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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 .

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