groups at @xmath240 ( e.g. , francis et al . 1996 ; pascarelle et al . 1996 ) probably do not pose a serious threat for the open - bubble inflation model , while massive clusters at @xmath241 ( e.g. , luppino & gioia 1995 ; pell et al . 1996 ) can easily be accommodated in the model . it should be noted that in adiabatic @xmath69 models normalized to fit the present small - scale observations , e.g. , fiducial cdm ( with a normalization inconsistent with that from the dmr ) , or mdm , or tilted cdm ( without a cosmological constant ) , it is quite difficult , if not impossible , to accommodate the above observational indications of early structure formation ( e.g. , ma & bertschinger 1994 ; ostriker & cen 1996 ) . with the recent improvements in observational capabilities , neoclassical cosmological tests hold great promise for constraining the world model . it might be significant that current constraints from these tests are consistent with that region of the open - bubble inflation model parameter space that is favoured by the large - scale structure constraints . these tests include the @xmath142 elliptical galaxy number counts test ( driver et al . 1996 ) , an early application of the apparent magnitude - redshift test using type ia supernovae ( perlmutter et al . 1996 ) , as well as analyses of the rate of gravitational lensing of quasars by foreground galaxies ( e.g. , torres & waga 1996 ; kochanek 1996 ) . it should be noted that these tests are also consistent with @xmath69 models , and plausibly with a time - variable cosmological constant " dominated spatially - flat model ( e.g. , ratra & quillen 1992 ; torres & waga 1996 ) , but they do put pressure on the flat-@xmath18 cdm model . smaller - scale cmb spatial anisotropy measurements will eventually significantly constrain the allowed range of model - parameter values . fig . 24 compares the 1-@xmath80 range of cmb spatial anisotropy predictions for a few representative open - bubble inflation ( as well as flat - space scale - invariant spectrum open ) models to available cmb spatial anisotropy observational data . from a preliminary comparison of the predictions of dmr two - year data normalized open - bubble inflation models to available cmb anisotropy observational data , ratra et al . ( 1995 ) concluded that the range of parameter space for the open - bubble inflation model that was favoured by the other observational data was also consistent with the small - scale cmb anisotropy data . this result was quantified by grs , who also considered open - bubble inflation models normalized to the @xmath1351-@xmath80 values of the dmr two - year data ( and hence considered open - bubble inflation models normalized at close to the dmr four - year data value , see figs . 5 and 6 of grs ) . grs discovered that ( given the uncertainties associated with the smaller - scale measurements ) the 1-@xmath80 uncertainty in the value of the dmr normalization precludes determination of robust constraints on model - parameter values , although the range of model - parameter space for the open - bubble inflation model favoured by the analysis here was found to be consistent with the smaller - scale cmb anisotropy observations , and @xmath93 open - bubble inflation models were not favoured by the smaller - scale cmb anisotropy observational data ( grs , figs . 5 and 6 ) . is favoured , but even at 1-@xmath80 @xmath242 is allowed this broad range is consistent with the conclusion of grs that it is not yet possible to meaningfully constrain cosmological - parameter values from the cmb anisotropy data alone . note also that hancock et al . ( 1996b ) do not consider the effects of the systematic shifts between the various dmr data sets , and also exclude a number of data points , e.g. , the four msam points and the max3 mup point ( which is consistent with the recent max5 mup result , lim et al . 1996 ) , which do not disfavour a lower value of @xmath1 for the open - bubble inflation model ( ratra et al . 1995 ; grs ) . ] a detailed analysis of the ucsb south pole 1994 cmb anisotropy data ( gundersen et al . 1995 ) by ganga et al . ( 1996a ) reaches a similar conclusion : at 1-@xmath80 ( assuming a gaussian marginal probability distribution ) the data favours open - bubble inflation models with @xmath243 , while at 2-@xmath80 the ucsb south pole 1994 data is consistent with the predictions of the open - bubble , flat-@xmath18 , and fiducial cdm inflation models . we have compared the dmr 53 and 90 ghz sky maps to a variety of open model cmb anisotropy angular spectra in order to infer the normalization of these open cosmogonical models . our analysis explicitly quantifies the small shifts in the inferred normalization amplitudes due to : ( 1 ) the small differences between the galactic- and ecliptic - coordinate sky maps ; ( 2 ) the inclusion or exclusion of the @xmath85 moment in the analysis ; and , ( 3 ) the faint high - latitude galactic emission treatment . we have defined a maximal 2-@xmath80 uncertainty range based on the extremal solutions of the normalization fits , and a maximal 1-@xmath80 uncertainty range may be defined in a similar manner . for this maximal 1-@xmath80 @xmath21 range the fractional 1-@xmath80 uncertainty , at fixed @xmath10 and @xmath2 ( but depending on the assumed cmb anisotropy angular spectrum and model - parameter values ) , ranges between @xmath244 and @xmath245 ( statistical and systematic ) uncertainty of bw ( footnote 4 , also see bunn , liddle , & white 1996 ) , @xmath246 , is smaller than the dmr four - year data 1-@xmath80 uncertainty estimated in , e.g. , g96 , wright et al . ( 1996 ) , and here . this is because we explicitly estimate the effect of all known systematic uncertainties for each assumed cmb anisotropy angular spectrum , and account for them , in the most conservative manner possible , as small shifts . ( in particular : we do not just account for the small systematic difference between the galactic- and ecliptic - frame maps ; we do not assume that any of the small systematic differences lead to model - independent systematic shifts in the inferred @xmath21 values ; and we do not add the systematic shifts in quadrature with the statistical uncertainty . ) since our accounting of the uncertainties is the most conservative possible , our conclusions about model - viability are the most robust possible . ] . ( compare this to the @xmath247 , 1-@xmath80 , uncertainty of eq . [ 19 ] . ) since part of this uncertainty is due to the small systematic shifts , the maximal 2-@xmath80 fractional uncertainty is smaller than twice the maximal 1-@xmath80 fractional uncertainty . for the largest possible 2-@xmath80 @xmath21 range defined above , the fractional uncertainty varies between @xmath248 and @xmath249 . note that this accounts for intrinsic noise , cosmic variance , and effects ( 1)(3 ) above . other systematic effects , e.g. , the calibration uncertainty ( kogut et al . 1996b ) , or the beamwidth uncertainty ( wright et al . 1994 ) , are much smaller than the effects we have accounted for here . it has also been shown that there is negligible non - cmb contribution to the dmr data sets from known extragalactic astrophysical foregrounds ( banday et al . 1996b ) . by analyzing the dmr maps using cmb anisotropy spectra at fixed @xmath1 but different @xmath2 and @xmath10 , we have also explicitly quantified the small shifts in the inferred normalization amplitude due to shifts in @xmath2 and @xmath10 . although these shifts do depend on the value of @xmath1 and the assumed model power spectrum , given the other uncertainties , it is reasonable to ignore these small shifts when normalizing the models considered in this work . we have analyzed the open - bubble inflation model , accounting only for the fluctuations generated during the evolution inside the bubble ( rp94 ) , including the effects of the fluctuations generated in the first epoch of spatially - flat inflation ( bgt ; yst ) , and finally accounting for the contribution from a non - square - integrable basis function ( yst ) . for observationally viable open - bubble models , the observable predictions do not depend significantly on the latter two sources of anisotropy . the observable predictions of the open - bubble inflation scenario seem to be robust it seems that only those fluctuations generated during the evolution inside the bubble need to be accounted for . as discussed in the introduction , a variety of more specific realizations of the open - bubble inflation scenario have recently come under scrutiny . these are based on specific assumptions about the vacuum state prior to open - bubble nucleation . in these specific realizations of the open - bubble inflation scenario there are a number of additional mechanisms for stress - energy perturbation generation ( in addition to those in the models considered here ) , including those that come from fluctuations in the bubble wall , as well as effects associated with the nucleation of a nonzero size bubble . while current analyses suggest that such effects also do not add a significant amount to the fluctuations generated during the evolution inside the bubble , it is important to continue to pursue such investigations both to more carefully examine the robustness of the open - bubble inflation scenario predictions , as well as to try to find a reasonable particle physics based realization of the open - bubble inflation scenario . as has been previously noted for other cmb anisotropy angular spectra ( g96 ) , the various different dmr data sets lead to slightly different @xmath21 normalization amplitudes , but well within the statistical uncertainty . this total range is slightly reduced if one considers results from analyses either ignoring or including the quadrupole moment . the dmr data alone can not be used to constrain @xmath1 over range @xmath14 in a statistically meaningful fashion for the open models considered here . it is , however , reasonable to conclude that when the quadrupole moment is excluded from the analysis , the @xmath9 model cmb anisotropy spectral shape is most consistent with the dmr data , while the quadrupole - included analysis favours @xmath12 ( for the open - bubble inflation model in the range @xmath250 ) . current cosmographic observations , in conjunction with current large - scale structure observations compared to the predictions of the dmr - normalized open - bubble inflation model derived here , favour @xmath202 . the large allowed range is partially a consequence of the current uncertainty in @xmath10 . this range is consistent with the value weakly favoured ( @xmath9 ) by a quadrupole - excluded analysis of the dmr data alone . it might also be significant that mild bias is indicated both by the need to reconcile these larger values of @xmath1 with what is determined from small - scale dynamical estimates , as well as to reconcile the smaller dmr - normalized @xmath251 $ ] values ( for this favoured range of @xmath1 ) with the larger observed galaxy number fluctuations ( e.g. , eq . [ 18 ] ) . in common with the low - density flat-@xmath18 cdm model , we have established that in the low - density open - bubble cdm model one may adjust the value of @xmath1 to accommodate a large fraction of present observational constraints . for a broad class of these models , with adiabatic gaussian initial energy - density perturbations , this focuses attention on values of @xmath1 that are larger than the range of values for @xmath10 inferred from the observed light - element abundances in conjunction with standard nucleosynthesis theory . whether this additional cdm is nonbaryonic , or is simply baryonic material that does not take part in standard nucleosynthesis , remains a major outstanding puzzle for these models . we acknowledge the efforts of those contributing to the @xmath0-dmr . @xmath0 is supported by the office of space sciences of nasa headquarters . we also acknowledge the advice and assistance of c. baugh , s. cole , j. garriga , t. kolatt , c. park , l. piccirillo , g. rocha , g. tucker , d. weinberg , and k. yamamoto . rs is supported in part by a pparc grant and kbn grant 2p30401607 . 1.fractional differences , @xmath253 , between the cmb spatial anisotropy multipole coefficients @xmath56 computed using the two boltzmann transfer codes ( and normalized to agree at @xmath254 ) . heavy type is for the open - bubble inflation model spectrum accounting only for perturbations that are generated during the evolution inside the bubble ( type [ 1 ] spectra above ) , and light type is for the open - bubble inflation model spectrum now also accounting for perturbations generated in the first epoch of inflation ( type [ 2 ] spectra ) . solid lines are for @xmath255 and dashed lines are for @xmath256 . these are for @xmath64 and @xmath65 . note that @xmath257 . 2.(a ) cmb anisotropy multipole coefficients for the open - bubble inflation model , accounting only for fluctuations generated during the evolution inside the bubble ( rp94 , solid lines ) , and also accounting for fluctuations generated in the first epoch of inflation ( bgt ; yst , dotted lines these overlap the solid lines , except at the lowest @xmath1 and smallest @xmath66 ) , for @xmath258 0.1 , 0.2 , 0.25 , 0.3 , 0.35 , 0.4 , 0.45 , 0.5 , 0.6 , 0.8 , and 1.0 , in ascending order . these are for @xmath121 gyr and @xmath8 . the coefficients are normalized relative to the @xmath259 amplitude , and different values of @xmath1 are offset from each other to aid visualization . in ( b ) are the set of cmb anisotropy spectra for the open - bubble inflation model , accounting only for fluctuations generated during the evolution inside the bubble ( rp94 ) , with @xmath255 and @xmath256 for the three different pairs of values ( @xmath62 , @xmath63 ) : ( @xmath260 gyr , @xmath261 ) , ( @xmath262 gyr , @xmath263 ) , and ( @xmath264 gyr , @xmath265 ) . spectra in the two sets are normalized to have the same @xmath259 , and @xmath63 increases in ascending order on the right axis . 3.cmb spatial anisotropy multipole coefficients for the flat - space scale - invariant spectrum open model ( w83 ) . conventions and parameter values are as in the caption of fig . 2 ( although only one set of spectra are shown in fig . 3a ) . fig . 4.cmb spatial anisotropy multipole coefficients for the open - bubble inflation spectrum , also accounting for both fluctuations generated in the first epoch of inflation and that corresponding to a non - square - integrable basis function ( yst , solid lines ) , and ignoring both these fluctuations ( rp94 , dotted lines ) . they are , in ascending order , for @xmath258 0.1 to 0.9 in steps of 0.1 , with @xmath64 and @xmath65 , normalized relative to the @xmath259 amplitude , and different values of @xmath1 are offset from each other to aid visualization . 5.cmb spatial anisotropy multipole coefficients , as a function of @xmath66 , for the various spectra considered in this paper , at @xmath255 and @xmath266 ( vertically offset ) . light solid and heavy solid lines show the open - bubble inflation cases accounting for ( type [ 2 ] spectra above ) and ignoring ( type [ 1 ] spectra , at @xmath256 these completely overlap the type [ 2 ] spectra ) fluctuations generated in the first epoch of inflation . dashed lines show the open - bubble inflation models , now also accounting for the contribution from the non - square - integrable basis function ( type [ 3 ] spectra ) . dotted lines show the flat - space scale - invariant spectrum open model spectra ( type [ 4 ] spectra ) . all spectra are for @xmath64 and @xmath65 . 6.likelihood functions @xmath79 ( arbitrarily normalized to unity at the highest peak at @xmath74 ) derived from a simultaneous analysis of the dmr 53 and 90 ghz ecliptic - frame data , ignoring the correction for faint high - latitude foreground galactic emission , and excluding the quadrupole moment from the analysis . these are for the @xmath64 , @xmath65 models . panel ( a ) is for the flat - space scale - invariant spectrum open model ( w83 ) , ( b ) is for the open - bubble inflation model accounting only for perturbations generated during the evolution inside the bubble ( rp94 ) , and ( c ) is for the open - bubble inflation model now also accounting for both the fluctuations generated in the first epoch of inflation and those corresponding to a non - square - integrable basis function ( yst ) . 7.likelihood functions @xmath79 ( arbitrarily normalized to unity at the highest peak near either @xmath93 or @xmath267 ) , derived from a simultaneous analysis of the dmr 53 and 90 ghz galactic - frame data , accounting for the faint high - latitude foreground galactic emission correction , and including the quadrupole moment in the analysis . conventions and parameter values are as for fig . 6 . fig . 8.ridge lines of the maximum likelihood @xmath21 value as a function of @xmath1 , for the open - bubble inflation model accounting only for fluctuations generated during the evolution inside the bubble ( type [ 1 ] spectra ) , for the eight different dmr data sets considered here , and for @xmath121 gyr , @xmath8 . heavy lines correspond to the case when the quadrupole moment is excluded from the analysis , while light lines account for the quadrupole moment . these are for the ecliptic - frame sky maps , accounting for ( dashed lines ) and ignoring ( solid lines ) the faint high - latitude foreground galactic emission correction , and for the galactic - frame maps , accounting for ( dot - dashed lines ) and ignoring ( dotted lines ) this galactic emission correction . the general features of this figure are consistent with that derived from the dmr two - year data ( grsb , fig . 2 ) . fig . 9.ridge lines of the maximum likelihood @xmath21 value as a function of @xmath1 , for the flat - space scale - invariant spectrum open model ( type [ 4 ] spectra ) , for the eight different dmr data sets , and for @xmath121 gyr , @xmath8 . heavy lines correspond to the ecliptic - frame analyses , while light lines are from the galactic - frame analyses . these are for the cases ignoring the faint high - latitude foreground galactic - emission correction , and either including ( dotted lines ) or excluding ( solid lines ) the quadrupole moment ; and accounting for this galactic emission correction , and either including ( dot - dashed lines ) or excluding ( dashed lines ) the quadrupole moment . the general features of this figure are roughly consistent with that derived from the dmr two - year data ( cayn et al . 1996 , fig . 3 ) . fig . 10.ridge lines of the maximum likelihood @xmath21 value as a function of @xmath1 , for the open - bubble inflation model now also accounting for both the fluctuations generated in the first epoch of inflation ( bgt ; yst ) and those from a non - square - integrable basis function ( yst ) , for the eight different dmr data sets considered here , and for @xmath64 , @xmath65 . heavy lines correspond to the cases where the faint high - latitude foreground galactic emission correction is ignored , while light lines account for this galactic emission correction . these are from the ecliptic frame analyses , accounting for ( dotted lines ) or ignoring ( solid lines ) the quadrupole moment ; and from the galactic - frame analyses , accounting for ( dot - dashed lines ) or ignoring ( dashed lines ) the quadrupole moment . the general features of this figure are consistent with that derived from the dmr two - year data ( yb , fig . 2 ) . fig . 11.ridge lines of the maximum likelihood @xmath21 value as a function of @xmath1 , for the two extreme dmr data sets , and two different cmb anisotropy angular spectra , showing the effects of varying @xmath62 and @xmath63 . heavy lines are for @xmath192 gyr and @xmath268 , while light lines are for @xmath123 gyr and @xmath124 . two of the four pairs of lines are for the open - bubble inflation model accounting only for fluctuations generated during the evolution inside the bubble ( type [ 1 ] spectra ) , either from the ecliptic - frame analysis without the faint high - latitude foreground galactic emission correction and ignoring the quadrupole moment in the analysis ( solid lines ) , or from the galactic - frame analysis accounting for this galactic emission correction and including the quadrupole moment in the analysis ( dotted lines ) . the other two of the four pairs of lines are for the flat - space scale - invariant spectrum open model ( type [ 4 ] spectra ) , either from the ecliptic - frame analysis without the faint high - latitude foreground galactic emission correction and ignoring the quadrupole moment in the analysis ( dashed lines ) , or from the galactic - frame analysis accounting for this galactic emission correction and including the quadrupole moment in the analysis ( dot - dashed lines ) . given the other uncertainties , the effects of varying @xmath62 and @xmath63 are fairly negligible . fig . 12.ridge lines of the maximum likelihood @xmath21 value as a function of @xmath1 , for the two extreme dmr data sets , for the four cmb anisotropy angular spectra models considered here , and for @xmath64 , @xmath65 . heavy lines are from the ecliptic - frame sky maps ignoring the faint high - latitude foreground galactic emission correction and excluding the quadrupole moment from the analysis , while light lines are from the galactic - frame sky maps accounting for this galactic emission correction and including the quadrupole moment in the analysis . solid , dotted , and dashed lines show the open - bubble inflation cases , accounting only for the fluctuations generated during the evolution inside the bubble ( type [ 1 ] spectra , solid lines ) , also accounting for the fluctuations generated in the first epoch of inflation ( type [ 2 ] spectra , dotted lines these overlap the solid lines except for @xmath269 and @xmath12 ) , and finally also accounting for the fluctuations corresponding to the non - square - integrable basis function ( type [ 3 ] spectra , dashed lines ) . dot - dashed lines correspond to the flat - space scale - invariant spectrum open model ( type [ 4 ] spectra ) . 13.conditional likelihood densities for @xmath21 , derived from @xmath79 ( which are normalized to be unity at the peak , for each dmr data set , cmb anisotropy angular spectrum , and set of model - parameter values ) . panel ( a ) is for the open - bubble inflation model accounting only for fluctuations generated during the evolution inside the bubble ( type [ 1 ] spectra ) , while panel ( b ) is for the flat - space scale - invariant spectrum open model ( type [ 4 ] spectra ) . the heavy lines are for @xmath255 , while the light lines are for @xmath256 . two of the four pairs of lines in each panel correspond to the results from the analysis of the galactic - frame maps accounting for the faint high - latitude foreground galactic emission correction and with the quadrupole moment included in the analysis , either for @xmath123 gyr and @xmath124 ( dot - dashed lines ) , or for @xmath192 gyr and @xmath268 ( dashed lines ) . the other two pairs of lines in each panel correspond to the results from the analysis of the ecliptic - frame maps ignoring this galactic emission correction and with the quadrupole moment excluded from the analysis , either for @xmath123 gyr and @xmath124 ( dotted lines ) , or for @xmath192 gyr and @xmath268 ( solid lines ) . given the other uncertainties , the effects of varying @xmath62 and @xmath63 are fairly negligible . 14.conditional likelihood densities for @xmath21 normalized as in the caption for fig . 13 . panel ( a ) is from the analysis of the ecliptic - frame maps ignoring the faint high - latitude foreground galactic emission correction and excluding the quadrupole moment from the analysis , while panel ( b ) is from the analysis of the galactic - frame maps accounting for this galactic emission correction and including the quadrupole moment in the analysis . these are for @xmath64 and @xmath65 . the heavy lines are for @xmath255 and the light lines are for @xmath256 . there are eight lines ( four pairs ) in each panel , although in each panel two pairs almost identically overlap . solid , dotted , and dashed lines show the open - bubble inflation cases , accounting only for the fluctuations generated during the evolution inside the bubble ( type [ 1 ] spectra , solid lines ) , also accounting for the fluctuations generated in the first epoch of inflation ( type [ 2 ] spectra , dotted lines these almost identically overlap the solid lines ) , and finally also accounting for the fluctuations corresponding to the non - square - integrable basis function ( type [ 3 ] spectra , dashed lines ) . dot - dashed lines correspond to the flat - space scale - invariant spectrum open model ( type [ 4 ] spectra ) . 15.projected likelihood densities for @xmath1 derived from @xmath79 ( normalized as in the caption of fig . panel ( a ) is for the open - bubble inflation model accounting only for the fluctuations generated during the evolution inside the bubble ( type [ 1 ] spectra ) , and panel ( b ) is for the flat - space scale - invariant spectrum open model ( type [ 4 ] spectra ) . two of the curves in each panel correspond to the results from the analysis of the galactic - frame maps accounting for the faint high - latitude foreground galactic emission correction and with the quadrupole moment included in the analysis , for @xmath123 gyr and @xmath124 ( dot - dashed lines ) and for @xmath192 gyr and @xmath268 ( dashed lines ) . the other two curves in each panel are from the analysis of the ecliptic - frame maps ignoring the galactic emission correction and excluding the quadrupole moment from the analysis , for @xmath123 gyr and @xmath124 ( dotted lines ) and for @xmath192 gyr and @xmath268 ( solid lines ) . 16.projected likelihood densities for @xmath1 derived from @xmath79 ( normalized as in the caption of fig . panel ( a ) is from the analysis of the ecliptic - frame sky maps ignoring the faint high - latitude foreground galactic emission correction and excluding the quadrupole moment from the analysis . panel ( b ) is from the analysis of the galactic - frame sky maps accounting for this galactic emission correction and including the quadrupole moment in the analysis . there are four curves in each panel , although in each panel two of them almost overlap . solid , dotted , and dashed lines show the open - bubble inflation cases , accounting only for the fluctuations generated during the evolution inside the bubble ( type [ 1 ] spectra , solid lines ) , also accounting for the fluctuations generated in the first epoch of spatially - flat inflation ( type [ 2 ] spectra , dotted lines these almost exactly overlap the solid lines ) , and finally also accounting for the fluctuations corresponding to the non - square - integrable basis function ( type [ 3 ] spectra , dashed lines ) . dot - dashed lines correspond to the flat - space scale - invariant spectrum open model ( type [ 4 ] spectra ) . these are for @xmath64 and @xmath270 . 17.marginal likelihood densities [ @xmath271 for @xmath1 , normalized to unity at the peak , for the open - bubble inflation model accounting only for fluctuations generated during the evolution inside the bubble ( rp94 ) , for the eight different dmr data sets , and for @xmath121 gyr , @xmath8 . panel ( a ) is from the ecliptic - frame analyses , and panel ( b ) is from the galactic - frame analyses . two of the four lines in each panel are from the analysis without the faint high - latitude foreground galactic emission correction , either accounting for ( dot - dashed lines ) or ignoring ( solid lines ) the quadrupole moment . the other two lines in each panel are from the analysis with this galactic emission correction , either accounting for ( dotted lines ) or ignoring ( dashed lines ) the quadrupole moment . 19.marginal likelihood densities for @xmath1 , for the open - bubble inflation model now also accounting for both the fluctuations generated in the first spatially - flat epoch of inflation and those that correspond to the non - square - integrable basis function ( yst ) , computed for @xmath64 and @xmath65 . conventions are as in the caption of fig . 20.marginal likelihood densities for @xmath1 ( normalized as in the caption of fig . panel ( a ) is for the open - bubble inflation model accounting only for the fluctuations generated during the evolution inside the bubble ( rp94 ) , while panel ( b ) is for the flat - space scale - invariant spectrum open model ( w83 ) . two of the lines in each panel are the results from the analysis of the galactic - frame data sets accounting for the faint high - latitude foreground galactic emission correction and with the quadrupole moment included in the analysis , for @xmath123 gyr and @xmath124 ( dot - dashed lines ) , and for @xmath192 gyr and @xmath268 ( dashed lines ) . the other two lines in each panel are the results from the analysis of the ecliptic - frame data sets ignoring this galactic emission correction and with the quadrupole moment excluded from the analysis , for @xmath272 gyr and @xmath124 ( dotted lines ) , and for @xmath273 gyr and @xmath268 ( solid lines ) . 21.marginal likelihood densities for @xmath1 ( normalized as in the caption of fig . 17 ) , computed for @xmath64 and @xmath65 . panel ( a ) is from the analysis of the ecliptic - frame sky maps ignoring the faint high - latitude foreground galactic emission correction and excluding the quadrupole moment from the analysis . panel ( b ) is from the analysis of the galactic - frame sky maps accounting for this galactic emission correction and including the quadrupole moment in the analysis . there are four lines in each panel , although in each panel two of the lines almost overlap . solid , dotted , and dashed curves are the open - bubble inflation cases , accounting only for the fluctuations generated during the evolution inside the bubble ( rp94 , solid lines ) , also accounting for the fluctuations generated in the first epoch of spatially - flat inflation ( bgt ; yst , dotted lines these almost identically overlap the solid lines ) , and finally also accounting for the fluctuations corresponding to the non - square - integrable basis function ( yst , dashed lines ) . dot - dashed curves correspond to the flat - space scale - invariant spectrum open model ( w83 ) . 22.fractional differences , @xmath274 , as a function of wavenumber @xmath49 , between the energy - density perturbation power spectra @xmath58 computed using the two independent numerical integration codes ( and normalized to give the same @xmath21 ) . the heavy curves are for the open - bubble inflation model spectrum accounting only for fluctuations that are generated during the evolution inside the bubble ( type [ 1 ] spectra ) , and the light curves are for the open - bubble inflation model spectrum now also accounting for fluctuations generated in the first epoch of inflation ( type [ 2 ] spectra ) . these are for @xmath255 ( solid lines ) and @xmath256 ( dashed lines ) , with @xmath64 and @xmath65 . 23.fractional energy - density perturbation power spectra @xmath58 as a function of wavenumber @xmath49 . these are normalized to the mean of the extreme upper and lower 2-@xmath80 @xmath21 values ( as discussed in 3.3 ) . panels ( a)(d ) correspond to the four different sets of ( @xmath62 , @xmath84 ) of tables 912 , and each panel shows power spectra for three different models at six values of @xmath1 . solid lines show the open - bubble inflation model @xmath58 accounting only for fluctuations generated during the evolution inside the bubble ( rp95 ) ; dotted lines are for the open - bubble inflation model now also accounting for fluctuations generated in the first epoch of inflation ( bgt ; yst ) ; and , dashed lines are for the flat - space scale - invariant spectrum open model ( w83 ) . starting near the center of the lower horizontal axis , and moving counterclockwise , the spectra shown correspond to @xmath258 0.1 , 0.2 , 0.3 , 0.45 , 0.6 , and 1 . note that at @xmath69 all three model spectra are identical and so overlap ; also note that at a given @xmath1 the open - bubble inflation model @xmath58 accounting for the fluctuations generated in the first epoch of inflation ( bgt ; yst , dotted lines ) essentially overlap those where this source of fluctuations is ignored ( rp95 , solid lines ) . panel ( a ) corresponds to @xmath123 gyr and @xmath124 , ( b ) to @xmath121 gyr and @xmath275 , ( c ) to @xmath192 gyr and @xmath276 , and ( d ) to @xmath121 gyr and @xmath122 ( normalized using the results of the dmr analysis of the @xmath123 gyr , @xmath277 models ) . panel ( e ) shows the three @xmath64 , @xmath65 open - bubble inflation spectra of table 13 at five different values of @xmath1 . the spectra are for the open - bubble inflation model accounting only for fluctuations generated during the evolution inside the bubble ( rp95 , solid lines ) , also accounting for fluctuations generated in the first epoch of inflation ( bgt ; yst , dotted lines ) , and also accounting for the contribution from the non - square - integrable basis function ( yst , dashed lines ) . starting near the center of the lower horizontal axis and moving counterclockwise , the models correspond to @xmath258 0.1 , 0.2 , 0.3 , 0.5 , and 0.9 . note that at a given @xmath1 the three spectra essentially overlap , especially for observationally - viable values of @xmath212 . the solid triangles represent the redshift - space da costa et al . ( 1994 ) ssrs2 + cfa2 ( @xmath278 mpc depth ) optical galaxies data ( and were very kindly provided to us by c. park ) . the solid squares represent the [ @xmath279 weighting ] redshift - space results of the tadros & efstathiou ( 1995 ) analysis of the @xmath133 qdot and 1.2 jy infrared galaxy data . the hollow pentagons represent the real - space results of the baugh & efstathiou ( 1993 ) analysis of the apm optical galaxy data ( and were very kindly provided to us by c. baugh ) . it should be noted that the plotted model mass ( not galaxy ) power spectra do not account for any bias of galaxies with respect to mass . they also do not account for nonlinear or redshift - space - distortion ( when relevant ) corrections nor for the survey window functions . it should also be noted that the observational data error bars are determined under the assumption of a specific cosmological model and a specific evolution scenario , i.e. , they do not necessarily account for these additional sources of uncertainty ( e.g. , gaztaaga 1995 ) . we emphasize that , because of the different assumptions , the different observed galaxy power spectra shown on the plots are defined somewhat differently and so can not be directly quantitatively compared to each other . 24.cmb anisotropy bandtemperature predictions and observational results , as a function of multipole @xmath66 , to @xmath280 . the four pairs of wavy curves ( in different linestyles ) demarcating the boundaries of the four partially overlapping wavy hatched regions ( hatched with straight lines in different linestyles ) in panel ( a ) are dmr - normalized open - bubble inflation model ( rp94 ) predictions for what would be seen by a series of ideal , kronecker - delta window - function , experiments ( see ratra et al . 1995 for details ) . panel ( b ) shows dmr - normalized cmb anisotropy spectra with the same cosmological parameters for the flat - space scale - invariant spectrum open model ( w83 ) . the model - parameter values are : @xmath177 , @xmath281 , @xmath282 , @xmath283 gyr ( dot - dashed lines ) ; @xmath72 , @xmath284 , @xmath8 , @xmath285 gyr ( solid lines ) ; @xmath256 , @xmath286 , @xmath287 , @xmath288 gyr ( dashed lines ) ; and , @xmath69 , @xmath289 , @xmath8 , @xmath290 gyr ( dotted lines ) for more details on these models see ratra et al . ( 1995 ) . for each pair of model - prediction demarcation curves , the lower one is normalized to the lower 1-@xmath80 @xmath21 value determined from the analysis of the galactic - coordinate maps accounting for the high - latitude galactic emission correction and including the @xmath291 moment in the analysis , and the upper one is normalized to the upper 1-@xmath80 @xmath21 value determined from the analysis of the ecliptic - coordinate maps ignoring the galactic emission correction and excluding the @xmath85 moment from the analysis . amongst the open - bubble inflation models of panel ( a ) , the @xmath72 model is close to what is favoured by the analysis of table 10 , and the @xmath256 model is close to that preferred from the analysis of table 11 . the @xmath177 model is on the edge of the allowed region from the analysis of table 12 , and the @xmath69 fiducial cdm model is incompatible with cosmographic and large - scale structure observations . a large fraction of the smaller - scale observational data in these plots are tabulated in ratra et al . ( 1995 ) and ratra & sugiyama ( 1995 ) . note that , as discussed in these papers , some of the data points are from reanalyses of the observational data . there are 69 detections and 22 2-@xmath80 upper limits shown . since most of the smaller - scale data points are derived assuming a flat bandpower cmb anisotropy angular spectrum , which is more accurate for narrower ( in @xmath66 ) window functions , we have shown the observational results from the narrowest windows available . the data shown are from the dmr galactic frame maps ignoring the galactic emission correction ( grski 1996 , open octagons with @xmath292 ) ; from firs ( ganga et al . 1994 , as analyzed by bond 1995 , solid pentagon ) ; tenerife ( hancock et al . 1996a , open five - point star ) ; bartol ( piccirillo et al . 1996 , solid diamond , note that atmospheric contamination may be an issue ) ; sk93 , individual - chop sk94 ka and q , and individual - chop sk95 cap and ring ( netterfield et al . 1996 , open squares ) ; sp94 ka and q ( gundersen et al . 1995 , the points plotted here are from the flat bandpower analysis of ganga et al . 1996a , solid circles ) ; bam 2-beam ( tucker et al . 1996 , at @xmath293 with @xmath294 spanning 16 to 92 , and accounting for the @xmath295 calibration uncertainty , open circle ) ; python - g , -l , and -s ( e.g. , platt et al . 1996 , open six - point stars ) ; argo ( e.g. , masi et al . 1996 , both the hercules and aries+taurus scans are shown note that the aries+taurus scan has a larger calibration uncertainty of @xmath296 , solid squares ) ; max3 , individual - channel max4 , and max5 ( e.g. , tanaka et al . 1996 , including the max5 mup 2-@xmath80 upper limit @xmath297k at @xmath298 , lim et al . 1996 , open hexagons ) ; msam92 and msam94 ( e.g. , inman et al . 1996 , open diamonds ) ; wdh13 and wdi , ii ( e.g. , griffin et al . 1996 , open pentagons ) ; and cat ( scott et al . 1996 cat1 at @xmath299 with @xmath294 spanning 351 to 471 , and cat2 at @xmath300 with @xmath294 spanning 565 to 710 , both accounting for calibration uncertainty of @xmath301 , solid hexagons ) . detections have vertical 1-@xmath80 error bars . solid inverted triangles inserted inside the appropriate symbols correspond to nondetections , and are placed at the upper 2-@xmath80 limits . vertical error bars are not shown for non - detections . as discussed in ratra et al . ( 1995 ) , all @xmath302 ( vertical ) error bars also account for the calibration uncertainty ( but in an approximate manner , except for the sp94 ka and q results from ganga et al . 1996a see ganga et al . 1996a for a discussion of this issue ) . the observational data points are placed at the @xmath66-value at which the corresponding window function is most sensitive ( this ignores the fact that the sensitivity of the experiment is also dependent on the assumed form of the sky - anisotropy signal , and so gives a somewhat misleading impression of the multipoles to which the experiment is sensitive see ganga et al . 1996a for a discussion of this issue ) . excluding the dmr points at @xmath303 , the horizontal lines on the observational data points represent the @xmath66-space width of the corresponding window function ( again ignoring the form of the sky - anisotropy signal ) . note that from an analysis of a large fraction of the data ( corresponding to detections of cmb anisotropy ) shown in these figures , grs ( figs . 5 and 6 ) conclude that all the models shown in panel ( a ) , including the fiducial cdm one , are consistent with the cmb anisotropy data .

cut - sky orthogonal mode analyses of the @xmath0-dmr 53 and 90 ghz sky maps are used to determine the normalization of a variety of open cosmogonical models based on the cold dark matter scenario . to constrain the allowed cosmological - parameter range for these open cosmogonies , the predictions of the dmr - normalized models are compared to various observational measures of cosmography and large - scale structure , viz . : the age of the universe ; small - scale dynamical estimates of the clustered - mass density parameter @xmath1 ; constraints on the hubble parameter @xmath2 , the x - ray cluster baryonic - mass fraction @xmath3 , and the matter power spectrum shape parameter ; estimates of the mass perturbation amplitude ; and constraints on the large - scale peculiar velocity field . the open - bubble inflation model ( ratra & peebles 1994 ; bucher , goldhaber , & turok 1995 ; yamamoto , sasaki , & tanaka 1995 ) is consistent with current determinations of the 95% confidence level ( c.l . ) range of these observational constraints . more specifically , for a range of @xmath2 , the model is reasonably consistent with recent high - redshift estimates of the deuterium abundance which suggest @xmath4 , provided @xmath5 ; recent high - redshift estimates of the deuterium abundance which suggest @xmath6 favour @xmath7 , while the old nucleosynthesis value @xmath8 requires @xmath9 . small shifts in the inferred @xmath0-dmr normalization amplitudes due to : ( 1 ) the small differences between the galactic- and ecliptic - coordinate sky maps , ( 2 ) the inclusion or exclusion of the quadrupole moment in the analysis , ( 3 ) the faint high - latitude galactic emission treatment , and , ( 4 ) the dependence of the theoretical cosmic microwave background anisotropy angular spectral shape on the value of @xmath2 and @xmath10 , are explicitly quantified . the dmr data alone do not possess sufficient discriminative power to prefer any values for @xmath1 , @xmath2 , or @xmath10 at the 95% c.l . for the models considered . at a lower c.l . , and when the quadrupole moment is included in the analysis , the dmr data are most consistent with either @xmath11 or @xmath12 ( depending on the model considered ) . however , when the quadrupole moment is excluded from the analysis , the dmr data are most consistent with @xmath13 in all open models considered ( with @xmath14 ) , including the open - bubble inflation model . earlier claims ( yamamoto & bunn 1996 ; bunn & white 1996 ) that the dmr data require a 95% c.l . lower bound on @xmath1 ( @xmath15 ) are not supported by our ( complete ) analysis of the four - year data : the dmr data alone can not be used to meaningfully constrain @xmath1 . # 10= 0.00em0 - 0 0.03em0 - 0 0.00em.04em0 - 0 0.03em.04em0 - 00 # 1;#2;#3;#4;#5 # 1 # 2 , # 3 , # 4 , # 5 # 1;#2;#3 # 1 # 2 , # 3 # 1;#2;#3 # 1 # 2 , # 3 # 1@xmath16 mit - ctp-2548 , kuns 1399 0.5 cm august 1996

You are an expert at summarizing long articles. Proceed to summarize the following text: the investigation of ordered double perovskite materials @xmath12 with @xmath3 an alkaline earth such as sr , ba , or ca and @xmath13 two different transition metals has been strongly stimulated by the discovery of a large room temperature magnetoresistive effect at low magnetic fields in sr@xmath6femoo@xmath1 @xcite . the fact that the double perovskites seem to be ferromagnetic metals with high curie temperatures @xmath4 of up to 635k @xcite and apparently have highly spin polarized conduction band makes these materials interesting for applications in spintronic devices such as magnetic tunnel junctions or low - field magnetoresistive sensors @xcite . however , the double perovskites are also of fundamental interest since both their basic physics and materials aspects are not well understood . it is evident that intensive research has been dedicated to both the variation of the metallic / magnetic ions on the @xmath14- and @xmath15-site as well as electron doping studies , where the divalent alkaline earth ions on the @xmath3-site are partially replaced by a trivalent rare earth ion such as la , in order to understand the electronic structure and the magnetic exchange in the double perovskites in detail . furthermore , these studies aimed for the tailoring and optimization of the magnetic properties of the double perovskites for their use in magnetoelectronic devices such as spin valves , magnetic information storage systems , or as sources for spin polarized electrons in spintronics . here , important aspects are the achievement of sufficiently high values for @xmath4 and the spin polarization to allow for the operation of potential devices at room temperature . along this line , in the compound sr@xmath6crreo@xmath1 a curie temperature of @xmath16k has been obtained @xcite . furthermore , in sr@xmath6femoo@xmath1 an increase in @xmath4 of about 70k has been reported as a result of electron doping by partial substitution of divalent sr@xmath9 by trivalent la@xmath10 @xcite . with respect to magnetoresistance , up to now large low - field magnetoresistive effects have been found not only in sr@xmath6femoo@xmath1@xcite , but also in many other double perovskites as for example sr@xmath6fereo@xmath1 @xcite , sr@xmath6crwo@xmath1 @xcite , and ( ba@xmath17sr@xmath18)@xmath19la@xmath20femoo@xmath1 @xcite . the origin of magnetism in the double perovskite still is discussed controversially . historically , the ferrimagnetism in the system sr@xmath6femoo@xmath1 has been explained in terms of an antiferromagnetic superexchange interaction between the mo@xmath21 ( @xmath22 ) spin and the fe@xmath10 ( @xmath23 ) spins @xcite . however , more recently moritomo _ et al . _ found a strong correlation between the room temperature conductivity and the curie temperature , implying that the mobile conduction electrons mediate the exchange interaction between the local fe@xmath10 spins @xcite . therefore , in analogy to the doped manganites it was tempting to explain the ferromagnetic coupling between the fe sites based on a double exchange mechanism , where the delocalized electron provided by the mo @xmath24 configuration plays the role of the delocalized @xmath25 electron in the manganites . however , as pointed out by sarma @xcite there are important differences between the manganites and the double perovskites . in the former , both the localized mn @xmath26 electrons and the delocalized mn @xmath25 electron reside at the same site and their spins are coupled ferromagnetically by a strong on - site hund s coupling . in the latter , the localized fe @xmath27 electrons ( fe@xmath10 : @xmath28 ) and the delocalized mo @xmath29 electron ( mo@xmath21 : @xmath30 ) nominally are at two different sites although the mo @xmath29 electron obtains a finite fe character by sizable hopping interaction . at first glance this seems to support a double exchange szenario . however , in sr@xmath6femoo@xmath1 according to band structure calculations the fe @xmath27 spin - up band is completely full making it impossible for another spin - up electron to hop between fe sites and thus forcing the delocalized electrons to be spin - down electrons . therefore , the hund s coupling strength , which provides the energy scale for the on - site spin coupling in the double exchange mechanism for the manganites , can not be invoked for sr@xmath6femoo@xmath1 . this demonstrates that in the double perovskites the antiferromagnetic coupling between the localized and the delocalized electrons , which according to the large @xmath4 is strong , must originate from another mechanism . such mechanism has been proposed by sarma @xcite for sr@xmath6femoo@xmath1 and extended to many other systems by kanamori and terakura @xcite . in this model , the hybridization of the mo @xmath29 ( @xmath26 ) and fe @xmath27 ( @xmath26 ) states plays the key role in stabilizing ferromagnetism at high curie temperatures @xcite . if the fe@xmath10 spins are ferromagnetically ordered , the hybridization between the fe @xmath27 ( @xmath26 ) and the mo @xmath29 ( @xmath26 ) states pushes up and down the mo @xmath31 and mo @xmath32 states , respectively . the essential point in this scenario , which is discussed in more detail below , is that the hybridized states are located energetically between the exchange - split fe @xmath33 and the fe @xmath34 levels . we note that in this model the magnetic moment at the mo site is merely induced by the fe magnetic moments through the hybridization between the fe @xmath27 and the mo @xmath29 states what can be considered as a magnetic proximity effect . in this sense the double perovskites should be denoted ferromagnetic and not ferrimagnetic @xcite . despite the recent progress in understanding the physics of the double perovskites there is still an open debate on the adequate theoretical modelling . in particular , the details of the interplay between structural , electronic , and magnetic degrees of freedom in the double perovskites is not yet clearly understood . recently , the @xmath4 in double perovskites was discussed to depend sensitively on the band structure and band filling in contrast to experimental results @xcite . however , since in experiments often changes in the number of conduction electrons achieved by partially replacing the divalent alkaline earth ions on the @xmath3- and @xmath35-site by trivalent rare earth ions are accompanied by significant steric effects , it is difficult to distinguish between doping and structural effects . it is therefore desirable to investigate both the influence of structural changes ( changes in the bond length and bond angles ) and the effect of carrier doping . here , we present a study of the effect of structural changes and doping in the double perovskite @xmath0crwo@xmath1 ( @xmath36 ) on the magnetotransport , the magnetic and optical properties together with band structure calculations . we note that a major difference between the system @xmath0crwo@xmath1 and the systems @xmath0femoo@xmath1 or @xmath0fereo@xmath1 is the fact that in the former the majority spin band is only partially filled ( cr@xmath10 , @xmath37 , only the cr @xmath27 ( @xmath26 ) levels are occupied ) , whereas in the latter the majority spin band is completely filled ( fe@xmath10 , @xmath28 , both the fe @xmath27 ( @xmath26 ) and fe @xmath27 ( @xmath38 ) levels are occupied ) . we discuss that despite this difference the magnetism in @xmath0crwo@xmath1 and @xmath0femoo@xmath1 is similar . for both systems the exchange gap is important for the magnetic exchange and half metallicity . however , whereas for the former system also the crystal field gap between the @xmath26 and the @xmath25 levels plays a key role , for the latter the crystal field gap is irrelevant . a key result of our study is that optimum magnetic properties such as half metallicity and high @xmath4 can only be obtained in the undoped compounds close to the ideal undistorted perovskite structure that is characterized by a tolerance factor @xmath39 of the perovskite unit cell . furthermore , we find that electron doping tends to decrease @xmath4 in @xmath0crwo@xmath1 . in our study , we have used both polycrystalline and epitaxial thin film samples . polycrystalline samples were prepared from stoichiometric mixtures of srco@xmath40 , baco@xmath40 , caco@xmath40 , cr@xmath6o@xmath40 , la@xmath6o@xmath40 , and wo@xmath40 with a purity ranging between 99.99% and 99.999% . these powders were thoroughly mixed , placed in al@xmath6o@xmath40 crucibles , and then repeatedly heated in a thermobalance under reducing atmosphere ( h@xmath6/ar : 5%/95% ) with intermediate grinding . the final sintering temperatures @xmath41 were increased from 1200@xmath42c for the first up to 1550@xmath42c for the final firing for the first series of samples . in order to increase the curie temperature we prepared a second series of undoped sr@xmath6crwo@xmath1 samples with more sintering steps and longer final sintering times at lower ( 1300@xmath42c ) temperatures . for this series we find the highest curie temperatures , which correspond to the values reported in literature . this most likely is caused by a slightly higher oxygen content in these samples compared to those sintered at higher temperatures in the same reducing atmosphere . the use of the thermobalance allowed to monitor the ongoing reaction process due to the associated weight loss of the samples . the exact oxygen content of the investigated samples has not been determined . however , due to the very similar preparation procedure for the samples used for the study of electron doping , we can assume a similar oxygen content . the polycrystalline samples were characterized by x - ray powder diffractometry to detect parasitic phases , e.g. the insulating compound srwo@xmath43 . interestingly , even in samples containing srwo@xmath43 no chromium containing parasitic phases could be detected . the most simple explanation for this observation is the high vapor pressure of cr@xmath6o@xmath40 resulting in a loss of chromium . however , no weight loss has been found in the thermogravimetric measurement . therefore , it is more likely that the formation of the mixed phase @xmath0cr@xmath44w@xmath45o@xmath46 compensates for the missing sr and w due to the srwo@xmath43 impurity phase . while the double perovskite sr@xmath6femoo@xmath1 can be grown as single crystal @xcite , our attempts to grow single crystals of sr@xmath6crwo@xmath1 by floating zone melting failed so far due to the high vapor pressure of cr@xmath6o@xmath40 at the melting point . epitaxial thin films of sr@xmath6crwo@xmath1 were fabricated in a uhv laser molecular beam epitaxy ( l - mbe ) system @xcite . the details of the thin film deposition have been described recently @xcite . [ tem ] shows a transmission electron micrograph of an epitaxial sr@xmath6crwo@xmath1 thin film grown on a srtio@xmath40 ( 001 ) substrate . the image shows that the thin film grows epitaxially with high crystalline quality . the @xmath47-axis oriented films have been analyzed using a high resolution 4-circle x - ray diffractometer . only ( @xmath48 ) peaks could be detected and typically , rocking curves of the ( 004 ) peak had a full width at half maximum ( fwhm ) of only 0.02 to 0.03@xmath49 , which is very close to the fwhm of the substrate peak . moreover , an afm analysis of the sr@xmath6crwo@xmath1 films showed that they have a very small surface roughness of the order of one unit cell @xcite . in this study the thin films were used for optical measurements in order to determine the band structure . it is well known that the tolerance factor @xmath50 determines the crystal structure of perovskites @xmath51o@xmath40 @xcite . only for @xmath50 close to unity a cubic perovskite structure is obtained . for @xmath52 a tilt and rotation of the oxygen octahedra is obtained compensating for the misfit of the ionic radii of the involved @xmath3 and @xmath14 cations . hence , the deviation of the tolerance factor from the ideal value @xmath53 can be used as a measure for the internal strain in perovskites induced by the different radii of the @xmath3 and @xmath14 cations . this can be seen from the definition of @xmath50 given by @xmath54 here , @xmath55 denotes the average ionic radius for the ions on the @xmath14-site . for @xmath56 , the strain is compensated by a tilt and rotation of the oxygen octahedra . this results in a deviation of the @xmath14-o-@xmath14 bond angles from the ideal value of @xmath57 . for @xmath58 the connecting pattern of the oxygen octahedra is rhombohedral , whereas it is orthorhombic for lower values of @xmath50 @xcite . a simple consequence of the deviation of the bond angles from @xmath57 is a decrease in the hopping amplitude because the electron transfer between the @xmath14 sites is via o @xmath59-states . this , in turn , results in a decrease of the one - electron bandwidth @xmath60 . for @xmath61 , a hexagonal structure is expected which is classified by the stacking sequence of the @xmath14o@xmath1 octahedra @xcite . .overview on the crystallographic properties of the investigated samples of the @xmath0crwo@xmath1 series with @xmath2 . the tolerance @xmath50 factor was calculated using the spuds simulation software @xcite . [ cols="^,^,^,^ " , ] [ bigtable ] fig . [ fig : res ] shows the temperature dependence of the resistivity for the double perovskites @xmath0crwo@xmath1 with @xmath62 . all samples show an increase of the resistivity with decreasing temperature . since the investigated samples are polycrystalline , the influence of grain boundaries plays an important role . hence , the observed semiconductor like resistivity vs temperature curves may be related to the grain boundary resistance , whereas the intrinsic resistance of the double perovskites may be metallic . although the intrinsic resistivity behavior can not be unambiguously derived from our measurements , the fact that @xmath63 for @xmath64 ( see inset of fig . [ fig : res ] ) provides significant evidence for a metallic behavior in the sr@xmath6crwo@xmath1 sample . here , @xmath65 is the electrical conductivity . the observed trend that the resistivity increases , if sr@xmath6 is replaced by ca@xmath6 and even more by ba@xmath6 , can be understood as follows : in contrast to the cubic perovskite sr@xmath6crwo@xmath1 , ca@xmath6crwo@xmath1 has a distorted perovskite structure with a @xmath14-o-@xmath14 bonding angle deviating from 180@xmath42 . this result in a reduction of the overlap between the relevant orbitals and , hence , the hopping amplitude . finally , the ba@xmath6crwo@xmath1 compound has the highest resistivity most likely due to its hexagonal structure ( see table [ table : samples ] ) . the magnetotransport properties of sr@xmath6crwo@xmath1 already has been discussed elsewhere @xcite . we found , that polycrystalline samples containing a large number of grain boundaries show a large negative low - field magnetoresistance , @xmath66/r(0)$ ] , of up to -41% at 5k . at room temperature , this effect is reduced to a few percent . the large grain boundary @xmath67 effect at low temperatures indicates that sr@xmath6crwo@xmath1 has a large spin polarization of the charge carries and due to its high curie temperature may be an interesting candidate for magnetoelectronic devices operating room - temperature . in fig . [ fig : f ] we plot the curie temperature @xmath4 , the saturation magnetization @xmath68 , and the ionic radii versus the tolerance factor for the series @xmath0crwo@xmath1 . it is evident that the curie temperature is largest for sr@xmath6crwo@xmath1 ( @xmath5k ) , whereas it is suppressed strongly for ca@xmath6crwo@xmath1 ( @xmath69k ) . we attribute this fact to the small ionic radius of ca@xmath9 , which results in @xmath70 and , in turn , in a distorted perovskite structure . this results in a reduction of the effective hopping interactions between cr @xmath27 and w @xmath71 states , leading to a reduced spin splitting of the conduction band @xcite . this naturally reduces the magnetic coupling strength and hence the @xmath4 . the saturation magnetization @xmath68 is known to depend strongly on the amount @xmath72 of antisites , with @xmath73 for no antisites and @xmath74 for 50% antisites or complete disorder . by simply assuming the presence of antiferromagnetically coupled cr and w sublattices , a maximum saturation magnetization of @xmath75 is expected for @xmath73 , which is decreasing to zero for @xmath74 . that is , this assumption leads to @xcite @xmath76 where @xmath68 is the saturation magnetization in units of @xmath77 ; @xmath78 and @xmath79 are the magnetic moments of the cr@xmath10 and w@xmath21 ions in units of @xmath80 , respectively . comparing the experimental data to this simple model prediction leads to a surprisingly good agreement for sr@xmath6crwo@xmath1 . the measured value of @xmath81f.u . is very close to the value of @xmath82f.u . expected from eq.([eq : msat ] ) for @xmath83 . [ fig : f]b also shows that the saturation magnetization of ca@xmath6crwo@xmath1 is larger than for sr@xmath6crwo@xmath1 despite the much lower curie temperature of the latter . the obvious reason for that is the lower amount of antisites in ca@xmath6crwo@xmath1 . from the measured value of @xmath84 we expect @xmath85f.u . for ca@xmath6crwo@xmath1 which is slightly larger than the measured value of @xmath86f.u .. the reason for the observation that the experimental @xmath87 value is below the one predicted by eq.([eq : msat ] ) most likely is the distorted crystal structure of ca@xmath6crwo@xmath1 . the magnetic properties of the ba@xmath6crwo@xmath1 compound are completely different from those of sr@xmath6crwo@xmath1 and ca@xmath6crwo@xmath1 . here , the large ionic radius of ba@xmath9 enlarges @xmath50 well above unity . this causes a structural phase transition towards a hexagonal structure , where the ferromagnetic interaction is strongly suppressed . therefore , not only @xmath4 is strongly suppressed ( @xmath88k , in contrast to @xmath89k for sr@xmath6crwo@xmath1 ) , but also the saturation magnetization ( @xmath90f.u . ) is close to zero . this clearly indicates the strong effect of the structural phase transition on the magnetic interaction . we note that we can exclude the possibility that the very small @xmath68 value is due to strong disorder . evidently , there should be almost complete disorder to suppress @xmath68 close to zero . however , in the ba@xmath6crwo@xmath1 compound the two different @xmath14 sites establish a certain amount of order , since almost all w ions occupy the @xmath91 site . whether the observed behavior is related to a canted antiferromagnetic phase or simply to the presence of minority phases has to be clarified . [ fig : acwomag ] shows that the magnetic interactions for all three compounds @xmath0crwo@xmath1 is ferromagnetic . however , the strongly reduced saturation magnetization of ba@xmath6crwo@xmath1 and the upturn in magnetization vs. temperature curve at low temperatures indicates that ferromagnetic interactions are small and that there may be paramagnetic regions in the sample . the substitution of sr by ca evidently results in a strong reduction of @xmath4 , however , the ca@xmath6crwo@xmath1 samples are still clearly ferromagnetic . again , the important point is that only the compound with @xmath39 has optimum magnetic properties with respect to applications in magnetoelectronics . we also have performed measurements of the coercive field @xmath92 . at low temperature we obtained @xmath93oe for the sr@xmath6crwo@xmath1 compound and @xmath94oe for the ca@xmath6crwo@xmath1 compound . furthermore , for the sr@xmath6crwo@xmath1 the coercive field was found to depend on the preparation conditions . for example , lowering the final firing temperature from 1550@xmath42c to 1300@xmath42c was found to increase @xmath92 from 450 to 1200oe . in agreement with the findings for the doped manganites @xcite , the coercive field increases with decreasing @xmath50 . this observation can be easily understood , since the induced structural distortions can effectively act as pinning centers for domain wall movement . in those cases where a large remanent magnetization and/or coercive field are important for applications , the use of double perovskites with reduced values of @xmath50 obtained by suitable substitution on the @xmath3-site may be desirable . comparing our results for the series @xmath0crwo@xmath1 with @xmath95 to other double perovskite compounds summarized in table [ bigtable ] , it is evident that the suppression of @xmath4 as a function of the deviation of the tolerance factor from its ideal value of @xmath53 is a general trend : it is only weak for the series @xmath0femoo@xmath1 with @xmath36 , where @xmath4 varies between 310k and 420k @xcite . however , it is also strong for the series @xmath0crreo@xmath1 with @xmath96k for sr@xmath6crreo@xmath1 and @xmath97k for ca@xmath6crreo@xmath1 @xcite . in general , a high curie temperature can only be realized in double perovskites of the composition @xmath98o@xmath1 having a tolerance factor close to unity . this is realized in the different systems for @xmath99 . for @xmath50 well below unity , the curie temperature is drastically reduced in agreement with what is found for the doped manganites @xcite . for the double perovskites , the system ( sr@xmath100ca@xmath101)@xmath6fereo@xmath1 is an exception of the general rule @xcite : here , the ca@xmath6fereo@xmath1 compound has the highest @xmath4 , although the tolerance factor decreases continuously from @xmath102 for sr@xmath6fereo@xmath1 to @xmath103 for ca@xmath6fereo@xmath1 on substituting sr by ca . we note , however , that ca@xmath6fereo@xmath1 is a unique material as it is a ferromagnetic insulator and that there may be an other mechanism causing the high ordering temperature @xcite . in fig . [ fig : tctoleranceall ] we have plotted the curie temperatures of the double perovskite materials listed in table [ bigtable ] versus their tolerance factor omitting the ferromagnetic insulator ca@xmath6fereo@xmath1 . despite the significant spread of data that may be partially caused be different sample quality , it is evident that a maximum curie temperature is obtained for the systems having a tolerance factor of @xmath8 . these systems have a cubic / tetragonal symmetry . as shown in fig . [ fig : tctoleranceall ] , for @xmath56 there is a transition to orthorhombic structures for @xmath104 , whereas for @xmath105 there is a transition to a hexagonal structure at @xmath106 . however , according to data from literature this transition seems to be smeared out as indicated by the shaded area . we briefly summarize the measured structural , transport and magnetic properties and discuss the underlying physics . recently , sarma _ et al . _ proposed an interesting model explaining the origin of the strong antiferromagnetic coupling between fe and mo in sr@xmath6femoo@xmath1 in terms of an strong effective exchange enhancement at the mo site due to a fe @xmath27-mo @xmath29 hybridization @xcite . kanamori and terakura extended this idea to explain ferromagnetism in many other systems , where nonmagnetic elements positioned between high - spin @xmath27 elements contribute to the stabilization of ferromagnetic coupling between the @xmath27 elements @xcite . the essence of this model is summarized in fig . [ fig : models]a for the case of sr@xmath6femoo@xmath1 . without any hopping interactions , the fe@xmath10 @xmath23 configuration has a large exchange splitting of the @xmath27 level in the spin - up and spin - down states and there is also a crystal field splitting @xmath107 into the @xmath26 and the @xmath25 states ( see fig . [ fig : models]a ) . the exchange splitting of the mo@xmath21 @xmath24 configuration ( better the mo-@xmath29-o-@xmath59 hybridized states ) is vanishingly small , however , there is a large crystal field splitting ( the @xmath25 states are several ev above the @xmath26 states and not shown in fig . [ fig : models]a ) . the interesting physics occurs on switching on hopping interactions , which result in a finite coupling between states of the same symmetry and spin . the hopping interaction not only leads to an admixture of the fe @xmath27 to the mo @xmath29 states , but more importantly to a shift of the bare energy levels . as shown in fig . [ fig : models]a , the delocalized mo @xmath26 spin - up states are pushed up , whereas the mo @xmath26 spin - down states are pushed down . this causes a finite spin polarization at the fermi level ( actually 100% in fig . [ fig : models]a ) resulting from the hopping interactions . this kinetic energy driven mechanism leads to an antiferromagnetic coupling between the delocalized mo @xmath29 and the localized fe @xmath27 electrons , since the energy is lowered by populating the mo @xmath29 spin - down band @xcite . the magnitude of the spin polarization derived from this mechanism obviously is governed by the hopping strength and the charge transfer energy between the localized and the delocalized states @xcite . the question now arises , whether the above model also applies for the @xmath0crwo@xmath1 compounds . the key concept of the model is the energy gain contributed by the spin polarization of the nonmagnetic element ( now w ) induced by the hybridization with the magnetic transition metal ( now cr ) . it has been pointed out by sarma @xcite that the underlying mechanism will always be operative , whenever the conduction band is placed within the energy gap formed by the large exchange splitting of the localized electrons at the transition metal site . as shown in fig . [ fig : models]b , this is also the case for the @xmath0crwo@xmath1 compounds : the w @xmath71 band resides in between the exchange split cr @xmath27 ( @xmath26 ) spin - up and spin - down bands . the schematic band structure of fig . [ fig : models]b has been confirmed by band structure calculations presented below . the only difference between the system @xmath0crwo@xmath1 and the systems @xmath0femoo@xmath1 or @xmath0fereo@xmath1 is the fact that in the former the majority spin band is only partially full . for cr@xmath10 , ( @xmath108 ) only the cr @xmath26 levels are occupied . in contrast , for fe@xmath10 ( @xmath28 ) both the fe @xmath26 and fe @xmath38 levels are occupied , that is , the majority spin band is completely filled . band structure calculations show that the crystal field splitting in the cr compounds ( @xmath109ev ) is slightly larger than in the fe compounds @xcite . on the other hand , the exchange splitting in the cr @xmath27 bands is somewhat smaller than for the fe @xmath27 bands due to the valence configuration cr @xmath110 with less electrons and weaker hund s coupling . taking these facts into account we have to split up the cr @xmath27 spin - up and spin - down band into two separate @xmath27 ( @xmath26 ) and @xmath27 ( @xmath38 ) bands with the fermi level lying in the gap between the bands as shown in fig . [ fig : models]b . indeed band structure calculations ( see below ) show that the cr @xmath27 ( @xmath25 ) spin - up band is about 0.5ev above the fermi level . however , the above mechanism still works as long as the w @xmath71 ( @xmath26 ) band is placed within the energy gap between the cr @xmath27 ( @xmath26 ) spin - up and the cr @xmath27 ( @xmath26 ) spin - down band . then , again the w @xmath71 ( @xmath26 ) levels would hybridize with the cr @xmath27 ( @xmath26 ) levels resulting in a negative spin polarization of the nonmagnetic element w @xmath71 ( @xmath26 ) and a stabilization of ferromagnetism and half - metallic behavior . we note that no hybridization takes place between the cr @xmath27 ( @xmath25 ) spin - up band and the w @xmath71 ( @xmath26 ) spin - up band due to the different symmetry of these levels . therefore , the exact position of the cr @xmath27 ( @xmath25 ) spin - up band is not relevant . we also note that due to the large crystal field splitting for the w @xmath71 band , the w @xmath27 ( @xmath25 ) band is several ev above the w @xmath27 ( @xmath26 ) band and not shown in fig . [ fig : models]b . summarizing our discussion we can state that the essential physical mechanism leading to ferromagnetism is very similar for the @xmath0crwo@xmath1 and the @xmath0femoo@xmath1 compounds . comparing fig . [ fig : models]a and b we see that the only difference is an upward shift of the @xmath27 bands for the @xmath0crwo@xmath1 compounds . we now discuss the experimental results in context with the models discussed above . we first discuss the strong dependence of @xmath4 on the tolerance factor , which in turn is intimately related to the bond angles . it is well known that in perovskite type transition metal oxides in general the increase of the @xmath14-o bond length and the deviation of the @xmath14-o-@xmath14 bond angle from @xmath57 has the effect of a reducing the hybridization matrix element @xmath111 . this is caused by the reduction of the overlap of the oxygen 2@xmath112 and transition - metal @xmath113 states @xcite , which in turn results in a reduction of the bare electron bandwidth @xmath60 . since the weakening of the hybridization causes a reduction of the energy gain stabilizing ferromagnetism , we expect a significant decrease of @xmath4 with increasing deviation of the tolerance factor from @xmath53 , or equivalently with an increasing deviation of the @xmath14-o-@xmath14 bond angle from @xmath57 . this is in good qualitative agreement with our results on the @xmath0crwo@xmath1 compound ( see fig . [ fig : f]a ) and the collected data plotted in fig . [ fig : tctoleranceall ] . theoretical models providing a quantitative explanation have still to be developed . it is known that for a @xmath57 bond angle , @xmath111 decreases with increasing bond length @xmath114 roughly as @xmath115 . therefore , the effective hopping amplitude between the transition metal ions @xmath14 has an even stronger dependence on @xmath114 . hence , a significant increase of @xmath111 and , in turn , @xmath4 is expected with a reduction of the bond length . applying this consideration to the investigated series @xmath0crwo@xmath1 we would expect the largest @xmath4 for the ca@xmath6crwo@xmath1 compound due to its smallest cell volume and , hence , shortest bond length . however , experimentally the largest @xmath4 was found for sr@xmath6crwo@xmath1 . this is caused by the fact that @xmath111 not only depends on the bond length but also strongly on the bond angle . for the cubic double perovskite sr@xmath6crwo@xmath1 with @xmath39 the bond angle has the ideal value of @xmath57 , whereas for the distorted double perovskite ca@xmath6crwo@xmath1 with @xmath56 the bond angle significantly deviates from this ideal value . for the double perovskites lattice effects come into play also for large tolerance factors @xmath116 , where in the hexagonal structure the formation of dimers suppresses strongly ferromagnetism and enhances antiferromagnetic interactions . furthermore , the magnetic interactions are also weakened due to the increased distance ( bond length ) between the magnetic ions . we finally would like to mention that in a recent work on @xmath0femoo@xmath1 , a linear correlation , @xmath117 , between @xmath4 and the bare one - electron bandwidth @xmath118 has been found by ritter _ et al . _ @xcite . summarizing our discussion we can state that our findings for the series @xmath0crwo@xmath1 can be extended to the double perovskites in general . the data summarized in table [ bigtable ] and fig . [ fig : tctoleranceall ] clearly indicated that for most double perovskites a maximum @xmath4 is obtained for a tolerance factor of @xmath8 corresponding to an about cubic perovskite structure with a bond angle close to @xmath57 . this optimum situation in most cases is realized in the sr@xmath119o@xmath1 compounds . the requirement @xmath39 for optimum @xmath4 in the double perovskite is different for the doped manganites . here , a maximum @xmath4 is achieved for compounds with @xmath120 , that is , for a significantly distorted perovskite structure . et al . _ @xcite have shown that the highest @xmath4 in doped manganites is obtained for @xmath121 ( in a more precise analysis by zhou _ et al . _ @xcite slightly larger values for @xmath50 have been derived based on a coordination number of 9 ) . for a tolerance factor below the optimum value , a dramatic decrease of @xmath4 has been found . whether or not the differences between the manganites and the double perovskites are related to the different mechanism stabilizing ferromagnetism in these compounds has to be clarified . we note , however , that the different behavior of the manganites and the double perovskites is likely to be related to different mechanisms . in manganites the polarization of the conduction band is driven by the hund s coupling ( intra atomic ) between the @xmath26 and the @xmath25 states , which is not sensitive to the details of the band structure . in contrast , in the double perovskites the spin polarization of the conduction band is itself dependent on the band structure and therefore is more intimately linked to it . the effect of carrier ( electron ) doping in sr@xmath6crwo@xmath1 was studied in a series of sr@xmath19la@xmath20crwo@xmath1 samples with @xmath123 , @xmath124 , @xmath125 and @xmath126 . in our experiments trivalent la@xmath10 is chosen to replace the sr@xmath9 ions because the ionic radius of la@xmath10 ( @xmath127 ) is similar to that of sr@xmath9 ( @xmath128 ) . as a result , there are only a small variations of the lattice parameters and the tolerance factor @xmath50 on changing the doping level from @xmath123 to @xmath129 . x - ray analysis showed that the lattice parameter slightly decreases from @xmath130 ( @xmath123 ) to @xmath131 ( @xmath129 ) as expected since the ionic radius of la@xmath10 is smaller than that of sr@xmath9 . however , the crystal structure of all samples remained cubic . this means that the structural changes are small on varying the doping level . in this way the effect of doping can be studied without being strongly influenced by structural effects . in fig . [ fig : la ] the magnetization curves of the series sr@xmath19la@xmath20crwo@xmath1 are shown for @xmath132k . all samples of this series have been fired between 1530 and @xmath133c in reducing atmosphere . for the undoped sample this results in a smaller curie temperature of about 390k as compared to about 460k for the samples fired at @xmath134c . the reason for that most likely is a slightly smaller oxygen content in the sample fired at a higher temperature . since a reduced oxygen content corresponds to an effective electron doping , for the doping series we compare only samples prepared under identical conditions . we find that the saturation magnetization @xmath68 decreases from @xmath135/f.u . for @xmath123 to @xmath136/f.u . for @xmath129 . as shown in the inset of fig . [ fig : la ] , with increasing doping level also the amount of antisites is increasing . at present , we only have a plausible explanation for the observed increase of the amount of antisites with increasing doping level , which so far could not be unambiguously proven . however , since la doping is expected to result in a reduction of the differences in the valence states of cr and w , it evidently results in a reduction of the differences in the ionic radii of cr and w ( @xmath137 and @xmath138 ) . therefore , increasing the doping level results in more similar ionic radii of cr and w paving the way for the creation of cr / w antisites . since the substitution of sr@xmath9 by la@xmath10 results both in electron doping and an increase of antisites , it is not possible to unambiguously attribute the measured decrease in @xmath68 to either the increasing doping level or the increase of antisites alone . as will be discussed in the following , for the series sr@xmath19la@xmath20crwo@xmath1 the measured reduction of @xmath68 most likely is caused by both electron doping and disorder . we first discuss the expected reduction of @xmath68 due to the increasing amount of antisites . we note that that several authors have found a reduction of @xmath68 following the increase of antisites @xcite . this behavior is consistent with simple monte carlo simulation studies @xcite . however , also more complicated models @xcite predict a reduction of @xmath68 with increasing amount of antisites due to charge transfer effects @xcite . in a first approach we can analyze our data using eq.([eq : msat ] ) . with the measured @xmath139 values for the amount of antisites we can calculate @xmath140 . we find that the calculated @xmath68 values are significantly larger than the measured ones . this suggests that the observed reduction of the saturation magnetization can not be explained by the increasing amount of antisites alone ( at least within the simple model yielding eq.([eq : msat ] ) ) . however , we also have to keep in mind that electron doping itself contributes to the reduction of @xmath68 . according to the illustration given in fig . [ fig : models]a it is evident that electron doping in the @xmath0femoo@xmath1 system increases the spin - down magnetic moment at the mo site , indicating that the electrons are filled into the mo @xmath141 band and thereby reduce @xmath68 @xcite . according to fig . [ fig : models]b , the same mechanism holds for the sr@xmath19la@xmath20crwo@xmath1 system . that is , our results suggest that the observed reduction of the saturation magnetization with increasing la@xmath10 substitution is caused both by an increase of the amount of antisites and an increase of the number of conduction electrons . summarizing our discussion of the saturation magnetization we would like to emphasize that a variation of the doping level in most cases is correlated with a variation of the amount of disorder , since doping contributes to a reduction of the difference of the valence states of cr and w ( fe and mo ) what in turn results in an increasing amount of antisites @xcite unfortunately , due to this fact the experimental situation is not completely clear . while in @xcite for sr@xmath6femoo@xmath1 and also in our study for sr@xmath6crwo@xmath1 la doping is clearly correlated with higher disorder , in @xcite the amount of antisites seems to be constant in sr@xmath19la@xmath20femoo@xmath1 for almost the whole doping series from @xmath123 to @xmath142 . for the system sr@xmath6crwo@xmath1 , the suppression of @xmath68 is stronger than for the system sr@xmath6femoo@xmath1 , probably due to the fact that the cr and w ions are easier to disorder because of their similar ionic radii . we finally note that in general both increasing doping and disorder can destroy the underlying half - metallic ferromagnetic state in double perovskites , thereby significantly populating / depopulating the different spin channels leading to a sharp decrease in @xmath68 @xcite . we next discuss the influence of la doping on the curie temperature . it is evident that an enhancement of @xmath4 by doping would be of great importance for possible applications of the double perovskites in spintronic devices . however , both the experimental data on the variation of @xmath4 with electron doping as well as the theoretical interpretation is controversial at present . in single crystals of sr@xmath19la@xmath20femoo@xmath1 moritomo _ et al . _ have found that @xmath4 does not change as a function of la doping @xmath143 @xcite . in contrast , navarro _ _ have reported a considerable increase of @xmath4 of about 70k in ceramic samples of sr@xmath19la@xmath20femoo@xmath1 investigating a wider doping range @xmath144 @xcite . in our study on the system sr@xmath19la@xmath20crwo@xmath1 we have found a reduction of @xmath4 of about 80k in the doping range @xmath145 as can be seen in fig . [ fig : la - tc ] . we note that part of the experimental discrepancies may be related to large error bars in the determination of @xmath4 . in particular , great care has to be taken over the determination of @xmath4 , since the transition from zero magnetization to finite magnetization is smeared out considerably due to finite applied magnetic fields and parasitic phases . it is evident from fig . [ fig : la - tc ] that in all our samples there are minor phases with optimum @xmath4 close to about 400k . however , it is also evident that la doping reduces @xmath4 of the major part of the sample . that is , different experimental results on @xmath4 may be in part related to different ways of measuring and analyzing the data . from the theoretical point of view one expects both an increase and decrease of @xmath4 with increasing doping . on the one hand , within the model presented in @xcite @xmath4 is expected to be rather reduced than enhanced by electron doping due to the fact that the possible energy gain by shifting electrons from spin - up band into the spin - down band is reduced @xcite . this , in turn , reduces the stability of the ferromagnetic phase in agreement with our results . however , the role of the increasing amount of antisites with increasing doping still has to be clarified . on the other hand , in a double exchange model the increase of the number of conduction electrons promoted by la doping is expected to enhance the double exchange interaction leading to an increase of @xmath4 . within this model the ferromagnetic interaction arises from the double exchange interaction between the localized moments on cr@xmath10 sites ( @xmath110 , @xmath146 ) mediated by itinerant electron provided by the w@xmath21 ions ( @xmath22 , @xmath147 ) . according to the band structure calculation presented below , the cr @xmath26 spin - up subband is completely filled and it is the electron in the @xmath26 spin - down subband of cr and w which mediates the double exchange interaction . in general , an increasing number of electrons in the spin - down subband is expected to strengthen the double exchange interaction leading to an increase of @xmath4 in conflict with our experimental findings . that is , the observed decrease of @xmath4 with increasing doping level seems to support the model presented in @xcite ( see also fig . [ fig : models ] ) . however , we have to take into account that by la doping besides the number of electrons we also increase the amount of disorder . the latter may weaken the double exchange interaction sufficiently to result in an effective decrease of @xmath4 also within a double exchange based model . further work is required to clarify this in more detail . the half - metallicity of ferromagnetic materials is one of the important ingredients required for applications . in order to obtain a conclusive picture regarding the spin polarization at the fermi level it is important to compare band structure calculations with experimental results . here , we compare the results of band structure calculations based on _ ab initio _ methods to experimental results on the optical reflectivity and transmissivity . in fig . [ fig : bandstructure ] the density of states of sr@xmath6crwo@xmath1 is plotted versus energy . the band structure has been calculated @xcite within the linear muffin - tin orbital ( lmto ) method using the atomic sphere approximation ( asa ) . a detailed discussion of this method can be found elsewhere @xcite . as a key result of the calculation we obtain a gap of about @xmath148ev in the spin - up band around the fermi level ( compare also fig . [ fig : models]b ) . this gap corresponds to gap between the crystal field split cr @xmath27 ( @xmath26 ) and cr @xmath27 ( @xmath38 ) subband . on the other hand , the spin - down states are available at the fermi - level . the broad band corresponds to the hybridized cr @xmath27 ( @xmath26 ) and w @xmath71 ( @xmath26 ) levels . that is , according to the band structure calculation we expect a half - metallic behavior for the double perovskite sr@xmath6crwo@xmath1 . a similar result but with slightly larger band gaps has been reported recently by jeng _ the result of the band structure calculation is in agreement with the model assumptions made above . the cr @xmath27 ( @xmath26 ) spin - up subband is completely filled and there is a gap to the cr @xmath27 ( @xmath38 ) spin - up subband lying about 0.5ev above the fermi level . the w @xmath71 ( @xmath26 ) spin - up band also is mainly above the fermi level . the hybridized cr @xmath27 ( @xmath26 ) spin - down states and the w @xmath71 ( @xmath26 ) spin - down states form a broad spin - down band with the fermi level lying in this band . this is fully consistent with the model presented in fig . [ fig : models ] . furthermore , the band structure calculation shows that la doping is expected to add electrons in the spin - down band . in order to verify the band structure calculations experimentally , we have performed optical reflection and transmission measurements of sr@xmath6crwo@xmath1 thin films with photon energies from 0.38ev to 7ev . the srtio@xmath40 substrates , on which the epitaxial quality of the films is high ( see fig . [ tem ] ) , are unfortunately transparent only in the range of photon energies of 0.20 - 3.2ev . therefore , we have also investigated strained epitaxial sr@xmath6crwo@xmath1 films on laalo@xmath40 substrates , which are transparent from 0.17 to 5.5ev , and polycrystalline sr@xmath6crwo@xmath1 films on mgal@xmath6o@xmath40 substrates , which are transparent from to 0.22 - 6.5ev , however have a larger lattice mismatch to sr@xmath6crwo@xmath1 . on each substrate , films with thicknesses of @xmath149 , 80 , and 320 nm were investigated . almost identical optical absorption spectra were obtained for the epitaxial films , suggesting that the observed optical features are indeed related to the sr@xmath6crwo@xmath1 films , as summarized in fig . [ fig : absorption ] . the dominant optical features , which are an absorption shoulder around 1ev , and a strong increase of the absorption above 4ev , are also found for the polycrystalline samples grown on mgal@xmath6o@xmath43 . the transmission @xmath150 at 4.6ev is around 0.1% for the films with @xmath151 nm , and around 30% for the films with @xmath149 nm . the error in the calculation of the absorption coefficient @xmath152/d $ ] with the reflection @xmath153 comes mostly from uncertainties in @xmath154 , which is sufficiently small for the 320 nm films at lower energies , and for the 30 nm films at higher energies . for the films with @xmath151 nm , reflectivity oscillations with a period @xmath155ev were observed at photon energies of 1.5 - 3.5ev , indicating a refractory index @xmath156 . this is consistent with the measured reflection data and @xmath157 in this energy range . as determined from fourier transform infrared ( ftir ) transmission measurements , the films remain transparent down to 0.2ev , however , @xmath153 is increasing significantly towards lower photon energies . the optical measurements agree fairly well with the band structure calculations . the increase of the absorption coefficient of sr@xmath6crwo@xmath1 above 4ev can most probably be attributed to a charge transfer transition between the p - like spin - up and -down oxygen bands at @xmath158ev below the fermi - level into the oxygen / metal bands at + 1ev above the fermi - level . the absorption shoulder around 1ev coincides roughly with the energy gap at the fermi level in the spin - up band , and is therefore probably caused by transitions from the cr spin - up @xmath26 states below the fermi - level into the oxygen / metal bands around + 1ev above the fermi - level . unfortunately , due to the substrate absorption below 0.2ev , no optical information is available in the infrared region which would allow a more definitive statement with respect to the density of states at the fermi level and the half - metallic character of sr@xmath6crwo@xmath1 . however , from the refractory index @xmath159 , one can estimate an optical conductivity of @xmath160@xmath161 at 1ev . comparing this result with recent measurements of sr@xmath6crreo@xmath1 @xcite , one can classify sr@xmath6crwo@xmath1 as a ( very ) bad half metal . this is also in agreement with recent transport measurements @xcite . we note , however , that the transport data may be ambiguous , since the conductivity of the sr@xmath6crwo@xmath1 thin film is comparable to that of the substrate . this is for example the case for srtio@xmath40 substrates , which obtain a conductive surface layer at the reducing atmosphere of the thin film deposition process @xcite . that is , more conclusive transport data are required to settle this issue . we have performed a detailed analysis of the structural , transport , magnetic and optical properties of of the double perovskite @xmath0crwo@xmath1 with @xmath2 . in agreement with band structure calculations the double perovskite sr@xmath6crwo@xmath1 is a half - metal with a high curie - temperature above 450k . the measured saturation magnetization of @xmath162f.u . is well below the optimum value of @xmath163f.u . due to 23% of cr / w antisites . the large amount of antisites most likely is caused by the similar ionic radii of the cr@xmath10 and w@xmath21 ions resulting in a low threshold for the formation of antisites . the substitution of sr by ba and ca in the double perovskite system @xmath0crwo@xmath1 showed that the maximum curie temperature is obtained for the compound with a tolerance factor close to one . this is in agreement with a large variety of data reported in literature on other double perovskite systems @xmath98o@xmath1 . the fact that the maximum @xmath4 is obtained for a tolerance factor close to one differs from the behavior of the doped manganites where a maximum @xmath4 was found for a tolerance factor of about 0.93 . the observation that a tolerance factor @xmath39 yields the highest @xmath4 has been explained within a model , where ferromagnetism is stabilized by the energy gain contributed by the negative spin polarization on the nonmagnetic w - site due to hybridization of the cr @xmath27 ( @xmath26 ) and the w @xmath71 ( @xmath26 ) states . this hybridization is weakened by bond angles deviating from @xmath57 or equivalently a tolerance factor deviating from unity . electron doping of sr@xmath6crwo@xmath1 by partial substitution of sr@xmath9 by la@xmath10 was found to decrease both the curie temperature and the saturation magnetization . the decrease in the saturation magnetization was found to be caused both by an increase in the amount of antisites and by increasing band filling . although the decrease of @xmath4 with increasing doping can be explained qualitatively within the same model , it seems to be in conflict with double exchange type models predicting an increase of @xmath4 with increasing band filling . however , an unambiguous conclusion can not be drawn at present . the reason for that is the fact that on increasing the doping level one also obtains an increasing amount of disorder . in order to clarify this issue it is required to study systems allowing for the variation of the doping level without changing disorder . this work was supported by the deutsche forschungsgemeinschaft and the bundesministerium fr bildung und forschung ( project 13n8279 ) . the authors acknowledge fruitful discussions with m. s. ramachandra rao . r. gross , j. klein , b. wiedenhorst , c. hfener , u. schoop , j. b. philipp , m. schonecke , f. herbstritt , l. alff , yafeng lu , a. marx , s. schymon , s. thienhaus , w. mader , in _ superconducting and related oxides : physics and nanoengineering iv _ , d. pavuna and i. bosovic eds . , spie conf . proc . * vol . 4058 * ( 2000 ) , pp . 278294 . for a review see : j. b. goodenogh , j. m. longo , in : k .- h . hellwege , o. madelung ( eds . ) , _ magnetic and other properties of oxides and related compounds _ , landolt - brnstein , new series , group iii , vol . * 4 * , springer , berlin , 1970 .

the structural , transport , magnetic and optical properties of the double perovskite @xmath0crwo@xmath1 with @xmath2 have been studied . by varying the alkaline earth ion on the @xmath3 site , the influence of steric effects on the curie temperature @xmath4 and the saturation magnetization has been determined . a maximum @xmath5k was found for sr@xmath6crwo@xmath1 having an almost undistorted perovskite structure with a tolerance factor @xmath7 . for ca@xmath6crwo@xmath1 and ba@xmath6crwo@xmath1 structural changes result in a strong reduction of @xmath4 . our study strongly suggests that for the double perovskites in general an optimum @xmath4 is achieved only for @xmath8 , that is , for an undistorted perovskite structure . electron doping in sr@xmath6crwo@xmath1 by a partial substitution of sr@xmath9 by la@xmath10 was found to reduce both @xmath4 and the saturation magnetization @xmath11 . the reduction of @xmath11 could be attributed both to band structure effects and the cr / w antisites induced by doping . band structure calculations for sr@xmath6crwo@xmath1 predict an energy gap in the spin - up band , but a finite density of states for the spin - down band . the predictions of the band structure calculation are consistent with our optical measurements . our experimental results support the presence of a kinetic energy driven mechanism in @xmath0crwo@xmath1 , where ferromagnetism is stabilized by a hybridization of states of the nonmagnetic w - site positioned in between the high spin cr - sites .

You are an expert at summarizing long articles. Proceed to summarize the following text: understanding the origin of life is one of the main challenges of modern science . it is believed that some basic prebiotic chemistry could have developed in space , likely tranferring prebiotic molecules to the solar nebula , which were finally delivered to earth . studies of the chemical composition of comets have indeed reported that these objects exhibit a rich chemistry in complex organic molecules ( or coms ) that are commonly detected in the ism ( see e.g. , * ? ? ? recently , the spacecraft rosetta studied the chemical composition of the comet 67p / churyumov - gerasimenko , and found 16 different coms of prebiotic interest such as glycoladehyde ( ch@xmath6ohcho ) , formamide ( nh@xmath6cho ) @xcite , and even the simplest amino acid glycine @xcite . interestingly , a new molecule was detected by the rosetta mission with a relatively high abundance compared to other coms present in the comet : the simplest isocyanate ( methyl isocyanate , ch@xmath0nco ; * ? ? ? * ) , which like nh@xmath6cho ( the simplest amide ) contains c , n , and o atoms . isocyanates are a family of prebiotic coms that could play an key role in the synthesis of aminoacid chains known as peptides @xcite . subsequent to the detection of ch@xmath0nco in comet 67p / churyumov - gerasimenko , this molecule was also detected in the massive hot molecular cores sgrb2 n @xcite and orion kl @xcite . however , ch@xmath0nco has not been detected so far toward solar - type protostars , the environment where we expect earth - like planets to form . iras 16293@xmath82422 ( hereafter iras16293 ) is located in the @xmath9 ophiuchi cloud complex at a distance of 120 pc @xcite , and it is considered an excellent low - mass protostar template for astrochemical studies ( see , e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? its molecular spectra is rich in coms which has allowed the detection of molecular species such as glycoladehyde and formamide ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , along with deuterated molecules @xcite , and exotic species such as ch@xmath0sh @xcite . iras16293 is a binary system with components a and b separated in the plane of the sky by 5@xmath10 ( @xmath7600 au ) , and whose masses are @xmath70.5 m@xmath11 @xcite . their emission exhibits line profiles with different linewidths of up to 8 km s@xmath12 for iras 16293 a and @xmath13 2 km s@xmath12 for iras 16293 b. the narrow emission of iras 16293 b , along with its rich com chemistry , makes this object the perfect target to search for new coms . in this letter , we report the first detection of ch@xmath0nco towards the solar - type protostar iras 16293@xmath82422 b carried out with the atacama large millimeter array ( alma ) . the comparison between the derived column density and our chemical modelling suggests that ch@xmath0nco can be formed via gas - phase reactions after the evaporation of hnco from dust grains . [ cols="^,^,^,^,^,^,^,^ " , ] in this model , hnco is formed mainly via hydrogenation on the surface of dust grains , and once the temperature of the dust reaches a temperature of @xmath7100 k , hnco is thermally desorbed and incorporated into the gas phase . the gas - phase chemistry formation of ch@xmath0nco can thus proceed ( see reactions above and * ? ? ? * ) , yielding a ch@xmath0nco abundance very close to the value measured toward iras16293 b ( of @xmath14 ) in this study . since no other mechanisms are required to explain the observed abundance of this molecule , we propose that gas phase formation may be the dominant mechanism for the production of ch@xmath0nco . we also stress that the predicted abundance of ch@xmath0nco does not depend on the maximum temperature assumed in our model ( we obtain the same results for t = 100 k and t = 250 k ) , as long as this temperature is high enough ( @xmath15100 k)to induce the full evaporation of the ices and the conversion of hnco into ch@xmath0nco alternatively , ch@xmath0nco could be produced by uv photoprocessing of hnco - containing interstellar ices , and subsequently be desorbed to the gas phase @xcite . this scenario is not taken into account in our chemical model , but could be needed if any ch@xmath0nco gas - phase destruction route were included , thus requiring an additional source of ch@xmath0nco molecules . we present the first detection of ch@xmath0nco toward a solar - type protostar . the derived abundance of 7@xmath210@xmath5 is similar to that measured in hot cores , although the column density ratios with other chemically related n- , c- , o - bearing species do not show an agreement overall . the ch@xmath0nco / nh@xmath6cho column density ratio is the only one that agrees best across all environments in which ch@xmath0nco has been detected so far ( hot corinos , hot cores and comet 67p / churyumov - gerasimenko ) . the observed ch@xmath0nco abundance in iras16293 b is well reproduced by a chemical model taking into account only the gas - phase formation of this species , although an origin in the solid phase remains to be explored . alma is a partnership of eso ( representing its member states ) , nsf ( usa ) and nins ( japan ) , together with nrc ( canada ) , nsc and asiaa ( taiwan ) , and kasi ( republic of korea ) , in co - operation with the republic of chile . the joint alma observatory is operated by eso , aui / nrao and naoj this research was partially financed by the spanish mineco under project aya2014 - 60585-p , by the italian ministero dell istruzione , universit e ricerca , through the grant progetti premiali 2012 ialma ( cup c52i13000140001 ) , and by the gothenburg centre for advanced studies in science and technology , where the re - calibration and re - imaging of all the alma archive data on iras16293 - 2422 was carried out as part of the gocas program `` origins of habitable planets '' . d . benefited from a fpi grant from spanish mineco . i.j .- s . acknowledges the financial support received from the stfc through an ernest rutherford fellowship ( proposal number st / l004801/1 ) . p . acknowledges partial support by the mineco under grants fis2012 - 39162-c06 - 01 , esp2013 - 47809-c03 - 01 , and esp2015 - 65597-c4 - 1 . altwegg , k. , balsiger , h. , bar - nun , a. , et al . 2015 , science , 347 , 6220 , id.1261952 + belloche , a. , menten , k.m . , comito , c. , et al . 2008 , a&a , 482 , 1 , 179 + beltrn , m. t. , codella , c. ; viti , s. , neri , r. , & cesaroni , r. 2009 , apj , 690 , 2 , l93 + bisschop , s.e . , jorgensen , j.k . , bourke , t.l . , bottinelli , s. , & van dishoeck , e.f . 2008 , a&a , 488 , 3 , 959 + biver , n. , bockele - morvan , d. , debout , v. , et al . 2014 , a&a , 516 , a109 + bottinelli , s. , ceccarelli , c. , neri , r. , et al . 2004 , apj , 617 , 1 , l69 + caux , e. , kahane , c. , castets , a. , et al . 2011 , a&a , 532 , a23 + cazaux , s. , tielens , a.g.g.m . , ceccarelli , c. , et al . 2003 , apj , 593 , 1 , l51 + cernicharo , j. , kisiel , z. , tercero , b. , et al . 2016 , a&a , 587 , l4 + coutens , a. , jorgensen , j. , van der wiel , m.h.d . , et al . 2016 , a&a , 590 , l6 + coutens , a. , vastel , c. , caux , e. , et al . 2012 , a&a , 539 , a132 + goesmann , f. , rosenbauer , h. , bredehft , j.h . , 2015 , science , 346 , 6247 , id.0689 + halfen , d.t . , ilyushin , v.v . , & ziurys , l.m . 2015 , apjl , 8212 , l5 + hasegawa , t. i. , herbst , e. , & leung , c. m. 1992 , , 82 , 167 hollis , j. m. , lovas , f. j. , & jewell , p. r. 2000 , apj , 540 , 2 , l107 + jorgensen , j.k . , bourke , t.l . , nguyen luong , q. , & takakuwa , s. 2011 , a&a , 534 , a100 + jorgensen , j.k . , favre , c. , bisschop , s.e . , bourke , t.l . , et al . 2012 , apjl , 757 , 1 , l4 + jorgensen , j.k . , van der wiel , m.h.d . , coutens , a. , et al . 2016 , a&a , 595 , a117 + kahane , c. , ceccarelli , c. , faure , a. , & caux , e. 2013 , apjl , 763 , 2 , l38 + kasten , w. , & dreizler , h. 1986 , z. natur . 41a , 637 + koput , j. 1986 , jmosp , 115 , 131 + kuan , y .- j . , huang , h .- c . , charnley , s.b . , et al . 2004 , 616 , 1 , l27 + loinard , l. , torres , r.m . , mioduszewski , a.j . , & rodrguez , l.f . 2008 , apjl , 675 , 1 , l29 + looney , l.w . , mundy , l.g . , & welch , w.j . 2000 , apj , 529 , 1 , 477 + lpez - sepulcre , a. , jaber , a.a . , mendoza , e. , et al . 2015 , mnras , 449 , 3 , 2438 + lykke , j.m . , coutens , a. , jorgensen , j. , et al . 2016 , in press . rivilla , v.m . , fontani , f. , beltrn , m.t . , et al . 2016 , apj , 826 , 2 + rivilla , v. m. and beltrn , m. t. , cesaroni , r. , et al . , 2016 , a&a , in print , arxiv:1608.07491 + roberts , j. f. , rawlings , j. m. c. , viti , s. , & williams , d. a. 2007 , , 382 , 733

we report the first detection of the prebiotic complex organic molecule ch@xmath0nco in a solar - type protostar , iras16293 - 2422 b. this species is one of the most abundant complex organic molecule detected on the surface of the comet 67p / churyumov - gerasimenko , and in the insterstellar medium it has only been found in hot cores around high - mass protostars . we have used multi - frequency alma observations from 90 ghz to 350 ghz covering 11 unblended transitions of ch@xmath0nco and 40 more transitions that appear blended with emission from other molecular species . our local thermodynamic equilibrium analysis provides an excitation temperature of 232@xmath141 k and a column density of ( [email protected])@xmath210@xmath3 @xmath4 , which implies an abundance of ( 7@xmath12)@xmath210@xmath5 with respect to molecular hydrogen . the derived column density ratios ch@xmath0nco / hnco , ch@xmath0nco / nh@xmath6cho , and ch@xmath0nco / ch@xmath0cn are @xmath70.3 , @xmath70.8 , and @xmath70.2 , respectively , which are different from those measured in hot cores and in comet 67p / churyumov - gerasimenko . our chemical modelling of ch@xmath0nco reproduces well the abundances and column density ratios ch@xmath0nco / hnco and ch@xmath0nco / nh@xmath6cho measured in iras16293 - 2422 b , suggesting that the production of ch@xmath0nco could occur mostly via gas - phase chemistry after the evaporation of hnco from dust grains .

You are an expert at summarizing long articles. Proceed to summarize the following text: quantum field theories have been widely accepted in the physics community , mainly because of their their well - tested predictions . one of the famous numbers predicted by quantum electrodynamics is the electromagnetic moment of the electron which has been tested up to a previously unencountered precision . unfortunately , quantum field theories are percepted with some suspicion by mathematicians . this is mainly due to the appearance of divergences when naively computing probability amplitudes . these _ infinities _ have to be dealt with properly by an apparently obscure process called renormalization . nevertheless , mathematical interest has been changing lately in favour of quantum field theories , the general philosophy being that such a physically accurate theory should have some underlying mathematically rigorous description . one of these interests is in the process of renormalization , and has been studied in the context of hopf algebras @xcite . of course , the process of renormalization was already quite rigorously defined by physicists in the early second half of the previous century however , the structure of a coproduct describing how to subtract divergence really clarified the process . one could argue though that since the elements in the hopf algebra are individual feynman graphs , it is a bit unphysical . rather , one would like to describe the renormalization process on the level of the 1pi green s functions , since these correspond to actual physical processes . especially for ( non - abelian ) gauge theories , the graph - by - graph approach of for instance the bphz - procedure is usually replaced by more powerful methods based on brst - symmetry and the zinn - justin equation ( and its far reaching generalization : the batalin - vilkovisky formalism ) . they all involve the 1pi green s functions or even the full effective action that is generated by them . the drawback of these latter methods , is that they rely heavily on functional integrals and are therefore completely formal . one of the advantages of bphz - renormalization is that if one accepts the perturbative series of green s function in terms of feynman graphs as a starting point , the procedure is completely rigorous . of course , this allowed the procedure to be described by a mathematical structure such as a hopf algebra . in this article , we prove some of the results on green s functions starting with the hopf algebra of feynman graphs for non - abelian gauge theories . we derive the existence of hopf subalgebras generated by the 1pi green s functions . we do this by showing that the coproduct takes a closed form on these green s functions , thereby relying heavily on a formula that we have previously derived @xcite . already in @xcite hopf subalgebras were given for any connected graded hopf algebra as solutions to dyson - schwinger equations . it turned out that there was a close relation with hochschild cohomology . it was argued by kreimer in @xcite that for the case of non - abelian gauge theories the existence of hopf subalgebras follows from the validity of the slavnov taylor identities _ inside _ the hopf algebra of ( qcd ) feynman graphs . we now fully prove this claim by applying a formula for the coproduct on green s functions that we have derived before in @xcite . in fact , that formula allowed us to prove compatibility of the slavnov taylor identities with the hopf algebra structure . this paper is organized as follows . in section 2 , we start by giving some background from physics . of course , this can only be a quick _ lifting of the curtain _ and is meant as a motivation for the present work . in section 3 , we make precise our setup by defining the hopf algebra of feynman graphs and introduce several combinatorial factors associated to such graphs . we put the process of renormalization in the context of a birkhoff decomposition . section 4 contains the derivation of the hopf algebra structure at the level of green s functions , rather then the individual feynman graphs . we will encounter the crucial role that is played by the so - called slavnov taylor identities . we start by giving some background from physics and try to explain the origin of feynman graphs in the perturbative approach to quantum field theory . we understand _ probability amplitudes for physical processes as formal expansions in feynman amplitudes _ , thereby avoiding the use of path integrals . we make this more explicit by some examples taken from physics . the interaction of the photon with the electron in quantum electrodynamics ( qed ) is described by the following expansion , @xmath0 here all graphs appear that can be built from the vertex that connects a wiggly line ( the photon ) to two straight lines ( the electron ) . the quartic gluon self - interaction in quantum chromodynamics is given by @xmath1 this expansion involves the gluon vertex of valence 3 and 4 ( wiggly lines ) , as well as the quark - gluon interaction ( involving two straight lines ) we shall call these expansions * green s functions*. of course , this names originates from the theory of partial differential equations and the zeroth order terms in the above expansions are in fact green s functions in the usual sense . we use the notation @xmath2 and @xmath3 for the green s function , indicating the external structure of the graphs in the above two expansions , respectively . from these expansions , physicists can actually derive numbers , giving the probability amplitudes mentioned above . the rules of this game are known as the feynman rules ; we briefly list them for the case of quantum electrodynamics . feynman rules for non - abelian gauge theories can be found in most standard textbooks on quantum field theory ( see for instance @xcite ) . + assigning momentum @xmath4 to each edge of a graph , we have : @xmath5 \parbox{30pt } { \begin{fmfgraph*}(30,10 ) \fmfleft{l } \fmfright{r } \fmf{plain , label=$k$}{l , r } \end{fmfgraph * } } \hspace{3mm}&= \frac{1}{\gamma^\mu k_\mu + m}\\[7 mm ] \parbox{30pt } { \begin{fmfgraph*}(30,30 ) \fmfset{wiggly_len}{4pt } \fmfset{wiggly_slope}{70 } \fmfleft{l } \fmfright{r1,r2 } \fmf{photon}{l , v } \fmf{plain}{r1,v } \fmf{plain}{v , r2 } \fmflabel{$k_1$}{l } \fmflabel{$k_2$}{r1 } \fmflabel{$k_3$}{r2 } \end{fmfgraph * } } \hspace{3 mm } & = -i e \gamma^\mu \delta(k_1+k_2 + k_3)\\[5mm]\end{aligned}\ ] ] here , @xmath6 is the electron charge , @xmath7 the electron mass and @xmath8 are @xmath9 dirac gamma matrices ; they satisfy @xmath10 . also , @xmath11 is an infrared regulator and @xmath12 is the so - called gauge fixing parameter . in addition to the above assignments , one integrates the above internal momenta @xmath4 ( for each internal edge ) over @xmath13 . consider the following electron self - energy graph + ( 150,60 ) according to the feynman rules , the amplitude for this graph is @xmath14 with summation over repeated indices understood . the alert reader may have noted that the above improper integral is actually not well - defined . this is the typical situation happening for most graphs and are the famous divergences in perturbative quantum field theory . this apparent failure can be resolved , leading eventually to spectacularly accurate predictions in physics . the theory that proposes a solution to these divergences is called _ renormalization_. this process consists of two steps . firstly , one introduces a _ regularization parameter _ that controls the divergences . for instance , in _ dimensional regularization _ one integrates in @xmath15 dimensions instead of in @xmath16 , with @xmath17 a complex number . adopting certain rules holds for complex dimension @xmath18 as well . indeed , using schwinger parameters , or , equivalently , the laplace transform , one can write @xmath19 as the integral over @xmath20 of @xmath21 . ] for this integration in complex dimensions , one obtains for instance for the above integral : @xmath22 where the @xmath23 on the left - hand - side is the graph and the @xmath23 on the right - hand - side is the gamma function from complex analysis . moreover , @xmath24 is a polynomial in the external momentum @xmath25 . the previous divergence has been translated into a pole of the gamma function at @xmath26 and we have thus obtained a control on the divergence . the second step in the process of renormalization is _ subtraction_. we let @xmath27 be the projection onto the pole part of laurent series in @xmath17 , i.e. , @xmath28 = \sum_{n<0 } a_n z^n\ ] ] more generally , we have a projection on the divergent part in the regularizing parameter . this is the origin of the study of rota - baxter algebras in the setting of quantum field theories @xcite . we will however restrict ourselves to dimensional regularization , which is a well suited regularization for gauge theories . for the above graph @xmath23 , we define the * renormalized amplitude * @xmath29 by simply subtracting the divergent part , that is , @xmath30 $ ] . clearly , the result is finite for @xmath31 . more generally , a graph @xmath23 might have subgraphs @xmath32 which lead to sub - divergences in @xmath33 . the so - called * bphz - procedure * ( after its inventors bogoliubov , parasiuk , hepp and zimmermann ) provides a way to deal with those sub - divergences in a recursive manner . it gives for the * renormalized amplitude * : [ bphz ] @xmath34 where @xmath35 is the so - called * counterterm * defined recursively by @xmath36\end{aligned}\ ] ] the two sums here are over all subgraphs in a certain class ; we will make this more precise in the next section . we now focus on a special class of quantum field theories quantum gauge theories which are of particular interest for real physical processes . without going into details on what classical gauge field theories are , we focus on the consequences on the quantum side of the presence of a classical gauge symmetry . such a gauge symmetry acts ( locally ) on the classical fields by * gauge transformations * and these transformations form a group , the gauge group . this group is typically infinite dimensional , since it consists of functions on space - time taking values in a lie group . for quantum electrodynamics this lie group is abelian and just @xmath37 , for quantum chromodynamics the theory of gluons and quarks it is @xmath38 . when ( perturbatively ) quantizing the gauge theory , one is confronted with this extra infinity . a way to handle it is by _ fixing the gauge _ , in other words , choosing an orbit under the action of the gauge group . all this can be made quite precise in _ brst - quantization_. although in this process the gauge symmetry completely disappears , certain identities between green s functions appear . this is a purely ` quantum property ' and therefore interesting to study . in addition , being identities between full green s functions , it is interesting with a view towards nonperturbative quantum field theory . for quantum electrodynamics , the identities are simple and linear in the green s functions : @xmath39 these are known as * ward identities * since they were first derived by ward in @xcite . the apparent mismatch between the number of external lines on the left and right - hand - side is resolved because the vertex graphs are considered at _ zero momentum transfer_. this means that the momentum on the photon line is evaluated at @xmath40 . for non - abelian gauge theories such as quantum chromodynamics ( qcd ) , the identities are quadratic in the fields and read : @xmath41 the dotted and straight line here corresponds to the ghost and quark , respectively . after their inventors , they are called the * slavnov taylor identities * @xcite . the importance of these identities lie in the fact that they are compatible with renormalization under the condition that gauge invariance is compatible with the regularization procedure . in fact , it turns out that dimensional regularization satisfies this requirement , see for instance section 13.1 of @xcite . as a consequence , the slavnov - taylor identities hold after replacing @xmath42 by @xmath43 or @xmath35 in the above formula . for instance , in the case of quantum electrodynamics one obtains the identity @xmath44 actually derived by ward , where @xmath45 and @xmath46 . for quantum chromodynamics on the other hand , one derives the formulae @xmath47 where the notation is as above : @xmath48 . the above formula can be readily obtained from the above slavnov taylor identities after replacing @xmath42 by @xmath35 . they are the key to proving renormalizability of non - abelian gauge theories , let us try to sketch this argument . first of all , the different interactions that are present in the theory can be weighted by a coupling constant . for example , in qcd there are four different interactions : gluon - quark , gluon - ghost , cubic and quartic gluon self - interaction . all of these come with their own coupling constants and gauge invariance ( or rather , brst - invariance ) requires them to be identical . in the process of renormalization , the coupling constants are actually not constant and depend on the energy scale . this is the _ running of the coupling constant _ and is the origin of the renormalization group describing how they change . for qcd , the four coupling constants @xmath49 are expressed in terms of the original coupling constant @xmath50 as @xmath51 we see that the slavnov taylor identities guarantee that _ the four coupling constants remain equal _ after renormalization . the above compatibility of renormalization with the slavnov taylor identities is usually derived using the zinn - justin equation ( or the more general bv - formalism ) relying heavily on path integral techniques . our goal in the next sections is to derive this result taking the formal expansion of the green s functions in feynman graphs as a starting point . we will work in the setting of the connes - kreimer hopf algebra of renormalization . we suppose that we have defined a ( renormalizable ) quantum field theory and specified the possible interactions between different types of particles . we indicate the interactions by vertices and the propagation of particles by lines . this leads us to define a set @xmath52 of vertices and edges ; for qed we have @xmath53 whereas for qcd we have , @xmath54 we stress for what follows that it is not necessary to define the set explicitly . a * feynman graph * is a graph built from vertices in @xmath55 and edges in @xmath56 . naturally , we demand edges to be connected to vertices in a compatible way , respecting the type of vertex and edge . as opposed to the usual definition in graph theory , feynman graphs have no external vertices , they only have external lines . we assume those lines to carry a labeling . an * automorphism * of a feynman graph is a graph automorphism leaving the external lines fixed and respects the types of vertices and edges . this definition is motivated by the fact that the external lines correspond physically to particles prepared for some collision experiment the interior of the graph and those lines are thus fixed . the order of the group of automorphisms @xmath57 of a graph @xmath23 is called its * symmetry factor * and denoted by @xmath58 . let us give two examples : @xmath59 for disconnected graphs , the symmetry factor is given recursively as follows . let @xmath60 be a connected graph ; we set @xmath61 with @xmath62 the number of connected components of @xmath23 that are isomorphic to @xmath60 . we define the * residue * @xmath63 of a graph @xmath23 as the vertex or edge the graph reduces to after collapsing all its internal vertices and edges to a point . for example , we have : @xmath64 henceforth , we will _ restrict to graphs with residue in @xmath43 _ ; these are the relevant graphs to be considered for the purpose of renormalization . for later use , we introduce another combinatorial quantity , which is the * number of insertion places * @xmath65 for the graph @xmath66 in @xmath23 . it is defined as the number of elements in the set of vertices and internal edges of @xmath23 of the form @xmath67 . for disconnected graphs @xmath68 , the number @xmath65 counts the number of @xmath69 of disjoint insertion places of the type @xmath70 . we exemplify this quantity by @xmath71 here , one allows multiple insertions of edge graphs ( i.e. a graph with residue in @xmath56 ) on the same edge ; the underlying philosophy is that insertion of an edge graph creates a new edge . for the definition of the hopf algebra of feynman graphs @xcite , we restrict to * one - particle irreducible * ( 1pi ) feynman graphs . these are graphs that are not trees and can not be disconnected by cutting a single internal edge . for example , all graphs in this paper are one - particle irreducible , _ except _ the following which is one - particle reducible : @xmath72 connes and kreimer then defined the following hopf algebra . we refer to the appendix for a quick review on hopf algebras . the hopf algebra @xmath73 of feynman graphs is the free commutative @xmath74-algebra generated by all 1pi feynman graphs , with counit @xmath75 unless @xmath76 , in which case @xmath77 , coproduct , @xmath78 where the sum is over disjoint unions of subgraphs with residue in @xmath43 . the antipode is given recursively by , @xmath79 two examples of this coproduct , taken from qed , are : @xmath80 the above hopf algebra is an example of a connected graded hopf algebra , i.e. @xmath81 , @xmath82 and @xmath83 indeed , the hopf algebra of feynman graphs is graded by the * loop number @xmath84 * of a graph @xmath23 ; then @xmath85 consists of rational multiples of the empty graph , which is the unit in @xmath73 , so that @xmath86 . one can enhance the feynman graphs with an external structure . this involves the external momenta on the external lines and can be formulated mathematically by distributions , see for instance @xcite . the case of quantum electrodynamics has been worked out in detail in @xcite . we now demonstrate how to obtain equation for the renormalized amplitude and the counterterm for a graph as a birkhoff decomposition in the group of characters of @xmath73 . let us first recall the definition of a birkhoff decomposition . we let @xmath87 be a loop with values in an arbitrary complex lie group @xmath88 , defined on a smooth simple curve @xmath89 . let @xmath90 be the two complements of @xmath35 in @xmath91 , with @xmath92 . a * birkhoff decomposition * of @xmath93 is a factorization of the form @xmath94 where @xmath95 are ( boundary values of ) two holomorphic maps on @xmath90 , respectively , with values in @xmath88 . this decomposition gives _ a natural way to extract finite values from a divergent expression_. indeed , although @xmath96 might not holomorphically extend to @xmath97 , @xmath98 is clearly finite as @xmath31 . we now look at the group @xmath99 of @xmath100-valued characters of a connected graded commutative hopf algebra @xmath73 , where @xmath100 is the field of convergent laurent series in @xmath17 . represented by @xmath73 in the category of commutative algebras . in other words , @xmath101 and @xmath102 are the @xmath100-points of the group scheme . ] the product , inverse and unit in the group @xmath102 are defined by the respective equations : @xmath103 for @xmath104 . we claim that a map @xmath105 is in one - to - one correspondence with loops @xmath93 on an infinitesimal circle around @xmath26 and values in @xmath106 . indeed , the correspondence is given by @xmath107 and to give a birkhoff decomposition for @xmath93 is thus equivalent to giving a factorization @xmath108 in @xmath102 . it turns out that for graded connected commutative hopf algebras such a factorization exists . let @xmath73 be a graded connected commutative hopf algebra . the birkhoff decomposition of @xmath87 ( given by an algebra map @xmath109 ) exists and is given dually by @xmath110\ ] ] and @xmath111 . the graded connected property of @xmath73 assures that the recursive definition of @xmath112 actually makes sense . in the case of the hopf algebra of feynman graphs defined above , the factorization takes the following form : @xmath113\\ \phi_+(\gamma)&=\phi(\gamma ) + \phi_-(\gamma ) + \sum_{\gamma \subsetneq \gamma } \phi_-(\gamma ) \phi(\gamma/\gamma)\end{aligned}\ ] ] the key point is now that the feynman rules actually define an algebra map @xmath114 by assigning to each graph @xmath23 the regularized feynman rules @xmath33 , which are laurent series in @xmath17 . when compared with equations one concludes that the algebra maps @xmath115 and @xmath116 in the birkhoff factorization of @xmath42 are precisely the renormalized amplitude @xmath43 and the counterterm @xmath35 , respectively . summarizing , we can write the bphz - renormalization as the birkhoff decomposition @xmath117 of the map @xmath118 dictated by the feynman rules . although the above construction gives a very nice geometrical description of the process of renormalization , it is a bit unphysical in that it relies on individual graphs . rather , as mentioned before , in physics the probability amplitudes are computed from the full expansion of green s functions . individual graphs do not correspond to physical processes and therefore a natural question to pose is how the hopf algebra structure behaves at the level of the green s functions . we will see in the next section that they generate hopf subalgebras , i.e. the coproduct closes on green s functions . in proving this , the slavnov taylor identities turn out to play an essential role . for a vertex or edge @xmath119 we define the * 1pi green s function * by @xmath120 where the sign is @xmath121 if @xmath122 is a vertex and @xmath123 if it is an edge . the restriction of the sum to graphs @xmath23 at loop order @xmath124 is denoted by @xmath125 . the coproduct takes the following form on the 1pi green s functions : @xmath126 with the sum over @xmath66 over all disjoint unions of 1pi graphs . the sketch of the proof is as follows first , one writes the coproduct @xmath127 as a sum of maps @xmath128 where these maps only detects subgraphs isomorphic to @xmath66 . one then proves the above formula for @xmath128 with @xmath66 a 1pi graph using simply the orbit - stabilizer theorem for the automorphism group of graphs . finally , writing @xmath129 in terms of @xmath128 and @xmath130 one proceeds by induction to derive the above expression . one observes that the coproduct does not seem to close on green s functions due to the appearance of the combinatorial factor @xmath65 . let us try to elucidate this and compute these factors explicitly . let @xmath131 be the number of vertices / internal edges of type @xmath122 appearing in @xmath23 , for @xmath119 . moreover , let @xmath132 be the number of connected components of @xmath66 with residue @xmath122 . since insertion of a vertex graph ( i.e. with residue in @xmath55 ) on a vertex @xmath133 in @xmath23 prevents a subsequent insertion at @xmath133 of a vertex graph with the same residue , whereas insertion of an edge graph ( i.e. with residue in @xmath56 ) creates two new edges and hence two insertion places for a subsequent edge graph , we find the following expression , @xmath134 indeed , the binomial coefficients arise for each vertex @xmath133 since we are choosing @xmath135 out of @xmath136 whereas for an edge @xmath6 we choose @xmath137 out of @xmath138 _ with repetition_. we claim that this counting enhances our formula to the following @xmath139 before proving this , we explain the meaning of the inverse of green s functions in our hopf algebra . since any green s function starts with the identity , we can surely write its inverse formally as a geometric series . recall that the hopf algebra is graded by loop number . hence , the inverse of a green s function at a fixed loop order is in fact well - defined ; it is given by restricting the above formal series expansion to this loop order . in the following , also rational powers of green s functions will appear ; they will be understood in like manner . let us simplify a little and consider a scalar field theory with just one type of vertex and edge , i.e. @xmath140 . we consider the sum @xmath141 naturally split into a sum over vertex and edge graphs . we have also inserted the above combinatorial expression for the number of insertion places . next , we write @xmath142 and try factorize the sum over @xmath143 into a sum over @xmath144 ( connected ) and @xmath145 . some care should be taken here regarding the combinatorial factors but let us ignore them for the moment . in fact , if we fix the number of connected components @xmath146 of @xmath143 in the sum to be @xmath147 we can write @xmath148 with @xmath144 a connected graph . here , we have simply inserted 1 , @xmath149 which follows directly from the definition of @xmath150 as the number of connected components of @xmath143 isomorphic to @xmath144 . now , by definition @xmath151 for a connected graph @xmath144 so that we obtain for the above sum @xmath152 by applying the same argument @xmath147 times . recall also the definition of the green s function @xmath153 from eq . . a similar argument applies to the edge graphs , leading to a contribution @xmath154 , with @xmath155 the number of connected components of @xmath156 . when summing over @xmath147 and @xmath155 , taking also into account the combinatorial factors , we obtain : @xmath157 the extension to the general setting where the set @xmath43 contains different types of vertices and edges is straightforward . an additional counting of the number of edges and numbers of vertices in @xmath23 gives the following relations : @xmath158 where @xmath159 is the number of lines ( of type @xmath6 ) attached to @xmath119 . for instance @xmath160 equals 2 if @xmath6 is an electron line and 1 if @xmath6 is a photon line . one checks the above equality by noting that the left - hand - side counts the number of internal half lines plus the external lines which are connected to the vertices that appear at the right - hand - side , taken into account their valence . with this formula , we can write eq . as @xmath161 this is still not completely satisfactory since it involves the number of vertices in @xmath23 which prevents us from separating the summation of @xmath23 from the other terms . we introduce the following notation for the fraction of green s functions above : @xmath162 with @xmath163 the total number of edges attached to @xmath133 . before we state our main theorem , let us motivate the definition of these elements in the case of qcd . in qcd , there are four vertices and the corresponding elements @xmath164 are given by , @xmath165 the combinations of the green s functions are identical to those appearing in formulas . indeed , as we will see in a moment , setting them equal in @xmath73 is compatible with the coproduct . although motivated by the study of the slavnov taylor identities in non - abelian gauge theories , the following result holds in complete generality . the ideal @xmath166 is a hopf ideal , i.e. @xmath167 let us write the above eq . in terms of the @xmath164 s : @xmath168 in this expression , @xmath169 appears with a certain power , say @xmath170 , and we can replace @xmath171 by @xmath172 as long as we add the term @xmath173 . this latter term can be factorized as @xmath174 times a certain polynomial in @xmath164 and @xmath169 and thus corresponds to an element in @xmath175 . as a result , we can replace all @xmath169 s by @xmath164 for some fixed @xmath133 modulo addition of terms in @xmath176 . the second step uses the following equality between vertices and edges : @xmath177 in terms of the loop number @xmath178 and residue @xmath122 of @xmath23 . the equality follows by an easy induction on the number of internal lines of @xmath23 ( cf . @xcite ) . finally , one can separate the sum over @xmath23 at a fixed loop order to obtain @xmath179 understood modulo terms in @xmath176 . from this one derives that @xmath180 lies in @xmath181 as follows . let us first find a more convenient choice of generators of @xmath175 . by induction , one can show that @xmath182 where @xmath183 is a ( formally ) invertible series in @xmath164 and @xmath169 . in fact , it starts with a nonzero term of order zero . by multiplying out both denominators in the @xmath164 and @xmath169 , we arrive at the following set of ( equivalent ) generators of @xmath175 @xmath184 with @xmath185 . a little computation shows that the first leg of the tensor product in the coproduct on these two terms coincide , using eq . . as a consequence , one can combine these terms to obtain an element in @xmath186 modulo the aforementioned terms in @xmath187 needed to arrive at . as a consequence , we can work on the * quotient hopf algebra * @xmath188 . suppose we work in the case of a non - abelian gauge theory such as qcd , with the condition that the regularization procedure is compatible with gauge invariance such as dimensional regularization ( see also @xcite ) . in such a case , the map @xmath118 defined by the ( regularized ) feynman rules vanishes on the ideal @xmath175 because of the slavnov taylor identities . hence , it factors through an algebra map from @xmath189 to the field @xmath100 . since @xmath189 is still a commutative connected hopf algebra , there is a birkhoff decomposition @xmath190 as before _ with @xmath35 and @xmath43 algebra maps from @xmath189 to @xmath100_. this is the crucial point , because it implies that both @xmath35 and @xmath43 vanish automatically on @xmath175 . in other words , both the counterterms and the renormalized amplitudes satisfy the slavnov taylor identities . in particular , the @xmath191 s are the terms appearing in eq . which coincide because @xmath192 . note also that in @xmath189 expression holds so that the coproduct closes on green s functions , i.e. they generate hopf subalgebras . as a corollary to this , we can derive a generalization of dyson s formula originally derived for qed @xcite . it provides a relation between the renormalized green s function written in terms of the coupling constant @xmath50 and the unrenormalized green s function written in terms of the bare coupling constant defined by @xmath193 for some @xmath194 . the following analogue of dyson s formula for qed holds in general , @xmath195 where @xmath196 . this follows from an application of @xmath197 to @xmath198 using eq . while counting the number of times the coupling constant @xmath50 appears when applying the feynman rules to a graph with residue @xmath122 and loop number @xmath178 . in fact , this number is @xmath199 which is also @xmath200 as noted before . for convenience , let us briefly recall the definition of a ( commutative ) hopf algebra . it is the dual object to a group and , in fact , there is a one - to - one correspondence between groups and commutative hopf algebras . let @xmath88 be a group with product , inverse and identity element . we consider the algebra of representative functions @xmath201 . this class of functions is such that @xmath202 . for instance , if @xmath88 is a ( complex ) matrix group , then @xmath203 could be the algebra generated by the coordinate functions @xmath204 so that @xmath205 are just the @xmath206th entries of the matrix @xmath50 . let us see what happens with the product , inverse and identity of the group on the level of the algebra @xmath207 . the multiplication of the group can be seen as a map @xmath208 , given by @xmath209 . since dualization reverses arrows , this becomes a map @xmath210 called the _ coproduct _ and given for @xmath211 by @xmath212 the property of associativity on @xmath88 becomes _ coassociativity _ on @xmath73 : @xmath213 stating simplify that @xmath214 . the inverse map @xmath221 , becomes the _ antipode _ @xmath222 , defined by @xmath223 . the property @xmath224 , becomes on the algebra level : @xmath225 where @xmath226 denotes pointwise multiplication of functions in @xmath73 . a hopf algebra @xmath73 is an algebra @xmath73 , together with two algebra maps @xmath227 ( coproduct ) , @xmath228 ( counit ) , and a bijective @xmath229-linear map @xmath230 ( antipode ) , such that equations are satisfied . if the hopf algebra @xmath73 is commutative , we can conversely construct a ( complex ) group from it as follows . consider the collection @xmath88 of multiplicative linear maps from @xmath73 to @xmath229 . we will show that @xmath88 is a group . indeed , we have the _ convolution product _ between two such maps @xmath231 defined as the dual of the coproduct : @xmath232 for @xmath233 . one can easily check that coassociativity of the coproduct ( eq . ) implies associativity of the convolution product : @xmath234 . naturally , the counit defines the unit @xmath6 by @xmath235 . clearly @xmath236 follows at once from eq . . finally , the inverse is constructed from the antipode by setting @xmath237 for which the relations @xmath238 follow directly from equation . with the above explicit correspondence between groups and commutative hopf algebras , one can translate practically all concepts in group theory to hopf algebras . for instance , a subgroup @xmath239 corresponds to a _ hopf ideal _ @xmath240 in that @xmath241 and viceversa . the conditions for being a subgroup can then be translated to give the following three conditions defining a hopf ideal @xmath175 in a commutative hopf algebra @xmath73 @xmath242

we describe the hopf algebraic structure of feynman graphs for non - abelian gauge theories , and prove compatibility of the so - called slavnov taylor identities with the coproduct . when these identities are taken into account , the coproduct closes on the green s functions , which thus generate a hopf subalgebra . graphs - leipzig

You are an expert at summarizing long articles. Proceed to summarize the following text: the mauritius radio telescope ( mrt ) @xcite is a fourier synthesis , t - shaped non - coplanar array operating at 151.5 mhz . the telescope was built to fill the gap in the availability of deep sky surveys at low radio frequencies in the southern hemisphere . the aim of the survey with mrt is to contribute to the database of southern sky sources in the declination ( @xmath3 ) range @xmath4 to @xmath5 , covering the entire right ascension ( @xmath6 ) , with a synthesised beam of @xmath7 and an expected point source sensitivity ( 1-@xmath0 ) of @xmath8 mjy beam@xmath9 . the _ zenith angle _ @xmath10 is given by @xmath11 , where , @xmath12 @xmath13 is the latitude of mrt . mrt has been designed to be the southern - sky equivalent of the cambridge 6c survey at 151.5 mhz @xcite . the next generation radio telescopes , like the low frequency array ( lofar ) and the murchison widefield array ( mwa ) , that are being built are low frequency arrays ; clearly indicating a renewed interest in metre - wavelength astronomy . the key astrophysical science drivers include acceleration , turbulence and propagation in the galactic interstellar medium , exploring the high redshift universe and transient phenomenon , as well as searching for the redshifted signature of neutral hydrogen from the cosmologically important epoch of reionisation ( eor ) . the surveys made using such arrays will provide critical information about foregrounds which will also provide a useful database for both extragalactic and galactic sources . mrt survey at 151.5 mhz is a step in that direction and , in addition , will provide the crucial sky model for calibration . imaging at mrt is presently done only on the meridian to minimise the problems of non - coplanarity . a two - dimensional ( 2-d ) image in @xmath6-@xmath14 coordinates is formed by stacking one - dimensional ( 1-d ) images on the meridian at different sidereal times . images of @xmath15 a steradian @xmath16 of the southern sky , with an rms noise in images of @xmath17 mjy beam@xmath9 ( 1-@xmath0 ) , were produced by @xcite . a suite of programs developed in - house was used to reduce @xmath18 hours of the survey data ( a quarter of the total @xmath19 hours observed over a span of @xmath20 years ) . the deconvolved images and a source catalogue of @xmath21 sources were published by @xcite . systematics in positional errors were found when the positions of sources common to mrt catalogue and the molonglo reference catalogue ( mrc ) @xcite were compared . @xcite treated the systematics in errors in @xmath6 and @xmath14 independently . by estimating two separate 1-d least - squares fits for errors in @xmath6 and @xmath14 the systematics were corrected only in the source catalogue . however , errors remained in the images which impede usefulness of mrt images for multi - wavelength analysis of sources . in addition , the source of errors was not investigated . at mrt , the visibility data is processed through several complex stages of data reduction specific to the array , especially , arising due to its non - coplanarity @xcite . it was therefore decided to correct for errors in the image domain and avoid re - processing the visibility data . this paper describes the application of 2-d homography , a technique ubiquitous in the computer vision and graphics community , to correct the errors in the image domain . homography is used to estimate a transformation matrix ( which includes rotation , translation and non - isotropic scaling ) that accounts for positional errors in the linearly gridded 2-d images . in our view , this technique will be of relevance to the new generation radio telescopes where , owing to huge data rates , only images after a certain integration would be recorded as opposed to raw visibilities @xcite . this paper also describes our investigations tracing the positional errors to errors in the array geometry used for imaging . our hypothesis on the array geometry , its subsequent confirmation endorsed by re - estimation of the array geometry and its effect on the images are also described . the rest of the paper is organised as follows . section [ s : poserror ] compares positions of sources common to mrt catalogue and mrc . the 2-d homography estimation is briefly described in section [ s : homography ] . section [ s : scheme ] presents the correction scheme and typical results . the re - estimation of mrt array geometry is described in section [ s : arraygeometry ] . finally , we summarise and present our conclusions in section [ s : conclusions ] . the positions of sources common to mrt catalogue and mrc were compared . we used mrc because of its overlap with mrt survey , its proximity in frequency compared to other reliable catalogues available and , comparable resolution @xmath22 . moreover , for sources of listed flux density @xmath23 jy ( at 408 mhz ) the catalogue is reported to be substantially complete and , the reliability is reported to be 99.9% @xcite . for our further discussions , errors in mrc source positions are considered random , without any systematics . about 400 bright sources common to the two catalogues and with flux density at 151.5 mhz greater than 5 jy ( @xmath24-@xmath0 ) were identified and their positions were compared . the sources were labelled as common if they lie within @xmath25 of each other . since mrc has a source density of @xmath26 source deg@xmath27 , the chances of considering two unrelated sources as common are extremely low . a flux threshold of 15-@xmath0 ensures a source population abundant to reliably estimate homography ( explained in next section ) . the positional errors in @xmath6 and @xmath3 show no systematics as a function of @xmath6 ( refer _ first rows _ of fig . [ f : sourcecomparison]a and [ f : sourcecomparison]b ) . for visualisation , the errors are shown in percentages of mrt beamwidths . the errors in @xmath6 and @xmath3 show a linear gradient as a function of @xmath14 . the errors in @xmath6 , plotted against @xmath14 , reach @xmath28 of the mrt beamwidth ( refer _ second row _ of fig . [ f : sourcecomparison]a ) . whereas , the errors in @xmath29 , plotted against @xmath14 , are significant and reach @xmath30 of mrt beamwidth . ( refer _ second row _ of fig . [ f : sourcecomparison]b ) . histograms in fig . [ f : sourcecomparison]c and fig . [ f : sourcecomparison]d show the distribution of errors in @xmath6 and @xmath3 , respectively . the histogram of errors in @xmath3 shows a broader spread compared to errors in @xmath6 . re - imaging , to correct for errors in the images , would involve re - reducing the @xmath31 hours of observed data . owing to the complexity involved it was decided to correct for the positional errors in the images , thus avoiding re - processing . the 2-d homography estimation technique was employed for correcting positional errors in images and is discussed in detail in the following section . the 2-d planar homography is a non - singular linear relationship between points on planes . given two sets of @xmath32 corresponding image points in projective coordinates , @xmath33 , homography maps @xmath34 to the corresponding @xmath35 @xcite . where , @xmath36 . the homography sought here is a non - singular @xmath37 matrix @xmath38 such that : @xmath39= \left [ \begin{array}{ccc } h^{}_{11}&h^{}_{12}&h^{}_{13}\\ h^{}_{21}&h^{}_{22}&h^{}_{23}\\ h^{}_{31}&h^{}_{32}&h^{}_{33 } \end{array}\right ] \left [ \begin{array}{c } x^{}_{k}\\y^{}_{k}\\1\end{array}\right ] . \label{eq : inhomog}\ ] ] where , @xmath40 and @xmath41 represent @xmath42 of @xmath32 corresponding mrt and mrc sources , respectively . in equation [ eq : inhomog ] , @xmath43 and @xmath44 are referred to as the _ homogeneous coordinates _ and are always represented one dimension higher than the dimension of the problem space . this is a commonly used representation in computer graphics . the simple reason is that with a @xmath45 matrix one can only _ rotate _ a set of 2-d points around the origin and _ scale _ them towards or away from the origin . a @xmath45 matrix is incapable of _ translating _ a set of 2-d points . the homogeneous coordinates allow one to express a translation as a multiplication . a single @xmath46 matrix , with homogeneous coordinates , can account for rotation , scaling and translation of 2-d coordinates . for example , from equation [ eq : inhomog ] , @xmath47 . notice , @xmath48 ( representing translation in @xmath6-dimension ) is simply being added to the normal dot product @xmath49 that together represents rotation and scaling . in homogeneous coordinates , the 2-d problem space is a plane hovering in the third dimension at a unit distance . a general homography matrix , for projective transformation , has 8 degrees - of - freedom ( dof ) . for our system , both errors in @xmath6 and @xmath3 have only @xmath14-dependency . therefore , a less general , 2-d affine transformation is sufficient . a 2-d affine transformation ( two rotations , two translations and two scalings ) requires 6-dof @xcite , therefore in @xmath38 , @xmath50 and @xmath51 are zero . since each 2-d point provides two independent equations , a minimum of 3 point correspondences are necessary to constrain @xmath38 in the affine space . a set of @xmath32 such equation pairs , contributed by @xmath32 point correspondences , form an over - determined linear system : @xmath52 @xmath53\mbox { , } \ ] ] @xmath54^{t}\mbox{and , } \ ] ] @xmath55^{t}.\ ] ] in equation [ e : system ] , @xmath56 represents transpose of a matrix . this system can be solved by least squares - based estimators . at this stage it is useful to consider the effect of using ( @xmath57)-coordinates to represent the brightness distribution on the celestial sphere . ideally , it is the directional cosines @xmath58 , with respect to the coordinates of the array , which represent the spherical coordinates in the sky . therefore , the image coordinates in which homography should in principle be estimated are @xmath59 . however , at mrt , for 1-d imaging on the meridian : @xmath60 therefore , @xmath14 is a natural choice for one of the coordinates and is indeed used in the present case . on the meridian , the directional cosine @xmath61 is zero . for small errors , @xmath62 , in @xmath61 , i.e. close to the meridian : @xmath63 @xmath64 [ e : sinza ] here , @xmath65 is the error in @xmath6 . equation [ e : sinza ] shows that an error in @xmath61 will lead to an error in @xmath6 with a @xmath66-dependence . the 2-d images of mrt are 1-d images on the meridian made at different sidereal times and stacked . therefore , positional errors both in @xmath6 and @xmath3 do not show systematics as a function of @xmath6 ( _ first rows _ of figs . [ f : sourcecomparison]a and [ f : sourcecomparison]b ) . we preferred @xmath42-representation because all mrt images were already generated in this coordinate system . this choice compelled us to seek solutions for errors in @xmath6 as a function of @xmath14 rather than @xmath66 . we plotted errors in @xmath6 against both @xmath66 and @xmath14 and obtained separate linear least - squares fits . the rms of residuals in both fits is @xmath67 of the beamwidth in @xmath6 . however , the rms of difference between the fitting functions @xmath66 and @xmath14 in the @xmath3 range of mrt @xmath68 is only @xmath69 of the beamwidth in @xmath6 . therefore , the random errors in the source positions are larger than the errors introduced by the preferred @xmath42-coordinates for @xmath34 and @xmath35 . in @xmath34 and @xmath35 , the @xmath6 ranges from 18 hours to 24 hours and the @xmath14 ranges from @xmath70 to @xmath71 ( corresponding to the declination range of @xmath72 to @xmath73 ) . moreover , in matrix @xmath74 ( refer equation [ e : system ] ) there are entries of 1 s & 0 s . such a matrix is ill - conditioned and in the presence of noise in the source positions , the solution for an over - determined system may diverge from the correct estimate @xcite . the effect of an ill - conditioned matrix is that it amplifies the divergence . a normalisation ( or pre - conditioning ) is therefore required . to obtain a good estimate of the transformation matrix we adopted the normalisation scheme proposed by @xcite . the normalisation ensures freedom on arbitrary choices of scale and coordinate origin , leading to algebraic minimisation in a fixed canonical frame . the homography matrix @xmath75 is estimated from normalised coordinates by the least - squares method using singular value decomposition ( svd ) . the matrix is then denormalised to obtain @xmath38 . the scheme is briefly described below : 1 . * normalisation of @xmath76 : * compute a transformation matrix @xmath77 , consisting of a translation and scaling , that takes points @xmath34 to a new set of points @xmath78 such that the centroid of the points @xmath78 is the coordinate origin @xmath79 , and their average distance from the origin is @xmath80 . * normalisation of @xmath81 : * compute a similar transformation matrix @xmath82 , transforming points @xmath35 to @xmath83 . * estimate homography : * estimate the homography matrix @xmath75 from the normalised correspondences @xmath84 using the algorithm described earlier in the main section . * denormalisation : * the final homography matrix is given by : @xmath85 [ f : scheme ] shows the block schematic of the correction scheme . at mrt , the full declination range for each sidereal hour range is divided into 4 _ zones _ ( refer _ second row _ in fig . [ f : sourcecomparison]a or [ f : sourcecomparison]b ) . each zone is imaged with different delay settings to keep the bandwidth decorrelation to @xmath86 . therefore , the 6 sidereal hours of images under consideration , have 24 images ( @xmath87 ) . using the population of common sources , there are four possible alternatives to correct mrt images by computing : 1 . 24 homography matrices - one for each image . 2 . 6 matrices - one for each sidereal hour . 3 . 4 matrices - one for each declination zone . 4 . a single homography matrix for the entire steradian . in principle , bright sources in each image ( @xmath88 ) can be used to independently estimate a homography matrix . our earlier experiments to correct each image independently showed that the homography matrices were similar . the plots of errors in @xmath6 and @xmath3 plotted against @xmath6 and @xmath14 ( refer to fig . [ f : sourcecomparison]a and [ f : sourcecomparison]b ) indicate that the errors are independent of the four delay zones and the range of @xmath6 . this implies that estimating a single homography matrix for the entire source population should suffice in representing the errors . the homography matrix estimated using @xmath89 common sources ( described in section [ s : poserror ] ) is : @xmath90 . \label{eq : matrixvalues}\ ] ] in the estimated homography matrix , @xmath91 indicates there is no correction required in @xmath6 as a function of @xmath6 . @xmath92 indicates mrt images should be corrected in @xmath6 with a @xmath14 dependence . the estimated correction is up to @xmath93 of the beam in @xmath6 , at the extreme ends of the @xmath14 range . similarly , @xmath94 indicates that there is no correction required in @xmath14 , as a function of @xmath6 . however , @xmath95 indicates that mrt images should be compressed in @xmath14 by a factor of 0.9990 ( which is @xmath1 part in 1000 ) . the values of @xmath48 and @xmath96 indicate that the zero cross - overs of errors in both @xmath6 and @xmath14 plotted against @xmath14 are close to the @xmath14 of the calibration source ( mrc1932 - 464 ) used for imaging . using equation [ eq : inhomog ] , the homography matrix is used to project each pixel from the images to a new position , effectively correcting for positional errors in images . [ f : sourcecomparisoncorrect ] shows positional errors in @xmath3 after homography - based correction . a comparison of these plots with fig . [ f : sourcecomparison ] demonstrate that homography has removed the systematics and the residual errors are within 10% of the beamwidth for sources above 15-@xmath0 , as expected . [ f : scatterplot]a and fig . [ f : scatterplot]b show scatter plots of errors in @xmath3 against errors in @xmath6 before and after correction , respectively . for visualisation , the errors are represented in percentages of respective mrt beamwidths . notice , after correction ( refer fig . [ f : scatterplot]b ) the scatter is almost circular as opposed to elliptical before correction ( refer fig . [ f : scatterplot]a ) . the rms before correction is @xmath97 of the beamwidth . after correction , the rms is reduced to @xmath98 of the beamwidth and , the systematic errors have been removed . [ f : contourplots]a and [ f : contourplots]b show mrt contours before and after correction , respectively , overlaid on sumss ( sydney university molonglo sky survey ) image @xcite , for a source around @xmath99 . the corrected mrt image contours in fig . [ f : contourplots]b overlap with the source in sumss image . [ f : contourplots]c and [ f : contourplots]d show similar comparison for a source around @xmath100 . notice fig . [ f : sourcecomparison]d , since the errors around @xmath100 are within 10% of the beamwidth , the contours in both figs . [ f : contourplots]c and [ f : contourplots]d show a good overlap as expected and homography has not applied perceivable correction to images at this declination . we have overlaid mrt contours on a number of extended sources at 843 mhz reported by @xcite . [ f : contourplotsextsources ] shows a typical overlay of mrt contours on sumss image of a region around the cluster abell 3667 . the overlay is perceivably satisfactory . the 2-d homography corrected the positional errors in the image domain . for imaging the remaining @xmath101 steradians of mrt survey , @xmath102 hours of data has to be reduced . ideally , for imaging the new regions , one would like to trace the source of these errors and correct them in the visibilities . in the following section we discuss how we traced the source of errors and corrected them in the visibility domain . this section describes our _ expansion - compression _ hypothesis for the source of errors in our images . the subsequent corrections we estimated and applied to eliminate the errors are also described . for meridian transit imaging , @xmath103 . the brightness distribution in the sky as a function of @xmath14 and the complex visibilities measured for different values of the north - south ( ns ) baseline vector component @xmath104 form a fourier pair @xcite . a scaling error of @xmath105 in @xmath106 will result in a scaling factor of @xmath107 in the @xmath104-component of the baseline vector . by positional error analysis it is clear that mrt images are stretched ( _ expanded _ ) in declination , i.e. , @xmath108 @xmath109 [ e : expcomp ] note , for images the 2-d homography estimated a correction ( _ compression _ ) factor , @xmath107 , of 0.9990 . this cued to the hypothesis that we have compressed the north - south baseline vectors . equation [ e : expcomp]b means , a baseline distance of @xmath110 m in the ns arm was wrongly measured as @xmath111 m ( 1 part in 1000 ) . similarly , a @xmath14-dependent correction in @xmath6 cued to possible @xmath104-component in the east - west ( ew ) baseline vectors . next , we describe the re - estimation of array geometry . we begin with a brief description of the mode of observations with mrt . mrt has 32 fixed antennas in the ew arm and 15 movable antenna trolleys in the ns arm . for measuring visibilities , the 15 ns trolleys are configured by spreading them over 84 m with an inter - trolley spacing of 6 m ( to avoid shadowing of one trolley by another ) . mrt measures different fourier components of the brightness distribution of the sky in 63 different configurations ( referred to as _ allocations _ ) to sample ns baselines every 1 m . therefore , effectively , there are 945 antenna positions ( 63 allocations * 15 antennas / allocation ) in the ns arm and a total of 30,240 ( 945 * 32 ) visibilities are used for imaging . a small error in a measuring scale of relatively shorter length is likely to build up systematically while establishing the geometry of longer baselines . this effect would be observed in the instrumental phases estimated using different calibrators . in principle , the instrumental phases estimated using two calibrators at different declinations , for a given baseline , should be the same , allowing for temporal variations in the instrumental gains . a non - zero difference in these estimates may be due to positional errors of the baseline or positions of calibrators . as mentioned earlier , our analysis of positional error in sources and the homography matrix cued to positional errors in baselines ( or antenna positions ) . the simple principle of astrometry @xcite was used to estimate errors in antenna positions and is discussed below . the observed visibility phase , @xmath112 , in a baseline with components @xmath113 , due to calibrator @xmath114 with direction cosines @xmath115 , is given by : @xmath116 where , @xmath117 represents true instrumental phases , @xmath118 represents ew antennas and @xmath119 represents ns antennas . for meridian transit imaging equation [ e : obsphasebasiceqn ] becomes : @xmath120 the instrumental phases , @xmath121 , estimated using the measured geometry are given by : @xmath122 here , @xmath123 and @xmath124 are errors in the assumed baseline vectors . @xmath121 are phases of complex baseline gains obtained in the process of calibration . equation [ eq : calphaserelation ] has three unknowns . to reduce the number of unknowns , one can eliminate the true instrumental phases by taking a difference @xmath125 between the instrumental phases estimated using two calibrators . this difference gives : @xmath126 } \nonumber \\ \lefteqn{\hspace{22 mm } + \delta w^{}_{ij } \left[\cos\left(za^{\mathcal{s}^{}_{1}}\right ) - \cos\left(za^{\mathcal{s}^{}_{2}}\right)\right ] . } \label{eq : diff1}\end{aligned}\ ] ] note , the @xmath127-components of the baseline vectors are short and non - cumulative measurements . therefore , in principle , one can consider @xmath124 as zero - mean random errors with no systematics . equation [ eq : diff1 ] in that case can be written as : @xmath128 . \label{eq : simplediff1}\ ] ] describing the system in terms of errors in antenna positions , as opposed to errors in baseline positions , equation [ eq : simplediff1 ] becomes : @xmath129 . \label{eq : diffantenna1}\ ] ] this equation is also not sufficient to solve for errors in the antenna positions as we have two unknowns and one equation . we set up another equation using a third calibrator source , @xmath130 , spaced away in declination from @xmath131 and @xmath132 : @xmath133 . \label{eq : diffantenna2}\ ] ] the equations [ eq : diffantenna1 ] and [ eq : diffantenna2 ] are a linear set of equations for one baseline . for the measurements in 63 allocations , the set of equations can be formulated in a matrix form and solved by svd - based least - squares estimator : @xmath134 where , the _ measurement vector _ @xmath135 is to be determined . here , @xmath136 . the measurement vector gives @xmath137 and @xmath138 estimates for 32 ew and 945 ns antenna locations , respectively . observation vector _ @xmath139 consists of two sub - matrices , @xmath140 and @xmath141 , formed using the left - hand - side of equations [ eq : diffantenna1 ] and [ eq : diffantenna2 ] , respectively . here , @xmath142 , i.e. , the total number of visibilities measured for imaging . therefore , @xmath143 . the _ data matrix _ @xmath144 . each row in the data matrix has only two non - zero elements , corresponding to a baseline formed by one ew and one ns antenna , making it very sparse . the observation vector is constructed from the gain tables of the array obtained using calibrators mrc0407 - 658 ( @xmath145 ) , mrc0915 - 118 ( @xmath146 ) and mrc1932 - 464 ( @xmath147 ) . the sensitivity per baseline at mrt is @xmath148 jy for a 1 mhz bandwidth and an integration time of one second . it takes @xmath149 minutes of time for sources at @xmath150 to transit a 2@xmath151 primary beamwidth of elements in the east - west array . this leads to a sensitivity per baseline ( including the non - uniform weighting due to primary beam ) of @xmath152 jy . the flux density of these three calibrators as seen by mrt is @xmath153 jy ; strong to get reliable calibration . further , the calibrators are unresolved and isolated from confusing sources and have well known measured positions @xcite . a plot of typical phase differences obtained using the pair of calibrators @xmath154 is shown in fig . [ f : phasediff ] . fig . [ f : error_estimate]a shows the estimated errors in 945 ns antenna positions . the errors show a gradient of 1 part in 1000 along the ns arm . this matches with the linear gradients in the phase differences estimated from the calibrators . the estimates in fig . [ f : error_estimate]b show alignment errors of the 32 antennas in the ew arm along the ns - direction . the fit shows a gradient of about 2 part in 10,000 . this indicates that the ew arm is mis - aligned from the true ew - direction . at one extreme end ( 1 km from the centre of the array ) of the ew arm the error is @xmath155 m , equivalent to an angular distance of @xmath156 from the centre of the array . this is the source of a small @xmath14-dependent error in @xmath6 that was observed in both positional error analysis and the homography matrix . further , our simulation of the synthesised beam in @xmath6 with old ew antenna positions and the corrected ew antenna positions indeed confirm this @xmath14-dependent error in @xmath6 . using the new antenna positions we have re - imaged one hour from the steradian and have also imaged a completely new steradian . we find no systematics in positional errors thus endorsing our re - estimated array geometry . the homography - based correction was able to correct for systematics in positional errors in the image domain and the errors are within 10% of the beamwidth for sources above 15-@xmath0 . the corrected images of one steradian are available for download at _ http://www.rri.res.in / surveys / mrt_. positional error analysis showed that uncorrected mrt images are stretched in declination by @xmath1 part in 1000 . this translates to a compression of the ns baseline vector , in the visibility domain . the analysis also showed a @xmath14-dependent error in @xmath6 . this cued towards possible errors in our estimation of the array geometry . by formulating a linear system , using instrumental phases estimated from three well separated calibrators whose positions are well known , the array geometry was re - estimated . the estimated error in the @xmath104-component of the ns baseline vectors is about 1 mm / m . in other words , the error is about half a wavelength at 150 mhz ( 1 m ) for a 1 km baseline . the estimates also show a small ( 2 part in 10,000 ) @xmath104-component in the purely ew baseline vectors . this indicates that the ew arm is mis - aligned and inclined at an angle of @xmath2 , to the true ew direction . these estimates match with the observed stretching of mrt images shown by both the positional error analysis and the homography matrix . using the new antenna positions we have re - imaged one hour from the steradian and have also imaged a completely new steradian . we find no systematics in positional errors . this endorses our re - estimated array geometry . re - imaging one steradian starting from visibilities would have been a very time consuming exercise . development of 2-d homography - based correction enabled us to correct for the positional errors in the image domain . in our view , this new technique will be of relevance to the new generation radio telescopes where , owing to huge data rates , only images after a certain integration would be recorded as opposed to raw visibilities . soobash daiboo acknowledges a phd bursary from the south african square kilometer array project . the authors would like to thank the anonymous referee for the constructive comments and suggestions

a steradian of the southern sky has been imaged at 151.5 mhz using the mauritius radio telescope ( mrt ) . these images show systematics in positional errors of sources when compared to source positions in the molonglo reference catalogue ( mrc ) . we have applied two - dimensional homography to correct for systematic positional errors in the image domain and thereby avoid re - processing the visibility data . positions of bright ( above 15-@xmath0 ) point sources , common to mrt catalogue and mrc , are used to set up an over - determined system to solve for the homography matrix . after correction the errors are found to be within 10% of the beamwidth for these bright sources and the systematics are eliminated from the images . this technique will be of relevance to the new generation radio telescopes where , owing to huge data rates , only images after a certain integration would be recorded as opposed to raw visibilities . it is also interesting to note how our investigations cued to possible errors in the array geometry . the analysis of positional errors of sources showed that mrt images are stretched in declination by @xmath1 part in 1000 . this translates to a compression of the baseline scale in the visibility domain . the array geometry was re - estimated using the astrometry principle . the estimates show an error of @xmath1 mm / m , which results in an error of about half a wavelength at 150 mhz for a 1 km north - south baseline . the estimates also indicate that the east - west arm is inclined by an angle of @xmath2 to the true east - west direction . [ firstpage ] surveys techniques : image processing astrometry techniques : interferometric telescope catalogues